The Canonical Operator in Many-Particle Problems and Quantum Field Theory 9783110762389, 9783110762709, 9783110762747, 2022934383

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Victor P. Maslov and Oleg Yu. Shvedov† The Canonical Operator in Many-Particle Problems and Quantum Field Theory

De Gruyter Expositions in Mathematics

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Edited by Lev Birbrair, Fortaleza, Brazil Victor P. Maslov, Moscow, Russia Walter D. Neumann, New York City, New York, USA Markus J. Pflaum, Boulder, Colorado, USA Dierk Schleicher, Bremen, Germany Katrin Wendland, Freiburg, Germany

Volume 71

Victor P. Maslov and Oleg Yu. Shvedov†

The Canonical Operator in Many-Particle Problems and Quantum Field Theory |

Mathematics Subject Classification 2020 Primary: 81Q20, 81T15, 81T70; Secondary: 53D12, 30H20 Authors Prof. Victor P. Maslov Moscow Inst. of Electron. & Mathematics Dept. of Applied Mathematics Moscow 109028 Russian Federation [email protected]

Oleg Yu. Shvedov†

ISBN 978-3-11-076238-9 e-ISBN (PDF) 978-3-11-076270-9 e-ISBN (EPUB) 978-3-11-076274-7 ISSN 0938-6572 Library of Congress Control Number: 2022934383 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2022 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

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In memory of Ludwig Faddeev

Preface to the English edition I have decided to dedicate the English translation of this book, written jointly with my late pupil, Oleg Shvedov, to the memory of the remarkable expert in mathematical physics and quantum theory, my close friend Ludwig Faddeev. The two of us always exchanged our new ideas and understood each other perfectly. In the present monograph, the notion known as the “Faddeev transformation” is substantially used. Here, I would like to cite a fragment of my article “My Dear Ludwig” published in the journal, Mathematical Notes, in 2017 (Vol. 101, no. 6). Although I knew that he was terminally ill, the news of Ludwig Faddeev’s death came as a shock to me. The floorboards spun under me, losing balance I tried and failed to find support from the walls of the room… We became friends long ago, in 1958. This is how Faddeev himself described it: “Concerning conferences, I was very lucky at the time—at the end of the 1950s functional analysis was flourishing, nationwide conferences on the subject were held every year in Moscow, Odessa, Kharkov. Participation in them was an excellent chance to learn from my elders and meet my contemporaries. As I mentioned before, in Odessa I met V. P. Maslov, we became friends and have supported each other ever since” (L. D. Faddeev, 30 years in mathematical physics, Trudy Mat. Inst. Steklov 176, 4–29, 1987) I am looking through his letters. And, for the first time, understand the meaning of the Russian saying “What is written by pen can’t be rubbed out by axe.” For instance, I love his handwriting—it reflects his character, as handwriting experts are correct in saying. Beautifully handwritten mathematical formulas—such a wonder, something I cannot do. I was the first who discerned, in his very first papers, his talent—to create a beautiful mathematical frame and fill it with physical meaning. Most important was mathematical beauty and elegance. He would always object to me whenever I mentioned his remarkable physical intuition, stressing that most important for him was the beauty of mathematical structure. In 1967, Faddeev visited me in Moscow and told me about his unpublished joint work with Victor Popov. His enthusiasm about the work was contagious. Its mathematical beauty was indescribable, while the underlying physics were so virtual, that the objects its authors had discovered became known as “Faddeev–Popov ghosts.” These “ghosts” continued to roam inside my house the morning after his departure, and it became clear to me that such beauty cannot be accidental. That night when Faddeev was describing his paper with Popov, he asserted that the Yang–Mills theory must be related to new particles. I am a (still) living witness of this remarkable utterance. The ideas of this work led to the award in 1999 of the Nobel Prize for clarifying the quantum structure of electroweak interactions. But the prize was awarded to two Dutch physicists, while the name of Faddeev, to the great surprise of the physics community, did not appear on the list of prizewinners under the pretext that Popov had died by then. Physicists were shocked and could not understand why Faddeev’s name had been thrown out of the short list for the prize. It is well known that Caton, when asked why a monument in his honor had not been erected during his lifetime, answered “precisely because people could then ask why a monument for Caton was not erected, rather than ask why it was erected.” In our time, it is not customary to construct monuments or to make memorial medals for living individuals. In that respect, Max Planck and Ludwig Faddeev are exceptions to the rule— medals in their name appeared during their lifetime. In the 1980s, we both invented new algebras with very general commutation relations. Faddeev asked me, “So who was the first one of us to discover these algebras?” I answered, “You https://doi.org/10.1515/9783110762709-201

VIII | Preface to the English edition

were, of course, because you found powerful applications to your version of these algebras in the theory of fields.” Indeed, Faddeev’s algebras developed rapidly, while my more general ones were mentioned in a paper by Faddeev and Takhtadzhyan, but did not have the resonance I had hoped for. Faddeev never changed his topic of study, and intuitively felt at what point of the front of science a breakthrough would be imminent, a point through which it would be possible to break away from stagnation. And the young scientific reserves would follow him into the breach! My reaction to his further discoveries was ecstatic or, as he wrote, “very enthusiastic”; I awaited impatiently for his new results, which grew exponentially with time. As for me, I was moving deeper and deeper into statistical physics, which he couldn’t stand, just as I couldn’t in my younger years, but later I overcame my aversion to the subject and rewrote the science of “sleazydynamics,” as Kapitsa and Leontovich called it. This is what Ludwig wrote to me when I was awarded the State Prize of Russian Federation for my work in thermodynamics: Dear Vitya, my congratulations for the long awaited prize. Had I learned about it in time, I would have come to Moscow to celebrate it with you. But I had already refused to come in order to avoid seeing those who are destroying our science… Concerning my scientific opinion, I am unable to professionally assess your recent work. Since my childhood, I have a strong aversion to statistical physics, which of course has been detrimental to me… I love you very much. At the time, Ludwig was moving toward a new breakthrough in field theory and particle theory. His life should not, and should never have ended then. After one of his earlier discoveries, which reflected the beauty of creation, he told me: “I am beginning to believe in God!” I am well familiar with this feeling: when a mathematical structure unexpectedly becomes clear—you cannot believe your eyes. It is so terribly unfair that life can suddenly end before you have had enough time to finish putting the structure together; and just when he had caught his second wind! Archimedes said, “Give me a point of support, and I will move the Earth!” But he did not say what happens when the point of support is lost. With the loss of my friend, I feel I have lost my point of support.

I first considered general algebras (i. e., certain nonlinear commutation relations) in the paper “Application of the method of ordered operators to obtain exact solutions” published in the journal Theoretical and Mathematical Physics in 1977 (Vol. 33, no. 2, pp. 185–209). Algebras with general commutation relations without Lie algebra conditions were considered in joint work with my colleagues V. Nazaikinskii and M.Karasev. Ludwig Faddev with coauthors, N. Reshetikhin and L. Takhtadzhyan, in the article “Quantization of Lie groups and Lie algebras” (Algebra i Analiz, 1989, Vol. 1, no. 1, pp. 178–206) introduced a special class of algebras that he called quantum analogues of Hopf algebras. In the galley proofs of the article, the authors added the following remark: “Algebras with nonstandard non-Lie commutation relations” were considered in a series of papers by V. P. Maslov, M. V. Karasev, and V. E. Nazaikinskii “Algebras with general commutation relations and their applications, I, II” in the (Journal of Soviet Mathematics, 1981, Vol. 15, pp. 167–273, 273–368). Hopf algebras have numerous applications as was noticed, in particular, by V. G. Drinfeld in the paper “Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang–Baxter equations” (Soviet Math Doklady, 1983, Vol. 268, no. 2. pp. 285–287). The quantum analogue of Hopf algebras constructed by Faddeev found spectacular applications as a new science, namely

Preface to the English edition

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quantum field theory. The more general quantum groups that I introduced do not have such spectacular applications, but they are more natural and simpler in the framework of the old science. What are their potential applications? That question remains open. In the English version of the present monograph, there are practically no changes or additions to the Russian edition of 2000, except that the bibliography has been expanded to include the most important later works of Oleg Shvedov, whose untimely death interrupted the series of papers developing the ideas of the book. In conclusion, let me express my deep gratitude to my translators, Alexander Stoyanovsky, Maria Shishkova, and Alexey Alimov. I am also grateful to my friend, Alexey Sossinsky, for his invariably highly qualified recommendations concerning English terminology and help in the formulation of my ideas in English. Special thanks are due to Vladimir Nazaikinskii for his scientific and style editing of the whole translation of the book.

Preface This book deals with the approximate solution of equations with a small parameter for functions whose number of arguments tends to infinity as the small parameter tends to zero. Such equations often arise in applications. At the same time, known asymptotic methods [13, 17, 24–27, 85, 86], only apply to equations for functions of a fixed number of arguments. In the present book, we develop a new asymptotic method, which permits one to construct approximations for functions of a large number of arguments. It turns out that various equations corresponding to various physical problems can be studied by one and the same method. In statistical physics, one often studies the problem of N classical particles moving in an external field and interacting with one another under the following assumptions: the external potential is of the order of unity, and the coefficient multiplying the interaction potential is of the order of 1/N. This physical problem corresponds to the multiparticle Liouville equation (equation (3.9) in Section 3.1) for a probability distribution density function depending on 6N + 1 arguments, one playing the role of time, 3N being the particle coordinates, and the remaining 3N being the momenta. Known methods for the analysis of this problem for large N [3, 11, 31, 46] are as follows. Instead of the problem of constructing the asymptotics for the N-particle probability distribution function, one considers the problem of constructing approximations to the so-called k-particle distribution obtained from the N-particle density by integration over all arguments except for 6k + 1 ones (see Section 3.1.4). These partial distributions satisfy the chain of Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) equations [7] (see Section 2.1.3). An analysis of this chain of equations permits one to construct the asymptotics of the partial distributions with k = const as N → ∞. These asymptotics are expressed via the solution of the Vlasov equation, well known in statistical physics. These asymptotics can be used to construct approximations to the mean values of observables of a special kind. It turns out that in general one cannot uniquely determine the approximations as N → ∞ to the mean values of observables of the general kind uniformly bounded with respect to N from the known approximations to the partial distributions. For computing such mean values, one must use an approximation to the N-particle distribution rather than to functions of finitely many arguments. The asymptotics of the N-particle density cannot be found by an asymptotic analysis of the BBGKY chain. To construct such asymptotics, one needs a completely new method. It is such method that is developed in the monograph. We construct the asymptotics of the solution of the Cauchy problem for the multiparticle Liouville equation. The asymptotic formula is expressed via not only the solution of the well-known Vlasov equation [52] but also the solutions of some new equations. The corresponding asymptotic formulas are given in Sections 3.1 and 3.4. https://doi.org/10.1515/9783110762709-202

XII | Preface These asymptotics lead to a new insight into the problem of chaos. The molecular chaos hypothesis has been known in statistical physics since Boltzmann. It says that particles are independent, i. e., the N-particle probability distribution density function is the product of one-particle functions. A weakened version of this hypothesis considered by Kac [14, 15] says that the finite rank correlation functions split into a product of one-particle functions for large N. This property was proved by the method of BBGKY chains. The method developed in our monograph permits one not only to confirm this result but also to construct an approximation to the N-particle density that equals a product of one-particle densities at the initial time. It turns out that the N-particle density is not close to a product of one-particle densities at the other times, i. e., the hypothesis on the preservation of chaos in the strong sense is not true. Similar approximations are also constructed for the Gibbs canonical temperature distribution in equilibrium statistical mechanics. The method developed in the book for constructing asymptotic solutions of multiparticle equations permits one to consider problems not only in classical statistical mechanics but also in many-particle quantum mechanics and quantum statistics. Such problems were also studied previously [3] by the BBGKY chain method [7] by the analogy with the problem of many classical particles; the approximations were constructed for variables expressible via the many-particle functions but not for the manyparticle functions themselves. These asymptotics are expressed via the solutions of the well-known self-consistent field equations (the Hartree and Hartree–Wigner equations). In quantum mechanics, the states of N particles at the fixed moment of time are given by N-particle wave functions—functions of N arguments that are the coordinates of the particles. These functions satisfy the Schrödinger evolution equation. If the external potential is of the order of unity and the interaction potential is O(1/N), then the evolution equation takes the form (3.1) in Section 3.1 and can be studied by the method developed in the book. The multiparticle Wigner equation can be studied by this method as well. The asymptotics of the many-particle function given in Section 3.2 and 3.3 are expressed via the solutions of not only the well-known self-consistent field equations but also some new equations. The chaos preservation problem in the quantum case is studied as well. The problem on the spectrum of a multiparticle Hamiltonian is very important in quantum mechanics. Bogoliubov [8] found the asymptotic spectrum for the case in which the external potential for a system of N particles is zero and the interaction potential depends only on the differences of coordinates of the particles. Bogoliubov also showed that this spectrum corresponds to the superfluidity phenomenon. The approach developed in the monograph permits one to construct families of asymptotic eigenvalues and eigenfunctions of the multi-particle Hamiltonian and to study the superfluidity problem in the presence of an external field. These families are constructed in Section 3.3.

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The problem of approximating the mean values of observables in the most general setting (for the case of an abstract Hamiltonian algebra) is considered in Section 3.5. The asymptotic method applicable to the construction of approximate solutions of the multiparticle Schrödinger, Liouville, and Wigner equations is developed in Chapter 2. The asymptotic formulas are constructed using the multiparticle canonical operator, which is introduced in Section 2.2. The proof of the asymptotic formulas is carried out in Section 2.7 (see also Section 5.3) in the following way: first, the formulas of commutation of the multiparticle canonical operator with other operators are used to prove that the asymptotic expressions constructed approximately satisfy the multiparticle equation, and then the inverse operator is estimated by the method in [24]. Although all the computations are given for the case in which the coefficient of the interaction potential tends to zero at the rate of 1/N as N → ∞, the remainder estimate can be carried out for an arbitrary coefficient, and in each specific case, one can estimate whether the error is large or not. Chapter 5 deals with one more type of multiparticle equations, equations with operator-valued symbol. These equations arise in the study of a system of k particles interacting with the N-particle system considered in Chapter 3. We consider the case in which k remains fixed, N tends to infinity, the mass of each of the k particles tends to zero at the rate of 1/N, the external potential for these particles is of the order of N, and the potential of interaction with each of the N particles is of the order of unity. Such a system cannot be studied even by the BBGKY chain method. In Section 5.3, we construct asymptotic solutions of such equations. It turns out that the self-consistent field equation for this system is not the well-known Hartree equation but an infinite set of new, more complicated (independent) equations. One solution of any of these equations corresponds to a family of asymptotic solutions of the multiparticle equation with operator-valued symbol. Further, it turns out that the chaos property does not hold for these types of equations even in the weak sense. At first sight, the construction of approximate solutions of multiparticle equations and the semiclassical approximation to equations of quantum mechanics are absolutely different problems. However, it turns out that there is an interesting relation between these two problems, which is also considered in the book. In each of the two cases, there exists a method for constructing asymptotics “in the weak sense,” which only permits constructing approximations to the mean values of observables of a special kind. These are the approach based on the Ehrenfest theorem in the theory of the semiclassical approximation and the approach based on the BBGKY chain in the many-body theory. These methods permit one to express the desired variables via the (exact) solutions of the “classical” equations, which are “simpler” than the original equations. In the semiclassical theory, this is a system of ordinary differential equations (the Hamiltonian system) instead of a partial differential equation, and in the case of many-particle theory, this is the self-consistent field equation for a function of finitely many arguments instead of an equation for a function of large number of arguments.

XIV | Preface However, the method for constructing asymptotics “in the weak sense” does not permit one to approximate the mean values of bounded observables of the general kind. For example, in semiclassical quantum mechanics there exist “wave packet” type approximate solutions of the Schrödinger equation. The Ehrenfest theorem permits one to determine the classical trajectory along which such a wave packet moves but does not say anything about how the wave packet shape changes in time. Similarly, in the many-body theory one cannot determine by the BBGKY method whether the multiparticle function is approximated by the product of one-particle functions. Thus, to construct asymptotics “in the strong sense,” one needs approaches different from the Ehrenfest theorem in semiclassical quantum mechanics and the BBGKY method in statistical physics. In quantum mechanics, the desired approach is provided by the complex germ theory, which permits one to determine not only the classical trajectory along which the wave packet moves but also the time evolution of the wave packet shape. For the latter, one needs to use the solutions of equations different from the classical Hamiltonian system. The method developed in the book for constructing the asymptotics of solutions of multiparticle equations is an analog of the complex germ theory. It also permits constructing approximations to the mean values of observables of the general kind and is based on the solutions of equations different from the well-known self-consistent field equations. In quantum mechanics, one considers not only systems semiclassical in all variables but also systems semiclassical in one group of variables and “quantum” in the remaining variables [21, 24]. The Hamiltonian system for such systems is “multivalued”: each of the eigenvalues of the operator-valued symbol of the Hamiltonian operator is a Hamiltonian function. The multiparticle systems considered in Chapter 5 are an analog of such quantum mechanical systems. There arises a question as to whether one can introduce general notions including those arising in semiclassical asymptotics and in the construction of approximate solutions of multiparticle equations as special cases. It turns out that such a generalization is possible. It is considered in Chapter 1; see also [112, 113]. One introduces the new notion of an abstract canonical operator, generalizing the canonical operator used in the theory of a complex germ at a point and the multiparticle canonical operator used in the construction of asymptotic solutions of multiparticle equations. It turns out that to each abstract canonical operator there corresponds a unique phase space with symplectic structure. This is the 2n-dimensional real vector space {(p, q)|p, q ∈ ℝn } with symplectic structure dp ∧ dq in the semiclassical theory and the unit sphere in the complex Hilbert space L2 (ℝ3 ) in the many-body problem. It turns out that one can find universal identities for asymptotic solutions of abstract equations. These identities can be applied to the asymptotics constructed by the traditional complex germ theory in finite-dimensional quantum mechanics as well as to the asymptotics obtained by our method for the solutions of multiparticle equations. These identities are given in Chapter 1 of the monograph. Using only these iden-

Preface | XV

tities (and not using even the evolution equations of motion), one can obtain a fairly broad class of results (in particular, many well-known results related to the traditional theory of the semiclassical approximation, including problems with operator-valued symbols) and also determine the spectrum of a multiparticle Hamiltonian, find out whether superfluidity takes place, whether the chaos conservation property in the strong sense holds, etc. In particular, The nonconservation of chaos turns out to be a straightforward consequence of the nonlinearity of the Vlasov equation. Note that a straightforward verification of the identities for equations with operator-valued symbols is a nontrivial problem. The identities in Chapter 1 permit one to define a formal asymptotic solution of the equations of motion as a function satisfying these identities hold. This definition does not use the equations of motion themselves. Hence, this notion can be applied to the development of a new approach to the asymptotic quantization problem, which has been intensively studied in the recent years [13, 16, 17, 57, 58, 83, 84, 125]. A similar approach can be used for the construction of new axioms of relativistic quantum field theory. All of the results in the book are obtained in two ways, by starting from the above mentioned universal identities (this method is described in detail) and by a straightforward substitution of the asymptotic formula into the equation (this method is described in less detail). Therefore, the monograph does not reproduce the authors’ papers [5, 32–45, 88–103, 110–113] but rather supplements them. In particular, the theory of the abstract canonical operator is illustrated by examples in Section 1.5 (the complex germ method in the semiclassical approximation theory) and Sections 2.3–2.6 (the construction of formal asymptotic solutions in the many-body problem). Chapter 4 generalizes the method proposed in Chapter 2 to a version that applies to systems with a variable number of particles. The asymptotics can be constructed heuristically in the following way: one can write the initial (second-quantized) equations via the creation–annihilation operators, choose a representation for these operators in which the second quantized equation looks like an infinite-dimensional generalization of the Schrödinger equation, and apply the theory of Lagrangian (isotropic) manifolds with complex germ [25, 27, 104] to the transformed equation. Unfortunately, the traditional interpretation of this theory is not very convenient for generalization to the infinite-dimensional case; hence, in Sections 4.3–4.4 we give a nontraditional exposition of the complex germ method on a Lagrangian manifold. This approach to the theory of Lagrangian manifolds with complex germ is based on the construction of asymptotics corresponding to isotropic manifolds in the form of superpositions of asymptotics of wave packet type. Formerly, superpositions of wave packet asymptotic solutions of the quantum equation were considered in the paper [106], and superpositions of Gaussian wave packets of fixed form were studied in the paper [81]. We consider superpositions of arbitrary wave packets in Sections 4.3–4.4. It turns out that the notion of Lagrangian manifold with complex germ can also be generalized to the general case of the abstract canonical operator. Section 4.5 (see

XVI | Preface also [113]) defines formal asymptotic solutions of the equations of motion corresponding to a Lagrangian manifold with complex germ. Section 4.7 provides a generalization of the theory of Lagrangian manifolds with a complex germ to the infinite-dimensional case. We construct the asymptotics in the Fock space corresponding to finite-dimensional isotropic manifolds with infinitedimensional complex germs. For a special choice of the one-dimensional isotropic manifold, the asymptotics constructed here becomes the asymptotics in Chapter 2. As one more example of application of the theory, we consider a problem with two types of particles, the number of particles of each type being large. We construct twodimensional invariant isotropic manifolds and the corresponding families of asymptotic eigenvalues and eigenfunctions. The approach applied to multiparticle equations can be called an infinitedimensional generalization of the theory of Lagrangian manifolds with complex germ. Note that the complex germ theory, which is often applied in the theory of differential equations for functions of finitely many arguments, has been well established only for the finite-dimensional case. In the monograph, we develop an infinite-dimensional generalization of the complex germ theory. One more application of the infinite-dimensional theory of Lagrangian manifolds with a complex germ is construction of asymptotics in quantum field theory. However, the rigorous justification of these asymptotics encounters the problem of ascribing mathematical meaning to quantum field theory itself [9, 78, 122, 123, 129, 134]. Without a solution of this problem, the approaches to which have only been developed in special cases [12, 50, 54, 55, 74, 78], it is impossible even to hope to justify the heuristic asymptotic formulas constructed in the monograph. In Chapter 6, we consider semiclassical field theory. In Section 6.2, we construct families of asymptotic eigenvalues and eigenfunctions in scalar field theory and in quantum electrodynamics. In a special case, the asymptotic eigenvalues become the eigenvalues obtained by the well-known method of quantization in a neighborhood of solitons; we also construct asymptotic eigenfunctions. Section 6.3 deals with the equations corresponding to a system of k particles interacting with a quantized field. It is shown that the operator-valued complex germ theory can be applied for certain relations between the parameters. The asymptotics constructed there cannot be obtained by methods such a quantization in a neighborhood of solitons [68, 69, 71, 79, 107] and the theory of processes in strong external fields [61, 73, 76]. In quantum field theory, there arise divergences. For their removal, one uses the renormalization procedure. We consider the question as to how these divergences appear in the leading order of semiclassical expansion. The methods in Chapter 2 can also be applied to systems of a large number of quantized fields. In Chapter 7 (see also [93]]), we construct spectral families for such systems, which cannot be found by known asymptotic methods (1/N-decomposition

Preface | XVII

and quantization in a neighborhood of solitons). We perform the spectrum renormalization procedure. The authors are sincerely grateful to I. Ya. Arefyeva, V. P. Belavkin, V. V. Belov, E. M. Vorobyev, V. L. Dubnov, V. V. Kucherenko, V. E. Nazaikinskii, O. N. Naida, Yu. B. Orochko, L. D. Faddeev, A. M. Chebotarev, V. M. Chetverikov, and D. V. Shirkov, with whom we discussed topics of this book at various times.

Contents Preface to the English edition | VII Preface | XI List of notation | XXVII 1 1.1 1.1.1 1.1.2 1.1.3 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.4.1 1.4.2 1.5 1.5.1 1.5.2

Abstract canonical operator and symplectic geometry | 1 Introduction | 1 Construction of approximate solutions of various equations: comparison of methods | 1 Abstract canonical operator and geometry of the phase space | 5 Problem of existence of equations of motion and the concept of asymptotic quantization | 8 Abstract canonical operator and induced geometric structures on the phase space | 9 Auxiliary notions | 9 Definition of an abstract canonical operator | 11 Well-definedness of the canonical operator—the action 1-form on the phase space | 12 Operator-valued 1-form induced by the canonical operator | 16 Canonical transformation of the abstract canonical operator | 18 Canonical and proper canonical transformations of the phase space | 21 Formal asymptotic solutions of the equations of motion | 21 Abstract complex germ and construction of formal asymptotic solutions of the equations of motion | 22 Definition of complex germ | 22 Properties of the abstract complex germ | 24 Canonical transformation of the abstract complex germ | 26 Canonical operator corresponding to a complex germ | 28 Asymptotics of the solution of the Cauchy problem | 30 Canonical transformations of the canonical operator and construction of asymptotics of the solution of the Cauchy problem | 30 Asymptotics of the solution of the Cauchy problem modulo O(εM/2 ) | 33 Theory of complex germ at a point in finite-dimensional quantum mechanics | 35 Definition of the canonical operator and checking the axioms | 35 Canonical and proper canonical transformations of the phase space | 36

XX | Contents 1.5.3 1.5.4 1.5.5 1.A 1.B 1.B.1 1.B.2 1.B.3 2 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.2 2.2.1 2.2.2 2.3 2.4 2.4.1 2.4.2 2.5 2.5.1 2.5.2 2.5.3 2.6 2.6.1 2.6.2

Complex germ | 37 Canonical transformations depending on time | 42 Commutation of the canonical operator with the Hamiltonian and proof of the asymptotic formula | 45 Classical and quantum mechanics: the main definitions [1, 20, 53] | 52 Some recollections from differential geometry | 54 Smooth mappings of Hilbert spaces | 54 Smooth manifolds and differential forms | 55 Universal covering, homology, and cohomology | 58 Multiparticle canonical operator and its properties | 61 Introduction | 61 Physical problems giving rise to the study of functions of large number of arguments | 61 Physical arguments leading to the choice of a norm in the space of functions of large number of arguments | 62 Method of BBGKY hierarchies | 64 Nonconservation of chaos for N-particle wave function: statement of the theorem | 66 Asymptotics of the solution of the Cauchy problem: statement of the theorem | 68 Definition of multiparticle canonical operator | 73 Examples of multiparticle functions satisfying the chaos property | 73 Definition and simplest properties of the multiparticle canonical operator | 74 Geometric structures on the one-particle space induced by the multiparticle canonical operator | 76 Canonical and proper canonical transformations of the manifold ℳ | 83 Canonical transformations of the phase space and their properties | 83 Proper canonical transformations | 87 Complex germ | 89 Complex germ corresponding to a Gaussian vector | 89 Canonical transformation of a complex germ | 94 Construction of the operator W | 95 Formal asymptotic solutions of the equations of motion | 97 Some examples of formal asymptotic solutions and the problem of conservation of chaos | 97 Invariant formal asymptotic solutions | 100

Contents | XXI

2.6.3 2.6.4 2.6.5 2.7 2.7.1 2.7.2 2.A 2.A.1 2.A.2 2.A.3 2.A.4 2.A.5 2.B 2.C 3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7

3.3 3.3.1 3.3.2 3.3.3 3.4

Equations for a canonical transformation and a complex germ depending on time | 102 Equations for the operator W | 104 Formal asymptotic solution satisfying a given initial condition | 108 Commutation of the canonical operator with the Hamiltonian and the main theorem | 112 Commutation of the canonical operator with other operators | 112 Asymptotics of the solution of the Cauchy problem | 119 Method of second quantization [6] | 120 Fock space and creation and annihilation operators | 120 Coherent states and their properties | 123 Generating functionals | 123 Norm of a Gaussian vector | 125 Linear canonical transformations | 127 Some properties of the unit sphere in the space L2 | 129 Proof of existence of the solutions of some equations | 133 Asymptotic solutions of the many-body problem | 139 Introduction | 139 Various physical applications of the method | 139 Probability density distribution: the choice of a norm | 143 Half-density function and applicability of the method in Chapter 2 | 146 Chaos conservation problem in systems of many classical particles | 148 Asymptotic formulas for the multiparticle density matrix | 149 Choice of a norm on the space of density matrices | 149 Half-density matrix and its properties | 150 Asymptotic solutions of the equation for the half-density matrix | 152 Representation via the germ creation and annihilation operators | 154 Hermitian property of the half-density matrix | 156 Half-density matrix for pure states | 156 Existence of solutions of certain auxiliary equations and asymptotics of solution of the Cauchy problem for the multiparticle Wigner equation | 158 Asymptotic solutions of the N-particle Schrödinger equation as N → ∞ and superfluidity | 161 Asymptotic solutions of the N-particle Schrödinger equation | 161 N. N. Bogoliubov’s superfluidity theory | 166 Generalization of the notion of superfluidity | 168 Asymptotics of solution of the N-particle Liouville equation and violation of the chaos conjecture for density function | 169

XXII | Contents 3.4.1 3.4.2 3.4.3 3.5

3.5.1 3.5.2 3.5.3 3.5.4 3.A 3.A.1 3.A.2

4 4.1 4.1.1 4.1.2

Existence of solutions of the Vlasov and Riccati equations | 169 Asymptotics of solution of the Cauchy problem for the N-particle Liouville equation | 174 Stationary asymptotic solutions of the many-body problem | 175 Asymptotic solutions of the equation corresponding to the uniformization of a functional on an abstract Hamiltonian algebra | 177 Definition of abstract Hamiltonian algebra | 178 Abstract half-density | 181 Tensor products of Hamiltonian algebras and of their representations. The notion of uniformization | 183 Construction of asymptotics of solution of the Cauchy problem | 186 Existence of solutions of the Hartree and Riccati equations | 188 Existence and uniqueness of solutions of a Hartree-type equation | 188 Existence of solutions of the variational equation and the Riccati equation | 189

Complex germ method in the Fock space | 193 Introduction | 193 Systems with variable number of particles | 193 Application of the complex germ method to systems with finitely many degrees of freedom | 194 4.1.3 Projection onto the N-particle subspace and Lagrangian manifolds with complex germ | 197 4.2 Complex germ at a point in the Fock space | 198 4.2.1 Auxiliary notions | 198 4.2.2 Checking the axioms of abstract canonical operator | 199 4.2.3 Canonical transformation of the phase space | 201 4.2.4 Proper canonical transformation of infinite-dimensional phase space | 202 4.2.5 Complex germ | 203 4.2.6 Formal asymptotic solutions of the equations of motion | 208 4.2.7 Equation for the operator W | 209 4.2.8 Commutation of the canonical operator with other operators | 210 4.2.9 Asymptotic solutions of the equations of motion | 213 4.2.10 Corrections to the asymptotic formula | 214 4.3 Superposition of wave packets in finite-dimensional quantum mechanics | 216 4.3.1 Statement of the problem | 216 4.3.2 Superposition of wave packets in the one-dimensional case | 217 4.3.3 Superposition of wave packets in the multidimensional case | 221

Contents | XXIII

4.3.4 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 4.4.6 4.4.7 4.5 4.5.1 4.5.2 4.5.3 4.6 4.6.1 4.6.2 4.6.3 4.7 4.7.1 4.7.2 4.7.3 4.7.4 4.7.5 4.7.6 4.7.7 4.A 4.A.1 4.A.2 4.A.3 4.A.4

Norm of the wave function corresponding to an isotropic manifold | 225 Canonical operator corresponding to a Lagrangian manifold with a complex germ | 228 Canonical operator corresponding to an isotropic manifold | 228 Case of a Lagrangian manifold of full dimension | 231 Holomorphic representation of distributions in 𝒮 ′ | 235 General case | 236 Complex germ in the holomorphic representation | 244 Definition of canonical operator on a Lagrangian manifold with complex germ | 252 Solution of the Cauchy problem: asymptotics corresponding to isotropic manifolds | 256 Superposition of wave functions corresponding to an abstract canonical operator | 259 Calculation of the norm as ε → 0 | 260 Canonical operator corresponding to an isotropic manifold | 262 Formal asymptotic solutions of the equations of motion corresponding to isotropic manifolds | 264 Specific features of statement of the Cauchy problem for a topologically invariant isotropic manifold | 265 Isotropic manifold diffeomorphic to a circle | 265 General case: main difficulties | 268 Definition of the operator 𝒦Γk | 270 Asymptotic formulas in the Fock space corresponding to finite-dimensional isotropic manifolds | 272 Verification of properties of the canonical operator | 272 Canonical operator corresponding to an isotropic manifold | 274 Isotropic manifold of special form and fixing the number of particles | 277 Case of two types of particles | 280 Complex germ | 282 Canonical transformation of complex germ and formal asymptotic solutions | 286 Asymptotic solutions of the Cauchy problem | 288 Separation of the cyclic variable and construction of the tunnel asymptotics | 289 Additive asymptotics: advantages and drawbacks | 289 Equation in the representation of generating functionals | 289 Definition of tunnel asymptotics | 291 Equation for the exponential factor, preexponential factor, and corrections | 291

XXIV | Contents 4.A.5 4.A.6 4.A.7 5 5.1 5.1.1 5.1.2 5.1.3 5.2 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.3.6 6 6.1 6.2 6.2.1 6.2.2 6.2.3 6.3 6.3.1 6.3.2 6.3.3 6.4 6.4.1 6.4.2

Equations of characteristics | 292 Relationship to the germ asymptotics | 295 Application to the multiparticle Schrödinger equation | 296 Asymptotic methods in problems with operator-valued symbol | 299 Problems with operator-valued symbol in quantum mechanics | 299 Physical problems resulting in equations with operator-valued symbol | 299 Analog of the Ehrenfest theorem for the operator-valued case. “Multivalence” of the Hamiltonian function | 301 Asymptotics of the solution of the Cauchy problem for equations with operator-valued symbol | 303 Abstract canonical operator in problems with operator-valued symbol | 313 Equations with operator-valued symbol in the many-particle problem | 316 Several examples of equations with operator-valued symbol | 316 Analog of the Hartree equation for the operator-valued case | 318 Nonpreservation of chaos for correlation functions in the operator-valued case | 321 Construction of asymptotics for the solution of the Cauchy problem | 323 Analysis of the system of recursive relations | 325 Case of the variable number of particles | 327 Semiclassical field theory in the Hamiltonian formalism | 331 Introduction | 331 Lagrangian manifolds with the complex germ in quantum field theory | 332 Scalar field theory: germ at a point | 332 Scalar field theory: germ on a manifold and quantization near solitons | 334 Scalar quantum electrodynamics | 335 Equations with operator-valued symbol in quantum field theory | 337 Scalar field | 337 Electromagnetic field | 339 Yang–Mills field | 342 Difficulties with the Hamiltonian field theory | 343 Difficulties with the Hamiltonian field theory in the functional representation | 343 Difficulties with the Hamiltonian field theory in the Fock representation | 347

Contents | XXV

6.5 6.5.1 6.5.2 6.6 6.6.1 6.6.2 6.6.3 6.7 6.7.1 6.7.2 6.8 6.8.1 6.8.2 6.8.3 6.8.4 6.9 6.9.1 6.9.2 6.9.3 6.9.4 7 7.1 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 7.3 7.3.1 7.3.2 7.3.3 7.3.4 7.3.5

General scheme of renormalization of the Hamiltonian field theory | 352 Conditions on the Hamiltonian and the initial data | 352 Axioms of regularized field theory | 355 Faddeev transformation and removal of the Stückelberg divergences | 358 Faddeev transformation | 358 S-matrix and nonuniqueness of the Faddeev transformation | 363 Faddeev transformation and particle decay rate | 365 Bogoliubov S-matrix and renormalization of the equations of motion | 367 Bogoliubov S-matrix and the equation of motion | 367 Bogoliubov S-matrix in the first order of perturbation theory | 370 Renormalization in semiclassical field theory | 371 Semiclassical approximation in regularized field theory | 371 Problem of removal of regularization in Hamiltonian formalism | 376 Condition on the Gaussian state vector | 380 Some conclusions | 383 Invariance of the conditions on the complex germ | 384 Transformation of the Riccati equation | 384 On operators with smooth Weyl symbol | 388 Study of the transformed equation | 392 Construction of an approximate solution of the Riccati equation | 397 Asymptotic methods for systems of a large number of fields | 403 Introduction | 403 O(N)-symmetric anharmonic oscillator as an analog of a many-field system | 404 Collective field method | 405 Classical equations in the second-quantized approach | 406 Relation between classical equations under various approaches | 406 Semiclassical wave functions under various approaches | 407 Asymptotic spectrum as N → ∞ | 410 Formalism of the third quantization and the semiclassical approximation | 412 Asymptotic methods in the theory of large number of fields on a lattice | 412 Classical equations in the third-quantized approach to the theory of large number of fields | 415 On the construction of asymmetric solutions | 417 φφχ model | 418 Case of spontaneous symmetry breaking | 419

XXVI | Contents 7.3.6 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.4.5 7.5 7.5.1 7.5.2 7.5.3 7.5.4

O(N)-asymmetric theories | 420 On renormalizations of the classical equations | 421 On regularization and renormalization | 421 On Gaussian and non-Gaussian solutions of classical equations | 422 On singularities of the Gaussian quadratic form | 423 Renormalization of the mass and the coupling constant | 424 Renormalization of the cosmological constant | 426 Asymptotic spectrum of the Hamiltonian of a large number of fields | 428 Stationary solutions of the Hartree-type equation | 428 Classical energies of various series | 430 Near-vacuum solutions of the N-field equation | 432 Other series of asymptotics | 437

Concluding remarks | 439 Bibliography | 443 Index | 447

List of notation Numbering of formulas, definitions, lemmas, theorems, etc. We use two-level numbering of formulas and three-level numbering of everything else (definitions, lemmas, theorems, etc.). For example: (2.6) is a reference to formula (6) in Chapter 2; Lemma 2.6.5 refers to Lemma 5 in Section 6 of Chapter 2.

Some notation We mean summation over repeated indices. – ab = ∑nk=1 ak bk is the scalar product of vectors a, b ∈ ℝn ; – |a|, a ∈ ℝn , is the length of a vector a ∈ ℝn , |a| = √a∗ a; 𝜕f 𝜕f – a 𝜕x = ∑nk=1 ak 𝜕x , a, x ∈ ℝn ; k

– – –

AT is the matrix transposed to a matrix A; A∗ is the matrix complex conjugate to a matrix A; A+ is the matrix Hermitian conjugate to a matrix A;

–

f : 𝒳 → 𝒴 ; 𝒳 → 𝒴 is a map f from a set 𝒳 to a set 𝒴 ;

– – – – – – – – – –

–

– – –

f

f

f : x 󳨃→ y, x 󳨃→ y, x ∈ 𝒳 , y ∈ 𝒴 is a map f from a set 𝒳 to a set 𝒴 ; 𝒳 × 𝒴 is the direct product of sets 𝒳 and 𝒴 ; 𝒳 N = 𝒳 × ⋅ ⋅ ⋅ × 𝒳 is the Nth power of a set X; Tr A is the trace of an operator A; ker A = {f ∈ H1 | Af = 0} is the kernel of a linear operator A : H1 → H2 ; Im A = {f ∈ H2 | ∃g ∈ H1 : f = Ag} is the image of a linear operator A : H1 → H2 ; ℒ/ℛ is the quotient space of a vector space ℒ by a subspace ℛ; [f ] is the equivalence class from ℒ/ℛ whose representative is f ∈ ℒ; E is the unity operator; (a, b) is the scalar product of vectors a, b of a Hilbert space; we adopt the system of notation in which the scalar product is linear with respect to the second argument but not with respect to the first one: (a, αb) = α(a, b); (αa, b) = α∗ (a, b); Lp (𝒳 → 𝒴 ) is the normed space whose elements are equivalence classes of functions 𝒳 → 𝒴 (𝒳 is a space with measure, 𝒴 is a normed space), for which the integral ‖f ‖p = ∫ ‖f (x)‖p dx exists (two functions are called equivalent if they differ on a set of zero measure); Lp (𝒳 ) ≡ Lp (𝒳 → ℂ); 𝒟(ℝn ) is the space of infinitely differentiable functions ℝn → ℂ with compact support; 𝒮 (ℝn ) is the space of infinitely differentiable functions ℝn → ℂ such that xi1 ⋅ ⋅ ⋅ xik 𝜕x𝜕 ⋅ ⋅ ⋅ 𝜕x𝜕 f ∈ L2 (ℝn ) for k, m = 0, 1, 2, . . . . j1

jm

https://doi.org/10.1515/9783110762709-203

1 Abstract canonical operator and symplectic geometry 1.1 Introduction In physics, one often meets evolution equations depending on a real parameter ε, i

d ε ψ (t) = H ε (t)ψε (t), dt

(1.1)

where the parameter t plays the role of time, ψε (t) belongs to certain Hilbert space ℋε , depending, in general, on the parameter ε, H ε is certain operator in this space. In the case when due to physical arguments the parameter ε can be treated as small, the problem arises in the construction of asymptotic solutions of equation (1.1) as ε → 0. In certain cases, one can manage to express such approximate solutions through (exact) solutions of “more simple” equations than equation (1.1). For instance, approximate solutions of equations of wave optics are expressed through solutions of more simple equations of geometric optics, asymptotic solutions of the quantum mechanical Schrödinger equation are expressed through solutions of the Hamiltonian system, and approximate solutions of the many particles problem are expressed, as we shall see in the forthcoming chapters, through solutions of the self-consistent field equations (the Vlasov equation in classical case and the Hartree equation in quantum case). In this chapter, we consider the main problems arising in the construction of asymptotic solutions, introduce some general notions, which will be used throughout the book, and illustrate them in the example of semiclassical approximation to the equations of finite-dimensional quantum mechanics. We shall establish certain identities (formulas (1.34) and (1.37)), which hold for asymptotic solutions of equation (1.1) in the general case of arbitrary spaces ℋε .

1.1.1 Construction of approximate solutions of various equations: comparison of methods One of the methods of construction of asymptotic solutions of equation (1.1) is the method of the complex germ at a point [25, 27], applicable if all of the spaces ℋε coincide with the space L2 (ℝn ), and the operators H ε depend on ε as 1 1 Hε = H ( 𝜕 ε −iε 𝜕x ,

2 ) x

(1.2)

(in the case when the function H is a polynomial, notation (1.2) means that the op𝜕 erators of multiplication by x are put to the left from the differentiation operators 𝜕x ; https://doi.org/10.1515/9783110762709-001

2 | 1 Abstract canonical operator and symplectic geometry for the definition of operator (1.2) in the general case, see [25] or Section 1.5). For simplicity, consider the case n = 1. The asymptotics for the solution of equation (1.1), constructed by the method of complex germ at a point, reads i

ε ψε (t) = e ε S(t) KP(t),Q(t) f (t) + O(√ε),

(1.3)

where P(t), Q(t) ∈ ℝ satisfy the Hamiltonian system ̇ = 𝜕H (P(t), Q(t)), Q(t) 𝜕P

̇ = − 𝜕H (P(t), Q(t)), P(t) 𝜕Q

(1.4)

the function S(t) is the action on the classical trajectory t

̇ ′ ) − H(P(t ′ ), Q(t ′ ))), S(t) = S(0) + ∫ dt ′ (P(t ′ )Q(t 0

the element f (t) ∈ L2 (ℝ) not depending on ε also satisfies a certain equation (see Secε tion 1.5), and by KP,Q we denote the operator from L2 (ℝ) to L2 (ℝ) of the form ε (KP,Q f )(x) =

1

ε1/4

i

e ε P(x−Q) f (

x−Q ), √ε

(1.5)

which will also be called the canonical operator. Note that the canonical operator, which is usually used in the theory of a complex germ at a point [4, 27], differs from operator (1.5) by a certain one-to-one isometric transformation (see Section 1.5). In this case, the parameter ε is, as a rule, denoted by ℏ and called the Planck constant. The element ψε (t) from L2 (ℝ) has the physical sense of the wave function describing the state of a quantum mechanical particle at the moment of time t, x plays the role of the coordinate of the particle, the operator of multiplication by x plays the ̂ the differentiation operator −iε 𝜕 plays the role of role of the coordinate operator Q, 𝜕x ̂ the momentum operator P, and the operator εH ε is the Hamiltonian operator (see Appendix 1.A). Consider the wave functions (1.3) at a certain fixed moment of time. They possess the following interesting property. The densities of probability distributions of coordinate and momentum are concentrated near the points Q and P, and have the width of order √ε. Indeed, for P = Q = 0, the wave function (1.2) has, up to a phase factor, the form f (x/√ε), and satisfies this property of probability distributions. In the general i case, the wave function (1.3) is obtained from the function e ε S f (x/√ε) by the action of i operators of multiplication by e ε Px and the shift of x by Q, which shifts the probability distribution density of momentum by P, and the probability distribution density of coordinate by Q. Hence, the mean values of the observables of a certain type, namely, of the monôn , are expressed as ε → 0 only through P and Q and do not depend on f ; it is ̂mQ mials P

1.1 Introduction

| 3

not difficult to show that they tend as ε → 0 to P m Qn . The analogous statement holds also for linear combinations of these monomials. Hence, formula (1.4) means that the Ehrenfest theorem holds: the Heisenberg equation for the mean values of functions of P, Q independent of ε goes as ε → 0 to the corresponding equations of classical mechanics (a wave packet can move only along a classical trajectory). On the other hand, for computing the limit as ε → 0 of mean values of general uniformly in ε bounded observables Oε one needs to use also the function f . For example, for the observable Oε = φ(

x−Q ), √ε

the limit of mean value as ε → 0 equals ε→0 󵄨 󵄨2 ⟨Oε ⟩ 󳨀→ ∫ dξφ(ξ )󵄨󵄨󵄨f (ξ )󵄨󵄨󵄨

and depends on f . In [32–37, 39], a method is developed in the construction of asymptotic solutions of equations describing a system of N particles, as N → ∞. These equations have the form (1.1). The parameter ε tending to zero in this case runs over a countable set of values {1/N}, where N is the natural number playing the role of the number of particles. The spaces ℋ1/N are in this case different and have the form L2 (ℝνN ), ν ∈ ℕ. In the case of the quantum system of N bosons moving in ν-dimensional space, equation (1.1) is the N-particle Schrödinger equation, and elements of the space ℋ1/N play the role of N-particle wave functions. In the case of N classical particles, equation (1.1) is the multi-particle Liouville equation, and elements of the space ℋ1/N play the role of N-particle semidensity functions (introduced in [37]), whose squares coincide with the N-particle function of density of probability distribution ρN (p1 , q1 , . . . , pN , qN ), where p1 , . . . , pN play the role of the momenta of particles, and q1 , . . . , qN play the role of the coordinates of particles. The asymptotics of solutions of equation (1.1) is constructed in the form i

ε ψε (t) = e ε S(t) Kφ(t) f (t) + O(√ε),

ε = 1/N.

(1.6)

Here, S(t) ∈ ℝ as in the previous case (1.3), φ(t) ∈ L2 (ℝν ) is certain function whose number of arguments does not tend to infinity as ε → 0 (which is equivalent to N → ∞), in contrast to the number of arguments of the function ψε (t); f (t) is in this case ε an element of certain Hilbert space ℱφ(t) (see Chapter 2), the operator Kφ(t) , which will also be called the canonical operator, is defined in Chapter 2. By O(√ε), we denote an element of the space ℋε such that the square of its norm in this space is of order O(√ε)

4 | 1 Abstract canonical operator and symplectic geometry as ε → 0. For S, φ, f also certain equations are obtained; in particular, the equation for φ coincides with the Hartree equation in the quantum case and reduces to the Vlasov equation in the classical case. As in the previous example, at each fixed moment of time all N-particle functions (1.6), corresponding to one function φ and various functions f , possess certain common property: the limits as ε → 0 of mean values of observables of special kind are determined uniquely by the function φ and do not depend on the function f . In particular, for the case of a system of N classical particles, this means that the so-called chaos property for k-particle distributions [14, 15] holds: ∫ dpk+1 dqk+1 ⋅ ⋅ ⋅ dpN dqN ρN (p1 , q1 , . . . , pk , qk ) N→∞ k=const

󳨀→ ρ(p1 , q1 ) ⋅ ⋅ ⋅ ρ(pk , qk ),

(1.7)

where the function ρ coincides with the square of the function φ and does not depend on f . The result (1.6) confirms the statement that the chaos property (1.7) is conserved in time and that ρ satisfies the Vlasov equation. Thus, we see that the property of chaos conservation is the analog of the quantum mechanical Ehrenfest theorem. On the other hand, the mean values of general observables bounded uniformly in N are not expressed uniquely as N → ∞ through φ, for computing the limits of these mean values as N → ∞ one needs also to know an element of the space ℱφ . In particular, for k tending to infinity in formula (1.7) the chaos property does not hold [37]. Thus, we see that for construction of asymptotic solutions of equation (1.1) in the examples listed above one introduces the following objects: (1) a phase space ℳ; (2) auxiliary Hilbert spaces ℱX , depending in general on an element X ∈ ℳ of the phase space but not depending on ε; (3) the canonical operator KXε —certain linear bounded uniformly in ε operator acting from the space ℱX to the space ℋε . It also turns out that the elements of the spaces ℋε of the form KXε f , where f ∈ ℱX , possess for fixed X the general property: the mean values of observables of special kind tend as ε → 0 to definite values independent on f , and for the observables of general kind, this limit depends on f . One poses the following Cauchy problem. Consider a solution ψε (t) of equation (1.1) with the initial condition of the following special form: i

ψε (0) = e ε S0 KXε 0 f0 .

1.1 Introduction

| 5

It can be proved that there exists a real function S(t), an element of the phase space X(t) ∈ ℳ, and an element f (t) of the space ℱX(t) such that i 󵄩ε ε→0 󵄩󵄩 ε S(t) ε 󵄩󵄩ψ (t) − e ε KX(t) f (t)󵄩󵄩󵄩 󳨀→ 0,

(1.8)

where we denote by ‖ ⋅ ‖ε the norm in the Hilbert space ℋε . The check of the property (1.8) is usually made as follows: (1) one considers the substitution of the element i

ε e ε S(t) KX(t) f (t)

of the space ℋε into equation (1.1); (2) it is shown that, under certain conditions on the element of the phase space X(t) and f (t) ∈ ℱX(t) the discrepancy is small as ε → 0; (3) the property (1.8) is checked by the method of an estimate of the inverse operator [24]. i

ε f (t) ∈ ℋε is an approximate solution of equation (1.1) Note that the element e ε S(t) KX(t) also in the case when S, X, f are smooth functions of √ε in a neighborhood of the point √ε = 0. Let us also note that for the problems with an operator valued symbol considered in Chapter 5, one should consider the substitution into equation (1.1) of the element of the space ℋε of the form i

ε (f0 (t) + √εf1 (t) + εf2 (t)). e ε S(t) KX(t)

1.1.2 Abstract canonical operator and geometry of the phase space It turns out that to each canonical operator satisfying certain conditions, one can assign certain geometric structures on the phase space. Let us illustrate this statement in the example of the canonical operator (1.5). Let us look how the canonical operator changes if one shifts the point (P, Q) of the phase space by a quantity of the order of ε. Formula (1.5) implies that ε ε (KP+εp,Q+εq f )(x) = e−iPq (KP,Q f )(x) + zε (x), ε→0

(1.9)

where ‖zε ‖L2 󳨀→ 0. Thus, the canonical operator is multiplied under such shift of element of the phase space by the exponential factor e−iPq up to higher corrections. The quantity under the exponent can be considered as the value of the action differential 1-form ω1 = PdQ on the tangent vector to the phase space at the point (P, Q), with the

6 | 1 Abstract canonical operator and symplectic geometry coordinates of the tangent vector (p, q). The differential of the 1-form ω1 is the 2-form dω1 = ω2 = dP ∧ dQ, providing a symplectic structure on the phase space. ε We can consider canonical operators such that the operators KXε and KX+εδX differ, up to higher corrections, only by a phase factor. Then to each such canonical operator one can assign an action 1-form ω1 on the phase space, and hence, its differential providing a symplectic structure on the phase space. Such 1-form arises also in the study of canonical operator (1.6), and also in some other cases. In modern mathematics, one develops the approach of geometric quantization [13, 16, 17] in which the problem is posed as follows: given a classical phase space with a fixed symplectic structure, and one considers the question on constructing quantum theory. In this chapter, we consider an approach in a sense inverse to geometric quantization; given a canonical operator, one assigns to it various geometric structures on the phase space. For example, in the case of construction of approximate solutions of the many-body problem, the canonical operator constructed in Chapter 2 using the considerably simple physical arguments, induces geometric structures on the phase space, which is the space of complex functions of one variable. Thus, in this chapter we assign to a family of asymptotic solutions of equation (1.1) (where ℋε not necessarily has the form (1.2)) a phase space with a symplectic structure. Let us now consider the shift of a point of the phase space by a quantity of order √ε. In the case of the canonical operator (1.5), it is not difficult to see that the following relation holds: ε ε KP+√εp,Q+√εq f = const KP,Q VP,Q [p, q]f ,

(1.10)

where VP,Q [p, q] is the operator in the space L2 (ℝ) = ℱP.Q of the form 1 𝜕

VP,Q [p, q] = ei(px−q i 𝜕x ) .

(1.11)

The quantity under the exponent (1.11) can be considered as the value of the operator valued 1-form Ω = x ⋅ dP −

1 𝜕 ⋅ dQ i 𝜕x

(1.12)

on the tangent vector with the coordinates (p, q). A similar operator valued 1-form on the phase space will be introduced in Section 1.2 also in the general case. We shall show that also in the general case the following commutation relations hold: [Ω(δx1 ), Ω(δx2 )] = iω2 (δx1 , δx2 )E,

(1.13)

where E is the unit operator, ω2 = dω1 (for the form (1.12), the check of relation (1.13) is easy).

1.1 Introduction

| 7

It turns out that knowing geometric structures ω1 and Ω induced by the canonical operator, one can establish certain interesting identities on S(t) and f (t) ∈ ℱX(t) without substitution of the element i

ε e ε S(t) KX(t) f (t)

(1.14)

of ℋε into equation (1.1). The main idea of derivation of the identities is the following. Let X(t) and X(t) + εδX(t) + O(ε2 ) be two solutions of the equation for an element of the phase space, and S(t) and S(t) + εδS(t) + O(ε2 ) be two solutions of the equation for S. Let us use the property ε ε KX(t)+εδX(t) f (t) = e−iω1 (δX(t)) KX(t) f (t) + O(√ε).

Let us substitute these solutions into formula (1.14). We obtain the following asymptotic solutions of equation (1.1): i

ε e ε S(t) KX(t) f (t)

and i

ε e ε S(t)+i(δS(t)−ω1 (δX(t))) KX(t) f (t).

Here, we have also taken into account that under the change of f (t) by a quantity of order ε, the element (1.14) of ℋε also changes by a quantity of order ε. If the solution of the Cauchy problem for equation (1.1) is unique, then δS(t) − ω1 (δX(t)) = const.

(1.15)

Let us consider condition (1.15) in the case of canonical operator (1.5) in more detail. Denote by U(t) the transformation taking the initial condition (P0 , Q0 ) for system (1.4) into the solution (P(t), Q(t)) of this system at the moment t, and define the function St (P0 , Q0 ) from the relation S(t) = S(0) + St (P0 , Q0 ). Property (1.15) means that dSt = P(t)dQ(t) − P0 dQ0 ,

(1.16)

where S(t), P(t), Q(t) are functions of (P0 , Q0 ). In particular, this implies that the transformation U(t) should preserve the 2-form ω2 . The function S(t) is defined up to an additive constant not depending on (P0 , Q0 ). Let us now consider the operators WX,t : ℱX → ℱU(t)X , taking the initial condition for an equation on f (t) into the solution of this equation. Let us study the following question: what properties of this operator can one obtain using the operator valued 1-form Ω introduced above? To this end, consider two solutions X(t) and

8 | 1 Abstract canonical operator and symplectic geometry X(t) + √εδX(t) + O(ε) of the equation for X(t) ∈ ℳ, differing by a quantity of order √ε, and let us use property (1.10). We obtain the following identity: WX(0),t VX(0) [δX(0)] = const VX(t) [δX(t)]WX(0),t .

(1.17)

In the case of the canonical operator (1.5), the statement (1.17) is equivalent to the following one: WP(0),Q(0),t (xdP0 −

1 𝜕 1 𝜕 dQ ) = (xdP(t) − dQ(t))WP(0),Q(0),t . i 𝜕x 0 i 𝜕x

(1.18)

We shall study in Sections 1.3 and 1.5, what the ambiguity is in defining the operator WX,t by relation (1.17). To investigate this question, it is useful to introduce the notion of an abstract complex germ (Section 1.3). It turns out that in many cases the operators WX,t are defined uniquely up to a multiplicative number constant. In the case of finitedimensional phase space ℳ, the operator WX,t satisfying the property (1.17) exists if and only if the transformation U(t) is canonical (i. e., for some S(t) the property (1.16) holds). In the infinite- dimensional case, for not any canonical transformation U(t) there exists such an operator WX,t . Thus, from the class of all canonical transformations, a class of proper canonical transformations is distinguished, for which such an operator WX,t exists (see Section 1.2). Note that for linear transformations of phase space, the notions of canonical and proper canonical transformation coincide with that introduced in Berezin’s book [6].

1.1.3 Problem of existence of equations of motion and the concept of asymptotic quantization In the text above, we assumed that the operators H ε in equation (1.1) are already given, and we discussed the problem of construction of approximate solutions of equation (1.1). However, one can pose the question also in another way. Assume that, given the spaces ℋε , the phase space ℳ, the Hilbert spaces ℱX , and the canonical operator KXε , we pose the question on what evolution operators Utε in the space ℋε can correspond to a given evolution operator Ut in the phase space ℳ, and how to construct the operator WX,t . Such a question could arise, for example, if we knew the Hamiltonian equations in classical mechanics but did not know the Schrödinger equation in quantum mechanics. The introduced geometric structures allow one to define the notion of a formal asymptotic solution of equations of motion (Section 1.2), without using the equations of motion themselves, and to construct the operator WX,t using the property (1.17) and without using equation (1.1) and commutation of canonical operator KXε with the operator (id/dt − H ε (t)). In some cases, not only the operator H ε is unknown, but the question on the choice of the space ℋε remains open. Such a situation arises, for instance, in quantum

1.2 Geometric structures on the phase space | 9

field theory, where also the similar small parameter ε arises. Up to now, it is unknown whether there exists a nontrivial model of relativistic quantum field theory in fourdimensional space-time [9]. Even for two-dimensional space-time, the construction of an example of such a theory is a nontrivial problem [12, 50]. In essence, at present in quantum field theory, only the postulated perturbation series has sense. In this respect, the question arises on postulation of asymptotic quantization. We can reformulate the main concepts of quantum field theory in terms of the spaces ℱX by analogy with quantum mechanics, in which the mean values of general observables bounded uniformly in ε are uniquely determined as ε → 0 by an element X of the phase space ℳ and an element of the space ℱX . In quantum field theory, the canonical operator has only formal sense, since the spaces ℋε are ill-defined. However, one can heuristically define the action 1-form ω1 and the operator valued 1-form Ω, and put them as the basis of the theory. In this case, the definition of operators WX,t from formula (1.17) still makes sense. Thus, the dynamics in the space ℋε , which has only formal sense in this case, is replaced by the dynamics in the space ℱX . Respectively, one can study the dynamics of physical variables. Besides the operators WX,t corresponding to translations in time, one can introduce by formula (1.17) operators W corresponding to other translations from the Poincaré group: space shifts, rotations, passing to uniformly rectilinearly moving counting system. One can also reformulate all axioms of quantum field theory in terms of spaces ℱX . This approach might be called asymptotic quantization of field theory. Let us now pass to the concrete definition of an abstract canonical operator.

1.2 Abstract canonical operator and induced geometric structures on the phase space In this section, we use notation from Appendix 1.B.

1.2.1 Auxiliary notions Let I be a subset of the set ℝ, containing zero as a limit point. Assume that for each ε ∈ I a Hilbert space ℋε is given, whose norm will be denoted by ‖ ⋅ ‖ε . Let also ℳ be a smooth manifold, and assume that for each X ∈ ℳ a Hilbert space ℱX is given, whose norm will be denoted by ‖ ⋅ ‖X . Let us also assume that for all X1 , X2 ∈ ℳ a symmetric distance function ρX1 ,X2 : ℱX1 × ℱX2 → ℝ is given, satisfying the following properties: (a) ρX1 ,X2 (f1 , f2 ) ≥ 0; ρX1 ,X2 (f1 , f2 ) = ρX2 ,X1 (f2 , f1 ); (b) ρX1 ,X3 (f1 , f3 ) ≤ ρX1 ,X2 (f1 , f2 ) + ρX2 ,X3 (f2 , f3 ); (c) ρX1 ,X2 (f1 + f1′ , f2 + f2′ ) ≤ ρX1 ,X2 (f1 , f2 ) + ρX1 ,X2 (f1′ , f2′ ); (d) ρX,X (f1 , f2 ) = ‖f1 − f2 ‖X .

10 | 1 Abstract canonical operator and symplectic geometry Note that in the case when all of the spaces ℱX are subspaces of a Hilbert space ℱ (in particular, if all of these spaces coincide), one can choose as the function ρ the distance function in the space ℱ , and the properties (a)–(d) will hold automatically. Let A1 (ε) and A2 (ε), ε ∈ I, be two families of linear operators bounded uniformly in ε, acting from the spaces ℱX1 (ε) and ℱX2 (ε) , respectively, to the space ℋε , and such ε→0

ε→0

that X1 (ε) 󳨀→ X, X2 (ε) 󳨀→ X. Let us say that these two families are equivalent, A1 (ε) ≈ A2 (ε), if there exists a dense subset 𝒟 of the space ℱX such that for any f ∈ 𝒟 and for any f1 (ε) ∈ ℱX1 (ε) , f2 (ε) ∈ ℱX2 (ε) with the property ε→0

ρX1 (ε),X (f1 (ε), f ) 󳨀→ 0,

ε→0

ρX2 (ε),X (f2 (ε), f ) 󳨀→ 0,

and the following holds: 󵄩󵄩 󵄩ε ε→0 󵄩󵄩A1 (ε)f1 (ε) − A2 (ε)f2 (ε)󵄩󵄩󵄩 󳨀→ 0. It is easy to check that this relation is indeed an equivalence relation, i. e., it is reflexive, symmetric, and transitive. Below in the proof of the equivalence of families of operators, we shall use the following lemma. Lemma 1.2.1. Let {f } ∈ ℱX be a complete system of vectors of the space ℱX , and assume ε→0

that for any f ∈ {f } there exist f1 (ε) ∈ ℱX1 (ε) , f2 (ε) ∈ ℱX2 (ε) such that ρX1 (ε),X (f1 (ε), f ) 󳨀→ 0, ε→0

ε→0

ρX2 (ε),X (f2 (ε), f ) 󳨀→ 0, ‖A1 (ε)f1 (ε) − A2 (ε)f2 (ε)‖ε 󳨀→ 0. Then A1 ≈ A2 .

Proof. Let f1′ (ε) ∈ ℱX1 (ε) , f2′ (ε) ∈ ℱX2 (ε) , ε→0

′ ρX1,2 (ε),X (f1,2 (ε), f ′ ) 󳨀→ 0,

where f ′ is an element of the linear span of {f }, f ′ = λ1 f (1) + ⋅ ⋅ ⋅ + λs f (s) ,

f (1) , . . . , f (s) ∈ {f },

λ1 , . . . , λs ∈ ℂ.

This linear span, due to completeness of {f }, forms a dense set 𝒟 in the space ℱX . By (i) the conditions of Lemma 1.2.1, there exist elements f1,2 (ε) ∈ ℱX1,2 (ε) such that ε→0

(i) ρX1,2 (ε),X (f1,2 (ε), f (i) ) 󳨀→ 0,

i = 1, . . . , s,

and 󵄩󵄩 󵄩󵄩ε s s 󵄩󵄩 󵄩 󵄩󵄩A1 (ε) ∑ λi f (i) (ε) − A2 (ε) ∑ λi f (i) (ε)󵄩󵄩󵄩 ε→0 󵄩󵄩 󵄩󵄩 󳨀→ 0. 1 2 󵄩󵄩 󵄩󵄩 i=1 i=1

1.2 Geometric structures on the phase space | 11

Since the operators A1 and A2 are bounded, we have 󵄩󵄩ε 󵄩󵄩 s 󵄩󵄩 󵄩󵄩 󵄩󵄩ε 󵄩󵄩󵄩 ′ (i) ′ ′ 󵄩󵄩 A (ε)(f (ε) − λ f (ε)) A (ε)f ≤ (ε) − A (ε)f (ε) ∑ 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 1 i 1 2 1 1 2 󵄩󵄩 󵄩󵄩 󵄩󵄩 i=1 󵄩 󵄩󵄩ε 󵄩󵄩 s 󵄩󵄩 󵄩󵄩 ′ (i) 󵄩 + 󵄩󵄩A2 (ε)(f2 (ε) − ∑ λi f2 (ε))󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 i=1 󵄩 󵄩 󵄩󵄩 󵄩󵄩ε s s 󵄩󵄩 󵄩󵄩 ε→0 (i) (i) 󵄩 + 󵄩󵄩A1 (ε) ∑ λi f1 (ε) − A2 (ε) ∑ λi f2 (ε)󵄩󵄩󵄩 󳨀→ 0, 󵄩󵄩 󵄩󵄩 i=1 i=1 󵄩 󵄩 since 󵄩󵄩 󵄩󵄩 s 󵄩󵄩 ′ 󵄩 ′ 󵄩󵄩f (ε) − ∑ λi f (i) (ε)󵄩󵄩󵄩 ≤ ρX1,2 (ε),X (f1,2 (ε), f ′ ) 󵄩󵄩 1,2 󵄩󵄩 1,2 󵄩󵄩 󵄩 i=1 󵄩X1,2 (ε) s

s

i=1

i=1

ε→0

(i) + ρX1,2 (ε),X (∑ λi f1,2 (ε), ∑ λi f (i) ) 󳨀→ 0.

Lemma 1.2.1 is proved.

1.2.2 Definition of an abstract canonical operator Definition 1.2.1. An abstract canonical operator KXε , ε ∈ I, X ∈ ℳ, is a linear operator from the space ℱX to the space ℋε , such that the following axioms hold. Axiom 1.2.1. Let X(ε) be a smooth curve on the manifold ℳ such that X(0) = X0 . Then, provided ε→0

ρX(ε),X (f (ε), f0 ) 󳨀→ 0,

f (ε) ∈ ℱX(ε) ,

f0 ∈ ℱX0 ,

the following relation holds: 󵄩󵄩 ε 󵄩ε ε→0 󵄩󵄩KX(ε) f (ε)󵄩󵄩󵄩 󳨀→ ‖f0 ‖X0 . Axiom 1.2.2. Let X1 (ε) and X2 (ε) be two equivalent curves on the manifold ℳ. Then KXε 1 (ε) ≈ KXε 2 (ε) . Axiom 1.2.3. Let X(ε) be a curve on the manifold ℳ, X(0) = X0 , and b be a tangent vector to the manifold ℳ at the point X0 with the coordinates β on the ith chart. Then there exists a complex function χX0 of b, smoothly depending on the coordinates of X0

12 | 1 Abstract canonical operator and symplectic geometry and on the components of b, such that ε ε KΦ (i)−1 (Φ(i) X(ε)+εβ) ≈ χX [b]KX(ε) . 0

(1.19)

Axiom 1.2.4. (a) Under the same conditions on X(√ε) and b, the following relation holds: ε −1 ε ̃ χΦ (i)−1 (Φ(i) X(√ε)+√εβ/2) (b/√ε)KΦ(i)−1 (Φ(i) X(√ε)+√εβ) ≈ KX(√ε) VX(√ε) [b]

(1.20)

for the certain linear operator VX(√ε) [b] acting on ℱX(√ε) . In formula (1.20), b̃ is the tangent vector to the manifold at the point Φ(i)−1 (Φ(i) X(√ε) + √εβ/2), which has the same coordinates in the ith chart as the vector b. (b) The set 𝒟X0 ⊂ ℱX0 consisting of all elements f ∈ ℱX0 such that the mapping b 󳨃→ VX0 [b]f is a smooth mapping T ℳX0 → ℱX0 , is dense in the space ℱX0 . ε→0

(c) For f (ε) ∈ ℱX(√ε) , ρX(ε),X0 (f (ε), f0 ) 󳨀→ 0, f0 ∈ 𝒟X0 , the following relation holds: ε→0

VX(√ε) [b]f (ε) 󳨀→ VX0 [b]f0 . 1.2.3 Well-definedness of the canonical operator—the action 1-form on the phase space Let us show that the numbers χX [b] and the operators VX [b] are determined uniquely from relations (1.19) and (1.20), and do not depend on the choice of charts. Lemma 1.2.2. 1. The complex function χX [b] is determined uniquely from property (1.19). 2. This function does not depend on the choice of the chart. 3. The following properties hold: 󵄨󵄨 󵄨 󵄨󵄨χX [b]󵄨󵄨󵄨 = 1,

χX [0] = 1,

χX [b1 + b2 ] = χX [b1 ]χX [b2 ].

Proof. 1. Suppose that there exist two functions χ 1 and χ 2 satisfying property (1.19). ε Then (χX1 0 [b] − χX2 0 [b])KX(ε) ≈ 0. Consider any function f (ε) ∈ ℱX(ε) such that ε→0

ρX(ε),X (f (ε), f ) 󳨀→ 0, ‖f ‖ ≠ 0. Axiom 1.2.1 implies that χ 1 = χ 2 . 2. For the proof of the second statement of the lemma, it suffices to check that ε ε KΦ (i)−1 (Φ(i) X(ε)+εβ) ≈ KΦ(j)−1 (Φ(j) X(ε)+εβ′ ) ,

(1.21)

where β and β′ are the coordinates of the vector b in the charts Φ(i) and Φ(j) . Property (1.21) follows from Axiom 1.2.2 and from equivalence of the curves Φ(i)−1 (Φ(i) X(ε)+ εβ) and Φ(j)−1 (Φ(j) X(ε) + εβ′ ).

1.2 Geometric structures on the phase space | 13

3. The property |χX0 [b]| = 1 follows from the definition of equivalence ≈ and Axiom 1.2.1. The property χX0 [0] = 1 is obvious. Further, let us apply Axiom 1.2.3 twice, changing first X(ε) by Φ(i)−1 (Φ(i) X(ε) + εβ1 ) and β by β2 , and then leaving X(ε) without change and changing β by β1 . We get ε ε KΦ (i)−1 (Φ(i) X(ε)+εβ +εβ ) ≈ χX [b1 ]KΦ(i)−1 (Φ(i) X(ε)+εβ ) 0 1

2

≈

ε χX0 [b1 ]χX0 [b2 ]KX(ε) .

2

On the other hand, ε ε KΦ (i)−1 (Φ(i) X(ε)+εβ +εβ ) ≈ χX [b1 + b2 ]KX(ε) . 0 1

2

This implies that χX0 [b1 + b2 ] = χX0 [b1 ]χX0 [b2 ]. Thus, all of the statements of Lemma 1.2.2 are checked. Corollary 1.2.1. The function χX [b] has the form X

χX [b] = e−iω1 [b] , where ω1 is a differential 1-form. Definition 1.2.2. The differential 1-form ω1 is called the action 1-form induced by the canonical operator. Remark 1.2.1. i 1. After multiplication of the canonical operator KXε by a function e ε α(X) , the differential form ω1 is changed by a full differential dα. 2. In the case of canonical operator (1.5), the 1-form ω1 has the form PdQ. Lemma 1.2.3. 1. The operators VX [b] are uniquely determined by formula (1.20). 2. The operators VX [b] do not depend on the choice of a chart. Proof. The first statement of the lemma is proved by the analogy with the first statement of the previous lemma. Let us prove the second statement. Denote by x(√ε), β the coordinates of the curve X(√ε) and the vector b in the ith chart, and by x′ (√ε), β′ the coordinates of this curve and this vector in the jth chart. The 1-form ω1 can be represented in two forms: ω1 = ∑ ω1k (x)dxk = ∑ ω′1k (x ′ )dxk′ . k

k

14 | 1 Abstract canonical operator and symplectic geometry Let us also use the relations dxk (b) = βk ,

dxk′ (b) = βk′ .

The statement of Lemma 1.2.3 will follow from the relation ε KΦ (i)−1 (x(√ε)+√εβ) exp{

β i ∑ ω1k (x(√ε) + √ε )βk } √ε k 2

ε ≈ KΦ (j)−1 (x ′ (√ε)+√εβ′ ) exp{

β′ i ∑ ω′1k (x ′ (√ε) + √ε )βk′ }. √ε k 2

To check this relation, let us use the following formulas: 𝜕x′

𝜕x′

(1) βk′ = ∑l 𝜕xk βl , ω1l = ∑k ω′1k 𝜕xk ; l l (2) Let xk (√ε) = x0.k + √εx1,k + εx2,k + o(ε). Then ′ xl′ (√ε) = x0,l +∑ k

+ ε(∑ k

𝜕xl′ x √ε 𝜕xk 1,k 𝜕xl′ 𝜕2 xl′ 1 x2,k + ∑ x x ) + o(ε); 𝜕xk 2 k,n 𝜕xk 𝜕xn 1,k 1,n

(3) Property 2 implies that (x(√ε) + √εβ) ≡ Φ(j) Φ(i)−1 (x(√ε) + √εβ) = x ′ (√ε) + √εβ′ + γ ′ , ′

where γk′ = ε ∑(βl l,n

2 ′ 𝜕2 xk′ 1 𝜕 xk x1,n + βl β ) + o(ε); 𝜕xl 𝜕xn 2 𝜕xl 𝜕xn n

(4) Properties 1 and 2 imply that β β′ 1 ∑ [ω1k (x(√ε) + √ε )βk − ω′1k (x ′ (√ε) + √ε )βk′ ] ε 2 2 k = ∑(x1,l + k,l

=∑

j,k,l

𝜕ω′1k ′ βl 𝜕ω1k )[ βk − β ] + O(√ε) 2 𝜕xl 𝜕xl l

ω′1j (x1,l

2 ′ βl 𝜕 xj + ) β + O(√ε) 2 𝜕xk 𝜕xl k

= ∑ ω′1j γj′ + O(√ε). j

(1.22)

1.2 Geometric structures on the phase space | 15

Hence, relation (1.22) is equivalent to the following one: ε ′ ′ ε KΦ (j)−1 (x ′ (√ε)+√εβ′ +γ ′ ) exp{i ∑ ω1j γj } ≈ KΦ(j)−1 (x ′ (√ε)+√εβ′ ) , j

which, in its turn, follows from Axiom 1.2.3. Lemma 1.2.3 is proved. Remark 1.2.2. 1. If the argument of the function χ in formula (1.20) would differ from Φ(i)−1 (Φ(i) X(√ε)+√εβ/2) and would equal, for instance, X(√ε), then the operators VX0 would depend on the choice of a chart, and Definition 1.2.1 would be incorrect. 2. In the case of canonical operator (1.5), operators VX [b] have the form (1.11). Let us now establish some properties of the operators VX [b]. Lemma 1.2.4. The operators VX [b] possess the following properties: 1. i VX [b1 + b2 ] = exp{− dωX1 (b1 , b2 )}VX [b1 ]VX [b2 ]. 2 2. 3.

(1.23)

The operator VX [0] equals unity. The operators VX [b] are unitary.

Proof. Axiom 1.2.4 implies: ε KΦ (i)−1 (x

0 +√εβ1 +√εβ2 )

≈ exp{−

β i ∑ ω (x + √εβ1 + √ε 2 )β2k } √ε k 1k 0 2

ε × KΦ (i)−1 (x

0 +√εβ1 )

≈ exp{−

VΦ(i)−1 (x0 +√εβ1 ) [b2 ]

β β i ∑[ω1k (x0 + √εβ1 + √ε 2 )β2k + ω1k (x0 + √ε 1 )β1k ]} √ε k 2 2

ε × KΦ (i)−1 (x ) VX [b1 ]VX [b2 ] 0 0 0

β β β β 𝜕ω i ≈ exp{− ∑ ω (x )(β + β2k ) − i ∑ 1k [β2k β1l + 2k 2l − 1k 1l ]} √ε k 1k 0 1k 𝜕x 2 2 l k,l ε × KΦ (i)−1 (x ) VX [b1 ]VX [b2 ]. 0 0 0

On the other hand, ε KΦ (i)−1 (x

0 +√εβ1 +√εβ2 )

≈ exp{−

i ∑ ω (x )(β + β2k )} √ε k 1k 0 1k

ε × KΦ (i)−1 (x ) VX [b1 + b2 ] exp{−i ∑ 0 0

k,l

𝜕ω1k (β + β2k )(β1l + β2l )}. 𝜕xl 1k

16 | 1 Abstract canonical operator and symplectic geometry Comparing the two obtained expressions, we obtain the formula 𝜕ω1l 𝜕ω i VX0 [b1 + b2 ] = exp{ ∑ β1k ( 1k − )β2l }VX0 [b1 ]VX0 [b2 ], 2 k,l 𝜕xl 𝜕xk

(1.24)

which is equivalent to formula (1.23). The fact that the operator VX [0] equals unity, follows directly from Axiom 1.2.4. Unitarity of operators VX [b] follows from Axiom 1.2.1, and their invertibility follows from the first statement of the lemma already proved (one can put in this statement b2 = −b, b1 = b, and vice versa). Thus, the operators VX [b] are unitary. Lemma 1.2.4 is proved.

