Quantum Mathematics II 9789819958832, 9789819958849

Citation preview

Springer INdAM Series  58

Michele Correggi Marco Falconi   Editors

Quantum Mathematics II

Springer INdAM Series Volume 58

Editor-in-Chief Giorgio Patrizio, Università di Firenze, Florence, Italy Series Editors Giovanni Alberti, Università di Pisa, Pisa, Italy Filippo Bracci, Università di Roma Tor Vergata, Rome, Italy Claudio Canuto, Politecnico di Torino, Turin, Italy Vincenzo Ferone, Università di Napoli Federico II, Naples, Italy Claudio Fontanari, Università di Trento, Trento, Italy Gioconda Moscariello, Università di Napoli Federico II, Naples, Italy Angela Pistoia, Sapienza Università di Roma, Rome, Italy Marco Sammartino, Università di Palermo, Palermo, Italy

This series will publish textbooks, multi-authors books, thesis and monographs in English language resulting from workshops, conferences, courses, schools, seminars, doctoral thesis, and research activities carried out at INDAM - Istituto Nazionale di Alta Matematica, http://www.altamatematica.it/en. The books in the series will discuss recent results and analyze new trends in mathematics and its applications. THE SERIES IS INDEXED IN SCOPUS

Michele Correggi . Marco Falconi Editors

Quantum Mathematics II

Editors Michele Correggi Dipartimento di Matematica Politecnico di Milano Milano, Italy

Marco Falconi Dipartimento di Matematica Politecnico di Milano Milano, Italy

ISSN 2281-518X ISSN 2281-5198 (electronic) Springer INdAM Series ISBN 978-981-99-5883-2 ISBN 978-981-99-5884-9 (eBook) https://doi.org/10.1007/978-981-99-5884-9 This work was supported by Istituto Nazionale di Alta Matematica “Francesco Severi”. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Preface

The two volumes Quantum Mathematics I and Quantum Mathematics II originate from the INdAM Intensive Period “INdAM Quantum Meetings (IQM22)”, which was held in Spring 2022 at the Department of Mathematics of Politecnico di Milano. The trimester was the perfect opportunity to restart the social aspects of research after the Covid restrictions, in the mathematical physics community working on the mathematical features of quantum mechanics. After almost two entire years of break due to the Covid-19 pandemic, the project was very successful since the first steps of the organization: almost all the invited scientists gladly accepted to participate, also showing the enthusiasm of meeting in person to form new collaborations as well as to renew existing ones. The main activities during IQM22 were: . A kick-off workshop at the beginning of March 2022 focusing on the topics of many-body quantum mechanics, quantum statistical mechanics and open quantum systems . A series of short courses given throughout the period March–May 2022 by Z. Ammari (Université de Rennes 1), C. Brennecke (Bonn Universität), J. Derezi´nski (University of Warsaw), M. Fraas (University of California Davis), M. Merkli (Memorial University of Newfoundland), F. Nier (Université Pais 13), N. Rougerie (ENS Lyon) . Thematic lectures and seminars; . A concluding workshop at the end of May 2022 focusing on the topics of semiclassical analysis, quantum field theory, nonlinear PDEs of quantum mechanics and their derivation More than 40 invited guests contributed to the activities of the period and gave rise to many fruitful collaborations among themselves and with the local members of the mathematical physics group. The participation of young researchers, postdocs and PhD students was very significant with more than 20 young people (some of which financially supported) showing interest in the scientific programme of the trimester and participating to the activities. Most of the contributions (talks, lectures, courses, etc.) were recorded and made available online. v

vi

Preface

All the contributions collected in these volumes are linked to IQM22, either as proceedings of its activities, or as brand new works originating and benefiting from the interactions occurred at IQM22. The main theme is the mathematics of quantum mechanics in a broad sense, but the present volume is focused on the following more specific topics: . Quantum Field Theory. The quantum theory of fields is one of the most challenging topics of quantum physics from the mathematical perspective. The focus of the contributions in this volume is on the rigorous understanding of key physical features concerning the evolution of fields, their ultraviolet behaviour and the vacuum polarization. . Open Quantum Systems. The effective description of small quantum systems interacting with an environment plays a crucial role in several areas of physics and in reconnecting theory and experiments. Thus, there is a strong motivation to mathematically understand the features of such systems and their modelling through evolutive maps generated by dissipative operators and superoperators. . Many-body quantum mechanics. In recent years, a lot of effort has been put into the investigation of the mathematics of many-body quantum systems, both bosonic and fermionic. We present here some reviews of the most recent advances on the bosonic side, but also some novel results for fermionic systems. To conclude, we express our gratitude to INdAM and its scientific board for the support to IQM22, which made possible the organization of a very fruitful intensive period and the involvement of a large number of young participants. Milano, Italy January 2023

Michele Correggi Marco Falconi

Contents

Part I Quantum Field Theory Dynamical Systems Involving Pseudo-Fermionic Operators and Generalized Quaternion Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yanga Bavuma and Francesco G. Russo An Evolution Equation Approach to Linear Quantum Field Theory . . . . . . Jan Derezi´nski and Daniel Siemssen

3 17

Renormalization of Spin–Boson Interactions Mediated by Singular Form Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Davide Lonigro The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom Plus Wall Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Marco Olivieri Part II Open Quantum Systems Asymptotic Dynamics of Open Quantum Systems and Modular Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Daniele Amato, Paolo Facchi, and Arturo Konderak Boson Quadratic GKLS Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Franco Fagnola Part III Many-Body Quantum Mechanics Energy Expansions for Dilute Bose gases from Local Condensation Results: A Review of Known Results . . . . . . . . . . . . . . . . . . . . . . . . . 199 Giulia Basti, Cristina Caraci, and Serena Cenatiempo Bogoliubov Theory for Ultra Dilute Bose Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Christian Brennecke

vii

viii

Contents

Thermodynamic Game and the Kac Limit in Quantum Lattices. . . . . . . . . . . 247 Jean-Bernard Bru, Walter de Siqueira Pedra, and Kauê Rodrigues Alves Uniform in Time Convergence to Bose–Einstein Condensation for a Weakly Interacting Bose Gas with an External Potential . . . . . . . . . . . . . 267 Charlotte Dietze and Jinyeop Lee Bogoliubov Theory for the Dilute Fermi Gas in Three Dimensions . . . . . . . . 313 Emanuela L. Giacomelli Bogoliubov Transformations Beyond Shale–Stinespring: Generic v ∗ v for Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Sascha Lill

.

Trial States for Bose Gases: Singular Scalings and Non-integrable Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Alessandro Olgiati

Part I

Quantum Field Theory

Dynamical Systems Involving Pseudo-Fermionic Operators and Generalized Quaternion Groups Yanga Bavuma and Francesco G. Russo

MSC (2020): Primary 81R05, 22E10; Secondary 22E70, 81R30, 81Q12

1 Introduction and Statement of the Main Result Bosons and fermions form two significant classes of fundamental particles which are involved in crucial topics of the physics of the matter. Roughly speaking a fermion is a particle that is associated with the structure of matter and a pseudo-fermion generalizes the notion of a fermion. The formalization involves some classical functional analysis and is modelled on a complex Hilbert space .H (endowed with the usual scalar product) via lowering and raising operators, which indeed lower or raise the eigenvalues of corresponding eigenstates. Pseudofermionic operators are then formalized as a pair of operators, say .a and .b, where the lowering operator is .a and the raising operator is .b, satisfying the canonical anticommutation relations (CAR) {a, b} = ab + ba = I,

.

and

{a, a} = {b, b} = 0.

(1)

Here .I denotes the identity operator and .b /= a† a priori, that is, .b is not necessarily the adjoint operator .a† of .a. In particular, if this happens, that is, if .b = a† and (1) are satisfied, .a is a fermionic operator (of adjoint .a† ). The terminology is quite standard among the theory of ladder operators, and more details can be found in [4, 6], but also in [10, 11, 22] from a more theoretical perspective. Note

Y. Bavuma Department of Mathematical Sciences, University of South Africa, Pretoria, South Africa F. G. Russo (O) Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, Cape Town, South Africa Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville, South Africa © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Correggi, M. Falconi (eds.), Quantum Mathematics II, Springer INdAM Series 58, https://doi.org/10.1007/978-981-99-5884-9_1

3

4

Y. Bavuma and F. G. Russo

also that {a, a} = 0 ⇐⇒ a2 = 0

and

.

{b, b} = 0 ⇐⇒ b2 = 0.

(2)

Now we are going to sketch a construction with ladder operators which can be found in [5, 7]. First of all, we formulate the existence of a nonzero vector .ϕ0 ∈ H such that .aϕ0 = 0, as well as a nonzero vector .W0 ∈ H such that .b† W0 = 0. We then introduce ϕ1 := bϕ0 ,

and

.

W1 := a† W0 ,

(3)

as well as the following non-self-adjoint operators N := ba,

and

.

N† = a† b† .

(4)

Then we introduce the following self-adjoint operators .Sϕ and .SW where .f ∈ H: Sϕ f = ϕ0 + ϕ1 , SW f = W0 + W1 .

.

(5)

We find the following equations which are satisfied: aϕ1 = ϕ0 ,

.

Nϕn = nϕn ,

.

b† W1 = W0

N† Wn = nWn , for n = 0, 1

(6) (7)

and one can argue that the normalizations of .ϕ0 and .W0 can be chosen such that = 1. Then we use the Kronecker delta .δk,n and find also that

.

= δk,n for k, n = 0, 1.

.

(8)

Looking at [4, 6, 7], or simply checking the conditions above, one can see that .Sϕ and .SW are bounded operators, but also strictly positive, self-adjoint, and invertible. They satisfy ||Sϕ || ≤ ||ϕ0 ||2 + ||ϕ1 ||2 ,

.

Sϕ Wn = ϕn ,

.

||SW || ≤ ||W0 ||2 + ||W1 ||2 , SW ϕn = Wn ,

(9) (10)

for .n = 0, 1, as well as .Sϕ = S−1 W and the following intertwining relations SW N = N† SW ,

.

Sϕ N† = NSϕ .

(11)

Dynamical Systems Involving Pseudo-Fermionic. . .