1.2.4 Operator-valued 1-form induced by the canonical operator Let us prove, by analogy with the Stone theorem (see, e. g., [18]), the following lemma. Lemma 1.2.5. For each b ∈ T ℳX , there exist self-adjoint operators ΩX (b) acting on the space ℱX such that VX [b] = exp{iΩX (b)}.

(1.25)

The set 𝒟X ⊂ ℱX defined in Axiom 1.2.4, belongs to the domain of each of the operators ΩX (b), which take 𝒟X into itself and are essentially self-adjoint on 𝒟X . For all f ∈ 𝒟X , b1 , b2 ∈ T ℳX , λ1 , λ2 ∈ ℝ, the following relations hold: ΩX (λ1 b1 + λ2 b2 )f = λ1 ΩX (b1 )f + λ2 ΩX (b2 )f , ΩX (b1 )ΩX (b2 )f − ΩX (b2 )ΩX (b1 )f =

−i dωX1 (b1 , b2 )f .

(1.26) (1.27)

Proof. Let us define the operator ΩX (b) on the area of definition 𝒟X as ΩX (b)f =

1 d 󵄨󵄨󵄨󵄨 󵄨 V [λb]f , i dλ 󵄨󵄨󵄨λ=0 X

f ∈ 𝒟X

(1.28)

(differentiation with respect to λ is well-defined due to definition of the set 𝒟X ). Let us show that ΩX (b)f ∈ 𝒟X , i. e., that the map 1 d 󵄨󵄨󵄨󵄨 b̃ 󳨃→ VX [b]̃ 󵄨 V [λb]f i dλ 󵄨󵄨󵄨λ=0 X

(1.29)

is infinitely differentiable. Formula (1.23) implies that the quantity in the right-hand side of formula (1.29) equals iλ 1 d 󵄨󵄨󵄨󵄨 󵄨󵄨 (VX [b̃ + λb]e 2 i dλ 󵄨󵄨λ=0

̃ dωX1 (b,b)

)f .

1.2 Geometric structures on the phase space | 17

̃ , the map (1.29) is also infinitely By the infinite differentiability of the map b̃ 󳨃→ VX [b]f differentiable. Hence, ΩX [b] : 𝒟X → 𝒟X . Let us now check relations (1.25) and (1.27). Let us show that ΩX (b1 + b2 )f = ΩX (b1 )f + ΩX (b2 )f . From formula (1.28), we have 1 d 󵄨󵄨󵄨󵄨 󵄨 V [λb + λb2 ]f i dλ 󵄨󵄨󵄨λ=0 X 1 1 d 󵄨󵄨󵄨󵄨 − i λ2 dωX1 (b1 ,b2 ) = f 󵄨 V [λb ]V [λb ]e 2 i dλ 󵄨󵄨󵄨λ=0 X 1 X 2 1 d 󵄨󵄨󵄨󵄨 1 d 󵄨󵄨󵄨󵄨 = 󵄨󵄨 VX [λb1 ]f + 󵄨 V [λb ]f = ΩX (b1 )f + ΩX (b2 )f . i dλ 󵄨󵄨λ=0 i dλ 󵄨󵄨󵄨λ=0 X 2

ΩX (b1 + b2 )f =

The property ΩX (αb)f = αΩX (b)f , α ∈ ℝ, is obvious. Relation (1.26) is checked. The quantity on the left-hand side of relation (1.27) equals 1 d 󵄨󵄨󵄨󵄨 1 d 󵄨󵄨󵄨󵄨 VX [α1 b1 ] V [α b ]f 󵄨󵄨 󵄨 i dα1 󵄨󵄨α1 =0 i dα2 󵄨󵄨󵄨α2 =0 X 2 2 1 d 󵄨󵄨󵄨󵄨 1 d 󵄨󵄨󵄨󵄨 VX [α2 b2 ] V [α b ]f − 󵄨󵄨 󵄨 i dα2 󵄨󵄨α2 =0 i dα1 󵄨󵄨󵄨α1 =0 X 1 1 αα d2 󵄨󵄨󵄨󵄨 =− {V [α b + α2 b2 ]2i sin( 1 2 ωX2 (b1 , b2 ))}f 󵄨 dα1 dα2 󵄨󵄨󵄨α1 ,α2 =0 X 1 1 2 = −idωX1 (b1 , b2 )f .

Relation (1.27) is checked. The symmetry of operators ΩX (b) is checked by differentiation of the equality (VX [λb]f , VX [λb]f ) = (f , f ) with respect to λ at λ = 0. Let us show that operators ΩX (b) are essentially self-adjoint on 𝒟X . By the well-known criterion of essential self-adjointness (see, e. g., [18]), it suffices to show that ker(Ω+X (b) ± iE) = {0}, i. e., if g ∈ ℱX satisfies the relation (g, ΩX (b)f ∓ if ) = 0

(1.30)

18 | 1 Abstract canonical operator and symplectic geometry for all f ∈ 𝒟X , then g = 0. In the case when relation (1.30) holds, the function (g, VX [λb]f )

(1.31)

of parameter λ satisfies the differential equation (

d ± 1)(g, VX [λb]f ) = 0, dλ

and equals const e∓λ . By unitarity of the operators VX [b], the constant vanishes, so that expression (1.31) equals zero for all f ∈ 𝒟X , whence also g = 0. Thus, the operator ΩX (b) is essentially self-adjoint on 𝒟X . Let us show that property (1.25) holds. Let f ∈ 𝒟X . Consider the family of vectors w(λ) ∈ 𝒟X of the form w(λ) = VX [λb]f − eiλΩX (b) f , satisfying the equation dw = iλΩX (b)w dλ

(1.32)

d (w(λ), w(λ)) = 0 following from and the initial condition w(0) = 0. By the property dλ equation (1.32), we obtain that w(λ) = 0 and, therefore, equality (1.25) holds on the dense subset 𝒟X . Since the operators occurring in this equality are bounded, it follows that the equality actually holds on the entire space ℱX . Lemma 1.2.5 is proved.

Definition 1.2.3. We shall say that the operator valued differential 1-form Ω is induced by the canonical operator.

1.2.5 Canonical transformation of the abstract canonical operator Let us now define the notion of canonical transformation of abstract canonical operator, which plays an important role in the construction of asymptotic solutions of equation (1.19). Definition 1.2.4. A canonical transformation of the abstract canonical operator KXε : ℱX → ℋε , X ∈ ℳ, is a quadruple (𝒰 ε , U, WX , S) consisting of (1) a unitary transformation 𝒰 ε : ℋε → ℋε of the space ℋε , (2) a smooth one-to-one map U : ℳ → ℳ for which the map U∗X : T ℳX → T ℳUX is also one-to-one, (3) a one-to-one transformation WX : ℱX → ℱUX , and (4) a function S : ℳ → ℝ, such that for any smooth curve X(√ε) on the phase space ℳ with X(0) = X0 , the following relation holds: ε

ε

i

𝒰 KX(√ε) ≈ e ε

S(X(√ε))

ε KUX(√ε) WX0 .

(1.33)

1.2 Geometric structures on the phase space | 19

Remark 1.2.3. In the case when the operator 𝒰 ε is the operator taking the initial condition of the Cauchy problem for equation (1.19) to the solution of this Cauchy problem at the moment of time t, knowing canonical transformation of the canonical operator allows one to construct approximation as ε → 0 for the element 𝒰 ε KXε f ∈ ℋε of the i

ε WX f . form e ε S(X) KUX

Let us establish some properties of the mapping WX . Lemma 1.2.6. The map WX is a linear operator preserving the norm. Proof. By property (1.33) and relations, i 󵄩󵄩 ε ε 󵄩ε ε→0 ′ ′ S(X) ε KUX WX (αf + α′ f ′ )󵄩󵄩󵄩 󳨀→ 0, 󵄩󵄩𝒰 KX (αf + α f ) − e ε i 󵄩ε ε→0 󵄩󵄩 ε ε S(X) ε KUX WX f 󵄩󵄩󵄩 󳨀→ 0, 󵄩󵄩α𝒰 KX f − αe ε 󵄩󵄩 ′ ε ε ′ 󵄩ε ε→0 ′ i S(X) ε KUX WX f ′ 󵄩󵄩󵄩 󳨀→ 0, 󵄩󵄩α 𝒰 KX f − α e ε

where f , f ′ ∈ ℱX , α, α′ ∈ ℂ, we obtain that 󵄩󵄩 ε ′ ′ ′ ′ 󵄩ε ε→0 󵄩󵄩KUX [WX (αf + α f ) − αWX f − α WX f ]󵄩󵄩󵄩 󳨀→ 0 (here we have used the triangle inequality). By Axiom 1.2.1, we have 󵄩󵄩 ′ ′ ′ ′󵄩 󵄩󵄩WX (αf + α f ) − αWX f − α WX f 󵄩󵄩󵄩X = 0, i. e., the operator WX is linear. Unitarity of the operator 𝒰 ε and Axiom 1.2.1 imply that 󵄩󵄩 ε ε 󵄩󵄩ε ε→0 󵄩󵄩𝒰 KX f 󵄩󵄩 󳨀→ ‖f ‖X ,

󵄩󵄩 ε 󵄩ε ε→0 󵄩󵄩KUX WX f 󵄩󵄩󵄩 󳨀→ ‖WX f ‖UX ,

f ∈ ℱX .

By property (1.19), this implies that ‖WX f ‖UX = ‖f ‖X , i. e., the operator WX preserves the norm. Lemma 1.2.6 is proved. Let us now study properties of the function S. Lemma 1.2.7. For b ∈ T ℳX , the following relation holds: X dSX [b] = ωUX 1 [U∗X b] − ω1 [b].

(1.34)

Remark 1.2.4. In the case of canonical operator (1.5), this formula turns into formula (1.16). Proof. Let XX0 ,b (ε) be a curve on the phase space ℳ, which starts with a point X0 and which is a representative of the equivalence class corresponding to a tangent vector b ∈ T ℳX0 . Formula (1.25) implies that ε

i

ε

𝒰 KXX

0 ,b

(ε)

ε ≈ e ε S[XX0 ,b (ε)] KUX X

0 ,b

(ε) WX0 ,

(1.35)

20 | 1 Abstract canonical operator and symplectic geometry and Axiom 1.2.3 implies that X0

KXε X

(ε) 0 ,b

≈ e−iω1

[b]

KXε 0 .

(1.36)

Acting on formula (1.36) from the left by the operator 𝒰 ε and applying formula (1.35) twice (for the curves XX0 ,b (ε) and X0 ), we obtain X0

e−iω1

[b]

ε

i ε

ε

UX0

𝒰 KX0 ≈ exp{ (S[XX0 ,b (ε)] − S[X0 ]) − iω

[U∗X0 b]}𝒰 ε KXε 0 .

The definition of differential of a function on a manifold implies the statement of Lemma 1.2.7. Let us now consider properties of the operator WX . Lemma 1.2.8. The following relation holds: VUX0 [U∗X0 b]WX0 = WX0 VX0 [b],

X0 ∈ ℳ,

b ∈ T ℳX0 .

(1.37)

Proof. Let Φ(i) be a chart covering the point X0 . Let us cover the manifold ℳ by one ′ ′ more chart Φ(i) , Φ(i) = Φ(i) U −1 . Smoothness of the map U implies that this chart is compatible with the other charts of the atlas. By Axiom 1.2.4, we have: −1 χΦ (i)−1 (Φ(i) X

0 +√εβ/2)

ε (b/̃ √ε)KΦ (i)−1 (Φ(i) X

0 +√εβ)

χ −1(i)′ −1 (i) (b̃ ′ /√ε)K ε (i)′ −1 (i) Φ (Φ X0 +√εβ) Φ (Φ X0 +√εβ/2)

≈ KXε 0 VX0 [b],

(1.38)

≈

(1.39)

ε KUX V [b]. 0 UX0

Here, b̃ ′ and b′ are the tangent vectors to the manifold ℳ at the points Φ(i) −1 (Φ(i) X0 + ′ √εβ/2) and UX0 whose coordinates in the chart Φ(i) coincide with β. Definition of the map U∗X0 directly implies that ′

b′ = U∗X0 b,

b̃ ′ = U∗,Φ(i)−1 (Φ(i) X0 +√εβ/2) b.̃

Let us act from the left on relation (1.38) by the operator 𝒰 ε and use relations (1.39), ′ definition of the chart Φ(i) , the previous lemma, and definition of the 1-form ω1 . We obtain that αε VUX0 [U∗X0 b]WX0 ≈ WX0 VX0 [b], where αε is the complex number equal to i αε = exp{ [S(Φ(i)−1 (Φ(i) X0 + √εβ)) − S(X0 )] ε (i)−1 (i) b̃ − i dSΦ (Φ X0 +√εβ/2) ( )}. √ε

1.2 Geometric structures on the phase space | 21

Denote S̃ = SΦ(i)−1 , x0 = Φ(i) X0 . We have ε→0 β 𝜕S̃ i ̃ ̃ αε = exp{ (S(x (x0 + √ε ))} 󳨀→ 1. 0 + √εβ) − S(x0 ) − √ε ∑ βi ε 𝜕x 2 i i

This implies property (1.37). Lemma 1.2.8 is proved. Lemma 1.2.9. The operator WX maps the set 𝒟X into 𝒟UX and satisfies the property WX ΩX (b)f = ΩUX (U∗X b) WX f ,

f ∈ 𝒟X ,

X ∈ ℳ,

b ∈ T ℳX .

(1.40)

Proof. Let us show that WX : 𝒟X → 𝒟UX . Let f ∈ 𝒟X . This means that the map b 󳨃→ VX [b]f is infinitely differentiable. Consider the map b 󳨃→ VUX [U∗X b]WX f = WX VX [b]f . It is also infinitely differentiable. By smoothness of the transformation U, this implies WX f ∈ 𝒟UX . Property (1.40) follows from the definition of operators Ω and from property (1.37). Lemma 1.2.9 is proved.

1.2.6 Canonical and proper canonical transformations of the phase space Formulas (1.34) and (1.37) do not use the operators 𝒰 ε . Hence, we can base on these formulas the definitions of canonical and proper canonical transformations of the phase space. Definition 1.2.5. A smooth one-to-one map U : ℳ → ℳ for which the operator U∗X , X ∈ ℳ, is invertible is called a canonical transformation if there exists a map S : ℳ → ℝ such that relation (1.34) holds. Definition 1.2.6. A canonical transformation U : ℳ → ℳ is said to be proper if there exists an invertible norm-preserving operator WX : ℱX → ℱUX , X ∈ ℳ such that relation (1.37) holds. Lemmas 1.2.7 and 1.2.8 show that a transformation U for which there exists a canonical transformation (𝒰 ε , U, WX , S) of the canonical operator, is a proper canonical transformation.

1.2.7 Formal asymptotic solutions of the equations of motion Let 𝒰tε be the operator in the space ℋε mapping the initial condition of the Cauchy problem for equation (1.1) into the solution of this Cauchy problem at the time mo-

22 | 1 Abstract canonical operator and symplectic geometry ment t. We have noted above that knowing the maps (𝒰tε , Ut , WX,t , St ) satisfying Definition 1.2.4, allows us to construct asymptotics of the solution of the Cauchy problem for equation (1.1) of the form i

e ε St (X) KUε t X WX,t f ,

(1.41)

where f is an element of ℱX not depending on t. On the other hand, in the previous section we have defined the maps WX,t and St without the use of operators 𝒰tε . Hence, we refer to an element of ℋε of the form (1.41) as a formal asymptotic solution of the equations of motion (1.1) if the maps St , WX,t satisfy Definitions 1.2.5 and 1.2.6, and the transformation Ut : ℳ → ℳ is a proper canonical transformation of the phase space. This definition does not use the equations of motion themselves. Definition 1.2.7. Let Ut be a proper canonical transformation of the phase space. An element (1.41) of ℋε is called a formal asymptotic solution of the equations of motion corresponding to this transformation if formulas (1.34) and (1.37) hold for the maps St and WX,t , respectively.

1.3 Abstract complex germ and construction of formal asymptotic solutions of the equations of motion 1.3.1 Definition of complex germ In this section, we consider the notion of an abstract complex germ, which is useful for the study of properties of operator W and for the construction of formal asymptotic solutions of equations of motion. Definition 1.3.1. A complex germ is a subspace 𝒢 ⊂ T ℳℂ X such that the set of all finite linear combinations of vectors of 𝒟X of the form ∗ ℂ ∗ Ωℂ X (ξ1 ) ⋅ ⋅ ⋅ ΩX (ξk )f0 ,

(1.42)

where ξ1 , . . . , ξk ∈ 𝒢 and f0 is an element of 𝒟X satisfying the relation Ωℂ X (ξ )f0 = 0

(1.43)

for all ξ ∈ 𝒢 , is a dense set in the space ℱX . Remark 1.3.1. If 𝒢 is a complex germ, then there exists at least one nonzero element f0 ∈ 𝒟X such that property (1.43) holds for all ξ ∈ 𝒢 .

1.3 Abstract complex germ and construction of solutions | 23

Let us illustrate Definition 1.3.1 by a simple example. Example. Consider the operator valued 1-form (1.12) induced by the canonical opera2 tor (1.5). In this case, the space T ℳℂ X is the space ℂ over the field ℂ, and the complexℂ ified operator valued 1-form Ω has the form 1 𝜕 , Ωℂ (P,Q) (b, c) = bx − c i 𝜕x

b, c ∈ ℂ.

Let us consider various subspaces of the space T ℳℂ X in this case, and find out which of them are complex germs in the sense of Definition 1.3.1. Let us first consider the subspace 𝒢 coinciding with T ℳℂ X . In this case, there does not exist an element f0 ∈ 𝒟X satisfying property (1.43). Indeed, for b = 0, c = 1, we have 𝜕f0 /𝜕x = 0, i. e., f0 = const and does not belong to L2 (ℝ). Hence, T ℳℂ X is not a complex germ. Let us now consider a one-dimensional subspace 𝒢 spanned by a vector (b, c), and find out in which case it is a complex germ. In this case, relation (1.43) goes to the following one: 1 𝜕 (bx − c )f0 = 0. i 𝜕x It implies that f0 has the form i b 2

f0 = αe 2 c x ,

(1.44)

α = const. Thus, we see that f ∈ L2 (ℝ) if and only if c ≠ 0, Im b/c > 0. It is not difficult to check that in this case f0 ∈ 𝒟X . Let us show that the system of vectors (1.42) is complete. Indeed, the vector (1.42), up to a multiplicative constant, coincides with k

(x −

k

b 2 b 2 bc∗ 1 c∗ 𝜕 1 c∗ 𝜕 ) ei 2c x = ei 2c x (x(1 − ∗ ) − ) 1. ∗ i b 𝜕x b c i b∗ 𝜕x

Since 1 − bc ≠ 0 (otherwise Im b/c = 0), the vector (1.42) equals the product of a polyb∗ c nomial of degree k (the coefficient before xk does not vanish) by the Gaussian exponent (1.44). Therefore, the set of all finite linear combinations of vectors (1.42) consists b 2 of the products of polynomials 𝒫 (x) by a Gaussian exponent, {𝒫 (x)ei 2c x }. According to [19], this system of functions is complete in L2 (ℝ). Thus, the one-dimensional subspace ∗

𝒢b,c = {(λb, λc) | λ ∈ ℂ}

is a complex germ if and only if Im b/c > 0 or 1 (bc∗ − b∗ c) > 0. i

(1.45)

24 | 1 Abstract canonical operator and symplectic geometry Note that the subspace of the space T ℳℂ X consisting of only zero is also a complex germ in the sense of Definition 1.3.1, because any element f0 ∈ 𝒟X satisfies relation (1.43), and hence, the set of all vectors (1.42) coincides with 𝒟X . Let us now define the notion of a complete complex germ, which distinguishes from all complex germs subspaces of “maximal possible” dimension. Definition 1.3.2. We say that a complex germ is complete if the element f0 ∈ 𝒟X is determined by relation (1.43) uniquely up to a number factor. Remark 1.3.2. 1. In the example considered above, a one-dimensional complex germ is complete, and the zero-dimensional one is not complete. 2. As we shall see in Section 1.5, in the case of finite-dimensional quantum mechanics Definition 1.3.2 coincides with the traditional definition of a complex germ [25, 27].

1.3.2 Properties of the abstract complex germ The definition of complex germ implies the following lemma. Lemma 1.3.1. Let 𝒢 be a complex germ. Then the following properties hold: for ξ1 , ξ2 ∈ 𝒢 , for ξ ∈ 𝒢 ,

ωℂ 2 (ξ1 , ξ2 ) = 0,

1 ℂ ω (ξ , ξ ∗ ) ≥ 0. i 2

(G1) (G2)

Proof. Property (1.27) implies that for ξ1 , ξ2 ∈ 𝒢 , 𝒢 ⊂ T ℳℂ X, ℂ ℂ − iωℂ 2,X (ξ1 , ξ2 )f0 = [ΩX (ξ1 ), ΩX (ξ2 )]f0 .

(1.46)

The right-hand side of this relation vanishes by property (1.43); hence, the left-hand side also vanishes. Property (G1) is checked. Further, consider the element of 𝒟X of the form Ωℂ X (ξ )f , ξ ∈ 𝒢 . By symmetry of the ℂ operators Ωℂ (Re ξ ) and Ω (Im ξ ), we have X X ∗ ℂ ∗ ℂ ℂ ∗ (Ωℂ X (ξ )f0 , ΩX (ξ )f0 ) = (f0 , ΩX (ξ )ΩX (ξ )f0 ).

(1.47)

The commutation relations (1.27) and property (1.43) imply that the right-hand side ∗ of relation (1.47) equals −iωℂ 2 (ξ , ξ )(f0 , f0 ), and the left-hand side of this relation is nonnegative. Property (G2) is checked. Lemma 1.3.1 is proved. Remark 1.3.3. In the case of the example considered in the previous section, property (G2) is equivalent to property (1.45).

1.3 Abstract complex germ and construction of solutions | 25

Denote by ℱX(0) the subspace of ℱX consisting of all elements f0 ∈ 𝒟X such that for all ξ ∈ 𝒢 property (1.43) holds. Denote also by 𝒟̃ X ⊂ 𝒟X the set of all finite linear ℂ ∗ combinations of vectors (1.42). Note that the operators Ωℂ X (ξ ) and ΩX (ξ ) map the set 𝒟̃ X into itself. ∗ Lemma 1.3.2. Let ωℂ 2 (ξ , ξ ) = 0, ξ ∈ 𝒢 . Then ℂ ∗ Ωℂ X (ξ )f = ΩX (ξ )f = 0

(1.48)

for f ∈ 𝒟̃ X . Proof. Let us first show that for any η ∈ 𝒢 , ∗ ωℂ 2,X (η, ξ ) = 0.

(1.49)

Proof of this relation is similar to the derivation of the Cauchy inequality. Let λ ∈ ℂ. By property (G2), we have ∗ ∗ −i ωℂ 2,X (ξ + λη, ξ + λ η) ≥ 0, ∗ and by assumption of the lemma, −i ωℂ 2,X (ξ , ξ ) = 0. For small |λ|, we obtain ∗ ∗ ℂ ∗ −i λωℂ 2,X (η, ξ ) − i λ ω2,X (ξ , η ) ≥ 0.

Considering the values of λ of the form δ, −δ, −iδ, iδ, where δ ∈ ℝ, we obtain that property (1.49) holds. Let f be of the form (1.42). Then by the commutation relations ∗ ℂ ℂ ∗ −i ωℂ 2,X (ξ , ξm ) = [ΩX (ξ ), ΩX (ξm )],

m = 1, . . . , k,

and by properties (1.43), (1.49), we have ℂ ∗ ℂ ∗ Ωℂ X (ξ )ΩX (ξ1 ) ⋅ ⋅ ⋅ ΩX (ξk )f0 k

∗ ℂ ∗ ℂ ∗ ℂ ∗ ℂ ∗ = ∑ (−iωℂ 2,X (ξ , ξm )ΩX (ξ1 ) ⋅ ⋅ ⋅ ΩX (ξm−1 )ΩX (ξm+1 ) ⋅ ⋅ ⋅ ΩX (ξk ))f0 = 0. m=1

∗ ℂ ∗ ̃ By property (G1), the operators Ωℂ X (ξ ) and ΩX (ξm ) commute on the domain 𝒟X ⊃ 𝒟X . Hence, it remains to check that ∗ Ωℂ X (ξ )f0 = 0,

(1.50)

∗ and this will imply that Ωℂ X (ξ )f = 0 for vectors f of the form (1.42). Property (1.50) follows from the relation: ∗ ℂ ∗ ℂ ℂ ∗ (Ωℂ X (ξ )f0 , ΩX (ξ )f0 ) = (f0 , [ΩX (ξ ), ΩX (ξ )]f0 ) = 0.

26 | 1 Abstract canonical operator and symplectic geometry Thus, property (1.48) holds for all elements of 𝒟X of the form (1.42), and hence, for all their finite linear combinations. Lemma 1.3.2 is proved. Lemma 1.3.2 allows one to consider the quotient space of a germ by the subspace 𝒢0 = {ξ ∈ 𝒢 | ω2 (ξ , ξ ) = 0}, ℂ

∗

(1.51)

and to introduce on it an operator valued 1-form. Indeed, the quotient space 𝒢 /𝒢0 consists of equivalence classes [ξ ]: η ∈ [ξ ] ⇐⇒ ξ − η ∈ 𝒢0 . Lemma 1.3.2 implies that the value of the operator valued 1-form ΩX (ξ ) considered on the domain 𝒟̃ X does not depend on the choice of a representative of the equivalence class.

1.3.3 Canonical transformation of the abstract complex germ Definition 1.3.3. The canonical transformation of a complex germ 𝒢 corresponding to ℂ a proper canonical transformation U of the phase space ℳ is the subspace U∗X 𝒢 ⊂ ℂ ℂ T ℳUX of the form {U∗X ξ , ξ ∈ 𝒢 }. ℂ Lemma 1.3.3. The subspace U∗X 𝒢 ⊂ T ℳℂ UX is a complex germ; if the germ 𝒢 is comℂ plete, then the germ U∗X 𝒢 is also complete.

Proof. Consider the system of vectors of the form ∗ ℂ ∗ WX Ωℂ X (ξ1 ) ⋅ ⋅ ⋅ ΩX (ξk )f0

(1.52)

which, due to invertibility and unitarity of the operator W, is complete in the space

ℱUX . By the commutation relations,

ℂ ℂ Ωℂ UX (U∗X ξ )WX = WX UX (ξ ),

ξ ∈ T ℳℂ X,

(1.53)

the vector (1.52) can be presented in the form ℂ ∗ ℂ ℂ ∗ Ωℂ UX (U∗X ξ1 ) ⋅ ⋅ ⋅ ΩUX (U∗X ξk )WX f . ℂ ∗ ℂ ℂ ℂ ℂ Since U∗X ξ1 = (U∗X ξ1 )∗ , U∗X ξ1 ∈ U∗X 𝒢 , and for all ξ ̃ ∈ U∗X 𝒢 , the vector WX f ∈ 𝒟UX satisfies the property

̃ Ωℂ UX (ξ )WX f = 0

(1.54)

ℂ by relation (1.53), then the subspace U∗X (𝒢 ) is a complex germ by Definition 1.3.1.

1.3 Abstract complex germ and construction of solutions | 27

ℂ Let us now assume that the germ 𝒢 is complete. Let us show that the germ U∗X 𝒢 ℂ is also complete, i. e., the element f ̃ ∈ 𝒟UX possessing the property ΩUX (ξ ̃ )f ̃ = 0 is determined uniquely up to a constant. Suppose that there exist two linearly independent elements f1̃ and f2̃ of the space 𝒟UX possessing this property. Then the elements WX−1 f1̃ , WX−1 f2̃ of the space 𝒟X possess property (1.43) and are also linearly indepenℂ dent, which contradicts the assumption of the lemma. Therefore, the germ U∗X 𝒢 is complete. Lemma 1.3.3 is proved.

Remark 1.3.4. If U is a canonical transformation of ℳ, which is not proper, then propℂ erties (G1) and (G2) also hold for the subspace U∗X 𝒢 of the space T ℳℂ UX , although ℂ U∗X 𝒢 is in general not a complex germ. Let us now state a sufficient condition for a canonical transformation to be proper. ℂ ℂ Lemma 1.3.4. Assume that the subspaces 𝒢 ⊂ T ℳℂ X and U∗X 𝒢 ⊂ T ℳUX are complex (0) germs, and that there exists an isometric invertible operator WX(0) : ℱX(0) → ℱUX . Then the canonical transformation is proper.

Proof. Let us define the operator WX on the dense subset 𝒟̃ X as follows: M

∗ ℂ ∗ WX ∑ αi Ωℂ X (ξ1,i ) ⋅ ⋅ ⋅ ΩX (ξk,i )f0,i i=1

M

ℂ ∗ ℂ ℂ ∗ (0) = ∑ αi Ωℂ UX (U∗X ξ1,i ) ⋅ ⋅ ⋅ ΩUX (U∗X ξk,i )WX f0,i , i=1

f0,i ∈ ℱX(0) ,

ξm,i ∈ 𝒢 .

(1.55)

ℂ Since U∗X 𝒢 is a complex germ, the elements of the space ℱUX of the form (1.55) form a complete system of vectors. Let us show that the operator W preserves the norm. To this end, it suffices to show that ∗ ℂ ∗ ℂ ∗ ℂ ∗ (WΩℂ X (ξ1 ) ⋅ ⋅ ⋅ ΩX (ξm )f0,1 , WΩX (η1 ) ⋅ ⋅ ⋅ ΩX (ηn )f0,2 ) ℂ ∗ ℂ ℂ ∗ (0) = (Ωℂ UX (U∗X ξ1 ) ⋅ ⋅ ⋅ ΩUX (U∗X ξm )W f0,1 , ℂ ∗ ℂ ℂ ∗ (0) Ωℂ UX (U∗X η1 ) ⋅ ⋅ ⋅ ΩUX (U∗X ηn )W f0,2 ).

It is not difficult to check this property using the equality ∗ ℂ (Ωℂ X (ξ )f , g) = (f , ΩX (ξ )g),

f , g ∈ 𝒟X ,

the commutation relations (1.46), and the assumption of the lemma concerning the operator W (0) . Note that correctness of definition of operator W automatically follows from conservation of the norm. Invertibility follows from completeness of the system (1.55). Lemma 1.3.4 is proved.

28 | 1 Abstract canonical operator and symplectic geometry 1.3.4 Canonical operator corresponding to a complex germ In this section, we shall assume that the space 𝒢 /𝒢0 , where 𝒢0 is of the form (1.51), is complete with respect to the inner product 1 ′ (n, ξ ∗ ), ([ξ ], [η]) = ωℂ i 2

η ∈ [η],

ξ ∈ [ξ ].

(1.56)

Note that this definition is correct (does not depend on the choice of representatives of the classes [ξ ], [η]), because property (1.49) holds for ξ ∈ 𝒢0 . Consider an orthonormal basis {[ei ]} with respect to the inner product (1.56) in the quotient space 𝒢 /𝒢0 , and denote by Λ̄ i , Λi the operators ∗ Λ̄ i = Ωℂ X (ei ),

Λi = Ωℂ X (ei )

(1.57)

with the domain 𝒟̃ X , which do not depend on the choice of representatives ei of the equivalence classes [ei ] by Lemma 1.3.2. Definition 1.3.4. The operators Λ̄ i and Λi are called the germ creation and annihilation operators. Lemma 1.3.5. The following commutation relations hold: [Λi , Λj ] = [Λ̄ i , Λ̄ j ] = 0,

[Λi , Λ̄ j ] = δij .

(1.58)

Proof. The proof follows directly from the commutation relations between the operators Ω, the definition of the inner product (1.56), and orthonormality of [ei ] with respect to this inner product. Denote by ℱn′ the space of sets of elements of the space ℱX(0) of the form {Φ(n) ∈ i ⋅⋅⋅i 1

n

ℱX(0) }, symmetric in i1 , . . . , in , for which the series ∑i1 ⋅⋅⋅in ‖Φ(n) ‖2 converges. Consider i ⋅⋅⋅i 1

the following inner product in ℱn′ :

(n) (n) (n) (Φ(n) 1 , Φ2 ) = ∑ (Φ1,i ⋅⋅⋅i , Φ2,i ⋅⋅⋅i ), 1

i1 ⋅⋅⋅in

n

1

n

n

(n) ′ Φ(n) 1 , Φ 2 ∈ ℱn .

(1.59)

Denote by ℱ ′ the direct sum of the spaces ℱn′ : ′

′

′

′

ℱ = ℱ0 ⊕ ℱ1 ⊕ ⋅ ⋅ ⋅ ⊕ ℱn ⊕ ⋅ ⋅ ⋅ ;

we shall denote the nth component of an element Φ ∈ ℱ ′ by Φ(n) . Remark 1.3.5. If the complex germ is complete, then the space ℱX(0) is one-dimensional; hence, the space ℱ ′ coincides with the Fock space.

1.3 Abstract complex germ and construction of solutions | 29

Denote by w𝒢,e : ℱ ′ → ℱX the operator of the form 1 ∑ Λ̄ i1 ⋅ ⋅ ⋅ Λ̄ in Φ(n) i1 ⋅⋅⋅in . √ n! n=0 i1 ⋅⋅⋅in ∞

(1.60)

w𝒢,e Φ = ∑

Let us consider some properties of operator w𝒢,e . Lemma 1.3.6. 1. The operator w𝒢,e preserves the norm: ‖w𝒢,e Φ‖X = ‖Φ‖. 2. The operator w𝒢,e is invertible. Proof. The symmetry of operators ΩX (ξ ), ξ ∈ T ℳX , implies that Λ+i = Λ̄ i on 𝒟̃ X . This implies that 1 ̄ ̄ (n) ∑ (Φ(m) i1 ⋅⋅⋅im , Λi1 ⋅ ⋅ ⋅ Λim Λj1 ⋅ ⋅ ⋅ Λjn Φj1 ⋅⋅⋅jn ). √ n!m! n,m=0 i1 ,...,im ∞

(w𝒢,e Φ, w𝒢,e Φ) = ∑

j1 ,...,jn

The commutation relations (Lemma 1.3.5) and the property Λi Φ(n) = 0 imply that j ⋅⋅⋅j 1

n

∞

(n) (w𝒢,e Φ, w𝒢,e Φ) = ∑ ∑ (Φ(n) j ⋅⋅⋅j , Φj ⋅⋅⋅j ). n=0 j1 ,...,jn

1

n

1

n

The first statement of the lemma is proved. Further, the completeness of the system of vectors (1.42) implies that the image of the operator w𝒢,e coincides with ℱX , and conservation of the norm implies that the kernel of this operator consists of only zero. Hence, the operator w𝒢,e is invertible. Lemma 1.3.6 is proved. Definition 1.3.5. The operator ′ε KX,𝒢,e = KXε w𝒢,e

is called the canonical operator corresponding to the germ 𝒢 and to the basis e. Let us provide a useful formula expressing the operator WX through the operators w𝒢,e . Lemma 1.3.7. For certain operator WX(0) preserving the norm and mapping the space (0) ℱX(0) onto ℱUX , the following relation holds: −1 WX = wU ℂ 𝒢,U ℂ e WX(0) w𝒢,e . ∗X

∗X

Proof. The proof follows from the definition (1.60) of the operator w𝒢,e and from the properties: −1 ℂ ℂ WX Ωℂ X (ei )WX = ΩUX (U∗X ei ),

30 | 1 Abstract canonical operator and symplectic geometry ∗ −1 ℂ ℂ ∗ WX Ωℂ X (ei )WX = ΩUX (U∗X ei ). (0) Corollary 1.3.1. Assume that an operator WX,t : ℱX(0) → ℱU(0)t X preserves the norm, is invertible, and satisfies the property, t (0) (0)−1 = ΩU t X (U∗X ξ ), WX,t ΩX (ξ )WX,t

ξ ∈ T ℳX ,

where U t is a canonical transformation ℳ → ℳ. Then the element of ℋε of the form KU′ εt X,U t

t ∗X 𝒢,U∗X e

(0) WX,t f

is a formal asymptotic solution of equations of motion. Corollary 1.3.2. In the case of a complete complex germ, ct KU′ εt X,U t

t ∗X 𝒢,U∗X e

f ∈ ℋε ,

ct ∈ ℂ,

is a formal asymptotic solution of equations of motion.

1.4 Asymptotics of the solution of the Cauchy problem 1.4.1 Canonical transformations of the canonical operator and construction of asymptotics of the solution of the Cauchy problem In Section 1.2, we studied properties of a canonical transformation of the canonical operator (𝒰 ε , U, WX , S) satisfying Definition 1.2.4. But the question on construction of canonical transformation was not considered there; we have only shown how to reconstruct the maps WX and S from the given map U and to construct formal asymptotic solutions of equations of motion. In this section, we shall consider the question on checking canonicity of the transformation (𝒰tε , Ut , WX,t , St )

(1.61)

in the case when the operator 𝒰tε has the form ε

𝒰t = e

−iH ε t

(1.62)

,

where H ε is a self-adjoint operator in the space ℋε . Since the operator (1.62) maps an initial condition ψε0 of the Cauchy problem into the solution of the Cauchy problem i

dψε (t) = H ε ψε (t), dt

ψε (0) = ψε0 ,

ψε (t) ∈ ℋε ,

(1.63)

1.4 Asymptotics of the solution of the Cauchy problem

| 31

then the canonical transformation (1.61) plays an important role in construction of asymptotics of the solution of the Cauchy problem. Definition 1.4.1. Any function ψεas (t) ∈ ℋε possessing the property 󵄩󵄩 ε 󵄩ε ε→0 ε 󵄩󵄩ψas (t) − ψ (t)󵄩󵄩󵄩 󳨀→ 0,

t ∈ [0, T],

(1.64)

for the solution function ψε (t) of the Cauchy problem (1.63) is called the asymptotics of the solution of the Cauchy problem (1.63) on [0, T]. Lemma 1.4.1. There exists a dense domain 𝒟 ⊂ ℱX such that the function i

ψεas (t) = e ε St (X) KUε t X WX,t f ,

f ∈ 𝒟,

is an asymptotics of the solution of the Cauchy problem (1.63) for the initial condition of the following special form: ψε0 = KXε f ,

f ∈ 𝒟.

(1.65)

Proof. The statement of this lemma directly follows from the property: i 󵄩󵄩 −iH ε t ε 󵄩ε ε→0 KX f − e ε St (X) KUε t X WX,t f 󵄩󵄩󵄩 󳨀→ 0, 󵄩󵄩e

f ∈ 𝒟,

which, in its turn, follows from Definition 1.2.4 of Section 1.2. Let us now pass directly to construction of canonical transformation (1.61). We must check that Definition 1.2.4 from Section 1.2 holds, i. e., that the following property holds: i 󵄩󵄩 −iH ε t ε 󵄩ε ε→0 KX(√ε) f (√ε) − e ε St (X(√ε)) KUε X(√ε) WX0 ,t f (√ε)󵄩󵄩󵄩 󳨀→ 0, 󵄩󵄩e t

f (√ε) ∈ ℱX(√ε) ,

ε→0

ρX(√ε),X (f (√ε), f ) 󳨀→ 0.

(1.66)

As a rule, in concrete cases the spaces ℱX are subspaces of a space ℱ , and the following commutation rule of the canonical operator with the Hamiltonian holds: (i

i i d d ε ε ε − H ε )e ε S(t,√ε) KX(t,√ε) = e ε S(t,√ε) KX(t,√ε) (i − H̃ S(t,√ε),X(t,√ε) ). dt dt

ε Here, H̃ S(t,√ε),X(t,√ε) are certain operators in the space ℱ . It also turns out that, under certain conditions on S(t, √ε), X(t, √ε) (in the case of theory of complex germ in finite-dimensional quantum mechanics they have the form of the Hamiltonian equations (1.5) on X and a condition on S), which determine St and Ut uniquely. The following relation holds:

󵄩󵄩 ̃ ε 󵄩 ε→0 󵄩󵄩HS(t,√ε),X(t,√ε) f − Hf̃ 󵄩󵄩󵄩 󳨀→ 0,

(1.67)

32 | 1 Abstract canonical operator and symplectic geometry where the operator H̃ does not depend on ε. Let us define the operator WX,t as the operator mapping the initial condition f0 ∈ ℱ of the Cauchy problem i

df = Hf̃ , dt

f0 = f (0),

into the solution f (t) of this problem. By relation (1.67), we have 󵄩󵄩 d 󵄩󵄩ε i 󵄩󵄩 󵄩 S(t,√ε) ε ε KX(t,√ε) WX0 ,t f0 󵄩󵄩󵄩 󵄩󵄩(i − H )e ε 󵄩󵄩 󵄩󵄩 dt i 󵄩󵄩 ε 󵄩ε ε→0 S(t,√ε) ε = 󵄩󵄩KX(t,√ε) e ε (H̃ S(t,√ε),X(t,√ε) f − Hf̃ )󵄩󵄩󵄩 󳨀→ 0. Definition 1.4.2. Any function Φε (t) ∈ ℋε such that the relation 󵄩󵄩 d 󵄩󵄩ε ε→0 󵄩󵄩 󵄩 ε ε 󵄩󵄩(i − H )Φ (t)󵄩󵄩󵄩 󳨀→ 0, 󵄩󵄩 󵄩󵄩 dt

t ∈ [0, T],

holds uniformly in t and 󵄩󵄩 ε ε 󵄩ε ε→0 󵄩󵄩Φ (0) − ψ0 󵄩󵄩󵄩 󳨀→ 0 is called an asymptotic solution of the Cauchy problem. The argument above implies that the function i

Φε (t) = e ε St (X(√ε)) KUε X(√ε) WX0 ,t f (√ε) ∈ ℋε t

is an asymptotic solution of the Cauchy problem (1.63) with the initial condition ε ψε0 = KX(√ε) f.

In order to check property (1.66) meaning canonicity of the transformation (1.61), it suffices to check that the following lemma holds. Lemma 1.4.2 ([24]). Let Φε (t) be an asymptotic solution of the Cauchy problem (1.63). Then Φε (t) is an asymptotics of the solution of this problem. Proof. Denote the discrepancy (i

d − H ε )Φε (t) dt

by δε , and the difference between the solution ψε (t) of the Cauchy problem (1.63) and the function Φε (t) by Δε (t). Equation (1.63) implies that this difference satisfies the equation (i

d − H ε )Δε (t) = −δε (t), dt

1.4 Asymptotics of the solution of the Cauchy problem

| 33

which can be represented in the form i

d ε ε (𝒰 Δ (t)) = −𝒰−t δε (t). dt −t ε

This equation implies that ε 𝒰−t Δε (t)

t

ε ′ = Δε (0) − ∫ dt ′ 𝒰−t ′ δε (t ). 0

ε Since the operator 𝒰−t is unitary, we have t

󵄩󵄩 󵄩ε 󵄩 ε 󵄩ε 󵄩 󵄩ε ′󵄩 ′ 󵄩ε ε→0 󵄩󵄩Δε (t)󵄩󵄩󵄩 = 󵄩󵄩󵄩𝒰−t Δε (t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩Δε (0)󵄩󵄩󵄩 + ∫ dt 󵄩󵄩󵄩δε (t )󵄩󵄩󵄩 󳨀→ 0. 0

Lemma 1.4.2 is proved.

1.4.2 Asymptotics of the solution of the Cauchy problem modulo O(ε M/2 ) Let us now consider construction of approximations of solutions of the Cauchy problem (1.63) up to O(εM/2 ), M ≥ 1. Definition 1.4.3. A function ψεas,M (t) ∈ ℋε satisfying the property 󵄩󵄩 ε 󵄩ε ε M/2 󵄩󵄩ψas,M (t) − ψ (t)󵄩󵄩󵄩 = O(ε ),

t ∈ [0, T],

T > 0,

(1.68)

is called the asymptotics of the solution of the Cauchy problem (1.63) up to O(εM/2 ) on [0, T]. Definition 1.4.4. Any function ΦεM (t) ∈ ℋε such that the relation 󵄩󵄩ε 󵄩󵄩 d 󵄩 󵄩󵄩 ε ε M/2 󵄩󵄩(i − H )ΦM (t)󵄩󵄩󵄩 = O(ε ) 󵄩󵄩 󵄩󵄩 dt

(1.69)

holds uniformly for t ∈ [0, T] and 󵄩󵄩 ε ε 󵄩ε M/2 󵄩󵄩ΦM (0) − ψ0 󵄩󵄩󵄩 = O(ε ) is called an asymptotic solution of the Cauchy problem up to O(εM/2 ). The statement analogous to Lemma 1.4.2 holds also for asymptotic solutions up to O(εM/2 ) of the Cauchy problem (1.63).

34 | 1 Abstract canonical operator and symplectic geometry Lemma 1.4.3. Let ΦεM (t) be an asymptotic solution of the Cauchy problem (1.63) up to O(εM/2 ) on [0, T]. Then ΦεM (t) is an asymptotic of the solution of this problem up to O(εM/2 ). Proof. The proof is similar to the proof of the previous lemma. As a rule, asymptotic solutions of the Cauchy problem up to O(εM/2 ) are constructed in the form i

ε ΦεM (t) = e ε S(t) KX(t) f√ε (t),

where (i

d ε − H̃ S(t),X(t) )f√ε (t) = O(εM/2 ), dt

(1.70)

and f√ε is certain polynomial in √ε of degree M − 1, f√ε (t) = f (0) (t) + ⋅ ⋅ ⋅ + ε

M−1 2

f (M−1) (t).

(1.71)

ε The operator H̃ S(t),X(t) is decomposed into an asymptotic series in √ε: ε H̃ S(t),X(t) = H̃ 0 (t) + √εH̃ 1 (t) + ⋅ ⋅ ⋅ + ε

where the operator ε−

M−1 2

M−1 2

H̃ M−1 (t) + O(εM/2 ),

O(εM/2 ) satisfies the condition 󵄩󵄩 − M−1 M/2 󵄩 ε→0 󵄩󵄩ε 2 O(ε )f 󵄩󵄩󵄩 󳨀→ 0,

f ∈ 𝒟,

on certain dense set 𝒟. Relation (1.70) holds if the recursive relations hold, which express f (M−1) (t) through the previous ones f (k) , k < M − 1, (i

d − H̃ 0 (t))f (M−1) (t) = g (M−1) (t), dt M−1

g (M−1) (t) = − ∑ H̃ k (t)f (M−1−k) (t). k=1

(1.72) (1.73)

If we denote by Wt the operator mapping the initial condition of the Cauchy problem, (i

d − H̃ 0 (t))f = 0, dt

f (0) = f0

1.5 Finite-dimensional theory of complex germ at a point | 35

into the solution of this problem at the time moment t, then the solution of equation (1.70) has the form t

f (M−1) (t) = Wt f0(M−1) + ∫ Wt Wτ−1 g (M−1) (τ)dτ,

(1.74)

0

where f0(M−1) is the initial condition for f (M−1) . Thus, using the canonical operator one can construct asymptotics of the solution of the Cauchy problem (1.63) with any degree of precision O(εM/2 ). Let us now proceed to illustration of the notion of canonical operator on the example of the construction of asymptotics of solutions of the Cauchy problem for equations of finite-dimensional quantum mechanics by the method of a complex germ at a point.

1.5 Theory of complex germ at a point in finite-dimensional quantum mechanics In this section, we shall illustrate the concepts developed in Section 1.2 on the example of the theory of a complex germ at a point in finite-dimensional quantum mechanics. 1.5.1 Definition of the canonical operator and checking the axioms Let us choose the space L2 (ℝn ) as the space ℋε . The elements of this space play the role of wave functions. The parameter ε, playing the role of the Planck constant, takes the values in the set (0, ε0 ]. As a smooth manifold ℳ, we choose the phase space ℝ2n of classical mechanics, whose elements will be denoted by (P, Q), P ∈ ℝn , Q ∈ ℝn . We choose all of the spaces ℱP,Q to be the same, ℱP,Q = L2 (ℝn ), so that one can choose the distance function in L2 (ℝn ) as the function ρ. ε Consider the operator KP,Q : ℱP,Q → ℋε of the form ε (KP,Q f )(x) =

1

εn/4

i

e ε P(x−Q) f (

x−Q ). √ε

(1.75)

Lemma 1.5.1. The Axioms 1.2.1–1.2.4 from Section 1.2 hold with χP,Q [p, q] = e−iPq ,

1 𝜕

(VP,Q [p, q]f )(x) = ei(px−q i 𝜕x ) f (x),

(p, q) ∈ T ℳP,Q .

(1.76)

Proof. In this case, one can introduce global coordinates on the manifold ℳ, and the spaces ℱP,Q are all the same; hence, the check of the axioms becomes rather simple. For checking Axiom 1.2.1, 󵄩󵄩 ε 󵄩󵄩 󵄩󵄩KP,Q f 󵄩󵄩 = ‖f ‖.

36 | 1 Abstract canonical operator and symplectic geometry For checking other axioms, note that the following equality holds: i

p

i

ε (KP+p,Q+q f )(x) = e− ε (P+ 2 )q e √ε

(px−q 1i

𝜕 ) 𝜕x

f (x).

The definition of equivalence ≈ (Section 1.2) implies that Axioms 1.2.2–1.2.4 hold. Lemma 1.5.1 is proved. Remark 1.5.1. 1. Lemma 1.5.1 implies that in this case the 1-form ω1 reads ωP,Q 1 [p, q] = Pq,

(p, q) ∈ T ℳP,Q = ℝ2n ,

and the operator valued 1-form Ω induced by the canonical operator reads ΩP,Q [p, q] = px − q 2.

1 𝜕 : L2 (ℝn ) → L2 (ℝn ). i 𝜕x

The set 𝒟X , which is the domain of the operators ΩP,Q [p, q], contains the space 𝒮 (ℝn ).

1.5.2 Canonical and proper canonical transformations of the phase space Let us now consider canonical and proper canonical transformations of the manifold ℳ ≡ ℝ2n . Let U be a smooth one-to-one map U : ℝ2n → ℝ2n for which the operator U∗X U

is invertible. According to the definition from Section 1.2, a transformation (P, Q) 󳨃→ (P ′ , Q′ ) is called canonical if there exists a function s : ℝ2n → ℝ such that n

n

′ ds = ∑ ( ∑ Pm k=1 m=1

′ n 𝜕Q′m ′ 𝜕Qm − Pk )dQk + ∑ Pm dP . 𝜕Qk 𝜕Pk k m.k=1

Lemma 1.5.2. The transformation U is canonical if and only if n

∑(

′ ′ 𝜕Pm 𝜕Q′m 𝜕Pm 𝜕Q′m − ) = δk,l , 𝜕Pl 𝜕Qk 𝜕Ql 𝜕Pk

∑(

′ ′ 𝜕Pm 𝜕Q′m 𝜕Pm 𝜕Q′m − ) = 0, 𝜕Ql 𝜕Qk 𝜕Qk 𝜕Ql

∑(

′ ′ 𝜕Pm 𝜕Q′m 𝜕Pm 𝜕Q′m − ) = 0, 𝜕Pl 𝜕Pk 𝜕Pk 𝜕Pl

m=1 n

m=1 n m=1

k, l = 1, . . . , n.

Proof. The proof follows from the equivalence of the two statements, dω = 0 and ω = ds, for a 1-form on the manifold ℝ2n .

1.5 Finite-dimensional theory of complex germ at a point | 37

According to the definition from Section 1.2, a canonical transformation U is called proper if for certain unitary operator W : ℱP,Q → ℱP′ ,Q′ the following property holds: VP′ ,Q′ [p′ , q′ ]W = WVP,Q [p, q];

(1.77)

here, n

p′m = ∑( l=1

n

′ 𝜕P ′ 𝜕Pm pl + m ql ), 𝜕Pl 𝜕Ql

′ qm = ∑( l=1

𝜕Q′m 𝜕Q′m pl + q ). 𝜕Pl 𝜕Ql l

(1.78)

Remark 1.5.2. 1. Property (1.77) can be represented in the form W (1

xm

𝜕 )W i 𝜕xm

2.

−1

f (x) =

𝜕Q′

− 𝜕P l

m

𝜕Q′l 𝜕Qm

xl

𝜕 ) f (x). i 𝜕xl

) (1

(1.79)

The condition of canonicity of a transformation (Lemma 1.5.2) can be written in the matrix form as follows: ( 𝜕P )T 𝜕P ′

( 3.

𝜕Pl′ 𝜕Pm ∑ ( 𝜕P′ l l=1 − 𝜕Q m n

−( 𝜕P )T 𝜕Q ′

−( 𝜕Q )T 𝜕P ′

( 𝜕Q )T 𝜕Q ′

−1

)

=

𝜕Q′ ( 𝜕Q′ 𝜕P 𝜕Q

𝜕Q′ 𝜕P ). 𝜕P ′ 𝜕P

(1.80)

In contrast to the infinite-dimensional case, in finite-dimensional quantum mechanics any canonical transformation is proper; the operator W is constructed below.

1.5.3 Complex germ Let αij , i, j = 1, . . . , n be a symmetric matrix of complex numbers such that the matrix Im αij consisting of their imaginary parts is positive definite. Definition 1.5.1. The Gaussian wave function corresponding to the matrix α is the following element of the space ℱP,Q : ψα (ξ ) =

i n 1 exp{ ∑ ξk αkl ξl }; n/4 2 k,l=1 (2π)

ε the element KP,Q ψα of the space ℋε is called a Gaussian wave packet.

(1.81)

38 | 1 Abstract canonical operator and symplectic geometry Remark 1.5.3. Function (1.81) belongs to the space L2 if and only if the matrix Im αij is positive definite; the square of the norm of function (1.81) equals (ψα , ψα ) =

1

√det(2 Im α)

(1.82)

,

where det(2 Im α) denotes the determinant of the matrix 2 Im α. Definition 1.5.2. The complex germ corresponding to the Gaussian wave function (1.81) ℂ 2n is the following subspace of the space (Tℝ2n P,Q ) (the complexified tangent space to ℝ at the point (P, Q)): n

󵄨󵄨 󵄨 󵄨󵄨

𝒢α = {(p1 , . . . , pn ; q1 , . . . , qn ) 󵄨󵄨󵄨 pi = ∑ αij qj }. j=1

Remark 1.5.4. The complex germ has the sense of the graph of the operator α corresponding to the matrix αij . Let us prove the following lemma. Lemma 1.5.3. The subspace 𝒢α is a complete complex germ in the sense of Definition 1.3.2. Proof. Proof of Lemma 1.5.3 uses the following statement. Lemma 1.5.4. 1. Let (p, q) ∈ 𝒢α . Then the operators Ωℂ P,Q (p, q) satisfy the property Ωℂ P,Q (p, q)ψα = 0. 2.

Suppose that for an element f0 ∈ ℱP,Q the following relation holds: Ωℂ P,Q (p, q)f0 = 0,

(p, q) ∈ 𝒢α .

Then there exists a complex number c such that f0 = cψα . Proof. Let f (ξ ) = g(ξ )ψα (ξ ). Then n

n

k=1

l=1

Ωℂ P,Q (p, q)f (ξ ) = ψα (ξ ) ∑ (pk ξk − ∑ αkl qk ξl + iqk

𝜕 )g(ξ ) 𝜕ξk

by the definition of ψα . Since (p, q) ∈ 𝒢α and the matrix αkl is symmetric, we have n

Ωℂ P,Q (p, q)f = ψα ∑ iqk k=1

𝜕g . 𝜕ξk

This expression vanishes if and only if g = const. Lemma 1.5.4 is proved.

1.5 Finite-dimensional theory of complex germ at a point | 39

Corollary 1.5.1. The statements of Lemma 1.3.1 hold. Indeed, in the proof of this lemma we used only existence of a vector f0 possessing property (1.43). Remark 1.5.5. The statements of Lemma 1.3.1 in this case read n

(2) (2) (1) ∑(p(1) i qi − pi qi ) = 0, i=1

1 n ∑(p q∗ − ql p∗l ) ≥ 0, i l=1 l l

(p(1) , q(1) ), (p(2) , q(2) ) ∈ 𝒢α ,

(1.83)

(p, q) ∈ 𝒢α .

(1.84)

They can be also easily checked directly from Definition (1.76): property (1.83) follows from symmetry of the matrix αij , and property (1.84) from the positive definiteness of the imaginary part of this matrix. We also see that equality in formula (1.84) is achieved if and only if (p, q) = 0. For the proof of Lemma 1.5.3, it suffices to check completeness of the system of vectors (1.42). To this end, similar to Subsection 1.3.4, let us consider the following inner product in the linear space ℂn over the field ℂ: (2) (2) (1) (1) (q(1) , q(2) ) = ωℂ 2 ((aq , q ), (aq , q )) . ′

∗

This bilinear form satisfies all the axioms of inner product. Consider an orthonormal (k) basis {C (k) ∈ ℝn } with respect to this inner product and the matrices Cmk = Cm , Bmk = n ∑l=1 αml Clk . By orthonormality, we have 1 + (C B − B+ C) = E, i

(1.85)

where E is the unity matrix, and by property (1.83) we have BT C = C T B. Let us introduce the operators (m)∗ (m)∗ Λ̄ m = Ωℂ ,C ), P,Q (B

(m) (m) Λm = Ωℂ , C ), P,Q (B

(1.86)

where B(m) and C (m) are the elements of ℂn with the components B(m) = Blm , Cl(m) = Clm . l Lemma 1.5.5. The following properties of operators (1.86) hold: [Λm , Λk ] = [Λ̄ m , Λ̄ k ] = 0,

[Λk , Λ̄ m ] = δkm .

Proof. It follows directly from properties (1.83), (1.84), and formula (1.86).

(1.87)

40 | 1 Abstract canonical operator and symplectic geometry Lemma 1.5.6. The operators xk and the following way:

1 𝜕 i 𝜕xk

are expressed through the operators (1.86) in

n

∗ xk = i ∑ (Ckm Λm − Ckm Λ̄ m ), m=1 n

1 𝜕 = i ∑ (Bkm Λ̄ m − B∗km Λm ). i 𝜕xk m=1 Proof. By definition of the operators Λm and Λ̄ m , we have n BT Λ ( ̄ m ) = ∑ ( +mk Bmk Λm k=1

T xk Cmk + ) (1 𝜕 ) . Cmk i 𝜕x k

Properties (1.83) and (1.85) can be represented in the matrix form as follows: BT B+

(

CT ) C+

C∗ −B∗

−1

= −i (

−C ). B

This implies the statement of Lemma 1.5.6. Corollary 1.5.2. There exists a finite linear combination of vectors (1.42), which is equal to A

A

x1 1 x2 2 ⋅ ⋅ ⋅ xnAn exp{

i n ∑ x α x }, 2 i,j=1 i ij j

where A1 , . . . , An are nonnegative integers. Proof. For the proof, it suffices to express the operators xi through Λi and Λ̄ i by the formulas of Lemma 1.5.6, and we use Lemmas 1.5.4 and 1.5.2. Corollary 1.5.3. The system of vectors (1.42) is complete. Proof. For the proof, it suffices to note that the product of any polynomial f (x) and a Gaussian exponent ψα belongs to the set of finite linear combinations of vectors (1.42), and we use a lemma from [19]. Proof. Corollary 1.5.3 implies the statement of Lemma 1.5.3. Lemma 1.5.3 and the results of Section 1.3 imply the following corollary. Corollary 1.5.4. Let U be a canonical transformation of the phase space. ℂ Then the subspace U∗(P,Q) 𝒢α satisfies properties (1.83) and (1.84).

1.5 Finite-dimensional theory of complex germ at a point | 41

Lemma 1.5.7. 1. The matrix α′ of the form α′ = (

2.

−1

𝜕P ′ 𝜕P ′ 𝜕Q′ 𝜕Q′ + α)( + α) 𝜕Q 𝜕P 𝜕Q 𝜕P

(1.88)

ℂ 𝒢α . satisfies the condition 𝒢α′ = U∗(P,Q) ′ The matrix α is symmetric, and its imaginary part is positive definite.

ℂ Proof. The condition 𝒢α′ = U∗(P,Q) 𝒢α follows directly from the definition of the subℂ spaces 𝒢α′ and U∗(P,Q) 𝒢α . The second statement of the lemma follows from Corollary 1.5.2 of Lemma 1.5.3.

Lemma 1.5.7 implies the following corollaries. Corollary 1.5.5. If U is a canonical transformation of the phase space, then the subspace ℂ U∗(P,Q) 𝒢α is a complete complex germ. Proof. This corollary is proved by applying Lemma 1.5.3 to the germ 𝒢α′ = U∗(P,Q) 𝒢α . ℂ

Corollary 1.5.6. Any canonical transformation in finite-dimensional quantum mechanics is proper. Proof. The proof follows from Corollary 1.5.2 of Lemma 1.5.7 and from Lemma 1.3.4. The operator W is constructed using the procedure given in the proof of Lemma 1.3.4. Let us define the canonical operator corresponding to the complex germ (1.45), and let us construct formal asymptotic solutions of equations of motion. Denote by ℱ = L2 (Z+n ) the space of all complex functions f ̃ of nonnegative integer arguments ν1 , . . . , νn such that ∞

∑

󵄨󵄨 ̃ 󵄨2 󵄨󵄨f (ν1 , . . . , νn )󵄨󵄨󵄨 < ∞.

ν1 ,...,νn =0

2 n Consider the following operator ω̃ B,C ℱ : ℱ → L (ℝ ):

̃ ω̃ B,C ℱ f =

∞

∑

ν1 ,...,νn =0

f ̃(ν1 , . . . , νn )

ν Λ̄ 1 1

√ν1 !

⋅⋅⋅

ν Λ̄ nn ψα . √νn !

Lemma 1.5.8. The operator ω̃ B,C ℱ possesses the following properties: B,C ̃ ̃ 1. ‖ω̃ ℱ f ‖ = √| det C|‖f ‖. 2 n 2. The operator ω̃ B,C ℱ is a one-to-one map of the space ℱ onto L (ℝ ).

(1.89)

42 | 1 Abstract canonical operator and symplectic geometry Proof. The first property follows from the relation: ̃ ̃ B,C ̃ (ω̃ B,C ℱ f , ωℱ f ) =

∞

󵄨2 󵄨󵄨 ̃ 󵄨󵄨f (ν1 , . . . , νn )󵄨󵄨󵄨 (ψα , ψα );

∑

ν1 ,...,νn =0

̃ ̃ hence, for a = √(ψα , ψα ) we have ‖ω̃ B,C ℱ f ‖ = a‖f ‖. Since 1 1 ∗ −1 2 Im α = (BC −1 − (BC −1 ) ) = C +−1 (C + B − B+ C)C −1 = (CC + ) , i i then, according to formula (1.82), (ψα , ψα ) = √det(CC + ) = | det C|; hence, a = √| det C|. The second statement of Lemma 1.5.8 follows from Lemma 1.5.3. Lemma 1.5.8 is proved. Remark 1.5.6. By Lemma 1.5.8, we have det C ≠ 0. Definition 1.5.3. The canonical operator corresponding to the germ 𝒢α and to the basis ε C (m) is the operator K̃ P,Q,B,C : ℱ̃ → L2 (ℝn ) of the form ε K̃ P,Q,B,C =

1

√| det C|

ε KB,C ω̃ B,C ℱ .

(1.90)

Remark 1.5.7. The canonical operator introduced in [4, 25, 27] is defined as ε KP,Q,B,C =

1 K ε ω̃ B,C , √det C B,C ℱ

(1.91)

and it differs from the operator (1.90) only by a phase factor. Denote by B′ and C ′ the matrices 𝜕P ′ B 𝜕Q )( ). 𝜕Q′ C 𝜕Q

𝜕P ′

B′ ( ′ ) = ( 𝜕P′ 𝜕Q C 𝜕P

Lemma 1.5.9. The operator W is determined by formula (1.77) uniquely up to a multiplicative number constant c; it reads B ,C W = cω̃ ℱ (ω̃ B,C ℱ ) . ′

′

−1

(1.92)

Proof. The proof follows from Lemma 1.3.7. 1.5.4 Canonical transformations depending on time Let us now consider canonical transformations depending on time. Consider a family U t : (P, Q) → (P t , Qt ) of canonical transformations of the phase space ℝ2n smoothly depending on t, and the corresponding operators WP,Q,t : ℱP,Q → ℱPt ,Qt . By canonicity

1.5 Finite-dimensional theory of complex germ at a point | 43

of the transformation U t , there exists a function St such that n

dSt = ∑(Pit dQti − Pi dQi ).

(1.93)

i=1

Differentiating this relation with respect to t and denoting n

H t = ∑ Pit Q̇ ti − Ṡ t , i=1

we obtain that the element (P t , Qt ) ∈ ℝ2n of the phase space that satisfies the Hamiltonian system 𝜕H t Q̇ ti = , 𝜕Pi

t

𝜕H , Ṗ it = − 𝜕Qi

i = 1, . . . , n,

(1.94)

where the arguments Pit , Qti of the function H t : ℝ2n → ℝ are omitted. Denote, as usual, 𝜕P t B 𝜕Q )( ). 𝜕Qt C 𝜕Q

𝜕P t

Bt ( t ) = ( 𝜕Pt 𝜕Q C 𝜕P

(1.95)

Definition from Section 1.2 directly implies the following lemma. Lemma 1.5.10. The function ψεt ∈ L2 (ℝn ) of the form i

t

ψεt = ct e ε S K̃ Pε t ,Qt ,Bt ,Ct f ̃,

(1.96)

where |ct | = 1 and f ̃ is an element of ℱ not depending on t, is a formal asymptotic solution of the equations of motion. Remark 1.5.8. Using the definition of the canonical operator and of the operators Λ̄ i , one can represent the formal asymptotic solution (1.96) in the following form: i

t

t

t

ψεt = ct e ε (S +P (x−Q )) f t (

x − Qt ), √ε

(1.97)

where f t (ξ ) =

1

∞

∑

√| det C t | ν1 ,...,νn =0 n

×

f ̃(ν1 , . . . , νn )

t∗ ∑ (Bt∗ k1 1 ξk1 + iCk1 1

k1 ,...,kn =1

× exp{

1 √ν1 ! ⋅ ⋅ ⋅ νn !

𝜕 t∗ 𝜕 ) ⋅ ⋅ ⋅ (Bt∗ ) kn n ξkn + iCkn n 𝜕ξk1 𝜕ξkn

i n ∑ ξ αt ξ }, 2 m,k=1 m mk k

αt = Bt (C t ) . −1

(1.98)

44 | 1 Abstract canonical operator and symplectic geometry Lemma 1.5.11. The matrices Bt , C t satisfy the system of equations in variations (i. e., the linearized system of equations) for the system (1.94): n 𝜕2 H t t 𝜕2 H t t −Ḃ tmk = ∑( Blk + C ), 𝜕Qm 𝜕Pl 𝜕Qm 𝜕Ql lk l=1

(1.99)

n 𝜕2 H t t 𝜕2 H t t t Blk + C ). Ċ mk = ∑( 𝜕Pm 𝜕Pl 𝜕Pm 𝜕Ql lk l=1

Proof. The proof directly follows from formula (1.95). Let us now derive an equation for the function f t (ξ ). Lemma 1.5.12. The element f t ∈ L2 (ℝn ) of the form (1.98) satisfies the equation n

if ̇ t = ∑ [ m,k=1

+

1 𝜕2 H t 1 𝜕 1 𝜕 1 𝜕2 H t 1 𝜕 1 𝜕 + (ξ + ξ ) 2 𝜕Pm 𝜕Pk i 𝜕ξm i 𝜕ξk 2 𝜕Qm 𝜕Pk m i 𝜕ξk i 𝜕ξk m

1 𝜕2 H t ξ ξ ]f t + H1 (t)f t , 2 𝜕Qm 𝜕Qk m k

H1 (t) ∈ ℝ.

(1.100)

−1 Proof. Denote the operator iẆ P,Q,t WP,Q,t by Ht,2 . Then

if ̇t = Ht,2 f t .

(1.101)

Consider the following operators Λtm and Λ̄ tm : n

t Λtm = ∑ (Btkm ξk + iCkm k=1 n

𝜕 −1 ) = WP,Q,t Λ0m WP,Q,t , 𝜕ξk

−1 t∗ 𝜕 ) = WP,Q,t Λ̄ 0m WP,Q,t . Λ̄ tm = ∑ (Bt∗ km ξk + iCkm 𝜕ξk k=1

(1.102)

Let us differentiate relations (1.102) with respect to t. According to system (1.99), we have Λ̇ t [Ht,2 , Λ0m ] −1 ( ̇m t ) = −iWP,Q,t ([H , Λ̄ 0 ]) WP,Q,t ̄ Λm t,2 m n

BtT = ∑ ( ml Bt+ ml l,s=1

𝜕2 H t

tT − 𝜕P 𝜕Q −Cml l s t+ ) ( 𝜕2 H t −Cml 𝜕Ql 𝜕Qs

2

t

− 𝜕P𝜕 H𝜕P l

𝜕2 H t 𝜕Ql 𝜕Ps

s

) (1

ξs

𝜕 ). i 𝜕ξs

Using formulas (1.102) once again, we obtain 𝜕2 H t

n − 𝜕P 𝜕Q [Ht,2 , ξl ] l s −i( 1 𝜕 ) = ∑ ( 𝜕2 H t [Ht,2 , i 𝜕ξ ] s=1 l

𝜕Ql 𝜕Qs

2

t

− 𝜕P𝜕 H𝜕P l

𝜕2 H t 𝜕Ql 𝜕Ps

s

) (1

ξs

𝜕 ). i 𝜕ξs

(1.103)

1.5 Finite-dimensional theory of complex germ at a point | 45

These relations determine the operator Ht,2 uniquely up to a summand proportional to the unity operator. It is not difficult to check that the operator in square brackets in formula (1.100) satisfies condition (1.103); hence, formula (1.101) implies formula (1.100). Lemma 1.5.12 is proved. 1.5.5 Commutation of the canonical operator with the Hamiltonian and proof of the asymptotic formula In this section, we shall show that function (1.97) is indeed an asymptotic solution of the equations of motion. Consider the Cauchy problem i

𝜕ψεt = Htε ψεt , 𝜕t i

ψεt ∈ L2 (ℝn ),

ψε0 = e ε (S0 +P0 (x−Q0 )) f0 (

(1.104)

x − Q0 ), √ε

where the operator H ε in the space L2 (ℝn ) is defined as 1 1 Htε = H t ( 𝜕 ε −iε 𝜕x ,

2 1 t ) + H(1) ( 𝜕 x −iε 𝜕x ,

2 ); x 1

2

here to each real function Htε from certain class one assigns the operator H( −iε 𝜕 , x ) in 𝜕x

L2 (ℝn ) by the following rule: 1 𝜕 −iε 𝜕x ,

(H (

i 2 1 ) f ) (x) ≡ ∫ H(p, x)fε̃ (p)e ε px dp, n/2 x (2πε)

(1.105)

where fε̃ denotes the ε-Fourier transform of the function f , fε̃ (p) =

i 1 ∫ dy e− ε py f (y). (2πε)n/2

(1.106)

Remark 1.5.9. 1. We shall assume below that the functions H have the form ̄ Q), H(P, Q) = G(P, Q) + h(P,

(1.107)

where G is a polynomial, and h̄ ∈ 𝒮 (ℝ2n ). Then the space 𝒮 (ℝn ) belongs to the 1 2 domain of the operators H( −iε 𝜕 , x ). Indeed, in the case G = 0 integral (1.105) 𝜕x takes the form i 1 ̄ x)f ̃ (p), ∫ dp e ε px h(p, ε n/2 (2πε)

46 | 1 Abstract canonical operator and symplectic geometry

2.

and it belongs as a function of x to the space 𝒮 (ℝn ), due to rapid decrease of h̄ and fε̃ . For the case h̄ = 0, see the next remark. If the function H(P, Q) is a monomial, H(P, Q) = Pi1 ⋅ ⋅ ⋅ Pil Qj1 ⋅ ⋅ ⋅ Qjm , 1

i1 , . . . , il , j1 , . . . , jm ∈ {1, 2, . . . , n},

2

then the operator H( −iε 𝜕 , x ) reads 𝜕x

1 𝜕 −iε 𝜕x ,

H(

2 𝜕 𝜕 ) ⋅ ⋅ ⋅ (−iε ). ) = xj1 ⋅ ⋅ ⋅ xjm (−iε 𝜕xi1 𝜕xil x

Let us now consider commutation of the operator i 𝜕t𝜕 − Htε with the canonical operator. Lemma 1.5.13. For f ∈ 𝒮 (ℝn ), the following relation holds: 1 H( 𝜕 −iε 𝜕x ,

2 ε ) K ε f (x) = KP,Q g(x), x P,Q

1 g(x) = H ( P − i√ε 𝜕ξ𝜕 ,

2 ) f (ξ ). Q + √εξ

(1.108)

Before proving this lemma, let us illustrate it in the case when the function H is 1 2 a polynomial. In this case, it suffices to check formula (1.108) for operators H( −iε 𝜕 , x ) of the form −iε 𝜕x𝜕 and xk , where k = 1, . . . , n. We have

𝜕x

k

x−Q ) √ε x − Qk 1 i x−Q = n/4 e ε P(x−Q) (Qk + √ε k )f ( ), √ε √ε ε

ε xk KP,Q f (x) =

−iε

1

xk e εn/4

i P(x−Q) ε

f(

𝜕 1 εi P(x−Q) x − Q 𝜕 ε KP.Q f (x) = −iε e f( ) √ε 𝜕xk 𝜕xk εn/4 1 i x−Q 𝜕f = n/4 e ε P(x−Q) (Pk − i√ε𝜕k )f ( ), 𝜕k f (ξ ) ≡ . √ε 𝜕ξ ε k

Let us proceed to the proof of Lemma 1.5.13 in the general case. Proof. Consider the function H(

1 𝜕 −iε 𝜕x ,

2 ε ) KP,Q f (x). x

(1.109)

1.5 Finite-dimensional theory of complex germ at a point | 47

We have: ε f (x) equals 1. The ε-Fourier transform of the function Ψ(x) = KP.Q Ψ̃ ε (p) =

i 1 ε f (y), ∫ dy e− ε py KP.Q n/2 (2πε)

and after the change of variables y = Q + ξ √ε and applying the canonical operator, it takes the form p−P 1 , ) Ψ̃ ε (p) = e− ε pQ f ̃( √ε εn/4 i

where f ̃ is the Fourier transform (1.106) of the function f for ε = 1. 2. Function (1.109) equals i

∫ H(p, x)e ε p(x−Q) f ̃(

dp p−P 1 ) , √ε (2πε)n/2 εn/4

̃ and after the change of variables p = P + π√ε it takes the form 1

εn/4

i dπ̃ ̃ e ε P(x−Q) ∫ H(P + π̃ √ε, Q + ξ √ε)eiπξ f ̃(π)̃ , (2π)n/2

where ξ = (x − Q)/√ε. This implies formula (1.108). Lemma 1.5.13 is proved. Let us now consider commutation of the operator i 𝜕t𝜕 with the canonical operator. Lemma 1.5.14. The following relation holds: i where g t (ξ ) = ( ε1 (Ṡ t − P Q̇ t ) −

i 𝜕 εi St ε e KPt ,Qt f t = e ε St KPεt ,Qt g t , 𝜕t

1 ̇t Pξ √ε

−

i ̇t 𝜕 Q 𝜕ξ √ε

+ i 𝜕t𝜕 )f t (ξ ).

Proof. The proof immediately follows from definition of the canonical operator. Assume now that f t satisfies equation (1.100), t

St = S0 + ∫(P τ Q̇ τ − H τ (P τ , Qτ )) dτ, 0

where (P τ , Qτ ) is a solution of the system (1.94), ct = 1, and ψεt has the form (1.97). Lemmas 1.5.13 and 1.5.14 imply the following corollary. Corollary 1.5.7. The following relation holds: (i

i 𝜕 − Htε )ψεt = e ε St KPε t ,Qt g t , 𝜕t

48 | 1 Abstract canonical operator and symplectic geometry 2

t

t where g t = (H1 (t) − 2i ∑nm=1 𝜕P𝜕 H𝜕Q − H(1) (P t , Qt ))f t − δt , δt = ( ε1 H̃ t (P t − i√ε 𝜕ξ𝜕 , Qt + √εξ ) + m m H̃ t (P t − i√ε 𝜕 , Qt + √εξ ))f t ; here, we have denoted by H̃ t the following function: (1)

𝜕ξ

H̃ t (P, Q) = H t (P, Q) − H t (P t , Qt ) d 1 d2 󵄨󵄨󵄨 − (α + α2 2 )󵄨󵄨󵄨 H t (P t + α(P − P t ), Qt + α(Q − Qt )), dα 2 dα 󵄨󵄨α=0 and t t t H̃ (1) (P, Q) = H(1) (P, Q) − H(1) (P t , Qt ).

Below we shall assume that the following condition holds: t Im H(1) =−

1 n 𝜕2 H t , ∑ 2 m=1 𝜕Pm 𝜕Qm

and let us choose the function H1 (t) in the form t H1 (t) = H(1) (P t , Qt ) +

i n 𝜕2 H t (P t , Qt ). ∑ 2 m=1 𝜕Pm 𝜕Qm

(1.110)

Let us show that ψεt is an asymptotic solution of the Cauchy problem for equation (1.104). To this end, let us use the following lemma. ̃ Lemma 1.5.15. Let f ∈ S(ℝn ) and let the following estimate hold as (P, Q) → (P,̃ Q): M H(P, Q) = O(((P − P)̃ 2 + (Q − Q)̃ 2 ) ),

(1.111)

where M is a nonnegative integer. Then 󵄨󵄨 1 󵄨󵄨 󵄨󵄨H ( 𝜕 󵄨󵄨 , −iε 󵄨 𝜕x

󵄨󵄨 2 󵄨󵄨 ε M f ) KP, ̃ Q̃ 󵄨󵄨󵄨 = O(ε ). x 󵄨󵄨

(1.112)

Proof of this lemma uses several auxiliary statements. Lemma 1.5.16. For f ∈ 𝒮 (ℝn ), l+s 󵄩󵄩 ̂ ̂ − Q̃ ) ⋅ ⋅ ⋅ (Q ̂ − Q̃ )K ε̃ ̃ f 󵄩󵄩󵄩 = O(ε 2 ), ̂ i − P̃ i )(Q 󵄩󵄩(Pi1 − P̃ i1 ) ⋅ ⋅ ⋅ (P j1 j1 js js P,Q 󵄩 l l ̂ =x . ̂ k = −iε 𝜕 , Q P k k 𝜕xk

(1.113)

ε Proof. By the commutation rule (see Lemma 1.5.13) and by the property ‖KP, ̃ Q̃ g‖ = ‖g‖, the expression in the left-hand side of formula (1.113) can be represented in the

1.5 Finite-dimensional theory of complex germ at a point | 49

form e

l+s 2

󵄩󵄩 1 𝜕 󵄩󵄩 l+s 1 𝜕 󵄩󵄩 󵄩 ⋅⋅⋅ ξj1 ⋅ ⋅ ⋅ ξjs f 󵄩󵄩󵄩 = O(ε 2 ). 󵄩󵄩 󵄩󵄩 i 𝜕ξi 󵄩 i 𝜕ξ 󵄩 il 1

Lemma 1.5.16 is proved. Remark 1.5.10. Thus, the heuristical derivation of formula (1.112) is to decompose the ̃ and to check formula (1.112) for function H into the Taylor series as (P, Q) → (P,̃ Q), each summand of the series. Lemma 1.5.17. Let H be an infinitely differentiable function with compact support, and let M

H(P, Q) = O((P 2 + Q2 ) )

(1.114)

as (P, Q) → 0, where M is a nonnegative integer. Then the following estimate holds: 1 εM

󵄩󵄩 1 󵄩󵄩 󵄩󵄩H ( 𝜕 󵄩󵄩 −iε , 󵄩 𝜕x

󵄩󵄩 2 󵄩 ε ) K0,0 f 󵄩󵄩󵄩󵄩 < C, x 󵄩󵄩

(1.115)

where C is a constant independent of ε. Proof. Since the ε-Fourier transform of the function f (x/√ε) has the form f ̃(p/√ε), f ̃ ∈ 𝒮 (ℝn ), formula (1.105) implies that (H (

1 𝜕 −iε 𝜕x ,

i 2 dp 1 p ε ) K0,0 f ) (x) = ∫ n/4 H(p, x)e ε px f ̃( ). n/2 √ε x (2πε) ε

(1.116)

For the proof of the lemma, it suffices to prove that 󵄨󵄨 i Cl p dp 󵄨󵄨󵄨 󵄨󵄨 px M , 󵄨󵄨∫ H(p, x)e ε f ̃( ) n/2 󵄨󵄨󵄨 ≤ ε 󵄨󵄨 √ε ε 󵄨󵄨 (1 + x 2 /ε)l

(1.117)

where Cl is a constant dependent on l and independent on ε. Indeed, for l sufficiently large, the function (1 + x2 /ε)−l belongs to L2 (ℝn ), and this implies estimate (1.115). To check property (1.117), let us extract the factor (P 2 + Q2 )M from the function H, M ̃ H(P, Q) = (P 2 + Q2 ) H(P, Q),

then the function H̃ is also infinitely differentiable and has compact support. Consider the change of variables P = √επ in the integral (1.117). Formula (1.117) is equivalent to the following one: M l 󵄨󵄨 󵄨󵄨 i x2 x2 󵄨 󵄨󵄨 ̃ √ πx 2 󵄨󵄨∫ H(π ε, x)(π + ) (1 + ) f ̃(π)e √ε dπ 󵄨󵄨󵄨 ≤ Cl . 󵄨󵄨 󵄨󵄨 ε ε

(1.118)

50 | 1 Abstract canonical operator and symplectic geometry i

For proof of formula (1.118), let us represent functions of type f (x)e √ε

i πx 𝜕 f ( √ε )e √ε , i 𝜕π

the form

πx

in the form

integrate by parts, and reduce the left-hand side of relation (1.118) to i

M

πx

̃ √ε, x)]. ∫ dπe √ε (π 2 − Δπ ) (1 − Δπ )l [f ̃(π)H(π

(1.119)

By boundedness of all derivatives of the function H and by rapid decrease of the function f ̃, expression (1.119) is bounded uniformly in ε. This implies inequality (1.117), and hence, the statement of Lemma 1.5.17. Lemma 1.5.18. Let the function H(P, Q) vanish for |P|2 + |Q|2 ≤ C and satisfy property (1.107). Then for any L > 0, 1 εL

󵄩󵄩 1 󵄩󵄩 󵄩󵄩H ( 𝜕 󵄩󵄩 , −iε 󵄩 𝜕x

󵄩󵄩 2 󵄩 ε→0 ε ) K0,0 f 󵄩󵄩󵄩󵄩 󳨀→ 0. x 󵄩󵄩

(1.120)

Proof. First of all, note that by equality (1.116) for the proof of the lemma, it suffices to check that l 󵄨󵄨 󵄨󵄨 1 i p x2 󵄨 󵄨󵄨 px 󵄨󵄨 L ∫ H(p, x)e ε f ̃( )(1 + ) dp󵄨󵄨󵄨 ≤ Cl . 󵄨󵄨 √ε ε 󵄨󵄨 ε

(1.121)

Representing the left-hand side of this relation in the form l

∫ i

L

2 2 2 i dp ̃(p/√ε)H(p, x)(1 + x ) ( x + p ) e ε px , f ε ε ε (x2 + p2 )L

(1.122)

i

𝜕 ε px using relation xe ε px = εi 𝜕p e and integrating by parts, we obtain that expression (1.122) is no greater than L f ̃( p )H(p, x) 󵄨 󵄨󵄨 󵄨󵄨 p2 √ε 󵄨 󵄨󵄨. ∫ dp󵄨󵄨󵄨(1 − εΔp )l ( − εΔp ) 󵄨󵄨 ε (x 2 + p2 )L 󵄨󵄨󵄨

By property x2 + p2 ≥ C for H ≠ 0, by property (1.107) and by rapid decrease of f ̃, this expression, for L sufficiently large, is bounded uniformly in ε. This implies the statement of Lemma 1.5.18. Proof of Lemma 1.5.15. First, let us check the statement of the lemma for P̃ = Q̃ = 0. Since any function H is represented as a sum of two functions, one of which satisfies the conditions of Lemma 1.5.17, and the other satisfies the conditions of Lemma 1.5.18, we obtain that for the M integer the estimate (1.115) holds. Extracting from function H finite number of summands of its Taylor decomposition and using Lemma 1.5.16, we obtain statement (1.112) for P̃ = Q̃ = 0.

1.5 Finite-dimensional theory of complex germ at a point | 51

ε Let us now consider the case of arbitrary P,̃ Q.̃ By the property ‖K0,0 f ‖ = ‖f ‖ = which, together with Lemma 1.5.13, implies the relation

ε ‖KP, ̃ Q̃ f ‖

󵄩󵄩 1 󵄩󵄩 󵄩󵄩H ( 𝜕 󵄩󵄩 −iε 𝜕x , 󵄩

󵄩󵄩 󵄩󵄩 󵄩󵄩 1 2 2 󵄩󵄩 󵄩󵄩 󵄩󵄩 ε ε ) KP, 󵄩󵄩 = 󵄩󵄩󵄩KP,̃ Q̃ H (P̃ − i√ε 𝜕 , Q̃ + √εξ ) f 󵄩󵄩󵄩 ̃ Q̃ f 󵄩 x 󵄩󵄩 󵄩󵄩 󵄩󵄩 𝜕ξ 󵄩󵄩 󵄩 󵄩󵄩 1 2 󵄩 ε H( ̃ = 󵄩󵄩󵄩󵄩K0,0 ) f 󵄩󵄩 P − i√ε 𝜕ξ𝜕 , Q̃ + √εξ 󵄩󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 2 󵄩 󵄩󵄩 ε 󵄩󵄩 , = 󵄩󵄩󵄩󵄩H ( ̃ ) K f 𝜕 0,0 󵄩󵄩 P − iε 𝜕x , Q̃ + x 󵄩󵄩 󵄩

relation (1.112) holds for arbitrary P,̃ Q.̃ Lemma 1.5.15 is proved. Lemma 1.5.15 and Corollary 1.5.2 of Lemma 1.5.14 imply the following theorem. Theorem 1.5.1. Function ψεt of the form (1.97) is an asymptotic solution of the Cauchy problem for equation (1.104). Remark 1.5.11. 1. If the operator Htε is independent of t and self-adjoint, then Section 1.4 implies that ψεt is an asymptotic of the solution of the Cauchy problem. 2. The statement of Theorem 1.5.1 holds also in the case when P and Q are smooth functions of √ε. Thus, in the case of a self-adjoint operator Htε independent of t, we have shown that the set of maps ε

(𝒰tε = e−iH t , Ut , WX,t , St ) 3.

is a canonical transformation of the canonical operator. Asymptotic solutions of equation (1.104) up to O(εM/2 ) can be constructed in the following way. Consider the function i

t

ψεt = e ε S KPε t ,Qt f t (ε), substitute it into equation (1.75), obtain the following equation on f t (ε): (i

𝜕 𝜕 t − H(1) (P t − i√ε , Qt + √εξ ) 𝜕t 𝜕ξ

1 𝜕 − [H t (P t − i√ε , Qt + √εξ ) − H t (P t , Qt ) ε 𝜕ξ −

𝜕H t t t 𝜕 𝜕H t t t (P , Q )(−i√ε ) − (P , Q )√εξ ])f t (ξ ) = 0, 𝜕P 𝜕ξ 𝜕Q

52 | 1 Abstract canonical operator and symplectic geometry decompose the operator in the left-hand side of this equation into the asymptotic series in √ε, and solve this equation by perturbation theory (Section 1.4).

Appendix 1.A. Classical and quantum mechanics: the main definitions [1, 20, 53] In this Appendix, we consider: (a) classical (deterministic) mechanics; (b) classical statistical mechanics; (c) quantum mechanics; (d) quantum statistical mechanics. In all of the cases, we recall the following notions: (a) the state space; (b) observables and their norms; (c) the mean value of an observable in a given state; (d) the evolution operator. We consider also examples corresponding to various physical systems. 1. The state space in classical mechanics is the vector space ℝ2n . We shall denote the first n components of an element of ℝ2n by p = (p1 , . . . , pn ), and the remaining n components by q = (q1 , . . . , qn ). The variables (p1 , . . . , pn ) play the role of momenta of the particles, and (q1 , . . . , qn ) play the role of coordinates of the particles. An observable O

O is a real continuous bounded function ℝ2n 󳨀→ ℝ, and the norm of an observable O is ‖O‖ = sup(p,q)∈ℝ2n |O(p, q)|. The mean value of an observable O in a state (p, q) is the number O(p, q). The evolution operator Ut is the operator mapping the initial condition (p(0) , q(0) ) of the Cauchy problem into the solution of the Cauchy problem t q̇ m =

𝜕H , 𝜕pm

ṗ tm = −

𝜕H , 𝜕qm

(0) qm (0) = qm ,

pm (0) = p(0) m ;

(1.A.1)

t here, the derivatives of the function H are taken at the point ptm , qm , m = 1, . . . , n. The system (1.A.1) is called the Hamiltonian system, and the function H is called the Hamiltonian function. 2. The state space in classical statistical mechanics is the space of real functions from L1 (ℝ2n ); the notion of observable remains the same. Note that for ρ(p, q) ∈ L1 (ℝ2n ), ‖ρ‖ = 1, ρ > 0, the function ρ plays the role of density of probability distribution; p = (p1 , . . . , pn ) are the momenta of the particles, q = (q1 , . . . , qn ) are the

Appendix 1.A. Classical and quantum mechanics: the main definitions [1, 20, 53]

| 53

coordinates of the particles. The mean value of an observable O in a state ρ is ⟨O⟩ = ∫ dp dq O(p, q)ρ(p, q). The evolution operator Ut′ is the operator taking each function ρ ∈ L1 (ℝ2n ) to the function (Ut′ ρ)(p, q) = ρ(Ut−1 (p, q)), where Ut is the operator considered in the previous example. 3. The state space in quantum mechanics is a Hilbert space ℋ; its elements will be also called state vectors or wave functions. An observable O is a bounded self-adjoint operator in the space ℋ. By ‖O‖, let us denote, as usual, ‖O‖ =

sup

Ψ∈ℋ,‖Ψ‖=1

‖OΨ‖.

The mean value of an observable O in a state Ψ ∈ ℋ is ⟨O⟩ = (Ψ, OΨ).

(1.A.2)

̂ ̂ t = e−iHt/ℏ ̂ is a The quantum mechanical evolution operator is the operator U , where H self-adjoint operator in ℋ called the Hamiltonian operator, and ℏ is a positive number called the Planck constant. 4. The state space in quantum statistical mechanics is the space of Hermitian trace class operators acting in the space ℋ. If ρ is a positive semidefinite operator and Tr ρ = 1, then it is also called the density matrix. A pure state is an element of the state space, which is the operator of projection PΨ onto an element Ψ ∈ ℋ. Note that the density matrix w1 PΨ1 + ⋅ ⋅ ⋅ + wn PΨn , w1 , . . . , wn > 0, w1 + ⋅ ⋅ ⋅ + wn = 1, not corresponding to a pure state, has the following physical sense: the system is in the state Ψ1 with the probability w1 , . . . , in the state Ψn with the probability wn . Define the notion of observable as in the previous case; the mean value of an observable O in a state ρ is Tr Oρ: it coincides with (1.A.2) for ρ = PΨ . The evolution operator is the operator taking the operator ρ0 to the operator ρt of the form

̂ t ρ0 U ̂ t−1 , ρt = U ̂ t is defined in the previous example. where the operator U Let us now consider examples of classical and quantum mechanical systems. 1. A particle of mass m acting in the n-dimensional space in the exterior potential U. The state space in classical mechanics is ℝ2n ; the function H has the form H(p, q) =

p2 + U(q). 2m

54 | 1 Abstract canonical operator and symplectic geometry ̂ reads In quantum mechanics, the state space is L2 (ℝn ), and the operator H 2

̂ = − ℏ Δ + U(x), H 2m ̂ k = −iℏ𝜕/𝜕xk in the space L2 (ℝn ) is where Δ is the Laplace operator. The operator p called the operator of the kth projection of momentum, and the operator q̂k of multiplication by xk is called the operator of the kth coordinate. Note that the operator ̂ is obtained from the Hamiltonian function H by change of the momenta p by the H operators of momenta, and the coordinates q by the operators of coordinates. 2. The system of N different particles with masses mi , each of which moves in the exterior potential U in the n-dimensional space and which interact with one another; the interaction potential equals V. The state space in classical mechanics is ℝ2nN , and the function H reads n

H(p, q) = ∑( i=1

p2i + U(qi )) + ∑ V(qi , qj ). 2mi 1≤i 0, issuing from a point X, X1 (0) = X2 (0) = X, are said to be equivalent, X1 ∼ X2 , if for a certain chart covering the point X, and the following relation holds: Φ(i) X1 (α) − Φ(i) X2 (α) α→0 󳨀→ 0. α A tangent vector to ℳ at a point X is an equivalence class of curves. The coordinates β of a tangent vector b to a manifold ℳ at a point X in a chart Φ(i) are the following elements of the coordinate space ℋ: β=

d 󵄨󵄨󵄨󵄨 (i) 󵄨 Φ X(α), dα 󵄨󵄨󵄨α=0

where X(α) is a representative of the equivalence class corresponding to the vector b. Note that the coordinates β are transformed under the passage from a chart to chart, Φ(j) Φ(i)−1 : Uij → Uji , in the following way: β′ = X̃ ′ (X)β, where ̃ X(X) = Φ(j) Φ(i)−1 X. The sum of two tangent vectors b1 and b2 at a point X ∈ ℳ is the tangent vector b whose components in some chart Φ(i) are equal to the sums of the components of vectors b1 and b2 ; the product of a tangent vector by a number λ ∈ ℝ is the tangent vector λb whose coordinates are λ times greater than coordinates of the vector b. It is not difficult to show that the definition of the sum of tangent vectors and of a product by a number do not depend on the choice of a chart. Denote the vector space of tangent vectors at a point X by T ℳX . In the book, we sometimes identify, for the sake of brevity, points of a manifold ℳ with their coordinates in certain chart. To each smooth map of manifolds Y : ℳ → 𝒩 , one can assign the induced map of tangent spaces Y∗X : T ℳX → T 𝒩YX , X ∈ ℳ, as follows. Let b ∈ T ℳX , and let X(α) be a representative of the equivalence class of curves on the manifold ℳ corresponding

Appendix 1.B. Some recollections from differential geometry | 57

to the vector b. Consider the curve y(α) = YX(α) on the manifold 𝒩 . It is not difficult ̃ to check that two equivalent curves X(α) ∼ X(α) on ℳ correspond to two equivalent curves on 𝒩 . This equivalence class is an element of T 𝒩YX , which we denote by Y∗X b. A real differential 1-form on ℳ is a linear map T ℳX → ℝ, and a real differential 2-form is a bilinear skew-symmetric map T ℳX ×T ℳX → ℝ. The definition of complex differential forms is obtained from the definition of real ones by replacing ℝ with ℂ. Let us define the notion of operator-valued differential 1-form. A map Ω taking each element b ∈ T ℳX to an operator ΩX (b) with domain 𝒟X independent of b in a Hilbert space ℋ1 is called by an operator valued 1-form if for f ∈ 𝒟X , b1 , b2 ∈ T ℳX , λ1 , λ2 ∈ ℝ, the following relation holds: ΩX (λ1 b1 + λ2 b2 )f = λ1 ΩX (b1 )f + λ2 ΩX (b2 )f . A function f : ℳ → ℝ is called smooth if the functions f (i) Φ(i)−1 : Ui → ℝ, where f is the restriction of the function f to the set Wi , are infinitely differentiable. The differential of a function f : ℳ → ℝ is the differential 1-form mapping a tangent vector b at a point X into the number (i)

d 󵄨󵄨󵄨󵄨 󵄨 f (X(α)), dα 󵄨󵄨󵄨α=0

df X (b) =

where X(α) is a representative of the equivalence class of curves corresponding to the vector b. Let Φ(i) (X) = (X1 , X2 , . . .). Then any differential 1-form can be represented as a superposition of differentials: dim ℋ

ω1 = ∑ ω1,k dXk . k=1

The functions ω1,k are called the components of the 1-form ω1 in the chart Φ(i) . The exterior product of 1-forms ω′1 and ω′′ 1 is the 2-form, ′ ′′ ′ ′′ (ω′1 ∧ ω′′ 1 )(b1 , b2 ) = ω1 (b1 )ω1 (b2 ) − ω1 (b2 )ω1 (b1 ).

The differential of a 1-form ω1 is the 2-form, dim ℋ

dω1 = ∑

k,l=1

𝜕ω1,k dXl ∧ dXk . 𝜕Xl

Let us now define the notion of complexified tangent space. The complexified tangent space T ℳℂ X is the vector space of ordered pairs (a; b) of elements of T ℳX , which we also denote by a + ib; a, b ∈ T ℳX . Define the sum and multiplication by a complex

58 | 1 Abstract canonical operator and symplectic geometry number as (a1 + ib1 ) + (a2 + ib2 ) ≡ (a1 + a2 ) + i(b1 + b2 ),

(λ1 + iλ2 )(a + ib) ≡ (λ1 a − λ2 b) + i(λ2 a + λ1 b).

Define the involution in the space T ℳℂ X as (a + ib)∗ ≡ a − ib. ℂ The complexification ωℂ 1 of a real (complex) differential 1-form ω1 is the linear map ω1 given by

ωℂ 1 (a + ib) = ω1 a + iω1 b. Similarly, one defines also complexification of an operator valued differential 1-form.

1.B.3 Universal covering, homology, and cohomology In this section, we restrict ourselves to considering finite-dimensional manifolds whose any two points can be joined by a smooth curve. Two curves on a manifold ℳ, X0 (τ), and X1 (τ), τ ∈ [0, 1], with coinciding endpoints, X0 (0) = X1 (0), X0 (1) = X1 (1), are called homotopic if there exists a smooth d d X(α, 0) = dα X(α, 1) = 0, X(0, τ) = X0 (τ), function X : [0, 1] × [0, 1] → ℳ such that dα X(1, τ) = X1 (τ). The universal covering of a manifold ℳ is the set of homotopy equivalence classes of curves starting at certain fixed point of the manifold ℳ. To each point of the universal covering, one uniquely assigns a point of the manifold ℳ. To any differential 1-form ν on ℳ such that dν = 0, one can assign a function s on the universal covering such that ds = ν. Let us now recall the definition of fundamental group. The product of closed curves X1 (τ) and X2 (τ) is the following curve: X1 (2τ),

X3 (τ) = {

τ ≤ 1/2,

X2 (2τ − 1), τ ≥ 1/2.

The fundamental group π1 (ℳ) is the set of homotopy equivalence classes of closed curves with the fixed endpoint, endowed with the product operation introduced above. The quotient group of this group by the set s

{∏ αi βi αi−1 βi−1 , αi , βi ∈ π1 (ℳ)} i=1

Appendix 1.B. Some recollections from differential geometry | 59

is called the homology group H1 (ℳ) of the manifold ℳ. This group is Abelian, and its dimension is called the Betti number of the manifold ℳ. Two differential forms ν1 and ν2 , such that dν1 = dν2 = 0 on ℳ, are called equivalent if their difference is a full differential, ν1 − ν2 = dS. The space of equivalence classes is called the cohomology space H 1 (ℳ) of the manifold ℳ. The following de Rham theorem holds: the dimension of this space coincides with the Betti number of the manifold ℳ.

2 Multiparticle canonical operator and its properties 2.1 Introduction 2.1.1 Physical problems giving rise to the study of functions of large number of arguments In this chapter, we shall consider construction of approximations for functions whose number of arguments tends to infinity. The problem in the study of functions of large number of arguments often arises in statistical physics both in the classical and in the quantum case. These functions describe a state of a system of a large number of particles. For instance, in quantum mechanics of N particles in ν-dimensional space, a state of the system is described by a wave function of N arguments x1 , . . . , xN , where the parameters x1 , . . . , xN play the role of coordinates of particles and belong to ℝν . If all of the N particles are identical bosons, then the wave function of N arguments must be symmetric. If these N particles are of k types: N1 bose-particles of the first type, . . . , Nk bose-particles of the kth type, then the wave function depending on the arguments x1(1) , . . . , xN(1) ; . . . ; x1(k) , . . . , xN(k) , is symmetric with respect to each set of arguments 1

k

x1(l) , . . . , xN(l) , for l = 1, . . . , k. This case can be considered by analogy with the case when l all the particles are identical. In the case when the particles have also inner degrees of freedom (like spin), then the variables x are sets (x,̃ j), where x̃ ∈ ℝν are coordinates of the particles, and the discrete index j enumerates the states corresponding to the inner degree of freedom (e. g., in the case of particles with spin, the index j is the projection of spin onto a distinguished space axis). Finally, we can consider also quantum systems of a large number of particles on a discrete lattice, describing a state of the system by a function ψN (i1 , . . . , iN ), where the parameters ik are the numbers of units of the lattice in which the particles are situated, and belong to the set of integers for the case of a infinite lattice and to a finite set in the case of a finite lattice. In the examples of systems of N bosons given above, the state space ℋ1/N is the following. Let 𝒳 be a space with a measure μ. Denote by 𝒳 N the Nth power of the space 𝒳 , 𝒳 N = 𝒳 × ⋅ ⋅ ⋅ × 𝒳 , with the measure, which is the Nth power of the measure μ, μN = μ ⊗ ⋅ ⋅ ⋅ ⊗ μ. Let us choose the number 1/N as the parameter tending to zero, where N ≥ N0 is a natural number, and as the space ℋ1/N , let us choose the space of complex symmetric functions ψN (x1 , . . . , xN ) from L2 (𝒳 N ), xi ∈ 𝒳 .

https://doi.org/10.1515/9783110762709-002

62 | 2 Multiparticle canonical operator and its properties Let us consider the evolution operators Ut1/N of the form Ut1/N = e−iHN t , where HN is the operator in L2 (𝒳 N ), which acts on a function ψ ∈ L2 (𝒳 N ) as follows: (HN ψ)(x1 , . . . , xN ) P0

=∑

p=1

1

N p−1

1 ∑ p! 1≤i =⋅⋅⋅ ̸ =i̸ 1

∫ dyi1 ⋅ ⋅ ⋅ dyip H (p) (xi1 , . . . , xip ; yi1 , . . . , yip ) p ≤N

× ψ(x1 , . . . , xi1 −1 , yi1 , xi1 +1 , . . . , xip −1 , yip , xip +1 , . . . , xN );

(2.1)

here, P0 ∈ ℕ, in the right-hand side of formula (2.1) one replaces p of N arguments xi1 , . . . , xip of the function ψ by yi1 , . . . , yip , H (p) are certain distributions separately symmetric with respect to the arguments xi1 , . . . , xip and with respect to the arguments yi1 , . . . , yip , satisfying the additional condition H

(p)

(x1 , . . . , xp ; y1 , . . . , yp ) = H (p) (y1 , . . . , yp ; x1 , . . . , xp );

(2.2)

bar means complex conjugation. In particular, in the case of a quantum system of N identical spinless boseparticles, situated in exterior field and interacting with one another, the operator HN is the Hamiltonian operator (1.A.3), in which pi are the momenta operators. Its kernel indeed has the form (2.1) for the appropriate choice of distributions H (p) , if the coefficient before the interaction potential between the particles is proportional to the inverse number of particles, i. e., if V = N1 ν, and ν does not depend on N. Note that all the constructions of this chapter remain valid if instead of the space of symmetric functions from L2 (𝒳 N ) one considers the Nth symmetrized power of any Hilbert space with involution.

2.1.2 Physical arguments leading to the choice of a norm in the space of functions of large number of arguments In this chapter, we shall consider normed spaces ℱ and the canonical operator mapping them into the space ℋ1/N ; this canonical operator satisfies the axioms listed in Section 1.2. In particular, we shall construct the asymptotics as N → ∞ of the solution of the following Cauchy problem: i

dψN (t) = HN ψN (t), dt

ψN (0) = ψ(0) N ,

ψN (t) ∈ ℋ1/N , i

(2.3)

in the case when the initial condition is chosen in the form e ε S0 KXε 0 f0 , i. e., a function ψas N (t, x1 , . . . , xN ) such that the difference between it and between the solution of the Cauchy problem (2.3) is small as N → ∞ with respect to the norm in the space

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L2 (𝒳 N ): 󵄨 󵄨󵄨2 N→∞ ∫ dx1 ⋅ ⋅ ⋅ dxN 󵄨󵄨󵄨ψas N (t, x1 , . . . , xN ) − ψN (t, x1 , . . . , xN )󵄨󵄨 󳨀→ 0, where by ∫ dx1 ⋅ ⋅ ⋅ dxN we denote the integral with respect to the measure μN . As a motivation of the choice of the norm in the space L2 (𝒳 N ) as the norm in the space ℋ1/N , let us give the following lemma. Lemma 2.1.1. Let AN be operators, bounded uniformly in N, in the space L2 (𝒳 N ), ‖AN ‖ < (2) 2 N C, and let Ψ(1) N and ΨN be elements of the space L (𝒳 ) such that N→∞ 󵄩󵄩 (1) (2) 󵄩 󵄩󵄩ΨN − ΨN 󵄩󵄩󵄩L2 (𝒳 N ) 󳨀→ 0,

N→∞ 󵄩󵄩 (1) 󵄩󵄩 󵄩󵄩ΨN 󵄩󵄩L2 (𝒳 N ) 󳨀→ const ≠ 0,

N→∞

(1) and (Ψ(1) N , AN ΨN ) 󳨀→ A.

N→∞

(2) Then also (Ψ(2) N , AN ΨN ) 󳨀→ A.

Proof. We have 󵄨󵄨 (2) 󵄩 (2) (2) (1) (1) 󵄨 (1) 󵄩 󵄨󵄨(ΨN , AN ΨN ) − (ΨN , AN ΨN )󵄨󵄨󵄨 ≤ const󵄩󵄩󵄩ΨN − ΨN 󵄩󵄩󵄩L2 (𝒳 N ) N→∞

(2) by conditions of the lemma. Hence, also (Ψ(2) N , AN ΨN ) 󳨀→ A. The lemma is proved. N→∞

(2) Remark 2.1.1. If one replaces the condition ‖Ψ(1) N − ΨN ‖L2 (𝒳 N ) 󳨀→ 0 by the following N→∞

N→∞

(2) one: ‖Ψ(1) N − cN ΨN ‖L2 (𝒳 N ) 󳨀→ 0 for some cN ∈ ℂ, |cN | 󳨀→ 1, then the statement of Lemma 2.1.1 will also hold.

The operators AN have the sense of observables in the N-particle quantum me(1) (2) (2) chanics, and the expressions (Ψ(1) N , AN ΨN ) and (ΨN , AN ΨN ) play the role of the mean (1) (2) values of these observables in the states ΨN and ΨN . Lemma 2.1.1 means that, knowing the wave function ψN up to an unknown function χN small with respect to the norm in L2 (𝒳 N ) as N → ∞, one can nevertheless determine uniquely the limits as N → ∞ of the mean values of general observables bounded uniformly in N. If the norm was chosen in another way, and the discrepancy was small with respect to another norm, then these mean values would be determined nonuniquely as N → ∞. All of the constructions of the present chapter are based on this choice of the norm in the space of N-particle functions as the norm of the space L2 (𝒳 N ). The asymptotic formulas of this chapter will not generally hold if the norm is chosen in another way. However, in some cases the arguments similar to Lemma 2.1.1 lead to another choice of the norm in the state space. Indeed, besides quantum mechanics of a system of many particles, one can consider also classical statistical mechanics of a large number of particles. A state of the system is given in this case by a function of 2N arguments p1 , q1 , . . . , pN , qN , where the arguments p1 , . . . , pN , belonging to ℝν , play the

64 | 2 Multiparticle canonical operator and its properties role of momenta of particles, and the arguments q1 , . . . , qN from ℝν play the role of coordinates of particles. This function plays the role of probability distribution. The arguments similar to the ones given above, lead to the choice of a norm as the norm in the space L1 (ℝ2νN ). Construction of an asymptotic formula approximating the solution of the Cauchy problem in this case is performed in Chapter 3 according to the following scheme: (a) one considers the N-particle half-density function equal to the square root of the N-particle density function; (b) it is shown that the discrepancy for the half-density function is small with respect to the norm in the space L2 (ℝ2νN ) if and only if the discrepancy for the density function is small with respect to the norm in the space L1 (ℝ2νN ); (c) using the methods proposed in the present chapter, one constructs the asymptotic formula for the half-density function. Similar method of reducing the problem of constructing asymptotic formula as N → ∞ with the small discrepancy with respect to a norm different from the norm in the space L2 (𝒳 N ), to construction of asymptotics in the space L2 (𝒳 N ), is constructed in the next chapter for the case of quantum statistical mechanics of a system of N particles. The states of such system are described by density matrices, which are Hermitian positive definite operators in the space L2 (ℝνN ). These density matrices can be given either by their kernels (functions of 2N variables each of which belongs to ℝν ), or by their symbols (Blokhintsev–Wigner density matrices) also depending of 2N variables, N of which play the role of coordinates of particles, and N are the momenta.

2.1.3 Method of BBGKY hierarchies One of earlier methods of study of many particle systems is the method of study of the so-called BBGKY hierarchy of equations, obtained by Bogoliubov, Born, Green, Kirkwood, and Yvon by convolution of multiparticle equations for the density matrix in the quantum case, or for the density of probability distribution in the classical case. Using this method, one can show, in particular, that if the k-particle correlation functions Rk,N (t; x1 , . . . , xk ; y1 , . . . , yk ) = ∫ dxk+1 ⋅ ⋅ ⋅ dxN ψN (t; x1 , . . . , xk ; xk+1 , . . . , xN )ψ∗N (t; y1 , . . . , yk ; xk+1 , . . . , xN ) are represented at the initial moment of time as N → ∞, k = const as the product, N→∞ k=const

Rk,N (0; x1 , . . . , xk ; y1 , . . . , yk ) 󳨀→ φ0 (x1 ) ⋅ ⋅ ⋅ φ0 (xk )φ∗0 (y1 ) ⋅ ⋅ ⋅ φ∗0 (yk ), 󵄨 󵄨2 ∫󵄨󵄨󵄨φ0 (x)󵄨󵄨󵄨 dx = 1,

2.1 Introduction

| 65

then at the moment of time t this property is also conserved, Rk,N (t; x1 , . . . , xk ; y1 , . . . , yk ) N→∞ k=const

󳨀→ φ(x1 , t) ⋅ ⋅ ⋅ φ(xk , t)φ∗ (y1 , t) ⋅ ⋅ ⋅ φ∗ (yk , t),

(2.4)

and the function φ satisfies the analog of the Hartree equation: i

P

0 𝜕φ(x, t) 1 = ∑ ∫ H (p) (x, x2 , . . . , xp ; y1 , . . . , yp ) 𝜕t (p − 1)! p=1

× φ∗ (x2 , t) ⋅ ⋅ ⋅ φ∗ (xp , t)φ(y1 , t) ⋅ ⋅ ⋅ φ(yp , t)dx2 ⋅ ⋅ ⋅ dxp dy1 ⋅ ⋅ ⋅ dyp

(2.5)

and the initial condition φ(x, 0) = φ0 (x). The analog of the result (2.4) for a system of many classical particles is discussed in the next chapter. Based on what was said above, one can assume that property (2.4) holds also for k = N. This assumption can be called the quantum analog of chaos conservation conjecture in the strong sense. It turns out that this conjecture, in general, is not true. Indeed, in the next chapter we shall show that property (2.4) does not imply that the N-particle wave function is close, with respect to the norm, to a product of oneparticle wave functions, i. e., that there exist functions ψN for which property (2.4) N→∞

holds, but for any cN ∈ ℂ, |cN | 󳨀→ 1, one has

󵄨 󵄨2 N→∞ ∫ dx1 ⋅ ⋅ ⋅ dxN 󵄨󵄨󵄨ψN (x1 , . . . , xN ) − cN φ(x1 ) ⋅ ⋅ ⋅ φ(xN )󵄨󵄨󵄨 ↛ 0.

(2.6)

This means that there exist observables AN bounded uniformly in N such that their mean values in the states ψN and ψ′N = φ(x1 ) ⋅ ⋅ ⋅ φ(xN ) are different as N → ∞. One has the following lemma inverse to Lemma 2.1.1. Lemma 2.1.2. Assume that for any cN ∈ ℂ, 󵄩󵄩 ′ 󵄩 N→∞ 󵄩󵄩ψN − cN ψN 󵄩󵄩󵄩 ↛ 0. Then for certain set AN of operators bounded uniformly in N, the following relation holds: N→∞

(ψN , AN ψN ) − (ψ′N , AN ψ′N ) ↛ 0. Proof. Consider, for instance, the following operators AN : AN χ = (ψN (ψN , ψ′N ) − ψ′N )(ψN (ψN , ψ′N ) − ψ′N , χ), for χ ∈ L2 (𝒳 N ). Property (2.6) implies the statement of Lemma 2.1.2. Property (2.4) means only that the mean values of observables of special kind HN /N, where HN has

66 | 2 Multiparticle canonical operator and its properties the form (2.1), have the following limit as N → ∞: P0

1 ∫ dx1 ⋅ ⋅ ⋅ dxp dy1 ⋅ ⋅ ⋅ dyp H (p) (x1 , . . . , xp ; y1 , . . . , yp ) p! p=1 ∑

× φ(y1 ) ⋅ ⋅ ⋅ φ(yp )φ∗ (x1 ) ⋅ ⋅ ⋅ φ∗ (xp ). In this chapter, we construct an asymptotics of the solution of the Cauchy problem, which allows one to compute the mean values as N → ∞ of observables bounded uniformly in N, and not only of the observables of special kind (2.1). These asymptotics are expressed not only through the solution of the Hartree-type equation (2.5), but also through solutions of other equations.

2.1.4 Nonconservation of chaos for N-particle wave function: statement of the theorem Let us state some theorems, which will be proved in this chapter. First of all, let us consider the following question. Suppose that a function ψN (t) ∈ L2 (𝒳 N ) satisfies equation (2.3). Does the property 󵄨 󵄨2 N→∞ ∫ dx1 ⋅ ⋅ ⋅ dxN 󵄨󵄨󵄨ψN (t, x1 , . . . , xN ) − cN (t)φ(x1 , t) ⋅ ⋅ ⋅ φ(xN , t)󵄨󵄨󵄨 󳨀→ 0, cN (t) ∈ ℂ,

(2.7)

hold at the time moment t if it holds at the initial time moment? Theorem 2.1.1 shows that the answer to this question turns out to be negative. Before stating the theorem, let us state an auxiliary lemma. Let H (p) (x1 , . . . , xp ; y1 , . . . , yp ), p = 1, . . . , P0 , be the kernels of bounded operators acting in the space L2 (𝒳 p ), and let φ0 ∈ L2 (𝒳 ), ∫ dx |φ0 (x)|2 = 1. Lemma 2.1.3. There exists a unique solution of equation (2.5) belonging for each t to the space L2 (𝒳 ) and satisfying the initial condition φ(x, 0) = φ0 (x). Proof. Proof of this lemma is given in Appendix 2.C. Assume that for some t0 ∈ ℝ and for some functions χ1 , χ2 ∈ L2 (𝒳 ) such that ∫ χ1,2 (x)φ∗ (x, t0 )dx = 0, the following relation holds: P0

1 ∫ H (p) (x1 , x2 , . . . , xp ; y1 , . . . , yp )χ1∗ (x) (p − 2)! p=2 ∑

× χ2∗ (x)φ∗ (x3 , t0 ) ⋅ ⋅ ⋅ φ∗ (xp , t0 )

× φ(y1 , t0 ) ⋅ ⋅ ⋅ φ(yp , t0 )dx1 ⋅ ⋅ ⋅ dxp dy1 ⋅ ⋅ ⋅ dyp ≠ 0.

(2.8)

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| 67

Theorem 2.1.1. There does not exist an interval (t1 , t2 ) such that t0 ∈ (t1 , t2 ) and for all t ∈ (t1 , t2 ) relation (2.7) holds. Proof. The proof of this theorem follows from the asymptotic formula for the solution of the Cauchy problem for equation (2.3). Theorem 2.1.1 implies that in “general position” (i. e., when relation (2.8) holds) property (2.4) for k = N is indeed not true. Let us consider an heuristical derivation of the statement of Theorem 2.1.1. Suppose the contrary. Then the product of one-particle wave functions will be an asymptotics of the solution of the Cauchy problem (2.3). In other words, the N-particle wave function equal, at the initial moment of time, to the product ψN (0, x1 , . . . , xN ) = φ0 (x1 ) ⋅ ⋅ ⋅ φ0 (xN ), at the moment of time t has the form ψN (t, x1 , . . . , xN ) = cN (t)φt (x1 ) ⋅ ⋅ ⋅ φt (xN ) + zNt (t, x1 , . . . , xN ),

(2.9)

where φt (x) ≡ φ(x, t) is the solution of the Cauchy problem for the Hartree-type equation (2.5), and 󵄨 󵄨2 N→∞ ∫ dx1 ⋅ ⋅ ⋅ dxN 󵄨󵄨󵄨zNt (t, x1 , . . . , xN )󵄨󵄨󵄨 󳨀→ 0.

(2.10)

In this case, one can expect that property (2.9) will be true for some other function zNt satisfying property (2.10), also in the case when we shift the solution of the Hartreetype equation (2.5) by a quantity of order 1/√N. In order to make the square of the norm of the wave function, equal to (∫ dx|φ0 (x)|2 )N , remain still of order unity, let us impose the following additional constraint on the value of shift of the initial condition for equation (2.5), φ0 → φ0 +

eia δφ , √N 0

namely, (φ0 , δφ0 ) = 0. The phase factor eia is extracted for convenience of the argument below. Since equation (2.5) is transformed under the change of the initial condition considered above is as follows: φt → φt +

𝜕φt ia 𝜕φt ia ∗ 1 1 ( e δφ0 + e δφ0 ) + O( ). ∗ √N 𝜕φ0 𝜕φ0 N

68 | 2 Multiparticle canonical operator and its properties According to what was said above, one can expect that the wave function of the form cN (t, eia δφ0 , e−ia δφ∗0 )(φt (x1 ) + × ⋅ ⋅ ⋅ × (φt (xN ) +

𝜕φt ia 𝜕φt ia ∗ 1 ( e δφ0 + e δφ0 )(x1 )) √N 𝜕φ0 𝜕φ∗0

𝜕φt ia 𝜕φt ia ∗ 1 ( e δφ0 + e δφ0 )(xN )) √N 𝜕φ0 𝜕φ∗0

(2.11)

will be also an asymptotic solution of equation (2.3). The wave function (2.11) pos𝜕φ sesses the following interesting property. Due to relation 𝜕φ∗0 ≡ 0, which holds at the 0

initial moment of time, at t = 0 formula (2.11) does not contain summands with e−ia . On the other hand, at the moment t this summand reads e−ia (aN (t)φt (x1 ) ⋅ ⋅ ⋅ φt (xn ) + cN (t)

1 √N

N

× ∑ φt (x1 ) ⋅ ⋅ ⋅ φt (xi−1 )g t (xi )φt (xi+1 ) ⋅ ⋅ ⋅ φt (xN )), i=1

where g t =

𝜕φt δφ∗0 . 𝜕φ∗0

(2.12)

Function g t can be represented in the form g t = αφt + g⊥t , where

(φt , g⊥t ) = 0. The contribution of the summand containing g⊥t into formula (2.12) has the following form, up to a normalizing factor of order unity: 1 N t ∑ φ (x1 ) ⋅ ⋅ ⋅ φt (xi−1 )g⊥t (xi )φt (xi+1 ) ⋅ ⋅ ⋅ φt (xN ). √N i=1

(2.13)

It will be shown in the next section that the norm of function (2.13) is of order unity, and does not tend to zero as N → ∞ if g⊥t ≠ 0. By linearity of equation (2.3) function (2.12) containing the summands of formula (2.11) proportional to e−ia , is an asymptotic solution of equation (2.3). But this solution vanishes at the initial moment of time, and at the moment of time t, in general, it does not vanish. We obtain a contradiction. Thus, the assumption (2.9) is, in general, not true. Hence, we have heuristically obtained the statement of Theorem 2.1.1.

2.1.5 Asymptotics of the solution of the Cauchy problem: statement of the theorem As we noted in the previous section, the product of one-particle wave functions is not, in general, an asymptotic solution of equation (2.3) as N → ∞. In this section, we shall provide a more complicated formula, which is an asymptotic of the solution of equation (2.3).

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| 69

Let us introduce the following notation. Let φ ∈ L2 (𝒳 ), ∫ dx|φ(x)|2 = 1, R ∈ L2 (𝒳 2 ), and (a) R(x, y) = R(y, x); (b) the following relation holds: ∫ R(x, y)φ∗ (y) = −φ(x);

(2.14)

(c) the operator in the space L2 (𝒳 ) with the kernel M(x, y) = R(x, y) + φ(x)φ(y)

(2.15)

has the norm less than 1. Denote by ΦN,φ,R the following element of the space L2 (𝒳 N ): [N/2]

ΦN,φ,R (x1 , . . . , xN ) = ∑

l=0

1 M(xi1 , xi2 ) ∑ (2N)l l! 1≤i1 =⋅⋅⋅ ̸ =i̸ 2l ≤N

× ⋅ ⋅ ⋅ × M(xi2l−1 , xi2l ) ∏ φ(xi ). i=i̸ 1 ,...,i2l

(2.16)

It will be shown in the next section that the conditions above imply that the norm of the wave function (2.16) is of order unity as N → ∞. It turns out that if an N-particle function has the form (2.16) at the initial moment of time, then it has up to corrections of order O(1/√N), the similar form cN (t)ΦN,φt ,Rt

(2.17)

at the moment of time t. Since for R(x, y) = −φ(x)φ(y), function (2.16) coincides with the product of one-particle wave functions; the many-particle wave function of this form at the initial moment of time will have the form (2.16) at the moment t with the function Rt (x, y), in general, not equal to −φt (x)φt (y); hence, property (2.9) will not hold in this case. Let us write out the equation satisfied by function Rt . Denote P0

1 ∫ dx1 ⋅ ⋅ ⋅ dxp dy1 ⋅ ⋅ ⋅ dyp H (p) (x1 , . . . , xp ; y1 , . . . , yp ) p! p=1

H[φ∗ , φ] = ∑

× φ∗ (x1 ) ⋅ ⋅ ⋅ φ∗ (xp )φ(y1 ) ⋅ ⋅ ⋅ φ(yp ).

70 | 2 Multiparticle canonical operator and its properties Consider the following equation: iṘ t (x, y) =

δ2 H

δφ∗ (x)δφ∗ (y) +∫

+∫

δ2 H

δφ∗ (y)δφ(z)

δ2 H

δφ∗ (x)δφ(z)

Rt (z, y)dz

Rt (x, z)dz

+ ∫ dz1 dz2 Rt (x, z1 )Rt (y, z2 )

δ2 H , δφ(z1 )δφ(z2 )

(2.18)

where the arguments φt∗ , φt of the functional H are omitted. Lemma 2.1.4. Assume that a function R0 ∈ L2 (𝒳 2 ) satisfies conditions (a)–(c). Then there exists a solution of equation (2.18), belonging to L2 (𝒳 2 ), satisfying conditions (a)–(c) and the initial condition R0 = R0 . Proof. The proof is given in Appendix 2.C. Let us state the theorem on asymptotics of the solution of the Cauchy problem. Let ψN (t; x1 , . . . , xN ) be the solution of equation (2.3) satisfying the initial condition ψN (0; x1 , . . . , xN ) = ΦN,φ0 ,R0 (x1 , . . . , xN ); such solution exists and is unique due to boundedness of operators with kernels H (p) (x1 , . . . , xp ; y1 , . . . , yp ). Let φt be the solution of the Cauchy problem for equation (2.5), and Rt be the solution of the Cauchy problem for equation (2.18). Denote t

S = ∫ dτ(i ∫ dxφτ∗ (x) t

0

dφτ (x) − H[φτ∗ , φτ ]), dτ

t

δ2 H i Rτ (x, y)}. c = exp{− ∫ dτ ∫ dxdy 2 δφ(x)δφ(y) τ

(2.19)

0

Theorem 2.1.2. The following relation holds: t 󵄨 󵄨2 N→∞ ∫ dx1 ⋅ ⋅ ⋅ dxN 󵄨󵄨󵄨ψN (t; x1 , . . . , xN ) − ct eiNS ΦN,φt ,Rt (x1 , . . . , xN )󵄨󵄨󵄨 󳨀→ 0.

Proof. This theorem follows from the results of Section 2.7. The proof is based on substitution of asymptotic formula (2.17) into equation (2.3). However, as shown in Chapter 1, many properties of asymptotic formulas of type (2.17) can be studied heuristically, using the concept of abstract canonical operator. In particular, in Section 1.5 we have shown that using this concept one can derive the asymptotic formula for the quantum mechanical Schrödinger equation without substitution of asymptotics

2.1 Introduction

| 71

as ℏ → 0 into the equation, but using only the classical Hamiltonian equations. The asymptotics is computed this way up to a multiplicative constant. In Sections 2.2–2.6, we apply the concepts of Chapter 1 for the case of construction of asymptotic solutions of equation (2.3). In Section 2.2, we introduce the notion of multiparticle canonical operator, and in Section 2.3, we check the correctness of this definition, and introduce geometric structures on the phase space. It turns out that in this case, the phase space is the unit sphere in the space L2 (𝒳 ), i. e., the set of complex functions φ : 𝒳 → ℂ such that ∫ dx|φ(x)|2 = 1. In Sections 2.4–2.5, we apply the definitions of canonical transformation, proper canonical transformation, and abstract complex germ to this case. It turns out that the transformation mapping the initial condition for equation (2.5) into the solution of this equation, is a proper canonical transformation depending on time. To this proper canonical transformation, one assigns, according to Definition 1.2.7, formal asymptotic solutions of equations of motion, which are determined up to multiplicative constants depending on time. In particular, the wave functions of type (2.17) are examples of such formal asymptotic solutions of equations of motion, which will be considered in Section 2.6. Thus, the method of Sections 2.2–2.6 allows one to obtain heuristically the asymptotic formula (2.17) without using equations (2.3), but using only equation (2.5) for function φ. This method is a generalization of the heuristic argument given at the end of the previous section, which allows one to conclude that the product of one-particle functions is not mapped to a product. Besides function (2.17), one can also construct other asymptotic solutions of the equations of motion. Indeed, asymptotics (2.17) are in a sense similar to the Gausi t sian wave packets ct e ℏ S KPℏt ,Qt ψtα in usual quantum mechanics (where ψα is of the form (1.81)). However, besides Gaussian wave packets, the method of a complex germ in quantum mechanics allows one to construct also other asymptotics, using the germ creation and annihilation operators (1.86). Similar operators can be introduced also in the case considered in the present chapter, thus obtaining new approximate solutions of equation (2.3) as N → ∞. Consider the following operators in the space L2 (𝒳 N ): 1 N Â ±φ,F,G ψ(x1 , . . . , xN ) = ∑ ∫ dyi A±φ,F,G (xi , yi ) √N i=1 × ψ(x1 , . . . , xi−1 , yi , xi+1 , . . . , xN ), where φ, F, G ∈ L2 (𝒳 ), and the functions A±φ,F,G (x, y) read A+φ,F,G (x, y) = −i(G⊥∗ (x)φ∗ (y) − φ(x)F⊥∗ (y)),

A−φ,F,G (x, y) = −i(F⊥ (x)φ∗ (y) − φ(x)G⊥ (y)), F⊥ = F − φ(φ, F),

G⊥ = G − φ(φ, G).

72 | 2 Multiparticle canonical operator and its properties Let F(x) = ∫ dy R(x, y)G(y).

(2.20)

We shall prove in this chapter that the norm of the functions  −φ,F,G ΦN,φ,R is small as N → ∞, and the norm of the functions  + ΦN,φ,R is of order unity. It turns out that φ,F,G

the operators  ±φ,F,G can be used for construction of new asymptotic solutions of equation (2.3). Let us write out the system of equations for the functions F, G: i −i

δ2 H δ2 H 𝜕F t (x) = ∫ dy( ∗ F t (y) + Gt (y)), ∗ 𝜕t δφ (x)δφ(y) δφ (x)δφ∗ (y)

𝜕Gt (x) δ2 H δ2 H = ∫ dy( F t (y) + Gt (y)). 𝜕t δφ(x)δφ(y) δφ(x)δφ∗ (y)

(2.21)

This system can be formally obtained as follows. One should consider the system consisting of the Hartree-type equation (2.5) and the adjoint equation; then one should consider the system in variations (the linearized system) for this system, and, finally, replace the variations δφ and δφ∗ with independent functions F and G. Lemma 2.1.5. For any initial condition (F0 , G0 ) ∈ L2 (𝒳 ) × L2 (𝒳 ), there exists a solution for the Cauchy problem (2.21). If condition (2.20) holds at the initial moment of time, then it holds also at the other moments of time. Proof. The existence of the solution of system (2.21) follows from boundedness of operators in the space L2 (𝒳 ) with the kernels: δ2 H , δφ∗ (x)δφ(y)

δ2 H , δφ∗ (x)δφ∗ (y)

δ2 H , δφ(x)δφ(y)

δ2 H , δφ(x)δφ∗ (y)

which, in its turn, follows from boundedness of operators in the space L2 (𝒳 p ) with the kernels H (p) (x1 , . . . , xp ; y1 , . . . , yp ). Condition (2.20) at the moment of time t follows from system (2.21) and equation (2.18). Lemma 2.1.5 is proved. Let (F1t , G1t ), . . . , (Fkt , Gkt ) be k solutions of the system (2.21) satisfying, at the initial moment of time (and hence at any other moment), condition (2.20). Consider the solution of equation (2.3) satisfying the initial condition ψN (0) = Â +φ

0 0 0 ,F1 ,G1

⋅ ⋅ ⋅ Â +φ

0 0 0 ,Fk ,Gk

ΦN,φ0 ,R0 .

Theorem 2.1.3. The following relations hold: 1. ‖ψN (t)‖ = O(1); t N→∞ 2. ‖ψN (t) − ct eiNS Â + t t t ⋅ ⋅ ⋅ Â + t t t ΦN,φt ,Rt ‖ 󳨀→ 0. φ ,F1 ,G1

φ ,Fk ,Gk

2.2 Definition of multiparticle canonical operator

| 73

This theorem is proved by substitution of the asymptotic formula into equation (2.3) (see Section 2.7). Note that these asymptotics can be also obtained heuristically (see Section 2.6). Note also that in this chapter we shall also construct asymptotic formulas for solution ψN of equation (2.3) up to O(N −M/2 ), where M is an arbitrary natural number, (1) 2 N i. e., functions ψ(0) N , ψN , ⋅ ⋅ ⋅ ∈ L (𝒳 ) such that the following formula holds: 󵄩󵄩 󵄩 1 (1) 1 󵄩󵄩 󵄩 (0) (M−1) 󵄩 −M/2 ). 󵄩󵄩ψN − ψN − ψN − ⋅ ⋅ ⋅ − M−1 ψN 󵄩󵄩󵄩 = O(N 󵄩󵄩 󵄩 N 󵄩 2 N We have required the operators with kernels H (p) (x1 , . . . , xp ; y1 , . . . , yp ) in L2 (𝒳 p ) to be bounded. In the concrete examples, which will be considered in Chapter 3, this assumption usually does not hold. However, it is necessary only for the proof of the existence of the solutions of the Cauchy problems for equations (2.3), (2.5), (2.18), and the system (2.21). All of the results of the present chapter are true if solutions of these Cauchy problems exist but the operators with kernels H (p) (x1 , . . . , xp ; y1 , . . . , yp ) are unbounded. Hence, in Chapter 3 we separately prove the existence of the solutions of equations (2.3), (2.15), (2.18), and (2.21). Let us now proceed to the definition of the multiparticle canonical operator.

2.2 Definition of multiparticle canonical operator In the previous section, we have chosen the state space ℋ1/N to be the space of symmetric functions from L2 (𝒳 N ), where 𝒳 is a space with measure. Let us now concretize the choice of the phase space ℳ and the canonical operator. 2.2.1 Examples of multiparticle functions satisfying the chaos property Let us first consider some examples of functions satisfying property (2.4) with (φ, φ) = 1. 1. The wave function, which is the product of one-particle wave functions, ψN (x1 , . . . , xN ) = φ(x1 ) ⋅ ⋅ ⋅ φ(xN ),

(2.22)

obviously satisfies property (2.4). 2. Let g be an element of L2 (𝒳 ) orthogonal to φ, (g, φ) = 0. Let us consider the following element of the space ℋ1/N : ψN (x1 , . . . , xN ) =

1 (g(x1 )φ(x2 ) ⋅ ⋅ ⋅ φ(xN ) √N + φ(x1 )g(x2 )φ(x3 ) ⋅ ⋅ ⋅ φ(xN ) + ⋅ ⋅ ⋅ + φ(x1 ) ⋅ ⋅ ⋅ φ(xN−1 )g(xN )).

(2.23)

74 | 2 Multiparticle canonical operator and its properties Note that despite the N summands in the sum and the overall factor 1/√N, the norm of this function is of order unity as N → ∞ due to condition (g, φ) = 0, and not of order √N. On the one hand, the wave function (2.23) satisfies property (2.4), and one the other hand, it is not close to the product of one-particle functions, since functions (2.22) and (2.23) are orthogonal to one another. Function (2.23) has the following physical sense: N − 1 particles in the system of N particles have the wave function φ, and one particle has the wave function g orthogonal to φ. 3. Consider a generalization of the previous example. Denote by ℱφ the subspace of the Fock space ℱ (see Appendix 2.A), which is the set of all elements (f0 , f1 , f2 , . . .) ∈ ℱ such that ∫ dx1 φ∗ (x1 )fn (x1 , . . . , xn ) = 0.

(2.24)

It is obvious that by symmetry of the function fn , we could contract it with the function φ also with respect to any other argument, and not only with respect to x1 . A generalization of example (2.23) is the element of the space ℋ1/N of the form N

√p! p √ p=0 N

(Kφ,N f )(x1 , . . . , xN ) = ∑

∑

1≤i1 0. 2m v

168 | 3 Asymptotic solutions of the many-body problem As ε → 0, this condition goes to condition (3.70), since for κ ≠ 0 it holds automatically, and for κ = 0 it coincides with condition (3.70). Let us now recall the well-known Landau argument [22] showing that in this example superfluidity arises. In the state with νi = 0 almost all particles of Bose gas move with the velocity p/m, i. e., the gas flows with this velocity as a whole. Presence of friction shows in dissipation of energy of the Bose gas. If one considers the influence of walls as a factor leading to a weak perturbation, then this process arises mostly by transitions between the states satisfying the chaos condition for the k-particle correlation functions (2.4) (the function φ has the form (3.64)); in the opposite case, the matrix element will be exponentially small. Hence, friction is possible if among the βk found above there are negative numbers, βk < 0. If p∗ =

inf (

̃ λ=2π n/L λ=0 ̸

2

2π 1 √ λ2 ( + Vλ ) − Vλ2 ) ≥ , |λ| 2m L

(3.71)

then for |p| < p∗ all βλ > 0; hence, Landau’s argument implies that one has superfluidity. If, as in the case of condition (3.67), one multiplies the potential V by εN and considers the limit as L → ∞, Ld /N = const, and after that the limit as ε → 0, then condition (3.71) will also go to the Bogoliubov condition (3.70). 3.3.3 Generalization of the notion of superfluidity Let us now show that phenomena analogous to superfluidity arise also in the case when U ≠ 0, i. e., when solutions of the system in variations (3.60) and of the Hartree equation (3.55) have the form different from const eipx . This corresponds to the physical situation when elementary excitations are not characterized by certain momentum. To this end, consider the following example. Let 𝒳 = T 1 × ℝ1 , assume that the potential U depends only on the coordinate x2 , and the potential V depends only on d2 the coordinate x1 . Consider the case when the operator  = − 21 dx 2 + U(x2 ) has purely 2

discrete spectrum. Then the Hartree equation (3.55) has the following solution: φ(x) = L−1/2 eipx1 χ0 (x2 ),

p=

2πj , L

j ∈ ℤ,

(3.72)

for Ω=

p2 + V0 + E0 . 2m

(3.73)

Denote by χn the normalized to unity eigenfunctions of the operator A corresponding to eigenvalues En numbered in the order of increasing.

3.4 Asymptotics of solution of the Liouville equation

| 169

The solutions of system (3.60) satisfying conditions (3.61)–(3.63) have the form ̃ F (λ,k) , G(λ,k) , λ = 2π n/L, ñ ≠ 0, ñ ∈ ℤ, k ∈ ℤ+ , i

̃

F (λ,0) (x1 , x2 ) = F (λ)0 (x1 )χ0 (x2 )e ℏ (βλ0 −Ω)t , i

̃

G(λ,0) (x1 , x2 ) = G(λ)0 (x1 )χ0∗ (x2 )e ℏ (βλ0 +Ω)t , F (λ,k) (x1 , x2 ) = 0, i

̃

G(λ,k) (x1 , x2 ) = L−d/2 e ℏ [(p+λ)χ1 +(βλk +Ω)t] χk∗ (x2 ),

k = 1, . . . , ∞;

here, the functions F (λ) , G(λ) coincide with the ones used in the previous example, and the numbers β̃ λk equal β̃ λ0 = βλ ,

(p + λ)2 p2 β̃ λk = Ek − E0 + − . 2m 2m

Thus, for arising of superfluidity in this case, one should require, besides condition (3.71), also the following condition: E1 − E0 >

p2 . 2m

This argument leads to the following definition. Definition 3.3.1. Assume that the conditions of the corollary of Lemma 3.3.2 hold, and one has βα > 0, α = 1, . . . , ∞. Then the wave function ψN,M is called the wave function corresponding to a superfluid state. Remark 3.3.1. The main state of the N-particle Hamiltonian also satisfies this definition. Thus, we have generalized the notion of superfluidity to the case U ≠ 0.

3.4 Asymptotics of solution of the N-particle Liouville equation and violation of the chaos conjecture for density function 3.4.1 Existence of solutions of the Vlasov and Riccati equations In this section, we consider construction of asymptotic solutions of equation (3.9). Let p, q ∈ ℝν , and let ρ0 (p, q) be a function from 𝒮 (ℝ2ν ) such that ρ0 (p, q) > 0,

∫ ρ0 (p, q) dp dq = 1.

(3.74)

Denote by 𝒮2 the space of all functions M(p1 , q1 ; p2 , q2 ), p1 , q1 , p2 , q2 ∈ ℝν , such that

170 | 3 Asymptotic solutions of the many-body problem (a) M is symmetric with respect to permutation of pairs of arguments (p1 , q1 ) and (p2 , q2 ); (b) for fixed (p1 , q1 ), M as a function of (p2 , q2 ) belongs to the space 𝒮 (ℝ2ν ); (c) the operator M in the space L2 (ℝ2ν ) whose kernel is the function M, is a Hilbert– Schmidt operator with the norm less than unity. Let us assign to each pair of functions ρ0 and M0 ∈ 𝒮2 , M0 √ρ0 = 0, the following element of L2 (ℝ2νN ): ρN,ρ0 ,M0 (p1 , q1 ; . . . ; pN , qN ) = ((Kρ0 ,N ΦM0 )(p1 , q1 ; . . . ; pN , qN )) [N/2]

≡(∑

l=0

2

1 M0 (pi1 , qi1 ; pi2 , qi2 ) ⋅ ⋅ ⋅ ∑ (2N)l l! 1≤i1 =⋅⋅⋅ ̸ =i̸ 2l ≤N 2

× M0 (pi2l−1 , qi2l−1 ; pi2l , qi2l ) ∏ √ρ0 (pi , qi )) , i=i̸ s s=1,...,2l

(3.75)

and consider it as the initial condition for the Cauchy problem for equation (3.9). Before proving the theorem, let us prove some auxiliary statements. Lemma 3.4.1. Let U and V be infinitely differentiable, 󵄨󵄨 𝜕2 U 󵄨󵄨 󵄨󵄨 𝜕2 V(q, q′ ) 󵄨󵄨 󵄨󵄨 𝜕2 V(q, q′ ) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨+󵄨 󵄨+󵄨 󵄨 < C, 󵄨󵄨 𝜕qi 𝜕qj 󵄨󵄨󵄨 󵄨󵄨󵄨 𝜕qi 𝜕qj 󵄨󵄨󵄨 󵄨󵄨󵄨 𝜕qi 𝜕q′ 󵄨󵄨󵄨 j

C = const.

(3.76)

Then on any interval t ∈ [0, T] there exists a solution of equation (3.9) satisfying initial condition (3.75). Proof. The solution of the Cauchy problem for equation (3.9) has the form ρtN (p1 , q1 ; . . . ; pN , qN )

= ρ0N (𝒫1t (p1 , q1 ; . . . ; pN , qN ), 𝒬t1 (p1 , q1 ; . . . ; pN , qN ); . . . ; t

t

𝒫N (p1 , q1 ; . . . ; pN , qN ), 𝒬N (p1 , q1 ; . . . ; pN , qN )),

where 𝒫kt , 𝒬tk is the solution of the Cauchy problem 𝜕𝒫kt 𝜕HN t t = (𝒫 , 𝒬 ; . . . ; 𝒫Nt , 𝒬tN ), 𝜕t 𝜕𝒬k 1 1

0

𝜕𝒬tk 𝜕H = − N (𝒫1t , 𝒬t1 ; . . . ; 𝒫Nt , 𝒬tN ), 𝜕t 𝜕𝒫k

𝒫k (p1 , q1 ; . . . ; pN , qN ) = pk ,

0

𝒬k (p1 , q1 ; . . . ; pN , qN ) = qk ,

3.4 Asymptotics of solution of the Liouville equation

| 171

which exists on an arbitrary finite interval t ∈ [0, T] by conditions of Lemma 3.4.1 and by the compressing maps principle (see, e. g., [25, Theorem 8.4 of Chapter 1]). Lemma 3.4.2. Assume that conditions (3.76) hold, and the derivatives of higher orders of U and V grow at infinity no faster than a polynomial. Then the Vlasov equation (3.13) has a solution on any finite interval t ∈ [0, T] belonging to 𝒮 (ℝ2ν ) and satisfying the initial condition ρ(p, q, 0) = ρ0 (p, q). Proof. According to [26], the solution of the Cauchy problem for the Vlasov equation is determined from the condition ρ(𝒫 (w, z, t), 𝒬(w, z, t), t) = f0 (w, z), where 𝒫 , 𝒬 is the solution of the Cauchy problem 𝜕H 𝜕𝒫 (w, z, τ) = − 0 (𝒫 (w, z, τ), 𝒬(w, z, τ)) 𝜕τ 𝜕q 𝜕V 󵄨󵄨󵄨󵄨 −∫ (q , 𝒬(w′ , z ′ , τ))ρ0 (w′ , z ′ )dw′ dz ′ , 󵄨 𝜕q1 󵄨󵄨󵄨q1 =𝒬(w,z,τ) 1 Q(w, z, 0) = z,

𝒫 (w, z, 0) = w,

𝜕H0 𝜕𝒬 (w, z, τ) = (𝒫 (w, z, τ), 𝒬(w, z, τ)), 𝜕τ 𝜕p

H0 (p, q) =

(3.77)

p2 + U(q). 2m

Solvability of the Cauchy problem (3.77) in the class of continuous functions of w and z growing as |w|, |z| → ∞ no faster than a polynomial, is proved by applying TheY

orem 8.4 of Chapter 1 of [25] to the Banach space of continuous maps ℝ2ν 󳨀→ ℝ2ν , Y(w, z) = (𝒫 (w, z), 𝒬(w, z)), with the norm ‖Y‖ = sup w,z

|𝒫 (w, z)| + |𝒬(w, z)| . |w2 | + |z 2 | + 1

It is also not difficult to check that 𝒫 , 𝒬 is also the solution of the following Cauchy problem for a system of ordinary differential equations: 𝜕H̃ 𝜕H̃ (𝒫 , 𝒬, τ), 𝒬̇ = (𝒫 , 𝒬, τ), 𝜕𝒬 𝜕𝒫 = w, 𝒬|τ=0 = z,

𝒫̇ = − 𝒫 |τ=0

(3.78)

̃ q, τ) = H0 (p, q) + ∫ V(q, 𝒬(w′ , z ′ , τ))ρ0 (w′ , z ′ )dw′ dz ′ . H(p, The existence and uniqueness of solution of system (3.78) with the initial conditions 𝒫 |τ=0 = 𝒫t , 𝒬|τ=0 = 𝒬t can be proved similarly. Smoothness of solution with respect to the parameters 𝒫t and 𝒬t follows from smoothness of H.̃ The equation for derivatives of 𝒫 , 𝒬 with respect to the parameters 𝒫t , 𝒬t implies that these derivatives grow at

172 | 3 Asymptotic solutions of the many-body problem infinity no faster than a polynomial. Hence, ρ(𝒫t , 𝒬t , t) = ρ0 (w(𝒫t , 𝒬t ), z(𝒫t , 𝒬t )), where w and z are smooth functions growing at infinity no faster than a polynomial. Hence, the solution of the Vlasov equation belongs to 𝒮 (ℝ2ν ). Lemma 3.4.2 is proved. Let us consider the function R0 (p1 , q1 ; p2 , q2 ) = M0 (p1 , q1 ; p2 , q2 ) − √ρ0 (p1 , q1 )ρ0 (p2 , q2 ),

(3.79)

and choose it as the initial condition for the Riccati equation (3.14). Lemma 3.4.3. For M0 ∈ 𝒮2 , M0 √ρ = 0, there exists a unique solution Rt of equation (3.14) satisfying initial condition (3.79) and such that the function M t (p1 , q1 ; p2 , q2 ) = Rt (p1 , q1 ; p2 , q2 ) + √ρ(p1 , q1 , t)ρ(p2 , q2 , t)

(3.80)

belongs to 𝒮2 . Proof. For proof of this lemma, consider the operator Γt : L2 (ℝ2ν ) → L2 (ℝ2ν ), uniquely determined from the condition ξ (p, q) = (Γt ξ )(𝒫̃t (p, q), 𝒬̃ t (p, q)), where 𝒫̃t , 𝒬̃ t is the solution of the Cauchy problem 𝒫̇̃t = −

𝜕Wt , 𝜕𝒬̃

𝒬̇̃ t =

𝒫̃t

m

,

𝒫̃0 = p,

𝒬̃ 0 = q,

also uniquely determined for t ∈ ℝ by the compressing maps principle. Denote H(φ∗ , φ) = i{∫ dp dq φ∗ (p, q)(

𝜕H0 𝜕H0 𝜕 𝜕 (p, q) − (p, q) ) 𝜕q 𝜕p 𝜕p 𝜕q

1 ∫ dp1 dq1 dp2 dq2 φ∗ (p1 , q1 )φ∗ (p2 , q2 ) 2 𝜕V(q1 , q2 ) 𝜕 𝜕V(q1 , q2 ) 𝜕 ×( + )φ(p1 , q1 )φ(p2 , q2 )}. 𝜕q1 𝜕p1 𝜕q2 𝜕p2 × φ(p, q) +

It is not difficult to note that equation (3.14) has the form (2.91).

(3.81)

3.4 Asymptotics of solution of the Liouville equation

| 173

Consider the following system of the form (2.92): 𝜕2 H 𝜕2 H −iζṫ = ζ + η, t 𝜕φ𝜕φ∗ 𝜕φ𝜕φ t

𝜕2 H 𝜕2 H iη̇ t = ζ + η, t 𝜕φ∗ 𝜕φ∗ 𝜕φ∗ 𝜕φ t

(3.82)

where the derivatives of H are taken at the point φ = φ∗ = √ρt . The definition of the operator Γt implies that the system (3.82) can be represented in the form ⋅

T Γt ζt ) =( 1 T2 Γt ηt

(

Γζ Γζ T2 )( t t) ≡ T ( t t), Γt ηt T1 Γt ηt

(3.83)

where T1 and T2 are the operators in L2 (ℝ2ν ) with the kernels uniquely determined from the conditions T1 (𝒫̃t (p, q), 𝒬̃ t (p, q); 𝒫̃t (p′ , q′ ), 𝒬̃ t (p′ , q′ ))

𝜕V(q, q′ ) 𝜕 𝜕V(q, q′ ) 𝜕 )√ρ(p, q, t)ρ(p′ , q′ , t), − 𝜕q 𝜕p 𝜕q′ 𝜕p′ T2 (𝒫̃t (p, q), 𝒬̃ t (p, q); 𝒫̃t (p′ , q′ ), 𝒬̃ t (p′ , q′ )) =(

=(

𝜕V(q, q′ ) 𝜕 𝜕V(q, q′ ) 𝜕 + )√ρ(p, q, t)ρ(p′ , q′ , t). 𝜕q 𝜕p 𝜕q′ 𝜕p′

It is not difficult to note that T1 , T2 ∈ 𝒮 (ℝ4ν ). Consider the operator Z1t Z2t

Zt = (

Z2t ) Z1t

mapping the initial condition ( ΓΓ0 ηζ ) for system (3.83) into the solution ( ΓΓt ηζ ) of this 0 t system. This operator exists by boundedness of the operators T1 and T2 , and by Lemma 2.6.6 it equals Zt ( 1t Z2

𝜕φ∗t

Z2t 𝜕φ∗ t ) = Γt ( 𝜕φt Z1 ∗ 𝜕φ

𝜕φ∗t 𝜕φ ) Γ−1 t . 𝜕φt 𝜕φ

By boundedness of the operator T, the series T n−1 n=1 n! ∞

S=∑

converges to certain bounded operator, so that Z = E + TS = E + ST.

174 | 3 Asymptotic solutions of the many-body problem Since T1 and T2 are Hilbert–Schmidt operators, Z1t − E and Z2t are also Hilbert–Schmidt operators; their kernels (Z1t − E)(p1 , q1 ; p2 , q2 ),

Z2t (p1 , q1 ; p2 , q2 ),

considered as functions of one pair of arguments (e. g., (p2 , q2 )) for fixed values of the other pair of arguments (p1 , q1 ), belong to 𝒮 (ℝ2ν ). The kernels of the operators and

𝜕φt 𝜕φ∗

possess the same property. Since, according to Lemmas 2.6.5 and 2.5.4, we have Rt = (

𝜕φt 𝜕φ

−E

𝜕φt 𝜕φt 0 𝜕φ∗t 𝜕φ∗t 0 + R )( ∗ + R ) , 𝜕φ∗ 𝜕φ 𝜕φ 𝜕φ −1

then for fixed values of p2 , q2 the kernel of the operator Rt , as a function of p1 , q1 , belongs to 𝒮 (ℝ2ν ). By symmetry of the kernel Rt , the similar statement holds also for fixed values of the other pair of arguments. The property ‖M t ‖ < 1 has been already proved in Chapter 2. Hence, M t ∈ 𝒮2 . Lemma 3.4.3 is proved. 3.4.2 Asymptotics of solution of the Cauchy problem for the N-particle Liouville equation Denote t

1 ct = exp{ ∫ dt ′ dp1 dq1 dp2 dq2 √ρ(p1 , q1 , t ′ )ρ(p2 , q2 , t ′ ) 2 0

×(

′ 𝜕V(q1 , q2 ) 𝜕 𝜕V(q1 , q2 ) 𝜕 + )Rt (p1 , q1 ; p2 , q2 )}. 𝜕q1 𝜕p1 𝜕q2 𝜕p2

Theorem 3.4.1. Let ρtN be the solution of equation (3.9) satisfying initial condition (3.75). Then: 1. N→∞

∫ dp1 dq1 ⋅ ⋅ ⋅ dpN dqN ρtN (p1 , q1 ; . . . ; pN , qN ) 󳨀→ C > 0, 2. 󵄨 ∫ dp1 dq1 ⋅ ⋅ ⋅ dpN dqN 󵄨󵄨󵄨ρtN (p1 , q1 ; . . . ; pN , qN ) 󵄨 N→∞ − ct2 ρN,ρt ,M t (p1 , q1 ; . . . ; pN , qN )󵄨󵄨󵄨 󳨀→ 0.