5

With this construction in mind, we have just shown that it is possible to introduce two fermionic operators .N and .N† , having eigenvalues 0 and 1, and eigenvectors respectively .Fϕ = {ϕ0 , ϕ1 } and .FW = {W0 , W1 }. Moreover .a and .b† are lowering operators for .Fϕ and .FW respectively; then .b and .a† are raising operators for .Fϕ and .FW respectively. Moreover the operators .Sϕ and .SW map .Fϕ in .FW and vice versa, and intertwine as per (11). In fact .Fϕ and .FW are biorthogonal and linearly independent in .H, providing examples of Riesz bases in .H; this is a typical notion of functional analysis which can be found in [4, 6, 10, 11, 22]. Looking at [7–9, 21], the well known Pauli matrices are ( X=

.

) 01 , 10

( Y =

0 −i i 0

)

( and

Z=

) 1 0 , 0 −1

(12)

and are involutions with respect to the product of matrices, that is, X2 = Y 2 = Z 2 = I.

.

(13)

Moreover (12) satisfy the following rules (XY )4 = (iZ)4 = I, (Y Z)4 = (iX)4 = I, (ZX)4 = (iY )4 = I,

.

(14)

and form the finite group of order 16 {I, −I, X, Y, Z, −X, −Y, −Z, iX, iY, iZ, −iX, −iY, −iZ, iI, −iI}.

.

(15)

As indicated in [7, Lemma 3.6], we may introduce the symbols u = XY, x = Y and y = XY Z

.

(16)

and rewrite (15) via the group presentation given by P = .

.

(17) The advantage of (17) is due to a topological decomposition of P via fundamental groups of space of orbits of spheres, found recently in [7]. This cannot be detected if we look only at (15) along with (12), (13) and (14). Note also that the generalized quaternion group is the finite 2-group (of order .2n ) n

Q2n = ,

(18)

6

Y. Bavuma and F. G. Russo

where the usual quaternion group .Q8 is obtained for .n = 3. Again it is not immediate to see that in (15) there is a copy of .Q8 ; in fact the set {I, −I, iY, −iY, iZ, −iZ, iI, −iI}

.

(19)

satisfies the rules in (18) when .v = iZ and .w = iY (and .n = 3). The relevance of the above notions is due to the following recent result: Theorem 1.1 (See [7], Theorem 1.2) There are two dynamical systems .S and .T involving pseudo-fermionic operators with groups of symmetries respectively P and .Q8 but with the same Hamiltonian .HS = HT . In particular, there exist dynamical systems admitting larger groups of symmetries, whose size does not affect the dynamical aspects of the system. Nöether’s Theorem illustrates the relevance of symmetries in dynamical systems (see [1]) motivating interactions between topological and algebraic methods in mathematical physics since a long time ago. Among other things Theorem 1.1 gives a realization of the group .Q8 in terms of pseudo-fermionic operators, satisfying a precise mathematical model, which is close to the model of the quantum harmonic oscillator. This was proposed originally in [14] and works well for the two level atom interacting with an electromagnetic field. The model presents an effective non-self-adjoint Hamiltonian, whose dynamics are analyzed in connection with pseudo-hermitian systems. Then [3] proved that the main information from [14] can be formalized in terms of pseudo-fermionic operators, adding a new element to the literature [2, 13, 15]. Theorem 1.1 actually shows that it is possible to rewrite the same effective Hamiltonian of [14] in terms of Pauli matrices, or, from another point of view, that there exists a realization of the Pauli group in pseudofermionic operators which are associated to a precise dynamical system in quantum field theory. We will see that the same dynamical system turns out to represent all generalized quaternion groups; this our main result. Theorem 1.2 There exists a dynamical system .S possessing a non self-adjoint Hamiltonian .Heff describing the two level atom interacting with an electromagnetic field, and .Heff can be decomposed in pseudo-fermionic operators .a and .b which realize .Q2n for all .n ≥ 3. We shall mention that the main ideas for the proof of Theorem 1.1 involve the presence of a technical decomposition in topology and algebra, called central product of groups, for the group P . This turns out to influence the simpler analytic expression of .HT with respect to .HS , so we will mention relations between the notion of central product of groups and pseudo-fermionic operators in the present work. The organization of the material is as follows: Sect. 2 recalls some crucial facts on the structure of central product of groups in connection with the dynamical systems involved in Theorem 1.2. Section 3 recalls a 2-dimensional complex representation of (18), which is known in finite group theory, but was never connected to the

Dynamical Systems Involving Pseudo-Fermionic. . .

7

theory of the pseudo-fermionic operators before. Then Sect. 3 end with the proof of Theorem 1.2, where the complex representation of the generalized quaternion groups play a fundamental role. Conclusions are enclosed in the final part, Sect. 4, illustrating a reformulation of a conjecture which appears in [7–9].

2 Some Recent Results on Pseudo-Fermionic Operators First of all we should be careful to denote commuting relations at the level of groups, or at the level of ladder operators. The following example can be found in [6, Chapter 3.5] and deals with two complex matrices ) ( ) ( 01 β −β 2 .a = , b= 1 −β 00

(20)

with .β non-zero complex scalar. These two matrices can be visualized as pseudofermionic operators. In particular one can compute the Lie commutator ab − ba = I

.

(21)

but this is different from the group commutator a−1 b−1 ab

(22)

b /= a† = a−1 .

(23)

.

since a priori .

In the first case we consider the structure of a Lie algebra on .C4 , using a commutator with respect to the operation of the Lie bracket in .C4 , while in the second case we are looking at the multiplicative structure in .C4 thinking of its elements as matrices and using the usual row by columns matrix product. A well known connection between pseudo-fermions and fermions is below: Proposition 2.1 (See [6], Theorem 3.5.1) Let .c and .T = T† be two operators on † .H such that c and .c satisfy CAR and in addition .T be positive. Then the operators −1 .a = TcT and .b = Tc† T−1 are pseudo-fermionic. Viceversa given two operators .a and .b on .H satisfying CAR, it is possible to construct two operators .c and .T with the above properties.

8

Y. Bavuma and F. G. Russo

According to [16], a finite group G is an (internal) central product of its subgroups H and K, if G = H K = {hk | h ∈ H, k ∈ K}

(24)

.

and the commutator subgroup [H, K] =

(25)

.

is trivial. In this situation G is briefly denoted by G=H ◦K

(26)

.

and turns out that both H and K are normal subgroups in G; moreover .H ∩ K < Z(H ) ∩ Z(K), where Z(G) = {g1 ∈ G | g1 g2 = g2 g1 ∀g2 ∈ G}

(27)

.

denotes the center of G. Compare (26) with [16, Theorem 5.3] and [18, Satz 9.10, Kapitel I, §9] and [20, pp. 140–141] for more information on the notion of a central product of two finite groups. We will also refer to the notion of connected sum .# between two manifolds (i.e.: two compact connected locally Euclidean topological spaces); this is another well known notion in algebraic topology (see [17, p. 257], or [19, p. 79]). The fundamental group of a (path connected) space X is denoted by .π(X) and its computation is made in [17, 19] for large classes of topological spaces such as polyhedra, surfaces, spheres and CW-complexes. The last notion that we need to mention is that of action of a group on a topological space X, and again this is widely used in the literature, see [19, Chapter 5]. Essentially this is a map .(g, x) ∈ G × X → g · x ∈ X such that .1 · x = x and .(g1 g2 ) · x = g1 · (g2 · x) for all .x ∈ X and .g1 , g2 ∈ G, but often one requires that this map is properly discontinuous, or that X is a G-space (see always [19, Chapter 5]) because one wants that the action of G on X preserves topological properties such as Hausdorff, compactness and connectivity. The space of orbits of X under the action of G is denoted by the set .X/G of all equivalence classes .G · x of elements of X. It turns out that .X/G can be topologized with the quotient topology induced by the natural projection .x ∈ X |→ G · x ∈ X/G. Moreover most of the properties of X pass to .X/G, provided the action is “good enough”. These are classical topics of algebraic topology in [17, 19]. For instance, the usual sphere { S = (x1 , x2 , . . . , xd .xd+1 ) ∈ R

.

d

d+1

|

d+1 E i=1

} xi2

=1

(28)

Dynamical Systems Involving Pseudo-Fermionic. . .

9

is a Riemannian manifold of dimension d where d is a non-negative integer (see [12, Chapter 5]) and not all finite groups act on .Sd in a properly discontinuous way, or, without non-trivially fixed points. This the case of the cyclic group of order .pm , denoted by m

Z(pm ) =

.

(29)

multiplicatively, and of .Q8 as well. In fact it turns out that the Pauli group can be decomposed in central product P = Q8 ◦ Z(4),

.

(30)

identifying on the common cyclic group of order two and to this decomposition we may associate a topological decomposition in terms of spaces of orbits of spheres of dimension three. The following result helps to understand the peculiarities of P . Theorem 2.2 (See [7], Theorem 1.1) There exist two compact path connected spaces of orbits .U = S3 /Q8 and .V = S3 /Z(4) such that the following conditions hold: (i). .U ∪ V is a compact path connected space with .U ∩ V /= ∅, .π(U ∩ V ) cyclic of order 2 and .P ∼ = π(U ∪ V )/N for some normal subgroup N of .π(U ∪ V ); (ii). .U #V is a Riemannian manifold of .dim(U #V ) = 3 and .P ∼ = π(U #V )/L for some normal subgroup L of .π(U #V ). Both in case (i) and (ii), P is central product of .π(U ) and .π(V ). Theorem 1.1 motivated the investigation for new families of groups, opening the following conjecture. Conjecture 2.3 (See [7], Conjecture 6.1) Groups of the form .A = Q8 ◦ B, where B is an abelian group containing at most one element of order 2, may have a construction of the Hamiltonian (in terms of pseudo-fermionic operators) as in Theorem 1.1. Looking at the proof of Theorem 1.1, Conjecture 2.3 is true with .B = Z(4) and A = P . Then central products motivated additional investigation at a topological level, finding results of the following type:

.

Theorem 2.4 (See [9], Theorem 1.3) Assume that .G = H ◦ K is a finite group which may be written as central product of its subgroups H and K. If H and K satisfy the following conditions : (i). contain no subgroups isomorphic to .Z(p) × Z(p) with p prime, (ii). contain at most one element of order 2, then there exists a Riemannian manifold X and a surjective homomorphism of groups from the fundamental group .π(X) of X to G.

10

Y. Bavuma and F. G. Russo

The proof of Theorem 2.4 uses essentially the sames ideas of [7, Proof of Theorem 1.1], generalizing to spheres of dimension .d ≥ 3. In fact the thesis of Theorem 2.4 may be also rephrased by saying that G is isomorphic to an appropriate quotient of .π(X), that is, there are two compact path connected spaces of orbits, U and V (again realized by actions of finite groups on spheres) such that .π(X)/L = π(U #V )/L for some normal subgroup L of .π(X). The surjective homomorphism of groups of Theorem 2.4 deals precisely with the construction of the quotient group .π(X)/L which turns out to be isomorphic to G.