(3.84)

3.4 Asymptotics of solution of the Liouville equation

| 175

Proof. This theorem immediately follows from Lemma 3.1.2 and from the results of the previous chapter. Corollary 3.4.1. Let V ≠ 0. Then for M0 = 0 there does not exist an interval (T1 , T2 ) on which the integral 󵄨 ∫ dp1 dq1 ⋅ ⋅ ⋅ dpN dqN 󵄨󵄨󵄨ρtN (p1 , q1 ; . . . ; pN , qN ) 󵄨 − ρ(p1 , q1 , t) ⋅ ⋅ ⋅ ρ(pN , qN , t)󵄨󵄨󵄨

(3.85)

tends to zero as N → ∞. Remark 3.4.1. Corollary 3.4.1 means that the initial condition for the Liouville equation in the form of the product of one-particle functions goes, at the moment t, to a N-particle distribution not close with respect to the norm in L1 as N → ∞ to a product of one-particle distribution functions. Thus, the chaos conservation conjecture for density function is refuted. 3.4.3 Stationary asymptotic solutions of the many-body problem In Corollary 3.4.2 below, we consider stationary asymptotic solutions of the Liouville equation. Corollary 3.4.2. Let ρ(p, q, t) and Rt be solutions of equations respectively (3.13) and (3.14) not depending on t. Then for solution of equation (3.9) satisfying initial condition (3.75), the following relation holds: 󵄨 ∫ dp1 dq1 ⋅ ⋅ ⋅ dpN dqN 󵄨󵄨󵄨ρtN (p1 , q1 ; . . . ; pN , qN ) 󵄨 N→∞ − ρ0N (p1 , q1 ; . . . ; pN , qN )󵄨󵄨󵄨 󳨀→ 0.

(3.86)

Indeed, due to conservation of the norm of ρtN in the space L1 , the constant ct in formula (3.84) is in this case identically equal to unity. Remark 3.4.2. Usually one considers [28–30] the following stationary solutions of the Vlasov equation: 2

ρ(p, q) =

p + W(q)) u( 2m ′2

∫ dp′ dq′ u( p2m + W(q′ ))

(3.87)

,

where W(q) satisfies the equation ′2

W(q) = U(q) +

∫ dp′ dq′ V(q, q′ ) u( p2m + W(q′ )) ′2

∫ dp′ dq′ u( p2m + W(q′ ))

.

(3.88)

176 | 3 Asymptotic solutions of the many-body problem As a rule, the function u(E) is chosen in the form 1

−1

u(E) = (λ + e θ (E−μ) ) ,

μ ∈ ℝ,

θ > 0,

λ ∈ {−1, 0, 1}.

Integral equation (3.88) is in this case called an equation with jumping nonlinearity. As shown in [28–30], the one-particle distribution (3.87) can be obtained in the semiclassical approximation from the Bose distribution for λ = −1 and the Fermi distribution for λ = 1. For λ = 1, θ → 0, function (3.87) goes to the function ρ(p, q) = const θ(μ − W(q) −

p2 ), 2m

and equation (3.88) goes to equation of Thomas–Fermi type. To conclude this section, let us consider an example of construction of stationary solution of the Riccati-type equation (3.14). Let 1

ρ(p, q) =

p2

e− θ ( 2m +W(q)) 1

p′

2

∫ dp′ dq′ e− θ ( 2m +W(q )) ′

, 1

W(q) = U(q) +

p′

2

′

∫ dp′ dq′ V(q, q′ )e− θ ( 2m +W(q )) 1

p′

2

′ ∫ dp′ dq′ e− θ ( 2m +W(q ))

.

In this case, stationary solution of equation (3.14) can be represented in the form ̃ 1 , q2 ), R(p1 , q1 ; p2 , q2 ) = √ρ(p1 , q1 )ρ(p2 , q2 )R(q where the symmetric function R̃ satisfies the equation 𝜕 ̃ 1 , q2 ) + 1 ∫ dp′ dq′ V(q1 , q′ )ρ(p′ , q′ )R(q ̃ ′ , q2 ) {R(q 𝜕q1 2θ +

1 V(q1 , q2 )} = 0. 2θ

The expression under the sign of derivative with respect to q1 is thus a function of q2 , which we denote by ν(q2 ). The product of this function by √ρ(p1 , q1 )ρ(p2 , q2 ) reads ν(q2 )√ρ(p1 , q1 )ρ(p2 , q2 ) = −{M(p1 , q1 ; p2 , q2 ) +

1 ∫ dp′ dq′ 2θ

3.5 Asymptotic solutions of the uniformization equation

| 177

× √ρ(p1 , q1 )ρ(p′ , q′ )V(q1 , q′ )M(p′ , q′ ; p2 , q2 ) +

1 (V(q1 , q2 ) − ∫ V(q1 , q′ )ρ(p′ , q′ )dp′ dq′ )√ρ(p1 , q1 )ρ(p2 , q2 )}. 2θ

Multiplying this equation by √ρ(p1 , q1 ), integrating over p1 , q1 , and using the following property of M: ∫ M(p1 , q1 ; p2 , q2 )√ρ(p1 , q1 )dp1 dq1 = 0, one can find the function ν(q2 ) from the obtained equation, and then also find the operator M with the kernel M(p, q; p′ , q′ ), M = −(E +

−1

1 1 √ρ 𝒱̃ √ρ) √ρ 𝒱̃ √ρ; 2θ 2θ

here, √ρ denotes the operator of multiplication by √ρ(p, q), and 𝒱̃ denotes the integral operator with the kernel ′

′

′

′

′

′

𝒱̃ (p1 , q1 ; p2 , q2 ) = V(q1 , q2 ) − ∫ dp dq ρ(p , q )(V(q1 , q ) − V(q , q2 ))

+ ∫ dp′ dq′ dp′′ dq′′ ρ(p′ , q′ )ρ(p′′ , q′′ )V(q′ , q′′ ).

3.5 Asymptotic solutions of the equation corresponding to the uniformization of a functional on an abstract Hamiltonian algebra In the previous sections of this chapter, we considered construction of asymptotic solutions of the multiparticle Schrödinger, Liouville, and Wigner equations. The norm with respect to which these asymptotics should be close to exact solutions, was chosen in such a way that, knowing the approximate solution, one could find the limit as N → ∞ of an arbitrary function bounded uniformly in N for which this limit exists. It was shown also that the problem of construction of such asymptotics for multiparticle density function (matrix) can be reduced to the problem of construction of approximation for the N-particle half-density function (matrix) with respect to the norm in the space L2 . It turns out [92] that all the examples studied in the previous sections can be considered from the unified viewpoint, if one introduces the notion of abstract Hamiltonian algebra [3, 109] whose particular cases are the algebras of observables in classical and quantum mechanics. We introduce the notion of abstract half-density whose particular cases are the notions of half-density matrix and half-density function as well

178 | 3 Asymptotic solutions of the many-body problem as the notion of wave function. We construct asymptotic solutions of the equation corresponding to uniformization [3] of a functional on an abstract Hamiltonian algebra. The particular cases of this equation are the multiparticle Liouville and Wigner equations.

3.5.1 Definition of abstract Hamiltonian algebra Definition 3.5.1 ([3, 109]). A Hamiltonian algebra 𝒜 is a vector space 𝒜 equipped with mappings π : 𝒜 × 𝒜 → 𝒜, λ : 𝒜 × 𝒜 → 𝒜, and j : 𝒜 → 𝒜 (which we also denote by π(A, B) ≡ AB, λ(A, B) ≡ {A, B}, and j(A) ≡ A+ ) such that the following axioms hold. Axiom 3.5.1. For any A, B, C ∈ 𝒜, α, β ∈ ℂ, one has (a) A(αB + βC) = αAB + βAC, {A, αB + βC} = α{A, B} + β{A, C}; (b) (A+ )+ = A, (αA + βB)+ = α∗ A+ + β∗ B+ ; (c) (AB)+ = B+ A+ , {A, B} = {A+ , B+ }; (d) {A, BC} = {A, B}C + B{A, C}; (e) A(BC) = (AB)C; (f) {A, B} = −{B, A}; {{A, B}, C} + {{B, C}, A} + {{C, A}, B} = 0. Axiom 3.5.2. There exists an element I ∈ 𝒜, called the unity, such that AI = IA = A, {A, I} = 0 for any A ∈ 𝒜. Axiom 3.5.3. For any A, B ∈ 𝒜 and for certain h ∈ ℝ, one has AB − BA = ih{A, B}. Remark 3.5.1. Since we allow the case h = 0, one cannot replace Axiom 3.5.3 by {A, B} = ih1 (AB − BA). also

Denote by ℒ the space of all complex linear functionals on the algebra 𝒜. Denote ∗

+

+

ℒ+ = {ρ ∈ ℒ | ρ(I) = 1, ρ(A ) = (ρ(A)) , ρ(A A) > 0 ∀A ∈ ℒ},

and assume that ℒ+ ≠ 0. Denote by 𝒜F ⊂ 𝒜 the set of all elements A ∈ 𝒜 such that supρ∈ℒ+ |ρ(A)| < ∞. Consider the following functional p : 𝒜F → ℝ: 󵄨 󵄨 p(A) = sup 󵄨󵄨󵄨ρ(A)󵄨󵄨󵄨. ρ∈ℒ+

It turns out that this functional is a seminorm.

3.5 Asymptotic solutions of the uniformization equation

| 179

Lemma 3.5.1. The following relations hold: p(A + B) ≤ p(A) + p(B),

p(αA) = |α|p(A),

A, B ∈ 𝒜F ,

α ∈ ℂ.

Proof. The proof is obvious. Let {A ∈ 𝒜F | p(A) ≠ 0} ≠ 0. Denote by ℒF the space of all linear functionals 𝒜F → ℂ such that supp(A)=1 |ρ(A)| < ∞. Introduce the following norm in ℒF : 󵄨 󵄨 ‖ρ‖ = sup 󵄨󵄨󵄨ρ(A)󵄨󵄨󵄨, p(A)=1

ρ ∈ ℒF ,

and let us check that axioms of a norm hold. Lemma 3.5.2. The functional ‖ ⋅ ‖ : ℒF → ℝ satisfies the relations: 1. ‖αρ‖ = |α| ⋅ ‖ρ‖, ‖ρ1 + ρ2 ‖ ≤ ‖ρ1 ‖ + ‖ρ2 ‖, ‖ρ‖ ≥ 0, ρ, ρ1 , ρ2 ∈ ℒF , α ∈ ℂ. 2. ‖ρ = 0‖ ⇐⇒ ρ = 0. Proof. The first statement is obvious. Let us prove the second statement. Since for ρ = 0 the norm ‖ρ‖ is evidently also equal to zero, it suffices to check the inverse statement. Let ‖ρ‖ = 0. This means that for any A ∈ 𝒜F such that p(A) ≠ 0, one has the relation ρ(A) = 0. It suffices to check that ρ(B) = 0 for p(B) = 0. Lemma 3.5.1 implies that p(A + B) ≤ p(A) + p(B) = p(A),

p(A) ≤ p(A + B) + p(B) = p(A + B).

Hence, p(A) = p(A + B) ≠ 0 and, therefore, ρ(B) = ρ(A + B) − ρ(A) = 0. Lemma 3.5.2 is proved. Remark 3.5.2. Hamiltonian algebra 𝒜 plays the role of the algebra of observables. The elements of ℒ+ describe the various states of the system. The quantity ρ(A) is the mean value of the variable A in the state ρ. Observables of the form A+ A are usually called nonnegative. The property ρ(A+ A) ≥ 0 for elements of ℒ+ means that mean values of nonnegative observables are also nonnegative. The functional p(A) has the following physical sense: it is the maximal possible mean value of an observable A. The norm ‖ρ‖ plays the following role: if the quantity ‖ρ1 − ρ2 ‖ is small, then the difference between the mean values ρ1 (A) − ρ2 (A) of any observable A such that p(A) = 1 is also small. Let us give examples of Hamiltonian algebras. Example 3.5.1. Denote by 𝒜cn the algebra of smooth functions A(p1 , q1 , . . . , pn , qn ) on ℝ2νn . Define the algebraic operations in the following way: (AB)(X) = A(X)B(X),

A+ (X) = A∗ (X), n

{A, B}(p1 , q1 , . . . , pn , qn ) = ∑( i=1

X = (p1 , q1 , . . . , pn , qn ),

𝜕A 𝜕B 𝜕A 𝜕B − )(p1 , q1 , . . . , pn , qn ). 𝜕qi 𝜕pi 𝜕pi 𝜕qi

(3.89)

180 | 3 Asymptotic solutions of the many-body problem It is not difficult to check that Axiom 3.5.1 holds. The element I(X) ≡ 1 plays the role of unity; hence, the Axiom 3.5.2 is also true. Finally, Axiom 3.5.3 is true for h = 0. Note that the algebra considered in this example is the algebra of observables of classical statistical mechanics. Lemma 3.5.3. In this case, the functional p(A) has the form 󵄨 󵄨 p(A) = sup 󵄨󵄨󵄨A(X)󵄨󵄨󵄨. 2νn X∈ℝ

Proof. Let |A(X)| ≤ C. Then for some observable B and for any ϕ ∈ [0, 2π), one has 2C − A(X)eiϕ − A+ (X)e−iϕ = B+ B. Thus, for any ρ ∈ ℒ+ we have 0 ≤ ρ(CI) − Re(ρ(A)eiϕ ), i. e., |ρ(A)| ≤ C. Hence, p(A) ≤ supX∈ℝ2νn |A(X)|. Let us now consider the element ρX0 (A) = A(X0 ) of the space ℒ+ . It is not difficult to note that p(A) ≥ |ρX0 (A)| = |A(X0 )|. This implies the statement of Lemma 3.5.3. Let us give examples of elements of the space ℒF . (A) To any function ρ ∈ L1 (ℝ2νn ), one can assign an element ρ ∈ ℒF of the form ρ(A) = ∫ dX ρ(X)A(X),

X ∈ ℝ2νn .

(3.90)

We have ‖ρ‖ = ∫ dX |ρ(X)|. (B) An element of the space ℒF of the form ρX0 (A) = A(X0 ) can be also formally written in the form (3.90), if we mean by ρ the distribution ρ(X) = δ(X − X0 ). We have ‖ρ‖ = 1. Note that the function ρ(X) in formula (3.90) plays the role of density of probability distribution. Example 3.5.2. Let 𝒜qn be the algebra of bounded operators in L2 (ℝνn ). Let us introduce the following algebraic operations: π is the usual product of operators, j is Hermitian conjugation, λ(A, B) =

1 (AB − BA), iℏ

where ℏ ≠ 0. The role of the unity I is played in this case by the unity operator. As in the previous example, it is not difficult to check Axioms 3.5.1–3.5.3. The functional p(A) is in this example the usual operator norm p(A) = ‖A‖ = sup ‖Aφ‖. ‖φ‖=1

3.5 Asymptotic solutions of the uniformization equation

| 181

Checking this property is similar to the proof of Lemma 3.5.3: the operator CE − Aeiϕ − A+ e−iϕ can be represented as B+ B. Note also that the elements of the space ℒF play the role of density matrices.

3.5.2 Abstract half-density Let us now generalize the notion of half-density to the case of an abstract Hamiltonian algebra. Definition 3.5.2. A representation of a Hamiltonian algebra 𝒜 on a Hilbert space ℋ is a pair of mappings Π : 𝒜F × 𝒟 → 𝒟 and Λ : 𝒜F × 𝒟 → 𝒟, 𝒟 ⊂ ℋ (which we also denote by Π(A, φ) ≡ Aφ ≡ ΠA φ and Λ(A, φ) ≡ {A, φ} ≡ ΛA φ, A ∈ 𝒜F , φ ∈ ℋ) such that the following axioms hold. Axiom 3.5.4. The maps ΠA and ΛA are linear operators in ℋ with a common dense domain 𝒟 ⊂ ℋ. Axiom 3.5.5. ΠI = E, ΛI = 0. Axiom 3.5.6. For any A, B ∈ 𝒜F , φ, χ ∈ 𝒟, the following relations hold: (a) ΠαA+βB = αΠA + βΠB , ΛαA+βB = αΛA + βΛB ; (b) {A, Bφ} = {A, B}φ + B{A, φ}; (c) ΠAB = ΠA ΠB , Λ{A,B} = ΛA ΛB − ΛB ΛA ; + + (d) (φ, ΠA χ) = (ΠA φ, χ), (φ, ΛA χ) = −(ΛA φ, χ). The elements of the space ℋ will be also called abstract half-densities. Let us assign to each element φ ∈ 𝒟 the following element of the space ℒF : ρφ (A) = (φ, ΠA φ).

(3.91)

It turns out that if the norm ‖φ1 − φ2 ‖ is small, then the norm ‖ρφ1 − ρφ2 ‖ is also small. Lemma 3.5.4. The following relation holds: ‖ρφ1 − ρφ2 ‖ ≤ ‖φ1 − φ2 ‖(‖φ1 ‖ + ‖φ2 ‖).

(3.92)

󵄨 󵄨 ‖ρφ1 − ρφ2 ‖ = sup 󵄨󵄨󵄨(φ1 , ΠA φ1 ) − (φ2 , ΠA φ2 )󵄨󵄨󵄨.

(3.93)

Proof. We have p(A)=1

Note that ρφ ∈ ℒ+ for ‖φ‖ = 1. Hence, in this case |ρφ (A)| = |(φ, ΠA φ)| ≤ 1. Thus, ‖ΠA ‖ ≤ 1 for p(A) = 1. Therefore, formula (3.93) implies formula (3.92). Lemma 3.5.4 is proved.

182 | 3 Asymptotic solutions of the many-body problem Let ρ ∈ ℒF . Denote by {H, ρ} the following functional: {H, ρ}(A) = ρ({A, H}). Definition 3.5.3. Let H ∈ 𝒜, H = H + . The abstract Liouville equation is the following equation: dρt = {H, ρt }, dt

ρt ∈ ℒF ,

t ∈ ℝ,

(3.94)

and the abstract equation for the half-density is the following equation: dφt = {H, φt }, dt

φt ∈ ℋ .

(3.95)

Lemma 3.5.5. Let φt satisfy equation (3.95). Then ρφt satisfies equation (3.94). Proof. Let A ∈ 𝒜. We have d ρ t (A) = (φt , ΠA ΛH φt ) + (ΛH φt , ΠA φt ) dt φ = (φt , [ΠA , ΛH ]φt ) = (φt , Π{A,H} φt ) = ρφt ({A, H}). This implies equation (3.94). Lemma 3.5.5 is proved. Let us give examples of representations of abstract Hamiltonian algebras. Example 3.5.3. Consider the Hamiltonian algebra 𝒜cn . Define the operations ΠA B ≡ AB, ΛA B ≡ {A, B}, where A ∈ 𝒜cn , B ∈ L2 (ℝ2νn ), using relations (3.89). Let 𝒟 = 𝒮 (ℝ2νn ). It is not difficult to check that Axioms 3.5.4–3.5.6 hold. Note also that the element ρφ of the space ℒF has the form 󵄨 󵄨2 ρφ (X) = 󵄨󵄨󵄨φ(X)󵄨󵄨󵄨 . For example, if φ is real, then the function ρ equals the square of the function φ. We see that in this case the notion of abstract half-density coincides with the notion of half-density function introduced above for the case of classical statistical mechanics. Example 3.5.4. Consider the Hamiltonian algebra 𝒜qn . Let ℋ be the Hilbert space L2 (ℝνn ). Define the operators ΠA , ΛA as ΠA φ = Aφ,

ΛA φ =

1 Aφ. iℏ

(3.96)

It is not difficult to check that Axioms 3.5.4–3.5.6 hold also for this case. In this example, abstract half-density plays the role of the wave function, and the corresponding

3.5 Asymptotic solutions of the uniformization equation

| 183

density matrix ρφ is proportional to the projection operator onto the one-dimensional subspace spanned by the vector φ ∈ L2 (ℝνn ). Example 3.5.5. Consider the same Hamiltonian algebra 𝒜qn . Let us choose the Hilbert space ℋ as the space of Hilbert–Schmidt operators acting in L2 (ℝνn ). Introduce the inner product in ℋ as (φ, χ) = Tr φ+ χ. Let ΠA φ = Aφ,

ΛA φ =

1 [A, φ]. iℏ

(3.97)

Density matrix ρφ has the form ρφ = φφ+ . In the case when abstract half-density φ is Hermitian, it coincides with the half-density matrix defined in Section 3.2. One can note that different (Schrödinger and Wigner) equations are considered from one and the same viewpoint: they are abstract equations for half-density corresponding to different representations of one and the same Hamiltonian algebra.

3.5.3 Tensor products of Hamiltonian algebras and of their representations. The notion of uniformization Let us define the notion of the tensor product. For simplicity, let us consider the case when one can assign to the elements A ∈ 𝒜, in a one-to-one manner, functions AY on certain space 𝒴 with measure, and algebraic operations are written as follows: (AB)X = ∫ dY dZ dXYZ AY BZ , (A+ )X = A∗X ,

X, Y, Z ∈ 𝒴 ,

{A, B}X = ∫ dY dZ fXYZ AY BZ , A, B ∈ 𝒜,

(3.98)

where f and d are, in general, distributions 𝒴 × 𝒴 × 𝒴 → ℂ. For the sake of brevity, we shall denote integrals of type (3.98) by dXYZ AY BZ , fXYZ AY BZ , i. e., we shall integrate over repeating indices. Define the tensor product 𝒜⊗N = 𝒜 ⊗ ⋅ ⋅ ⋅ ⊗ 𝒜 as the algebra of functions AY1 ⋅⋅⋅YN : 𝒴 × ⋅ ⋅ ⋅ × 𝒴 = 𝒴 N → ℂ, Y1 , . . . , YN ∈ 𝒴 . Define the algebraic operations in the following way: N

{A, B}X1 ⋅⋅⋅XN = ∑ dX1 Y1 Z1 ⋅ ⋅ ⋅ dXp−1 Yp−1 Zp−1 p=1

× fXp Yp Zp dXp+1 Zp+1 Yp+1 ⋅ ⋅ ⋅ dXN ZN YN AY1 ⋅⋅⋅YN BZ1 ⋅⋅⋅ZN , (AB)X1 ⋅⋅⋅XN = dX1 Y1 Z1 ⋅ ⋅ ⋅ dXN YN ZN AY1 ⋅⋅⋅YN BZ1 ⋅⋅⋅ZN , (A+ )X ⋅⋅⋅X = A∗X1 ⋅⋅⋅XN , 1

N

A, B ∈ 𝒜⊗N ,

Xi , Yi , Zi ∈ 𝒴 .

184 | 3 Asymptotic solutions of the many-body problem It is easy to check the axioms of a Hamiltonian algebra for 𝒜⊗N . Note that for this check, it is necessary to use Axiom 3.5.3 for the algebra 𝒜. Let γ : ℒ → ℝ be a polynomial functional of the form K0

1 ρX1 ⋅ ⋅ ⋅ ρXk WX(k)⋅⋅⋅X 1 k k! k=1

γ[ρ] = ∑

(3.99)

for some observables W (k) ∈ 𝒜⊗k . Definition 3.5.4 ([3]). An ε-uniformization of the functional (3.99) is a set of observables H (N) ∈ 𝒜⊗N of the form K0

εk−1 k! k=1

HX(N)⋅⋅⋅X = ∑ 1

N

∑

1≤j1 =⋅⋅⋅ ̸ =j̸ k ≤N

WX(k)⋅⋅⋅X j1

jk

∏ IXp .

p=j̸ 1 ,...,jk

(3.100)

Let us now consider tensor products of representations. Without loss of generality, one can assume that the Hilbert space ℋ is represented as L2 (𝒳 ) for some space 𝒳 with measure. We shall denote elements of ℋ by φx , x ∈ 𝒳 , and integrals of type ∫ dx φ∗x χx will be denoted by φ∗x χx . Let (L2 (𝒳 ), Π, Λ) be a representation of the Hamiltonian algebra 𝒜, and let the operations Π, Λ have the form (Aφ)x = bXxy AX φy ,

{A, φ}x = cXxy AX φy

for some distributions b, c : 𝒴 × 𝒳 × 𝒳 → ℂ. Consider the Hilbert space ℋ⊗N = ℋ ⊗ ⋅ ⋅ ⋅ ⊗ ℋ = L2 (𝒳 N ) and the following algebraic operations: (Aφ)x1 ⋅⋅⋅xN = bX1 x1 y1 ⋅ ⋅ ⋅ bXN xN yN AX1 ⋅⋅⋅XN φy1 ⋅⋅⋅yN , N

{A, φ}x1 ⋅⋅⋅xN = ∑ bX1 x1 y1 ⋅ ⋅ ⋅ bXp−1 xp−1 yp−1

(3.101)

p=1

× cXp xp yp βXp+1 xp+1 yp+1 ⋅ ⋅ ⋅ βXN xN yN AX1 ⋅⋅⋅XN φy1 ⋅⋅⋅yN , where βXxy = bXxy − iℏcXxy ,

Xi ∈ 𝒴 ,

xi , yi ∈ 𝒳 .

It is not difficult to note that Axioms 3.5.4–3.5.6 indeed hold for operations (3.101). The corresponding representation is called the tensor product of N representations (L2 (𝒳 ), Π, Λ).

3.5 Asymptotic solutions of the uniformization equation

| 185

Let N be fixed, ε = 1/N, and let H be of the form (3.100). Then abstract equation (3.95) for half-density takes the form i

d N,t φ dt x1 ⋅⋅⋅xN K0

=∑

k=1

k

1

N k−1

× icSq xl

∑ WS(k)⋅⋅⋅S bS1 xl

∑

1

1≤l1 0 for ξ ∈ 𝒢M , ξ ≠ 0. Proof. The first property follows from the symmetry of the matrix Mij , and the second one follows from the property ‖M‖ < 1. Let us show that Definition 4.2.2 agrees with the concepts of Section 1.3. To this end, let us introduce one more inner product in the space l2 , ∞ 1 (ξ1 , ξ2 )′ = ωℂ ((ξ , Mξ ), (ξ , Mξ )) = ξ1,i (E − M + M)ij ξ2,j , ∑ 2 1 1 i 2 2 i,j=1

4.2 Complex germ at a point in the Fock space

| 205

and let {G(i) ∈ l2 , i = 1, . . . , ∞} be an orthonormal basis with respect to this inner product. (i) (i) Denote Gm = Gmi , F = MG, Fm = Fmi . By orthonormality of the basis, we have G+ G − F + F = E,

(4.27)

and by property (g1) we have GT F = F T G. Denote Λk = Ωℂ (G(k) , F (k) ), Λ̄ k = Ωℂ (F (k)∗ , G(k)∗ ), k = 1, . . . , ∞. Definition of the 1-form Ωℂ implies that ∞

Λk = −i ∑ (Fmk a+m − Gmk a−m ), m=1 ∞

∗ + ∗ − Λ̄ k = −i ∑ (Gmk am − Fmk am ). m=1

Lemma 4.2.7. 1. The operators Λα and Λ̄ α satisfy the relations [Λα , Λβ ] = [Λ̄ α , Λ̄ β ] = 0,

[Λα , Λ̄ β ] = δαβ ,

α, β = 1, . . . , ∞,

(4.28)

and the property Λα ψM = 0, 2. 3.

α = 1, . . . , ∞.

(4.29)

Let an element f0 ∈ ℱφ satisfy the property Λα f0 = 0, α = 1, . . . , ∞. Then f0 = cψM for some constant c. The system of vectors Λ̄ α1 ⋅ ⋅ ⋅ Λ̄ αp ψM is complete in the space ℱ .

Proof. The proof is similar to Lemma 2.5.3. Corollary 4.2.1. The subspace 𝒢M is a complete complex germ in the sense of Definition 1.3.2. The inverse statement is also true. Lemma 4.2.8. Let 𝒢 be a subspace of the space TMφℂ , which is a complete complex germ. Then it has the form (4.26) for some matrix M corresponding to a Hilbert–Schmidt operator with the norm less than unity. Proof. Let us first show that the subspace 𝒢 is the graph of an operator. Consider the operator 𝒫 : 𝒢 → l2 of the form 𝒫 (ζ , η) = ζ ,

(ζ , η) ∈ 𝒢 ,

(4.30)

206 | 4 Complex germ method in the Fock space and let us show that the image of the operator 𝒫 coincides with l2 . Suppose the contrary, then there exists a nonzero element λ from the orthogonal complement to Im 𝒫 . Let f0 be an element of Dφ such that Λα f0 = 0 for α = 1, . . . , ∞, i. e., ∞

∑ (ζm a−m − ηm a+m )f0 = 0,

m=1

(ζ , η) ∈ 𝒢 .

(4.31)

∗ + Consider the vector ∑∞ m=1 λm am f0 ∈ Dφ , which also satisfies condition (4.31) due to the choice of λ and the commutation relations between a±m . According to Definition 1.3.2, the element f0 is determined up to a constant; hence, ∞

∗ + ( ∑ λm am − γ)f0 = 0 m=1

for some γ ∈ ℂ. This implies ∞

∞

m=1

k=1

∗ + am − γ)f0 , ( ∑ λk∗ a+k − γ)f0 ) 0 = (( ∑ λm ∞

∞

∞

m,k=1

k=1

m=1

= (f0 , ∑ [λm a−m , λk∗ a+k ]f0 ) + (( ∑ λk a−k − γ)f0 , ( ∑ λm a−m γ)f0 ). But the first summand in this formula is positive, and the second one is nonnegative. We obtain a contradiction. Hence, the image of the operator 𝒫 of the form (4.30) coincides with l2 . Further, according to Lemma 1.3.1, the following relations hold: ζ1 η2 − ζ2 η1 = 0, ζζ − ηη ≥ 0, ∗

∗

(ζ1 , η1 ), (ζ2 , η2 ) ∈ 𝒢 ,

(ζ , η) ∈ 𝒢 .

(4.32) (4.33)

Let us show that the kernel of the operator 𝒫 consists of only zero. Assume the contrary: 0 ≠ (ζ , η) ∈ 𝒢 , 𝒫 (ζ , η) = ζ = 0. This means that the element (0, η) of the space l2 × l2 belongs to 𝒢 , which contradicts to relation (4.33). Thus, the operator 𝒫 (4.30) is a one-to-one map. Hence, the germ 𝒢 has the form (4.26) for some matrix M. Property (4.32) implies symmetry of M, and property (4.33) implies the property ‖M‖ < 1. Let us show that M is a Hilbert–Schmidt operator. Property (4.31) implies ∞

g (m) = (a−m − ∑ Mmn a+n )f0 = 0. n=1

(m) The first component g1,k of this element g (m) of the Fock space reads (m) g1,k = √2(f0 )2,mk − Mmk (f0 ),0 = 0.

4.2 Complex germ at a point in the Fock space

| 207

2 Hence, ∑∞ m,k=1 |Mmk | < ∞, i. e., M is a Hilbert–Schmidt operator. Lemma 4.2.8 is proved. ℂ Definition 4.2.3. The canonical transformation of the germ 𝒢M is the germ U∗φ 𝒢M coñ sisting of all vectors (ζ , η)̃ such that

ζ̃ =

𝜕φ̃ ∗ 𝜕φ̃ ∗ ζ + η, 𝜕φ∗ 𝜕φ

η̃ =

𝜕φ̃ 𝜕φ̃ ζ+ η, ∗ 𝜕φ 𝜕φ

(4.34)

where (ζ , η) ∈ 𝒢M . Lemmas 4.2.8 and 1.3.3 imply the following corollary. ℂ ̃ ) | ζ ∈ l2 }, where the Corollary 4.2.2. The complex germ U∗φ 𝒢M has the form {(ζ , Mζ matrix

𝜕φ̃ 𝜕φ̃ 𝜕φ̃ ∗ 𝜕φ̃ ∗ M̃ = ( ∗ + M)( ∗ + M) 𝜕φ 𝜕φ 𝜕φ 𝜕φ

−1

is symmetric and corresponds to a Hilbert–Schmidt operator with the norm less than unity. Consider the operator wF,G : ℱ → ℱ mapping an element (f (0) , fi(1) , . . . , fi(p) , . . .) ⋅⋅⋅i 1

1

p

of ℱ , where i1 , i2 , . . . = 1, . . . , ∞, into the following element of ℱ : ∞ 1 Λ̄ ⋅ ⋅ ⋅ Λ̄ αp ∑ fα(p) 1 ⋅⋅⋅αp α1 p=0 √p! α1 ,...,αp =1 ∞

wF,G f = ∑ ×

1

√4 det(G+ G)

exp{

1 ∞ + ∑ a M a+ }Φ(0) . 2 m,n=1 m mn n

Lemma 4.2.9. 1. The operator wF,G is unitary. 2. The following relations hold: −1 wF,G a+k wF,G = Λ̄ k ,

3.

−1 wF,G a−k wF,G = Λk .

For some number constant c, the following relation holds: −1 W = cwF,̃ G̃ wF,G ,

(4.35)

208 | 4 Complex germ method in the Fock space where 𝜕φ̃ ∗ G̃ (i) ( ̃ (i) ) = ( 𝜕φ 𝜕φ̃ F

∗

𝜕φ∗

𝜕φ̃ ∗ G(i) 𝜕φ ) ( ). ̃ 𝜕φ F (i) 𝜕φ

Proof. The proof of the first statement is similar to proof of Lemma 2.5.5. The second statement follows from definition of creation and annihilation operators, and the third statement is a corollary of Lemma 1.3.7. Definition 4.2.4. The canonical operator corresponding to the germ 𝒢M and to the baF,G sis {G(i) } is the following operator Kφ,ε : ℱ → ℱ: F,G Kφ,ε = Kφε wF,G .

4.2.6 Formal asymptotic solutions of the equations of motion Ut

Let φ 󳨃󳨀→ φt be a family of proper canonical transformations of the space l2 , and let Wφ,t : ℱ → ℱ be the corresponding maps W. By canonicity of the transformation U t , there exists a function St : l2 → ℝ such that ∞

t ∗ −idSt = ∑ (φ∗t k dφk − φk dφk ). k=1

According to Definition 1.2.7, the formal asymptotic solution of equations of motion is the following element of ℱ depending on t: i

t

e ε S Kφε t Wφ,t f0 ,

f0 ∈ ℱ .

(4.36)

F,G The element (4.36) can be represented through the operator Kφ,ε . Consider the transt formation, corresponding to U ,

(G(i) , F (i) ) 󳨃→ (G(i)t , F (i)t ) of basis on the complex germ G(i)t =

𝜕φt∗ (i) 𝜕φt∗ (i) F , G + 𝜕φ∗ 𝜕φ

F (i)t =

𝜕φt (i) 𝜕φt (i) F . G + 𝜕φ∗ 𝜕φ

(4.37)

Lemma 4.2.8 implies the following corollary. Corollary 4.2.3. The element i

t

t

t

Ψε (t) = ct e ε S KφFt ,ε,G f ,

ct ∈ ℂ,

󵄨󵄨 t 󵄨󵄨 󵄨󵄨c 󵄨󵄨 = 1,

(4.38)

4.2 Complex germ at a point in the Fock space

| 209

of ℱ , where f does not depend on t, is a formal asymptotic solution of equations of motion. Let us now consider equations for φt , F t , Gt . Lemma 4.2.10. Let U t be a family of canonical transformations of the space l2 . 1. There exists a function Ht : l2 → ℝ such that iφ̇ t =

𝜕Ht ∗t t (φ , φ ), 𝜕φ∗

and one has i ∞ ∗t t Ṡ t = ∑ (φt∗ φ̇ t − φtk φ̇ t∗ k ) − Ht (φ , φ ). 2 k=1 k k 2.

Elements G(k)t , F (k)t of l2 satisfy the system −iĠ (k)t = iḞ (k)t =

𝜕2 Ht (k)t 𝜕2 Ht (k)t G + F , 𝜕φ𝜕φ∗ 𝜕φ𝜕φ 𝜕2 Ht 𝜕2 Ht (k)t G(k)t + F . ∗ ∗ 𝜕φ 𝜕φ 𝜕φ∗ 𝜕φ

(4.39)

Proof. The proof is similar to proof of Lemmas 2.6.4 and 2.6.6.

4.2.7 Equation for the operator W Let us now consider the element f (t) = Wφ,t f0 ∈ ℱ . Lemma 4.2.11. The element f (t) ∈ ℱ satisfies the equation ∞ 𝜕2 Ht + 𝜕2 Ht − 1 − 𝜕2 Ht − 1 if ̇(t) = ∑ [ a+i aj + a+i a + a a ] ∗ ∗ 2 𝜕φi 𝜕φj 𝜕φ∗i 𝜕φj j 2 i 𝜕φi 𝜕φj j i,j=1

× f (t) + H1 (t)f (t),

(4.40)

where H1 (t) is the real number expressed through the constant ct as follows: i(log ct ) − ⋅

∞ 𝜕2 Ht 1 t Re ∑ Mmn = H1 (t). 2 𝜕φ 𝜕φ m n m,n=1

(4.41)

210 | 4 Complex germ method in the Fock space Proof. Formula (4.40) is proved completely similar to Lemma 2.6.7. For proof of the formula (4.41), let us choose the element f0 as follows: 1

f0 =

√4 det(G+ G)

exp{

1 ∞ + ∑ a M a+ }Φ(0) . 2 m,n=1 m mn n

Then f =

ct

√4 det((Gt )+ (Gt ))

exp{

1 ∞ + t + ∑ a M a }Φ(0) . 2 m,n=1 m mn n

This function satisfies equation (4.40). From this, one can find H1 (t). Substitution of f (t) into equation (4.40) yields i(log

ct

⋅

√det[(Gt )+ (Gt )] 4

) =

𝜕2 Ht 1 ∞ M t + H1 . ∑ 2 m,n=1 𝜕φm 𝜕φn mn

(4.42)

By the relation (log det z)⋅ = Tr zż −1 , we have (log det[(Gt ) Gt ]) +

⋅

= Tr(Ġ t Gt−1 + Ġ t+ (Gt+ ) ) −1

∞

=i ∑ ( m,n=1

𝜕2 Ht 𝜕2 Ht 𝜕2 Ht 𝜕2 Ht t M − δ − δ + M ∗t ). 𝜕φm 𝜕φ∗n mn 𝜕φm 𝜕φn mn 𝜕φ∗m 𝜕φn mn 𝜕φ∗m 𝜕φ∗n mn

Substituting this formula into formula (4.42), we obtain formula (4.41). Lemma 4.2.11 is proved. 4.2.8 Commutation of the canonical operator with other operators Let us now show that the constructed formal asymptotic solutions of equations of motion are indeed asymptotic solutions of equation (2.A.7). Let us assign to each function l2 → ℝ of the form P0

Ht (φ∗ , φ) = ∑

∑

m,n=1 i1 ,...,im ,j1 ,...,jn ∈ℕ

Hi1 ⋅⋅⋅im j1 ⋅⋅⋅jn (t)φ∗i1 ⋅ ⋅ ⋅ φ∗im φj1 ⋅ ⋅ ⋅ φjn

the following operator in the space ℱ : P

0 1 Ht (√εa+ , √εa− ) = ∑ ∑ ε m,n=1 i ,...,i ,j ,...,j 1

×ε

m+n −1 2

m 1

n ∈ℕ

Hi(m,n) ⋅⋅⋅i j ⋅⋅⋅j (t) 1

m 1

a+i1 ⋅ ⋅ ⋅ a+im a−j1 ⋅ ⋅ ⋅ a−jn .

n

4.2 Complex germ at a point in the Fock space |

211

Here, P0 ∈ ℕ, Hi(m,n) are sets of numbers separately symmetric in i1 , . . . , im and in 1 ⋅⋅⋅im j1 ⋅⋅⋅jn j1 , . . . , jn , such that (n,m) Hi(m,n)∗ ⋅⋅⋅i j ⋅⋅⋅j = Hj ⋅⋅⋅j i ⋅⋅⋅i . 1

m 1

n

1

n 1

m

Consider the operator 1 Hε,t = Ht (√εa+ , √εa− ) + H1,t (√εa+ , √εa− ) + ⋅ ⋅ ⋅ ε + εk−1 Hk,t (√εa+ , √εa− ), k ∈ ℕ, and the equation iΨ̇ ε (t) = Hε,t Ψε (t),

Ψε (t) ∈ ℱ .

(4.43)

Without loss of generality, one can assume that k ≤ P0 − 1 (otherwise this inequality can be achieved by increasing P0 ). d , a±k , and Kφε . Let us establish some properties of the operators dt Lemma 4.2.12. The following relations hold: 1. i

i t d i i ∞ t∗ t d εi St ε t e Kφt = e ε S Kφε t [i − Ṡ t + ∑ (φ φ̇ − φ̇ t∗ k φk ) dt dt ε 2ε k=1 k k

+

i ∞ + t ∑ (a φ̇ − a−k φ̇ t∗ k )]; √ε k=1 k k

2. a+k Kφε t = Kφε t (a+k +

φ∗k ), √ε

a−k Kφε t = Kφε t (a−k +

φk ). √ε

Proof. By formula (4.14), we have Kφε t+Δt = Kφε t +φ̇ t Δt+o(Δt) = Kφε t (1 + Δt( +

1 ∞ + t ∑ (a φ̇ − a−k φ̇ t∗ k ) √ε k=1 k k

i ∞ t∗ t t ∑ (φ φ̇ − φ̇ t∗ k φk )) + o(Δt)). 2ε k=1 k k

Hence, i

i t d εi St ε d i i ∞ t∗ t e Kφt = e ε S Kφε t (i − Ṡ t + ∑ (φ φ̇ − φtk φ̇ t∗ k ) dt dt ε 2ε k=1 k k

+

i ∞ + t ∑ (a φ̇ − a−k φ̇ t∗ k )). √ε k=1 k k

212 | 4 Complex germ method in the Fock space The first statement of the lemma is proved. The second statement follows from the formula eA Be−A = B + [A, B] valid for operators A and B whose commutator is proportional to the unit operator. Lemma 4.2.12 is proved. Denote H̃ t (φ∗ , φ) = Ht (φ∗ , φ) − Ht (φt∗ , φt ) −

𝜕Ht t∗ t 𝜕Ht t∗ t (φ , φ )(φ∗ − φt∗ ) − (φ , φ )(φ − φt ). ∗ 𝜕φ 𝜕φ

(4.44)

Lemma 4.2.12 implies the following corollary. Corollary 4.2.4. Assume that the following condition holds: i ∞ t Ṡ t = ∑ (φt∗ φ̇ t − φtk φ̇ t∗ k )−H , 2 k=1 k k

iφ̇ t =

𝜕Ht t∗ t (φ , φ ). 𝜕φ∗

(4.45)

Then (i

i t i t d d − Hε,t )e ε S Kφε t = e ε S Kφε t (i − H̃ ε,t ), dt dt 1 H̃ ε,t = H̃ t (φt∗ + √εa+ , φt + √εa− ) ε + H1,t (φt∗ + √εa+ , φt + √εa− )

+ ⋅ ⋅ ⋅ + εk−1 Hk,t (φt∗ + √εa+ , φt + √εa− ).

(4.46)

(4.47)

Remark 4.2.4. 1. The operator H̃ ε,t is a polynomial function of the creation and annihilation operators, and the coefficient before each summand contains nonnegative power of ε. Indeed, by definition of the function H̃ t all the summands containing ε−1 and ε−1/2 vanish. 2. If conditions (4.45) did not hold, then the right-hand side of equation (4.46) would contain summands of order ε−1 and ε−1/2 . i

t

Corollary 4.2.5. The element e ε S Kφε t f (t) of ℱ is an asymptotic solution of equation (4.43) if and only if 󵄩󵄩 d 󵄩󵄩 ε→0 󵄩 󵄩󵄩 󵄩󵄩(i − H̃ ε,t )f (t)󵄩󵄩󵄩 󳨀→ 0. 󵄩󵄩 󵄩󵄩 dt

(4.48)

Proof. The proof follows from formula (4.41) and from unitarity of the canonical operator Kφε t .

4.2 Complex germ at a point in the Fock space |

213

4.2.9 Asymptotic solutions of the equations of motion Let us make use of the following lemma. Lemma 4.2.13. Let O1 , . . . , Ok be operators (in general, unbounded) acting in a normed space ℱ . Then each of the operators O1 , . . . , Ok can be represented as a finite linear combination of the operators 1/2 k/2 O′m = εm O1 + ⋅ ⋅ ⋅ + εm Ok ,

where m = 1, . . . , K, K ≥ k, εi ≠ εj for i ≠ j. Proof. This lemma follows from nondegeneracy of the matrix ε11/2 ( ... εk1/2

⋅⋅⋅ .. . ⋅⋅⋅

ε1k/2 .. ) . . εkk/2

Corollary 4.2.6. Let an element f ∈ ℱ belong to the domain of each of the operators Oε = ε1/2 O1 + ⋅ ⋅ ⋅ + εk/2 Ok ,

(4.49)

for ε = ε1 , . . . , εK , K ≥ k, εi ≠ εj for i ≠ j. Then ε→0

‖Oε f ‖ 󳨀→ 0. Proof. Indeed, Lemma 4.2.13 implies that f belongs to the domain of all the operators ε→0

Ok , therefore, ‖Oε f ‖ 󳨀→ 0.

Corollary 4.2.7. Assume that: i t (1) for 2P0 + 1 values of ε > 0, the element Ψ(t) = e ε S Kφε t f (t) ∈ ℱ belongs to the domain of the operator Hε,t , (2) the element f (t) ∈ ℱ satisfies equation (4.40) for H1 = H1,t (φ∗t , φt ). Then the element Ψ(t) ∈ ℱ is an asymptotic solution of equations of motion (4.48). Proof. In this case, the canonical operator Kφε t is invertible; hence, the fact that Ψ(t) belongs to the domain of Hε,t implies that f (t) belongs to the domain of the operator (Kφε t ) Ht Kφε t . −1

(4.50)

214 | 4 Complex germ method in the Fock space After multiplication of the operator (4.49) by ε3/2 the operator (4.50) takes the form (4.49), so that 1 1 −1 H̃ ε = (Kφε t ) Ht Kφε t − O1 − O. √ε 2 ε Since equation (4.40) for f holds, property (4.48) goes to the following one: 󵄩󵄩 ̃ 󵄩 ε→0 󵄩󵄩(Hε − O3 )f (t)󵄩󵄩󵄩 󳨀→ 0. This condition is indeed satisfied by Corollary 4.2.6; the operator H̃ ε − O3 also has the form (4.49). Therefore, property (4.48), and hence, Corollary 4.2.7, is proved. Corollary 4.2.8. Assume that ct satisfies condition (4.42) and condition 1 of Corollary 4.2.7 holds. Then the function i

t

Ψ(t) = ct e ε S Kφε t wF t ,Gt f0 ∈ ℱ is an asymptotic solution of equation (4.43). Remark 4.2.5. Similar arguments hold also in the case when φt depends on ε and ε→0

φt (ε) 󳨀→ φt .

4.2.10 Corrections to the asymptotic formula Let us now consider construction of asymptotic solutions of equation (4.43) up to O(εM/2 ), M > 1. Consider operator (4.47), and let us represent it as a polynomial in √ε: H̃ ε,t = H̃ t(0) (a+ , a− ) + √εH̃ t(1) (a+ , a− ) + ⋅ ⋅ ⋅ + εP0 /2−1 H̃ t

(P0 −2)

(a+ , a− ),

(4.51)

and consider the following element of ℱ : fεM (t) = f0 (t) + ε1/2 f1 (t) + ⋅ ⋅ ⋅ + ε

M−1 2

fM−1 (t).

(4.52)

k = 0, . . . , M − 1,

(4.53)

Lemma 4.2.14. Assume that fk (t) satisfies the equation ifk̇ (t) − H̃ t(0) fk (t) = gk (t), k

gk (t) = ∑ H̃ t(m) fk−m , m=1

4.2 Complex germ at a point in the Fock space |

215

and elements fk (t), k = 0, . . . , M − 1, belong to the domain of all the operators H̃ t(n) , n = 0, . . . , P − 1. Then the following element of ℱ : i

t

S ε M ε Kφt fε (t) ΨM ε (t) = e

satisfies the property 󵄩󵄩 d 󵄩󵄩 󵄩󵄩 󵄩 M M/2 󵄩󵄩(i − Hε,t )Ψε (t)󵄩󵄩󵄩 = O(ε ). 󵄩󵄩 dt 󵄩󵄩

(4.54)

Proof. By the already established commutation relations (4.46) and by unitarity of the operator Kφε t , we have 󵄩󵄩 d 󵄩󵄩 󵄩󵄩 󵄩󵄩 d 󵄩󵄩 󵄩 󵄩 󵄩 M M 󵄩󵄩(i − Hε,t )Ψε (t)󵄩󵄩󵄩 = 󵄩󵄩󵄩(−i + H̃ ε,t )fε (t)󵄩󵄩󵄩 󵄩󵄩 dt 󵄩󵄩 󵄩󵄩 󵄩󵄩 dt 󵄩󵄩P0 +M−3 󵄩󵄩 M 󵄩󵄩 󵄩󵄩 = 󵄩󵄩󵄩 ∑ εL/2 ∑ H̃ t(m) fL−m (t)󵄩󵄩󵄩 = O(εM/2 ), 󵄩󵄩 󵄩󵄩󵄩 L=M m=1 󵄩 where we have taken into account relation (4.53). Lemma 4.2.14 is proved. Let us now consider the construction of solutions of equation (4.53). Lemma 4.2.15. The solution of equation (4.53) satisfying the initial condition fk (0) = fk,0 reads t

fk (t) = wF t ,Gt [fk,0 − i ∫ dτ wF−1τ ,Gτ gk (τ)].

(4.55)

0

Proof. Consider the following element of ℱ : χk (t) = wF t ,Gt fk (t). Equation (4.54) implies iẇ F t ,Gt χk (t) + iwF t ,Gt χk̇ (t) − Ht(0) wF t ,Gt χk (t) = gk (t).

(4.56)

If gk = 0, then as we have seen above, this equation has a solution as an element of the Fock space χk ∈ ℱ independent of t. This means that the first and the third summand in the left-hand side of formula (4.56) cancel. Hence, t

χk (t) =

χk(0)

− i ∫ wF τ ,Gτ gk (τ) dτ. 0

This implies the statement of Lemma 4.2.15.

216 | 4 Complex germ method in the Fock space Thus, we have constructed asymptotic solutions of equation (2.A.7) with any power of precision in ε1/2 .

4.3 Superposition of wave packets in finite-dimensional quantum mechanics 4.3.1 Statement of the problem During the previous three chapters, we considered asymptotics obtained by means of abstract canonical operator (Section 1.2), which generalizes the notion of canonical operator in the theory of complex germ at a point. On the other hand, besides the germ at a point one often uses the method of a complex germ on an isotropic manifold [4, 25, 27] whose particular case is the method of canonical operator on a Lagrangian manifold [24], allowing one to construct, in the case of equations of quantum mechanics, rapidly oscillating solutions of the form i φ(x) exp{ S(x)} ε

(4.57)

(ε is the parameter of semiclassical decomposition). These solutions are of order O(1) in certain domain of finite measure of the space ℝn . It is such asymptotics that are usually considered in physical literature as semiclassical ones [21]. In the subsequent sections of this chapter, we shall consider the following questions. Can one construct wave functions of type (4.57) and check that they are asymptotic solutions based only on the results of the theory of a germ at a point? How, using the analogy between the abstract canonical operator and the theory of germ at a point, can one construct the analog of the canonical operator on a Lagrangian manifold in the case of a general abstract canonical operator? How related is this generalization of a canonical operator on a Lagrangian manifold with the traditional definition [24]? We shall also consider construction of asymptotics corresponding to isotropic manifolds with a complex germ, which have the form (4.57) on some surface in ℝn , which is the projection of an isotropic manifold in the phase space ℳ = ℝ2n onto the subspace ℝn . The study of the questions above is based on the following observation. Let (P(τ), Q(τ)) ∈ ℝ2n be a smooth k-dimensional surface in the phase space, let f (τ) ∈ ℱP(τ),Q(τ) , and let S be a real function on ℝk . Similar to [82], let us consider the superposition of wave functions constructed by the method of a germ at a point: ∫

dτ εi S(τ) ε e KP(τ),Q(τ) f (τ). εk/4

(4.58)

In Subsection 4.3.2, we shall consider this integral in the one-dimensional case, and in Subsection 4.3.3 in the multidimensional case. It will turn out that this integral is not

4.3 Superposition of wave packets in quantum mechanics | 217

exponentially small only in the case when on some subset of the set ℝk the following relation holds: n 𝜕Qj 𝜕S = ∑ Pj , 𝜕τl j=1 𝜕τl

(4.59)

which means the condition of isotropy of the surface (P(τ), Q(τ)): n

∑( j=1

𝜕Pj 𝜕Qj 𝜕τl 𝜕τs

−

𝜕Pj 𝜕Qj 𝜕τs 𝜕τl

) = 0.

(4.60)

Hence, we can restrict ourselves by isotropic manifolds. We shall show that integral (4.58) indeed coincides with the wave function constructed by means of the traditional method of a canonical operator on a Lagrangian manifold with a complex germ [4, 25]. However, when defining the wave function in the form (4.58), there are no difficulties with focal points; hence, there is no need in covering the Lagrangian manifold by charts each of which has a one-to-one projection onto one of the planes of the form P1 = ⋅ ⋅ ⋅ = Ps = Qs+1 = ⋅ ⋅ ⋅ = Qn = 0, and writing out the canonical operator separately in each chart. In Subsection 4.3.4, we shall consider the computation of the norm of the wave function (4.58) in the space L2 (ℝn ) as ε → 0. It will turn out that “in the general position” this norm is of order O(1) as ε → 0 just because of the factor ε−k/4 in formula (4.58). However, on certain subspace S0 of vector functions f (τ) ∈ L2 (ℝn ) this norm tends to zero as ε → 0. This means that we can introduce an equivalence relation on {f (τ) ∈ L2 (ℝn )}, calling two functions equivalent if their difference belongs to the subspace S0 , and consider the canonical operator on the corresponding quotient space. This procedure is considered in the next section. In this section, we shall restrict ourselves by considering isotropic manifolds diffeomorphic to ℝk on which one can introduce global coordinates. The general case is considered in Section 4.6. 4.3.2 Superposition of wave packets in the one-dimensional case Consider the wave function (4.58) in the one-dimensional case. Definition of canonical operator (1.5) implies that it has the form Ψε (x) = ∫ where f (τ, ⋅) ∈ L2 (ℝ).

i x − Q(τ) dτ exp{ (S(τ) + P(τ)(x − Q(τ))}f (τ, ), 1/2 √ε ε ε

(4.61)

218 | 4 Complex germ method in the Fock space Let us compute asymptotics of integral (4.61) in the case when the smooth curve (P(τ), Q(τ)), τ ∈ ℝ, has a one-to-one projection onto the line Q = 0. In this case, one can assume without loss of generality that the parameter τ is chosen so that Q(τ) = τ. Let us fix the point x. Lemma 4.3.1. Let f ∈ 𝒮 (ℝ2 ). 1. 2.

For S′ (x) ≠ P(x), one has the relation For S′ (x) = P(x), one has

ε→0 1 Ψε (x) 󳨀→ εm

ε→0

i

0 ∀m.

2

1 ′′

Ψε (x)e− ε S(x) 󳨀→ ∫ dα f (x, α)e−iα (P (x)− 2 S ′

(x))

.

Corollary 4.3.1. If in a neighborhood of the point x the relation S′ (x) = P(x) holds, then i

Ψε (x) = ∫ dα f (x, α)e− 2 S

′′

(x)α2 + εi S(x)

+ z ε (x),

ε→0

z ε (x) 󳨀→ 0.

(4.62)

Remark 4.3.1. 1. The condition S′ (x) = P(x) is a particular case of (4.59). 2. A heuristical derivation of the statement of Lemma 4.3.1 consists in the replacement τ = x − α√ε and the development of the expression under the integral into a power series with respect to √ε. For proof of Lemma 4.3.1, let us first prove auxiliary statements. Lemma 4.3.2. Let f (τ, ⋅) ∈ 𝒮 (ℝ). Then there exists a continuous function ϕ : ℝ → ℝ such that the integral ∫ ϕ(α) dα converges and |f (τ, α)| < ϕ(α) for τ ∈ [x − a, x + a]. Proof. The function (α2 + 1)f (τ, α) is bounded for τ ∈ [x − a, x + a] as an element of 𝒮 (ℝ) for fixed τ. Hence, f (τ, α) < c/(α2 + 1). The integral of the function in the right-hand side of this inequality indeed converges. Lemma 4.3.2 is proved. Proof of Lemma 4.3.1. Note first that the expression 1 εm

∫ |x−τ|>a

dτ εi (S(τ)+P(τ)(x−τ)) x−τ e f (τ, ) √ε ε1/2

tends to zero as ε → 0 for any m, because it does not exceed εM−m−1/2

∫ |x−τ|>a

× ((

(τ2 2

dτ + 1)((x − τ)2 + ε)M M

x−τ x−τ ) + 1) (τ2 + 1)f (τ, ), √ε √ε

4.3 Superposition of wave packets in quantum mechanics | 219

which, in its turn, tends to zero as ε → 0 for M > m + 1/2 due to rapid decrease of f . Hence, Ψε (x) = 𝒵 ε (x)

εm

∫ |x−τ|≤a

dτ εi (S(τ)+P(τ)(x−τ)) x−τ f (τ, e ) + 𝒵 ε (x), 1/2 √ε ε

(4.63)

ε→0

󳨀→ 0.

Consider the following change of variables in integral (4.63): τ = x − α√ε, after which this integral takes the form i ∫ dα exp{ (S(x − α√ε) + α√εP(x − α√ε))}f ̄(x − α√ε, α), ε

(4.64)

ℝ

where f (τ, α), for τ ∈ [x − a, x + a], f ̄(τ, α) = { 0, for τ ∈ ̸ [x − a, x + a]. Extracting from the expression under the integral in formula (4.64), the factor i i α[P(x) − S′ (x)]}, exp{ S(x) + √ε ε one can represent integral (4.64) in the form i

i

e ε S(x) ∫ dα e √ε

α[P(x)−S′ (x)]

g√ε,x (α),

(4.65)

ℝ

where the function g√ε,x (α) satisfies, by Lemma 4.3.2, the estimate 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨g√ε,x (α)󵄨󵄨󵄨 = 󵄨󵄨󵄨f ̄(x − α√ε, α)󵄨󵄨󵄨 < ϕ(α),

(4.66)

∫ℝ ϕ(α) dα converges, and as ε → 0, one has ε→0

2

1 ′′

g√ε,x (α) 󳨀→ f (x, α)e−iα (P (x)− 2 S ′

(x))

.

(4.67)

Note that the estimate similar to (4.66) holds also for derivatives of g with respect to α (the proof is similar). This implies that i

∫ dα e √ε ℝ

α[P(x)−S′ (x)] (2m+1) g√ε,x (α)

< C2m+1 ,

220 | 4 Complex germ method in the Fock space where C2m+1 is a constant independent of ε. After integrating by parts, we obtain that for S′ (x) ≠ P(x), the expression i

∫ dα e √ε

α[P(x)−S′ (x)]

g√ε,x (α)

ℝ

1 εm+1/2

is bounded uniformly in ε. This implies the first statement of Lemma 4.3.1. For proof of the second statement of the lemma, let us use the estimate (4.66), relation (4.67), and the following Lebesgue theorem (see, for instance, [19]). Theorem 4.3.1 (The Lebesgue theorem). Assume that a sequence of functions {fn } on a space 𝒳 with measure dx converges almost everywhere to a function f (x), and for all n one has 󵄨󵄨 󵄨 󵄨󵄨fn (x)󵄨󵄨󵄨 ≤ ϕ(x), where ϕ is an integrable function on 𝒳 . Then the function f is also integrable on 𝒳 and n→∞

∫ dx fn (x) 󳨀→ ∫ dx f (x). 𝒳

𝒳

Applying this theorem for any sequence εn → 0 to formula (4.65), we obtain for S (x) = P(x) the second statement of Lemma 4.3.1. Lemma 4.3.1 is proved. ′

Thus, integral formula (4.61) indeed gives us a wave function (4.62) of the form i

φ(x)e ε S(x) + O(√ε)

(4.68)

in the case when the surface (P(τ), Q(τ)) has a one-to-one projection onto the line P = 0. Remark 4.3.2. ̇ 1. In the case when there exist focal points (at which Q(τ) = 0), the wave function (4.61) does not have the form (4.68) near these points. This difficulty is well known in the theory of canonical operator on a Lagrangian manifold [24]. In the approach considered here, we can define the wave function near the focal points using integral formula (4.61), which makes sense also at the focal points. This definition coincides, up to higher corrections, with the usual canonical operator [24]. This can be checked without difficulty by considering the Fourier transform of the function (4.61) and applying similar arguments to it. 2. Formula (4.62) implies that different functions f can yield the same functions Ψε up to higher corrections. Hence, we can introduce an equivalence relation in 𝒮 (ℝ)

4.3 Superposition of wave packets in quantum mechanics | 221

calling two functions from 𝒮 (ℝ) equivalent if their difference Δf satisfies the condition i

∫ dα e− 2 αS

′′

(x)

Δf (x, α) = 0.

Then the wave function (4.61) is uniquely determined (up to higher corrections) by the equivalence class.

4.3.3 Superposition of wave packets in the multidimensional case Let us now consider the wave function (4.58) in the case of a k-dimensional smooth surface without self-intersections in 2n-dimensional nonsingular phase space. Definition of the canonical operator implies that the wave function (4.58) reads Ψε (x) = ∫ ℝk

dτ ε

k+n 4

i

e ε (S(τ)+P(τ)(x−Q(τ))) f (τ,

x − Q(τ) ). √ε

(4.69)

We shall consider functions f (τ, ξ ), which vanish for |τ| > a for some a > 0, are infinitely differentiable with respect to τ and ξ , and for each τ belong to the space 𝒮 (ℝn ) as functions of ξ . Denote the space of all such functions f by Sk,n . The wave function (4.69) is exponentially small in the case when x depends on √ε so that |x−Q(τ)| ≥ a > 0, due to rapid decrease of f at infinity. Therefore, integral (4.69) is not exponentially small only if there exists τ̄ ∈ ℝk such that x = Q(τ)̄ + ξ (√ε)√ε,

(4.70)

where ξ (√ε) is a smooth function of √ε. Denote ξ (0) = ξ . Let us compute asymptotics of the integral (4.69) in the case when x has the form (4.70), and the projection surface Q(τ) of the manifold (P(τ), Q(τ)) onto the Q-plane is nondegenerate. We shall show that the wave function (4.69) coincides in this case with the wave function given by the canonical operator on a Lagrangian manifold with complex germ [25]. In the general case, this is also true: we can consider the Fourier transform of function (4.69) with respect to s coordinates and compute in a similar way the asymptotics of this Fourier transform. On the other hand, the integral formula (4.69) already gives us a well-defined wave function, therefore, expression (4.69) can be considered as an alternative definition of canonical operator on a Lagrangian manifold with complex germ. Lemma 4.3.3. Let the vectors 𝜕Q 𝜕Q ̄ ..., (τ), (τ)̄ 𝜕τ1 𝜕τk be linearly independent, and let f ∈ Sk,n .

222 | 4 Complex germ method in the Fock space 1.

If condition (4.59) is not satisfied, then ε→0 1 ε Ψ (x) 󳨀→ 0 εm

2.

∀m.

(4.71)

If condition (4.59) is satisfied in a neighborhood of τ,̄ then Ψε (x) = Xε (τ,̄ ξ )

1

ε

n−k 4

i

e ε (S(τ)+P(τ)(x−Q(τ))) , ̄

n

ε→0

Xε (τ,̄ ξ ) 󳨀→ X(τ,̄ ξ ) = ∫ dα f (τ,̄ ξ − ∑ l=1

ℝk k

× exp{i ∑ αl l=1

̄

̄

(4.72)

𝜕Q α) 𝜕τl l

𝜕P i k 𝜕P 𝜕Q ξ − ∑ αl αm }, 𝜕τl 2 l,m=1 𝜕τl 𝜕τm

ξ ∈ ℝn . (4.73)

Remark 4.3.3. 1. Formula (4.73) can be represented in the form k

X(τ,̄ ξ ) = ∫ dα exp{i ∑ αl ( ℝk

2.

l=1

𝜕P 𝜕Q 1 𝜕 ξ− )}f (τ,̄ ξ ). 𝜕τl 𝜕τl i 𝜕ξ

(4.74)

Formula (4.73) implies that integral formula (4.69) indeed yields wave functions coinciding with the ones constructed in the method of a complex germ on an isotropic manifold, because the wave function (4.73) does not vanish only if its argument x has the distance of order √ε from the projection of the isotropic manifold onto the Q-plane. In the proof of Lemma 4.3.3, we shall use the following lemma.

Lemma 4.3.4. Let Φ(α, √ε) be a real smooth function uniformly bounded in ε together with the derivatives of all orders with respect to α for τ̄ + α√ε ∈ supp f , α ∈ ℝk , √ε ∈ ℝ. Then the function g(α, √ε) = eiΦ(α,√ε) f (τ̄ + α√ε,

x − Q(τ̄ + α√ε) ) √ε

and its derivatives satisfy the relation 󵄨󵄨 󵄨󵄨 𝜕m 󵄨󵄨 󵄨 g(α, √ε)󵄨󵄨󵄨 ≤ ϕm (α), 󵄨󵄨 󵄨󵄨 𝜕αi ⋅ ⋅ ⋅ 𝜕αi 󵄨󵄨 1 m

m = 0, . . . , ∞,

where functions ϕm are such that ∫ ϕm (α) dα converge.

(4.75)

4.3 Superposition of wave packets in quantum mechanics | 223

Proof. Let us show that the functions 𝜕m g M (α, √ε)(|α|2 + 1) 𝜕αi1 ⋅ ⋅ ⋅ 𝜕αim are uniformly bounded, i. e., do not exceed certain constant independent on √ε, but possibly dependent on M and m. This will imply the statement of the lemma. By uniform boundedness in √ε of derivatives of all orders of the function Φ in the ball |τ| < a covering the support of the function f (⋅, α) for all α, by infinite differentiability of the function f , and by boundedness of its derivatives in the ball |τ| < α. It suffices to show that 󵄨󵄨 x − Q(τ̄ + α√ε) 󵄨󵄨󵄨󵄨 M 󵄨󵄨 2 )󵄨󵄨 < C 󵄨󵄨(|α| + 1) f (τ̄ + α√ε, 󵄨󵄨 󵄨󵄨 √ε

(4.76)

for any function f (τ, ξ ) rapidly decreasing as a function of ξ for fixed τ and vanishing for |τ| > a. The Lagrange theorem and formula (4.70) imply that k x − Q(τ̄ + α√ε) 𝜕Q (1) (τ )αa , = ξ (ε) + ∑ √ε 𝜕τ a a=1

where |τ(1) | < a. By linear independence of the vectors 𝜕Q/𝜕τi in a neighborhood of the point τ and by rapid decrease of f , this implies the estimate (4.75). Lemma 4.3.4 is proved. Proof of Lemma 4.3.3. For proof of Lemma 4.3.3, note first that if B is any ball covering the point τ, then integration over ℝn \ B in formula (4.69) yields, by rapid decrease of f , a wave function satisfying property (4.71). Hence, without loss of generality one can assume that the function f in formula (4.69) vanishes outside the ball B. Further, consider the following change of variables in the integral (4.69): τ = τ̄ + α√ε,

α ∈ ℝk .

Then formula (4.72) holds for Xε (τ, ξ ) = ∫ dα exp{

i n 𝜕S 𝜕Q √ε), ̄ (τ))}g(α, ∑ α ( (τ)̄ − P(τ)̄ √ε l=1 l 𝜕τl 𝜕τl

where g(α, √ε) is of the form (4.75), and k 1 𝜕S ̄ a √ε) Φ(α, √ε) = {(S(τ̄ + α√ε) − S(τ) − ∑ (τ)α ̄ ε 𝜕 a=1 τa k

𝜕Q ̄ a √ε) (τ)α ̄ 𝜕 a=1 τa

̄ + P(τ)(Q( τ)̄ − Q(τ̄ + α√ε) − ∑

(4.77)

224 | 4 Complex germ method in the Fock space

̄ τ)̄ − Q(τ̄ + α√ε))} + (P(τ̄ + α√ε) − P(τ))(Q( ̄ + √εξ (√ε)(P(τ̄ + α√ε) − P(τ)). Smoothness of functions S, P, Q implies that function Φ satisfies the conditions of Lemma 4.3.4. Assume that condition (4.59) is not satisfied. Then for some i 𝜕S 𝜕Q −P ≠ 0. 𝜕τi 𝜕τi By Lemma 4.3.4, the integral ∫ dα exp{

𝜕m i k 𝜕Q 𝜕S −P )} g(α, √ε) ∑ αl ( √ε l=1 𝜕τl 𝜕τl (𝜕αi )m

is bounded uniformly in ε; after integrating by parts we obtain that property (4.71) holds. The second statement of Lemma 4.3.3 follows from the conditions f (τ̄ + α√ε,

k x − Q(τ̄ + α√ε) ε→0 𝜕Q αa , ) 󳨀→ f (τ,̄ ξ − ∑ √ε 𝜕τ a a=1 ε→0

k

Φ(α, √ε) 󳨀→ ∑ αl l=1

𝜕P i k 𝜕P 𝜕Q ξ − ∑ αl αm , 𝜕τl 2 l,m=1 𝜕τl 𝜕τm

and from the Lebesgue theorem. Lemma 4.3.3 is proved. Remark 4.3.4. We see that, as in the previous case, after the change of function f in formula (4.69) by a function Δf satisfying the condition k

∫ dα exp{i ∑ αl ( ℝk

l=1

𝜕P 𝜕Q 1 𝜕 ξ− )}Δf (τ,̄ ξ ) = 0, 𝜕τl 𝜕τl i 𝜕ξ

(4.78)

the wave function (4.69) changes by a small amount. Hence, we can introduce an equivalence relation on 𝒮 (ℝn+k ), calling two functions equivalent if their difference satisfies condition (4.78). Definition 4.3.1. A smooth surface X(τ) = (P(τ), Q(τ)) ∈ ℝ2n , τ ∈ ℝk , is called an isotropic manifold diffeomorphic to ℝk if (a) X(τ1 ) ≠ X(τ2 ) for τ1 ≠ τ2 ; (b) for any τ ∈ ℝk , the vectors 𝜕X/𝜕τi are linearly independent; (c) condition (4.60) holds.

4.3 Superposition of wave packets in quantum mechanics | 225

4.3.4 Norm of the wave function corresponding to an isotropic manifold In this section, we shall consider the asymptotics of the inner product of two wave functions Ψε1,2 (x) = ∫

dτ εi S(τ) ε e (KP(τ),Q(τ) f1,2 (τ))(x), εk/4

f1,2 (τ) ∈ 𝒮 (ℝk ),

(4.79)

without the assumption that there are no focal points. In formula (4.79) (P(τ), Q(τ)) is an isotropic manifold, and the smooth function S(τ) satisfies condition (4.59). By the relation, i

(KPε1 ,Q1 f1 , KPε1 ,Q1 f2 ) = e 2ε (P1 +P2 )(Q1 −Q2 )

i

× ∫ dξ f1∗ (ξ )e √ε

[(P2 −P1 )ξ −(Q2 −Q1 ) 1i

𝜕 𝜕ξ

]

f2 (ξ ),

(4.80)

which follows directly from the definition of the canonical operator (1.5). We have (Ψε1 , Ψε2 ) = ∫

dτ1 dτ2 εi Φ(τ1 ,τ2 ) e εk/2 i

× ∫ dξ f1∗ (τ1 , ξ )e √ε

[(P(τ2 )−P(τ1 ))ξ −(Q(τ2 )−Q(τ1 )) 1i

𝜕 𝜕ξ

]

f2 (τ2 , ξ ),

(4.81)

where 1 Φ(τ1 , τ2 ) = S(τ2 ) − S(τ1 ) + (P(τ1 ) + P(τ2 ))(Q(τ1 ) + Q(τ2 )). 2 Lemma 4.3.5. Let f1 , f2 ∈ Sk,n . Then as ε → 0 we have ε→0

(Ψε1 , Ψε2 ) 󳨀→ ∫ dτdξ ∫ dα f1∗ (τ, ξ ) k

× exp{i ∑ αl ( l=1

𝜕P 𝜕Q 1 𝜕 )}f2 (τ, ξ ). ξ− 𝜕τl 𝜕τl i 𝜕ξ

(4.82)

Proof. Let us first show that the contribution of the domain |τ1 − τ2 | > C into integral (4.81) is exponentially small. To this end, let us first note that for |τ1 − τ2 | > C and τ1 , τ2 ∈ supp f1 ∩ supp f2 one has 󵄨󵄨 󵄨2 󵄨 󵄨2 󵄨󵄨P(τ2 ) − P(τ1 )󵄨󵄨󵄨 + 󵄨󵄨󵄨Q(τ2 ) − Q(τ1 )󵄨󵄨󵄨 > C1 ; this follows from finiteness of support of functions f1 and f2 , from absence of selfintersections of the surface (P(τ), Q(τ)), and from the fact that the continuous function |P(τ2 ) − P(τ1 )|2 + |Q(τ2 ) − Q(τ1 )|2 achieves its minimal value. Exponential smallness of the expression under the integral follows from the following Lemma 4.3.6.

226 | 4 Complex germ method in the Fock space Lemma 4.3.6. Let 󵄨2 󵄨2 󵄨 󵄨󵄨 󵄨󵄨P(τ1 , τ2 )󵄨󵄨󵄨 + 󵄨󵄨󵄨Q(τ1 , τ2 )󵄨󵄨󵄨 ≥ C1 . Then for any M we have 󵄨󵄨 i 1 󵄨󵄨󵄨󵄨 [P(τ1 ,τ2 )ξ −Q(τ1 ,τ2 ) 1i 𝜕ξ𝜕 ] 󵄨󵄨 ∗ √ε dξ f (τ , ξ )e f (τ , ξ ) ∫ 󵄨 󵄨󵄨 ≤ C2 . 1 2 2 1 󵄨󵄨 εM 󵄨󵄨󵄨

(4.83)

Proof. Denote by f2̃ the Fourier transform of the function f2 with respect to the variable ξ, f2 (τ2 , ξ ) = ∫ dπ f2̃ (τ2 , π)e−iπξ . Then the expression in the left-hand side of formula (4.83) takes the form 󵄨󵄨 i 1 󵄨󵄨󵄨󵄨 󵄨 (Pξ −Qπ) ∗ f1 (τ1 , ξ )f2̃ (τ2 , π)e−iπξ 󵄨󵄨󵄨 󵄨󵄨∫ dξdη e √ε M 󵄨󵄨 ε 󵄨󵄨 󵄨 󵄨󵄨 (|P 2 | + |Q2 |) i 󵄨 (Pξ −Qπ) ∗ ̃ (τ , π)e−iπξ 󵄨󵄨󵄨󵄨 √ε dξdη e f (τ , ξ ) f ≤ 󵄨󵄨󵄨 ∫ 1 2 2 1 󵄨󵄨 󵄨󵄨 C M εM 󵄨 1 M 2 2 󵄨󵄨 󵄨 i 𝜕 1 󵄨󵄨 󵄨 (Pξ −Qπ) 𝜕 ( 2 + 2 ) f1∗ (τ1 , ξ )f2̃ (τ2 , π)e−iπξ 󵄨󵄨󵄨 ≤ M 󵄨󵄨󵄨∫ dξdη e √ε 󵄨󵄨 𝜕ξ 𝜕π C1 󵄨󵄨 M 󵄨󵄨󵄨 𝜕2 󵄨󵄨󵄨 1 𝜕2 ≤ M ∫ dξdπ 󵄨󵄨󵄨( 2 + 2 ) f1∗ (τ1 , ξ )f2̃ (τ2 , π)e−iπξ 󵄨󵄨󵄨. 𝜕π C1 󵄨󵄨 𝜕ξ 󵄨󵄨

(4.84)

Here, the arguments τ1 , τ2 of the functions P, Q are omitted. By smoothness of the functions f1 and f2̃ , by their finiteness with respect to the first argument and rapid decrease with respect to the second argument, expression (4.84) is bounded uniformly in τ1 , τ2 . Lemma 4.3.6 is proved. Consider the change of variables τ2 = τ,

τ1 = τ + α√ε,

and note that by Lemma 4.3.6 it suffices to consider integral (4.81) over the set |τ2 −τ1 | < C, i. e., to multiply the expression under the integral in formula (4.81) by θ(C − |α|√ε). Let us use the following lemma. Lemma 4.3.7. The following estimate holds: 󵄨󵄨 i 󵄨󵄨 [P(τ+α√ε)−P(τ)]ξ ∗ 󵄨󵄨∫ dξ f1 (τ + α√ε, ξ )e √ε 󵄨󵄨 󵄨󵄨 Q(τ + α√ε) − Q(τ) 󵄨 × f2 (τ, ξ − )θ(C − |α|√ε)󵄨󵄨󵄨 < ϕ(α), 󵄨󵄨 √ε where ϕ(α) is an integrable function.