3 Representations of Generalized Quaternion Groups Now we report a result of technical nature, concerning generalized quaternion groups. Lemma 3.1 The group .Q2n contains no subgroups isomorphic to .Z(2) × Z(2), has at most one element of order 2 and can be written as the product of two normal subgroups .Q2n = H K with .H ∩ K /= 1, but the condition .[H, K] = 1 is not always satisfied. In particular, .Q16 cannot be decomposed in central product of two proper normal subgroups. Proof First of all [20, 5.3.6] shows that .Q2n has exactly one subgroup of order 2. This means that it has at most one element of order 2. Again [20, 5.3.6] shows that .Q2n cannot have subgroups isomorphic to .Z(2) × Z(2), because if so, then it would have more than one element of order two, which is a contradiction. It remains to show that there are normal subgroups H and K such that .Q2n = H K and .H ∩ K /= 1. Looking at (18) and at its normal subgroups, we introduce H = and K = .

.

(31)

Of course, H ∩ K ⊇ Z(Q2n ) = /= 1,

(32)

moreover H and K are normal subgroups such that .H = Q2n−1 and .K = Z(2n−1 ). On the other hand, .Q2n = = so every element of .Q2n can be written as product of an element of H by another of K. Note that w −1 v −1 wv ∈ [H, K] ⊆ H ∩ K

(33)

w −1 (w −1 vw)wv = w −2 vw 2 v = w −2 w 2 vv = v 2 /= 1.

(34)

.

and (18) show .

Dynamical Systems Involving Pseudo-Fermionic. . .

11

Finally we inspect for .n = 4 the lattice of the normal subgroups of .Q16 and find that the proper normal subgroups of .Q16 are either the subgroups of K, or H , or 2 .L = . Incidentally note that .H = L = Q8 . Assume that there are normal subgroups A and B such that .A ◦ B = Q16 . These should be of the forms which we have just mentioned. Let’s discuss the possible options: if .A = B = H , then .A ◦ B = Q8 ◦ Q8 hasn’t order 16, so we exclude it. If .A = H and B is a cyclic subgroup of K, we could identify only on a common cyclic subgroup of order two. If .B = K has order 8, then .A ◦ B hasn’t again order 16, so we exclude it, but if .B ⊆ K has order 4, then we get .A ◦ B = P /= Q16 , to be excluded again. The case of B cyclic of order two brings to .Q8 ◦ B = Q8 , which should be excluded by reasons of order again. Up to the isomorphism .H = L, there are no more cases. We conclude that .Q16 cannot be decomposed in central product of two proper normal subgroups. u n Corollary 3.2 Let G be a finite p-group. Then the conditions (i) and (ii) of Theorem 2.4 are true if and only if G is cyclic, or generalized quaternion. u n

Proof This follows from Lemma 3.1.

The following result helps us to visualize generalized quaternion groups via the group .GL2 (C) of non-singular .(2 × 2) complex matrices. 2π i

Lemma 3.3 Considering the .2n−1 th root of the unity .e 2n−1 , the generalized quaternion group .Q2n possesses the following faithful complex representation .ϕ, defined on the generators of (18) by v ∈ Q2n |−→ ϕ(v) =

.

( 2π i e 2n−1 0

w ∈ Q2n |−→ ϕ(w) =

.

) 0

−2π i

e 2n−1

∈ GL2 (C)

( ) 0 −1 ∈ GL2 (C) 1 0

(35)

(36)

and extended uniquely to a general element of .Q2n . In particular, Q2n =

.

/( 2π i e 2n−1 0

) 0

−2π i

e 2n−1

( )\ 0 −1 , ⊆ GL2 (C). 1 0

Proof This can be found in [16, 18]. We can prove our main result, that is, Theorem 1.2.

(37) u n

12

Y. Bavuma and F. G. Russo

Proof Considering the mathematical model proposed in [3, 14], one can start from the Schrödinger’s Equation noting that ˙ .i o(t) = Heff o(t),

Heff

1 = 2

(

−iδ ω ω iδ

) .

(38)

Here .δ is a real quantity, related to the decay rates for the two levels, while the complex parameter .ω characterizes the radiation-atom interaction. We look for a decomposition of the Hamiltonian in terms of ladder operators which are † appropriate to our context of study. Note that .Heff /= Heff and write .ω = |ω|eiθ . Then introduce the operators (

) −|ω| −e−iθ (O + iδ) , |ω| eiθ (O − iδ) ( ) 1 −|ω| e−iθ (O − iδ) , b= |ω| 2O −eiθ (O + iδ) a=

.

1 2O

(39)

where O=

/

.

|ω|2 − δ 2 ,

(40)

which we will assume here to be real and strictly positive. We have that ab + ba = I,

.

and

a2 = b2 = 0,

(41)

so we are sure that .a and .b are pseudo-fermionic operators. Moreover, .Heff can be written in terms of these two operators as ) 1 = O ba − I . 2 (

Heff

.

(42)

Note that the above computations are known from [3, 7, 14], so there is nothing of new until this point. Now consider Lemma 3.3 and ϕ(v) =

.

( 2π i e 2n−1 0

) 0

−2π i

e 2n−1

( ) 0 −1 and ϕ(w) = 1 0

(43)

Dynamical Systems Involving Pseudo-Fermionic. . .

13

and the matrices μ1 =

.

( ) 1 −|ω| −iδe−iθ =b+a |ω| O −iδeiθ

( μ2 = i

0 e−iθ eiθ 0

) = i(b − a)

(44)

( ) 1 iδ −|ω|e−iθ = ab − ba. .μ3 = −iδ O −|ω|eiθ One can easily check (see [7, Eq. 5.2 and 5.3]) that (44) satisfy (12), (13) and (14), so they are Pauli operators, decomposed ad hoc by pseudo-fermionic operators. On the other hand, we may use Euler Formulas for the complex numbers and write (44) in terms of a linear combination of (12), getting ⎧ ) ( ) ) ( ( ⎪ |ω| iδ sin θ iδ cos θ ⎪ ⎪ Y − Z, X − . μ = 1 ⎪ ⎪ O O O ⎪ ⎪ ⎨ μ2 = (i cos θ ) X + (i sin θ ) Y, ⎪ ⎪ ) ( ) ) ( ( ⎪ ⎪ iδ |ω| sin θ |ω| cos θ ⎪ ⎪ ⎪ Y+ Z. X− ⎩μ3 = − O O O

(45) (46) (47)

Again in terms of (12) we may rewrite (35) and (36), getting (

2π 2n−1

ϕ(v) = cos

.

)

( I + i sin

2π 2n−1

) Z

(48)

and ϕ(w) = −iY.

.

(49)

Looking at (47), we multiply both sides by i, getting ( iμ3 = −

.

) ( ) δ |ω| Z, ((i cos θ )X + (i sin θ )Y ) − O O

(50)

but now we recognize .μ2 in the right side of the equation above, hence ( iμ3 = −

.

) ( ) |ω| δ μ2 − Z O O

(51)

and isolating Z we get ( Z=−

.

) (ω) iO μ2 μ3 − δ δ

(52)

14

Y. Bavuma and F. G. Russo

hence we replace (52) in (48), finding ( ϕ(v) = cos

.

(

2π . = cos 2n−1

2π 2n−1 )

) I+

) ) ( ( O ωi 2π 2π sin n−1 μ3 − sin n−1 μ2 δ δ 2 2

(53)

) ) ( ( O ω 2π 2π I + sin n−1 (ab − ba) + sin n−1 (b − a). δ δ 2 2

Regarding (36) we note from (14) that .X = iZY , so we may replace this expression of X in (46), getting μ2 = (− cos θ ) ZY + (i sin θ ) Y = ((− cos θ ) Z + (i sin θ ) I)Y.

(54)

det((− cos θ ) Z + (i sin θ ) I) = (− cos θ + i sin θ ) (cos θ + i sin θ )

(55)

.

Since .

.

= −e−iθ eiθ = −1,

we are sure that the matrix .((− cos θ ) Z +(i sin θ ) I) is invertible, hence (54) implies Y = ((− cos θ ) Z + (i sin θ ) I)−1 · μ2

.

(( .

(( .

=

=

(56)

) ) )−1 ( |ω| cos θ i cos θ O μ2 + (i sin θ ) I μ3 + ( · μ2 δ δ

) ( ) )−1 i|ω| cos θ i cos θ O (ab − ba) + ( (b − a) + (i sin θ ) I) · i(b − a) δ δ

so we conclude by (49) that .ϕ(w) is as above, up to the multiplicative factor .−i. The result follows, since (35) and (36) are in terms of (44) but also in terms of (39).

4 Conclusions We end reporting a series of additional considerations which we noted during our investigations. The classification of finite extra-special p-groups should be recalled here (p is an arbitrary prime). Lemma 4.1 (See [20]) If p is odd, any nonabelian p-group of order .p3 must be isomorphic either to E1 = ,

.

Dynamical Systems Involving Pseudo-Fermionic. . .

15

which is called nonabelian p-group of order .p3 and exponent p, or to E2 = , 2

.

which is called nonabelian p-group of order .p3 and exponent .p2 . In fact one can see that .E1 has no elements of order .p2 , while .E2 has it. Lemma 4.2 (See [20]) Any nonabelian 2-group of order 8 either is isomorphic to the dihedral group D8 =

.

or to the quaternion group Q8 = .

.

A finite p-group G, that is, a finite group G of prime power order, is extraspecial, if .Z(G) = [G, G] = Z(p), see [16, 18, 20]. Heisenberg groups, when realized with coefficients in .Z(p), turn out to be extraspecial p-groups of order .p3 for any odd prime p (see [16, 18, 21]). Now we report their main classification: Lemma 4.3 (See [20], Exercises 6 and 7) A nonabelian extraspecial p-group G of .|G| = p2n+1 has .Z(G) = Z(p) and .G/Z(G) = Z(p) × . . . × Z(p) = Z(p)2n . Moreover, if .p = 2, then G is the central product of .D8 ’s or a central product of .D8 ’s and a single copy of .Q8 . If .p > 2, then either G has exponent p, or else it is a central product of .E1 ’s and a single .E2 . From Lemma 4.3, there are only two types of extraspecial 2-groups of order 21+2n . In particular, .Q32 is different from the extraspecial 2-groups .Q8 ◦ Q8 = D8 ◦ D8 and .D8 ◦ Q8 of order 32, which are the only two types of extraspecial 2-groups of this order. In fact generalized quaternion groups are not extraspecial 2-groups in general. This brought us to study the following problem, which is apparently new in the literature and of course related to Conjecture 2.3 in a wider context:

.

Problem 4.4 Finding complex representations of finite p-groups G in terms of (39) and (42) (or eventually of generalizations of (39) and (42)). Then studying the constants of the motion in dynamical systems possessing groups of symmetries of the type of G. Acknowledgments The first author (Y.B.) thanks University of South Africa (South Africa) for the financial support given by Harry Bopape, Research Support Directorate. The second author (F.G.R.) thanks the Gruppo Nazionale di Fisica Matematica of the I.N.d.A.M. (Italy) and National Research Foundation of South Africa for grants no. 118517 and 113144.

16

Y. Bavuma and F. G. Russo

References 1. Arnold, V.I., Mathematical Methods of Classical Mechanics. Springer, Berlin (1978) 2. Attia, M., Koussa, W., Maamache, M.: Pseudo-fermionic coherent states with time-dependent metric. J. Math. Phys. 61(4), 042101 (2020) 3. Bagarello, F.: Linear pseudo-fermions. J. Phys. A 45, 444002 (2012) 4. Bagarello, F.: Pseudo-Bosons and Their Coherent States. Springer, Berlin (2022) 5. Bagarello, F., Russo, F.G.: A description of pseudo-bosons in terms of nilpotent Lie algebras. J. Geom. Phys. 125, 1–11 (2018) 6. Bagarello, F., Gazeau, J.-P., Szafraniec, F.H., Znojil, M.: Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects. John Wiley and Sons, Hoboken (2015) 7. Bagarello, F., Bavuma, Y., Russo, F.G.: Topological decompositions of the Pauli group and their influence on dynamical systems. Math. Phys. Anal. Geom. 24, 16 (2021) 8. Bavuma, Y.: The relevance of the Pauli group in dynamical systems with pseudo-fermions. Ph.D. Thesis, University of Cape Town, 2021. Online at: http://hdl.handle.net/11427/35685 9. Bavuma, Y.: A short note on the topological decomposition of the central product of groups. Trans. Comb. 11(3), 123–129 (2022). https://doi.org/10.22108/TOC.2022.130505.1908 10. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics. I. Springer, Berlin (1979) 11. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics. II. Springer, Berlin (1981) 12. Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. AMS, Providence (2001) 13. Cherbal, O., Maamache, M.: Time-dependent pseudofermionic systems and coherent states. J. Math. Phys. 57(2), 022102 (2016) 14. Cherbal, O., Drir, M., Maamache, M., Trifonov, D.A.: Fermionic coherent states for pseudoHermitian two-level systems. J. Phys. A 40, 1835–1844 (2007) 15. Cherbal, O., Ighezou, F., Maamache, M., Zenad, M.: Ladder invariants and coherent states for time-dependent non-Hermitian Hamiltonians. Int. J. Theor. Phys. 59, 1214–1226 (2020) 16. Gorenstein, D.: Finite Groups. Chelsea Publishing Company, New York (1980) 17. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) 18. Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967) 19. Kosniowski, C.: A First Course in Algebraic Topology. Cambridge University Press, Cambridge (1980) 20. Robinson, D.: A Course in the Theory of Groups. Springer, Berlin (1980) 21. Rocchetto, A., Russo, F.G.: Decomposition of Pauli groups via weak central products. Preprint, 2020. arXiv:1911.10158 22. Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. Birkhäuser, Basel (1990)

An Evolution Equation Approach to Linear Quantum Field Theory ´ Jan Derezinski and Daniel Siemssen

1 Introduction The wave equation or, more generally, the Klein–Gordon equation on curved Lorentzian manifolds is one of the classic topics of linear partial differential equations [2, 15, 26]. One could expect that it is difficult to find new important concepts in this subject. However, the present paper analyzes a few natural objects associated to the Klein–Gordon equation, which we believe are rather fundamental and to a large extent were overlooked in the mathematical literature until quite recently. These objects include the (in-out) Feynman and anti-Feynman inverse (or propagator), and various well-posed realizations of the Klein–Gordon operator. A proof of the existence of the Feynman and anti-Feynman propagators under rather mild assumptions is probably the main result of our paper. The Feynman and anti-Feynman propagators play a central role when we compute the scattering operator in Quantum Field Theory (QFT) on curved spacetimes. In fact, the Feynman propagator is associated to each internal line of a Feynman diagram. The Feynman propagator is thus crucial in the global approach to QFT, involving the whole spacetime, when one wants to compute the scattering operator. Our results are thus complementary to a large mathematical literature devoted to QFT on curved spacetimes involving the local approach [1, 6, 21, 22].

J. Derezi´nski (O) Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Warszawa, Poland e-mail: [email protected] D. Siemssen Department of Mathematics and Informatics, University of Wuppertal, Wuppertal, Germany © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Correggi, M. Falconi (eds.), Quantum Mathematics II, Springer INdAM Series 58, https://doi.org/10.1007/978-981-99-5884-9_2

17

18

J. Derezi´nski and D. Siemssen

1.1 Propagators on the Minkowski Spacetime As indicated above, the main motivation of our paper comes from curved spacetimes. However, it is natural to start from the flat Minkowski spacetime. Let us recall the most important “propagators” or “two-point functions” used in QFT, known from many textbooks, e.g., Appendix 2 of Bogoliubov–Shirkov [5] and Appendix C of Bjorken–Drell [4]: . the forward/backward or retarded/advanced propagator G∨/∧ (x, y) :=

.

1 (2π)4

f p2

e−i(x−y)·p dp, + m2 ± i0 sgn(p0 )

(1a)

. the Feynman/anti-Feynman propagator GF/F (x, y) :=

.

1 (2π)4

f

e−i(x−y)·p dp, + m2 ∓ i0

p2

(1b)

. the Pauli–Jordan propagator i .G (x, y) := (2π)3

f

PJ

e−i(x−y)·p sgn(p0 )δ(p2 + m2 ) dp,

(1c)

. the positive/negative frequency or particle/antiparticle bisolution G(±) (x, y) :=

.

1 (2π)3

f

e−i(x−y)·p θ (±p0 )δ(p2 + m2 ) dp.

(1d)

Mathematically, (1a), (1b) are distinguished inverses and (1c), (1d) are distinguished bisolutions of the Klein–Gordon operator −o + m2 . We will call them jointly “propagators”. They satisfy a number of identities: GPJ = G∨ − G∧.

.

(2a)

= iG(+) − iG(−) , .

(2b)

GF − GF = iG(+) + iG(−) , .

(2c)

GF + GF = G∨ + G∧ , .

(2d)

GF = iG(+) + G∧ = iG(−) + G∨ , .

(2e)

GF = −iG(+) + G∨ = −iG(−) + G∧ .

(2f)

An Evolution Equation Approach to Linear Quantum Field Theory

19

Let us describe applications of these propagators to quantum field theory. We will restrict ourselves to scalar fields. We will consider both basic formalisms for scalar free fields—the neutral or real and charged or complex formalism. In the neutral formalism the basic object is a self-adjoint operator-valued ˆ distribution on spacetime φ(x) satisfying the Klein–Gordon equation ˆ (−o + m2 )φ(x) = 0.

.

Here are various “two-point functions” of these fields: . commutation relations ˆ ˆ [φ(x), φ(y)] = −iGPJ (x, y)1,

.

. vacuum expectation of products of fields ˆ φ(y)o) ˆ (o | φ(x) = G(+) (x, y),

.

. vacuum expectation of direct/reverse time-ordered products of fields ( ) ˆ φ(y)}o ˆ o | T{φ(x) = −iGF (x, y), ) ( ˆ φ(y)}o ˆ = iGF (x, y). o | T{φ(x)

.

In the charged formalism the field is non-self-adjoint. It will be denoted with a ˆ different letter: ψ(x). It also satisfies the Klein–Gordon equation ˆ (−o + m2 )ψ(x) = (−o + m2 )ψˆ ∗ (x) = 0.

.

The “two-point functions” of the charged field are slightly more rich than in the neutral case: ˆ [ψ(x), ψˆ ∗ (y)] = −iGPJ (x, y)1, .

.

(4a)

ˆ ψˆ ∗ (y)o) = G(+) (x, y), . (o | ψ(x)

(4b)

ˆ (o | ψˆ ∗ (x)ψ(y)o) = G(−) (x, y), .

(4c)

( ) ˆ ψˆ ∗ (y)}o = −iGF (x, y), . o | T{ψ(x)

(4d)

( ) ˆ ψˆ ∗ (y)}o = iGF (x, y). o | T{ψ(x)

(4e)

We will use the name classical propagators as the joint name for GPJ , G∨ and The functions G(+) , G(−) , GF and GF express vacuum expectation values, therefore they will be jointly called non-classical propagators. G∧ .

20

J. Derezi´nski and D. Siemssen

1.2 The Klein–Gordon Equation on a Curved Spacetime and Its Classical Propagators In the first part of the introduction (until Sect. 1.10) we will discuss the Klein– Gordon equation and related objects in a purely mathematical setting, without a direct reference to classical or quantum fields. Consider a globally hyperbolic manifold M equipped with a metric tensor g = [gμν ] and its inverse g −1 = [g μν ], an electromagnetic potential A = [Aμ ] and a scalar potential Y . Throughout most of the introduction we will assume that g, A, Y are smooth—this assumption will not be necessary in the rest of our paper. Let Dμ := −i∂μ . Our paper is devoted to the Klein–Gordon operator: 1

1

1

K := −|g|− 4 (Dμ − Aμ )|g| 2 g μν (Dν − Aν )|g|− 4 − Y.

.

(5)

(Note that we use the so-called half-density formalism, see Sect. 7.1.) The equation Ku = 0

(6)

.

will be called the (homogeneous) Klein–Gordon equation. It has been shown by many authors that there exist unique distributions G∨ and ∧ G on M × M with the following properties. If f ∈ Cc∞ (M), then ∨

u (x) =

.

f

∨

G (x, y)f (y) dy,

∧

u (x) =

f

G∧ (x, y)f (y) dy

satisfy Ku∨ = Ku∧ = f

.

and supp u∨ , supp u∧ are contained in the future, resp. past causal shadow of supp f . We can also generalize the Pauli–Jordan propagator (1c) by using the identity (2a): GPJ = G∨ − G∧ .