(4.85)

4.3 Superposition of wave packets in quantum mechanics | 227

By finiteness of functions f1 and f2 with respect to the first argument and by the Lebesgue theorem, this lemma will imply the statement of Lemma 4.3.5, because as ε → 0, ε→0 1 Φ(τ + α√ε, τ) 󳨀→ 0, ε

and i

f1∗ (τ + α√ε, ξ )e √ε

[(P(τ+α√ε)−P(τ))ξ −(Q(τ+α√ε)−Q(τ)) 1i k

ε→0

× f2 (τ, ξ ) 󳨀→ f1∗ (τ, ξ ) exp{i ∑ αl ( l=1

𝜕 𝜕ξ

]

𝜕Q 1 𝜕 𝜕P ξ− )}f2 (τ, ξ ). 𝜕τl 𝜕τl i 𝜕ξ

Proof of Lemma 4.3.7. The expression in the left-hand side of formula (4.85) can be represented in the form 󵄨󵄨 k 󵄨󵄨 󵄨󵄨∫ dξ f ∗ (τ + α√ε, ξ ) exp{i ∑ αl 𝜕P (τ)} ̃ 1 󵄨󵄨 𝜕τl 󵄨󵄨 l=1 k

× f2 (τ, ξ − ∑ l=1

󵄨󵄨 󵄨 𝜕Q ̃ l )θ(C − |α|√ε)󵄨󵄨󵄨󵄨, (τ)α 󵄨󵄨 𝜕τl 󵄨

|τ̃ − τ| < C.

(4.86)

For C > |α|√ε, one has the relation k

∑(( l=1

2

2

𝜕P 𝜕Q ) + ( ) )(τ)̃ ≥ C̃ > 0. 𝜕τl 𝜕τl

By Lemma 4.3.6, function (4.86) does not exceed certain function rapidly decreasing as |α| → ∞. This implies the statement of Lemma 4.3.7. Lemma 4.3.7, and hence, Lemma 4.3.5, is proved. Lemma 4.3.5 implies the following corollary. ε→0

Corollary 4.3.2. (Ψε , Ψε ) 󳨀→ 0 if and only if k

∫ dα exp{i ∑ αl ( l=1

𝜕Q 1 𝜕 𝜕P ξ− )}f (τ, ξ ) = 0. 𝜕τl 𝜕τl i 𝜕ξ

(4.87)

Proof. Part “if” follows from formula (4.82). For proof of part “only if” note that the ε→0

Cauchy–Schwarz inequality implies that (Ψε1 , Ψε ) 󳨀→ 0. Then formula (4.82) implies that k

∫ dτ dξ f1 (τ, ξ ) ∫ dα exp{i ∑ αl ( l=1

𝜕P 𝜕Q 1 𝜕 ξ− )}f (τ, ξ ) = 0 𝜕τl 𝜕τl i 𝜕ξ

for any f1 ∈ Sk,n . This implies formula (4.87). Corollary 4.3.2 is proved.

(4.88)

228 | 4 Complex germ method in the Fock space

4.4 Canonical operator corresponding to a Lagrangian manifold with a complex germ In this section, we shall introduce the notion of complex germ on an isotropic manifold and consider the canonical operator corresponding to the complex germ on a manifold, similar to the procedure from Sections 1.3, 1.5 for a germ at a point. We shall also show that the canonical operator corresponding to a Lagrangian manifold with a complex germ, indeed yields asymptotic solutions of the Cauchy problem.

4.4.1 Canonical operator corresponding to an isotropic manifold Let Γk be a k-dimensional isotropic manifold (P(τ), Q(τ)) diffeomorphic to ℝk , in 2ndimensional phase space. Denote by ℒ the linear space of functions Ψε (x), ε ∈ (0, ε0 ], x ∈ ℝn , which belong 2 to L (ℝn ) for each ε. Consider the operator 𝒦Γk : Sk,n → ℒ,

assigning to each function f (τ, ξ ) from the space Sk,n the element of ℒ of the form (4.69): (𝒦Γk )ε f (x) = ∫ ℝk

dτ ε

k+n 4

i

e ε (S(τ)+P(τ)(x−Q(τ))) f (τ,

x − Q(τ) ). √ε

(4.89)

Consider the subspace ℛ of the space ℒ consisting of all functions Ψε (x) such that 󵄩󵄩 ε 󵄩󵄩ε ε→0 󵄩󵄩Ψ 󵄩󵄩 󳨀→ 0.

(4.90)

Denote by ℒ/ℛ the quotient space of the linear space ℒ by this subspace, consisting of all equivalence classes [f ] of elements of ℒ: two elements of ℒ are called equivalent if their difference belongs to the subspace ℛ. As we have seen in the previous section, 𝒦Γk f ∈ ℛ if and only if relation (4.87) holds. Denote by S0 the subspace of the space Sk,n , consisting of all functions f for which relation (4.87) holds, and by Sk,n /S0 the corresponding quotient space. Consider the operator 𝕂Γk : Sk,n /S0 → ℒ/ℛ, assigning to each equivalence class from Sk,n /S0 an element of ℒ/ℛ in the following way: one chooses a representative of the equivalence class from Sk,n /S0 , which is

4.4 Lagrangian manifold with complex germ |

229

mapped by the operator 𝒦Γk into a representative of the equivalence class from ℒ/ℛ, 𝕂Γk [f ] = [𝒦Γk f ],

(4.91)

where f is a representative of the equivalence class [f ] ∈ Sk,n /S0 . Definition (4.91) does not depend on the choice of a representative of the class by Lemma 4.3.5. Definition 4.4.1. The operator 𝕂Γk is called the canonical operator corresponding to the isotropic manifold Γk . Introduce the inner product in the quotient space Sk,n /S0 in the following way: ([f1 ], [f2 ]) = ∫ dτ dξ f1∗ (τ, ξ ) ∫ dα × exp{iαa (

𝜕P 𝜕Q 1 𝜕 ξ− )}f2 (τ, ξ ). 𝜕τa 𝜕τa i 𝜕ξ

(4.92)

Definition of the subspace S0 implies that definition of the inner product (4.92) is correct, i. e., does not depend on the choice of representatives f1 and f2 of the equivalence classes [f1 ] and [f2 ]. The results of the previous section mean that for any Ψε1 ∈ 𝕂Γk [f1 ], Ψε2 ∈ 𝕂Γk [f2 ] one has the relation ε→0

(Ψε1 , Ψε2 ) 󳨀→ ([f1 ], [f2 ]).

(4.93)

′ Denote by Sk,n the space of linear functionals on Sk,n , whose elements will also be called distributions. For the study of the structure of the quotient space Sk,n /S0 , it is convenient to consider the map JΓk

′ , Sk,n /S0 󳨀→ Sk,n

assigning to each equivalence class [f ] a linear functional JΓk [f ] mapping an element f1 ∈ Sn,k into ∫ dτ dξ dα f1 (τ, ξ ) exp{iαa (

𝜕Q 1 𝜕 𝜕P ξ− )}f (τ, ξ ). 𝜕τa 𝜕τa i 𝜕ξ

This formula can be also written out in the form JΓk [f ](τ, ξ ) = ∫ dα exp{iαa (

𝜕P 𝜕Q 1 𝜕 ξ− )}f (τ, ξ ), 𝜕τa 𝜕τa i 𝜕ξ

(4.94)

where the equality is understood as equality of distributions. By definition of equivalence class, the distribution (4.94) does not depend on the choice of representative in an equivalence class. Hence, the map JΓk

′ Sk,n /S0 󳨀→ Im JΓk ⊂ Sk,n

230 | 4 Complex germ method in the Fock space is a one-to-one map from the quotient space Sk,n /S0 onto certain subspace of the space ′ of distributions Sk,n . It turns out that elements of the set Im JΓk are distributions g(τ, ξ ), belonging for each τ to the space 𝒮 ′ (ℝn ) and continuously depending on τ. One has the following lemma. Lemma 4.4.1. Let f ∈ Sk,n , χ ∈ 𝒮 (ℝn ). Then the integral ∫ dα ∫ dξ χ(ξ ) exp{iαa (

𝜕Q 1 𝜕 𝜕P ξ− )}f (τ, ξ ) 𝜕τa 𝜕τa i 𝜕ξ

(4.95)

converges, is a smooth function of τ, and defines a continuous linear functional of the element χ ∈ 𝒮 (ℝn ). Proof. First of all, note that integral (4.4.1) can be represented as integral over α of the expression ∫ dξ dπ exp{iαa (

𝜕P 𝜕Q ξ− π)} 𝜕τa 𝜕τa

i 𝜕P 𝜕Q × exp{iπξ − αa α }χ(ξ )f ̃(τ, π), 2 𝜕τa 𝜕τb b

(4.96)

where f ̃ is the Fourier transform of the function f with respect to the second argument, f (τ, ξ ) = ∫ dπ eiπξ f ̃(τ, π). By conditions of the lemma, integral (4.96) converges, and by the Lebesgue theorem one can go to the limit in this integral with respect to τ. The arguments similar to proof of Lemma 4.3.6 show that integral (4.96) does not exceed C/(|α|2 + 1)M for any M, where C is a constant independent of α. Hence, integral of expression (4.96) with respect to α converges and is continuous in τ. Similarly, one checks infinite differentiability of expression (4.4.1) with respect to τ. Let us check continuity of functional (4.4.1) with respect to χ. Let a sequence χn converge to zero in the topology of 𝒮 (ℝn ), i. e., 󵄨󵄨 󵄨󵄨 n→∞ 𝜕l 󵄨 󵄨 maxn 󵄨󵄨󵄨ξi1 ⋅ ⋅ ⋅ ξim χn (ξ )󵄨󵄨󵄨 󳨀→ 0 󵄨󵄨 𝜕ξj1 ⋅ ⋅ ⋅ 𝜕ξjl ξ ∈ℝ 󵄨󵄨

(4.97)

for any i1 , . . . , im , j1 , . . . , jl = 1, . . . , n. Let us show that the quantity (4.4.1) equal to integral of the quantity (4.96) over α, also tends to zero. The product of the quantity (1 + αa (

M

𝜕Q 𝜕Q 𝜕P 𝜕P + )α ) 𝜕τa 𝜕τb 𝜕τa 𝜕τb b

4.4 Lagrangian manifold with complex germ |

231

by integral (4.96) does not exceed 𝜕P 𝜕Q i α }χ(ξ )f ̃(τ, π) ∫ dξ dπ exp{iπξ − αa 2 𝜕τa 𝜕τb b × (1 −

𝜕Q 𝜕2 𝜕P 𝜕2 ξ− π)}. + ) exp{iαa ( 𝜕τa 𝜕τa 𝜕ξ 2 𝜕π 2

Integrating by parts and using property (4.97), we obtain that the absolute value of quantity (4.96) does not exceed an (1 + αa (

M

𝜕P 𝜕P 𝜕Q 𝜕Q + )α ) , 𝜕τa 𝜕τb 𝜕τa 𝜕τb b

n→∞

an 󳨀→ 0.

(4.98)

𝜕P 𝜕P 𝜕Q 𝜕Q Since the matrix 𝜕τ + 𝜕τ is positive definite, integral of (4.98) does not exceed a 𝜕τb a 𝜕τb const ⋅ an and tends to zero. This implies continuity of the functional with respect to χ. Lemma 4.4.1 is proved.

One has also the following property of elements. Lemma 4.4.2. For g ∈ Im JΓk , (

𝜕P 𝜕Q 1 𝜕 ξ− )g = 0, 𝜕τa 𝜕τa i 𝜕ξ

a = 1, . . . , k.

(4.99)

Proof. For the proof, it suffices to shift the parameter α in formula (4.94) by a constant.

Note also that we can define the inner product of two elements g1 , g2 ∈ Im JΓk as (g1 , g2 ) = (JΓ−1k g1 , JΓ−1k g2 ). Below we shall consider the operator ′ 𝕂Γk JΓ−1k : Im JΓk ⊂ Sk,n → ℒ/ℛ.

4.4.2 Case of a Lagrangian manifold of full dimension Let us now proceed to the study of the structure of the set Im JΓk , introduce parametrization of this set, and establish the relation between operator (4.91) and the traditional canonical operator. Let us start with the case of n-dimensional isotropic surface, i. e., a Lagrangian manifold [24]. The general case will be considered below.

232 | 4 Complex germ method in the Fock space Let us first consider the case when the Lagrangian manifold has a one-to-one projection onto the Q-plane, i. e., det(

𝜕Qi ) ≠ 0, 𝜕τj

i, j = 1, . . . , n.

(4.100)

𝜕Qi . 𝜕τj

(4.101)

Denote Bij =

𝜕Pi , 𝜕τj

Cij =

′ Lemma 4.4.3. Let g ∈ Im JΓn ⊂ Sk,n . Then

g(τ, ξ ) =

φ(τ) i n exp{ ∑ ξj (BC −1 )jl (τ)ξl }, √det C 2 j,l=1

(4.102)

where φ ∈ 𝒟(ℝn ). Remark 4.4.1. 1. The factor (det C)−1/2 is extracted for simplification of the formulas below. 2. In this case, the matrix BC −1 is real. Proof. By property (4.99), n

∑(ξl Blj − l=1

1 𝜕 C )g = 0. i 𝜕ξl lj

This implies that the function g, up to a factor depending only on τ, reads exp{

i n ∑ ξ (BC −1 )jl (τ)ξl }. 2 j,l=1 j

Smoothness of function φ follows from formula (4.94) and Lemma 4.4.1, and finiteness follows from finiteness of the function f with respect to τ. Hence, φ ∈ 𝒟(ℝn ). Lemma 4.4.3 is proved. Remark 4.4.2. In the case when the manifold Γn has focal points in which condition (4.100) does not hold, the distribution g has at these points the form different ̇ from (4.102). Indeed, consider the one-dimensional case. Then for Q(τ) ≠ 0 one has formula (4.102), g(τ, ξ ) =

χ(τ) √|Q|̇

i Ṗ

ξ2

e 2 Q̇ ,

4.4 Lagrangian manifold with complex germ |

233

̇ and for Q(τ) = 0 one has g(τ, ξ ) = χ(τ)√

2π δ(ξ ). |P|̇

Here, the function χ is not continuous. The boundary conditions for it can be obtained using Lemma 4.4.1: the distribution g is continuous with respect to the parameter τ. Since as Q̇ → 0, one has 1 √|Q|̇

e

i Ṗ 2 ξ 2 Q̇

2π iπ/4 {√ |P|̇ δ(ξ )e , 󳨀→ { 2π −iπ/4 , {√ |P|̇ δ(ξ )e

̇ Q→0

Ṗ Q̇ Ṗ Q̇

> 0, < 0,

the function χ is multiplied by eiπ/2 after passing a focal point. In the next section, we shall consider the holomorphic representation for distributions, which allows one to consider the cases of absence and presence of focal points by one and the same procedure. The statement inverse to Lemma 4.4.3 is also valid. Lemma 4.4.4. Let g have the form (4.102). Then g ∈ Im JΓn . Proof. Consider the element f ∈ Sn,n which is a representative of an equivalence class [f ] ∈ Sn,n /S0 and which has the form f (τ, ξ ) = c exp{−

1 n ∑ ξ β ξ }, 2 l,s=1 l ls s

where c and βls are smooth finite functions of τ, βls = βsl , and the real part of the matrix βls is positive definite. Formula (4.94) implies that i

T

1

T

JΓn [f ](τ, ξ ) = c ∫ dα eiξa Bab αb − 2 αa Bab Cbc αc e− 2 (ξa −αb Cba )βac (ξc −Ccd αd ) = c√

i −1 (2π)n e 2 ξa (BC )ab ξb ; T + iB C)

det(C T βC

(4.103)

note that by isotropy of the manifold, the matrix BT C is symmetric. Formula (4.103) implies that for c = φ√

det(C T βC + iBT C) , (2π)n det C

(4.104)

function (4.103) coincides with function (4.102). Hence, any function of the form (4.102) belongs to Im JΓn . Lemma 4.4.4 is proved. Let us now find the inner product of elements (4.102) of the space Im JΓn .

234 | 4 Complex germ method in the Fock space Lemma 4.4.5. Let g1 , g2 ∈ Im JΓn and g1,2 =

i −1 φ1,2 e 2 ξ (BC )ξ . √det C

Then (g1 , g2 ) = ∫ dτ φ∗1 (τ)φ2 (τ).

(4.105)

Proof. According to formula (4.92), the inner product (g1 , g2 ) equals 1

(g1 , g2 ) = ∫ dτ dξ c1∗ e− 2 ξa βab ξb = ∫ dτ

∗

i −1 φ2 e 2 ξa (BC )ab ξb √det C

c1∗ φ2 (2π)n (2π)n det C √ = ∫ dτ φ2 c1∗ √ , ∗ −1 √det C det(β − iBC ) det(C T β∗ C − iC T B)

where c1 is related to φ1 by formula (4.104). This implies formula (4.105). Lemma 4.4.5 is proved. Thus, in this case (k = n) one can uniquely assign to an element of the quotient space Sk,n /S0 a function φ ∈ 𝒟(ℝn ); denote the corresponding one-to-one map by IΓn , IΓn : Sn,n /S0 → 𝒟(ℝn ),

(4.106)

which is uniquely determined by the property IΓn [f ](τ) 2i ξa (BC−1 )ab ξb . e √det C

JΓn [f ](τ, ξ ) =

(4.107)

Consider the operator 𝕂Γn IΓ−1n : 𝒟(ℝn ) → ℒ/ℛ. Lemma 4.3.3 implies that i

(𝒦Γn f )ε (x) = e ε S(τ) (JΓn f )(τ,̄ 0) + O(√ε), ̄

where Q(τ)̄ = x. This implies that the function Ψε (x) =

i φ(τ)̄ ̄ e ε S(τ) √det C

belongs to the equivalence class 𝕂Γn IΓ−1n φ ∈ ℒ/ℛ. Thus, in the case when the Lagrangian manifold has a one-to-one projection onto the Q-plane, the operator 𝕂Γn IΓ−1n coincides with the traditional canonical operator [24].

4.4 Lagrangian manifold with complex germ |

235

In the case of a general Lagrangian manifold, define the operator (4.106) as follows: IΓ−1n φ = [f ],

f (τ, ξ ) = φ(τ)√

det(Cβ + iB) − 21 ξa βab ξb ; e (2π)n

in the case of absence of focal points, it coincides with the operator given above. Hence, the canonical operator 𝕂Γn IΓ−1n coincides with the traditional canonical operator at each chart covering the manifold Γn and diffeomorphically projecting onto the Q-plane. It is not difficult to check by passing to the Fourier transform that this statement is true also in the case when a chart projects onto the plane P1 = ⋅ ⋅ ⋅ = Ps = Qs+1 = ⋅ ⋅ ⋅ = Qn = 0. Therefore, in the general case the operator 𝕂Γn IΓ−1n also coincides with the traditional canonical operator [24]. 4.4.3 Holomorphic representation of distributions in 𝒮 ′ For the study of the general case, it is convenient to introduce holomorphic presentation for distributions. Note that presentation of vectors of Hilbert spaces in the form of analytic functionals is often used in theory of second quantization [6] and in quantum field theory [124]. It turns out that in such presentation many difficulties related to focal points do not arise. Let us assign to each element f ∈ 𝒮 ′ being a linear functional on 𝒮 , its value on 2 2 √ the function e 2zξ −ξ /2 multiplied by e−z /2 : fz (z) = (𝒵 f )(z) = ∫ dξ e z = (z1 , . . . , zn ), The factor e−z

2

/2

√2zξ −ξ 2 /2−z 2 /2

f (ξ ),

zi ∈ ℂ.

(4.108)

is introduced to simplify formulas below.

Lemma 4.4.6. Function (4.108) is analytic. Proof. Let us show that the function fz (z)ez g(z + Δz) − g(z) = ∫ dξ 2Δzξe + ∫ dξ (e

2

/2

√2zξ −ξ 2 /2

√2Δzξ

= g(z) is analytic. We have

f (ξ )

− 1 − √2Δzξ )e

√2zξ −ξ 2 /2

f (ξ ),

Δz ∈ ℂn .

The first summand in this formula has the form aΔz, a ∈ ℂn , and the second one is of order o(|Δz|) by continuity of the functional f . This implies the statement of Lemma 4.4.6.

236 | 4 Complex germ method in the Fock space Let us now show that if fz = gz then f = g. To this end, it suffices to prove the following lemma. Lemma 4.4.7. Let 𝒵 f = 0. Then f = 0. Proof. The conditions of the lemma imply that 2

fz (ia)e−a /2 = ∫ dξeiaξ f (ξ )e−ξ

2

/2

(4.109)

vanishes. But the quantity (4.109) considered as a distribution of a, coincides with the Fourier transform of the distribution f (ξ )e−ξ

2

(4.110)

/2

from the space 𝒮 ′ (ℝn ). Hence, distribution (4.110) also vanishes. Hence, its value on any infinitely differentiable finite function equals zero, and since the operator of mul2 tiplication by eξ /2 maps 𝒟(ℝn ) to 𝒟(ℝn ), we have ∫ φ(ξ )f (ξ )dξ = 0,

φ ∈ 𝒟(ℝn ).

(4.111)

Since the set 𝒟(ℝn ) is dense in 𝒮 (ℝn ) (see, for instance, [18]), by continuity of the functional f we obtain that its value on any element of 𝒮 equals zero, i. e., f = 0. Lemma 4.4.7 is proved. Example 4.4.1. The function g(τ, ξ ) =

χ(τ) √|Q|̇

i Ṗ

ξ2

e 2 Q̇ ,

considered in the previous section, corresponds to the following function gz (z): χ(τ)

̇ 2π ( ̇ Q Ṗ − 21 )z 2 e Q−i . ̇ Q̇ √|Q|̇ 1 − P/

From this, one sees directly that the passage of Q̇ through zero yields change of the phase of the function by π/2.

4.4.4 General case In this section, we shall consider the structure of the quotient space Sk,n /S0 in the general case. Let us first find out what analytical in z functions g(τ1 , . . . , τk , z1 , . . . , zn ) belong to the space Im 𝒵 JΓk . Lemma 4.4.7 implies that different equivalence classes [f ] ∈ Sk,n /S0 correspond to different functions 𝒵 JΓk [f ]. Hence, the functions 𝒵 JΓk [f ]

4.4 Lagrangian manifold with complex germ |

237

yield a parameterization of the quotient space Sk,n /S0 . Thus, we can consider one more realization of the canonical operator 𝕂Γk (𝒵 JΓk )−1 : Im 𝒵 JΓk → ℒ/ℛ. Let us prove several auxiliary statements. Let us assign a function Φ(τ) = (Φ1 (τ), . . . , Φn (τ)) of the form Φl (τ) =

Ql (τ) + iPl (τ) √2

(4.112)

to an isotropic manifold (P(τ), Q(τ)), τ ∈ ℝk , in the phase space. Lemma 4.4.8. Let a, b ∈ ℂn , f ∈ 𝒮 (ℝn ). Then a−b

eaz−b 𝜕z fz (z)[e √2 𝜕

ξ − a+b √2

𝜕 𝜕ξ

f ]z .

Proof. We have ab

eaz−b 𝜕z fz (z) = ∫ dξeaξ − 2 e−b 𝜕z e 𝜕

After the change ξ = η − ∫ dη e

𝜕

a+b , √2

this expression becomes

2 +η2

η a−b +b √2

√2zη− z

2

e

2 −a2 4

f (η −

2

2

√2zξ − ξ − z 2 2

f (ξ ).

a−b a+b η− a+b 𝜕 ) = [e √2 √2 𝜕η f ]z (z). √2

The proof of Lemma 4.4.8 is complete. Corollary 4.4.1. Let f ∈ Sk,n . The following relations hold: JΓk [f ]z (τ, z) = ∫ dα exp{αa (

𝜕Φ∗ 𝜕 𝜕Φ z− )}fz (τ, z), 𝜕τa 𝜕τa 𝜕z

a = 1, k.

Proof. To prove this corollary, it suffices to apply Lemma 4.4.8 for a=

𝜕Φ , 𝜕τa

b=

𝜕Φ∗ , 𝜕τa

a = 1, k.

Corollary 4.4.2. The following relation holds: (

𝜕Φ 𝜕Φ∗ 𝜕 z− )J k [f ]z (τ, z) = 0. 𝜕τa 𝜕τa 𝜕z Γ

Proof. This assertion immediately follows from Lemma 4.4.2. Lemma 4.4.9. 1. The vectors

𝜕Φ , 𝜕τa

a = 1, k, are linearly independent.

(4.113)

238 | 4 Complex germ method in the Fock space 2.

There exists a real nondegenerate k × k matrix λab such that 𝜕Φ = λab χ (b) 𝜕τa

(4.114)

and the vectors χ (b) are mutually orthogonal, ‖χ (b) ‖ = 1. 3.

det λ = √det 𝜕Φ 𝜕τ

∗

a

𝜕Φ . 𝜕τb

Proof. Assume the converse. Then, for some numbers ca ∈ ℂ not all of which are zero, ∗ 𝜕Φ 𝜕Φ we have ca 𝜕τ = 0, i. e., cb∗ 𝜕Φ c = 0. Since the manifold (P(τ), Q(τ)) is isotropic, it 𝜕τb 𝜕τa a a follows that: cb∗ (

𝜕Q 𝜕Q 𝜕P 𝜕P + )c = 0. 𝜕τb 𝜕τa 𝜕τb 𝜕τa a

This contradicts the requirement that the manifold ΓK does not have singular points. The first assertion of Lemma 4.4.9 is proved. To prove the second assertion, we apply the usual orthogonalization process χ (j) = Since the matrix k

𝜕Φ∗ 𝜕Φ 𝜕τa 𝜕τb

j−1

𝜕Φ 𝜕Φ (i) − ∑ χ (i)∗ χ . 𝜕τj i=1 𝜕τj

is real, which in turn follows from the fact that the manifold

−1 𝜕Φ for some real nondegenerate matrix λ−1 . This Γ is isotropic, we have χ (a) = λab 𝜕τb implies the second assertion of the lemma. The third assertion follows from the relation:

𝜕Φ∗ 𝜕Φ ∗ ∗ = χ (c)∗ λac λbd χ (d) = λac λbd δcd = (λT λ)ab . 𝜕τa 𝜕τb The proof of Lemma 4.4.9 is complete. Consider an orthonormal basis χ (i) , i = 1, n, whose first k components have already been determined by formula (4.114). We denote z (i) = χ (i) z.

(4.115)

With each function z we associate a function of the numbers z (1) , . . . , z (n) expressed by formula (4.115). Since the quantity in the exponent in formula (4.113) has the form αa λab (z (b) −

𝜕 ), 𝜕z (b)

4.4 Lagrangian manifold with complex germ |

239

it follows that, after the change α = λ−1 β, we have dβ βa (z (a) − 𝜕(a) ) 𝜕z e fz (τ, z (1) , . . . , z (n) ). det λ

JΓk [f ]z (τ, z (1) , . . . , z (n) ) = ∫

(4.116)

Lemma 4.4.10. For the function gz analytic in z to be representable in the form (4.116), it is necessary and sufficient that 1 (a) (a)

gz (τ, z (1) , . . . , z (n) ) = e 2 z

z

y(τ, z (k+1) , . . . , z (n) ).

(4.117)

Proof. To prove the necessity, we note that (4.116) implies (z (a) − 𝜕/𝜕z (a) )gz = 0, and hence, formula (4.117) holds. Conversely, choosing f independent of z (1) , . . . , z (k) , we see that any function (4.117) can be represented in the form (4.116). The proof of Lemma 4.4.10 is complete. Now we consider the requirements that should be imposed on the function y in formula (4.117) so that gz ∈ Im JΓk . For this, we introduce several notions and prove some auxiliary assertions. Definition 4.4.2. An analytic function f (z1 , . . . , zn ) =

∞

∑

ν1 ,...,νn =0

fν1 ⋅⋅⋅νn

ν

z1 1 √ν1 !

ν

⋅⋅⋅

znn √νn !

is said to be quasipolynomial if, for any A1 , . . . , An > 0, the series ∞

A

∑

ν1 ,...,νn =0

ν1 1 ⋅ ⋅ ⋅ νnAn |fν1 ⋅⋅⋅νn |2

(4.118)

converges. Now we derive several criteria for a function to be quasipolynomial. Lemma 4.4.11. For an analytic function, n 1 ∑ f i1 ⋅⋅⋅iN zi1 ⋅ ⋅ ⋅ ziN √ N=0 N! i1 ,...,iN =1 ∞

f (z1 , . . . , zn ) = ∑

(f i1 ⋅⋅⋅iN is a number set symmetric with respect to i1 , . . . , iN ) to be quasipolynomial, it is necessary and sufficient that, for any A, the series ∞

∑ NA

N=0

converges.

n

󵄨 󵄨2 ∑ 󵄨󵄨󵄨f i1 ⋅⋅⋅iN 󵄨󵄨󵄨

i1 ,...,iN =1

(4.119)

240 | 4 Complex germ method in the Fock space Proof. It follows from the representation for the function f in the form of series that f i1 ⋅⋅⋅iN = √

ν1 ! ⋅ ⋅ ⋅ νn ! fν1 ⋅⋅⋅νN , N!

where ν1 is the number of units in the sequence i1 , . . . , iN and νn is the number of indices n in this sequence. Therefore, NA

∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨f i1 ⋅⋅⋅iN 󵄨󵄨󵄨 = (ν1 + ⋅ ⋅ ⋅ + νn )A |fν1 ⋅⋅⋅νn |2 .

i1 ,...,iN =1

Nevertheless, condition (4.119) means that ∞

∑

(ν1 + ⋅ ⋅ ⋅ + νn )A |fν1 ⋅⋅⋅νn |2 < ∞.

ν1 ,...,νn =0

(4.120)

This condition is equivalent to condition (4.118). The proof of the lemma is complete. Corollary 4.4.3. Let U be a unitary n × n matrix, and let f (z) be a quasipolynomial function. Then the function f (Uz), where (Uz)i = Uij zj , is also quasipolynomial. Proof. Indeed, the quantity (4.119) is invariant with respect to unitary transformations. Lemma 4.4.12. For a function 𝒵 f to be quasipolynomial, it is necessary and sufficient that f ∈ 𝒮 (ℝn ). The proof of Lemma 4.4.12 is based on Lemma 4.4.13. Lemma 4.4.13. The functions κn (x) = (

x − 𝜕/𝜕z 1 −x2 /2 ) e √2 n!

(4.121)

satisfy the relations 󵄨 󵄨 maxn 󵄨󵄨󵄨κn (x)󵄨󵄨󵄨 < cn1/4 . x∈ℝ

Proof. To prove the lemma, we note that ∞

∑ κn (ξ )

n=0

z2 ξ 2 zn = exp{√2zξ − − }. n! 2 2

(4.122)

4.4 Lagrangian manifold with complex germ

| 241

Therefore, κn (ξ ) = √n! ∮ C

2 2 dz √2zξ − z − ξ }, exp{ 2 2 2πiz n+1

(4.123)

where C is an arbitrary contour bypassing the origin anticlockwise. We choose the contour C as follows: z = √neiφ , φ ∈ [0, 2π). The integral (4.123) then becomes 2π

2 2iφ dφ n! √2neiφξ − ne − ξ }. √ ∫ exp{ nn 2πeinφ 2 2 0

Since the real part of the exponent is equal to n (ξ − √2n cos φ)2 n − ≤ , 2 2 2 we have n! n/2 4 󵄨󵄨 󵄨 󵄨󵄨κn (ξ )󵄨󵄨󵄨 ≤ √ n e = √2πn(1 + O(1/n)). n The proof of Lemma 4.4.13 is complete. Proof of Lemma 4.4.12. Let f ∈ 𝒮 (ℝn ). Using formulas (4.122) and (4.108), we obtain fz,ν1 ⋅⋅⋅νn = ∫ dx1 ⋅ ⋅ ⋅ dxn κν1 (x1 ) ⋅ ⋅ ⋅ κνn (xn )f (x1 , . . . , xn ). The properties f ∈ L2 (ℝn ) and ∫ dxκν2 (x) = 1 imply that the series ∞

∑

ν1 ,...,νn =0

|fz,ν1 ⋅⋅⋅νn |2

converges. Further, we note that ∫ dx1 ⋅ ⋅ ⋅ dxn κν1 (x1 ) ⋅ ⋅ ⋅ κνn (xn ) M

× (−

xn2 n x12 𝜕2 1 𝜕2 − ⋅ ⋅ ⋅ − + + ⋅ ⋅ ⋅ + − ) f (x1 , . . . , xn ) 2 𝜕x12 2 2 2 𝜕xn2

= (ν1 + ⋅ ⋅ ⋅ + νn )2 fz,ν1 ⋅⋅⋅νn .

Therefore, the series (4.120) also converges. Thus, the function fz is quasipolynomial. Further, assume that the function fz is quasipolynomial. Then the series f (x1 , . . . , xn ) =

∞

∑

ν1 ,...,νn =0

fz,ν1 ⋅⋅⋅νn κν1 (x1 ) ⋅ ⋅ ⋅ κνn (xn )

242 | 4 Complex germ method in the Fock space converges absolutely and uniformly by Lemma 4.4.13, and hence, the function f is continuous. Note that (xi f )z =

zi + 𝜕/𝜕zi fz , √2

(

z − 𝜕/𝜕zi 𝜕 f) = i fz , √2i 𝜕xi z

and since the multiplication by zi and the differentiation take quasipolynomial functions into quasipolynomial ones, we see that the function xi f ,

𝜕 f 𝜕xi

belong to L2 (ℝn ) and are continuous. Similar reasonings also hold for the functions xi1 ⋅ ⋅ ⋅ xil

𝜕 𝜕 ⋅⋅⋅ f. 𝜕xj1 𝜕xjn

Therefore, we have f ∈ 𝒮 (ℝn ). The proof of Lemma 4.4.12 is complete. Let us introduce the norm ‖fz ‖ as ‖f ‖ in the space L2 (ℝn ), where fz = 𝒵 f . We note 2 that it coincides with ∑∞ ν1 ,...,νn =0 |fν1 ⋅⋅⋅νn | . Lemma 4.4.14. The function y in formula (4.117) is quasipolynomial in z and finite and smooth in τ. Proof. Since f ∈ 𝒮 (ℝn ), the function fz in formula (4.116) is quasipolynomial. We consider the integral (4.116) for the function fz of the form fz =

∞

∑

ν1 ,...,νn =0

fν1 ⋅⋅⋅νn (τ)

z (1)ν1 z (n)νn . ,..., √ν1 ! √νn !

Calculating the integral (4.116) for each term of this series, we obtain a nonzero value only if all νa , a = 1, k, are even, i. e., νa = 2μa . We have JΓk [f ]z =

(2π)n/2 det λ × ×

∞

∑

μ1 ,...,μk ,νk+1 ,...,νn =0

1 (a) (a)

e2z

z

f2μ1 ⋅⋅⋅2μk ,νk+1 ⋅⋅⋅νn

Γ(μ1 + 1/2)2μ1 +1/2 ⋅ ⋅ ⋅ Γ(μk + 1/2)2μk +1/2 √(2μ1 )! ⋅ ⋅ ⋅ (2μk )!

z (k+1)νk+1 z (n)νn ⋅⋅⋅ . √νk+1 ! √νn !

This formula implies the assertion of Lemma 4.4.14. The proof of Lemma 4.4.14 is complete.

4.4 Lagrangian manifold with complex germ

| 243

Corollary 4.4.4. Let g ∈ Im 𝒵 JΓk . Then there exists a function f ∈ 𝒮k,n such that (a) fz is independent of z (1) , . . . , z (k) , (b) formula (4.116) holds, and this function is unique. Remark 4.4.3. 1. Property (a) of Corollary 4.4.4 can be reformulated as 𝜕Φ∗ 𝜕 f = 0. 𝜕τa 𝜕z z 2.

(4.124)

The function fz is expressed in terms of gz = JΓk [f ]z as 1/2

(

(2π)k 1 𝜕Φ −1 𝜕Φ ) fz = gz exp{− z R z}, det R 2 𝜕τa ab 𝜕τb

Rab =

𝜕Φ 𝜕Φ∗ . 𝜕τa 𝜕τb

Corollary 4.4.5. For a function g(τ, z), τ ∈ ℝk , z ∈ ℝn , to be smooth in τ and analytic in z and to belong to Im 𝒵 JΓk , it is necessary and sufficient that the following two conditions be satisfied: ∗ 𝜕 𝜕Φ z − 𝜕Φ )g = 0, (a) ( 𝜕τ 𝜕τ 𝜕z a

a

𝜕Φ −1 𝜕Φ (b) the function g exp{− 21 z 𝜕τ Rab 𝜕τ z} is quasipolynomial. a

b

Thus, we have found the function class Im 𝒵 JΓk . Corollary 4.4.1, Lemma 4.4.8, and the definition of quotient space 𝒮k,n /𝒮0 imply the following assertion. Lemma 4.4.15. Let [f ] ∈ 𝒮k,n /𝒮0 . Then there exists a representative of equivalence class f ∈ [f ] such that 𝜕 𝜕Φ (x + )f (x) = 0, 𝜕τa 𝜕x

(4.125)

and such an element is unique. Proof. Let f ′ be an arbitrary representative [f ]. Then by formula (4.108), it is associated with the function fz′ , and by formula (4.116), with JΓk [f ]z . We consider the function fz , which is uniquely determined by Corollary 4.4.4 of Lemma 4.4.14. Since it satisfies condition (4.124), we obtain condition (4.125) by Lemma 4.4.8. The proof of Lemma 4.4.15 is complete. Thus, we have found one more parametrization of the space 𝒮k,n /𝒮0 in the form of elements of 𝒮k,n satisfying condition (4.125). Now we express the inner product (4.92) in terms of the function f . It turns out that formula (4.92) becomes significantly simpler.

244 | 4 Complex germ method in the Fock space Lemma 4.4.16. If f1 and f2 satisfy condition (4.125), then the relation holds: ([f1 ], [f2 ]) = √

(2π)k det 𝜕Φ 𝜕τ

∗

a

𝜕Φ 𝜕τb

∫ dτ dξf1∗ (τ, ξ )f2 (τ, ξ ).

(4.126)

Proof. Since exp{iαa (

𝜕Q 1 𝜕 𝜕Φ ξ − 𝜕/𝜕ξ 𝜕P ξ− )} = exp{αa ( )} √2 𝜕τa 𝜕τa i 𝜕ξ 𝜕τa 1 𝜕Φ∗ 𝜕Φ 𝜕Φ∗ ξ + 𝜕/𝜕ξ ) − αa × exp{−αa ( α }, √2 𝜕τa 2 𝜕τa 𝜕τb b

it follows from property (4.125) that formula (4.92) becomes 1 𝜕Φ∗ 𝜕Φ α }. ∫ dτ dξf1∗ (τ, ξ )f2 (τ, ξ ) ∫ dα exp{− αa 2 𝜕τa 𝜕τb b This implies the assertion of the lemma. Lemmas 4.4.15 and 4.4.16 imply the following corollary. Corollary 4.4.6. Assume that the set 𝒟k,n is dense in the subspace of the space 𝒮k,n consisting of all functions f satisfying property (4.125). Then the corresponding subset [𝒟k,n ] of the quotient space 𝒮k,n /𝒮0 , which consists of all equivalence classes [f ], f ∈ 𝒟k,n , is dense in 𝒮k,n /𝒮0 . 4.4.5 Complex germ in the holomorphic representation In this section, we consider another realization of the quotient space 𝒮k,n /𝒮0 by introducing the complex germ by analogy with Section 1.5. Since the equivalence class [f ] bijectively corresponds to the distribution JΓk [f ], we shall consider the geometric interpretation of precisely this distribution rather than of the function f defined in Lemma 4.4.15, which appears as a result of an additional condition of special form (4.125). To avoid the difficulties due to focal points, we consider the distribution JΓk [f ] in the holomorphic representation. We first study the geometric interpretation of Gaussian functions with respect to z. Definition 4.4.3. A Gaussian function is defined to be a function gz of the form gz (τ, ξ ) = c(τ) exp{

1 n ∑ z M (τ)zj }, 2 i,j=1 i ij

where c ∈ 𝒟(ℝk ) and Mij (τ) = Mji (τ) is a smooth function of τ.

(4.127)

4.4 Lagrangian manifold with complex germ |

245

Lemma 4.4.17. 1. For the Gaussian function (4.127) to be quasipolynomial, it is necessary and sufficient that the norm of all operators M(τ) in ℝn with the matrix Mij (τ) do not exceed the unity: 󵄩 󵄩󵄩 󵄩󵄩M(τ)󵄩󵄩󵄩 < 1. 2.

(4.128)

For gz = 𝒵 g, the quantity ∫ dξ |g(τ, ξ )|2 becomes |c|2 1 . √π n √det(E − MM ∗ )

Proof. We note that for g(τ, ξ ) = a exp{

i n ∑ ξ β ξ }, 2 i,j=1 i ij j

where Im β > 0 and g ∈ 𝒮 (ℝn ), the function (4.108) is equal to gz = √

1 n (2π)n a exp{ ∑ zi [(E − iβ)−1 (E + iβ)]ij zj } det(E − iβ) 2 i,j=1

and, by Lemma 4.4.12, is quasipolynomial. We choose β in the form β = i(E + M)−1 (E − M). If the norm of M is less than unity, then the imaginary part of the matrix β is positive definite. Thus, under condition (4.128), the function (4.127) is quasipolynomial. Further, we note that a=√

det(E − iβ) c, (2π)n

󵄨 󵄨2 ∫ dξ 󵄨󵄨󵄨g(τ, ξ )󵄨󵄨󵄨 = |a|2 √

(2π)n , det(β − β∗ )/i

whence, using the expression for β, we obtain the second assertion of the lemma. Conversely, assume that the function (4.127) is quasipolynomial. We show that ‖M‖ < 1. Assume the contrary, i. e., assume that, for some χ (1) such that ∑ni=1 χi(1) χi(1) = 1, 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∑ χ (1) Mij χ (1) 󵄨󵄨󵄨 ≥ 1. 󵄨󵄨 i j 󵄨󵄨 󵄨󵄨i,j=1 󵄨󵄨

(4.129)

Consider the orthonormal basis {χ (i) } in ℝn such that its first component coincides with χ (1) and introduce new variables z (i) by formula (4.115). By the corollary

246 | 4 Complex germ method in the Fock space of Lemma 4.4.11, the function of the transformed arguments is also quasipolynomial. We consider this function for z (2) = ⋅ ⋅ ⋅ = z (n) . It has the form γ n √(2n)! (z (1) )2n , 2n n! √(2n)! n=0 ∞

gz = c(τ) ∑

n

where γ = ∑ χi(1) Mij χj(2) . ij=1

2n

A |γ| (2n)! By the definition of quasipolynomial function, the series ∑∞ n=0 n 22n (n!)2 must converge for any A. But this it true only for |γ| < 1, which contradicts formula (4.129). The obtained contradiction proves Lemma 4.4.17.

Corollary 4.4.7. A function gz of the form (4.127) belongs to Im 𝒵 JΓk if and only if the following two conditions are satisfied: ∗ 𝜕Φ (a) M 𝜕Φ = 𝜕τ , 𝜕τ a

a

(b) the norm of the matrix M̃ ij = Mij −

𝜕Φi −1 𝜕Φj R 𝜕τa ab 𝜕τb

is less than unity.

Proof. This corollary follows from Corollary 4.4.2 of Lemma 4.4.16 and from Lemma 4.4.17. Now we consider the geometric interpretation of Gaussian function. We fix the parameter τ. Definition 4.4.4. The subspace of the space ℝ2n over the field ℂ of the form z

n

𝒢M = {(ζ , Mζ ) | ζ ∈ ℝ }

(4.130)

will be called a complex germ in the holomorphic representation corresponding to the function (4.127). Lemma 4.4.18. z 1. For all (ζ , η) ∈ 𝒢M , the relation holds: (ζ 2.

𝜕 − ηz)gz = 0. 𝜕z

(4.131)

The function gz is determined from condition (4.131) for each τ up to a multiplicative constant.

Proof. The proof is similar to that of Lemma 1.5.4. Lemma 4.4.19. For an n-dimensional subspace 𝒢 z ⊂ ℝn to be a complex germ in the holomorphic representation corresponding to an element of Im 𝒵 JΓk of the form (4.127), it is necessary and sufficient that the following conditions be satisfied:

4.4 Lagrangian manifold with complex germ

| 247

(a) the vectors (

𝜕Φ∗ 𝜕Φ , ) 𝜕τa 𝜕τa

(4.132)

belong to 𝒢 z , z (b) for (ζ1 , η1 ), (ζ2 , η2 ) ∈ 𝒢M , ζ1 η2 − ζ2 η1 = 0,

(4.133)

ξξ ∗ − ηη∗ ≥ 0,

(4.134)

z (c) for (ξ , η) ∈ 𝒢M ,

and the equality in this relation is attained if and only if the vector (ξ , η) is a linear combination of vectors (4.132). Proof. First, we prove the necessity. Property (a) follows from item (a) of Lemma 4.4.17, and property (b) follows from the symmetry of the matrix M. Let us verify property (c). We represent ξ in the form k

ξ = ∑ μa a=1

𝜕Φ∗ + ξ⊥ , 𝜕τa

𝜕Φ∗ ξ = 0. 𝜕τa ⊥

Then k

η = Mξ = ∑ μa a=1

𝜕Φ∗ 𝜕Φ∗ η⊥ = Mξ⊥ = 0. 𝜕τa 𝜕τa

𝜕Φ + η⊥ , 𝜕τa

We have ̃ ⊥. ξ ∗ ξ − η∗ η = ξ ∗ (E − M + M)ξ = ξ⊥∗ (E − M̃ + M)ξ

(4.135)

̃ < 1 (item (b) in the Corollary of Lemma 4.4.17), expression (4.135) is By relation ‖M‖ nonnegative and equal to zero only if ξ⊥ = 0. The necessity is proved. Let us prove the sufficiency. We show that the subspace 𝒢 z is uniquely projected on the plane {(ξ , 0)}. Indeed, assume that there exists a nonzero element of 𝒢 z whose projection on this plane is zero. This means that it has the form (0, η), which contradicts condition (4.134). Thus, the subspace 𝒢 z has the form (4.130) for some matrix M. ∗ 𝜕Φ It follows from property (a) that M 𝜕Φ = 𝜕τ ; property (b) implies the symmetry of the 𝜕τ a

a

248 | 4 Complex germ method in the Fock space matrix M, and property (c) implies the condition ξ ∗ (E − M + M)ξ > 0 𝜕Φ = 0, ξ ≠ 0, which means that the matrix E − M̃ + M̃ is positive definite, i. e., for ξ 𝜕τ a ̃ < 1. By the corollary of Lemma 4.4.17, the function (4.127) belongs to Im 𝒵 J k . The ‖M‖ Γ

proof of Lemma 4.4.19 is complete.

Now we introduce a basis on the germ and an operator w by analogy with Secz z tion 1.5. Let us consider the following equivalence relation on 𝒢M . Two vectors in 𝒢M will be called equivalent if their difference is a linear combination of vectors (4.132). On the corresponding quotient space, the bilinear form (4.134) satisfies all axioms of inner product, and hence, we can choose an orthonormal basis on the quotient space of the form (G(α) , F (α) ),

α = k + 1, n.

We also write G(α) =

𝜕Φ∗ , 𝜕τa

F (α) =

𝜕Φ , 𝜕τa

a = 1, k.

(4.136)

Let us consider n × n matrices F and G of the form Fij = Fi , Gij = Gi . Properties (4.133), (4.134) can be represented as (j)

GT F = F T G,

G+ G − F + F = L

(j)

(4.137)

and by L we denote a diagonal matrix whose first k diagonal elements are zero, and the other such elements are equal to 1. Definition 4.4.5. Operators of the form Λzα = i(G(α)

𝜕 − F (α) z), 𝜕z

𝜕 Λ̄ zα = −i(G(α)∗ z − F (α)∗ ), 𝜕z

α = k + 1, n,

(4.138)

will be called germ operators of creation and annihilation. It follows from Lemma 4.4.18 that Λzα gz = 0 if gz is the Gaussian function to which the germ corresponds. By 𝒮 ,̃ we denote the space of complex functions f ̃ of integer nonnegative arguments νk+1 , . . . , νn and τ ∈ ℝn such that relation (4.120) is satisfied. Assume that, for each τ, there is a given complex germ satisfying properties (a)–(c) of Lemma 4.4.19

4.4 Lagrangian manifold with complex germ

| 249

and a basis is chosen on this germ according to the above described procedure. We consider an operator w̃ F,G : 𝒮 ̃ → Im 𝒵 JΓk of the form (w̃ F,G f ̃)(τ, z) =

∞

∑

νk+1 ,...,νn =0

(Λ̄ zk+1 )νk+1

fν̃ k+1 ,...,νn

√νk+1 !

⋅⋅⋅

(Λ̄ zn )νn √νn !

1

e 2 zi (FG

−1

)ij (τ)zj

.

(4.139)

Lemma 4.4.20. The operator w̃ F,G satisfies the following properties: 1. 1

󵄩󵄩 ̃ F,G ̃󵄩󵄩2 󵄩󵄩w f 󵄩󵄩 =

√π n (2π)k

| det G|‖f ̃‖2 ,

(4.140)

where ‖f ̃‖2 = 2.

∞

∑

νk+1 ,...,νn =0

∫ dτ|fν̃ k+1 ,...,νn |2 ;

(4.141)

the operator w̃ F,G is a bijective mapping of the space S̃ onto Im 𝒵 JΓk . The proof of this lemma is based on the following assertions. As in the proof of Lemma 4.4.19, we represent G(α) and F (α) in the form k

G(α) = ∑ μ(α) a a=1 k

F (α) = ∑ μ(α) a a=1

G⊥(α)

𝜕Φ∗ + G⊥(α) , 𝜕τa 𝜕Φ + F⊥(α) , 𝜕τa

(4.142)

𝜕Φ 𝜕Φ∗ = F⊥(α) = 0. 𝜕τa 𝜕τa

We denote Azα = i(G⊥(α)

𝜕 − F⊥(α) z), 𝜕z

𝜕 Ã zα = −i(G⊥(α)∗ z − F⊥(α)∗ ). 𝜕z

Lemma 4.4.21. (a) The relation Λ̄ zα 𝒵 JΓk 𝒵 −1 = 𝒵 JΓk 𝒵 −1 Ā zα holds. (b) If f satisfies condition (4.124), then Ā zα f also satisfies this condition. Proof. The operator Λ̄ zα can be represented as a sum of the operator Ā zα and a linear ∗ 𝜕Φ 𝜕 combination of the operators 𝜕τ z − 𝜕Φ . Assertion (a) of Lemma 4.4.21 holds by 𝜕τa 𝜕z a ∗ z ̄ the commutativity of the operators A and 𝜕Φ z − 𝜕Φ 𝜕 , which follows from one of α

𝜕τa

𝜕τa 𝜕z

the properties of germ and Corollary 4.4.2 of Lemma 4.4.8. Assertion (b) follows from property (4.142). The proof of the lemma is complete.

250 | 4 Complex germ method in the Fock space Corollary 4.4.8. The relation w̃ F,G f ̃ = 𝒵 JΓk 𝒵 −1 f holds, where f =

∞

∑

νk+1 ,...,νn =0

×√

fν̃ k+1 ,...,νn (τ)

(Ā zk+1 )νk+1 √νk+1 !

(Ā zn )νn

⋅⋅⋅

√νn !

det R 1 n exp{ ∑ z M̃ (τ)zj }, 2 i,j=1 i ij (2π)k

𝜕Φi −1 𝜕Φj M̃ ij = (FG−1 )ij = R , 𝜕τa ab 𝜕τb

Rab =

(4.143)

𝜕Φ∗ 𝜕Φ . 𝜕τa 𝜕τb

Proof. Corollary 4.4.8 immediately follows from Lemma 4.4.21 and Remark 4.4.3(2) after Lemma 4.4.14. Corollary 4.4.9. 󵄩󵄩 ̃ F,G ̃󵄩󵄩2 󵄩󵄩w f 󵄩󵄩 = ∫ dτ√

det( 𝜕Φ 𝜕τ

∗

π n (2π)k

𝜕Φ ) 𝜕τb

∞

󵄨󵄨 ̃ 󵄨2 󵄨󵄨fνk+1 ,...,νn (τ)󵄨󵄨󵄨 . + ̃ ̃ det(E − M M) νk+1 ,...,νn =0 a

∑

Proof. The proof follows from Lemmas 4.4.17 and 4.4.17. Proof of Lemma 4.4.20. The first assertion of Lemma 4.4.20 follows from Corollary 4.4.2. Indeed, E − M̃ + M̃ = (G+ ) (G+ G − F + F + G+ −1

𝜕Φ∗ −1 𝜕Φ R G)G−1 . 𝜕τa ab 𝜕τb

We note that, in this formula, G can be replaced by the matrix G⊥ , which has the form G⊥(a) = G(a) , where G⊥(a) is defined by formula (4.142). In this case, G⊥ = det G. Therefore, ̃ = | det G|−2 det(G+ G⊥ − F + F⊥ + G+ det(E − M̃ + M) ⊥ ⊥ ⊥

𝜕Φ∗ −1 𝜕Φ R G ). 𝜕τa ab 𝜕τb ⊥

Note that the matrix (G⊥+ G⊥ − F⊥+ F⊥ )αβ is equal to δαβ for α, β > k, and to zero if at least one of the indices α, β does not exceed k. On the other hand, the matrix (G⊥+

𝜕Φ∗ −1 𝜕Φ R G ) 𝜕τa ab 𝜕τb ⊥ cd

is different from zero only for c, d ≤ k and is equal to 𝜕Φ∗ 𝜕Φ . 𝜕τc 𝜕τd

4.4 Lagrangian manifold with complex germ | 251

This implies det 𝜕Φ 𝜕τ

∗

̃ = det(E − M̃ M) +

𝜕Φ 𝜕τb | det G|2 a

.

The first assertion of Lemma 4.4.20 is proved. To prove the second assertion of Lemma 4.4.20, we note that f ̃ = 0 for w̃ F,G f ̃ = 0. It remains to verify that the image of the mapping w̃ F,G coincides with Im 𝒵 JΓk . For this, by Corollary 4.4.2 of Lemma 4.4.15, it is necessary to verify that (a) any function f (4.143) be quasipolynomial and (b) any quasipolynomial function f satisfying condition (4.124) have the form (4.143). We use the change (4.115). To verify assertion (a), we note the following. Let f =

∞

∑ l1 ,...,ln =0

f (l1 , . . . , ln )

l

z11 √l1 !

l

⋅⋅⋅

znn . √ln !

Then the quantity ∞

∑ (l1 + ⋅ ⋅ ⋅ + ln )2M

l1 ,...,ln =0

coincides with the squared norm of the function n

M

𝜕 ) f. (∑ z 𝜕z (i) i=k+1 (i)

(4.144)

Expressing z (i) and 𝜕/𝜕z (i) via Azα and Ā zα and applying formula (4.143), it is easy to show that the squared norm of the function (4.144) has a finite norm. Let us verify property (b). First, we note that any function f with a finite norm satisfying condition (4.124) can be represented in the form of series (4.143), where the function f ̃ satisfies condition (4.141) that the norm is finite; for this, it suffices to express z (i) in terms of Azα and Ā zα and to use the commutation relation [Azα , Ā zβ ] = δαβ ,

[Azα , Azβ ] = [Ā zα , Ā zβ ] = 0

(4.145)

and the property Azα exp{

1 n ∑ z M̃ z } = 0. 2 i,j=1 i ij j

(4.146)

252 | 4 Complex germ method in the Fock space We show that f ̃ ∈ S.̃ For this, by commutation relations (4.145) and property (4.146), it M ̄ suffices to show that the norm of (∑∞ α=k+1 Aα Aα ) f is finite. But this assertion follows from the fact that f is quasipolynomial. The proof of Lemma 4.4.20 is complete.

4.4.6 Definition of canonical operator on a Lagrangian manifold with complex germ The results obtained in the preceding section allow us to define a canonical operator on a Lagrangian manifold with complex germ. Definition 4.4.6. A set of n-dimensional subspaces 2n

𝒢P(τ),Q(τ) ⊂ T ℳP(τ),Q(τ) = ℝ ℂ

will be called a complex germ on an isotropic manifold Γk if the following axioms are satisfied: (1) the vectors (

𝜕P 𝜕Q , ), 𝜕τa 𝜕τa

a = 1, k,

(4.147)

belong to 𝒢P(τ),Q(τ) , (2) p1 q2 = p2 q1 for (p1 , q1 ), (p2 , q2 ) ∈ 𝒢P(τ),Q(τ) , (3) for (p, q) ∈ 𝒢P(τ),Q(τ) , 1 ̄ ≥ 0, (pq̄ − pq) i and the equality in this relation is attained if and only if the vector (p, q) is a linear combination of vectors (4.147). Definition 4.4.7. The set of an isotropic manifold and a complex germ on it will be called a Lagrangian manifold with complex germ. Lemma 4.4.22. The n-dimensional subspace 𝒢P(τ),Q(τ) is a complex germ in the sense of Definition 4.4.5 if and only if a subspace ℝ2n of the form {(q − ip, q + ip) | (p, q) ∈ 𝒢P(τ),Q(τ) } is a complex germ in the holomorphic representation. Proof. The proof follows from Lemma 4.4.19 and Definition 4.4.5. Lemma 4.4.22 allows us to introduce the canonical operator corresponding to a germ on an isotropic manifold.

4.4 Lagrangian manifold with complex germ |

253

Let us consider a basis (B(i) , C (i) ) on the germ 𝒢P(τ),Q(τ) of the form B(i) (τ) =

F (i) (τ) − G(i) (τ) , √2i

C (i) (τ) =

F (i) (τ) + G(i) (τ) , √2

i, j = 1, n,

and, as usual, write Bij = Bi and Cij = Ci . By Lemma 4.4.22, the vector (B(i) , C (i) ) ∈ ℝ2n belongs to the germ 𝒢P(τ),Q(τ) , and (j)

B(a) =

𝜕P , 𝜕τa

C (a) =

(j)

𝜕Q , 𝜕τa

i(B+ C − C + B) = L,

BT C = C T B.

(4.148)

Properties (4.148) are similar to properties (4.136) and (4.137). ̃ We consider the operator wΓB,C k : 𝒮k → 𝒮n,k /𝒮0 that can uniquely be determined from the property w̃ F,G =

√| det G| √4 π n (2π)k

B,C

𝒵 JΓk wΓk .

By Lemma 4.4.20, we have the following assertion. Lemma 4.4.23. The operator wΓB,C k is a bijective mapping preserving the norm. Lemmas 4.4.8 and 4.4.18 imply the following assertion. Lemma 4.4.24. For an element [f0 ] of the quotient space 𝒮k,n /𝒮0 to satisfy the condition, Ωℂ P,Q (p, q)JΓk [f0 ] = 0,

(4.149)

for all (p, q) ∈ 𝒢P(τ),Q(τ) , it is necessary and sufficient that, for a function φ ∈ 𝒟(ℝk ), [f0 ] = wΓB,C k (φ, 0, 0, . . .). B,C Now we calculate the explicit form of the operators wΓB,C k and JΓk wΓk . For this, we use Corollary 4.4.8 of Lemma 4.4.21 and formula (4.139). As shown in item 2, in contrast to the holomorphic function 𝒵 JΓk wΓB,C k f , the distri-

bution JΓk wΓB,C k f is singular at focal points (where det C = 0). Therefore, we consider only the case det C ≠ 0. Definition 4.4.8. Operators of the form (α)∗ (α)∗ Λ̄ α = Ωℂ , C ), P,Q (B

(α) (α) Λα = Ωℂ P,Q (B , C ),

will be called germ operators of creation and annihilation.

α = k + 1, n,

254 | 4 Complex germ method in the Fock space Lemma 4.4.25. For det C ≠ 0, the operator JΓk wΓB,C k has the form (JΓk wΓB,C k f )(τ, ξ ) =

∞

∑

νk+1 ,...,νn =0

fνk+1 ⋅⋅⋅νn (τ)

νk+1 Λ̄ k+1

√νk+1 !

⋅⋅⋅

ν Λ̄ nn g0 (τ, ξ ), √νn !

i

g0 (τ, ξ ) =

e 2 arg det(C−iB) 1 i n exp{ ∑ ξi (BC −1 )ij ξj }. √det C 2 i,j=1 √4 (2π)n−k

(4.150)

Proof. Lemma 4.4.8 readily implies Λ̄ zα 𝒵 = 𝒵 Λ̄ α ,

Λzα 𝒵 = 𝒵 Λα .

Therefore, it is sufficient to verify formula (4.150) for f ∈ 𝒮 ̃ such that only their zero component is different from zero, i. e., to verify the relation 𝒵 g0 =

√4 π n (2π)k

exp{

| det G|

i n ∑ z (FG−1 )ij zj }. 2 i,j=1 i

(4.151)

which can easily be verified by using formula (4.108) as in the proof of Lemma 4.4.17. The proof of Lemma 4.4.25 is complete. No problem with focal points arises when we consider an element of 𝒮k,n /𝒮0 of the form wΓB,C k f and, therefore, we consider the general case. We write B(α) ⊥ =

F⊥(α) − G⊥(α) , √2i

C⊥(α) =

F⊥(α) + G⊥(α) , √2

α = k + 1, ∞,

where the functions G⊥ and F⊥ are defined by formula (4.142) (α)∗ (α)∗ Ā α = Ωℂ P,Q (B⊥ , C⊥ ),

(α) (α) Aα = Ωℂ P,Q (B⊥ , C⊥ ).

Lemma 4.4.26. The operator wΓB,C k has the form ̃ (wΓB,C k f )(τ, ξ ) = [

∞

∑

νk+1 ,...,νn =0

fν̃ k+1 ⋅⋅⋅νn (τ)

νk+1 Ā k+1

√νk+1 !

⋅⋅⋅

ν Ā nn f0 (τ, ξ )], √νn !

(4.152)

where f0 (τ, ξ ) =

1 √4 π n (2π)k × exp{−

√

det R ̃ det G|̃ det(E + M)|

1 n ̃ −1 (E − M)] ̃ ξj }. ∑ ξ [(E + M) ij 2 i,j=1 i

(4.153)

4.4 Lagrangian manifold with complex germ | 255

Proof. Lemma 4.4.8 implies Ā zα 𝒵 = 𝒵 Ā α and Azα 𝒵 = 𝒵 Aα , and the relation 𝒵 f0 =

√4 π n (2π)k √(2π)k

√

det R 1 n exp{ ∑ zi M̃ ij zj } | det G| 2 i,j=1

which implies the assertion of the lemma can be verified directly. The proof of Lemma 4.4.26 is complete. Remark 4.4.4. 1. An element of 𝒮k,n in square brackets in formula (4.152), which is a representation of the equivalence class [wΓB,C k f ] ∈ 𝒮k,n /𝒮0 is chosen so as to satisfy condition (4.125). 2. The operators Ā in formula (4.152) can be replaced by Λ,̄ but then this element of 𝒮k,n does not satisfy condition (4.125). 3. The choice of the function f0 in the form (4.153) is also not unique; it becomes unique if we impose additional condition (4.125). 4. We note that the direct verification of the fact that the operator JΓk takes the function (4.152) to the function (4.150) requires rather cumbersome calculations [45]. Definition 4.4.9. An operator of the form ̃ 𝕂Γk wΓB,C k : 𝒮 → ℒ/ℛ

(4.154)

will be called the canonical operator corresponding to the Lagrangian manifold with complex germ and to the basis (B(i) , C (i) ). Remark 4.4.5. 1. Lemma 4.4.26 implies that the explicit formula for the operator 𝕂Γk wΓB,C can be k written as ε

− (𝕂Γk wΓB,C k f ) (x) = ∫ dτε

f (τ, ξ ) =

2.

k+n 4

∞

i

e ε (S(τ)+P(τ)(τ−Q(τ))) f (τ,

fν̃ k+1 ⋅⋅⋅νn (τ)

∑

νk+1 ,...,νn =0

νk+1 Λ̄ k+1

√νk+1 !

⋅⋅⋅

x − Q(τ) ) + o(1), √ε

ν Λ̄ nn f0 (τ, ξ ), √νn !

where f0 is any function from 𝒮k,n such that 𝒵 JΓk f0 has the form (4.151) (one of such functions has the form (4.153)). It follows from Lemma 4.4.25 that if there are no focal points (at which det C = 0), then the canonical operator (4.154) can be represented as ε

i

(S(τ)+P(τ)(τ−Q(τ))) ε g(τ, (𝕂Γk wΓB,C k f ) (x) = e ̄

̄

̄

x − Q(τ)̄ ) + o(1), √ε

256 | 4 Complex germ method in the Fock space i

g(τ, ξ ) =

∞ 1 e 2 arg det(C−iB) fνk+1 ⋅⋅⋅νn (τ) ∑ n−k √det C (2π) 4 νk+1 ,...,νn ν ν Λ̄ k+1 Λ̄ n i n × k+1 ⋅ ⋅ ⋅ n exp{ ∑ ξi (BC −1 )ij ξj } 2 i,j=1 √νk+1 ! √νn !

(4.155)

for x − Q(τ)̄ = O(√ε). Definition (4.155) differs from the traditional one in that it contains the factor i e 2 arg det(C−iB) . In the case where the isotropic manifold is not diffeomorphic to ℝk , this factor results in a change of the quantization condition (see Section 4.6). 𝜕 4. The mean value of the observable O(−iε 𝜕x , x) in the state (4.155) as ε → 0 has the form

3.

∫ dτO(P(τ), Q(τ))

∞

∑

󵄨󵄨 󵄨2 󵄨󵄨fνk+1 ⋅⋅⋅νn (τ)󵄨󵄨󵄨 .

νk+1 ,...,νn =0

4.4.7 Solution of the Cauchy problem: asymptotics corresponding to isotropic manifolds In this section, we show that the canonical operator defined in item 1 indeed gives the asymptotics of the solution of the Cauchy problem for equation (1.104), i

𝜕ψεt = Htε ψεt , 𝜕t

ψεt ∈ L2 (ℝn ),

ψε0 ∈ 𝕂Γk [f0 ],

(4.156)

0

where Γk0 is the isotropic manifold (Pt (τ), Qt (τ)), τ ∈ ℝk , in the phase space, [f0 ] ∈ 𝒮k,n /𝒮0 , and the operator Htε is defined as in Section 1.5, 1

1

1 𝜕 2 𝜕 2 H = H(−iε , x) + H(1) (−iε , x). ε 𝜕x 𝜕x ε

By Γkt , we denote the manifold (Pt (τ), Qt (τ)) satisfying the Hamiltonian system (1.94) for each τ, and by 𝒮t (τ) we denote a function of the form 𝒮t (τ) = 𝒮0 (τ) + ∫{Pt ′ (τ)Q̇ t ′ (τ) − Ht ′ (Pt ′ (τ), Qt ′ (τ))} dt , ′

(4.157)

where 𝒮0 (τ) is determined by formula (4.88). Lemma 4.4.27. The relation holds 𝜕Q (τ) 𝜕𝒮t − Pt (τ) t = 0, 𝜕τa 𝜕τa

a = 1, k.

(4.158)

4.4 Lagrangian manifold with complex germ | 257

Proof. Obviously, this assertion holds at the initial time. To verify equation (4.158) at other times, it suffices to show that the derivative with respect to t of the left-hand side of formula (4.158) is zero; this assertion can be proved using formula (4.157) and the Hamiltonian system (1.94). The proof of Lemma 4.4.27 is complete. It follows from Lemma 4.4.27 that the manifold Γkt is also isotropic. We show that any element of equivalence class i

e ε 𝒮t (0) 𝕂Γk [ft ] ∈ ℒ/ℛ t

is an asymptotics of the Cauchy problem. For this, it suffices to show that i 󵄩󵄩 ε 󵄩 𝒮 (0) ε 󵄩󵄩ψt (x) − e ε t (𝒦Γkt ft ) (x)󵄩󵄩󵄩 → 0. ε→0

(4.159)

First, we show that (𝒦Γk ft )ε is an asymptotic solution of equation (4.156). As in Sect tion 1.5, we let Ht,2 denote an operator of the form n

(Ht,2 f )(τ, ξ ) = ∑ [ m,l=1

1 𝜕 1 𝜕 1 𝜕2 Ht − 2 𝜕Pm 𝜕Pl i 𝜕ξm i 𝜕ξl

1 𝜕2 Ht 1 𝜕 1 𝜕 1 𝜕2 Ht (ξm + ξm ) + ξ ξ ]f (τ, ξ ) 2 𝜕Qm 𝜕Pl i 𝜕ξl i 𝜕ξl 2 𝜕Qm 𝜕Ql m l + H1 (t)f (τ, ξ ), (4.160) +

where H1 (t) has the form (1.110) and the derivatives of the function Ht are calculated at the point Pt (τ), Qt (τ). Theorem 4.4.1. Let [(i

𝜕 − Ht,2 )ft ] = 0. 𝜕t

(4.161)

Then 󵄩󵄩 𝜕 󵄩ε i 󵄩󵄩 󵄩 ε󵄩 ε 𝒮 (0) 󵄩󵄩(i − Ht )e ε t (𝒦Γkt ft ) 󵄩󵄩󵄩 → 0. 󵄩󵄩 𝜕t 󵄩󵄩 ε→0

(4.162)

Remark 4.4.6. 1. Condition (4.161) means that ∫ dα exp{iαa (

𝜕P 𝜕Q 1 𝜕 𝜕 ξ− )}(i − Ht,2 )ft = 0 𝜕τa 𝜕τa i 𝜕ξ 𝜕t

(4.163)

and is weaker than the requirement i

𝜕ft = Ht,2 ft . 𝜕t

(4.164)

258 | 4 Complex germ method in the Fock space 2.

It is easy to verify that the operators 𝜕P 𝜕Q 1 𝜕 ξ− 𝜕τa 𝜕τa i 𝜕ξ

and i

𝜕 − Ht,2 𝜕t

commute with each other, and hence, condition (4.161) is equivalent to the condition (i

𝜕 − Ht,2 )[ft ] = 0. 𝜕t

(4.165)

Proof of Theorem 4.4.1. We consider the left-hand side of (4.162). By the definition of operator 𝒦 and Lemma 4.4.27, it is equal to ∫

i dτ 𝜕 (i − Htε )e ε 𝒮t (τ) KPεt (τ),Qt (τ) ft (τ). k/4 𝜕t ε

(4.166)

According to the results of Section 1.5, the integrand in formula (4.166) can be expanded in an asymptotic series in √ε so that the integral (4.166) becomes ∫

dτ εi 𝒮t (τ) ε 𝜕 e KPt (τ),Qt (τ) [(i − Ht,2 )ft (τ)ε1/2 gt,1 (τ) 𝜕t εk/4 + ⋅ ⋅ ⋅ + εM/2 gt,M (τ) + O(ε

M−1 2

)].

Since 󵄩󵄩 dτ i M+1 󵄩 M+1 k 󵄩󵄩 󵄩󵄩 𝒮 (τ) ε − 󵄩󵄩∫ k/4 e ε t KPt (τ),Qt (τ) O(ε 2 )󵄩󵄩󵄩 ≤ const ε 2 4 󵄩󵄩 󵄩󵄩 ε and for a sufficiently large M, this quantity tends to zero as ε → 0; it follows from Lemma 4.3.5 that the integral (4.166) tends to zero if condition (4.161) is satisfied. The proof of Theorem 4.4.1 is complete. Corollary 4.4.10. If H ε is a self-adjoint operator, then it follows from Section 1.4 that relation (4.159) is satisfied. Corollary 4.4.11. Under the same conditions, any element i

ψεt,as (x) ∈ e ε 𝒮t (0) 𝕂Γk [ft ]

(4.167)

t

is an asymptotic solution of the Cauchy problem (4.156). Corollary 4.4.12. By the definition of canonical operator on a Lagrangian manifold with complex germ, the element (4.167) becomes i

t

t

0

0

ct e ε 𝒮t (0) 𝕂Γk wΓBk ,C (wΓBk ,C ) [f ], t

t

0

−1

󵄨󵄨 t 󵄨󵄨 󵄨󵄨c 󵄨󵄨 = 1.

4.5 Abstract canonical operator

| 259

4.5 Superposition of wave functions corresponding to an abstract canonical operator It turns out that, under certain conditions, the scheme developed in the preceding two sections for the case of finite-dimensional quantum mechanics, which permits obtaining wave functions corresponding to Lagrangian manifolds with complex germ, can also be generalized to the general case of abstract canonical operator. We consider an abstract canonical operator (an analog of the canonical operator for the germ at a point) and demand that the following additional conditions be satisfied: (a) all spaces ℱX coincide, i. e., ℱX = ℱ , (b) for ‖X1 − X2 ‖ > C1 , C1 > 0, the following condition is satisfied on the everywhere dense set 𝒟̃ ⊂ ℱ : 1 󵄨󵄨 ε 󵄨 ε 󵄨(K f1 , KX2 f2 )󵄨󵄨󵄨 < C2 , ε M 󵄨 X1

∀M > 0,

f1 , f2 ∈ 𝒟̃ ,

(c) on the same set 𝒟̃ , ‖X − X1 ‖ 󵄨󵄨 ε 󵄨 ε ) 󵄨󵄨(KX1 f1 , KX2 f2 )󵄨󵄨󵄨 ≤ ϕM ( 2 √ε for X1 , X2 from a neighborhood of a point X ∈ ℳ; here, ‖X2 − X1 ‖ is understood as the distance between the images of points X1 and X2 in a certain chart, and the function ϕM which in general depends on X, f1 , and f2 satisfies the condition M 󵄨󵄨 2 󵄨 󵄨󵄨(α + 1) ϕM (α)󵄨󵄨󵄨 < CM .

These properties are satisfied for the case of canonical operator corresponding to the germ at a point in finite-dimensional quantum mechanics and in the Fock space but are not satisfied in the case of multiparticle canonical operator (Chapter 2), because the quantity 1/N 1/N (Kφe f) iα f , Kφ

is of order O(1). In this section, we consider an element of the space ℋε of the form (𝒦Γk ,s f )ε = ∫

dτ εi s(τ) ε e KX(τ) f (τ), εk/4

(4.168)

f (τ) ∈ ℱ is a vector-valued function ℝk → ℱ such that ∫ dτ‖f (τ)‖2 < ∞, and Γk = {X(τ) ∈ ℳ} is a smooth k-dimensional surface in the phase space without singularities, s: ℝk → ℝ. We formulate the notion of isotropic manifold in the general case and show that, as in the case of Section 4.3, the element (4.168) is not exponentially small

260 | 4 Complex germ method in the Fock space only in the case of isotropic manifold; we calculate the norm of the element (4.168) as ε → 0 and determine the analog of the canonical operator 𝕂Γk corresponding to an isotropic manifold.

4.5.1 Calculation of the norm as ε → 0 Before we formulate a lemma on the limit as ε → 0 of the expression ε

((𝒦Γk ,s f )ε , (𝒦Γk ,s f )ε ) ,

(4.169)

we calculate it on the heuristic level of rigor. Expression (4.169) can be written as ∫

dτ1 dτ2 2i (s(τ1 )−s(τ2 )) ε ε e (KX(τ2 ) f (τ2 ), KX(τ f (τ1 )). 1) εk/4

(4.170)

By property (b), the integrand in formula (4.170) is exponentially small for τ1 ≠ τ2 , and hence, the main contribution to the integral (4.170) is made by the domain of integration τ1 − τ2 ∼ √ε. In this case, we can apply Axiom 1.2.4 of abstract canonical operator (Section 1.2), introduce the notation τ1 = τ2 + α√ε, τ2 = τ, pass to the limit as ε → 0, τ, α = const, and integrate the obtained expression. For simplicity, we identify the elements of the phase space with their coordinates in a chart. We have ε KX(τ+α√ε) f (τ + α√ε) ≈ K ε

f (τ)

2

𝜕X X αa + ε2 αa αb 𝜕τ𝜕 𝜕τ X(τ)+√ε 𝜕τ a

≈ exp{−

a

b

iαa X+ 21 √ε 𝜕τ𝜕Xa αa 𝜕X i 𝜕2 X ω1 ] − αa αb ωX1 [ ]} [ √ε 𝜕τa 2 𝜕τa 𝜕τb

ε × KX(τ) VX(τ) [

𝜕X α ]f (τ), 𝜕τa a

(4.171)

1 1 𝜕s 1 𝜕2 s (s(τ + α√ε) − s(τ)) ≈ αa + α α √ε 𝜕τa ε 2 𝜕τa 𝜕τb a b (the summation is carried out over the repeated indices from 1 to k) and the measure dτ1 dτ2 /εk/2 turns into dτdα. Thus, the integrand behaves as ε → 0 as const × i 𝜕s 𝜕X exp{ √ε αa ( 𝜕τ − ωX1 [ 𝜕τ ])}, and hence, the integral (4.170) is exponentially small for a

a

𝜕X ≠ ωX1 [ 𝜕τ ]. Therefore, we shall consider only the surfaces for which the following a condition is satisfied: 𝜕s 𝜕τa

𝜕s 𝜕X = ωX1 [ ]. 𝜕τa 𝜕τa X

Definition 4.5.1. A smooth manifold ℝk → ℳ such that the vectors k

(4.172) 𝜕X 𝜕τa

∈ T ℳX(τ) , a =

1, k, are linearly independent for all τ ∈ ℝ is called an isotropic manifold in the phase

4.5 Abstract canonical operator | 261

space diffeomorphic to ℝk if there exists a function s: ℝk → ℝ such that relation (4.172) is satisfied. The function s is called an action on an isotropic manifold. The following lemma takes place. Lemma 4.5.1. Let Γk = {X(τ), τ ∈ ℝk } be an isotropic manifold. Then ε

i

ε ε f (τ + α√ε)) e ε (s(τ+α√ε)−s(τ)) (KX(τ) f (τ), KX(τ+α√ε)

→ (f (τ), VX(τ) [

ε→0

𝜕X α ]f (τ)). 𝜕τa a

Proof. Assume that the manifold (τ) is isotropic. Then formula (4.171) implies α X+ 1 √ε 𝜕X αa 𝜕X 1 𝜕2 s 1 𝜕s αa + αa αb − a ω1 2 𝜕τa [ ] √ε 𝜕τa √ε 2 𝜕τa 𝜕τb 𝜕τa 1 𝜕2 X − αa αb ωX1 [ ] → 0 2 𝜕τa 𝜕τb ε→0

and, therefore, 󵄩󵄩ε 󵄩󵄩 i 𝜕X 󵄩󵄩 󵄩󵄩 (s(τ+α√ε)−s(τ)) ε ε √ α ]f (τ) f (τ + α ε)e − K V [ K 󵄩󵄩 → 0. 󵄩󵄩 X(τ+α√ε) X(τ) X(τ) 󵄩󵄩 ε→0 󵄩󵄩 𝜕τa a Axiom 1.2.1 implies the assertion of the lemma. The proof of Lemma 4.5.1 is complete. Thus, one can expect that the following relation is satisfied: ε

((𝒦Γk ,s f )ε , (𝒦Γk ,s f )ε ) → ∫ dτdα(f (τ), VX(τ) [ ε→0

𝜕X α ]f (τ)). 𝜕τa a

(4.173)

Lemma 4.3.4 is a special case of this relation. Now we shall prove relation (4.173). By Sk , we denote the set of all finite functions f : ℝk → ℱ such that the following relation holds for an integrable function ϕτ (α) and a constant C: ε ε f (τ)) ≤ ϕτ (α), (KX(τ+α√ε) f (τ + α√ε), KX(τ)

|α√ε| < C,

∫ ϕτ (α) dα < ∞.

(4.174)

Lemma 4.5.2. The set Sk is dense in the space L2 (ℝk → ℱ ) of all finite functions f : ℝk → ℱ for which the integral ∫ ‖f (τ)‖2 dτ converges. Proof. It suffices to prove that if a function f takes finitely many values which, in turn, belong to 𝒟̃ , then relation (4.174) holds. Indeed, the set of such functions f is dense in L2 (ℝ→ ℱ ). Property (4.174) for such functions follows from property (b). Lemma 4.5.2 is proved.

262 | 4 Complex germ method in the Fock space Lemma 4.5.3. Let f ∈ Sk . Then relation (4.173) holds. Proof. This lemma is proved by analogy with Section 4.3. First, we show that the contribution of the domain |τ1 − τ2 | > C (C is an arbitrary positive number) to the integral (4.170) is exponentially small. Indeed, in this case, ‖X1 − X2 ‖ > C1 and, therefore, one can apply property (b). Since f is finite, it follows that the contribution of the domain |τ1 − τ2 | > C to the integral indeed does not exceed εM , where M is an arbitrary positive number. Now we consider the contribution of the domain |τ1 − τ2 | < C. In this case, it follows from property (4.174) that the Lebesgue theorem can be applied, and Lemma 4.5.1 implies property (4.173). Lemma 4.5.3 is proved. Corollary 4.5.1. The value of (4.169) tends to zero as ε → 0 if and only if ∫ dαVX(τ) [

𝜕X α ]f (τ) = 0; 𝜕τa a

(4.175)

the equality is here understood in the generalized sense: ∫ dα(f1 (τ), VX(τ) [

𝜕X α ]f (τ)) = 0 𝜕τa a

∀f1 ∈ Sk .

Proof. Proof is similar to that of Corollary 4.3.2 of Lemma 4.3.5. Corollary 4.5.2. For f1 , f2 ∈ Sk , ε

((𝒦Γk f1 )ε , (𝒦Γk f2 )ε ) → ∫ dτdα(f1 (τ), VX(τ) [ ε→0

𝜕X α ]f (τ)). 𝜕τa a 2

4.5.2 Canonical operator corresponding to an isotropic manifold In the phase space ℳ, let Γk be a k-dimensional isotropic manifold X(τ), τ ∈ ℝk , diffeomorphic to ℝk . Let s(τ) denote the action on this isotropic manifold. Let ℒ be a linear space {ψε ∈ ℋε , ε ∈ I}. Then the operator 𝒦Γk : Sk → ℒ

associates each function f ∈ Sk with an element of ℒ, because for each ε, it takes f ∈ Sk to an element of ℋε . We shall consider a subspace ℛ of the space ℒ consisting of all sets of elements ψε ∈ ℋε , ε ∈ I, such that 󵄩󵄩 ε 󵄩󵄩ε 󵄩󵄩ψ 󵄩󵄩 → 0. ε→0

(4.176)

We let ℒ/ℛ denote the quotient space of the linear space ℒ with respect to the subspace ℛ. By S0 , we denote the subspace of the space Sk consisting of all f ∈ Sk satisfying relation (4.175), and by Sk /S0 we denote the corresponding quotient space.

4.5 Abstract canonical operator | 263

We consider the operator 𝕂Γk : Sk /S0 → ℒ/ℛ of the form 𝕂Γk [f ] = [𝒦Γk f ].

(4.177)

Definition 4.5.2. The operator 𝕂Γk will be called a canonical operator corresponding to an isotropic manifold. On the quotient space Sk /S0 , we introduce the inner product ([f1 ], [f2 ]) = ∫ dα(f1 (τ), VX(τ) [

𝜕X α ]f (τ)). 𝜕τa a 2

(4.178)

By Corollary 4.5.1 of Lemma 4.5.3, this definition is well-posed. We note that the inner product ([f1 ], [f2 ]) has the following property: for ψ1 ∈ 𝕂Γk [f1 ] and ψ2 ∈ 𝕂Γk [f2 ], (ψε1 , ψε2 ) → ([f1 ], [f2 ]). ε→0

As in Section 4.4, we can consider the one-to-one mapping JΓk

Sk /S0 → Im JΓk ⊂ Sk′ of the form JΓk [f ] = ∫ dαVX(τ) [

𝜕X α ]f (τ) 𝜕τa a

and introduce the inner product (JΓk [f1 ], JΓk [f2 ]) = ([f1 ], [f2 ]) and the operator 𝕂Γk JΓ−1k [f ]: Im JΓk ⊂ Sk′ → ℒ/ℛ. The following lemma is proved by analogy with a similar assertion in Section 4.4. Lemma 4.5.4. For g ∈ Im JΓk ∩ 𝒟, Ω(

𝜕X )g = 0. 𝜕τa

Remark 4.5.1. Relation (4.179) is understood in the generalized sense.

(4.179)

264 | 4 Complex germ method in the Fock space 4.5.3 Formal asymptotic solutions of the equations of motion corresponding to isotropic manifolds Let Ut be a proper canonical transformation of the phase space. In Section 1.2, we introduced formal asymptotic solutions of equations of motion of the form i

e ε St (X) KUε t X WX,t f ∈ ℋε .

(4.180)

In this section, we consider a superposition of formal asymptotic solutions of equations of motion of the form (4.168) and study its properties. We assume that Sk is invariant with respect to the action of the operator WX,t . Assume that X(τ), τ ∈ ℝk , is an isotropic manifold in the phase space that is diffeomorphic to ℝk and S0 (τ) is an action on it. Consider an element of ℋε of the form dτ εi (St (X(τ))+S0 (τ)) ε e KUt X(τ) WX(τ),t f . εk/4

ψε (t) = ∫

(4.181)

Lemma 4.5.5. The manifold Xt (τ) = Ut X(τ) is isotropic and the function St (τ) = St (X(τ)) + S0 (τ) is an action on this manifold. Proof. We have 𝜕St 𝜕S 𝜕X = dStX(τ) [ ]+ 0 𝜕τa 𝜕τa 𝜕τa U X(τ)

= ω1 t

[U∗X

𝜕S 𝜕X 𝜕X X (τ) 𝜕X (τ) ] − ωX1 [ ] + 0 = ω1 t [ t ]. 𝜕τa 𝜕τa 𝜕τa 𝜕τa

Here, we used Lemma 1.2.7. The proof of Lemma 4.5.5 is complete. Definition 4.5.3. A canonical transformation of a manifold Γk of the form X(τ), τ ∈ ℝk , is a manifold Γkt of the form Ut X(τ), τ ∈ ℝk . Lemma 4.5.5 implies the following corollary. Corollary 4.5.3. For an element ψε (t) ∈ ℋε (4.181), ψε (t) = (𝒦Γk ,St WX(τ),t f )ε . t

Lemma 4.5.6. Let f ∈ S0 . Then WX(τ),t f ∈ S0 .

4.6 Specific features of the Cauchy problem

| 265

Proof. The proof follows from Definition (4.175) of subspace S0 and the relation VUt Xt (τ) [

𝜕Xt (τ) 𝜕X(τ) α ]W = WX(τ),t VXt (τ) [ α ], 𝜕τa a X(τ),t 𝜕τa a

which follows from Lemma 1.2.8. Let us consider a function of the form ψε (t) = 𝒦Γk ,St WX(τ),t ft , t

(4.182)

where all ft belong to the same equivalence class [f ] ∈ Sk /S0 . Definition 4.5.4. A function of the form (4.182) will be called a formal asymptotic solution of equations of motion corresponding to an isotropic manifold.

4.6 Specific features of statement of the Cauchy problem for a topologically invariant isotropic manifold In Sections 4.3–4.5, we considered a superposition of elements of the space ℋε , which correspond to finite-dimensional isotropic manifolds diffeomorphic to ℝk . In this section, we consider a generalization of the scheme proposed in Sections 4.3–4.5 to the case of manifolds that are not diffeomorphic to ℝk .

4.6.1 Isotropic manifold diffeomorphic to a circle We first illustrate the main problems in such a generalization with a simple example, where the manifold is diffeomorphic to a circle S1 . Let X(τ) ∈ ℳ be a smooth function on the circle S1 . We can consider this function as a smooth function on the line ℝ1 satisfying the periodicity condition X(τ + 2π) = X(τ).

(4.183)

As in the preceding section, we let S denote the set of functions f : S1 → ℱ such that the property (4.174) is satisfied. Let f ∈ S. We define an analog of the operator 𝒦 (formula (4.168)). As was shown in the preceding section, for the norm of the function (4.168) be not too small, it is necessary to require that the function s satisfy the condition ds dX = ωX1 [ ]. dτ dτ

(4.184)

266 | 4 Complex germ method in the Fock space It follows from condition (4.184) that the function s can be defined only on the universal covering ℝ1 of the manifold S1 ; since the integral 2π

∫ ωX1 [ 0

dX ]dτ dτ

(4.185)

is in general different from zero, a function s(τ) of the form τ

s(τ) = ∫ ωX1 [ 0

dX ]dτ + const dτ

is in general not periodic with period 2π. Nevertheless, we define the canonical operator in the form (4.168), where the integral is taken over any interval of the form [τ0 , τ0 + 2π] with arbitrary τ0 . This is well posed only if the integrand is 2π-periodic in τ, i. e., not for all values of ε but only for the values satisfying the condition s(τ + 2π) − s(τ) = 2πN, ε where N is an integer. Thus, we must choose the range I of the parameter ε in the form 2π

1 dX {ε = ∫ ωX1 [ ]dτ, N ∈ ℤ, N ≠ 0} 2πN dτ

(4.186)

0

if the integral (4.185) is different from zero, and in the form I = [0, ε0 ) if the integral (4.185) is equal to zero. Since the scheme considered in the preceding section can be used without changes here if the integral (4.185) is equal to zero, we restrict our consideration to the case where the integral (4.185) is nonzero. Note that if ωX1 is the 1-form of action in the finite-dimensional classical mechanics, ω1 = PdQ, then condition (4.186) can be represented as 1 ∮ PdQ = N. 2πε

(4.187)

By ℒ, we denote the linear space of sets of elements ψε ∈ ℋε , ε ∈ I; the set I has the form (4.186). We consider the operator 𝒦Γ1 : S → ℒ

of the form 2π

τ

0

0

dτ i dX ε f (τ). (𝒦Γ1 f ) = ∫ 1/4 exp{ 1/4 ∫ ωX1 [ ]dτ}KX(τ) dτ ε ε ε

(4.188)

4.6 Specific features of the Cauchy problem

| 267

All formulas obtained in the preceding section remain valid for the operator (4.188); one only must bear in mind that the range of ε is discrete and the parameter τ varies in the interval [0, 2π). We can also introduce the equivalence of ℛ in the space ℒ, subspace S0 ⊂ S, operator 𝕂Γ1 defined by formula (4.177), and determine an analog of the formal asymptotic solution of equations of motion (4.181), ψεas

2π

τ

0

0

i dX dτ = ∫ 1/4 exp{ (∫ ωX1 [ ]dτ + St (X(τ)))}KUε t X(τ) ft (τ), ε dτ ε

(4.189)

where ε ∈ I and ft (τ) = WX(τ),t f (τ). An analog of Corollary 4.4.11 of Theorem 4.4.1 holds in the case of finite-dimensional quantum mechanics. Lemma 4.6.1. Let ψε (t) be a solution of the Cauchy problem for equation (4.89) satisfying the initial condition ψε (0) = ψεas (0), where ε ∈ I, H ε is a self-adjoint operator, and ψεas (t) belongs to the domain of this operator. Then 󵄩󵄩 ε 󵄩ε ε 󵄩󵄩ψ (t) − ψas (t)󵄩󵄩󵄩 → 0. ε→0

(4.190)

Note that ε in formula (4.190) ranges in the set (4.186). We see that to pose the Cauchy problem, it is necessary to impose quantization conditions of the form (4.187) on ε. The following question arises: Is it possible to change the form of these conditions? To answer this question, we change the definition of the operator 𝒦Γ1 (4.188) by multiplying the integrand by e−iβτ : 2π

τ

0

0

i dX dτ ε f (τ). (𝒦Γ1 f ) = ∫ 1/4 exp{ ∫ ωX1 [ ]}dτ e−iβτ KX(τ) ε dτ ε ε

(4.191)

For the integrand to be a periodic function of τ with period 2π, the parameter ε must satisfy a condition different from (4.186), namely, 2π

{ε =

1 dX ∫ ωX1 [ ]dτ, N ∈ ℤ, N + β ≠ 0}. 2π(N + β) dτ

(4.192)

0

This means that, in the specific case ω1 = PdQ, condition (4.187) is replaced by a condition of the form 1 ∮ PdQ = N + β. 2πε In particular, we can consider the value β = 1/2 and obtain the Bohr–Sommerfeld quantization condition (see, e. g., [21]), which is usually used in one-dimensional quantum mechanics.

268 | 4 Complex germ method in the Fock space When the operator 𝒦Γ1 is defined in the form (4.191). All assertions mentioned above, in particular, Lemma 4.6.1, remain valid; it is only necessary to replace the set of the form (4.186) by a set of the form (4.192). Is it possible to deal without the quantization conditions? It turns out that this is possible. Indeed, we choose the parameter β in formula (4.191) in the form 2π

βε = {

1 dX ∫ ωX1 [ ]dτ}; 2πε dτ

(4.193)

0

here, {a} denote the fractional part of a number a, i. e., the difference a − [a], where [a] is the greatest integer that does not exceed a. In this case, the integrand in formula (4.191) is a periodic functions of τ and, therefore, the operator 𝒦Γ1 is well-defined. Lemma 4.6.1 also holds in this case, and the range I of the parameter τ has the form [0, ε0 ). To complete this section, we consider formula (4.191) for β = βε in finitedimensional quantum mechanics 2π

τ

2π

0

0

0

i 1 dτ (𝒦Γ1 f ) (x) = ∫ 1/4 exp{ ∫ PdQ − iτ{ ∫ PdQ}} ε 2πε ε ε

1 x − Q(τ) × exp{ P(τ)(x − Q(τ))}f (τ, ). √ε ε

(4.194)

The integrand in this formula is a periodic function in any case.

4.6.2 General case: main difficulties Now we consider the general case. Let Λk be a k-dimensional smooth manifold, and let X be a smooth mapping of this manifold into the phase space: X: Λk → ℳ.

(4.195)

We let τa , a = 1, k, denote the coordinates of points τ ∈ Λk in a certain chart. Definition 4.6.1. The union of a manifold Λk and a smooth mapping (4.195) such that 𝜕X X(τ) ≠ X(τ′ ) for τ ≠ τ′ and the vectors 𝜕τ ∈ T ℳX(τ) are linearly independent for all a

τ ∈ Λk will be called an isotropic manifold Γk in the phase space if ωX2 [

𝜕X 𝜕X , ] = 0. 𝜕τa 𝜕τb

(4.196)

4.6 Specific features of the Cauchy problem

| 269

Remark 4.6.1. In the case Λk = ℝk , Definition 4.6.1 is equivalent to Definition 4.5.1. Indeed, formula (4.172) implies (4.196), because 0=

𝜕2 s 𝜕s 𝜕 X 𝜕X 𝜕 X 𝜕X 𝜕X 𝜕X − = ω [ ]− ω [ ] = ωX2 [ , ]. 𝜕τa 𝜕τb 𝜕τb 𝜕τa 𝜕τa 1 𝜕τb 𝜕τb 1 𝜕τa 𝜕τa 𝜕τb

Conversely, assume that condition (4.196) is satisfied. Then the integral τ

∫ ωX1 [ 0

𝜕X ]dτa 𝜕τa

(4.197)

over a curve connecting the origin with the point τ is independent of the shape of this curve by formula (4.196), and hence, it can be taken as the function s. In the general case, formula (4.196) is not equivalent to formula (4.172), because the integral of the 1-form on Λk , ντ [a] = ωX1 [X∗ a],

a ∈ TΛkτ ,

(4.198)

over a curve connecting the points τ0 and τ, τ

∫ ν,

(4.199)

τ0

which coincides with the integral (4.197) in the case Λk = ℝk , depends on the shape of the curve. So the integral of the 1-form over a closed curve on Λk is, in general, different from zero. Therefore, to obtain a well-posed definition of the form (4.168) in the case of an isotropic manifold that is not diffeomorphic to ℝk , it is necessary to require that the integral of the 1-form ν over any closed contour γ be equal to 2πεN(γ), ∮ ν = 2πεN(γ),

N(γ) ∈ ℤ.

(4.200)

Then the quantity e

i s(τ) ε

τ

,

where s(τ) = ∫ ν, τ0

is uniquely determined on the manifold Λk . For contours γ and γ ′ on Λk such that there exists a two-dimensional surface on Λk with boundaries on these contours, the integrals (4.200) coincide. Therefore, it suffices to impose quantization conditions (4.200) on l basis cycles of the cohomology group H1 (Λk ) of the manifold Λk ; here, l is the di-

270 | 4 Complex germ method in the Fock space mension of H1 (Λk ) (the Betti number of the manifold Λk ), ∮ ν = 2πεNi ,

Ni ∈ ℤ.

γi

(4.201)

We note that, in the case considered in Section 4.3, these quantization conditions that turn into ∮ PdQ = 2πεNi γi

can be imposed only if the integrals ∮γ ν are commensurable, i. e., the ratio of two i such integrals ∮γ ν i

∮γ ν j

=

Ni Nj

(4.202)

is a rational number. Thus, it may seem that the canonical operator cannot at all be defined on an isotropic manifold if the ratio of integrals (4.201) is irrational. This difficulty was overcome in [4, 25] as follows: the canonical operator is considered not on a single isotropic manifold but on an l-parametric family of manifolds from which, for each ε, it is necessary to choose isotropic manifolds that satisfy the quantization condition (4.201). In this case, to construct an asymptotics of the solution of the Cauchy problem for the Schrödinger equation as ℏ → 0, we must determine not a k-parametric family of solutions of the Hamiltonian equations but a (k + 1)-parametric family, which is much more difficult from the computational standpoint. But it turns out that, by analogy with the definition of the canonical operator (4.194), which does not use the quantization conditions, one can obtain a well-conditioned definition of canonical operator in the general case without imposing any quantization conditions on ε at all.

4.6.3 Definition of the operator 𝒦Γk Let Γk be an isotropic manifold (4.195). We consider a 1-form ν of the form (4.198) on the manifold Λk . By ν1 , . . . , νl , we denote the basis 1-forms of the l-dimensional cohomology space H 1 (Λk ). Without loss of generality, we can assume that ∮γ νm = δjm ; j

otherwise, one can satisfy this relation by redefining the basis. By the definition of space H 1 , the 1-form ν can be represented as ν = ds + κ1 ν1 + ⋅ ⋅ ⋅ + κl νl ,

κ1 , . . . , κl ∈ ℝ,

(4.203)

4.6 Specific features of the Cauchy problem | 271

and this representation is unique. By M k , we denote a universal covering of the manifold Λk . Let s1 , . . . , sl be functions of M k such that dsi = νi . We denote βm = {

κm } 2πε

(4.204)

so that the function exp{

l i l ∑ κm sm (τ) − 2πi ∑ βm sm (τ)} ε m=1 m=1

on M k depends only on the projection of the element M k on Λk , and thus can be considered as a function on Λk . The following structures are necessary to define the operator 𝒦: (a) an isotropic manifold Γk in the phase space, which is a union of the manifold Λk X

and the embedding Λk → ℳ; (b) a measure dσ on the manifold Λk ; (c) basis 1-forms ν1 , . . . , νl of the l-dimensional cohomology space H 1 (Λk ) such that ∮γ νm = δjm ; j

(d) a point τ0 ∈ Λk .

By Sk , we denote the set of all functions f : Λk → ℱ such that property (4.174) is satisfied for τ ∈ Λk , and by ℒ we denote the linear space of sets of elements Ψε ∈ H ε , ε ∈ (0, ε0 ]. We define an operator 𝒦Γk : Sk → ℒ

as (𝒦Γk f )ε = ∫

l κm dσ i ε ̄ + 2πi ∑ sm (τ)( ̄ s( τ) − βm )}KX(τ) f (τ), exp{ k/4 ε 2πε ε m=1

(4.205)

where τ

s(τ)̄ = ∫ ds, τ0

τ̄

sm (τ)̄ = ∫ νm , τ0

the 1-form ds is defined by formula (4.203), and τ̄ is an arbitrary point of universal covering M k whose projection coincides with the point τ ∈ Λk .

272 | 4 Complex germ method in the Fock space Remark 4.6.2. Formula (4.205) implies ε

((𝒦Γk f1 )ε , (𝒦Γk f2 )ε ) → ∫ dσ ∫ dα ε→0

𝒟σ 𝒟τ

× (f1 (τ), VX(τ) (

𝜕X α )f (τ)), 𝜕τa a 2

(4.206)

and the proof is similar to that of Corollary 4.5.2 of Lemma 4.5.3. We let S0 denote the subspace of the space Sk consisting of all elements of Sk satisfying property (4.175). We define a canonical operator 𝕂Γk : Sk /S0 → ℒ/ℛ by analogy with Section 4.5 (formula (4.177)). We note that Theorem 4.4.1 and Corollary 4.4.11 of it also hold in the case of isotropic manifolds of general form.

4.7 Asymptotic formulas in the Fock space corresponding to finite-dimensional isotropic manifolds In this section, we apply the above developed concept to the canonical operator Kεφ : ℱ → ℱ , φ ∈ l2 , of the form Kεφ = exp{

1 ∞ + ∑ (a φ − a−k φ∗k )}. √ε k=1 k k

(4.207)

We first verify the properties of the operator Kεφ listed in Section 4.5. 4.7.1 Verification of properties of the canonical operator Let us consider the subset 𝒟̃ ⊂ ℱ . By ‖Φn ‖, we denote the norm of the nth component of the element Φ = (Φ0 , Φ1 , . . .) ∈ ℱ (Φn is a symmetric function of n arguments i1 , . . . , in ∈ ℕ in L2 (ℕ)). By 𝒟̃ , we denote the set of all elements Φ ∈ ℱ such that, for any M, ∞

∑ nM ‖Φn ‖2 < ∞.

n=0

(4.208)

In particular, 𝒟̃ contains vectors of the form i

0 1 (i) + am }Φ(0) , ∑ 𝒫 (i) (a+ ) exp{ a+l Mls(i) a+s + jm 2 i=1

(4.209)

4.7 Asymptotic formulas in the Fock space | 273

where 𝒫 (i) is a polynomial in creation operators and the matrix M (i) corresponds to the Hilbert–Schmidt operator with norm less than 1, j(i) ∈ l2 . Lemma 4.7.1. Properties (b) and (c) in Section 4.5 are satisfied for the set 𝒟̃ . Proof. By the Baker–Hausdorff formula, we have 󵄨 󵄨 󵄨 󵄨󵄨 ε 󵄨 󵄨 1 (φ φ∗ −φ∗ φ ) 󵄨 󵄨 ε ε ε 󵄨󵄨(Kφ1 f1 , Kφ2 f2 )󵄨󵄨󵄨 = 󵄨󵄨󵄨e 2ε 1l 2l 1l 2l 󵄨󵄨󵄨 󵄨󵄨󵄨(f1 , Kφ2 −φ1 f2 )󵄨󵄨󵄨 = 󵄨󵄨󵄨(f1 , Kφ2 −φ1 f2 )󵄨󵄨󵄨. We show that the following inequality is (globally) satisfied: ‖φ‖ 󵄨󵄨 󵄨 ε ), 󵄨󵄨(f1 , Kφ f2 )󵄨󵄨󵄨 ≤ ϕM ( √ε

M

ϕM (α)(α2 + 1) < const,

(4.210)

which will imply the assertion of the lemma. We write φ/√ε = χ. Then inequality (4.210) implies the following lemma. Lemma 4.7.2. For χ ∈ l2 , the following relation holds: + ∗ − M󵄨 󵄨 (χ ∗ χ + 1) 󵄨󵄨󵄨(f2 , eχi ai −χi ai f1 )󵄨󵄨󵄨 > CM ,

where CM is a constant independent of χ. Proof. It suffices to show that the quantity + ∗ − L󵄨 󵄨 (χj∗ χj ) 󵄨󵄨󵄨(f2 , eχi ai −χi ai f1 )󵄨󵄨󵄨

is bounded. We shall use the identity 1/2

(χj∗ χj ) (f2 , eχi ai −χi ai f1 ) = (f2 , [ +

∗ −

=

χj∗ a−j

; eχi ai −χi ai ]f1 ) +

∗ −

√χl∗ χl χj a+j + ∗ − f2 , eχi ai −χi ai f1 ) ( ∗χ √χl l

− (f2 , eχi ai −χi ai +

∗ −

χj∗ a−j √χl∗ χl

f1 ).

Applying this identity 2L times and using the unitarity of the operator eχi ai −χi ai and the property +

∗ −

󵄩󵄩 a±j χj 󵄩󵄩2 ∞ 󵄩󵄩 󵄩 f 󵄩󵄩 ≤ ∑ (n + 1)‖fn ‖2 , 󵄩󵄩 󵄩󵄩 √χ ∗ χ 󵄩󵄩󵄩 n=0 l l

we obtain the assertion of Lemma 4.7.2. The proof of Lemma 4.7.2, and hence, of Lemma 4.7.1 is complete.

274 | 4 Complex germ method in the Fock space Lemma 4.7.3. Assume that the relation ∞

󵄩2 󵄩 ∑ nM 󵄩󵄩󵄩f (τ)̄ 󵄩󵄩󵄩 ≤ CM

(4.211)

n=0

holds for all M. Then condition (4.174) is satisfied. Proof. The proof of this lemma is similar to that of Lemma 4.7.1. Instead of the set Sk , it is convenient to consider the subset S̃k , which is also dense f in L (ℝk → ℱ ) and consists of all smooth functions ℝk → D̃ with compact support 2

such that quantity (4.211) is bounded for any function together will all of its derivatives. Thus, we have verified the assumptions in Section 4.5. Now we shall apply the results of this section to construct a canonical operator on an isotropic manifold.

4.7.2 Canonical operator corresponding to an isotropic manifold φ

Let Γk be a k-dimensional isotropic manifold: ℝk → l2 . We consider the case of a manifold diffeomorphic to ℝk ; the general case is considered according to Section 4.6. The property of being isotropic means that 𝜕φ 𝜕φ∗ 𝜕φ 𝜕φ∗ = . 𝜕τa 𝜕τb 𝜕τb 𝜕τa

(4.212)

By s(τ), we denote a real function of the form τ

𝜕φ 𝜕φ∗ 1 s(τ) = ∫ (φ∗ − φ∗ )dτa , 2 𝜕τa 𝜕τa

(4.213)

0

and by ℒ we denote the linear space of functions [0, ε0 ) → ℱ . Let us consider an operator 𝒦Γk : S̃k → ℒ of the form (𝒦Γk f )ε = ∫

dτ ε1 s(τ) √ε1 (φj a+j −φ∗j a−j ) e f (τ), e ε3/4

f (τ) ∈ ℱ .

(4.214)

As was shown in Section 4.5 (Lemma 4.5.3 and its Corollary), for f ∈ S̃0 , i. e., for ∫ dα exp{iαa (

𝜕φi + 𝜕φ∗i − a − a )}f (τ) = 0, 𝜕τa i 𝜕τa i

we have 󵄩󵄩 ε󵄩 󵄩󵄩(𝒦Γk f ) 󵄩󵄩󵄩 → 0. ε→0

(4.215)

4.7 Asymptotic formulas in the Fock space | 275

Therefore, we can consider a subspace ℛ ⊂ ℒ of the form 󵄩 󵄩 Ψε ∈ ℛ ⇐⇒ 󵄩󵄩󵄩Ψε 󵄩󵄩󵄩 → 0, ε→0

the corresponding quotient space ℒ/ℛ, and the canonical operator 𝕂Γk : S̃k /S̃0 → ℒ/ℛ of the form (4.177). We consider the structure of the quotient space S̃k /S̃0 in more detail. For this, we consider the mapping JΓk : S̃k /S̃0 → Im JΓk ⊂ S̃k′ which, in this case, has the form JΓk [f ] = ∫ dα exp{iαa (

𝜕φi + 𝜕φ∗i − a − a )}f (τ), 𝜕τa i 𝜕τa i

(4.216)

and f (τ) is a representative of the class [f ] ∈ S̃k /S̃0 . Although JΓk [f ] does not belong to the Fock space, we can associate it with the analytic generating functional JΓk [f ](τ, z) = ∫ dα(Cz , exp{αa (

𝜕φi + 𝜕φ∗i − a − a )}f (τ)), 𝜕τa i 𝜕τa i

(4.217)

where z ∈ ℂ, Cz is a coherent state. We note that a reasoning similar to Lemma 4.7.2 proves the convergence of this integral. Now we investigate which functionals correspond to the elements of Im JΓk , i. e., we study the structure of the quotient space S̃k /S̃0 . First, we prove several auxiliary assertions. Lemma 4.7.4. The functional (4.217) can be expressed in terms of the generating functional f (τ, z) corresponding to the vector f (τ) ∈ ℱ as follows: JΓk [f ](τ, z) = ∫ dα exp{αa (

𝜕φ∗i 𝜕 𝜕φi z− )}f (τ, a). 𝜕τa 𝜕τa 𝜕zi

Proof. The proof follows from the relation: (Cz , F(a+ , a− )Ψ) = F(z,

𝜕 )(Cz , Ψ) 𝜕z

and the definition of generating functional. Lemma 4.7.5. The vectors

𝜕φ 𝜕τa

∈ l2 , a = 1, k, are linearly independent.

Proof. The proof completely repeats the proof of Lemma 4.4.9.

276 | 4 Complex germ method in the Fock space At each point τ, we consider an orthonormal basis in the space l2 the first k of whose components χ1 , . . . χk are basis vectors in the subspace spanned by the vectors 𝜕φ/𝜕τa . We associate each element z ∈ l2 with a set of numbers ζi = (χi , z), then we have ∞ z = ∑∞ i=1 ζi χi . We define f (τ, ζ1 , ζ2 , . . .) = f (τ, ∑i=1 ζi χi ) and introduce a similar relation for JΓk [f ](τ, ζ1 , ζ2 , . . .). Since 𝜕φ = λab χb , 𝜕τa we have JΓk [f ](τ, ζ1 , ζ2 , . . .) = ∫ dα exp{αa λab (ζb −

𝜕 )}f (τ, ζ1 , . . . , ζk , ζk+1 , . . .). 𝜕ζb

(4.218)

We note that the functional JΓk [f ](τ, ζ1 , . . . , ζk , ζk+1 , . . .) satisfies the following condition for fixed τ, ζ1 , . . . , ζk : 󵄨 󵄨2 ∫ ∏ dζα∗ dζα 󵄨󵄨󵄨JΓk [f ]󵄨󵄨󵄨 exp{− ∑ ζα∗ ζα } < ∞. α≥k+1

α≥k+1

(4.219)

Indeed, since the element f of the Fock space satisfies condition (4.219) and the integration in formula (4.218) does not concern the variables ζk+1 , ζk+2 , . . . , we obtain formula (4.219) for JΓk [f ]. Further, we note that the functional JΓk [f ] satisfies the equation (ζb −

𝜕 )J k = 0, 𝜕ζb Γ

b = 1, k,

which is a consequence of Lemma 4.5.4. This implies that the set Im JΓk is contained in the set of functionals of the form 1

{JΓk [f ](τ, ζ1 , ζ2 , . . .) = e 2 ζb ζb g(τ, ζk+1 , ζk+2 , . . .)},

(4.220)

where g smoothly depend on τ. Choosing the function f in (4.218) in the form f (τ, ζ1 , . . . , ζk , ζk+1 ) =

det λ g(τ, ζk+1 , ζk+2 , . . .), (2π)k/2

(4.221)

we see that any element of the set (4.220) belongs to Im JΓk . Thus, we have found two parametrizations of the quotient space S̃k /S̃0 : (a) each element of the quotient space S̃k /S̃0 is bijectively associated with a functional (4.220) satisfying an additional condition (Lemma 4.5.4); (b) each element of S̃k /S̃0 is bijectively associated with a set of vectors f (τ) ∈ ℱ of the form (4.221), which is a representative of this equivalence class and satisfies the

4.7 Asymptotic formulas in the Fock space | 277

additional conditions 𝜕φi − a f (τ) = 0, 𝜕τa i

(4.222)

and, as we just showed, f (τ) is uniquely defined by this condition. We present a useful formula for the inner product of two elements of the quotient space S̃k /S̃0 in representation (b): ([f1 ], [f2 ]) = ∫ dτ(f1 (τ), f2 (τ)) where det λ = √det

𝜕φ∗ 𝜕φ , f , and f2 𝜕τa 𝜕τb 1

(2π)k/2 , det λ

(4.223)

satisfy condition (4.222). Formula (4.223) follows

directly from Definition (4.178). We also note that, after the transition from variables ζ back to variables z, formula (4.221) becomes f (τ, z) = √

1 𝜕φ −1 𝜕φ det R exp{− z R z}JΓk [f ](τ, z), k 2 𝜕τa ab 𝜕τb (2π)

Rab =

𝜕φ∗ 𝜕φ . 𝜕τa 𝜕τb

(4.224)

This formula can easily be obtained from Lemma 4.7.4 and formula (4.222). Thus, we have defined canonical operator on an isotropic manifold, which takes an element of the quotient space S̃k /S̃0 to an equivalence class in ℒ/ℛ whose representative is the element (4.213).

4.7.3 Isotropic manifold of special form and fixing the number of particles It turns out that, choosing an isotropic manifold of a special form and a special dependence f (τ), one can obtain an element (𝒦Γk f )ε such that only one of its components is different from zero, and the number of this component is of order of 1/ε. Thus, we show that the multiparticle canonical operator considered in Chapter 2 is, in fact, a special case of the theory of isotropic manifolds in the infinite-dimensional case. We note that the theory of germ at a point considered in Section 4.2 gives asymptotics such that all of their components are different from zero. We consider a one-dimensional isotropic manifold Γk diffeomorphic to a circle φ(τ) = φeiτ ,

τ ∈ [0, 2π).

(4.225)

We use the example to illustrate different methods for choosing the quantization conditions, which were proposed in Section 4.6.1.

278 | 4 Complex germ method in the Fock space (a) Quantization condition of the form (4.186): φ∗ φ 1 = N, ∮ φ∗ dφ = 2πεi ε

N ∈ ℤ,

(4.226)

and f is a single-valued function on the circle. In this case, the parameter ε runs through the set of discrete values (4.226) and the function s has the form s(τ) = −φ∗ φτ.

(4.227)

We choose a function f (τ) so that its sth component depends on τ as fs (τ) = fs eisτ ,

(4.228)

where fs is a symmetric function in L2 (ℕs ). Lemma 4.7.6. Let φ∗ φ = 1, and let φ∗i a−i f = 0.

(4.229)

Then with an accuracy up to a coefficient, the 1/εth component of the vector (𝒦Γ1 f )ε coincides with Kφ,N f , N −N √N √N e Kφ,N f , N!

(4.230)

where Kφ,N is the multiparticle canonical operator introduced in Chapter 2 and the other components of the vector (𝒦Γ1 f )ε are zero. Remark 4.7.1. 1. Property (4.229) is a special case of property (4.222). 2. The number N in quantization condition (4.226) plays the role of the number of particles. Proof. Let us verify formula (4.230) for the coherent state Cg eiτ . We have 2π

(𝒦Γ1 Cg eiτ )ε = ∫ 0

dτ − εi − 2ε1 √ε1 φl a+l eiτ gl a+l (0) e e e Φ , ε1/4

where Φ(0) is the vacuum vector; here, we used the formula 1

e √ε

(φl (τ)a+l −φ∗l (τ)a−l )

1

f (τ) = e √ε

φl (τ)a+l − 2ε1 φ∗l (τ)φl (τ)

f (τ),

(4.231)

4.7 Asymptotic formulas in the Fock space | 279

which follows from property (4.229). To calculate the integral over τ in formula (4.231), 1

we expand the exponential e √ε

(φl (τ)+gl )a+l

2π

in a power series. Using the property,

2π, ν = 0,

∫ dτeiντ = {

0,

0

ν ≠ 0, ν ∈ ℤ,

we obtain (𝒦Γ1 Cg eiτ )ε =

N

g N 1/4 e−N/2 N N/2 1 ((φl + l )a+l ) Φ(0) . √N! √N! √N

Using formula (2.A.4), we obtain formula (4.230) for coherent states. It follows from the completeness of coherent states that formula (4.230) holds for any f of the form (4.228). The proof of the lemma is complete. Remark 4.7.2. If we assume that the sth component of f depends on τ not as in (4.228) but as fs (τ) = fs ei(s+1)τ , then not the Nth but the (N − 1)th component of the vector (𝒦Γ1 f )ε is different from zero. So we see that not the integer N in formula (4.226) but the number N − 1 has the meaning of the number of particles. Thus, the physical meaning of the quantization condition is determined not only by the these conditions but also by the choice of the function f . (b) We consider the quantization condition of the form ε(N + β) =

1 ∮ φ∗ dφ = φ∗ φ. 2πi

(4.232)

In this case, the function f in the definition of canonical operator (4.214) must be multiplied by e−iβτ (formula (4.191)). If f has the form (4.228), then the integer N in formula (4.232) again has the meaning of the total number of particles. Moreover, a variation in the quantization condition does not change formula (4.230) in the leading order in ε (it is only multiplied by the multiplier (1 + O(ε)) independent of f ). The difference is that not the vector (𝒦Γ1 f )1/N that is at all undefined by condition (4.232) but the vector (𝒦Γ1 f )1/(N+β) will approximately coincide with the vector all of whose components are zero except for the Nth component given by formula (4.230). (c) We can at all avoid choosing the quantization conditions and define the canonical operator in the form similar to (4.194), 2π

(𝒦Γ1 f )ε = ∫ 0

dτ − 2ε1 −iτ[ ε1 ]+ √ε1 φl a+l eiτ e e f (τ), ε1/4

(4.233)

where [1/ε] is the integral part of 1/ε. In this case, if we choose f in the form (4.228), then the [1/ε]th component of (𝒦Γ1 f )ε is different from zero and coincides with (4.230) and the other components of (𝒦Γ1 f )ε are zero.

280 | 4 Complex germ method in the Fock space Thus, we showed that, in the case of a special choice of isotropic manifold and the dependence of f on τ, the canonical operator (𝒦Γk f )ε actually coincides with the multiparticle canonical operator introduced in Chapter 2. If the operator Hε,t in (4.43) contains only the terms with equally many creation and annihilation operators, then we obtain the case considered in Chapter 2, because the number of particles remains unchanged. The following statements of the Cauchy problem are possible. (a) Quantization condition (4.226) is chosen. In this case, ε takes the values 1/N, the initial condition in the Cauchy problem Ψε (0) is the Fock space element (𝒦Γ1 f )ε , and only its 1/ε = Nth component is different from zero and equal to Kφ0 ,N f0 . The solution of the Cauchy problem Ψε (t) also has only the Nth component that is different from zero, while the other components are zero. The function Ψεas (t) i

t

is constructed. It is an element (𝒦Γ1 ft )ε e ε S ∈ ℱ and only its Nth component is different from zero. We shall prove that 󵄩󵄩 ε 󵄩 ε 󵄩󵄩Ψas (t) − Ψ (t)󵄩󵄩󵄩 → 0. ε→0

Note that the squared norm in this formula coincides with the squared norm in the space L2 (ℕN ) and the parameter ε in the Hamiltonian can be replaced with 1/N. (b) Quantization condition (4.232) is chosen, ε takes the values 1/(N + β). The initial condition of the Cauchy problem, just as the solution of this problem, has only the 1/ε − β = Nth component different from zero, and the parameter ε in the Hamiltonian can be replaced with 1/(N + β). (c) No quantization condition is chosen. In this case, ε takes any arbitrary value, and the exact and approximate solutions of the Cauchy problem have only the [1/ε]th component that is different from zero. Now we consider the case where the wave function f (τ) satisfies additional condition (4.229). By the results obtained above, for any function f (τ) ∈ ℱ , there exists a unique function f ̃(τ) ∈ ℱ satisfying condition (4.229) and such that ε󵄩 󵄩󵄩 ε ̃ ε ε 󵄩󵄩(𝒦Γ1 f ) − (𝒦Γ1 f ) 󵄩󵄩󵄩 → 0. ε→0

(4.234)

4.7.4 Case of two types of particles Now we consider the case of two types of particles. Namely, we assume that the space 𝒳 with measure dx can be represented as the direct product ℕ×{1, 2}, and thus the creation and annihilation operators a±A,i have two indices the first of which corresponds to the number of the particle type, and the second, to the number of the degree of freedom.

4.7 Asymptotic formulas in the Fock space | 281

If the Hamiltonian contains only the terms with the same number of creation and annihilation operators for each type of particles, the subspace of the Fock space consisting of elements of the form ∞ 1 ∑ √N1 !N2 ! i1 ,...,iN ,j1 ,...,jN

=1 2

1

Φi ⋅⋅⋅i1

(N ,N2 ) a+ 1 N1 j1 ⋅⋅⋅jN2 1,i1

⋅ ⋅ ⋅ a+1,iN a+2,j1 ⋅ ⋅ ⋅ a+2,jN Φ(0) 1

2

(4.235)

is invariant with respect to evolution. The vectors (4.235) have the physical meaning of states in which the number of particles of the first type is equal to N1 and the number (N ,N ) of particles of the second type is equal to N2 . The function Φi ⋅⋅⋅i1 2j ⋅⋅⋅j that is separately 1

N1 1

N2

symmetric in the indices i and in the indices j plays the role of the wave function. To construct asymptotics of the form (4.235), we can consider a two-dimensional isotropic manifold of the form φ1,i (τ1 , τ2 ) = φ1,i eiτ1 ,

φ2,i (τ1 , τ2 ) = φ2,i eiτ2 ,

τ1 , τ2 ∈ (0, 2π⌋.

(4.236)

We note that, in this case, the quantization conditions of the form (4.226) cannot be imposed in general, because they have the form ∞

∑ φ∗1,i φ1,i = εN1 , i=1

∞

∑ φ∗2,i φ2,i = εN2 i=1

(4.237)

∞ ∗ ∗ and, in the case of incommensurable quantities α1 = ∑∞ i=1 φ1,i φ1,i and α2 = ∑i=1 φ2,i φ2,i , cannot be satisfied. Therefore, we shall use the concept described in Section 4.6.3, which does not employ the quantization conditions. We choose an isotropic manifold Γ2 , which is the embedding of a two-dimensional torus Λ2 into the phase space l2 of the form (4.236); we take the measure dσ on the torus in the form dτ1 dτ2 , the basis forms of the space H 1 (Λ2 ) in the form dτ1 /2π and dτ2 /2π, and the point τ(0) at which τ1 and τ2 are zero. By Section 4.6.3, the canonical operator can be written as

∫ Λ2

dτ1 dτ2 −iτ1 [ αε1 ]−iτ2 [ αε2 ] √ε1 (φ1,l a+1,l eiτ1 −φ∗1,l a−1,l e−iτ1 ) e e √ε 1

× e √ε

(φ2,l a+2,l eiτ2 −φ∗2,l a−2,l e−iτ2 )

f (τ1 τ2 ).

(4.238)

If f (τ1 τ2 ) ∈ ℱ is chosen in the form, f (a+1,⋅ eiτ1 , a+2,⋅ eiτ2 )Φ(0) (by a+1,⋅ we denote the set of operators a+1,1 , a+1,2 , . . . ), then the integral (4.237) gives the element of the space ℱ at which the number of particles of the first type is equal to

282 | 4 Complex germ method in the Fock space [α1 /ε] and the number of particles of the second type is equal to [α2 /ε]. Formula (4.238) can be used for any ε. A similar argument can be used to study the case of k types of particles; in this case, it is necessary to consider a k-dimensional isotropic manifold.

4.7.5 Complex germ Now we return to finite-dimensional isotropic manifolds of general form. We consider an element [f ] of the quotient space S̃k /S̃0 associated with the functional JΓk [f ](τ, z) of Gaussian form JΓk [f ](τ, z) = c(τ) exp{

1 ∞ ∑ z M (τ)zj }, 2 i,j=1 i ij

(4.239)

where c(τ) and Mij (τ) are smooth finite functions. Lemma 4.7.7. For the functional (4.239) to belong to the space Im JΓk , it is necessary and sufficient that the following conditions be satisfied: (a) M

𝜕φ∗ 𝜕φ = ; 𝜕τa 𝜕τa

(4.240)

(b) the operator M̃ associated with a matrix of the form 𝜕φi −1 𝜕φj Rab , M̃ ij = Mij − ∑ 𝜕τ 𝜕τb a a,b=1

Rab =

𝜕φ∗ 𝜕φ , 𝜕τa 𝜕τb

(4.241)

is the Hilbert–Schmidt operator with norm less than 1. If the above listed conditions are satisfied, then the squared norm of the element Im JΓk (4.239) has the form ∫ dτ

|c|2 ̃ √det(E − M̃ + M)

√

1 𝜕φ∗ 𝜕φ . (2π)k 𝜕τa 𝜕τb

Proof. By the results of Section 4.7.2, the functional (4.239) belongs to the set Im JΓk if and only if the following two conditions are satisfied: (a) (

𝜕φ 𝜕φ∗ 𝜕 z− )J k [f ](τ, z) = 0; 𝜕τa 𝜕τa 𝜕z Γ

(4.242)

4.7 Asymptotic formulas in the Fock space | 283

(b) the functional exp{

1 ∞ ∑ z M̃ (τ)zj }, 2 i,j=1 i ij

where M̃ ij has the form (4.241), corresponds to an element of the Fock space. These two conditions are equivalent to the first item in the lemma. To verify the second assertion in the lemma, it suffices to use formula (4.223) and the results of Appendix 2.A. Lemma 4.7.7 is proved. It turns out that the Gaussian functional (4.239) can be interpreted geometrically in terms of complex germ, i. e., the graph of the operator M(τ). Definition 4.7.1. The complex germ corresponding to the Gaussian functional (4.239) is τ the set of subspaces 𝒢M(τ) , τ ∈ ℝk , of complexified tangent spaces T ℳℂ φ(τ) , τ

2

𝒢M(τ) = {(ζ , M(τ)ζ ) | ζ ∈ l }.

Lemma 4.7.8. The complex germ satisfies the following properties: (g1) the vectors (

𝜕φ∗ 𝜕φ , ), 𝜕τa 𝜕τa

a = 1, k,

(4.243)

τ belong to 𝒢M(τ) ; τ (g2) for (ζ1 , η1 ), (ζ2 , η2 ) ∈ 𝒢M(τ) ,

ζ1 η2 − ζ2 η1 = 0; τ (g3) for (ζ , η) ∈ 𝒢M(τ) ,

ζ ∗ ζ − η∗ η ≥ 0, and the equality is attained if and only if the vector (ζ , η) can be represented as a linear combination of vectors (4.243). Proof. Property (g1) follows from formula (4.240). Property (g2) follows from the symmetry of the matrix M. To verify property (g3), we write ζ as k

𝜕φ∗ μa + ζ⊥ , a=1 𝜕τa

ζ =∑

ζ⊥

𝜕φ = 0, 𝜕τa

a = 1, k.

284 | 4 Complex germ method in the Fock space Then k

𝜕φ μa + Mζ⊥ , 𝜕τ a a=1

η=∑

𝜕φ∗ Mζ = 0. 𝜕τa ⊥

Therefore, ζ ∗ ζ − η∗ η = ζ⊥∗ (E − M + M)ζ⊥ .

(4.244)

̃ ⊥ , it follows that the quantity (4.244) is positive for ζ⊥ ≠ 0. The property Since Mζ⊥ = Mζ (g3) is verifies. The proof of Lemma 4.7.8 is complete. Lemma 4.7.9. Assume that an operator of the form ηm zm − ζm 𝜕z𝜕 takes the funcm τ tional (4.239) to zero. Then (ζ , η) ∈ 𝒢M(τ) . Proof. The proof is obvious. Now we introduce germ operators of creation and annihilation. For this, we consider the subspace of the space l2 , which is spanned by the vectors 𝜕φ∗ /𝜕τa , a = 1, k, and the orthogonal complement ℋΓk of this space. In ℋΓk , we introduce another inner product as ̃ 2 ). (ξ1 , ξ2 )′ = (ξ1 , (E − M̃ + M)ξ

(4.245)

̃ −1 are bounded, and hence, the space ℋ k The operators E − M̃ + M̃ and (E − M̃ + M) Γ with inner product (4.245) is complete, i. e., it is a Hilbert space. We consider {Gα′ , α = k + 1, ∞}, which is a basis in ℋΓk orthonormal with respect to the inner product (4.245). 𝜕φ Since Gα′ ∈ ℋΓk , we see that Gα′ 𝜕τ = 0. The set of the functions Gα′ and the functions a Ga = 𝜕φ∗ /𝜕τa , a = 1, k, is a basis in l2 . Instead of the functions Gα′ we can consider elements of l2 of the form Gα = Ga + λαa 𝜕φ∗ /𝜕τa , where λαa are elements of l2 , for each a = 1, k. We denote (Gi )m ≡ Gmi , Fi = MGi , (Fi )m ≡ Fmi . Properties (g2) and (g3) can be written as G+ G − F + F = L

(4.246)

(Lαβ is equal to δαβ for α, β > k and equal to zero if at least one of the indices does not exceed k), GT F = F T G.

(4.247)

When deriving property (4.246), we took into account that the basis Gα′ is orthonormal with respect to the inner product (4.245).

4.7 Asymptotic formulas in the Fock space

| 285

We introduce operators Λα and Λ̄ α as ∞

Λα = −i ∑ (Fmα zm − Gmα m=1 ∞

𝜕 ), 𝜕zm

(4.248)

𝜕 ∗ ∗ Λ̄ α = −i ∑ (Gmα zm − Fmα ). 𝜕z m m=1 Lemma 4.7.10. The operators Λα and Λ̄ α satisfy the relations [Λα , Λβ ] = [Λ̄ α , Λ̄ β ] = 0,

[Λα , Λ̄ β ] = δαβ ,

α, β ≥ k + 1,

and the property Λα JΓk [f ] = 0,

where JΓk [f ] has the form (4.239).

Proof. The proof follows from formulas (4.239), (4.246), (4.247), and (4.248). ̃ ̃ ̃ We consider an operator wΓF,G k : Sk → Sk /S0 uniquely determined from the relation F,G F,G = JΓk w k , where the operator w̃ k : S̃k → Im JΓk has the form

w̃ ΓF,G k

Γ

Γ

(w̃ ΓF,G k χ) =

∞ 1 χ (n) Λ̄ ⋅ ⋅ ⋅ Λ̄ αn ΦM , ∑ √det G+ G n=0 √n! α ,...,α =k+1 α1 ⋅⋅⋅αn α1 1 n

1

∞

∑

(4.249)

where χ = (χ (0) , χα(1)1 , χα(2) , . . .) ∈ S̃k is a set of functions χα(n) symmetric with respect to 1 α2 1 ⋅⋅⋅αn

α1 , . . . , αn = k + 1, ∞ for which the series (4.208) converges.

̃ Lemma 4.7.11. The operator w̃ ΓF,G k is a one-to-one mapping of Sk onto Im JΓk , and it preserves the norm. Proof. Without loss of generality, we can assume that Gα 𝜕τ

𝜕φ a

2

= 0 and the vectors

𝜕φ /𝜕τa form a subspace of l of the form (α1 , . . . , αk , 0, 0, . . .). Otherwise, this can be 𝜕φ 𝜕φ∗ 𝜕 attained by adding linear combinations of operators of the form 𝜕τ z − 𝜕τ 𝜕z to the a a 2 ̄ operators Λα and redefining the basis in the space l . In this case, the element Im J k ∗

Γ

is associated with the element √4 (2π)k

∞ 1 χ (n) Ā ⋅ ⋅ ⋅ Ā αn ∑ √4 det G+ G n=0 √n! α ,...,α =k+1 α1 ⋅⋅⋅αn α1 1 n

×√

∞

∑

det R 1 ∞ exp{ ∑ a+ M̃ a+ }Φ(0) 2 α,β=k+1 α αβ β (2π)k

(4.250)

286 | 4 Complex germ method in the Fock space of the quotient space, which satisfies additional condition (4.222). In formula (4.250), the operators Ā α have the form ∞

∗ + ∗ − Ā α = −i ∑ (Gβα aβ − Fβα aβ ), β=1

∞

Aα = −i ∑ (Fβα a+β − Gβα a+β ). β=1

(4.251)

We note that, by the definition in Appendix 2.A in Chapter 2, the transformation (4.251) is a linear proper canonical transformation. It is easy to see that such transformations indeed take S̃k into S̃k . But the vector (4.250) coincides up to a constant precisely with the canonical transformation of the vector χ ∈ S̃k . Therefore, the mapping (4.249) is one-to-one. To verify that the norm is preserved, it suffices to consider the transformed vacuum vector √4 (2π)k √4 det G+ G

√

1 ∞ det R exp{ ∑ a+ M̃ a+ }Φ(0) , 2 α,β=k+1 α αβ β (2π)k

whose squared norm, by formula (4.223), has the form √det R 1 . ̃ √det G+ G det(E − M̃ + M)

(4.252)

The matrix G+ G consists of two blocks; namely, if the indices of this matrix coincide 𝜕φ∗ 𝜕φ with a, b = 1, k, then (G+ G)ab = 𝜕τ 𝜕τ ; if the indices coincide with α, β, where α, β ≥ a a ̃ −1 . The matrix elements (G+ G)aα are zero, and hence, k + 1, then (G+ G)αβ = (E − M̃ + M) αβ

the quantity (4.252) is equal to 1. This implies that the norm is preserved. Lemma 4.7.11 is proved.

4.7.6 Canonical transformation of complex germ and formal asymptotic solutions Now we consider the canonical transformation of complex germ. Let U be a proper canonical transformation of the phase space. We consider the superposition of formal asymptotic solutions of the equations of motion of the form (𝒦Γk Wf )ε = ∫ t

− dτ εi St (τ) √ε1 (φtl a+l −φ∗t l al ) e e Wt f (τ) k/4 ε

(4.253)

and note that, instead of an element Wt f (τ) ∈ S̃k , we can choose any representative of the corresponding equivalence class of S̃k /S̃0 . Thus, we have defined an operator Wt in the space S̃k /S̃0 of the form Wt [f ] = [Wt f ].

(4.254)

4.7 Asymptotic formulas in the Fock space | 287

By Lemma 4.5.6, this operator is a well-defined. We shall express the operator Wt in terms of the operator wΓF,G k introduced in the preceding subsection. We first consider how the operator Wt acts on the element of the quotient space S̃k /S̃0 corresponding to the Gaussian function (4.239) and into which subspace the complex germ is taken by a canonical transformation. It turns out that the direct calculation of the functional JΓ̃k W[f ] (here Γ̃ k is the transformed isotropic manifold) is rather cumbersome in this case. We simplify the calculations by using the fact that, by Lemma 4.7.4, the functional JΓ̃k W[f ] is Gaussian and apply Lemma 4.7.9. We determine the transformed elements of l2 : 𝜕φ̃ ∗ 𝜕φ̃ ∗ G̃ i = Gi + F, ∗ 𝜕φ 𝜕φ i

𝜕φ̃ 𝜕φ̃ F̃i = G + F. 𝜕φ∗ i 𝜕φ i

(4.255)

Lemma 4.7.12. The relations WΛα = Λ̃ α W and W Λ̄ α = Λ̄̃ α W hold. Proof. The proof follows from the definition of the operator W. Corollary 4.7.1. The functional JΓ̃k W[f ] has the form c′ (τ) exp{

1 ∞ ∑ z M ′ (τ)zj }, 2 i,j=1 i ij

where the operator Mij′ has graph of the form {(G̃ i ζi , F̃i ζi ), ζ ∈ l2 }. Proof. The proof follows from Lemmas 4.7.9 and 4.7.12. Corollary 4.7.2. The operator W can be represented as W = cwΓF,̃kG (wΓF,G , k ) ̃ ̃

−1

|c| = 1.

Proof. This formula follows from Lemma 4.7.12, Corollary 4.7.1, and the definition of operator W. We have thus constructed the formal asymptotic solutions Ψεas (t) = ∫

− l l dτ εi St (τ) √ε1 (φtl a+l −φ∗t l al ) t e e c wΓFk,G f0 . k/4 t ε

(4.256)

Remark 4.7.3. In the example considered in Section 4.7.3, the method developed here agrees well with the method developed in Chapter 2.

288 | 4 Complex germ method in the Fock space 4.7.7 Asymptotic solutions of the Cauchy problem In this section, we show that the formal asymptotic solutions (4.256) indeed satisfy equation (4.43). As in Section 4.2, we consider the operator 1 Hε,t = Ht (√εa+ , √εa− ) + H1,t (√εa+ , √εa− ) + ⋅ ⋅ ⋅ ε + εk−1 Hk,t (√εa+ , √εa− ). We write ∞ 2 𝜕2 Ht + 1 + 𝜕 Ht − a + a H2,t = ∑ [ a+i i ∗ 𝜕φ∗ j ∗ 𝜕φ aj 2 𝜕φ 𝜕φ j j i i i,j=1

𝜕2 Ht − 1 + a−i a ] + H1 (t). 2 𝜕φi 𝜕φj j

(4.257)

d − H2,t )f t ] = 0, dt

(4.258)

Lemma 4.7.13. Let [(i and Hε,t f t ∈ S̃k . Then 󵄩󵄩 d 󵄩ε 󵄩󵄩 󵄩 t ε󵄩 1/2 󵄩󵄩(i − Hε,t )(𝒦Γkt f ) 󵄩󵄩󵄩 = O(ε ). 󵄩󵄩 dt 󵄩󵄩 Proof. The proof is similar to that of Lemma 4.2.14. d Remark 4.7.4. By [(i dt − H2,t )f t ], we denote the equivalence class in S̃k /S̃0 whose representative is

(i

d − H2,t )f t . dt

Lemma 4.7.14. To satisfy condition (4.257), it is necessary and sufficient that (i

d − H2,t )JΓk [f t ] = 0. dt

Proof. The proof follows from the definition of operator JΓk . Lemmas 4.7.13 and 4.7.14 imply the following theorem.

Appendix 4.A. Separation of cyclic variable and construction of tunnel asymptotics | 289

Theorem 4.7.1. Under the conditions of Lemma 4.7.13, the following relation holds for element (4.256) of ℱ : 󵄩󵄩 d 󵄩󵄩 󵄩󵄩 󵄩 1/2 ε 󵄩󵄩(i − Hε,t )Ψas (t)󵄩󵄩󵄩 = O(ε ) 󵄩󵄩 󵄩󵄩 dt for t

𝜕2 Ht ′ 1 c = exp{− ∫[Re Tr( M(t ′ )) + H1 (t ′ )]dt ′ }. 2 𝜕φ𝜕φ t

0

Thus, the function (4.256) is indeed an approximate solution of equation (4.43). To prove Theorem 4.7.1, it suffices to use the definition of operator w and the argument similar to that in the proof of Lemma 4.2.11.

Appendix 4.A. Separation of the cyclic variable and construction of the tunnel asymptotics 4.A.1 Additive asymptotics: advantages and drawbacks Throughout the book, we have constructed functions ψas,t such that the difference ε between such functions and the solution ψtε of the Cauchy problem of equation (1.1) is small as ε → 0. We showed that, as ε → 0, the constructed asymptotics can be used to calculate the mean values of general observables that are uniformly bounded in ε with an arbitrary accuracy of O(ε1/2 ) if we consider the leading asymptotics and with an arbitrary accuracy of O(εM/2 ) if we take into account the first M −1 corrections to the leading asymptotics. In particular, one can determine asymptotics of the probabilities of any events if these probabilities do not tend to zero as ε → 0. The leading asymptotics of mean values, which are exponentially small, as ε → 0 cannot be obtained from the constructed approximate solutions, because the error is in this case a power-law function of ε and exceeds the corresponding mean value. For instance, an example of exponentially small mean value can be the probability of the event that all particles are gathered in a certain domain of the space. In this section, we consider the tunnel asymptotics, which can also be used to calculate the exponentially small mean values.

4.A.2 Equation in the representation of generating functionals To construct the tunnel asymptotics, we consider equation (4.2) in the representation of Berezin generating functionals (Appendix 2.A) in which the elements of Fock space

290 | 4 Complex germ method in the Fock space are associated with functionals of infinitely many variables a∗1 , a∗2 , . . . ∈ ℂ, the creation operators a+i are the operators of multiplication by a∗i , the operators a−i are the differentiation operators 𝜕/𝜕a∗i , and the vacuum vector of the Fock space is associated with the functional identically equal to 1. In this representation, we consider equation (2.3) with the set 𝒳 of the form 𝒵+ = {0, 1, 2, . . . }. We thus consider the case where the operator (4.1) contains only terms with equally many creation and annihilation operators and elements of the Fock space that + − are eigenvectors of the operator of the number of particles ∑∞ k=0 ak ak with eigenvalue N, ε = 1/N. These elements of the Fock space are associated with the functionals Φt , which vary by λN times as the argument a∗ varies by λ times. This means that Φt (a∗0 , a∗1 , a∗2 , . . . ) = Φt (1,

a∗1 a∗2 a∗ a∗ , ∗ , . . . ) = Φ̃ t ( 1∗ , ∗2 , . . . ). ∗ a0 a0 a0 a0

In this representation, equation (4.3) becomes 1 𝜕 i 𝜕Φt )Φt , = H(a∗ , N 𝜕t N 𝜕a∗

(4.A.1)

where NH(a∗ ,

L

∞ 0 𝜕 𝜕 1 𝜕 1 H (l) a∗ ⋅ ⋅ ⋅ a∗il ∗ ∗ . )=∑ ∑ ∗ l−1 i1 ⋅⋅⋅il j1 ⋅⋅⋅jl i1 N 𝜕a 𝜕a 𝜕aj N j l=1 i ,...,i ,j ,...,j =0 1

l 1

l

1

l

Lemma 4.A.1. The functional Φ̃ t satisfies the equation (

i 𝜕 1 ∞ 𝜕 1 𝜕 − H(1, ζ1 , ζ2 , . . . ; 1 − ∑ ζβ , , . . .)) N 𝜕t N β=1 𝜕ζβ N 𝜕ζ2 × Φ̃ t (ζ1 , ζ2 , . . .) = 0.

(4.A.2)

Proof. The proof directly follows from the definitions of the functional Φ̃ t and equation (4.A.1). Remark 4.A.1. 1. If we pass from the functional Φ̃ t to an element of the Fock space, then the obtained equation is an equation of type (4.2) with the parameter ε replaced with 1/N. 2. If we apply the complex germ method to the obtained equation at a point of the Fock space (Section 4.2), then the results obtained by this method coincide with the results obtained in Chapter 2.

Appendix 4.A. Separation of cyclic variable and construction of tunnel asymptotics | 291

4.A.3 Definition of tunnel asymptotics Now we shall construct tunnel asymptotics for stationary solutions of equations (4.A.1) and (4.A.2). Definition 4.A.1. The generating functional ΦNt = e−iNEN t eNS[a

∗

χ

/N] (N)

(4.A.3)

[a∗ /N]

is called the tunnel asymptotics of the solution of equation (4.A.1) with an accuracy of O(1/N M+1 ) if e−NS[a

∗

/N]

(−

1 𝜕 1 i 𝜕 + H(a∗ , )) = O( M+2 ). N 𝜕t N 𝜕a∗ N

(4.A.4)

Remark 4.A.2. We consider the inner product (ΦNt , ΦNt ) and represent it as continual integral, where a∗ /√N = z ∗ is replaced with a/√N = z, ∞

(ΦNt , ΦNt ) = ∫ Π dzdz ∗ χN∗ (z)χN (z ∗ ) exp{N(S(z ∗ ) + S∗ (z) − ∑ zi∗ zi )}. i=1

If χN is a polynomial in 1/N, then the main contribution to this integral is given by a small neighborhood of the maximum point of the function ∞

S(z ∗ ) + S∗ (z) − ∑ zi∗ zi .

(4.A.5)

i=1

4.A.4 Equation for the exponential factor, preexponential factor, and corrections Now we consider the construction of the functions S and χ. We write l0

∞

Πp = ∑

∑

l=1 i1 ,...,il ,j1 ,...,jl =0

×

∑ 1≤k1 nm, is possible. The Hamiltonian of this model is the sum of the free Hamiltonian H0 = ∫ dkωk A+k A−k + ∫ dkΩk B+k B−k ,

(6.100)

where ωk = √k + m2 , Ωk = √k + M 2 , and the interaction Hamiltonian λH1 = λ ∫ dxφn (x)Φ(x),

(6.101)

which is expressed in terms of the fields 1

φ(x) =

d/2

Φ(x) =

(2π) 1

(2π)d/2

dk [A+ e−ikx + A−k eikx ], √2ωk k dk [B+ e−ikx + B−k eikx ], ∫ √2Ωk k ∫

(6.102)

which correspond to particles of mass m and M. The term in the quantum Hamiltonian corresponding to the process “1 particle of mass M → n particles of mass m” has the form ̂ 1n = ∫ H 1

dk1 fk1 /Λ d/2

(2π)

√2ωk1

⋅⋅⋅

f kn fkn /Λ d/2

(2π)

dpfp/Λ

√2ωkn (2π)d/2 √2Ωp

+ + − ̃ × Ld g(L(k 1 + ⋅ ⋅ ⋅ + kn − p))ak1 ⋅ ⋅ ⋅ akn bp ,

(6.103)

and can be expressed in terms of the creation and annihilation operators of a±k and b±k particles of mass m and M. After applying the Faddeev transformation and removing the regularization, the interaction Hamiltonian assumes the form H̄ 11n =

1 (2n)

d (n−1) s

∫

sk1

√2ωk1

⋅⋅⋅

dkn

1

√2ωkn √f Ωk1 +⋅⋅⋅kn

× a+k1 ⋅ ⋅ ⋅ a+kn bk1 +⋅⋅⋅+kn Φk1 ,...,kn ,

(6.104)

where Φk1 ⋅⋅⋅kn = 1 + α(k1 ⋅ ⋅ ⋅ kn )(ωk1 + ⋅ ⋅ ⋅ ωkn − Ωk1 +⋅⋅⋅kn ).

(6.105)

366 | 6 Hamiltonian semiclassical field theory As in the previous section, nonuniqueness of the function α corresponds to the nonuniqueness in the choice of the Faddeev transformation. Assume that initially the state of the system corresponds to a single particle of mass M with the wave function Ψ0p : Ψ0 = ∫ dpΨ0p b+p Φ(0) . Now in the lowest order of perturbation theory, the representation of the interaction the state of the system at time t, t

Ψ = (1 − iλ ∫ dτeiH0 τ H̃ 11n e−iH0 τ + ⋅ ⋅ ⋅)Ψ0 t

0

will contain the component corresponding to n particles of mass m with the wave function Ψtk1 ⋅⋅⋅kn =

1 (2n)

d (n−1) 2

1 √2ωk1

⋅⋅⋅

1

2

√2ωkn √2Ωk1 +⋅⋅⋅+kn t

× Ψ0k1 +kn √n!Φk1 ⋅⋅⋅kn (−iλ ∫ dτe−i(ωk1 +⋅⋅⋅ωkn −Ωk1 +⋅⋅⋅+kn )τ ). 0

The probability density function over the momenta k1 ⋅ ⋅ ⋅ kn of the system of n particles has the form 󵄨 󵄨2 Pkt 1 ⋅⋅⋅kn = 󵄨󵄨󵄨Ψtk1 ,...,kn 󵄨󵄨󵄨 1 1 1 1 = ⋅⋅⋅ d(n−1) 2ω 2ω 2Ω (2n) k1 k k1 +kn t 󵄨 (ωk1 + ⋅ ⋅ ⋅ + ωkn − Ωk1 +⋅⋅⋅kn ) 󵄨󵄨󵄨2 sin 󵄨 󵄨 󵄨2 󵄨 2 󵄨󵄨 . × 󵄨󵄨󵄨Ψ0k1 +kn 󵄨󵄨󵄨 |Φk1 ⋅⋅⋅kn |2 λn!󵄨󵄨󵄨 󵄨 󵄨󵄨 (ωk1 + ⋅ ⋅ ⋅ ωkn − Ωk1 +⋅⋅⋅+kn )/2 󵄨󵄨

(6.106)

For α = 0, in view of Stückelberg divergences, for sufficiently large d the total probability of finding n particles at time t is infinite. Hence, in this problem the application of the Faddeev transformation also avoids the appearance of Stückelberg divergences in intermediate expressions. To find the decay rate, one introduces the quantity Γtk1 ⋅⋅⋅kn =

Pkt 1 ⋅⋅⋅kn t

󳨀󳨀󳨀󳨀→ Γk1 ⋅⋅⋅kn , t→∞

(6.107)

6.7 Bogoliubov S-matrix and renormalization of the equations of motion

| 367

which is called the disintegration probability per unit time. As t → ∞, since 1 󵄨󵄨󵄨󵄨 sin 󵄨 t 󵄨󵄨󵄨 ξ

2

tξ 󵄨2 2 󵄨󵄨󵄨

󵄨󵄨 → 2πδ(ξ ) 󵄨󵄨

the quantity (6.107) does not depend on the choice of the Faddeev transformation Γk1 ⋅⋅⋅kn =

λn!

1 1 1 ⋅⋅⋅ 2ω 2ω 2Ω (2π) k1 kn k1 +kn 󵄨󵄨 0 󵄨󵄨2 × 󵄨󵄨Ψk1 +kn 󵄨󵄨 δ(ωk1 + ⋅ ⋅ ⋅ + ωkn − Ωk1 +kn ). d(n−1)

(6.108)

So, when considering the process of decay of a particle with momentum P, 󵄨󵄨 0 󵄨󵄨2 󵄨󵄨Ψp 󵄨󵄨 → δ(p − P), formula (6.108) assumes the standard form Γk1 ⋅⋅⋅kn =

λn!

d(n−1)

1 1 1 ⋅⋅⋅ 2ωk1 2ωkn 2ΩP

(2π) × δ(k1 + ⋅ ⋅ ⋅ + kn − P)δ(ωk1 + ⋅ ⋅ ⋅ + ωkn − ωP ).

Thus, the approach based on the Faddeev transformation is capable of producing the known results of quantum field theory with avoidance of Stückelberg divergences on intermediate stages of calculation.

6.7 Bogoliubov S-matrix and renormalization of the equations of motion Another approach to the construction of the Schrödinger equation in quantum field theory free from Stückelberg divergences is based on the use of the concept of Bogoliubov interaction switching. Instead of the field theory with interaction constant independent of the space-time coordinates, one considers the nonphysical theory with the interaction constant ξ (x, t). For such theory, one considers the S-matrix, which is a functional of the function ξ (⋅), for which no Stückelberg divergences appear after renormalization. Choosing the concept of an S-matrix as an original concept of the theory, one can define the Schrödinger equation [10, 62].

6.7.1 Bogoliubov S-matrix and the equation of motion Let us show that from the Bogoliubov S-matrix one can recover the evolution operator. For simplicity, we consider the field theory on a torus.

368 | 6 Hamiltonian semiclassical field theory We will consider the interaction switching function of ξ (t) that depends only on time. To the classical field theory, there corresponds the Hamiltonian H0 + λξ (t)H1 ,

(6.109)

where H0 denotes the free Hamiltonian and H1 is the interaction Hamiltonian. In Bogoliubov’s approach, it is postulated that to the classical Hamiltonian (6.109) there corresponds some operator S[ξ (⋅)] (called the S-matrix) constructed via the Bogoliubov–Parasyuk R-operation. In the axiomatic approach, the construction of the S-matrix was considered in [10, 133]. The Bogoliubov–Parasyuk theorem can be also verified in the Hamiltonian approach [122, 123]. In this case, it is necessary to consider the ultraviolet regularization of the Hamiltonian (6.109), add to it the counterterms, and carry out Wick quantization. Under this approach, at each time the counterterm Hamiltonian depends on the value of the function ξ and its derivatives at this time. After that, one studies the S-matrix of this theory, which under a suitable choice of the counterterms becomes regular as Λ → ∞ in each order of perturbation theory. To construct the evolution operator, we denote by ξ0 (τ) the smooth function, which vanishes for τ < −T2 , increases from 0 to 1 for −T2 < τ < −T1 , and which is equal to 1 for τ > −T1 , where −T1 < 0. Consider two smooth interaction switching functions ξ0 (t), { { { ξ1t (τ) = {1, { { {ξ0 (t − τ),

τ > t,

ξ0 (τ − t),

τ < t,

ξ2t (τ) = {

ξ0 (t − τ),

τ < 0,

0 < τ < t,

τ > t,

to which, in accordance with the Bogoliubov–Parasyuk theorem, there corresponds S-matrices which are also regular if the regularization is removed. Consider the product Ũ t = S+ [ξ2t ]S[ξ1t ]

(6.110)

and investigate its relation to the evolution operator in the regularized theory with the Hamiltonian ̂Λ = H ̂0 + λξ (t)H ̂Λ + H ̂ Λ [λξ (⋅)]. H 1 ct

(6.111)

6.7 Bogoliubov S-matrix and renormalization of the equations of motion

| 369

Denoting the evolution operator as t2

̂0 + λξ (t)H ̂Λ + H ̂ Λ [λξ (⋅)])}, UtΛ1 ,t2 [ξ (⋅)] = T exp{−i ∫ dt(H 1 cl t1

we also introduce the evolution operator in the representation of the interaction: t2

̂ ̂Λ + H ̂ Λ [λξ (⋅)])e−iĤ0 t }. Ũ tΛ1 t2 [ξ (⋅)] = T exp{−i ∫ dteiH0 t (λξ (t)H 1 cl t1

These operators are related by ̂ ̂ UtΛ1 ,t2 [ξ (⋅)] = e−iH0 t2 Ũ tΛ1 t2 [ξ (⋅)]eiH0 t .