.

Thus the classical propagators GPJ , G∨ and G∧ are well defined (and also well known) for general Klein–Gordon equations on globally hyperbolic manifolds.

1.3 Pseudounitary Structure We will denote by Wsc the space of smooth space-compact functions M e x |→ u(x) ∈ C solving (6).

An Evolution Equation Approach to Linear Quantum Field Theory

21

Let u, v ∈ Wsc . Then it is easy to see that ( ) 1 1 j μ (x; u, v) := − u(x)g μν (x)|g| 4 (x) Dν − Aν (x) |g|− 4 (x)v(x)

.

( ) 1 1 − Dν − Aν (x) |g|− 4 (x)u(x)g μν (x)|g| 4 (x)v(x). is a conserved current, that is ∂μ j μ = 0. Hence a Hermitian form on Wsc called sometimes the charge f (u | Qv) :=

.

S

j μ (x, u, v) dsμ (x)

(7)

does not depend on the choice of a Cauchy surface S (where dsμ (x) denotes the natural measure on S times the normal vector). Note that many authors instead of the charge prefer to use the symplectic form on Wsc given by the imaginary part of (7). The charge form Q is not positive definite: it contains vectors with positive and negative charge. The space Wsc can be decomposed in many ways in a direct sum of a maximally positive space and a maximally negative space, both orthogonal to one another in the sense of the charge form. Every such a decomposition can be encoded with help of an admissible involution S• : an operator on Wsc satisfying S•2 = 1,

.

(u | QS• v) = (S• u | Qv)

is positive.

(8)

As every involution, S• , determines a pair projections, so that ||•(+) + ||•(−) = 1,

.

S• = ||•(+) − ||•(−) .

The ranges of ||•(+) and ||•(−) are Q-orthogonal and maximally positive, resp. negative. They will be used to define Fock states in QFT.

1.4 Non-Classical Propagators on Curved Spacetimes In the literature it is often claimed that it makes no sense to ask for distinguished non-classical propagators on generic spacetimes. The main message of our paper disputes this statement. We will argue that for a large class of non-stationary spacetimes there exist physically relevant distinguished non-classical propagators. We will consider two types of spacetimes. 1. the slab geometry case, where M can be identified with [t− , t+ ] × E, where [t− , t+ ] is a finite interval describing time;

22

J. Derezi´nski and D. Siemssen

2. the unrestricted time case, where M is a globally hyperbolic spacetime with no boundary, equipped with the Klein–Gordon equation which is asymptotically stationary and stable in the future and in the past. By the asymptotic stationarity we will mean that one can identify M with R × E such that g, A and Y converge as t → ±∞ to limiting values sufficiently fast. Often, for simplicity we will just assume that there exists T > 0 such that for ±t > T the Klein–Gordon operator is stationary. By stability we mean the positivity of the Hamiltonian. Thus in Case (2) we assume that for large t the Hamiltonian is positive. Case (2) is probably more interesting both physically and mathematically. Nevertheless, it is instructive to compare Case (1) with Case (2). In our opinion the assumption of asymptotic stationarity and stability in Case (2) is natural from the physical point of view. Asymptotic stationarity is a necessary condition to apply the ideas of scattering theory, which is the main means of extracting useful information from QFT. Stability is satisfied in typical physics applications. Non-classical propagators are associated with special boundary conditions in the past and future. After quantization, they will be used to encode Fock representations of Canonical Commutation Relation. In this subsection we will describe them without a reference to quantum fields, as a part of an operator-theoretic analysis of the Klein–Gordon operator. Consider first Case (1). In order to define non-classical propagators at time t+ and t− we fix a pair of admissible involutions S+ , resp. S− . They lead to corresponding (+) (−) projections ||± and ||± . The choice of S± is to a large extent arbitrary, although one can argue that those satisfying the so-called Hadamard property [14, 31, 34] are more physical than the others. In Case (2) it is natural to select the admissible involutions S± given by the sign of the generator of the dynamics at t → ±∞. (Recall that we assume that (+) the evolution is asymptotically stationary and stable.) Thus ||± is the projection (−) onto “in/out positive frequency modes” and ||± onto “in/out negative frequency modes”. (+) (−) The projections ||± and ||± naturally define two pairs of bisolutions of the (+) (−) and G± . The identity (2b) now splits into two Klein–Gordon equation, G± independent identities (+)

(−)

GPJ = iG± − iG± ,

.

(9)

It is less obvious that the Feynman and anti-Feynman propagators also possess natural unique generalizations. The Feynman propagator GF can be described as (+) the inverse of the Klein–Gordon operator corresponding to the Cauchy data in ||+ (−) for t = t+ or t → +∞, and in ||− for t = t− or t → +∞. The anti-Feynman propagator GF is the inverse of the Klein–Gordon operator corresponding to the (−) (+) Cauchy data in ||+ for t = t+ or t → +∞, and in ||− for t = t− or t → −∞.

An Evolution Equation Approach to Linear Quantum Field Theory

23

Clearly, in Case (1) GF and GF depend on the choice of S+ , S− . In Case (2) they are defined uniquely. Note that GF are GF are sometimes called the in-out, resp. out-in Feynman propagators [16], to distinguish them from some other, non-canonical proposals, such as those mentioned below in (20). We will sometimes use these terms to stress their physical meaning. However, in our opinion, when one writes the Feynman propagator using the definite article the, there should be no doubt that GF is meant. Note that in the generic case the relations (2) are not satisfied, except for (2a) and the two versions of (2b), see also (9).

1.5 Well-Posedness/Self-Adjointness of the Klein–Gordon Operator Formally, the Feynman and anti-Feynman propagators are inverses of the Klein– Gordon operator. One can ask whether this can be interpreted in a more precise operator-theoretic sense. We will see that this is often true, however the situation is quite different in Case (1) and (2). It is easy to see that the Klein–Gordon operator K is Hermitian (symmetric) on, say, Cc∞ (M). In Case (1) K is obviously not essentially self-adjoint—it possesses many extensions parametrized by boundary conditions at t = t+ and t = t− . The admissible involutions S+ and S− determine special boundary conditions that lead to closed realizations K F = (K F )∗ , so that we have GF = (K F )−1 ,

.

GF = (K F )−1 = (GF )∗ .

Note that K F and K F are not self-adjoint. Clearly, they are invertible, and hence well-posed. 1 In Case (2) there seems to be no need for boundary conditions and one can expect that K is often essentially self-adjoint on Cc∞ (M). Suppose that K is essentially self-adjoint and let us denote by K s.a. its self-adjoint extension. Then we can expect that GF = lim (K s.a. − ie)−1 ,

.

e-0

GF = lim (K s.a. + ie)−1 , e-0

(10)

in the sense of quadratic forms on an appropriate weighted space, e.g., −s L2 (M) with s > 12 .

1 An

operator which has a non-empty resolvent set is called well-posed, see [12]. For instance, self-adjoint operators are well-posed.

24

J. Derezi´nski and D. Siemssen

The above conjectures are obviously true on the Minkowski space. They also hold in the stationary case. In the absence of the electrostatic potential this is straightforward, with the electrostatic potential it requires some work, see [8]. There exists also recent interesting papers by Vasy [33] and Taira–Nakamura [27– 29], where all these properties are proven for some classes of spacetimes, mostly assuming the asymptotic Minkowskian property and non-trapping conditions. The question of the self-adjointness of the Klein–Gordon operator is beyond the scope of our paper. It is much more difficult to answer and most probably requires additional assumptions (like non-trapping conditions). However, as the analysis of our paper shows, the Feynman inverse is well-defined for essentially all asymptotically stable and stationary spacetimes. Note that if asymptotic stability and stationarity does not hold, but K can be interpreted as a self-adjoint operator, then one can try to use (10) as the definition of the Feynman/anti-Feynman propagators.

1.6 Reduction to a 1st Order Equation for the Cauchy Data In order to compute non-classical (actually, also classical) propagators, it is useful to convert the Klein–Gordon equation into a 1st order evolution equation on the phase space describing Cauchy data. To this end, we fix a decomposition M = I × E, where I = [t− , t+ ] or I = R. We assume that M is Lorentzian and E is Riemannian. We will use Latin letters for spatial indices. We introduce h = [hij ] = [gij ], h−1 = [hij ],

.

β j := g0i hij , α 2 := g0i hij gj 0 − g00 . Note that [hij ], [hij ] are positive definite and α 2 > 0. Set 1

1

1

L := |g|− 4 (Di − Ai )|g| 2 hij (Dj − Aj )|g|− 4 + Y,

.

i i W := β i Di − A0 + β i Ai + |g|−1 |g|,0 − β i |g|−1 |g|,i . 4 4 Then the Klein–Gordon operator and the charge can be written as 1 K = (D0 + W ∗ ) 2 (D0 + W ) − L, α f ) 1 ( D0 + W (t, x-) v(t, x-) dx u(t, x-) 2 (u | Qv) = α (t, x-) E f ( ) 1 v(t, x-) dx. + D0 + W (t, x-) u(t, x-) 2 α (t, x-) E .

(11)

An Evolution Equation Approach to Linear Quantum Field Theory

25

Therefore, the Klein–Gordon equation Ku = 0 can be rewritten as a 1st order equation for the Cauchy data on E: ( ) ∂t + iB(t) w = 0,

.

where ] ] [ [ B11 (t) B12 (t) W (t) α 2 (t) := , .B(t) = B21 (t) B22 (t) L(t) W ∗ (t) ] [ ] [ w1 u ( ) := . w= w2 −α −2 − i∂t + W (t) u The current preserved by the dynamics is given by the matrix [ Q=

.

] 01 . 10

It is natural to introduce the classical Hamiltonian ] [ L(t) W ∗ (t) .H (t) = QB(t) = W (t) α 2 (t) and the Cauchy data operator M := ∂t + iB(t).

.

(The notational clash with the occasionally appearing manifold M should cause no confusion.) We will say that an operator E is a bisolution/inverse or Green’s operator of M if it satisfies MEw = 0,

EMw = 0,

resp. MEw = w,

EMw = w

.

[ for a large class of functions t |→ w(t) = be written as a 2 × 2 matrix [

] w1 (t) . An inverse/bisolution of M can w2 (t)

] E11 (t, s) E12 (t, s) .E(t, s) = . E21 (t, s) E22 (t, s)

26

J. Derezi´nski and D. Siemssen

If we set G(t, s) := iE12 (t, s),

.

(12)

then G is formally a bisolution/inverse of the Klein–Gordon operator: KGu = 0,

GKu = 0,

KGu = u,

GKu = u,

.

resp.

for a large class of spacetime functions x |→ u(x).

1.7 Stationary Case The theory of propagators for the Klein–Gordon equation greatly simplifies in the stable stationary case. More precisely, suppose for the moment that M = R × E and that g, A and Y , hence also B(t) =: B and H (t) =: H , do not depend on time t. Assume also that B is stable, which means that H is positive and has a zero nullspace. First of all, the Cauchy data can then be organized in a Hilbert space. Actually, using the Hamiltonian H and the generator B, one can construct a whole scale of natural Hilbert spaces Wλ , λ ∈ R, which can be used to describe the Cauchy data. Among them three have a special importance. The energy space, W 1 , has the scalar 2 product given by the Hamiltonian H . There is also the dual energy space, which we denote by W− 1 , with the scalar product given by (QH Q)−1 . Finally, interpolating 2 between W 1 and W− 1 , we obtain the dynamical space W0 , which in addition to 2 2 the scalar product has a natural pseudo-unitary structure given by the charge Q, and which is then used for quantization. The operator B can be interpreted as self-adjoint on all members of the scale Wλ . Therefore, the dynamics is simply defined as e−itB and preserves the scale Wλ . Then we can define the propagators on the level of the Cauchy data as follows: E PJ (t, s) := e−i(t−s)B ,

.

E ∨ (t, s) := θ (t − s)e−i(t−s)B , E ∧ (t, s) := −θ (s − t)e−i(t−s)B , E (+) (t, s) := e−i(t−s)B 1[0,∞[ (B), E (−) (t, s) := e−i(t−s)B 1[−∞,0[ (B),

An Evolution Equation Approach to Linear Quantum Field Theory

27

( ) E F (t, s) := e−i(t−s)B θ (t − s)1[0,∞[ (B) − θ (s − t)1]−∞,0] (B) , ( ) E F (t, s) := e−i(t−s)B θ (t − s)1]−∞,0] (B) − θ (s − t)1[0,∞[ (B) . At least formally, E ∨ , E ∧ , E F , E F are inverses and E PJ , E (+) , E (−) are bisolutions of M. Then we set • G• := iE12 ,

.

G(+) :=

(+) E12 ,

• = PJ, ∨, ∧, F, F, . G(−) :=

(−) −E12 ,

(13a) (13b)

(hence we use (12) or its minor modifications) obtaining the generalizations of the propagators from the Minkowski space to the general stationary case. Note that in the stationary case all the identities (2) still hold.

1.8 Evolution on Hilbertizable Spaces In the generic situation the generator B(t) depends on time. This leads both to technical and conceptual problems. First, in order to do functional analysis we need topology. However the Hilbert spaces Wλ are no longer uniquely defined. It seems reasonable to assume that Cauchy data are described by elements of a certain nested pair of Hilbertizable spaces W1 ⊂ W0 , which does not change throughout the time. (A Hilbertizable space is a space with a topology of a Hilbert space, but without a fixed scalar product.) We devote the whole Sect. 3 to a construction of cousins of all propagators described in Sect. 1.7 in the setting of an evolution on Hilbertizable spaces (without assuming the existence of a charge form preserved by the dynamics). The construction of the dynamics in the stationary case was straightforward. Constructing the evolution generated by a time-dependent generator B(t) is much more technical. To this end we use an old result of Kato [25]. In this approach one assumes that the Cauchy data are described by a nested pair of Hilbertizable spaces, and the generators are self-adjoint with respect to certain time-dependent scalar products compatible with both Hilbertizable structures. Besides, one needs to make some technical assumptions, which essentially say that the generator of the evolution does not vary too much in time, so that all the time it acts in the same nested pair of Hilbertizable spaces. Using this evolution it is easy to define E ∨ , E ∧ , E PJ , which are the analogs of classical propagators on the level of the Cauchy data operator. In order to define “non-classical” propagators we need to choose the incoming and outgoing “particle/antiparticle projections”, which as we discussed in Sect. 1.4 are determined by specifying involutions S± . This leads to a straightforward defini-

28

J. Derezi´nski and D. Siemssen (+)

(−)

tion of “in/out particle bisolutions” E± and “in/out antiparticle bisolutions” E± , which are two distinguished analogs of particle and antiparticle bisolutions. (The plus/minus in the parentheses correspond to particles/antiparticles; the plus/minus without parentheses correspond to the future/past.) What is more interesting, we can also try to define a natural Feynman and anti-Feynman propagator, denoted E F , E F . In the general Hilbertizable setting the existence of these propagators is not guaranteed and requires an extra condition that we call the asymptotic complementarity.

1.9 Pseudo-Unitary Dynamics As discussed above, the evolution of Cauchy data for the Klein–Gordon equation preserves the charge form—a natural Hermitian indefinite scalar product. On the technical level it is convenient to assume that the charge form is compatible with the Hilbertizable structure. More precisely, we need the structure of a Krein space. Note, in parenthesis, that we prefer to work in the complex setting of a Krein space instead of the real setting of a symplectic space, perhaps more common in the literature. If the dynamics commutes with the complex conjugation, e.g., if there are no electromagnetic potentials, then by restricting our dynamics to the real space we can go back to the real symplectic setting. In Sect. 4 we discuss propagators in the context of a pseudo-unitary evolution on a Krein space. We note an important property of Krein spaces: every maximally positive subspace is complementary to every maximally negative subspace. By this property, if the boundary conditions are given by admissible involutions (see (8)), then the condition of asymptotic complementarity is automatically satisfied. Therefore the in-out Feynman and anti-Feynman propagator always exist. The existence of these two propagators under rather general conditions is probably the main result of our paper. To sum up, in the context of an evolution on Krein spaces we are able to define the whole family of “propagators” on the level of the Cauchy data operator: E ∨ , E ∧ , E PJ , E± , E± , E F , E F .

.

(+)

(−)

(14)

1.10 Abstract Klein–Gordon Operator Let L(t), α(t) and W (t) be time-dependent operators on a Hilbert space K. We assume that L(t) is positive, α(t) is positive and invertible, plus some additional technical assumptions. By an abstract Klein–Gordon operator we mean an operator

An Evolution Equation Approach to Linear Quantum Field Theory

29

of the form ( ) 1 ( ) Dt + W (t) − L(t), K := Dt + W ∗ (t) 2 α (t)

.

(15)

acting on the Hilbert space L2 (I, K) = L2 (I ) ⊗ K. In our applications, K is the Hilbert space L2 (E) (the space of functions on a spacelike Cauchy surface). Besides, α 2 (t) is related to the metric tensor, W (t) consists mostly of A0 , and L(t) is a magnetic Schrödinger operator on E. The usual Klein–Gordon operator (5) on the Hilbert space L2 (M) = L2 (I ) ⊗ L2 (E) has the form of (15), as discussed in Sect. 1.6, see (11). The operator (15) is second order in t. It can be viewed as a 1-dimensional magnetic Schrödinger operator with operator-valued potentials. To describe its Cauchy data, under our assumptions it is natural to introduce the scale of Hilbertizable spaces λ

1

λ

1

Wλ := L(t)− 2 − 4 K ⊕ L(t)− 2 + 4 K,

.

(16)

where usually |λ| ≤ 12 . Note that W0 has a natural Krein structure. Using the formalism of Sect. 4 we construct various propagators (14). Then, using a slight extension of (13), we pass from the Cauchy data propagators to spacetime propagators: • G• := iE12 ,

.

(+) (+) := E±,12 G± ,

• = PJ, ∨, ∧, F, F; (−) (−) := −E±,12 G± .

Feynman and anti-Feynman inverses of the Klein Gordon operator can be viewed as some special inverses of its well-posed realizations. In Case (1) the Klein–Gordon operator is Hermitian (symmetric) but not selfadjoint. It possesses many well-posed realizations defined by boundary conditions. In particular, each Feynman-type and anti-Feynman-type inverse defines a certain well-posed realization. Note that these realizations are always non-self-adjoint. In Case (2) the situation is more difficult and not fully understood. Clearly, the Feynman and anti-Feynman inverses are not bounded operators. It is natural to conjecture that under quite general conditions they are boundary values of a certain distinguished self-adjoint realization of the abstract Klein–Gordon operator. This conjecture has been partly proven in [33] and [27–29].

1.11 Bosonic Quantization Let us now describe applications of the above mathematical analysis of the Klein– Gordon equation to Quantum Field Theory.

30

J. Derezi´nski and D. Siemssen

Various authors use different formalisms when introducing quantum fields. These formalisms are essentially equivalent, however it may often be difficult for the reader to translate the concepts from one formalism to another. Therefore, we start our discussion with some remarks about various approaches to quantization. We restrict ourselves to linear bosonic theories. Bosonic quantization can be divided into two steps: 1. First we choose an algebra of observables satisfying canonical commutation relations corresponding to a classical phase space. 2. Then we select a representation of this algebra on a Hilbert space. The first step can be presented in several ways, which superficially look differently. In particular, we can use the real or complex formalism: . The real or neutral formalism starts from a real space equipped with an antisymmetric form (it does not have to be symplectic, that is, non-degenerate) ˆ and leads to a self-adjoint field o. . The complex or charged formalism starts from a complex space equipped with a Hermitian form (sometimes called the charge) and leads to a pair of non-selfˆ w ˆ ∗. adjoint fields w, The neutral formalism is in a sense more general, since every charged particle can be understood as a pair of neutral particles in the presence of a U (1) symmetry. In both the real and the complex approach, we can use the one-component formalism or the two-component formalism. In the two-component formalism we split the fields into “positions” and “momenta”. This splitting is typical for Quantum Field Theory. Thus we can distinguish four formalisms of bosonic quantization, which can be summarized in the following table: Real (or neutral) fields

Complex (or charged) fields

1-component formalism

ˆ ˆ ' )] = iwωw ' [o(w), o(w ω is an antisymmetric form on a real space

ˆ ˆ ∗ (w ' )] = (w | Qw ' ) [w(w), w Q is a Hermitian form on complex space

2-component formalism

ˆ [φ(u), πˆ (v)] = i

ˆ [ψ(u), ηˆ ∗ (v)] = i(u | v) ∗ [ψˆ (u), η(v)] ˆ = i(v | u)

is a bilinear scalar product on a real space

(· | ·) is a sesquilinear scalar product on a complex space

If the number of degrees of freedom is finite, by the Stone-von Neumann Theorem all irreducible representations of the CCR over a symplectic space are equivalent. If the number of degrees of freedom is infinite this is not true, and we have to select a representation. Usually this is done by fixing a state on the algebra of observables and going to the GNS representation. In most applications to QFT

An Evolution Equation Approach to Linear Quantum Field Theory

31

one chooses a pure quasi-free state, and then this representation naturally acts on a bosonic Fock space. There are several ways to describe pure quasi-free states (called also Fock states). As we mentioned above, in our paper these states are described by admissible involutions on the phase space, see (8). In the real formalism one needs to assume in addition that S• is anti-real (i.e., S• = −S• ).