(6.112)

Let us investigate some properties of the above operators: 1. Consider the translation of the interaction switching function: ξl (τ) = ξ0 (τ − t). The following relation holds: UtΛ1 t2 [ξt ] = UtΛ1 +T,t2 +T [ξt+T ],

(6.113)

Ũ tΛ1 t2 [ξt ] = e−iH0 t Ũ tΛ1 +t2 ,t2 +T [ξt+T ]eiH0 T .

(6.114)

which can also be written as

Denoting ξt̃ (τ) = ξ0 (t − τ), we find that Λ Λ Λ S[ξ1 ] = Ũ t,+∞ [ξt̃ ]Ũ 0,t [l]Ũ −∞,0 [ξ0 ],

Λ Λ S[ξ2 ] = Ũ t,+∞ [ξt̃ ]Ũ −∞,t [ξt ].

The operator (6.110) assumes the form + Λ Λ Λ Ũ t = (Ũ −∞,t [ξt ]) Ũ 0,t [l]Ũ −∞,0 [ξ0 ].

370 | 6 Hamiltonian semiclassical field theory From (6.112) and (6.114), we obtain + Λ Λ Λ e−iH0 t Ũ t = (Ũ −∞,0 [ξ0 ]) Ũ 0,t [l](Ũ −∞,0 [ξ0 ]).

(6.115)

The left-hand side of (6.115) is regular as Λ → ∞. Λ 2. The operator U0,t [l] coincides with the evolution operator in the original theory. Hence, as the transformation TΛ , which appears in formulas (6.69) and (6.71), one can take the operator Λ TΛ = Ũ −∞,0 [ξ0 ].

(6.116)

This shows that the condition on the initial data (6.69) can be expressed in terms of the evolution operator in the theory with gradual interaction inclusion. Note that the function ξ0 appearing in (6.116) is not unique. This nonuniqueness is similar to that appearing in the construction of the Faddeev transformation in perturbation theory. Ultraviolet Stückelberg divergences can be removed by using the transformation (6.116). In the field theory in an infinite volume, there appear also volume divergences that can be removed by composition of transformation (6.116) and the Faddeev transformation.

6.7.2 Bogoliubov S-matrix in the first order of perturbation theory Let us illustrate formula (6.116) for the transformation that determines the condition on the initial data on an example of the first nontrivial order of perturbation theory in the model Φ3 . In this case, T

Λ,L

0

= 1 − iλ ∫ dτξ0 (τ)eiH0 τ H1Λ,L e−iH0 τ + ⋅ ⋅ ⋅ . −∞

This choice of the transformation T Λ,L corresponds to the choice of the operator A1Λ,L in formula (6.77) in the form 0

A1Λ,L = −i ∫ dτξ0 (τ)eiH0 τ H1Λ,L e−iH0 τ . −∞

So, the functions α1,0 and α1,1 , which appear in (6.88) and (6.89), can be expressed in terms of the Fourier transform of the discontinuous function 0

0

ξ0̃ (ω) = ∫ ∫ dτξ0 (τ)eiωτ , −∞ −∞

(6.117)

6.8 Renormalization in semiclassical field theory | 371

which is equal to ξ0 (τ) for τ < 0 and zero for τ > 0, as follows: α1.0 (k1 , k2 , k3 ) = −iξ0̃ (ωk1 + ωk2 + ωk3 ), α1,1 (k1 , k2 , k3 ) = −iξ0̃ (ωk1 + ωk2 − ωk3 ). The Fourier transform (6.117) as ω → ∞ has the form ξ0̃ (ω) = (iω)−1 + O(ω−∞ ), and hence, operator (6.91) does not contain Stückelberg divergences. The operator (6.90) also does not contained ultraviolet Stückelberg divergences, but as L → ∞ there appear volume divergences, and hence, in order to take the limit over an infinite volume, it is also necessary to consider an additional transformation of the operator H̃ that removes the volume divergences.

6.8 Renormalization in semiclassical field theory In this section, we consider the application of semiclassical methods to the theory of scalar field with Hamiltonian depending on the small parameter λ in the following way: 1 m2 2 1 1 φ (x) + Vint (√λφ(x))]. H = ∫ dx[ π 2 (x) + (∇φ)2 (x) + 2 2 2 λ

(6.118)

6.8.1 Semiclassical approximation in regularized field theory According to Section 6.5, consider the regularized quantum field theory, in which the classical Hamiltonian is written as the sum 1 (0) √ H [π λ, φ√λ] λ Λ,L 1 1 int √ = H0 [π √λ, φ√λ] + HΛ,L [π λ, φ√λ], λ λ

0 HΛ,L =

of the free Hamiltonian 1 m2 2 1 H0 [Π(⋅), Φ(⋅)] = ∫ dx[ Π2 (x) + (∇Φ)2 (x) + Φ (x)] 2 2 2 and the interaction Hamiltonian x int HΛ,L [Π(⋅), Φ(⋅)] = ∫ dxg( )Vint (ΦΛ (x)). L

(6.119)

372 | 6 Hamiltonian semiclassical field theory According to the proof of the Bogoliubov–Parasyuk theorem on the removal of divergences from the Bogoliubov S-matrix, to the Hamiltonian functional (6.119) one should add the counterterm Hamiltonian, which depends on the small parameter λ, as follows: ct (n+1) √ HΛ,L = ∑ λn HΛ,L [π λ, φ√λ]. n⩾0

The specific form of the counterterms will be specified later. The quantum Schrödinger equation for the vector ΨtΛ,L,λ of the Fock space ℋ has the form i

d t (n) √ ̂(⋅)]ΨtΛ,L,λ . Ψ = ∑ λn−1 HΛ,L [ λπ̂ (⋅), √λφ dt Λ,L,λ n⩾0

(6.120)

In (6.120), we take the Wick ordering of operators: the classical fields φ(x) and momenta π(x) are expressed via (6.53) in terms of A±k , which are later replaced by the creation and annihilation operators a±k . ̂(⋅)√λ, in terms of which the HamilThe commutator of the operators π̂ (⋅)√λ and φ tonian is expressed, is proportional to the small parameter λ, and hence, semiclassical methods and, in particular, the theory of complex germ, can be applied as λ → 0. To study the Bogoliubov S-matrix, we will consider a more general case, when (0) the regularized Hamiltonian HΛ,L depends explicitly on time and has the form 1 (0) √ 1 1 int √ H [π λ, φ√λ] = H0 [π √λ, φ√λ] + ξ (t) HΛ,L [π λ, φ√λ], λ Λ,L λ λ

(6.121)

where ξ (t) is some smooth function of t. For ξ (t) = 1, the functional (6.121) becomes (6.119). Consider the unitary transformation i

i

λ UΦ,Π,S = e λ S e √λ

∫ dx[Π(x)̂ φ(x)−Φ(x)̂ π (x)]

,

where S is a real number, Φ(⋅), Π(⋅) are real functions, and π̂ (x) = ̂(x) = φ

i

d/2

(2π) 1

(2π)d/2

∫ dk√

ωk + [a (k)e−ikx − a− (k)eikx ], 2

∫ dk√

1 [a+ (k)e−ikx + a− (k)eikx ]. 2ωk

(6.122)

6.8 Renormalization in semiclassical field theory | 373

From the canonical commutation relations, we have the following properties of the operators: Φ(x) , √g Π(x) , = π̂ (x) + √λ

λ λ ̂(x)UΦ,Π,S ̂(x) + (UΦ,Π,S ) φ =φ −1

λ λ (UΦ,Π,S ) π̂ (x)UΦ,Π,S −1

(6.123)

where 1 1 Dt = − (Ṡ t + ∫ dx(Π̇ t (x)Φt (x) − Φ̇ t (x)Πt (x))) λ 2 1 d + ∫ dx(Φ̇ t (x)π̂ (x) − Π̇ t (x) − Π̇ t (x)) + . √λ dt Consider the element Ψt = UΦλ t ,Πt ,St Yt

(6.124)

of the space ℋ, where the vector Y t has limit as λ → 0. Substituting vector (6.124) in equation (6.120) and using the commutation rules (6.123), we get the equation for Y t , ̂(⋅)]Y t , iDt Y t ∑ λn−1 H (n) [Π(⋅) + √λπ̂ (⋅), Φ(⋅) + √λφ n⩾0

(6.125)

where for brevity we omitted the indexes Λ, L. Equation (6.125) contains the singular (as λ → 0) terms of the order of O(1/λ) and O(1/√λ). Equating these terms to zero, we get the relation for the action 1 Ṡ t = ∫ dx(−Π̇ t (x)Φt (x) + Φ̇ t (x)Πt (x)) − H (0) [Π(⋅), Φ(⋅)] 2

(6.126)

and the Hamiltonian system δH (0) , Φ̇ t (x) = δΠ(x)

δH (0) ̇ Π(x) = . δΦ(x)

(6.127)

If relations (6.126) and (6.127) are satisfied, equation (6.125) reduces as λ → 0 to i

d t Y = H2 [π̂ (⋅), φ(⋅)]Y t , dt

(6.128)

̂2 = H2 [π̂ (⋅), φ ̂(⋅)] corresponds to the functional where the operator H 1 δ2 H (0) δ2 H (0) 1 δ2 H (0) H2 [π(⋅), φ(⋅)] = π π+π φ+ φ φ + H (1) . 2 δΠδΠ δΠδΦ 2 δΦδΦ

(6.129)

374 | 6 Hamiltonian semiclassical field theory ̂(⋅)] is the same as in H[π̂ (⋅), φ(⋅)]. In (6.129), The ordering of the operators in H2 [π̂ (⋅), φ the arguments of Φ(⋅), Π(⋅) and the functionals H (0) and H (1) are omitted, and for inte2 (0) H δ2 H (0) π(y)dy are denoted as π δδΠδΠ π. grals of the type ∫ π(x) δΠ(x)δΠ(y) Equation (6.128) is the Schrödinger equation with Hamiltonian quadratic with respect to the creation and annihilation operators ̂2 = ∫ dx[ 1 a+ (x)(A ̂ t a+ )(x) + a+ (x)(B ̂ t a− )(x) + 1 a− (x)(A ̂ t∗ a− )(x)] + γ t . H 2 2 ̂ t and B ̂ t are operators in space of functions of the variable x, which depend on Here, A time in the case if the classical solution is nonstationary and γ t is a real function. The Schrödinger equation with quadratic Hamiltonian can be solved explicitly. In particular, equation (6.128) has the Gaussian-type solutions 1 ̂t a+ (x))Φ(0) } ct exp{ ∫ dxa+ (x)(M 2

(6.130)

̂t with the norm smaller than 1 with time-depending constant ct and the operator M satisfying the equation: 2 (0) 1 ̂̇ t = 1 (1 + M ̂t ) 1 δ H ̂t ) − 1 (1 − M ̂ t )ω ̂t ). ̂(1 − M iM (1 + M √ω 2 2 ̂ δΦδΦ √ω ̂

(6.131)

The equation for the ct has the form 1 δ2 H (0) 1 i ̂)M̃ t ] − iH (1) , −ω (ln ct ) = − Tr[( √ω 4 ̂ δΦδΦ √ω ̂ 2

2

(6.132)

H δ H where δδΦδΦ is the operator with kernel δΦ(x)δΦ(y) . It is known that expression (6.130) defines an element of the Fock space, which ̂0 as has limit if the regularization is removed if and only if the limit of the operator M Λ, L → ∞ is a Hilbert–Schmidt operator, because the squared Hilbert–Schmidt norm of this operator plays the role of the probability of the presence of two particles in the system. ̂t is not necessarily a Hilbert–Schmidt operaIn turns out that in the limit theory M 0 ̂ M ̂ has this property. This means that the probability of creation of a pair tor even if M of particles is infinite. Hence, in this case the Schrödinger equation has no solutions lying in the Fock space. This example served as one of the arguments in favor of the conjecture that the evolution transformation in quantum field theory is nonunitary and that at different time instants one should use different representations of the canonical commutation relations. We will show that, at least in leading order of the semiclassical expansion, there is no need to consider nonunitary transformations of the evolution. It turns out that (0)

(0)

6.8 Renormalization in semiclassical field theory | 375

in general elements of the Fock space of the quadratic theory are not in a one-to-one correspondence with elements of the space ℋren of the renormalized theory. It will be shown that the generalized vector (6.130) corresponds to an element of the space ℋren ̂0 is a Hilbert–Schmidt operator, but rather under not in the case when the operator M a more involved condition, which will be derived in this section. After the transformation, ̃ + M) ̃ −1 √ω ̃(1 − M)(1 ̃, R = i √ω at =

ct

1/2

̃ (det(1 + M)) (det Im R)1/4

(6.133)

equation (6.131) reduces to the form δ2 H (0) = 0, Ṙ + R2 + δΦδΦ

(6.134)

and equation (6.132) becomes δ2 H (0) 1 1 ′ i(ln at ) = − Tr(Im R − √−Δ + m2 − ) − H (1) . 2 2√−Δ + m2 δΦδΦ

(6.135)

Note that transformation (6.133) corresponds to the transition from the Fock representation to the Schrödinger representation, in which the field operators are diagonal. The solution of equation (6.128), for which the initial condition is represented in the form 1 ̂0 a+ (x))Φ(0) }, P(a+ ) exp{ ∫ dxa+ (x)(M 2 where P(a+ ) is a polynomial in creation and annihilation operators, is constructed using the gem creation operators ̂(x) − qt (x)π̂ (x)], Λ[pt , qt ] = ∫ dx[pt (x)φ

(6.136)

d ̂2 if which commute with the operator i dt −H

ṗ t (x) = −

δH2 (pt (⋅), qt (⋅)) , δq(x)

q̇ t (x) =

δH2 (pt (⋅), qt (⋅)) , δp(x)

and which map in this case a solution of equation (6.128) to a solution.

(6.137)

376 | 6 Hamiltonian semiclassical field theory 6.8.2 Problem of removal of regularization in Hamiltonian formalism In the Hamiltonian quantum field theory, the condition on the dependence of the initial datum for the Schrödinger equation ΨΛ,L = TΛ,L Φ

(6.138)

on the regularization parameters Λ and L is posed so that the operators t −1 t WΛ,L = TΛ,L UΛ,L TΛ,L

(6.139)

would be regular as Λ, L → ∞. In this case, we consider the leading order of the semiclassical expansion. Let us give the definition of an operator regular as Λ, L → ∞ in leading order of the semiclassical expansion. In accordance with the previous section, semiclassical initial conditions depend singularly on the parameter λ and have the form (6.124). It is natural to t say that the operator WΛ,L is regular as Λ, L → ∞ in leading order of semiclassics if t the parameters involved in the semiclassical asymptotics of the vector WΛ,L ΨλΛ,L have the limit as Λ, L → ∞. This means that, for any pair of functions Φ0 (x), Π0 (x) from the Schwartz space, there exist a pair of functions Φ0Λ,L (x), Π0Λ,L (x) from this space and a number St such that the operator t VΛ,L,λ = (UΦλ t

t t Λ,L ,ΠΛ,L ,SΛ,L

t ) WΛ,L,λ UΦλ 0 ,Π0 ,0 −1

(6.140)

has the strong limit as λ → 0 and the limit operators lim (UΦλ t

g→0

t t Λ,L ,ΠΛ,L ,SΛ,L

t ) WΛ,L,λ UΦλ 0 ,Π0 ,0 −1

(6.141)

strongly converge as Λ, L → ∞. If this property is satisfied, we will say that condition (6.138) is invariant under evolution in leading order of the semiclassical expansion. Let us construct the transformation TΛ,L by using the method of Bogoliubov S-matrix, which was considered in the previous section. Consider, as before, the infinitely differentiable function ξ0 (τ), τ ⩽ 0, which vanishes for τ < −T1 , which is equal to 1 for −T2 < τ < 0, and which increases from 0 to 1 on the interval τ ∈ (−T1 , T2 ). Consider the operator x 1 Λ,L Hint = ξ (τ) ∫ dxg( ) Vint (√λφΛ (x)) L λ √ ̂(⋅), √λπ̂ (⋅), τ, ξ (⋅)). + Hct ( λφ

(6.142)

Here, Hct denotes the coutertem Hamiltonian independent of τ for τ ∈ (−T2 , 0) and vanishing for τ < −T1 . To ensure the invariance of the class (6.138) of vectors in the

6.8 Renormalization in semiclassical field theory | 377

leading term of the semiclassical expansion, it suffices to consider the terms with n = 0, 1 in (6.120). The explicit form of Hct will be given in the next section. As an operator involved in condition (6.138), consider the operator 0

Λ,L TΛ,L = T exp{−i ∫ dτeiH0 τ Hint (τ)e−iH0 τ },

(6.143)

−∞

which transforms the initial condition of the Cauchy problem i

dX τ Λ,L = eiH0 τ Hint (τ)e−iH0 τ X τ , dτ

X −∞ = X

(6.144)

to the solution of this problem for τ = 0. The function ξ0 is compactly supported, and hence, the right-hand side of equation (6.144) vanishes for τ < −T1 . Hence, the initial condition for τ = −∞ can be replaced by the initial condition for τ = −T0 < −T1 . Let us find in the semiclassical approximation the vector (6.138) and the operator (6.139). We need some auxiliary notation. Let S and i ln a be real numbers, let Φ(x), Π(x), q(x), p(x) be smooth rapidly decreasing functions, R be the operator with a symmetric kernel and positively definite imaginary part. t ,t We denote by 𝒰ξ2(⋅)1 the mapping that sends the initial conditions of the Cauchy problems for equations (6.126), (6.127), (6.135), (6.137), and (6.134) for t = t1 , pt1 = p,

qt1 = q,

Φt1 = Φ,

Πt1 = Π,

Rt1 = R,

St1 = S,

at1 = a

to the solutions of these equations for t = t2 , t ,t

(pt2 , qt2 , Φt2 , Πt2 , Rt2 , St2 , at2 ) = 𝒰ξ2(⋅)1 (pt1 , qt1 , Φt1 , Πt1 , Rt1 , St1 , at1 ), where ξ (⋅) is the interaction switching function involved in (6.121). To construct the vector TΛ,L X −∞ as g → 0, we make the substitution Ψt = e−iH0 τ X τ in equation (6.144). The Cauchy problem (6.144) is reduced to the form i

dΨτ Λ,L = [H0 + Hint (τ)]Ψτ , dτ

ψ−T0 = eiH0 T0 X

−∞

.

We chose the initial condition X −∞ in the form g Xg−∞ = UΦ,Π,0 cΛ[p1 (⋅), g(⋅)] ⋅ ⋅ ⋅ Λ[pk (⋅), qk (⋅)]

1 × exp{ ∫ dxa+ (x)(Ma+ )(x)}Φ(0) , 2

(6.145)

378 | 6 Hamiltonian semiclassical field theory where M is a Hilbert–Schmidt operator with norm smaller than 1, Φ, Π, c, M are regular as Λ, L → ∞, and the operator M in the limit Λ, L → ∞ also remains a Hilbert–Schmidt operator. In accordance with the previous section, the asymptotic solution of the Cauchy problem (6.145) for τ = 0 has the form Ψ0g = UΦg ′ ,Π′ ,S′ c′ Λ[p′1 (⋅), g1′ (⋅)] ⋅ ⋅ ⋅ Λ[p′k (⋅), q(⋅)] 1 × exp{ ∫ dxa+ (x)(M ′ a+ )(x)}Φ(0) , 2

(6.146)

where the parameters from formula (6.146), which depend on Λ, L, have the form (p′j , q′j , Φ′ , Π′ , R(M ′ ), S′ , a(c′ )) 0,−T

−T0 ,0

= 𝒰ξ (⋅) 0 𝒰0

(pj , qj , Φ, Π, R(M), 0, a(c)),

j = 1, k,

and R(M) and a(c) denote the quantities from (6.133). Equations (6.126), (6.127), and (6.137) are regular as Λ, L → ∞, and hence, the parameters Φ′ , Π′ , S′ , p′j , qj′ also have finite limits if the regularization is removed. The operator M ′ is a solution of equation (6.131) satisfying the initial condition −T0 M for t = −T0 , whose limit as Λ, L → ∞ is a Hilbert–Schmidt operator. However, the limit as Λ, L → ∞ of the operator M ′ may fail to be a Hilbert–Schmidt operator. So, the Gaussian vector (6.131) of the regularized theory 1 g UΦ,Π,0 const exp{ ∫ dxa+ (x)(Ma+ )(x)}Φ(0) 2

(6.147)

corresponds to an element of the renormalized Fock space not only when M is a Hilbert–Schmidt operator in the case the regularizations are removed, but when ̃ for t = 0, becomes a Hilbert– the solution of equation (6.131), which coincides with M Schmidt operator for t < −T1 . The constant in (6.147), in general, is not regular as Λ, L → ∞ and can be determined from the following condition: the solution of equation (6.132), which coincides with this constant for t = 0, has for t < −T1 the limit as Λ, L → ∞. Let us reformulate this assertion in terms of the operator R. Assume that the or̃ M ̃ = M(x ̃ 2 , − i1 𝜕 ), decreases as M(x, ̃ k) = O(|k|−d/2−δ ) dered symbol of the operator M, 𝜕x as t → ∞. For δ > 0, the operator M is a Hilbert–Schmidt operator, but this is not so for δ ⩽ 0. From (6.133), it follows that the symbol of the operator R behaves at infinity as R(x, k) = i√k2 + m2 + O(|k|−d/2−δ+1 ).

(6.148)

6.8 Renormalization in semiclassical field theory | 379

Making Λ, L → ∞, equation (6.134) becomes ′′ Ṙ + R2 + (−Δ + m2 + ξ (t))Vint (Φc (t, x)) = 0.

(6.149)

This result can be formulated as follows. Proposition 6.8.1. The vector (6.147) of the regularized theory corresponds to a vector of the renormalized state space if the solution of equation (6.149), which is R for t = 0, satisfies for t < −T1 property (6.148) for δ > 0. If property (6.148) for t < −T1 is satisfied for δ > 0, then expression (6.147) does not correspond to any vector of the renormalized state space. In the next section, we will examine equation (6.149) and formulate explicitly the condition of this proposition. Let us study the problem of strong convergence of the operator (6.141) on a dense domain. Any vector of the Fock space can be approximated arbitrarily well by finite linear combinations of the vectors Y = cΛ[p1 (⋅), q1 (⋅)] ⋅ ⋅ ⋅ Λ[pk (⋅), qk (⋅)]Φ(0) . Let us study the convergence of the vector t lim VΛ,L,λ Y

(6.150)

λ→0

as Λ, L → ∞. The semiclassical asymptotics of the vector t t TΛ,L UΦg 0 ,Π0 ,0 Y WΛ,L,g UΦg 0 ,Π0 ,0 Y = (TΛ,L )−1 UΛ,L

is constructed in accordance with the previous section. This asymptotics have the form UΦg ′ ,Π′ ,S′ Y ′ , where Y ′ = c′ Λ[p′1 (⋅), q1′ (⋅)] ⋅ ⋅ ⋅ Λ[p′k (⋅), qk′ (⋅)]

1 × exp{ ∫ dxa+ (x)(M ′ a+ )(x)}Φ(0) . 2

(6.151)

The parameters involved in the vector (6.151) are as follows: (p′j , q′j , Φ′ , Π′ , R(M ′ ), S′ , a(c′ )) 0,−T0 ,0

= 𝒰0

t,−T

−T0 ,0

𝒰ξ (⋅) 0 𝒰0

(pj , qj , Φ0 , Π0 , R(0), 0, a(c)),

j = 1, k,

t the function ξ (t) for t > 0 extends as ξ (t) = 1. If ξ (t) = Φ′ , ΠtΛ,L = Π′ , SΛ,L = S′ , then the vector (6.150) coincides with (6.151). Since p′j , qj′ are regular as Λ, L → ∞, it follows that in order to verify that vector (6.151) is regular as Λ, L → ∞ it suffices to show that:

380 | 6 Hamiltonian semiclassical field theory (i) M ′ is a Hilbert–Schmidt operator; (ii) c′ has limit as Λ, L → ∞. Requirement (ii) is equivalent under condition (i) to the requirement that the quantity ln a′ = ln a(c′ ) be finite. Condition (i) means that property (6.154) holds for the symbol of the operator R′ = R(M ′ ). Let us examine equation (6.149). We will show that these conditions are satisfied for a certain choice of the counterterm H (1) .

6.8.3 Condition on the Gaussian state vector Consider equation (6.149). We let R(x, k) denote the ordered symbol of the operator 𝜕 ). The indexes 1 and 2 mean that the differentiation operators R : R = R(x2 , − i1 𝜕x stand to the left of the operators of multiplication by x: R ∫ dk f (k)eikx ≡ ∫ dk R(x, k)f (k)eikx .

(6.152)

In this section, we will study the behavior of the symbol R(x, k) as k → ∞. Consider the asymptotic expansion of this symbol with respect to 1/ωk , ωk = √k2 + m2 , as k → ∞: Rtm (x, k/ωk ) , ωm m⩾1 k

Rt (x, k) = ℛt (x, k) = iωk + ∑

(6.153)

where Rtm (x, n) are smooth functions rapidly decreasing in x at infinity together with the derivatives of all orders. We substitute expansion (6.153) in equation (6.149) and j find under what conditions the right-hand side vanishes up to O(1/ωk ), where j is an arbitrary given number. To this end, we will use formula for the ordered symbol of the product of operators. 𝜕 ̂ = B(x2 , − i1 𝜕 ) be some operators. Then their product  B ̂ has Let  = A(x2 , − i1 𝜕x ), B 𝜕x 2 1 𝜕 ̂ ̂ the ordered symbol A ∗ B, AB = A ∗ B(x , − i 𝜕x ), which can be represented in one of the two equivalent forms: 2

1

(A ∗ B)(x, k) = A(x, k − i (A ∗ B)(x, k) = ∫

dz dp (2π)d

𝜕 )B(x, k), 𝜕x

A(x, k − p)B(x + z, k)eipz .

To derive (6.155), it suffices to use the relation ̂ ∫ dpf (p)eipx = ∫ dkeikx ∫ dz dp ei(p−k) B(y, p)f (p), B (2π)d

(6.154) (6.155)

6.8 Renormalization in semiclassical field theory | 381

which follows from the Fourier inversion formula and formula (6.152), and apply to it the operator A.̂ Formula (6.154) follows from (6.155). Expanding formally the right-hand side of (6.155) in a series with respect to the differentiation operator, we get (−i)l 𝜕l A(x, k) 𝜕l B(x, k) , l! 𝜕ki1 ⋅ ⋅ ⋅ 𝜕kil 𝜕xi1 ⋅ ⋅ ⋅ 𝜕xil l⩾0

(A ∗ B) = ∑

(6.156)

where summation over the repeated indexes is implied. If the functions A and B have the form, sA =

Atl (x, k/ωk ) m ωk 1

,

B=

Btl (x, k/ωk ) m

ωk 2

the series (6.156) is an asymptotic expansion with respect to −m −m −l O(ωk 1 2 )

, 1 , ωk

because the lth term

of (6.156) is of the order of as k → ∞. The asymptotic formulas (6.156) can be justified by applying the stationary-phase method to integral (6.155). Using (6.156) and (6.153), substituting the symbol ℛ ∗ ℛ of the operator ℛ2 in equation (6.155), and equating to zero the terms of the order of O(1/ωsk ), s ⩾ 0, we get the relation ′′ 2iRt1 (x, n) + ξ (t)Vint (Φt (x)) = 0

for s = 0, and Ṙ s (x, n) + 2iRts+1 (x, n) +

+

∑

l+m=s+1, i,m⩾1

∑

ωl−1 k

m+n+l=s m,n⩾l, l⩾0

l t (−i) 𝜕′ ωk 𝜕 Rm (x, n) l! 𝜕ki1 ⋅ ⋅ ⋅ 𝜕kil 𝜕xi1 ⋅ ⋅ ⋅ 𝜕xil

Rtm (x, k/ωk ) 𝜕t Rtn (x, n) (−i)l m+1 𝜕l ωk = 0, l! 𝜕ki1 ⋅ ⋅ ⋅ kil ωm 𝜕xi1 ⋅ ⋅ ⋅ 𝜕xil k

(6.157)

for s ⩾ 1; in this formula, we have n = k/ωk . In the derivation of formula (6.157), we used the assumption that Ṙ ts has the same order as Rts . From the recurrent relation (6.157), one can express Rts+l in terms of the previous orders of perturbation theory with respect to 1/ωk , Rt1 , . . . , Rts . Moreover, the function Rts+1 is uniquely determined by the values of the fields Φt (⋅) and the momenta Πt (⋅) at the same time instant t and by the values of the function ξ and its derivatives at time t. For ξ (t) = 0, i. e., for t < −T1 , the above coefficients of the asymptotics vanish, so that Rt = iωk for these values of t. For t > −T2 , the asymptotics do not depend on ξ .

382 | 6 Hamiltonian semiclassical field theory From the recurrent relations (6.157), we have the first several functions Rts : i ′′ Rt1 (x, n) = ξ (t)Vint (Φt (x)), 2 𝜕 1 𝜕 ′′ Rt2 (x, n) = − ( + n )ξ (t)Vint (Φ′ (x)). 4 𝜕t 𝜕x Note that the first derivative of Φt with respect to time is expressible in terms of the momentum Πt , and the time derivatives of Φt of higher order, which enter the succeeding functions Rs , can be expressed from the motion equations in terms of Φt and Πt . We have constructed the asymptotics as |k| → ∞ only for one solution of equation (6.149), which for t < T1 corresponds to vacuum, i. e., it is equal to iωk . Consider the construction of the asymptotics of solution of equation (6.149) satisfying the initial condition (6.148). Substituting the operator with kernel Rt = ℛt + τt in equation (6.149), where ℛt is the already constructed solution (6.153) of equation (6.149), we have the equation r ṫ + 2iωk r t = 0 in leading order of 1/ωk . Its solution is r T = r −T0 exp{−2iωk (t + T0 )}; the corrections to this solution will be constructed in the next section. So, Proposition 6.8.1 can be reformulated as follows. Proposition 6.8.2. Formula (6.149) corresponds to a vector of the renormalized state space if and only if R0 coincides, up to O(|k|−d/2+1−δ ), with the asymptotics of ℛ0 (see (6.153)), R0 = ℛ0 + O(|k|−d/2+1−δ ),

(6.158)

which is independent of the choice of the interaction switching function ξ (t) and depends only on the functions Π0 (x) and Φ0 (x). Let us verify the conditions imposed on R′ and a′ at the end of the previous section. The condition on R′ is immediate from the invariance for t = 0 of the asymptotic formula (6.153) under the change of the function ξ (t) by ξ (t − t0 ), where t0 = const. The quantity a′ is finite with the appropriate choice of the counterterm H (1) , which compensates the divergences appearing in the evaluation of the trace of the operator R. To find the explicit form of this counterterm, one can consider asymptotic solutions of the regularized equation (6.134), t

ℛΛ,L (x, k) = iωk +

∑

m⩾1, j⩾0

Rtmj (x, k/ωk , k/Λ) ωm Λj k

,

single out the diverging part of the trace dkdx Im Rtmj (x, k/ωk , k/Λ) 1 1 t , − Tr(Im ℛΛ,L − iωk ) ∼ − ∑ ∫ 2 2 ωm Λj k m=1,d, h=0,d

6.8 Renormalization in semiclassical field theory | 383

2

l δ H add it with the diverging part of the trace Tr( 4̂ ), and equate the sum to the ω (0)

δΦδΦ

counterterm H (1) . Since Rtmj is uniquely determined by the fields Φt (⋅), Πt (⋅) and the

value of the function ξ and of its derivatives at the point t, the counterterm H (1) is, for t > −T2 , a functional of fields and momenta. This verifies the requirements imposed on R′ and a′ .

6.8.4 Some conclusions So, we have obtained a criterion that expression (6.147) defines an element of the renormalized state space (Propositions 6.8.1 and 6.8.2). Note that various choices of the initial conditions for equation (6.131) have been considered in the literature. So, in papers on semiclassical field theory, which are not concerned with the problem of renormalization, it is usually implicitly assumed that the initial condition for equation (6.134) for t = 0 has the form (6.148). The papers [75] and [108] considered only the initial states which are proper for the Hamiltonian operator at the initial time. To these states, there correspond Gaussian functionals with the following singular part of the symbol of the operator Rt : (R ∗ R)(x, k) ∼ −(k2 + m2 + V ′′ (Φtc (x))).

(6.159)

Formulas (6.148) and (6.159) are obtained from our condition (6.158) only in the case if the asymptotics as |k| → ∞ is constructed up to O(1/|k|) and O(1/|k2 |), respectively, and this is true only for sufficiently small dimensions of the space. At the same time, condition (6.159) always holds for the static solutions of equations. Although we considered for simplicity only the case of the scalar theory, similar arguments can be carried out almost verbatim to more involved theories. So, in the case of the calibration theory, one can consider a Coulomb-type calibration and write the evolution equation in this calibration. After that, the “Bogoliubov interaction switching” can be applied as in the case of a scalar field. The case when the theory contains fermion fields in addition to boson fields (in particular, quantum electrodynamics) is also considered similarly. So, substitution (6.124), where the operator U contains only operators of boson fields, approximately satisfies the quantum Schrödinger equation also in the presence of fermions. In this case, the vector Y t satisfies the fermion Schrödinger equation with quadratic Hamiltonian, which like the boson equation, can be solved explicitly. The above condition on the operator R does not depend on the counterterms and is controlled only by the classical action. Hence, it applies both for renormalizable and nonrenormalizable theories. The coefficient of the exponential depends on the “one-loop” counterterm H (1) . This conclusion agrees with the arguments from the book [124], according to which,

384 | 6 Hamiltonian semiclassical field theory in the evaluation of a path integral via the stationary-phase method, the tree diagrams determine the exponent, and the one-loop ones, the preexponent. It is frequently implied that the problem of divergences and renormalizations in the field theory should be taken into account only in the consideration of looped Feynman diagrams, while there are no divergence issues in the study of tree-diagrams. In our study, we see that in classical field theory one should take into account the problem of divergences, because the condition on the initial data for equation (6.134), which is independent of the counterterms was derived from the analysis of this problem We have considered only the leading order of semiclassics. The study of the problem of initial conditions for higher orders of expansion is similar. However, here one should take into account that, in addition to ultraviolet divergences, higher orders of expansion also involve volume divergences related to creation of particles from vacuum in accordance with Haag’s theorem. So, in place of transformation (6.143), one should consider the composition of transformation (6.143) and the Faddeev transformation.

6.9 Invariance of the conditions on the complex germ In this section, we adduce a mathematical proof of the fact that the condition imposed on the quadratic form appearing in the Gaussian state vector is invariant under evolution. ̂ t acting in the space of functions L2 (ℝd ) is of Equation (6.149) on the operator R the following type: ̂t dR ̂ 2 + (−Δ + m2 + ft̂ ) = 0. +R t dt

(6.160)

In equation (6.160), ft̂ is the operator of multiplication by the function ft : ℝd → ℝ, which depends on the real parameter t. Differentiation of an operator depending on the parameter t will be understood in the sense of the uniform topology. We say that ̇ the operator  t is the derivative of an operator  t if 󵄩 󵄩󵄩  ̇ 󵄩󵄩 󵄩󵄩 t+δt −  t −  t 󵄩󵄩󵄩 󳨀󳨀󳨀󳨀→ 0. 󵄩󵄩 󵄩󵄩 δt→0 󵄩󵄩 δt Let us investigate some properties of equation (6.160).

6.9.1 Transformation of the Riccati equation ̂ in the space L2 (ℝd ) satisfying the conditions: We denote by ℒ the class of operators R

6.9 Invariance of the conditions on the complex germ

| 385

̂ −1/2 , where ω = √−Δ + m2 , is bounded; (i) the operator ω−1/2 Rω ̂ is symmetric; (ii) the kernel of the operator R (iii) for some operator of multiplication by the constant c > 0, the operator 1i ω−1/2 × ̂ −R ̂ ∗ )ω−1/2 − c is positively definite. (R ̂ ∈ ℒ. We set Let ℛ ̂−ℛ ̂∗ ). Γ = i−1 (ℛ Since the operator Γ − cω is positively definite (this follows from property (iii)), the operator Γ has bounded inverse Γ−1 . Moreover, ‖Γ−1 ‖ ⩽ (cm)−1 . ̂ is symmetric, so is the operator Γ−1 , and hence, Since the kernel of the operator ℛ −1 Γ is bounded and self-adjoint. According to the theory of functions of bounded selfadjoint operators, we construct the bounded operator 1/2

Γ−1/2 = (Γ−t ) . Lemma 6.9.1. The operators ω1/2 Γ−1/2 and Γ−1/2 ω1/2 are bounded and have bounded inverses. Proof. The operator Γ−1/2 ω1/2 satisfying 󵄩󵄩 −1/2 1/2 󵄩󵄩2 1/2 −1 1/2 1/2 −1/2 −1/2 −1/2 −1 󵄩󵄩Γ ω g 󵄩󵄩 = (ω g, Γ ω g) = (ω g, ω (ω Γω ) g) −1 −1 󵄩 󵄩 = (g, (ω−1/2 Γω−1/2 ) g) ⩽ 󵄩󵄩󵄩(ω−1/2 Γω−1/2 ) 󵄩󵄩󵄩‖g‖2 is defined on the domain of the operator ω1/2 . The operator ω−1/2 Γω−1/2 has the bounded inverse by virtue of (iii), and moreover, −1/2 ‖ω Γω−1/2 ‖ ⩽ c−1 . So, the operator A = Γ−1/2 ω−1/2 is bounded and extends to the entire Hilbert space. Therefore, the operator A+ = ω1/2 Γ−1/2 , which is uniquely determined from the relation ω−1/2 A+ = Γ−1/2 , is bounded. Let us show that the operators AA+ and A+ A have bounded inverses. The estimate (f , A+ , Af ) = (f , ω1/2 Γ−1 ω1/2 f ) = ((ω−1/2 Γω−1/2 ) f , ω−1/2 Γω−1/2 (ω−1/2 Γω−1/2 ) f ) −1 󵄩2 󵄩 󵄩 󵄩−2 ⩾ c󵄩󵄩󵄩(ω−1/2 Γω−1/2 ) f 󵄩󵄩󵄩 ⩾ c‖f ‖2 󵄩󵄩󵄩ω−1/2 Γω−1/2 󵄩󵄩󵄩 −1

implies that the operator ω−1/2 Γω−1/2 is invertible.

−1

386 | 6 Hamiltonian semiclassical field theory Next, by virtue of (i), (iii), the bounded self-adjoint operator ω−1/2 Γω−1/2 satisfies the estimates 󵄩 󵄩 c(f , f ) ⩽ (ω−1/2 f , Γω−1/2 f ) ⩽ 󵄩󵄩󵄩ω−1/2 Γω−1/2 󵄩󵄩󵄩(f , f ). Hence, on the dense domain of the operator ω, we have 󵄩 󵄩 c(g, ωg) ⩽ (g, Γg) ⩽ 󵄩󵄩󵄩ω−1/2 Γω−1/2 󵄩󵄩󵄩(g, ωg). As a result, (f , AA+ f ) = (f , Γ−1/2 ωΓ−1/2 f ) ⩾

(f , f ) . |ω−1/2 Γω−1/2 |

So, the operators AA+ and A+ A are invertible and, therefore, (see [6]) so is the operator A. Lemma 6.9.1 is proved. Lemma 6.9.1 permits one to introduce the unbounded operator Γ−1/2 defined on the domain of the operator ω−1/2 by the formula Γ1/2 = (ω1/2 Γ−1/2 ) ω1/2 . −1

̂ the operator R, ̂ Consider the transformation associating with the operator M ̂=ℛ ̂ − M) ̂ −1 Γ1/2 . ̂ + iΓ1/2 M(1 R

(6.161)

̂ ∈ ℒ if and only if M ̂ is an operator with symmetric kernel and norm Lemma 6.9.2. R smaller than 1. ̂ be an operator with symmetric kernel and norm smaller than 1. Let us Proof. 1. Let M ̂ show that R ∈ ℒ. The operator ̂ −1/2 = ω−1/2 ℛ ̂ − M) ̂ −1/2 Γ1/2 ω−1/2 ̂ω−1/2 + iω−1/2 Γ1/2 M(1 ω−1/2 Rω ̂ ∈ ℒ. From (6.161) and since is bounded since so is the operator Γ1/2 ω−1/2 and since ℛ ̂ is symmetric, it follows that so is the kernel of the operathe kernel of the operator M ̂ Consider the operator tor R. 1 −1/2 ̂ ̂ ∗ −1/2 ω (R − R )ω i

= ω−1/2 Γ1/2 (1 − M)−1 (1 − MM + )(1 − M + ) Γ1/2 ω−1/2 . −1

The estimate (f , (1 − MM + )f ) ⩾ (f , f )(1 − ‖M‖2 )

(6.162)

6.9 Invariance of the conditions on the complex germ

| 387

implies that (f ,

1 −1/2 ̂ ̂ ∗ −1/2 −1 󵄩2 󵄩 ω (R − R )ω f ) ⩾ 󵄩󵄩󵄩(1 − M + ) Γ1/2 ω−1/2 f 󵄩󵄩󵄩 (1 − ‖M‖2 ) i ⩾

‖f ‖2 (1 − ‖M‖2 ) = c1 ‖f ‖2 ‖ω−1/2 Γ−1/2 (1 − M + )‖2

̂ ∈ ℒ. for some c1 > 0. So, R ̂ ̂ is symmetric. 2. Let R ∈ ℒ. From (6.161), it follows that the kernel of the operator M ̂ Let us verify that ‖M‖ < 1. We denote by 1 ̂ −ℛ ̂)ω−1/2 ω1/2 Γ−1/2 A = Γ−1/2 ω1/2 ω−1/2 (R i ̂ by the relation the bounded operator related to M M = A(1 + A)−1 . The operator 1 ̂ −ℛ ̂)ω−1/2 ω1/2 Γ−1/2 1 + A + A+ = Γ−1/2 ω1/2 ω−1/2 (R i is positive definite by property (iii) and Lemma 6.9.1, 1 + A + A+ ⩾ c1 > 0. For the operator M, we have 1 − M + M = (1 + A+ ) (1 + A + A+ )(1 + A)−1 ⩾ c2 > 0. −1

Hence, ‖M| < 1. Lemma 6.9.2 is proved. ̂t , which depends on t ∈ ℝ and lies in the class ℒ for Consider the operator ℛ ̂ω−1/2 and ω−1/2 Γ1/2 are differentiable. Consider each t, such that the operators ω−1/2 ℛ the t-dependent transformation (6.161). Lemma 6.9.3. ̂t . ̂t ω−1/2 is differentiable if and only if so is the operator M 1. The operator ω−1/2 ℛ 2. The following property holds: d ̂ ̂2 R + Rt + (−Δ + m2 + ft̂ ) dt t

⋅

̂ −1/2 )−1 (iM ̂ − Zt − Yt M ̂t − M ̂t Y ∗ − M ̂t Z ∗ M ̂t ) = (−Γ1/2 t t t − Mt Γt t

×

(Γ−1/2 t

−

̂t )−1 . Γ−1/2 M t

388 | 6 Hamiltonian semiclassical field theory Here, the derivative is understood in the sense d̂ ̂ t ω−1/2 )ω1/2 , R = ω1/2 (ω−1/2 R dt t and Zt = Γ−1/2 ( t

d̂ ̂ 2 −1/2 ℛ + ℛt + (−Δ + m + ft̂ ))Γ , dt t

i Yt = [(Γ1/2 ω−1/2 )ω1/2 Γ−1/2 + (Γ−1/2 ω1/2 )ω−1/2 Γ−1/2 ] t t t 2 t 1 d ̂ ̂∗ ̂∗ ℛ ̂ + (−Δ + m2 + ft̂ )]Γ−1/2 . (ℛ + ℛ ) + ℛ + Γ−1/2 [ t t 2 dt The first assertion of the lemma is a corollary of the conditions imposed on the ̂ and theorems on differentiation of the product of operators and the inverse operator ℛ operator. To verify equation (6.162), it suffices to use formulas for R: ̂ − MΓ−1/2 ℛ ̂∗ ), R = (Γ−1/2 − MΓ−1/2 ) (Γ−1/2 ℛ −1

̂Γ−1/2 − ℛ ̂∗ Γ−1/2 M)(Γ−1/2 − Γ−1/2 M) . R = (ℛ −1

̂ satisfies equation (6.160) if and only if Corollary 6.9.1. The operator R ̂t = Zt + Yt M ̂t + M ̂t Y ∗ + M ̂t Z ∗ M ̂t . iM t t

(6.163)

6.9.2 On operators with smooth Weyl symbol We denote by 𝒜0 the class operators  in the space L2 (ℝd ) such that the operators of the form [ωα1 [ωα2 ⋅ ⋅ ⋅ [ωαn A]̂ ⋅ ⋅ ⋅]] are bounded for any n and α1 , . . . , αn ∈ [0, 1]. Example 6.9.1. ω−α ∈ 𝒜0 for α ⩾ 0. Lemma 6.9.4. ̂ ∈ 𝒜0 . Then  + B ̂ ∈ 𝒜0 ,  B ̂ ∈ 𝒜0 . 1. Let A,̂ B ̂ 2. Let A ∈ 𝒜0 . Then (a) [ωα A]̂ ∈ 𝒜0 for α ∈ [0, 1],  + ,  T ∈ 𝒜0 , ̂ −β ∈ 𝒜0 for β ∈ ℝ (b) ωβ Aω 3. Let  ∈ 𝒜0 and let the operator  −1 be bounded. Then  −1 ∈ 𝒜0 .

(6.164)

6.9 Invariance of the conditions on the complex germ

| 389

Proof. The first assertion is clear, since the commutators of the form (6.164) for the ̂ can be expressed in terms of the similar product and sum of the operators  and B ̂ ̂ commutators for the operators A and B. Assertion 2(a) also follows from the boundedness of the operators (6.164). To verify 2(b), note in view of the above properties that the operators ̂ −α = [ωα A]ω ̂ −α + A,̂ ωα Aω

̂ α = ω−α [Aω ̂ α ] + Â ω−α Aω

̂ −nα ∈ 𝒜0 , ω−nα Aω ̂ nα ∈ 𝒜0 . lie in the class 𝒜0 for α ∈ [0, 1]. It follows that ωnα Aω β β Since, for sufficiently large natural n, either n or − n lie in the interval [0, 1], we have ̂ −β ∈ 𝒜0 . To prove the third assertion of the lemma, we represent the commutator ωβ Aω [ωα ,  −1 ] in the form [ωα ,  −1 ] = − −1 [ωα , A]̂  −1 . In a similar way, we can represent the other commutators for the operator  −1 in terms of the commutators for the operator A.̂ Lemma 6.9.4 is proved. Let us show that operators with smooth Weyl symbols belongs to the class 𝒜0 . Recall the definition of Weyl ordering. Under this approach, to each function of the form ̃ β), A(x, k) = ∫ dαdβeiαk+iβx A(α,

α, β, x, k ∈ ℝd ,

(6.165)

there corresponds the operator ̂ ̃ β), Â = ∫ dαdβeiαk+iβ̂x A(α,

(6.166)

̂ = (k̂ ⋅ ⋅ ⋅ k̂ ) of the ̂ = (x̂1 ⋅ ⋅ ⋅ x̂d ) and k which is expressed in terms of the operators x 1 d form x̂m f (x) = xm f (x),

𝜕 k̂m f (x) = −i f (x), 𝜕xm

x ∈ ℝd .

̂+ = x ̂, Function (6.165) is called the Weyl symbol of the operator (6.166). We have x + T T T ̂ ̂ ̂ ̂ ̂ ̂ ̂ =x ̂ , k = −k, and so the property A = A means that k = k, x A(x, k) = A(x, −k), and the Hermitian property  + =  is equivalent to saying that the Weyl symbol A∗ (x, k) = A(x, k) is real.

390 | 6 Hamiltonian semiclassical field theory Lemma 6.9.5. Let the function A(x, k) be infinitely differentiable and the functions (x2 + 1)(−

l

𝜕2 + 1) A(x, k) 𝜕x2

(6.167)

be bounded. Then the operator  with the Weyl symbol A(x, k) is bounded. Proof. We represent the function A as the Fourier integral (6.165). The operator (6.166) can be written as 1

̂ ̃ β). Â = ∫ dβeiβ̂x ∫ dαeiαk e 2 αβ A(α,

For the norm of the operator A,̂ we have the estimate 󵄩󵄩 󵄩󵄩 ̂ β) ̂ ⩽ ∫ dβ󵄩󵄩󵄩∫ dαeiα(k+ ̂ β)󵄩󵄩󵄩, 2 A(α, ‖A‖ 󵄩󵄩 󵄩󵄩 󵄩 󵄩

(6.168)

̂ is since the norm of the operator ei√βx is 1. Since the norm of the operator φ(k) max |φ(k)|, estimate (6.168) takes the form 󵄨󵄨 󵄨󵄨 ̂ ⩽ ∫ dβ max󵄨󵄨󵄨∫ dαeikx A(α, ̃ β)󵄨󵄨󵄨. ‖A‖ 󵄨 󵄨󵄨 k 󵄨󵄨 󵄨

(6.169)

We have ̃ β) = ∫ dαeikx A(α, =

1

(2π)d 1 d

(2π)

× (−

∫ dxA(x, k)e−iβx ∫

(x2 n

m dx 2 m (x + 1) + 1)

−n 𝜕 ) A(x, k)(β2 + 1) e−iβx . 𝜕x2

(6.170)

The quantity (6.170) for m > d/2 is majorized by const (β2 + 1)−1 , and the integral (6.169) converges for d/2. This verifies that the operator  is bounded. To prove that  ∈ 𝒜0 , we investigate the Weyl symbol of the product of operators. ̂ and C ̂ be two operators with Weyl symbols C (x, k) and C (x, k). We denote Let C 1 2 1 2 ̂C ̂ by (C1 ∗ C2 )(x, k) the Weyl symbol of the product of the operators C 1 2 . Let us express it in terms of the functions C1 and C2 . From (6.166) using the Baker–Hausdorff formula, we obtain ̂ (α1 +α2 )k+i(β x 1 +β2 )̂ ̂C ̂ C 1 2 = ∫ dα1 dβ1 dα2 dβ2 e i

(α β −α2 β1 ) ̃ (α , β )C ̃ 2 1 2 ×C . 1 1 1 2 (α2 , β2 )e

(6.171)

6.9 Invariance of the conditions on the complex germ

| 391

The Weyl symbol of the operator (6.171) is obtained by replacing the operator ̂ ei(α1 +α2 )k+i(β1 +β2 )̂x by the function ei(α1 +α2 )k+i(β1 +β2 )x : (C1 ∗ C2 )(x, k) = ∫ dα1 dβ1 dα2 dβ2 ei(α1 +α2 )k+i(β1 +β2 )x i

̃ (α , β )e 2 (α1 β2 −α2 β1 ) . ×C 1 1 2 ̃ and C ̃ in terms of C and C and using the inverse Fourier Expressing the function C 1 2 1 2 transform, we get dp dq dx′ dk′

(C1 ∗ C2 )(x, k) = ∫

2d

(2π)

C1 (x −

p q , k+ ) 2 2

× C2 (x′ , k′ )eip(k−k )+iq(x−x ) . ′

(6.172)

′

Lemma 6.9.6. Under the hypotheses of Lemma 6.9.5, Â ∈ 𝒜0 . ̂ = [ωα , A]̂ satisfies the conditions of Proof. It suffices to verify that the commutator B ̂ has the form Lemma 6.9.5. According to (6.172), the Weyl symbol of the operator B B(x, k) = ∫

dqdy (2π)d

[ωαk+q/2 − ωαk−q/2 ]A(y, k)eiq(x−y) .

Integrating by parts, we get m

(x2 + 1) (− =∫

n

𝜕2 + 1) B(x, k) 𝜕x dqdyeiq(x−y) n′

m′

(2π)d (q2 + 1) (y2 + 1)

F(k, q, y),

(6.173)

where the function 𝜕2 F(k, q, y) = (y + 1) (− 2 + 1) 𝜕y 2

m′

× [(y + i

n+n′

n′

A(y, k)(q2 + 1)

2 m α ωk+q/2 − ωαk−q/2 𝜕 ) + 1] n′ 𝜕q (q2 + 1)

is bounded. This shows that B satisfies the condition of Lemma 6.9.5. An induction shows that the condition of Lemma 6.9.5 is satisfied for the Weyl symbols of operators (6.165). Hence, the operators (6.165) are bounded. Lemma 6.9.6 is proved. ̂ m ∈ 𝒜0 . We denote by 𝒜m the class of operators  in the space L2 (ℝd ) such that Aω Remarks. ̂ m ∈ 𝒜0 and ωm  ∈ 𝒜0 are equivalent by Lemma 6.9.4. 1. The properties Aω

392 | 6 Hamiltonian semiclassical field theory 2.

The property  ∈ 𝒜m means that symbol of the operator  decreases as ω−m k as |k| → ∞.

Lemma 6.9.7. Let the function A(x, k) be infinitely differentiable and let the functions (x2 + 1)′ × (− 𝜕x𝜕 2 + 1)n A(x, k)ωm k be bounded for l, n > 0. ̂ Then the operator A with the Weyl symbol A(x, k) lies in the class 𝒜m . ̂ = ωm  has the form Proof. The symbol of the operator B B(x, k) = ∫

dqdy (2π)d

iq(x−y) ωm . k+q/2 A(y, k)e

(6.174)

The function ωm k+q/2 can be written as 𝜕l ωm 1 k q q ∑ i1 ⋅⋅⋅ il l 𝜕k ⋅ ⋅ ⋅ 𝜕kil 2 i i1 ,...,il 1 l=0 n

ωm k+q/2 = ∑

1

+ ∫ dξ 0

𝜕n+1 ωm (1 − ξ )n k+ξ q/2 q ⋅ ⋅ ⋅ qin+1 . ∑ n!2n+1 i ⋅⋅⋅i 𝜕ki1 ⋅ ⋅ ⋅ 𝜕kin+1 i1 1

n+1

The integral (6.174) of the first n + 1 terms can be explicitly evaluated. Substituting the last term in integral (6.174) we get an expression, which can be estimated as in the previous lemma. Now Lemma 6.9.7 follows.

6.9.3 Study of the transformed equation The error of the solution of equation (6.160) can be estimated by considering equation (6.163) and the use of the following lemma. Lemma 6.9.8. Let ω−1/2 Γ1/2 ∈ 𝒜0 . Then ̂ ∈ 𝒜m ⇔ R ̂ −ℛ ̂ ∈ 𝒜m−1 . M This fact follows from the results of the previous section and formula (6.161). Lemma 6.9.9. Let the operators Yt and Zt be bounded and let Yt = Yt+ . Then the solution of the Cauchy problem for equation (6.163) exists and is unique in ̂ with symmetric kernel and norm smaller than 1. the class of operators M Proof. Consider the auxiliary Cauchy problem for the system of equations i

ut Y d = ( t ) = ( t∗ v −Zt dt

Zt ut ∗) ( t ) . −Yt v

(6.175)

6.9 Invariance of the conditions on the complex germ

| 393

Since the operators Yt and Zt are bounded, the solution of the Cauchy problem (6.175) exists and is unique. The solution operator At B∗t

Bt , A∗t

which transforms the initial condition of the Cauchy problem for equation (6.175) in the solution of this problem, ut A ( t ) = ( ∗t v Bt

Bt u0 ) ( ), A∗t v0

satisfies the equation i

d At ( dt B∗t

Yt Bt ∗) = ( At −Zt∗

Zt At ∗) ( ∗ −Yt Bt

Bt ) A∗t

(6.176)

and the initial equation A0 = 1,

B0 = 0.

(6.177)

The solution of the Cauchy problem (6.176)–(6.177) can be written as the series ∞ Bt Ant ∗ ) = ∑ ( n∗ At Bt n=0

At B∗t

(

Bnt ), An∗ t

(6.178)

which converges in the operator-valued norm and whose coefficients are determined from the recurrent relations t

A∗t ( n∗ Bt

Bnt Y ) = −i ∫ dτ ( τ∗ An∗ −Z t τ

A0t ( 0∗ Bt

B0t ) A0∗ t

0

1 =( 0

Zτ An−1 τ ∗ ) ( n−1 ∗ −Yτ (Bτ )

Bn−1 τ ∗ ), (An−1 τ )

0 ). 1

That the series (6.178) converges follows from the estimates: ‖At ‖n ⩽

cn t n , n!

‖Bt ‖n ⩽

cn t n n!

for some constant c, which can be verified by induction. To show that the solution of the Cauchy problem for equation (6.163) is unique, consider the bounded operator Tt = Mt (A∗t + B∗t M0 ) − (Bt + At M0 ).

(6.179)

394 | 6 Hamiltonian semiclassical field theory From (6.163) and (6.175), it follows that iṪ t = (Yt + Mt Zt∗ )Tt ,

(6.180)

T0 = 0.

(6.181)

and for t = 0, we have

The operator Yt + Mt Zt∗ are bounded, and hence, an application of the contraction mapping principle gives the uniqueness of the solution of the Cauchy problem for equation (6.180) in the class of bounded operators. Consequently, Tt = 0, Mt (A∗t + B∗t M0 ) = Bt + At M0 .

(6.182)

For a proof of uniqueness, it suffices to verify that the operator A∗t + B∗t M0 has the bounded inverse. To verify the existence of the inverse, consider the matrix (6.178). It is easily checked that At B∗t

Bt ) A∗t

−1

(

A−n t B−n∗ t

∞

= ∑( n=0

B−n t ), A−n∗ t

where A−n t ( −n∗ Bt

t

B−n A−n+1 t τ ) = i dτ ( ∫ A−n∗ (B−n+1 )∗ t τ 0

B−n+1 Yτ τ −n+1 ∗ ) ( (Aτ ) −Zτ∗

Zτ ). −Yτ∗

As a result, At B∗t

(

Bt ) A∗t

−1

=(

A+t −B+t

−BTt ). ATt

Hence, the matrix (6.178) is the matrix of the canonical transformation [6]. Consequently, A has a bounded inverse. Therefore, −1 −1 + 1 − (A−1 t Bt )(At Bt ) = At (At ) +

−1

> 0.

So, ‖A−1 t Bt ‖ < 1. Hence, the operator (A∗t + B∗t M0 )

−1

∗−1 = (1 + A∗−1 t Bt M0 ) At −1

is bounded. This proves that the Cauchy problem for equation (6.163) has a unique solution.

6.9 Invariance of the conditions on the complex germ

| 395

For a proof of the existence, it suffices to consider the operator Mt = (Bt + At M0 )(A∗t + B∗t M0 ) . −1

(6.183)

By direct substitution, we see that the operator Mt satisfies the equation (6.163). Lemma 6.9.9 is proved. Lemma 6.9.10. Let the operators Yt − ω and Zt be bounded and let Yt = Yt+ . Then the solution of the Cauchy problem for equation (6.163) exists and is unique in the class of operators M with symmetric kernel and norm smaller than 1. Proof. The transformation ̃ −iωt M = e−iωt Me reduces equation (6.163) to the form ̃̇ = Z̃ + Y ̃M ̃+M ̃Y ̃∗ + M ̃Z̃ ∗ M, ̃ iM

(6.184)

where Z̃ = eiωt Zeiωt ,

̃ = eiωt (Y − ω)e−iωt . Y

For equation (6.184), the conditions of Lemma 6.9.9 are satisfied. Now the result of Lemma 6.9.10 follows. ̂ 0 ∈ ℒ. Lemma 6.9.11. Let R Then the solution of the Cauchy problem for equation ̂ −1/2 )ω1/2 + R ̂ 2 + (−Δ + M 2 + ft̂ ) = 0 ω1/2 (ω−1/2 Rω

(6.185)

exists and is unique in the class ℒ. ̃t = iω, the conditions of For a proof of this lemma, it suffices to note that, for ℛ Lemma 6.9.10 are satisfied and then employ Lemma 6.9.3. We will say that the family of operators {At }, t ∈ (t1 , t2 ), lies in 𝒜0 if operators (6.165) are uniformly bounded with respect to t. Here, it is implied that {At } ∈ 𝒜m if {ωm At } ∈ 𝒜0 . Lemma 6.9.12. Let {Xt }, {Wt } ∈ 𝒜0 , and let Ct be the operator satisfying the equation iĊ t = Xt Ct + Wt .

(6.186)

Then {Ct } ∈ 𝒜0 for C0 ∈ 𝒜0 . Proof. We first note that, under the conditions of the lemma, Ct is a bounded operator. Now the assertion is immediate from the contraction mapping principle.

396 | 6 Hamiltonian semiclassical field theory Let us show that the conditions of Lemma 6.9.12 are also satisfied for the commutator [ωα , Ct ]. Indeed, it satisfies the equation i[ωα , Ct ] = Xt [ωα , Ct ] + [ωα , Xt ]Ct + [ωα , Wt ]. Hence, [ωα , Ct ] is a bounded operator. By induction, Ct ∈ 𝒜t . Lemma 6.9.12 is proved. Lemma 6.9.13. Let {Xt } ∈ 𝒜0 , {Wt } ∈ 𝒜m , Ct be the solution of equation (6.186). Then Ct ∈ 𝒜m for C0 ∈ 𝒜0 . For a proof, it suffices to note that the operator Ct ωm satisfies the conditions of Lemma 6.9.12. Lemma 6.9.14. For {Yt } ∈ 𝒜0 and {Zt } ∈ 𝒜m , for an operator satisfying equation (6.176) and the initial condition (6.177), the relations {At } ∈ 𝒜0 , {Bt } ∈ 𝒜m hold. Proof. From Lemma 6.9.12, we have {(

At B∗t

Bt )} ∈ 𝒜0 . A∗t

Hence, {At } ∈ 𝒜0 . Since iḂ t = Yt Bt + Zt A∗t , it follows from Lemma 6.9.13 that {Bt } ∈ 𝒜m . Lemma 6.9.14 is proved. Lemma 6.9.15. Under the hypotheses of Lemma 6.9.14, if M0 ∈ 𝒜m , then Mt ∈ 𝒜m . Proof. From Lemma 6.9.4 and the definition of the class Am , we have ωm Bt ∈ 𝒜0 , A∗t ∈ 𝒜0 ,

ωm At ω−m ∈ 𝒜0 , B∗t M0 ∈ 𝒜0 ,

ωm M0 ∈ 𝒜0 ,

(A∗t + B∗t M0 )

−1

∈ 𝒜0 .

Hence, from (6.183) it follows that ωm Mt ∈ 𝒜0 . Lemma 6.9.15 is proved. Lemma 6.9.16. For {Yt − ω} ∈ 𝒜0 , {Zt } ∈ 𝒜m , M0 ∈ 𝒜m ⇒ Mt ∈ 𝒜m . The proof is similar to that of Lemma 6.9.10. The following result is a consequence of the above lemmas. ̂ 0 ∈ ℒ, Γ1/2 ∈ 𝒜0 , ̂t ∈ ℒ, R Lemma 6.9.17. Let ℛ t −1/2 (Γ1/2 ) ∈ 𝒜0 , t ω

Γt − ω ∈ 𝒜0 ,

⋅

2

2

̂+ℛ ̂ + (−Δ + M + ft̂ ) ∈ 𝒜m−1 . ℛ t

6.9 Invariance of the conditions on the complex germ

| 397

Then ̂0 − ℛ ̂t − ℛ ̂0 ∈ Am−1 ⇒ R ̂t ∈ 𝒜m−1 . R

(6.187)

6.9.4 Construction of an approximate solution of the Riccati equation ̂t ∈ ℒ, depending on t ∈ ℝ as a parameter, so as to satisfy the Let us find an operator ℛ conditions of Lemma 6.9.17. We will seek this operator in the form ̂t = Â t + iω ℛ

̂t 1/4 B

e ω1/2 eBt ω1/4 , ̂

(6.188)

̂ t satisfy the conditions where the bounded operators  t and B  Tt =  +t =  t ,

̂T = B ̂+ = B ̂t. B t t

(6.189)

̂ t ω−1/4 ‖ < ∞. Then the operator ℛ ̂t lies in the class ℒ. Lemma 6.9.18. Let ‖ω1/4 B ̂t is symmetric follows from properProof. The fact that the kernel of the operator ℛ ̂ω−1/2 , we use the estities (6.189). To verify the boundedness of the operator ω−1/2 ℛ mate 󵄩󵄩 −1/2 ̂ −1/2 󵄩󵄩 󵄩󵄩 −1/2 ̂ −1/2 󵄩󵄩 󵄩󵄩 −1/4 B̂ t 1/4 󵄩󵄩2 e ω 󵄩󵄩 . 󵄩󵄩ω ℛl ω 󵄩󵄩 ⩽ 󵄩󵄩ω At ω 󵄩󵄩 + 󵄩󵄩ω The property ‖ω−1/2  t ω−1/2 ‖ < ∞ follows from the boundedness of the operators ω−1/2 ̂ and  t . For the norm of the operator ω−1/4 eB1 ω1/4 , we have ∞ −1/4 1/4 1󵄩 󵄩󵄩 −1/4 B̂ t 1/4 󵄩󵄩 ̂ t ω1/4 󵄩󵄩󵄩 = e‖ω B̂ t ω ‖ < ∞. e ω 󵄩󵄩 ⩽ ∑ 󵄩󵄩󵄩ω−1/4 B 󵄩󵄩ω 󵄩 n! n=0

̂t ω−1/2 is bounded. This shows that the operator ω−1/2 ℛ Let us study the positive definiteness of the operator 1 −1/2 ̂ ̂∗ )ω−1/2 − c. ω (ℛt − ℛ t i To this end, it suffices to verify the identity 2(f , ω−1/4 eBt ω1/4 ⋅ ω1/4 eBt ω−1/4 f ) ⩾ c(f , f ), ̂

̂

which is equivalent to the boundedness of the operator (ω1/4 eBt ω−1/4 ) . ̂

−1

(6.190)

398 | 6 Hamiltonian semiclassical field theory Similar to (6.190), 1/4 ̂ 1/4 󵄩󵄩 1/4 B̂ t −1/4 −1 󵄩󵄩 ) 󵄩󵄩 ⩽ e‖(ω Bt w )‖ < ∞. 󵄩󵄩(ω e ω

̂t ∈ ℒ. This verifies the claim ℛ ̂t of To fulfill the conditions of Lemma 6.9.17, one has to construct an operator ℛ the form (6.188) such that the symbol of the operator d ̂ ̂t )2 + (−Δ + m2 + ft̂ ) ℛ + (ℛ dt t decreases sufficiently and rapidly as k → ∞. To address the Hermitian properties, it will be convenient to invoke the Weyl calculus of operators. Assume that the functions C1 and C2 have power-like behavior at infinity. For |k| → ∞, the asymptotics of symbol (6.172) can be studied as follows. Consider the function (C1 ∗ C2 )(x, Λk) for k = const, Λ → ∞. Integral (6.172) is transformed to (C1 ∗ C2 )(x, Λk) = ∫ dpdqdp′ dq′ (

2d

Λ p q ) C1 (x − , Λ(k + )) 2π 2 2

× C2 (x + q, Λ(k + p))e−iΛ[pp +qq ] . ′

(6.191)

′

For Λ → ∞, the stationary-phase method is applied to integral (6.191), for which the error estimate is considered in [24]. The asymptotics of symbol (6.172) is written as (C1 ∗ C2 )(x, k) = C1 (x +

󵄨 i 𝜕 i 𝜕 ′ ′ 󵄨󵄨󵄨 , p − )C (x , k ) 󵄨󵄨 ′ 2 󵄨󵄨x′ =x 2 𝜕k′ 2 𝜕x′

k =k

and can be expanded, as |k| → ∞, in the asymptotic series π

𝜕m+n C1 ei 2 (m−n) m+n m!n!2 𝜕ki1 ⋅ ⋅ ⋅ 𝜕kim 𝜕xj1 ⋅ ⋅ ⋅ 𝜕xjn m,n=0 ∞

(C1 ∗ C2 )(x, k) = ∑ ×

𝜕m+n C2 . 𝜕xi1 ⋅ ⋅ ⋅ 𝜕xim 𝜕kj1 ⋅ ⋅ ⋅ 𝜕kjm

(6.192)

̂t as |k| → Let us find the asymptotics of the expansion of the symbol of the operator ℛ ∞, for which the conditions of Lemma 6.9.17 are satisfied, i. e., 2

ℛ̇ t + ℛt ∗ ℛt + ω + ft (x) = O(

as |k| → ∞.

1 ) |k|m

(6.193)

6.9 Invariance of the conditions on the complex germ

| 399

We will search this expansion in the form m

m

ℛt = iωk + ∑ ℛt + O(|k|

−m−1

n=1

),

(6.194)

where n

ℛt (x, k) =

rtn (x, k/ωk ) . ωnk

Taking into account that, for C1 = C2 , in formula (6.192) one can retain only the terms with even m + n, ∞

ℛ∗ℛ= ∑ ∑

l=0 m+n=2l

(−1)l 𝜕m+n ℛ 𝜕m+n ℛ , 22l m!n! 𝜕ki1 ⋅ ⋅ ⋅ 𝜕kim 𝜕xj1 ⋅⋅⋅ 𝜕xjn 𝜕xi1 ⋅ ⋅ ⋅ 𝜕xim 𝜕kj1 ⋅ ⋅ ⋅ 𝜕kjn

from (6.194) we get the following recurrent relations: 2iωk ℛ1t (x, k) + ft (x) = 0, l−1 ℛ̇ t (x, k)

+

|(l−1)/2|

+ 2i ∑

s=1

+

(6.195)

2iωk ℛlt (x, k) 𝜕2s ℛl−2s 𝜕2s ωk (−1)s t 2s (2s)!2 𝜕ki1 ⋅ ⋅ ⋅ 𝜕ki2π 𝜕xi1 ⋅ ⋅ ⋅ 𝜕xi2π

𝜕m+n ℛpt ei 2 (m−n) 2m+n n!m! 𝜕ki1 ⋅ ⋅ ⋅ 𝜕kim 𝜕xj1 ⋅ ⋅ ⋅ 𝜕xjn m+n+p+q=l−1 π

∑

p⩾1, q⩾1, m+n=2k⩾0 m+n

×

𝜕 ℛqt = 0. 𝜕xi1 ⋅ ⋅ ⋅ 𝜕xim 𝜕kj1 ⋅ ⋅ ⋅ 𝜕kjn

(6.196)

From these recurrent relations, the functions ℛlt can be determined uniquely; morêt is symmetric: over, the kernel of the operator ℛ l

l

ℛt (x, −k) = ℛt (x, k).

Let us show that the asymptotics of the Weyl symbols of the operators A and B as |k| → ̂. ∞ is uniquely determined by the above asymptotics of the symbol of the operator ℛ From (6.188), it follows that the symbol of the operator A coincides with the real ̂. From the recurrent relations (6.196), it follows part of the symbol of the operator ℛ that the functions ℛ2t , ℛ4t , . . . are real, and the functions ℛ1t , ℛ3t , . . . are purely imaginary. Hence, A=∑ l⩾1

rt2l (x, k/ωk ) ω2l k

.

400 | 6 Hamiltonian semiclassical field theory ̂ consider the function C such that To construct the symbol of the operator B, Im ℛt = C ∗ C + O(|k|−m ).

(6.197)

From (6.192), we get the recurrent relations on the coefficients of the expansion of C in the asymptotic series with respect to ω−2 k , C = ∑ Cn ,

Cn (x, k) =

n

cn (x, k/ωk ) ω2n−1/2 k

.

They have the form C0 = ω1/2 , k

1 t ℛ = i 2t−1

𝜕m+n Cp ei 2 (m−n) 2m+n n!m! 𝜕ki1 ⋅ ⋅ ⋅ 𝜕kim 𝜕xj1 ⋅ ⋅ ⋅ 𝜕xjn π

∑

m+n=2s⩾0 s+p+q=l; p,q⩾0 m+n

×

𝜕

Cq

𝜕xi1 ⋅ ⋅ ⋅ 𝜕xim 𝜕kj1 ⋅ ⋅ ⋅ 𝜕kjn

.

Now the coefficients Cp can be uniquely recovered. Next, in a similar way from the relation ̂ ω−1/4 eBt = ω−1/4 C t ̂

one can uniquely determine the asymptotics of the symbol of the operator eBt as |k| → ̂ t can be uniquely de∞. As a result, the asymptotics of the symbol of the operator B termined, ̂

Bt = ∑ l⩾t

bl (x, k/ωk ) . ω2l k

The above algorithm allows one to uniquely find the functions al and bl . ̂ t whose symbols can be expressed in terms of Consider the operators  t and B finitely many functions al and bl : m

At (x, k) = ∑ l=1

al (x, k/ωk ) , ω2l k

m

Bt (x, k) = ∑ l=1

b1 (x, k/ωk ) . ω2l k

̂ω−1/2 and ω−1/2 Γ1/2 From the above, it follows that in this case the operators ω−1/2 ℛ are differentiable, and so the conditions of Lemma 6.9.17 are satisfied. We have the following result.

6.9 Invariance of the conditions on the complex germ

| 401

Theorem 6.9.1. Assume that the initial condition for the Riccati equation is such that ̂ 0 ∈ ℒ, R

m−1

̂ 0 − iω − ∑ R ̂ 0 ∈ 𝒜m . R m n=1

Then m−1

̂ t − iω − ∑ R ̂ t ∈ 𝒜m . R n n=1

This verifies the invariance under evolution of the above conditions on the complex germ.

7 Asymptotic methods for systems of a large number of fields 7.1 Introduction In quantum field theory, one frequently encounters problems involving models with a large number of quantized fields [132]. In particular, by studying Feynman diagrams in quantum chromodynamics under the assumption that the number of quark colors tends to infinity [127, 128], one can establish qualitative properties of elementary particles, which are in good accord with experimental findings [131]. The study of other models (the theory of scalar fields, quantum electrodynamics) with a large number of particle types is also of great interest. In this chapter, we consider the so-called vector models in which the number of fields N tends to infinity. Expansions in powers of 1/N for such models frequently appear in the physical literature; various approaches to the 1/N-expansion have been proposed. Let us illustrate these methods on an example of the theory of N scalar fields. The classical Hamiltonian of this model is expressed in terms of the fields φ1 , . . . , φN and the momenta π1 , . . . , πN as follows: N 1 1 m2 HN = ∫ dx ∑ ( πa (x)πa (x) + ∇φa (x)∇φa (x) + φ (x)φa (x)) 2 2 a a=1 2

+

λ ∫ dxφa (x)φb (x)φb (x). 4N

(7.1)

The following methods of studying the corresponding quantum system were used in the literature: 1. Study of perturbation theory series with respect to λ/4N and identification of the terms of the order of 1 as λ/N → 0, N → ∞ [130]. 2. Application of the path integral method, the Laplace method expansions, the stationary-phase method, and the saddle point method [63, 66, 77]. 3. Use of the “collective field method” [80]. Here, one considers the wave functionals ΨN [φ1 (⋅), . . . , φN (⋅)], Ψ[φ1 (⋅), . . . , φN (⋅)] = Ψ[ψ(⋅, ⋅)],

(7.2)

that depend only on combinations of fields of the form ψ(x, y) = ∑ φa (x)φa (y). a

After this substitution, the Schrödinger equation reduces to the semiclassical form. https://doi.org/10.1515/9783110762709-007

404 | 7 Asymptotic methods for many-field systems 4. Study (see [59]) of the Heisenberg equations for the operators N

̂ y) = ∑ φ ̂a (x)φ ̂a (y), ψ(x, a=1 N

N

̂a (y), ξ̂ (x, y) = ∑ π̂a (x)φ a=1

̂ (x, y) = ∑ π̂a (x)π̂a (y). η a=1

5.

Investigation of the Heisenberg equations for the fields φa (x).