1.12 Classical Field Theory on Curved Spacetimes Let us now briefly describe linear Classical Field Theory on curved spacetimes. In the introduction, for brevity we restrict ourselves to complex scalar fields. In Sects. 6 and 7 we will consider also real scalar fields. The phase space of our system is Wsc introduced in Sect. 1.3. Then the field ψ and its complex conjugate ψ ∗ are interpreted as the linear, resp. antilinear functional on Wsc given by := u(x),

.

:= u(x),

u ∈ Wsc .

Clearly, the fields ψ and ψ ∗ satisfy the Klein–Gordon equation: Kψ(x) = Kψ ∗ (x) = 0.

.

(17)

As described in Sect. 1.2, the space Wsc is naturally a symplectic vector space. The Poisson bracket of the fields was first computed by Peierls and can be expressed by the Pauli–Jordan propagator: {ψ(x), ψ(y)} = {ψ ∗ (x), ψ ∗ (y)} = 0,

.

{ψ(x), ψ ∗ (y)} = −GPJ (x, y).

1.13 Quantum Field Theory on Curved Spacetimes Quantization of the classical theory defined by (17) is performed in two steps. ˆ First we replace the classical fields ψ(x), ψ ∗ (x) by ψ(x), ψˆ ∗ (x) interpreted as distributions with values in a ∗-algebra satisfying the following commutation relations, a generalization of the identity (4a): ˆ ˆ [ψ(x), ψ(y)] = [ψˆ ∗ (x), ψˆ ∗ (y)] = 0,

.

ˆ [ψ(x), ψˆ ∗ (y)] = −iGPJ (x, y)1.

32

J. Derezi´nski and D. Siemssen

They also satisfy the Klein–Gordon equation ˆ K ψ(x) = K ψˆ ∗ (x) = 0.

.

The second step consists in choosing a representation of the CCR. If the Klein–Gordon equation is stationary and stable, then there exists a natural Fock representation given by the so-called positive energy quantization [7, 8]. As we discussed above, in the stationary case all propagators described for the Minkowski space have a natural generalization and the relations (4) still hold. If the Klein–Gordon equation is not necessarily stationary and we selected the involutions S− and S+ (see Sect. 1.4), we consider the Fock representations corresponding to S− and S+ , the so-called in and out representations. As described (+) (−) above, we obtain two pairs of bisolutions G± and G± , which after quantization describe the vacuum expectation values of products of fields in the in and out Fock representation. More precisely, the identities (4b) and (4c) split into two identities: ˆ ψˆ ∗ (y)o± ) = G± (x, y), . (o± | ψ(x)

(18a)

(−) ˆ (o± | ψˆ ∗ (x)ψ(y)o ± ) = G± (x, y).

(18b)

.

(+)

After quantization the Feynman and anti-Feynman propagators satisfy slight modifications of the identities (4d) and (4e): ( ˆ ψˆ ∗ (y)}o− ) o+ T{ψ(x) = −iGF (x, y), . . (o+ | o− ) ( ˆ ψˆ ∗ (y)}o+ ) o− | T{ψ(x) = iGF (x, y). (o− | o+ )

(19a) (19b)

Strictly speaking, (19) are true if the Shale condition for the in and out states holds, so that the vectors o− and o+ belong to the same representation of CCR. If the Shale condition is violated, the left hand sides do not make sense. However, the right hand sides are well defined. This is an example of a renormalization in QFT: we are able to compute a quantity, which at the first sight is ill-defined. In a large part of the literature the so-called Hadamard property is considered to be the main criterion for a physically satisfactory state [14, 31, 34]. Note that if one assumes enough smoothness, then in Case 2 the two-point functions (+) (−) G± (x, y), G± (x, y) automatically satisfy the Hadamard property. This is a nontrivial fact proven by Gérard–Wrochna [19], see also [18]. In the slab geometry case we can choose non-Hadamard states for o− and o+ if we insist. However, there are good arguments saying that Hadamard states are “more physical” than the others. Actually, one could argue that the main argument for the Hadamard condition in Case (1) is the fact that by Gérard and Wrochna [19] it is automatic in the Case (2) scenario.

An Evolution Equation Approach to Linear Quantum Field Theory

33

1.14 Hadamardists and Feynmanists Oversimplifying and exaggerating, one can distinguish two approaches to QFT on curved spacetimes: let us call them the Hadamardist and Feynmanist approach. The Hadamardist approach is mostly represented by researchers with a mathematical or General Relativity background. It stresses that QFT should be considered in a local fashion, often restricting attention to a small causally convex region of a spacetime. In such a setting it is impossible to choose a distinguished state o on the algebra of observables in a locally covariant way, see e.g. [13, 21]. This approach stresses that one has a lot of freedom in choosing a state and argues that physical states should satisfy the so-called Hadamard property. In the modern mathematical literature this condition is usually described in an elegant but (abstract way with ˆ ψˆ ∗ (y)o. the help of the wave front set [31] of their two-point function o | ψ(x) Using this two-point function one can define the (formal) ∗-algebra of observables and perturbatively renormalize local polynomials of fields. A particularly clear explanation of this philosophy can be found in Apps. B and D1 of [21]. In the Feynmanist approach the main goal is usually to compute scattering amplitudes, cross sections, etc. see e.g. [10]. Such computations are typically based on path integrals and Feynman diagrams. Clearly, in this case it is indispensable to look at the spacetime globally, so that one can define an in and out state, and the assumption of asymptotic stability and stationarity in the past and future is rather natural. There is no need to worry about the Hadamard property. As we mentioned above, it is automatic under rather weak assumptions thanks to the results of Fulling–Narcowich–Wald [18] and Gérard–Wrochna [19]. The (in-out) Feynman propagator is needed to compute perturbatively the scattering operator in terms of Feynman diagrams. There is no contradiction between these two approaches and both have their philosophical merits. Our paper clearly belongs to the latter approach (in spite of being rather abstract mathematically). Actually, in Case (1) (the slab geometry case) it is natural to use the hybrid point of view, which reconciles the Hadamardist and Feynmanist philosophy. If we want to compute the scattering operator between time t− and t+ it is natural to choose S− and S+ both satisfying the Hadamard condition, and then to use the corresponding (in-out) Feynman propagator.

1.15 Comparison with Literature The basic formalism of pseudo-unitary (or symplectic) evolution equations described in this paper is of course contained more or less explicitly in all works on Quantum Field Theory in curved spacetime, including the standard textbooks [3, 17, 30, 35]. Surprisingly, however, its point of view is rarely fully exploited.

34

J. Derezi´nski and D. Siemssen

The (trivial) observation about the existence of two distinguished states on asymptotically stationary spacetimes can be found e.g. in [3], Sect. 3.3. It is rather obvious that they are the preferred states for many actual applications, such as the calculation of the scattering operator. In the well-known paper [11] Duistermat and Hörmander prove the existence of the Feynman parametrix, which is unique only up to a smoothing operator. However, the canonical in-out Feynman inverse defined rigorously in our paper is essentially absent from the mathematical literature, with a few recent exceptions [20, 27–29, 33]. Sometimes one considers another generalization of the Feynman propagator: (+/−) , one can introduce given a Hadamard state with the two-point functions G• GF• = iG•(+) + G∧ = −iG•(−) + G∨ ,

.

(20)

which is an inverse of the Klein Gordon operator, and can be called the Feynman inverse associated to the pair of two-point functions G•(+) , G•(+) , see e.g. [9] and [21] App. D1. Note, however, that GF• is non-unique and, more importantly, does not satisfy the relation (19a), which is the basis for perturbative calculations of the scattering operator. In the more physically oriented literature the in-out Feynman propagator, defined rigorously in our paper, is ubiquitous, even if implicitly. It essentially appears each time when the functional integration method is applied to compute scattering amplitudes, at least on asymptotically stationary spacetimes. More precisely, in order to compute perturbatively Feynman diagrams for the scattering operator one needs to associate the Feynman propagator to each line. It seems that this point is not sufficiently appreciated in a part of mathematically oriented literature. Let us quote from App. B of [21], which as we mentioned above, expresses the Hadamardist philosophy: “[the effective action] depends upon a choice of state [. . . ]. Here, the choice of state would enter the precise choice of the formal path-integral measure [Dφ].” In reality, typically the path-integral formalism yields a unique prescription for Feynman diagrams (which then need to be renormalized). This prescription naturally involves two states: the “in state” and the “out state”. It also determines uniquely the Feynman propagator, which should be associated to each line of the Feynman diagram. The formula (19a), which gives a physical meaning to the in-out Feynman propagator, can be found in the physical literature in various places, see, for instance, Equation (4.7) of [17] and the following equations. Until quite recently, the question of the self-adjointness of the Klein–Gordon operator was almost absent in the mathematical literature. There were probably two reasons for this. First, the question seemed difficult. The Klein–Gordon operator is unbounded from below and above, and the positivity is usually the major tool in functional analysis. The second reason is that this question at the first sight appeared not interesting physically. Indeed, the space L2 (M) is not the space of states of any

An Evolution Equation Approach to Linear Quantum Field Theory

35

reasonable quantum system and e−iτ K seems to have no meaning as a quantum evolution. However, in physical literature many researchers tacitly assume that the Klein– Gordon operator is self-adjoint, see e.g. [32] or Sect. 9 of [17]. The Feynman and anti-Feynman propagator are important for applications and, formally, assuming (10), they can be computed from (K ± i0)

.

−1

f

∞

= ±i

e∓itK dt,

(21)

0

Note that (21) presupposes that one can interpret K as a self-adjoint operator, so that (z − K)−1 can be defined for z /∈ R.

2 Preliminaries In this section we collect various basic mathematical definitions and facts that are useful in our paper. Most readers will find them rather obvious—nevertheless, they should be recorded. If A is an operator, then R(A), N(A), D(A) and σ (A) denote the range, the nullspace, the domain and the spectrum of A. B(W) denotes the space of bounded operators on a Banach space W.