In this chapter, we consider a different approach to systems with a large number of fields. In this approach, the classical equations are not equations for functions of finitely many arguments, but rather an equation for the functional, which is a nonlinear analogue of the Schrödinger equation of the quantum theory of a single field. The main idea of the present method is as follows. Similar to the representation of the Hamiltonian of the many-particle quantum theory in the second-quantized form in terms of the particle creation and annihilation operators, the Hamiltonian of the N-field theory can be written in the “third-quantized” form in terms of the creation and annihilation operators of the fields A± [φ(⋅)]. This equation can be attacked by semiclassical methods. The concept of third quantization was introduced in quantum gravitation and cosmology [72] in the study of processes with variable number of universes, and also in the string theory in the construction of the theory of string interaction. In the present chapter, we will show that the method of third quantization also applies to the ordinary quantum field theory. This method is capable of delivering new asymptotic solutions to the Schrödinger equation for a system of N fields.

7.2 O(N)-symmetric anharmonic oscillator as an analog of a many-field system In this section, we consider an analogue of a system of a large number of fields (the O(N)-symmetric anharmonic oscillator). In this model, the wave function depends on N coordinates x1 , . . . , xN , and the Hamiltonian has the form N

H = ∑ (− i=1

axi2 g N 2 2 1 𝜕2 + ) + ∑x x . 2 𝜕xi2 2 4N i,j i j

(7.3)

In physical literature, various approaches to the 1/N-expansion are illustrated, as a rule, on an example of a model (7.3). The analogue of the third-quantized approach for this model is the asymptotic method discussed in Chapter 3. The results obtained by this method are compared with those obtained using the collective field method.

7.2 O(N)-symmetric anharmonic oscillator as an analog of a many-field system | 405

7.2.1 Collective field method In applications of the collective field method to system (7.3) it is assumed, as in (7.2), that the wave function ψN (x1 , . . . , xN ) is also O(N)-symmetric, i. e., it depends only on R = √x12 + ⋅ ⋅ ⋅ + xN2 , ψN (x1 , . . . , xN ) = ψ(R). The surface area of the sphere depends on its radius, and so the total probability in this case is not of the form ∫ dR|ψ(R)|2 , but rather has the form R

󵄨 󵄨2 󵄨 󵄨2 ∫ dx1 ⋅ ⋅ ⋅ dxN 󵄨󵄨󵄨ψN (x1 , . . . , xN )󵄨󵄨󵄨 = CN ∫ dR RN−1 󵄨󵄨󵄨ψ(R)󵄨󵄨󵄨 0

for some R-independent constant CN . The asymptotics are considered with respect to the L2 -norm, and so we examine the change of the variable Φ = ΨR

N−1 2

.

(7.4)

The equation for the function Φ, which is a corollary of the Schrödinger equation 𝜕ψN = HN ψN , 𝜕t

(7.5)

𝜕Φ 1 𝜕2 Φ N 2 − 1 aR2 g 4 =− + Φ+ Φ+ R Φ. 2 2 𝜕t 2 𝜕R 2 4N 8R

(7.6)

i has the form i Rescaling

R = r √N, we reduce equation (7.6) to the semiclassical form i 𝜕Φ 1 𝜕2 Φ gr 4 1 1 ar 2 = − 2 2 + 2 (1 − 2 )Φ + Φ+ Φ, N 𝜕t 2 4 2N 𝜕r 8r N where 1/N is the analogue of the Planck constant.

(7.7)

406 | 7 Asymptotic methods for many-field systems 7.2.2 Classical equations in the second-quantized approach According to Chapter 2, the Hamiltonian (7.3) can be written in terms of the creation and annihilation operators a± (x) as 2 2 ̂ = ∫ dx a+ (x)(− 1 𝜕 + ax )a− (x) H 2 𝜕x 2 2 2

+

gε (∫ dx a+ (x)x 2 a− (x)) , 4

(7.8)

where ε = 1/N. The Schrödinger equation (7.5) can be considered as the following equation for a vector of the Fock space Ψ = (0, 0, . . . , ψN , . . .): i

dΨ = (ε−1 H(√εa+ , √εa− ) + H1 (√εa+ , √εa− ))Ψ; dt

here, H[φ∗ , φ] = ∫ dxφ∗ (x)(−

g 1 𝜕2 ax2 󵄨 󵄨2 )φ(x) + (∫ dxx2 󵄨󵄨󵄨φ(x)󵄨󵄨󵄨 ). + 2 2 𝜕x 2 4

The classical equation in this case (a Hartree-type equation) 𝜕φt (x) δH[φt∗ , φt ] = 𝜕t δφ∗ (x)

(7.9)

𝜕φt (x) 1 𝜕2 g ax 2 t 󵄨 󵄨2 = (− + )φ (x) + x 2 φ(t, x) ∫ dy y2 󵄨󵄨󵄨φt (y)󵄨󵄨󵄨 . 2 𝜕t 2 𝜕x 2 2

(7.10)

i is as follows: i

7.2.3 Relation between classical equations under various approaches Let us show that the classical equations of the collective field method can be derived from the Hartree equation (7.10). Consider the Gaussian ansatz i

φ(t, x) = ce 2 ax

2

(7.11)

for equation (7.10). The normalization factor ct can be found from the condition 󵄨 󵄨2 ∫ dx󵄨󵄨󵄨φ(x)󵄨󵄨󵄨 = 1.

(7.12)

7.2 O(N)-symmetric anharmonic oscillator as an analog of a many-field system | 407

Using (7.11), we get the equations iα d c=− , dt 2 dα g 2 +α +a+ = 0. dt 2 ln α i

(7.13) (7.14)

The transformation P=

Re α , √2 Im α

Q=

1 √2 Im α

(7.15)

reduces equation (7.14) to the Hamiltonian system dQ = P, dt

dP dU =− , dt dQ

(7.16)

which corresponds to the motion of particles in the potential U(Q) =

1 aQ2 gQ4 + + . 2 4 8Q2

(7.17)

But it is the Schrödinger equation (7.7), as obtained by the collective field method, which corresponds to the motion of a particle in potential (7.17), and hence, the classical equations for the collective field method have the form (7.16). So, the Gaussian ansatz reduces the classical equations of the second-quantized approach to the Hamiltonian system of the collective field method.

7.2.4 Semiclassical wave functions under various approaches Consider the semiclassical wave functions constructed by the complex germ method in the above approaches. In the collective field method, using the germ substitution t

t

t

Φ(r, t) = const eiNS eiNP (r−Q ) f (t, √N(r − Qt ))

(7.18)

in equation (7.7) we get the condition on St , 2

(P t ) − U(Qt ), Ṡ t = P t Q̇ t − 2 the classical Hamiltonian system (7.16), and the oscillation equation for the function f , (i

𝜕 1 𝜕2 1 + − U ′′ (Qt )ξ 2 )f = 0. 𝜕t 2 𝜕ξ 2 2

(7.19)

408 | 7 Asymptotic methods for many-field systems In the second-quantized approach, the substitution t

ΨN = eiNS Kφt ,N Y t , where Kφt ,N is the many-particle canonical operator (see Chapter 2 that gives the condition on St ), i ̇ − φ̇ ∗ (x)φ(x)) − H[φ∗ , φ], Ṡ t = ∫ dx(φ∗ (x)φ(x) 2

(7.20)

the Hartree equation (7.10), and the equation for the vector Y t , which can be conveniently written in the following form. Consider the generalized Fock vector Z t = δ(A+t + A−t )Y t ,

(7.21)

where A+t = ∫ dxφt (x)a+ (x),

A−t = (A+t ) . +

(7.22)

The vector Y t is uniquely determined from the generalized vector (7.21) under the additional condition A−t Y t = 0. The vector (7.21) satisfies the Schrödinger equation i

dZ t = H2 Z t dt

(7.23)

with the quadratic Hamiltonian H2 = ∫ dxa+ (x)(− +

1 𝜕 ω2 x 2 − + )a (x) 2 𝜕x2 2

g (∫ dx(a+ (x)x2 φ(x) + φ∗ (x)x 2 a− (x))), 4

(7.24)

where 󵄨 󵄨2 ω2 = a + g ∫ dxx2 󵄨󵄨󵄨φ(x)󵄨󵄨󵄨 . Let us now (heuristically) show that the wave function (7.18), which appears in the collective field method, can be represented in terms of the many-particle canonical operator with a special choice of φ and Y. Consider first as a vector Y the vacuum vector of the Fock space Y = Φ(0) . The action of the many-particle canonical operator on the vector Y produces the wave function, which is represented as the product of one-particle functions: ΦN (x1 , . . . , xN ) = φ(x1 ) ⋅ ⋅ ⋅ φ(xN ).

(7.25)

7.2 O(N)-symmetric anharmonic oscillator as an analog of a many-field system | 409

For the function φ of the form (7.11), the wave function (7.25) is O(N)-symmetric, Ψ = i 2 cN e 2 αR , and hence, the function (7.4) has the form 1 i Φ0 = const exp{N[ αr 2 + ln r]}r 1/2 . 2 2

(7.26)

It is known that the function Φ0 (r) = χ(r)eNS(r) can be approximated by a Gaussian wave packet [25] of the following form. The function S expands near the maximum Q of the real part S up to terms of the order of (x − Q)2 , and the function χ(r) is approximated by the constant χ(Q). In case (7.26), the corresponding Gaussian wave packet has the form (7.18) for P and Q of the form (7.15) and −2

f (ξ ) = e−(Q

−i Re α)ξ 2 /2

.

Consider now the wave function (7.18) for an arbitrary function f . Since the operator of multiplication by (r + Q) acts on the germ function (7.18) by multiplication by 2Q (in the zero approximation), the operator N 1/2 (r − Q) can be approximated by the operator N 1/2 (r 2 −Q2 ) . So, any function (7.18)) can be approximated by a function of the form 2Q g(N 1/2 (r 2 − Q2 ))Φ0 (r)

(7.27)

for some function g. The function (7.27) can be expressed in terms of the many-particle canonical operator g(N 1/2 ∫ dx x2 a+ (x)a− (x) − N 1/2 Q2 )Kφ,N Φ(0) .

(7.28)

Using the commutation rules of various combinations of the creation and annihilation operators with the operator Kφ,N , we can write (7.28) as Kφ,N g(B+ + B− )Φ(0) + O(N −1/2 ),

(7.29)

where B+ = ∫ dxx2 φ(x)ã + (x) = ∫ dxz(x)a+ (x), B− = (B+ ) , +

z(x) = (x2 − Q2 )φ(x).

Since the commutator of the operators B− and B+ is proportional to the identity operator, the vector Y = g(B+ + B− )Φ(0) can be written as Y = χ(B+ )Φ(0) .

(7.30)

410 | 7 Asymptotic methods for many-field systems Formula (7.30) means that the components of the vector Y have the form Yp = (ξ1 , . . . , ξp ) = cp z(ξ1 ) ⋅ ⋅ ⋅ z(ξp ).

(7.31)

So, with a special choice (7.11) for the function φ and the components of the vector Y (see (7.29)), any wave function of the form (7.18) can be written as (7.20). Therefore, the class of semiclassical wave functions in the second-quantized approach (7.20) includes all semiclassical wave functions of the collective field method. At the same time, for a different choice of φ and Y, the function (7.20) is not expressible in the form (7.18). Hence, the class of asymptotic solutions (7.20) is broader than (7.18). 7.2.5 Asymptotic spectrum as N → ∞ Let us examine the asymptotic spectrum of the Hamiltonian (7.3) as N → ∞. In the collective field method, the application of the complex germ theory at a point gives the following asymptotic spectrum of the operator (7.3): 1 El = NU(Q) + √U ′′ (Q)(l + ) + O(N −1 ), 2

(7.32)

where Q is the point of minimum of the potential U. The corresponding asymptotic eigenfunctions have the form (7.18). Consider now the asymptotic spectrum obtained by the complex germ method in the Fock space. In this approach, the series of asymptotic eigenvalues and eigenfunctions correspond to stationary stable solutions of the Hartree equation Ωφ(x) = (−

1 𝜕2 ax2 g 󵄨 󵄨2 + )φ(x) + x 2 φ(x) ∫ dy y2 󵄨󵄨󵄨φ(y)󵄨󵄨󵄨 , 2 2 𝜕x 2 2

(7.33)

which satisfy the normalization condition (7.12). Equation (7.33) is the Schrödinger equation for the one-dimensional harmonic oscillator (−

1 𝜕2 ω2 x 2 + − Ω)φ(x) = 0, 2 𝜕x2 2

(7.34)

with unknown frequency ω, whose square is determined from the condition 󵄨 󵄨2 ω2 = a + g ∫ dx x 2 󵄨󵄨󵄨φ(x)󵄨󵄨󵄨 . The eigenfunctions of the operator appearing in equation (7.34) have the form ΨK =

K

(A+ ) Ψ , √K 0

(7.35)

7.2 O(N)-symmetric anharmonic oscillator as an analog of a many-field system

| 411

and can be expressed in terms of the function Ψ0 = const e−ωx

2

/2

and the creation operators ω 1 𝜕 . A+ = √ x − √2ω 𝜕x 2 As the function φ, we consider the kth eigenfunction φ = ΨK .

(7.36)

The eigenvalue Ω of the Hartree equation has the form 1 Ω = ω(K + ). 2 Since x =

1 (A+ + A− ), where A− √2ω 2

= (A+ )+ , and [A− , A+ ] = 1, for the mean value of the

operator x we have

(ΨK , x2 ΨK ) =

1 1 (K + ). ω 2

Now equation (7.35) assumes the form ω2 = a +

1 g (K + ) ω 2

(7.37)

from which ω is determined uniquely. The series of asymptotic eigenvalues of the Hamiltonian (7.3) is expressible in terms of the spectrum of the system in variations 1 𝜕2 ω2 x2 + − Ω)F(x) 2 𝜕x2 2 g + x2 φ2 (x) ∫ dyy2 φ(y)(F + G)(y), 2 ω2 2 1 𝜕2 + − Ω)G(x) βG(x) = (− 2 𝜕x2 2 g + x2 φ2 (x) ∫ dyy2 φ(y)(F + G)(y). 2

−βF(x) = (−

(7.38)

For n − K ≠ 0, ±2, the matrix element (ΨK , x2 Ψn ) vanishes, and hence, among the solutions of system (7.38) we have Fn+ = 0,

Gn+ = Ψn ,

(7.39)

412 | 7 Asymptotic methods for many-field systems where βn = ω(n − K). System (7.38) also has solutions of the form 7K−2 = √K(K − 1)ΨK+2 ,

βK−2 = −2ω;

GK−2 = −√(K + 1)(K + 2)ΨK−2 ,

and in addition 7K+2 = − GK+2 =

√(K + 1)(K + 2) √K(K − 1) ΨK+2 − Ψ , βK+2 + 2ω βK+2 − 2ω K−2

√(K + 1)(K + 2) √K(K − 1) ΨK+2 + Ψ , βK+2 − 2ω βK+2 + 2ω K−2

βK+2 = √4ω2 +

g (2K + 1). ω

The system composed of the above functions G and the function φ is complete in L2 , the constructed solution of the Hartree equation is stable, and the asymptotic spectrum of the Hamiltonian (7.3) in the second-quantized approach is given by the expression (see Chapter 3) E n = Nε + ε0 +

∑

m⩾0, m=K ̸

βm nm + O(N −1 ),

(7.40)

where 2

1 g 1 ε = H[φ∗ , φ] = ωK (K + ) − (K + ) . 2 2 2 4ωK Note that, for K = 0 and n1 = n3 = n4 = ⋅ ⋅ ⋅ = 0, n2 = l, formula (7.40) becomes (7.32). So, the asymptotic spectrum in the second-quantized approach contains not only all energy levels found by the collective field method, but also new asymptotic eigenvalues of the Hamiltonian (7.3).

7.3 Formalism of the third quantization and the semiclassical approximation 7.3.1 Asymptotic methods in the theory of large number of fields on a lattice We illustrate the third-quantized approach to the theory of large number of fields on an example of the field theory on a lattice. In this approach, the d-dimensional space

7.3 Formalism of the third quantization and the semiclassical approximation

| 413

is approximated by a cubic lattice of M ×⋅ ⋅ ⋅×M points. The lattice spacing is l. The field φa (x) is approximated by the family of numbers φa (j1 , . . . , jd ), j1 , . . . , jd = 0, j1 , . . . , jd = 0, M − 1. The field derivative is approximated by the family of numbers 𝜕φa 1 (j , . . . , jd ) = (φa (j1 , . . . , js + 1, . . . , jd ) − φa (j1 , . . . , js , . . . , jd )); 𝜕xs 1 l

(7.41)

here, for js = M −1, in (7.41) we assume that js +1 = 0. The integrals of the form ∫ dxf (x) are approximated by the integral sums ld

M−1

∑

j1 ,...,jd =0

f (j1 , . . . , jd ).

(7.42)

The Poisson bracket of the variables ∫ dxφa (x)f (x) and ∫ dxπb (x)g(x) for a = b in the continuous theory has the form {∫ dxφa (x)f (x), ∫ dxπa (x)g(x)} = ∫ dxf (x)g(x).

(7.43)

In order that property (7.43) would be preserved also in the theory on a lattice, we require that {φa (j1 , . . . , jd ), πa (j1 , . . . , jd )} = l−d .

(7.44)

The classical Hamiltonian of the field theory on a lattice has the form 2

N 1 1 d 𝜕φ m2 2 2 H = ∑ ld { ∑ [ (πa (j)) + ∑ ( a ) (j) + (φa (j)) ] 2 s=1 𝜕xs 2 a=1 2 j

+

N λ 2 ( ∑ (φa (j)) )}, 4N a=1

(7.45)

where the summation is over integer vectors j = (j1 , . . . , jd ), j1 , . . . , jd = 0, M − 1, and 𝜕φ (7.41) is denoted by 𝜕x a (j). s The corresponding quantum Hamiltonian acts in the space of functions ΨN [φ1 (⋅), . . . , φN (⋅)],

(7.46)

depending on N ⋅ M d arguments φa (j1 , . . . , jd ),

a = 1, N,

j1 , . . . , jd = 0, M − 1.

This Hamiltonian is obtained from the Hamiltonian function (7.45) by replacing the fields φa (j) by the operators of multiplication by φa (j), and replacing the momenta

414 | 7 Asymptotic methods for many-field systems πa (x) by the differentiation operators π̂a (x) = l−d

1 𝜕 . i 𝜕φa (j)

The quantum Hamiltonian 2

N 𝜕2 1 d 𝜕φa m2 2 ̂ = ∑ { ∑ [− 1 H + ( ) (j) + (φa (j)) ] ∑ d 𝜕φ (j)𝜕 (j) 2 𝜕x 2 2l a a s a=1 s=1 j

+

N λ 2 ( ∑ (φa (j)) )}, 4N a=1

(7.47)

which is of the type considered in Chapter 2, is a multidimensional analogue of the O(N)-symmetric anharmonic oscillator, which was considered in the previous section. In order to apply the method considered in Chapter 2, we represent operator (7.47) in terms of the creation and annihilation operators. The space of functions (7.46) is considered as a subspace of the Fock space consisting of the vectors of the form Ψ0 Ψ1 |φ1 (⋅)| ( ) ⋅⋅⋅ ΨN [φ1 (⋅), . . . , φN (⋅)] ⋅⋅⋅

(7.48)

with symmetric components, in which the creation and annihilation operators A± [φ(⋅)], which are the operator-valued distributions of M d arguments φ(j1 , . . . , jd ), j1 , . . . , jd = 0, M − 1, are introduced. The operator (7.47) can be written in the form 𝜕2 ̂ = ∑{∫ ∏ dφ(j)A+ [φ(⋅)][− 1 H 2ld 𝜕φa (j)𝜕φa (j) j j 2

+

m2 1 d 𝜕φ 2 (j)) + (φ(j)) ]A− [φ(⋅)] ∑( 2 s=1 𝜕xs 2

+

λ 2 (∫ ∏ dφ(j)A+ [φ(⋅)](φ(j)) A− [φ(⋅)])}. 4N j

(7.49)

For N → ∞, the semiclassical methods of Chapter 2 can be applied to the Schrödinger equation with Hamiltonian (7.49).

7.3 Formalism of the third quantization and the semiclassical approximation

| 415

7.3.2 Classical equations in the third-quantized approach to the theory of large number of fields Consider the analogue of representation (7.49) in the continuous field theory. In this case, the Schrödinger equation is an equation for the functional depending on N functions φ1 (x), . . . , φN (x). The space of symmetric functionals of N functions is considered as a subspace of the Fock space, in which the operator-valued generalized functionals A± [φ(⋅)] are introduced by (∫ DφA+ [φ(⋅)X[φ(⋅)]Ψ) [φ1 (⋅), . . . , φ(⋅)] k

1 k = ∑ X[φa (⋅)]Ψk−1 [φ1 (⋅), . . . , φa−1 (⋅), φa+1 (⋅), . . . , φk (⋅)], √k a=1

(7.50)

(∫ DφA [φ(⋅)]X [φ(⋅)]Ψ)[φ1 (⋅), . . . , φk−1 (⋅)] −

∗

= √k ∫ DφX ∗ [φ(⋅)]Ψk [φ(⋅), φ1 (⋅), . . . , φk−1 (⋅)]. The Hamiltonian (7.1) can be written in the form H = ∫ DφA+ [φ(⋅)] ∫ dx(−

δ2 1 2 δφ(x)δφ(x)

1 m2 + ∇φ(x)∇φ(x) + φ(x)φ(x))A− [φ(⋅)] 2 2 ελ + ∫ DφDϕ ∫ dxφ2 (x)ϕ2 (x)A+ [φ(⋅)]A− [φ(⋅)]A+ [ϕ(⋅)]A− [ϕ(⋅)], 4

(7.51)

where ε = 1/N. The Heisenberg motion equations for the operators Φ±t [φ(⋅)] = √εeiHt A± [φ(⋅)]e−iHt do not contain the small parameter ε, i

d − 1 δ2 Φt (φ(⋅)) = ∫ dx(− dt 2 δφ(x)δφ(x)

1 m2 + ∇φ(x)∇φ(x) + φ(x)φ(x))Φ−t (φ(⋅)) 2 2 λ + ∫ DΦ ∫ dxφ2 (x)ϕ2 (x)Φ+t (ϕ(⋅))Φ−t (ϕ(⋅))Φ−t (φ(⋅)), 2

which is contained in the commutation relations [∫ DφΦ−t [φ(⋅)]X ∗ [φ(⋅)], ∫ DφΦ+t [φ(⋅)]Y[φ(⋅)]] = ε ∫ DφX ∗ [φ(⋅)]Y[φ(⋅)].

(7.52)

416 | 7 Asymptotic methods for many-field systems The analogue of the Hartree equation, which is obtained from equation (7.52) by replacing the operator-valued functionals Φ+t [φ(⋅)] and Φ−t [φ(⋅)] by the scalar functionals Φt [φ(⋅)] and Φ∗t [φ(⋅)], has the form i

1 δ2 d Φt [φ(⋅)] = ∫ dx(− dt 2 δφ(x)δφ(x) 1 m2 + ∇φ(x)∇φ(x) + φ(x)φ(x))Φt [φ(⋅)] 2 2 λ + ∫ Dϕ ∫ dxφ2 (x)ϕ2 (x)Φ∗t (ϕ(⋅))Φt (ϕ(⋅))Φt (φ(⋅)). 2

(7.53)

The classical equations can also be obtained by the complex germ method. The substitution in the Schrödinger equation, which corresponds to a complex germ at a point, has the form i Ψ = exp{ St }UΦ′ Y t , ε

(7.54)

where St is a real function, UΦ′ is a unitary operator of the form UΦ = exp{

1 ∫ Dϕ(Φ(φ(⋅))A+ (φ(⋅)) − Φ∗ (φ(⋅))A− (φ(⋅)))}, √ε

(7.55)

and Y t is a vector of the Fock space, which is regular as ε → 0. Substituting vector (7.55) in the Schrödinger equation dΨ 1 = H(√εA+ , √εA− )Ψ, dt ε

(7.56)

dSt i = − [(Φ̇ t , Φt ) − (Φt , Φ̇ t )] − H(Φ∗ , Φ) dt 2ε

(7.57)

i we get

to the leading order, the Hartree equation in the succeeding order, and the equation for the vector Y t , i

dY t = H2 Y t , dt

(7.58)

where H2 = ∫ DφA+ [φ(⋅)] × ∫ dx(−

1 δ2 2 δφ(x)δφ(x)

1 1 + (∇φΛ )2 (x) + φ2Λ (x)(m2 + (Φt , φ2Λ (x)Φt )))A− [φ(⋅)] 2 2 λ + ∫ dx[∫ Dφφ2Λ (x)(Φ∗ [φ(⋅)] + Φ[φ(⋅)]A+ [φ(⋅)])]. 4

(7.59)

7.3 Formalism of the third quantization and the semiclassical approximation

| 417

7.3.3 On the construction of asymmetric solutions The methods of Chapter 2 are capable of delivering asymptotic solutions of the Schrödinger equation with Hamiltonian (7.51), i

dΨN = HN ΨN , dt

(7.60)

which are symmetric functionals ΨN [φ1 (⋅), . . . , φN (⋅)] under permutations of the arguments φ1 (⋅), . . . , φN (⋅). It turns out that by using semiclassical methods one can construct asymptotic solutions of equation (7.60), which are symmetric only under permutations of N − k arguments. To construct such asymptotics, Ψ[χ1 (⋅), . . . , χk (⋅), φ1 (⋅), . . . , φN−k (⋅)], which are symmetric under permutations of the arguments φ1 , . . . , φN−k , we consider the Fock space consisting of columns of the form Ψ0 [χ1 (⋅), . . . , χk (⋅)] Ψ1 [χ1 (⋅), . . . , χk (⋅), φ1 (⋅)] ( ). ⋅⋅⋅ ΨN−k [χ1 (⋅), . . . , χk (⋅), φ1 (⋅), . . . , φN−k (⋅)] ⋅⋅⋅

(7.61)

The Hamiltonian (7.1) can be written as the sum φ

χ

HN−k = HN−k + H φχ + Hk of the Hamiltonian of the fields φ, N−k

φ

HN−k = ∑ ∫ dx(− α=1

+

1 δ2 1 m2 + ∇φ (x)∇φ (x) + φ (x)φα (x)) α α 2 δφα (x)δφα (x) 2 2 α

N−k λε ∫ dx ∑ φa (x)φa (x)φb (x)φb (x), 4 a,b=1

of the Hamiltonian of the fields χ, χ

k

HN−k = ∑ ∫ dx(− α=1

+

1 δ2 1 m2 + ∇χ (x)∇χ (x) + χ (x)χα (x)) α 2 δχ α (x)δχ α (x) 2 α 2 α

k λε ∫ dx ∑ χα (x)χα (x)χβ (x)χβ (x) 4 α,β=1

(7.62)

418 | 7 Asymptotic methods for many-field systems and of the interaction Hamiltonian H φχ =

k N−k λε ∫ dx ∑ χα (x)χα (x) ∑ φa (x)φα (x). 4 α=1 α=1

In the Fock space (7.61), one introduces the creation and annihilation operators A± [φ(⋅)]. The Hamiltonian of the field φ is represented in the form (7.51); the Hamiltonian of the field χ has the form (7.62), and the interaction Hamiltonian is H φχ =

k λε ∫ dx ∑ χα (x)χα (x) ∫ DφA+ [φ(⋅)]φα (x)A− [φ(⋅)]. 4 α=1

The complex germ theory can be applied as N → ∞, k = const. One considers the substitution (7.54) in the Schrödinger equation; moreover, in this case Y t is the vector (7.61). The equations for St and Φt coincide with those obtained above, and i

dY t = (H2 + Hχ )Y t , dt

(7.63)

where the operator H2 is given by (7.59), and k

Hχ = ∑ ∫ dx(− α=1

+

1 δ2 1 + ∇χα (x)∇χα (x) α α 2 δχ (x)δχ (x) 2

m2 + λ(Φt , φ2 (x)Φt ) χa (x)χα (x)). 2

(7.64)

7.3.4 φφχ model Consider the quantum theory of N scalar fields φ1 , . . . , φN interacting with the field χ. The Hamiltonian of this model has the form N

H = ∫ dx ∑ [− a=1

1 δ2 1 m2 + ∇φ (x)∇φ (x) + φ (x)φa (x) a 2 δφa (x)δφa (x) 2 a 2 a

1 δ2 1 M2 g − + ∇χ(x)∇χ(x) + χ(x)χ(x) + φ (x)φa (x)χ(x)]. √N a 2 δχ(x)δχ(x) 2 2 The third-quantized approach can also be applied to this theory. The Hamiltonian of the model can be represented in terms of the creation and annihilation operators

7.3 Formalism of the third quantization and the semiclassical approximation

| 419

A± [φ(⋅)] as follows: H = ∫ DφA+ [φ(⋅)] × ∫ dx[−

δ2 1 2 δφ(x)δφ(x)

m2 2 1 φ (x) + g √εφ2 (x)χ(x)]A− [φ(⋅)] + (∇φ)2 (x) + 2 2 + ∫ dx[−

1 δ2 1 M2 2 + (∇χ)2 (x) + χ (x)]. 2 δχ(x)δχ(x) 2 2

After rescaling the variables, A± √ε = Φ± ,

√εχ = Y,

we see that the Hamiltonian is proportional to 1/ε, the commutator between the operators Φ± [φ(⋅)] is of the order of ε, and the coefficient for each differentiation operator is of the order of ε. So, the semiclassical methods can also be applied to this model. The points of the classical phase space are tuples consisting of the complex-valued functional Φ[φ(⋅)], the classical field Y(x), and the momentum 𝒫 (x). The classical equations in this case are obtained via the germ substitution i

t

e ε S UΦ′ exp{

i 1 δ )}Y t ∫ dx(𝒫 (x)χ(x) − Y(x) √ε i δχ(x)

(7.65)

in the Schrödinger equation; they read as i

d 1 δ2 Φt [φ(⋅)] = ∫ dx[− dt 2 δφ(x)δφ(x)

1 m2 2 + (∇φ)2 (x) + φ (x) + gφ2 (x)Y(x)]Φt [φ(⋅)], 2 2 Ẏ = 𝒫 , −𝒫̇ = −ΔY + M 2 Y + g(Φ, φ(x)Φ).

7.3.5 Case of spontaneous symmetry breaking Consider the theory of a large number of fields with spontaneously broken symmetry N

H = ∫ dx ∑ (− a=1

+

1 δ2 1 + ∇φ (x)∇φa (x)) a 2 δφ (x)δφa (x) 2 a

N λ ( ∑ φa (x)φa (x) − Nv2 ). 4N a=1

(7.66)

420 | 7 Asymptotic methods for many-field systems This Hamiltonian can be written as H = ∫ DφA+ [φ(⋅)] ∫ dx[− +

δ2 1 1 + (∇φ)2 (x)]A− [φ(⋅)] 2 δφ(x)δφ(x) 2

ελ v2 ∫ dx(∫ DφA+ [φ(⋅)]φ2 (x)A− [φ(⋅)] − ), 4 ε

and hence, the methods considered in the present chapter can also be applied to this system. However, the germ substitution (7.54) gives symmetric functionals, and hence, corresponds to the case of unbroken symmetry. In order to construct asymmetric solutions, we relabel the field φN by χ and represent the Hamiltonian of the model as H = ∫ dx[−

1 δ2 1 + ∇χ(x)∇χ(x) 2 δχ(x)δχ(x) 2

N

+ ∑ (− a=1

1 δ2 1 + ∇φ (x)∇φa (x)) 2 δφa (x)δφa (x) 2 a 2

λ N−1 + ( ∑ φ (x)φa (x) + χ(x)χ(x) − Nv2 ) ]. 4N a=1 a

(7.67)

This Hamiltonian can be studied by analogy with the previous section. In this case, the elements of the classical phase space are triples consisting of a complex-valued functional Φ[φ(⋅)], a field Y, and a momentum 𝒫 .

7.3.6 O(N)-asymmetric theories The method considered in the present chapter can also be applied to a broader class of models than that covered by the 1/N-expansion methods. Consider the O(N)asymmetric model Φ4 N

H = ∑ ∫ dx(− a=1

δ2 1 2 δφa (x)δφa (x)

2

1 m2 g 4 + ∇φa (x)∇φa (x) + (φa )(x)) + (φa (x)) ) 2 2 4 +

N λε ∫ dx ∑ φa (x)φa (x)φb (x)φb (x). 4 a,b=1

An analogue of the Hartree equation in this case has the form u (x) 1 1 iΦ̇ t [φ(⋅)] = ∫ dx( π 2 (x) + (∇φ)2 (x) + t φ2 (x) + gφ4 (x))Φt [φ(⋅)], 2 2 2

(7.68)

7.4 On renormalizations of the classical equations | 421

where ut (x) = m2 + λ(Φt , φ2 (x)Φt ),

(Φt , Φt ) = 1.

So, the classical equation for this system of N fields is the one-field model Φ4 lying in an external self-consistent field, which depends on the functional Ψ. However, this model is not exactly solvable.

7.4 On renormalizations of the classical equations In the previous section, we considered the derivation of the classical equation (7.53) for the functional Φt [φ(⋅)]. However, we have not taken into account the problem of divergences and renormalizations that appears in quantum field theory. In this section, we will examine the divergences in equation (7.53).

7.4.1 On regularization and renormalization The quantum field theory can be regularized in different ways. In this regard, we note, in particular, the method of lattice regularization discussed at the beginning of the previous section. In the present section, we consider the following ultraviolet and infrared regularizations. In place of the field φ(x), one considers the regularized field φΛ (x), φΛ (x) = ∫ dyAΛ (x − y)φ(y),

(7.69)

where AΛ (x − y) → δ(x − y) as the regularization is removed. For the infrared regularization, consider the theory in a box of size L × L × ⋅ ⋅ ⋅ × L with periodic boundary conditions. After regularization, equation (7.53) is written as u (x) 1 1 iΦ̇ t [φ(⋅)] = ∫ dx( π 2 (x) + (∇φΛ )2 (x) + t φ2Λ (x))Φt [φ(⋅)], 2 2 2

(7.70)

where ut (x) = m2 + λ(Φt , φ2Λ (x)Φt ),

(Φt , Φt ) = 1.

(7.71)

Let us study the system of equations (7.70), (7.71), and consider the divergent expressions. As in the previous chapter, consider the renormalization procedure.

422 | 7 Asymptotic methods for many-field systems 7.4.2 On Gaussian and non-Gaussian solutions of classical equations Let us study equation (7.70). Since the function ut (x) is involved in equations of the classical field theory for model (7.70), we will require that this function be nonsingular: lim ut (x) < ∞.

(7.72)

Λ→∞

Consider the wave functional representable as the product of a polynomial and the Gaussian exponent ∑ ∫ dx1 ⋅ ⋅ ⋅ dxn fn (x1 , . . . , xn )φ(x1 ) ⋅ ⋅ ⋅ φ(xn ) n

1 × exp{ ∫ dxdyφ(x)R(x, y)φ(y)}. 2

(7.73)

Next, consider the divergences appearing in the solution of the Cauchy problem for equation (7.70). Note that the operator B+ [Pt , Qt ] = ∫ dx[φ(x)Pt (x) −

1 δ Q (x)] i δφ(x) t

(7.74)

d commutes with i dt −H, where H is the operator appearing on the right of equation (7.70) if

Q̇ t (x) = Pt (x), Ṗ t (x) + ∫ dy dzAΛ (x − y)(−Δy + ut (y))AΛ (y − z)Qt (z) = 0.

(7.75)

The initial condition (7.73) can be written as a linear combination of the functionals of the form B+ [P01 , Qt0 ], . . . , B+ [P0m , Qm 0 ]Φ,

(7.76)

where Φ is a Gaussian functional. The solution of equation (7.70) satisfying the initial condition (7.76) can be expressed in terms of the solutions (Pt1 , Q1t ), . . . , (Ptm , Qm t ) of system (7.75), t B+ [Pt1 , Q1t ], . . . , B+ [Ptm , Qm t ]Φ .

The equations (7.75) are regular as Λ → ∞, and hence, so are the operators (7.74) as Λ → ∞. So, singularities may appear only in the Gaussian functional Φt . Consider the Gaussian solution of equation (7.70): i ct exp{ ∫ dxdyφ(x)Rt (x, y)φ(y)}. 2

(7.77)

7.4 On renormalizations of the classical equations | 423

̂ the operator with the kernel R(x, y), We denote by R ̂ )(x) = ∫ dyR(x, y)f (y), (Rf ̂ the operator with the kernel AΛ (x, y). The Gaussian quadratic form and denote by A satisfies the nonlinear equation d̂ ̂ ̂ ̂ R + Rt Rt + A(−Δ + ut )Â = 0, dt t

(7.78)

where ut is the operator of multiplication by ut (x). The coefficient of the exponential ct satisfies the equation 1 ̂ d (ln ct ) = − Tr R t. dt 2 Rt :

(7.79)

Equation (7.78) can be considered as an equation for the symbol of the operator Ṙ t (x, k) + (Rt ∗ Rt )(x, k) + Ak ∗ (k2 + ut (x))Ak = 0;

(7.80)

𝜕 here, it is taken into account that the operator  can be represented as A(−i 𝜕x ), where A(k) = Ak is the Fourier transform of the function AΛ . The relation Ak → 1 holds as Λ → ∞.

7.4.3 On singularities of the Gaussian quadratic form In order to examine the ultraviolet divergences, consider the symbol of the operator Rt as k → ∞. We will use the formula for asymptotic expansion of the product of ordered symbols of operators (−i)l 𝜕l B(x, k) 𝜕l C(x, k) l 𝜕ki1 ⋅ ⋅ ⋅ 𝜕kil 𝜕xi1 ⋅ ⋅ ⋅ 𝜕xil l⩾0

(B ∗ C)(x, k) = ∑

(7.81)

with respect to 1/|k|. The third term on the left of (7.80) is of the order of |k|2 , and so one can expect that Rt ∼ |k| as |k| → ∞. By singling out the most singular term from Ri , Rt (x, k) = iAk √k2 + μ2 + rt (x, k), we reduce equation (7.80) to the form rṫ (x, k) + 2iAk √k2 + μ2 rt (x, k) + Ak (ut (x) − μ2 )Ak = O(|k|−1 ).

(7.82)

424 | 7 Asymptotic methods for many-field systems Thus, rt (x, k) is of the order of |k|−1 : rt (x, k) ∼

iAk (ut (x) − μ2 ) 2√k2

+

μ2

+ O(|k|−2 ).

(7.83)

Continuing further this iteration process, we obtain, as in the previous chapter, the expansion of the symbol Rt with respect to the powers of O(|k|−α ). As an initial condition, which is invariant under the evolution, we can take the following one: Rt (x, k) = Rsing (x, k) + Rreg t t (x, k);

(7.84)

here, Rsing is the above asymptotics of the solution of equation (7.78), and Rreg t t is of the order of O(|k|−α ).

7.4.4 Renormalization of the mass and the coupling constant Let us now verify condition (7.72). To this end, we examine the singularities of the mean value φ2Λ (x): ⟨φ2Λ (x)⟩ = (Φt , φ2Λ (x)Φt ). This matrix element can be written in the form ̂ ⟨φ2Λ (x)⟩ = ∫ DφP[φ(⋅)]φ2Λ (x) exp{−(φ, Im Rφ)},

(7.85)

where P is a polynomial functional φ(⋅) with smooth coefficient functions. This integral can be represented as P[

̂ δ δ2 ] ∫ dy dzAΛ (x − y)AΛ (x − z) ∫ Dφe(j,φ)−(φ, Im Rφ) . δj(⋅) δj(y)δj(z)

Evaluating the Gaussian integral, we have P[

1 ̂ −1 δ δ2 ] ∫ dy dzAΛ (x − y)AΛ (x − z) e 4 (j,(Im R) j) . δj(⋅) δj(yδj(z) 2

(7.86)

δ The only singularity of this expression is contained in the operator δjσj , and hence, the matrix element under consideration can be represented as a sum of the regular and

7.4 On renormalizations of the classical equations | 425

singular parts: 1 ̂ ̂ −1 A)(x, ̂ ⟨φ2Λ (x)⟩ = (A(Im R) x) + ⟨φ2Λ (x)⟩reg . 2 ̂ ̂ −1 A)(x, ̂ ̂ ̂ −1 A. ̂ Here, (A(Im R) y) denotes the kernel of the operator A(Im R) The symbol of the operator Rt has the asymptotics Rt (x, k) = iAk √k2 + μ2 (1 +

uℓ (x) − μ2 + O(|k|−3 )), 2(k2 + μ2 )

̂ ̂ −1 Â we have and hence, for the symbol of the operator A(Im R) ̂ ̂ −1 A)(x, ̂ (A(Im R) k) =

Ak √k2 + μ2

(1 −

ut (x) − μ2 + O(|k|−3 )). 2(k2 + μ2 )

̂ in terms of its symUsing the representation of the kernel of an arbitrary operator B bols, 1 ̂ B(x, x) = d ∑ B(x, p), L p the mean value (7.85) can be written, up to the finite quantity ⟨φ2 (x)⟩reg , in the form ⟨φ2Λ (x)⟩ =

Ak 1 ∑ 2Ld k √k2 + μ2 −

ut (x) − μ2 Ak + ⟨φ2 (x)⟩reg . ∑ 3/2 2 2 4Ld k (k + μ )

(7.87)

Substituting (7.87) in (7.71), we arrive at the following condition: m2 + λ[ −

Ak 1 ∑ 2Ld k √k2 + μ2

ut (x) − μ2 Ak ] + λ⟨φ2 (x)⟩reg = ut (x). ∑ d 2 + μ2 )3/2 4L (k k

(7.88)

Let us study (7.88). We set μ2 = m2 +

Ak λ . ∑ d 2L k √k2 + μ2

(7.89)

The sum in the right-hand side of equation (7.89) diverges as Λ → ∞, and m2 should be taken to be infinite. Moreover, μ2 should be finite. In the next section, it will be shown

426 | 7 Asymptotic methods for many-field systems that μ2 controls the spectrum of weakly excited states, and is the squared renormalized mass. Relation (7.88) can be represented as ut (x) − μ2 u (x) − μ2 Ak . = ⟨φ2 (x)⟩reg − t d ∑ 2 2 3/2 λ L k 4(k + μ ) The sum over k, which appears in this equality, diverges in the (3 + 1)-dimensional case. Hence, the parameter λ should be chosen so as to have Ak 1 1 1 + = < ∞. ∑ λ Ld k 4(k2 + μ2 )3/2 λR

(7.90)

The quantity λR plays the role of the renormalized coupling constant.

7.4.5 Renormalization of the cosmological constant Consider now the divergence in the phase factor in the functional Φt . Note that the asymptotic formula for the N-particle wave functional, which is expressed in terms of the many-particle canonical operator, involves both the functional Φ and the factor t t eiS , which appear in the formula for the combination of eiS Φt . t To remove the divergence in the phase factor in eiS Φt , the cosmological constant ℰ is added to the right-hand side of equation (7.70), 1 1 iΦ̇ t [φ(⋅)] = ∫ dx( π 2 (x) + (∇φΛ )2 (x) 2 2 u (x) + t φ2Λ (x) + ℰ )Φt [φ(⋅)]. 2

(7.91)

This classical equation corresponds to the N-field Hamiltonian augmented with the constant N ℰ . Equation (7.79) for the coefficient of the exponential has the form d 1 ̂ (ln ct ) = − Tr R t − i ∫ dxℰ . dt 2

(7.92)

From equations (7.57) and (7.92), it follows that at = ct eiSi satisfies the equation d 1 ̂ iλ 2 (ln at ) = − Tr R ∫ dx⟨λΛ (x)⟩ . t − i ∫ dxℰ + dt 2 4

(7.93)

7.4 On renormalizations of the classical equations | 427

̂ t . To this end, we expand the symbol R ̂ t up to O( 1 4 ), Let us find the singularity of Tr R |ω | k

where ωk = √k2 + μ2 ,

Rt (x, k) = iAk ωk + r1 + r2 + r3 + O(

1 ), |ωk |4

(7.94)

and rm = O( |ω1 | ). k Substituting expansion (7.94) in equation (7.78) and using formula (7.81), we get the following equalities. The terms of the order of O(1), O( |ω1 | ) and O( |ω1 |2 ) have the k k form 2iAk ωk r1 + Ak ∗ (u − μ2 )Ak = 0, 𝜕r 𝜕 (iA ω ) 1 = 0, 𝜕k k k 𝜕x 𝜕r 𝜕2 r1 1 𝜕2 𝜕 (iAk ωk ) = 0. r2̇ + iAk ωk r3 + r12 − i (iAk ωk ) 2 − 𝜕k 𝜕x 2 𝜕km 𝜕kn 𝜕xm 𝜕xn r1̇ + 2iAk ωk r2 − i

Since the trace of the operator can be expressed in terms of the integral of its symbol and since the integral of the derivative vanishes, we have Tr r2 = −

r d Tr 1 , dt 2iωk

Tr r3 =

r12 r1 d2 Tr − Tr . 2iωk dt 2 4ω2k

̂ t can be written as Hence, the singular part of the trace Tr R 2

2 2 ̂ sing = ∫ dx 1 ∑ Ak (iωk + i ut (x) − μ − i (ut (x) − μ ) ) + dF (Tr R) 2 ωk 8 dt Ld k ω3k

for some singular function F depending on the function ut (x) and its derivatives. From the singularity factor at , we can single out at = eF bt1 , where bt is regular. Note that the divergence in the coefficient of the exponential in the representation of functionals was also manifested in the previous chapter. This divergence disappears when changing to a different representation. So, the cosmological constant should be chosen so that the expression 2

−

1 i ut (x) − μ2 i (ut (x) − μ2 ) λ 2 A (iω + − ) + ⟨φ2Λ (x)⟩ − ε = −ε̄ ∑ k k 2 ωk 8 4 2Ld k ω3k

428 | 7 Asymptotic methods for many-field systems be finite. From (7.71), using ⟨φ2Λ (x)⟩ =

ut (x) − μ2 μ2 − m2 + λ λ

and employing (7.89) and (7.90), it follows that −ε̄ can be represented as −ε̄ = −ε +

1 1 2 1 2 2 (u − μ2 ) + (μ − m2 ) − d ∑ Ak ωk . 4λR 4λ 2L k

For ε=

1 2 1 2 (μ − m2 ) − d ∑ Ak ωk , 4λ 2L k

the divergences in the phase factor are canceled.

7.5 Asymptotic spectrum of the Hamiltonian of a large number of fields Consider the application of the method of Chapter 2 to the problem of construction of asymptotic stationary solutions equations of the quantum theory of a large number of fields. These asymptotic formulas correspond to stable solutions of the stationary Hartree-type equation, which is obtained from its nonstationary form via the substitution Φt = e−iΩt Φ0

(7.95)

and can be represented as 1 1 u(x) 2 φ (x)], ∫ dx[ π 2 (x) + (∇φΛ )2 (x) + 2 2 2 Λ

Φ = ΩΦ,

u(x) = m2 + λ(Φ, φ2Λ (x)Φ).

(7.96)

In this formula, φΛ is the regularized field (7.69). The system is considered on the d-dimensional torus under the periodicity condition. 7.5.1 Stationary solutions of the Hartree-type equation Consider some solutions of equation (7.96). Let us examine the first equation of system (7.96) for u(x) = M 2 .

7.5 Asymptotic spectrum of the Hamiltonian of a large number of fields | 429

We let 𝒦 denote the set of d-dimensional vectors of the form 2πn , n1 , . . . , nd ∈ Z}. L

d

𝒦 = {k ∈ R : k =

We also set Ak = ∫ dzA(z)eikz ,

φk = L−d/2 ∫ dxφ(x)eikx .

Now the Hamiltonian (7.96) transforms to the following form: H=

1 𝜕2 + A2k ω2k φk φ−k ], ∑ [− 2 k∈𝒦 𝜕φk 𝜕φ−k

(7.97)

where ωk = √k2 + M 2 . Consider the operators 1 𝜕 1 a+k = (√Ak ωk φk − ), 2 𝜕φ √Ak ωk −k a−k

𝜕 1 1 = (√Ak ωk φk + ), 2 √Ak ωk 𝜕φk

(7.98)

which satisfy the commutation relations [a−k , a+p ] = δkp ,

[a±k , a±p ] = 0.

(7.99)

The Hamiltonian (7.97) can be expressed in terms of operators (7.98) as 1 H = ∑ Ak ωk [a+k a−k + ]. 2 k The first equation of system (7.96) has the solution of the form Φ[φ(⋅)] = ca+k1 ⋅ ⋅ ⋅ a+kn Φ0 [φ(⋅)], where the normalized factor is determined from the condition (Φ, Φ) = 1, Φ0 [φ(⋅)] = exp{−

1 ∑ A ω φ φ }. 2 k∈𝒫 k k k −k

(7.100)

430 | 7 Asymptotic methods for many-field systems The unknown M 2 can be found from the second equation of system (7.96). We first consider the simplest case n = 0 and Φ = cΦ0 . For this case, in view of (7.98) we have φ(x) =

√Ak + −ikx 1 [a e + a−k eikx ]. ∑ Ld/2 k √2ωk k

(7.101)

Hence, by virtue of (7.99) and the condition a−k Φ0 = 0, (Φ(0) , φ2Λ (x)Φ(0) ) =

A 1 ∑ k . d L k 2ωk

(7.102)

Hence, M = μ by (7.89). In the general case, we have (Φ, φ2Λ (x)Φ) =

Aki Ak 1 1 n + . ∑ ∑ d d 2 2 L k 2√k + M L i=1 √k2 + M 2 i

(7.103)

Now M is determined from equation (7.96), which can be written as follows: Aki M 2 − μ2 1 1 1 1 n − . = d ∑ Ak ( )+ d ∑ 2 2 √ λ 2L k L i=1 √k2 + M 2 k +M √k2 + μ2 i For d = 1, 2, the left- and right-hand sides of this equation are finite. For d = 3, one should also take into account the renormalization of the coupling constant: M 2 − μ2 M 2 − μ2 1 1 1 = d ∑ Ak ( + ) − √k2 + M 2 √k2 + μ2 2(k2 + μ2 )3/2 λR 2L k +

Aki 1 n . ∑ Ld i=1 √k2 + M 2 i

7.5.2 Classical energies of various series According to the complex germ theory, to each stable solution of the Hartree equation there corresponds a series of asymptotic eigenvalues and eigenfunctionals of the N-field Hamiltonian. Moreover, the energy corresponding to each of the solutions of this series behaves as N → ∞ as follows: NH[φ∗ , φ] + O(1).

7.5 Asymptotic spectrum of the Hamiltonian of a large number of fields | 431

Let us find NH[φ∗ , φ] for each of the above solutions of the Hartree-type equation, which can be written as 1 m2 1 φ (x)]Φ(0) ) H(Φ∗ , Φ) = (Φ(0) , ∫ dx[ π 2 (x) + (∇φΛ )2 (x) + 2 2 2 Λ λ 2 + (Φ(0) , φ2Λ (x)Φ(0) ) . 4

(7.104)

From relations (7.96) and (7.102), we have E ∼ N(Ω −

2

A λ (∑ k ) ) + O(1). 4Ld k 2ωk

(7.105)

For an arbitrary n, we find that E λ 2 = Ω − ∫ dx(Φ, φ2Λ (x)Φ) . N 4 Hence, 2

n 1 E λLd M 2 − m2 = ∑ Ak √k2 + M 2 + ∑ Aki √k2i + M 2 − ( ), N 2 k 4 λ i=1

where we have used relation (7.96). For the difference of the energy E and the groundstate energy (7.105), we have E − E0 1 = ∑ Ak (√k2 + M 2 − √k2 + μ2 ) N 2 k n

+ ∑ Ak1 √k2i + M 2 − i=1

λLd 2 (M − μ2 )(M 2 + μ2 − 2m2 ), 4λ

inasmuch as M 2 + μ2 − 2m2 =

n Aki Ak 1 1 λ ( (∑ + ) + ). ∑ 2 Ld k 2 √k2 + M 2 √k2 + μ2 i=1 √k + M 2 i

As a result, E − E0 1 = ∑ Ak (√k2 + M 2 − √k2 + μ2 N 2 k −

M 2 − μ2 1 1 + ( )) √k2 + M 2 √k2 + μ2 4 n

+ ∑ Aki (√k2i + M 2 − i=1

M 2 − μ2 1 ). 4 2 √ki + M 2

(7.106)

432 | 7 Asymptotic methods for many-field systems For d ⩾ 5, the sum involved in relation (7.106) is divergent. This fact corresponds to the nonrenormalization of the theory Φ4 in the space-time of this number of dimensions.

7.5.3 Near-vacuum solutions of the N-field equation The asymptotic wave functional of the ground state is represented in terms of the many-particle canonical operator as follows: 1

[N/2]

ΨN [φ(⋅), . . . , φN (⋅)] = ∑

l=0

×

(2N)l l ∑

1⩽i1 =⋅⋅⋅ ̸ =i̸ 2l ⩽N

M[φi1 (⋅), φi2 (⋅)] ⋅ ⋅ ⋅ M[φi2l−1 (⋅), φi2l (⋅)]

× ∏ Φ[φi (⋅)]. i=i̸ 1 ,...,i2l

(7.107)

The functional M, which depends on two functions, satisfies the Riccati equation, which can be written in the following form. We introduce the functional R related to M as M[φ(⋅), ϕ(⋅)] = R[φ(⋅), ϕ(⋅)] + Φ[φ(⋅)]Φ[ϕ(⋅)] and which satisfies the condition ∫ R[φ(⋅), ϕ(⋅)]Φ∗ [ϕ(⋅)]Dϕ = −Φ[φ(⋅)].

(7.108)

According to Chapter 2, the equation for R has the form 2

∑ (∫ dx[− i=1

μ2 1 δ2 1 + (∇φi )(x) + φ2i (x)] − Ω) 2 δφi (x)δφi (x) 2 2

× R[φ1 (⋅), φ2 (⋅)] +

2 λ ∫ dx ∏ (φ2i (x)Φ[φi (⋅)] 2 i=1

+ ∫ Dφ̃ i φ̃ 2i (x)Φ∗ [φ̃ i (⋅)]R[φi (⋅), φ̃ i (⋅)]) = 0.

(7.109)

In order to solve the Riccati equation (7.109), consider the functionals depending on the single function Φk1 k2 =

1 + + (0) a a Φ [φ(⋅)]. √2 k1 k2

(7.110)

7.5 Asymptotic spectrum of the Hamiltonian of a large number of fields | 433

The solution of equation (7.109) satisfying condition (7.108) has the form R[φ1 (⋅), φ2 (⋅)] = −Φ[φ1 (⋅)]Φ[φ2 (⋅)] +

∑

k1 ,k2 ,p1 ,p2

ck1 k2 p1 p2 Φk1 k2 [φ1 (⋅)]Φp1 p2 [φ2 (⋅)].

(7.111)

Substitution (7.111) gives the following conditions on the coefficients ck1 k2 p1 p2 : ∑

k1 ,k2 ,p1 ,p2

(Ak1 ωk1 + Ak2 ωk2 + Ap1 ωp1 + Ap2 ωp2 )

× ck1 k2 p1 p2 Φk1 k2 [φ1 (⋅)]Φp1 p2 [φ2 (⋅)] +

λ ∫ dx[(φ21,Λ (x) − (Φ(0) , φ2Λ (x)Φ(0) ))Φ[φ1 (⋅)] 2

+ ∑ ck1 k2 k′1 k′2 Φk1 k2 [φ1 (⋅)](Φ(0) , φ2Λ (x)Φk′1 k′2 )] k′1 ,k′2

× [(φ22,Λ (x) − (Φ(0) , φ2Λ (x)Φ(0) ))Φ[φ2 (⋅)] + ∑ cp1 p2 p′1 p′2 Φp1 p2 [φ2 (⋅)](Φ(0) , φ2Λ (x)Φp′1 ,p′2 )] = 0. p′1 ,p′2

(7.112)

Note that the vector φ2Λ (x)Φ(0) contains only the vacuum and the two-frequency components, which are both nonzero. So, (φ22,Λ (x) − (Φ0 , φ2Λ (x)Φ0 ))Φ = ∑ Φkk k2 (Φk1 k2 , φ2Λ (x)Φ(0) ). k1 ,k2

Hence, expression (7.111) satisfies the equation (7.109) for (Ak1 ωk1 + Ak2 ωk2 + Ap1 ωp1 + Ap2 ωp2 )ck1 k2 p1 p2 +

λ ∫ dx ((Φk1 k2 , φ2 (x)Φ(0) ) 2

+ ∑ ck1 k2 k′1 k′2 (Φ(0) , φ2 (x)Φk′1 k′2 ))((Φp1 p2 , φ2 (x)Φ(0) ) k′1 ,k′2

+ ∑ cp′1 p′2 p1 p2 (Φ(0) , φ2 (x)Φp′1 p′1 )) = 0. p′1 ,p′2

Consider P ck1 k2 p1 p2 = ∑ α−k δ δ 1 ,p1 k1 +k2 +P p1 +p2 −P P∈𝒦

(7.113)

434 | 7 Asymptotic methods for many-field systems and evaluate the matrix elements involved in formula (7.113): (Φk1 k2 , φ2Λ (x)Φ(0) ) =

1 √Ak1 Ak2 −i(k1 +k2 )x e . Ld √2ωk ωk 1 2

Equation (7.113) reduces to (Ak1 ωk1 + AP−k1 ωP−k1 + Ap1 ωp1 + AP−p1 ωP−p1 )αkP1 p1 +

√Ap′1 AP−p′1 √Ak′1 AP−k′1 λ ∑ 2Ld k′ ,p′ √2ωp′ ωP−p √2ωk′ ωP−k′ 1 1 1 1 1 1

× (δkk′1 + αkP1 k′ )(δp1 p′1 + αpP1 p′ ) = 0. 1

(7.114)

1

So, the equation for the functional R[φ(⋅), ϕ(⋅)] reduces to the equation for the function P αkp of two discrete variables k, p, which depends on P as a parameter. We denote by P αP the operator with kernel αkp , denote by T P the operator of multiplication by

T P = Ak ωk + AP−k ωP−k , and denote by BP the operator with matrix element BPkp =

λ √Ak AP−k √Ap AP−p . 2Ld √2ωk ωP−k √2ωp ωP−p

Equation (7.114) assumes the form T P αP + αP T P + (1 + αP )BP (1 + αP ) = 0. The solution of the equation is written as αP = (1 + M P ) (1 − M P ), −1

where M P = (T P )

−1/2

2

1/2

1/2 1/2

[(T P ) + 2(T P ) BP (T P ) ] (T P )

−1/2

.

(7.115)

So, the solutions of equation (7.109) can be constructed if the operator involved in relation (7.115) is positive definite: 2

1/2

1/2 1/2

[(T P ) + 2(T P ) BP (T P ) ]

> 0.

(7.116)

7.5 Asymptotic spectrum of the Hamiltonian of a large number of fields | 435

This condition is a condition of stability of the solution of the Hartree-type equation. In this case, one can construct stationary asymptotic solutions of the Schrödinger equation for a system of N fields. Consider now other asymptotic eigenvalues of the Hamiltonian of a system of N fields front the series corresponding to the considered solution of the Hartree-type equation. This asymptotic spectrum can be expressed in terms of the eigenvalues of the system in variations 1 u(x) 2 1 φ (x)] − Ω)F[φ(⋅)] (∫ dx[ π 2 (x) + (∇φΛ )2 (x) + 2 2 2 Λ λ + ∫ dxφ2Λ (x) ∫ Dϕ ϕ2Λ (x)(F[ϕ(⋅)]Φ∗ [ϕ(⋅)] 2 + G[ϕ(⋅)]Φ[ϕ(⋅)])Φ[φ(⋅)] = −βF[φ(⋅)],

1 1 u(x) 2 (∫ dx[ π 2 (x) + (∇φΛ )2 (x) + φ (x)] − Ω)G[φ(⋅)] 2 2 2 Λ λ + ∫ dxφ2Λ (x) ∫ Dϕ ϕ2Λ (x)(F[ϕ(⋅)]Φ∗ [ϕ(⋅)] 2 + G[ϕ(⋅)]Φ[ϕ(⋅)])Φ∗ [φ(⋅)] = βG[φ(⋅)].

(7.117)

The functionals Fk1 ⋅⋅⋅kn = 0,

Gk1 ⋅⋅⋅kn = a+k1 ⋅ ⋅ ⋅ a+kn Φ∗ ,

β = Ak1 ωk1 + ⋅ ⋅ ⋅ + Akn ωkn

(7.118)

satisfy system (7.117) for n ≠ 0, 2. Indeed, the vector φ2Λ (x)Φ(0) contains only the vacuum and two-particle components. Hence, the matrix element (F + G, φ2Λ (x)Φ(0) ), which appears in formula (7.117), vanishes. The nontrivial solution of the system in variations has the form G = g0 Φ∗ + ∑ gk1 k2 Φ∗k1 k2 , k1 ,k2

F = f0 Φ + ∑ fk1 k2 Φk1 k2 , k1 ,k2

where Φk1 k2 is given by formula (7.110). From (7.117), we have the equations (Ak1 ωk1 + Ak2 ωk2 + β)fk1 k2 +

√Ak1 Ak2 √Ap1 Ap2 λ ∑ 2Ld p1 ,p2 √2ωk ωk √2ωp ωp 1 2 1 2

× (fp1 p2 δp1 +p2 ,k1 +k2 + gp,p2 δp1 +p2 +k1 +k2 ) = 0,

436 | 7 Asymptotic methods for many-field systems (Ak1 ωk1 + Ak2 ωk2 − β)gk1 k2 √Ak1 Ak2 √Ap1 Ap2 λ ∑ 2Ld p1 ,p2 √2ωk ωk2 √2ωp ωp 1 1 2

+

× (gp1 p2 δp1 +p2 ,k1 +k2 + fp1 p2 δp1 +p2 +k1 +k2 ) = 0. Consider the substitution fk1 k2 = f−k1 δk1 +k2 +P ,

gk1 k2 = gk1 δk1 +k2 −P .

From the resulting system, (Ak1 ωk1 + AP−k1 ωP−k1 + β)fk1 +

√Ak1 AP−k1 √Ap1 AP−p1 λ (fP1 + gp1 ) = 0, ∑ d 2L p1 ,p2 √2ωk ωP−k √2ωP ωP−p 1 1 1 1

(Ak1 ωk1 + AP−k1 ωP−k1 − β)gk1 +

√Ak1 AP−k1 √Ap1 AP−p1 λ (fp1 + gp1 ) = 0 ∑ 2Ld p1 ,p2 √2ωk ωP−k √2ωp ωP−p 1 1 1 1

one can express the functions fk and gk in terms of the eigenvalue β of the system in variations and the unknown constant aP , fk = − fk =

1 λ aP √Ak AP−k , d 2L √2ωk ωP−k Ak ωk + AP−k ωP−k + β

λ aP √Ak AP−k 1 ; 2Ld √2ωk ωP−k −Ak ωk − AP−k ωP−k + β

this constant is determined from the condition ap = ∑ p

fp + gp √2ωp ωP−p

√Ap AP−p .

The equation for β has the form Ap AP−p Ap ωp + AP−p ωP−p 1 1 = d∑ . 2 λ 2L p ωp ωP−p β − (Ap ωp + AP−p ωP−p )2

7.5 Asymptotic spectrum of the Hamiltonian of a large number of fields | 437

For d + 1 = 4, the left- and right-hand sides of this equation diverge as Λ → ∞. Using the definition of the renormalized coupling constant, we get the equation ωp + ωP−p 1 1 1 1 − ), = d (∑ 2 λR 2L p ωp ωP−p β − (ωp + ωP−p )2 2ω3p

(7.119)

which is regular as Λ → ∞. 7.5.4 Other series of asymptotics Consider the series of asymptotic eigenvalues of the N-field Hamiltonian that correspond to the above solution of the Hartree-type equation (7.100) for an arbitrary n. To this end, we need to study the spectrum of the system in variations (7.117). For (G∗ , φ2Λ (x)Φ(0) ) = 0, the system has a solution with F = 0. The nontrivial solutions are of the following form. We set X = G∗ . System (7.117) takes the form λ ∫ dx φ2Λ (x)((X, φ2Λ (x)Φ) 2 + (Φ, φ2Λ (x)F))Φ = −βF,

(HM − Ω)F +

λ ∫ dx φ2Λ (x)((Φ, φ2Λ (x)X) 2 + (F, φ2Λ (x)Φ))Φ = βX.

(HM − Ω)X +

(7.120)

Let K = k1 + ⋅ ⋅ ⋅ + kn . Consider the functional X with momentum P and the functional F ̂ = ∑ ka+ a− we have with momentum 2K − P; for the momentum of the operator P k k k ̂ = PX, PX

̂ = (2K − P)F. PF

Hence, (X, φ2Λ (x)Φ) = (X, e−iPx φ2Λ (0)eiPx Φ) = ei(K−P)x (X, φ2Λ (0)Φ), ̂

̂

and (Φ, φ2Λ (x)F) = ei(K−P)x (Φ, φ2Λ (0)F). Similarly, we can express the operator ∫ dx eikx φ2 (x) in terms of the projection operators ΠQ onto the subspace corresponding to the given momentum Q, ∫ dx ei(K−P)x φ2Λ (x)Φ = Ld Π2K−P φ2 (0)Φ.

438 | 7 Asymptotic methods for many-field systems Hence, the system in variations simplifies to read λLd a∗ Π2K−P φ2Λ (0)Φ = 0, 2 λLd a (HM − Ω − β)X + ΠP φ2Λ (0)Φ = 0, 2 a = (Φ, φ2Λ (0)X) + (F, φ2Λ (0)Φ). (HM − Ω + β)F +

As in the previous section, the nontrivial eigenvalue β can be expressed from the equation: 1 1 (Φ, φ2Λ (0)[(β − HM + Ω)−1 ΠP − (β + HM − Ω)−1 Π2K−P ]φ2Λ (0)Φ). = λ Ld

Concluding remarks The following results are obtained in the book: 1. We construct approximate solutions of equations of the type (1.1) describing the evolution of elements of various Hilbert spaces for various ε. 2. A physically interesting example of construction of such asymptotics is considered—namely, the construction of approximations to functions in which the number of arguments tends to infinity as the small parameter tends to zero and which satisfy equations of certain type. This problem is considered in Chapter 2. The asymptotic formulas approximate the exact solutions in the L2 -norm. Examples of such equations in physics include the many-particle Schrödinger, Liouville, and Wigner equations (see Chapter 3) as well as the equations of quantum field theory. However, the scope of the method developed in Chapter 2 is not exhausted by these applications: the method applies wherever there is a need to construct approximate solutions of equations for functions of a large number of arguments. 3. We show that the asymptotic solutions of the Schrödinger, Liouville, and Wigner equations as N → ∞ can be expressed not only in terms of the well-known selfconsistent field equations (the Hartree, Vlasov, and Hartree–Wigner equations), but also in terms of some new equations. 4. Asymptotic formulas for the many-particle equations with operator-valued symbol are constructed; for this case, new self-consistent field equations are obtained. 5. Approximations to the solution of the many-particle Liouville equation in the L1 -norm are constructed. To this end, the new concept of the half-density function (square root of the N-particle distribution) is introduced. It is shown that the construction of approximations for the half-density in the L2 -norm is equivalent to the construction of approximations for the probability distribution function in the L1 -norm. The asymptotic formula for the half-density can be constructed by our method. 6. We study the question as to whether approximate solutions of N-particle equations can be used instead of the exact ones in the computation of the limits as N → ∞ of the mean values of general observables uniformly bounded with respect to N. This problem is examined in the most general case of an arbitrary abstract Hamiltonian of the algebra of observables. The concept of abstract half-density is introduced; its special cases include the half-density function in the classical case and the square root of the density matrix in the quantum case. It is shown that if the difference between the exact and approximate half-densities tends to zero in the L2 -norm, then the limits as N → ∞ of the mean values of observables uniformly bounded with respect to N can be computed via the approximate half-density instead of the exact one. 7. The chaos preservation problem in the classical and quantum statistical mechanics is considered. It is shown that if the initial N-particle density function (wave function) splits into a product of one-particle density functions (wave functions), the N-particle function at time t cannot in general be approximated by a product of oneparticle functions as N → ∞. https://doi.org/10.1515/9783110762709-008

440 | Concluding remarks 8. It is well known that, for finite-order correlation functions, chaos is preserved in a system of N particles in an external field of the order of unity with pairwise interaction of intensity O(1/N). We show that, in general, chaos is not preserved even for correlation functions in the operator-valued case. 9. An approximate spectrum of the many-particle Hamiltonian operator in quantum mechanics is constructed. In a number of cases, this result has physically interesting corollaries. For example, Bogoliubov’s superfluidity theory [8] follows from the results in Chapter 2 as a special case. 10. Stationary asymptotic solutions of the Liouville equation are constructed. We show that they approximate the Gibbs canonical distribution from statistical physics. 11. The structure of the set of asymptotic solutions of equation (1.1) is studied. We find that, under some reasonable assumptions about the canonical operator (the axioms in Section 1.2), from which these asymptotic formulas are derived, to each equation of the type (1.1), one can assign an analog of the classical mechanics phase space for the Schrödinger equation and also introduce geometric structures on this phase space (the action 1-form and the operator-valued 1-form Ω; see Section 1.2). In particular, such geometric structures are introduced for the case of the many-particle Schrödinger, Liouville, and Wigner equations, for which the unit sphere in the complex space L2 is an analog of the classical phase space. The knowledge of geometric structures permits one to introduce the concept of complex germ (Section 1.3) and construct formal asymptotic solutions of the equations of motion without substituting these solutions into equation (1.1). 12. A general concept of complex germ is stated; it can be applied to finitedimensional quantum mechanics as well as second-quantized equations in the Fock space (in the infinite-dimensional case). Thus, the complex germ theory is extended to the infinite-dimensional case. 13. A new insight into the complex germ theory on a finite-dimensional isotropic manifold is given. (The theory of full-dimensional Lagrangian manifolds [24] is a special case of this.) We construct functions corresponding to a Lagrangian manifold with complex germ as an infinite superposition of wave functions corresponding to the germ at a point. The uniqueness of such an integral representation is studied. Unlike the traditional complex germ theory on finite-dimensional isotropic manifolds, our integral formulas can be directly extended to the case of the Fock space and even to the case of an arbitrary abstract canonical operator for the germ at a point (see Sections 1.2 and 1.3), whose special cases embrace all known examples of canonical operators for a germ at a point. 14. The problem of selection of quantization conditions is studied. The following approaches to this problem were considered earlier. (a) One considers a fixed isotropic manifold; the parameter ε → 0 ranges over a discrete set such that a quantization condition of the type (4.187) is satisfied. If the Betti number of the manifold is greater than 1, then this approach encounters the fol-

Concluding remarks |

441

lowing difficulty: one has to require the integrals ∮γ P dQ, j = 1, . . . , b, where {γj }bj=1 is j

a basis of independent cycles on the manifold, to be commensurable. This condition, of course, is not always satisfied in specific examples. (b) If one considers the canonical operator on a family of isotropic manifolds with complex germ, then for each ε, from this family one chooses an isotropic manifold satisfying the quantization conditions. This approach is not very convenient for applications; to construct the asymptotics, one should know the evolution of the entire family of isotropic manifolds with complex germ. In the book, we show that: (a) The numbers β in the quantization condition (4.192) can be chosen arbitrarily. (b) One can redefine the canonical operator so that the quantization conditions will not be needed at all. 15. An approach to constructing new axioms of relativistic quantum field theory is proposed. This problem is discussed in Section 1.1; in Section 6.2, we consider specific Lagrangian manifolds with a complex germ in various models of quantum field theory. In Section 6.3, some problems with an operator-valued symbol in quantum field theory are examined. 16. Although the entire book deals with additive asymptotics, we consider heuristic considerations concerning multiplicative asymptotics for second-quantized equations in Appendix 4.A. The analysis of these asymptotics leads to instanton effects. 17. The problem of divergences in the Hamiltonian formulation of quantum field theory is examined. It is well known that the Stückelberg divergence appears in this formalism in addition to the usual “loop” divergences of perturbation theory. To study the divergences, one should consider a theory involving ultraviolet and infrared regularization instead of the local field theory. The properties of relativistic invariance and locality should be restored as the regularization parameters tend to infinity. In the S-matrix theory, one imposes certain conditions on the dependence of the Hamiltonian on the regularization parameters. In the Hamiltonian theory, one should also impose conditions on the state vector. The specific form of the condition in perturbation theory can be found by using both Faddeev-type transformations, based on an iterative procedure, and the Bogoliubov S-matrix, which corresponds to a smooth onset of the interaction. It is shown that Stückelberg divergences are removed after these transformations. Some examples are given. 18. Conditions that should be imposed on the initial data in semiclassical field theory are considered. Since the theory of Lagrangian manifolds with complex germ is reduced to the complex germ theory at a point, one can be concerned only with the latter case. The Bogoliubov S-matrix is considered in the germ approximation. Conditions on the complex germ are obtained. 19. We prove that the condition on the complex germ is indeed invariant under evolution.

442 | Concluding remarks 20. A new approach to the theory of a large number of fields is developed. This approach is based on the representation of the corresponding many-field Hamiltonian in the third-quantized form in terms of the creation and annihilation operators of fields and the application of semiclassical methods. This method is capable of constructing a broader class of approximate solutions of many-field equations in comparison with the previously available methods. 21. The classical equations of the third-quantized formalization of the theory of N fields are renormalized. 22. The asymptotic spectrum of the theory of a large number of fields is constructed. O. Yu. Shvedov developed the ideas and methods of the monograph in his last papers [114–121].

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Index Abstract half-density 181 Action – 1-form 5 – induced by the canonical operator 13 – on the classical trajectory 2 Asymptotic solution of the Cauchy problem 32 – with accuracy O(εk ) 33 Asymptotics of the solution of the Cauchy problem 31 – with accuracy O(εk ) 33 BBGKY hierarchy 64 Bogoliubov S-matrix 367 Bosons 61 Canonical operator 2, 5 – abstract 11 – action 1-form 13 – canonical transformation 18 – complex germ 22 – operator-valued 1-form 16 – corresponding to an isotropic manifold 229, 274 – corresponding to germ 28, 42 – in Fock space 199 – in quantum mechanics 35, 42 – multiparticle 74 Canonical transformation – in multiparticle problems 83 – in quantum mechanics 41 – of complex germ 26 – of the abstract canonical operator 18 – of the phase space 21, 36 – proper 21, 37 Chaos property 4, 65, 148 Classical mechanics 52 Complex germ – abstract 22 – canonical transformation of 26 – complete 24 – creation and annihilation operators 28 – canonical transformation of 94 – in multiparticle problems 90 – in quantum mechanics 35 – on an isotropic manifold 252 Creation–annihilation operators 121 – germ 28 https://doi.org/10.1515/9783110762709-010

Curve on manifold 56 – equivalence 56 – homotopy 58 Differential 1-form 57 – exterior product 57 – operator-valued 6, 57 – induced by the canonical operator 16 Differential 2-form 57 Evolution equation 1 Evolution operator 53 Faddeev transformation 358 Fock space 121 – canonical transformation 127 – coherent state 123 – generating functional 124 Formal asymptotic solution 22 – in multiparticle problems 97 – invariant 100 Gaussian – vector 89 – complex germ of 90 – wave packet 37 – complex germ 38 Gaussian function 244 Haag’s theorem 331, 349 Half-density – function 64, 146 – matrix 150 Hamiltonian – function 52 – operator 53 Hamiltonian algebra 178 – uniformization 184 Hamiltonian system 2, 52 Hartree equation 141 Isotropic manifold 224, 261, 268 Lagrangian manifold with complex germ 252 Liouville equation 141 Observable 53 – mean value 53

448 | Index

Phase space 4 Quantization in a neighborhood of solitons 335 Quantum mechanics 52 – complex germ method in 1, 35 Regularization 352, 421 Renormalization 357, 424, 426 Scalar field 332, 337 Scalar quantum electrodynamics 335 Schrödinger equation 139 Second quantization method 120 Smooth manifold 56 – atlas 56 – Betti number 59 – cohomology space 59 – coordinate chart 55 – curve on 56 – de Rham theorem 59 – fundamental group 58 – homology group 59 – universal covering 58

State space 52 – density matrix 53 – mixed state 53 – pure state 53 Stückelberg divergence 351, 358 Superfluidity 168 Symplectic structure 6 Tangent space 56 – induced mappings 56 Tangent vector 56 Third quantization 412 Ultraviolet divergence 350 Universal covering 58 Vlasov equation 142 Volume divergence 350 Wigner equation 140 Yang–Mills field 342

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