2.1 Scales of Hilbert Spaces Suppose that W is a Hilbert space and A a positive invertible operator on W. Then one defines A−α W as the domain of Aα for α ≥ 0 and as its anti-dual for α < 0. We thus obtain a scale of nested Hilbert spaces A−α W, α ∈ R, with A0 W = W and A−α W continuously and densely embedded in A−β W for α ≥ β. By restriction/extension, the operator Aβ can be interpreted as a unitary from A−α W to A−α+β W. Often we simplify notation by writing Wα for A−α W, so that W0 = W and W1 = D(A). In practice, the starting point of a construction of a scale of Hilbert spaces is often not an operator A but a nested pair of Hilbert spaces. More precisely, suppose that (W, V) is a pair of Hilbert spaces, where V is densely and continuously embedded in W. Then there exists a unique invertible positive self-adjoint operator A on W with the domain V such that (v | v)V = (Av | Av)W ,

.

v ∈ V.

We can then define the scale Wα = A−α W, α ∈ R. Note that W = W0 and V = W1 .

36

J. Derezi´nski and D. Siemssen

We will often use the following facts: Proposition 2.1 Consider a scale of Hilbert spaces Wα , α ∈ R. 1. A ∈ B(Wα , Wβ ) implies A∗ ∈ B(W−β , W−α ). 2. Let α0 ≤ α1 , β0 ≤ β1 . If A ∈ B(Wα0 , Wβ0 ) can be restricted to an operator in B(Wα1 , Wβ1 ), then for τ ∈ [0, 1] it can be restricted to an operator in B(W(1−τ )α0 +τ α1 , W(1−τ )β0 +τβ1 ).

2.2 One-Parameter Groups Let W be a Banach space. Recall that a one-parameter group on W is a homomorphism R e t |→ R(t) ∈ B(W).

.

It is well-known that to every strongly continuous one-parameter group R(t) one can uniquely associate a densely defined operator B called the generator of R(t), so that R(t) = e−itB . It can be shown that D(B) is preserved by R(t) and the following equation is true: (∂t + iB)R(t)w = 0,

.

w ∈ D(B).

(22)

Suppose now that W is a Hilbert space. A unitary group on W is always of the 1 form e−itB , where B is a self-adjoint operator. Let := (B 2 +1) 2 . Note that e−itB preserves the scale Wα := −α W for α ≥ 0, and can be uniquely extended by continuity to Wα for α ≤ 0. For any α we have (∂t + iB)e−itB w = 0,

.

w ∈ W1+α ,

(23)

where the left-hand side of (23) is understood as an element of Wα . In practice, two choices of α are especially useful: α = 0 corresponds precisely to (22), and α = − 12 means that (23) is considered on the form domain of B. We will use both points of view when considering a natural setup for non-autonomous evolutions, see Theorem 3.10. If in addition B is invertible, then we can slightly modify the scale of Hilbert spaces by setting Wα := |B|−α W. Note that B is then unitary from Wα to Wα−1 .

An Evolution Equation Approach to Linear Quantum Field Theory

37

2.3 Hilbertizable Spaces Definition 2.2 Let W be a complex2 topological vector space. We say that it is Hilbertizable if it has the topology of a Hilbert space for some scalar product (· | ·)• on W. We will then say that (· | ·)• is compatible with (the Hilbertizable structure of) W. The subscript • serves ( ) as a placeholder for a name of a scalar product. The Hilbert space W, (· | ·)• will be occasionally denoted W• . The corresponding norm will be denoted || · ||• . In what follows W is a Hilbertizable space. Let (· | ·)1 , (· | ·)2 be two scalar products compatible with W. Then there exist constants 0 < c ≤ C such that c(w | w)1 ≤ (w | w)2 ≤ C(w | w)1 .

.

Let R be a linear operator on W. We say that it is bounded if for some (hence for all) compatible scalar products (· | ·)• there exists a constant C• such that ||Rw||• ≤ C• ||w||• .

.

Suppose that A is a (densely defined) operator on W. We say that it is similar to self-adjoint if there exists a compatible scalar product (· | ·)• such that A is selfadjoint with respect to (· | ·)• . Note that for such operators the spectral theorem can be applied. In particular, for any (complex-valued) Borel function f on the spectrum of A we can define f (A). Let Q be a sesquilinear form on W. We say that it is bounded if for some (hence for all) compatible scalar products (· | ·)• there exists C• such that |(v | Qw)| ≤ C• ||v||• ||w||• ,

.

v, w ∈ W.

Note that on Hilbertizable spaces we do not have a natural identification of sesquilinear forms with operators.

2.4 Interpolation Between Hilbertizable Spaces Definition 2.3 A pair of Hilbertizable spaces (W, V), where V is densely and continuously embedded in W, will be called a nested Hilbertizable pair. After fixing scalar products (· | ·)V,• and (· | ·)W,• compatible with V, resp. W, we can interpolate between the Hilbert spaces V• and W• obtaining a scale of Hilbert spaces Wα,• , α ∈ R, with V• = W1,• and W = W0,• . By

2 Analogous

definitions and results are valid for real Hilbertizable spaces.

38

J. Derezi´nski and D. Siemssen

complex interpolation, for α ∈ [0, 1] they do not depend on the choice of scalar products (· | ·)W,• and (· | ·)V,• as Hilbertizable spaces. Therefore, the family of Hilbertizable spaces Wα , α ∈ [0, 1], is uniquely defined. If R ∈ B(W) and its restriction to V belongs to B(V), then R restricts to B(Wα ) for 0 ≤ α ≤ 1.

2.5 From Complex to Real Spaces and Back To pass from a complex space to a real one, it is useful to have the notion of a conjugation: Let W be a complex space. An antilinear involution v |→ v on W will be called a conjugation. In the context of Hilbertizable spaces we always assume that conjugations are bounded. For an operator R on W we set Rv := Rv,

.

∗

R T := R .

If R satisfies R = ±R, it will be called real resp. anti-real. The real subspace of W is defined as WR := {w ∈ W | w = w}.

.

(24)

Conversely, to pass from a real space to a complex one, suppose now that Y is a real space. Then Y ⊗R C = CY will denote the complexification of Y (i.e., for every w ∈ W we can write w = wR + iwI with wR , wI ∈ Y), and we have the natural conjugation vR + ivI = vR − ivI .

2.6 Complexification of (Anti-)Symmetric Forms Let Y be a real space. Every symmetric form q on Y, and thus in particular every scalar product, extends to a Hermitian form on CY: ( ) vR + ivI | q(wR + iwI ) := + .

Note the property (v | qw) = (v | qw).

− i + i.

An Evolution Equation Approach to Linear Quantum Field Theory

39

Extending an antisymmetric form ω on Y to a Hermitian form on CY is slightly different: ( ) vR + ivI | Q(wR + iwI ) := − . (25) + i + i. Note the property (v | Qw) = −(v | Qw), which differs from the symmetric case above.

2.7 Realification of Hermitian Forms Let Q be a Hermitian form on W. We say that a conjugation · preserves Q if (v | Qw) = (v | Qw).

.

In that case, .

Re (v | Qw),

v, w ∈ WR ,

is a symmetric form on WR . Note that Im (v | Qw) = 0 on WR . Similarly, we say that a conjugation · anti-preserves Q if (v | Qw) = −(v | Qw).

.

(26)

In that case, .

Im (v | Qw),

v, w ∈ WR ,

is an antisymmetric form on WR . Note that Re (v | Qw) = 0 on WR .

2.8 Involutions Definition 2.4 We say that a pair (Z•(+) , Z•(−) ) of subspaces of a vector space W is complementary if Z•(+) ∩ Z•(−) = {0},

.

Z•(+) + Z•(−) = W.

Definition 2.5 An operator S• on W is called an involution, if S•2 = 1.

40

J. Derezi´nski and D. Siemssen

We can associate various objects with S• : ||•(±) :=

.

1 (1 ± S• ), 2

Z•(±) := R(||•(±) ).

(27)

(||•(+) , ||•(−) ) is a pair of complementary projections and (Z•(+) , Z•(−) ) is the corresponding pair of complementary subspaces. A possible name for Z•(+) is the positive space, and for Z•(−) is the negative space (associated with S• ). We will however prefer names suggested by QFT: Z•(+) will be called the particle space, and Z•(−) the antiparticle space. If W is Hilbertizable, we will usually assume that S• is bounded. Then so are ||•(+) and ||•(−) , moreover, Z•(+) and Z•(−) are closed.

2.9 Pairs of Involutions Suppose that S1 and S2 are two bounded involutions on W. Let (±)

||i

.

:=

1 (1 ± Si ), 2

(±)

Zi

:= R(||i(±) ),

i = 1, 2,

be the corresponding pairs of complementary projections and subspaces. The following operator can be defined by many distinct expressions: (+) (+) (−) (−) ϒ := 1 − (||1 − ||2 )2 = 1 − (||1 − ||2 )2.

(28a)

.

=

||1(+) ||2(+) (+)

= (||1

+ ||2(−) ||1(−) (−)

(+)

+ ||2 )(||2

=

||2(+) ||1(+) (−)

+ ||1(−) ||2(−).

(+)

+ ||1 ) = (||2

(−)

(28b) (+)

+ ||1 )(||1

= (||1(+) ||2(+) + ||2(−) )(||2(+) ||1(+) + ||1(−) ). (+)

(+)

= (||2 ||1 =

(−)

(+)

(+)

+ ||1 )(||1 ||2

(−)

+ ||2 ).

(−)

(28c) (28d) (28e)

1 1 (2 + S1 S2 + S2 S1 ) = (S1 + S2 )2 . 4 4 (+)

(−)

+ ||2 ).

(28f) (+)

Observe that ϒ commutes with ||1 , ||1 , ||2

(−)

and ||2 .

Proposition 2.6 The following conditions are equivalent: (i) ϒ is invertible. (+) (−) (+) (−) (ii) ||1 + ||2 and ||2 + ||1 are invertible. (+) (+) (−) (+) (+) (−) (iii) ||1 ||2 + ||2 and ||2 ||1 + ||1 are invertible.

An Evolution Equation Approach to Linear Quantum Field Theory

41 (+)

(−)

Moreover, if one of the above holds, then the pairs (Z1 , Z2 ) as well as (+) (−) (Z2 , Z1 ) are complementary. Proof (i) ⇐⇒ (ii) and (i) ⇐⇒ (iii) follow from (28c) and (28e) by the following easy fact: If R, S, T are maps such that R = ST = T S, then R is bijective if and only if both T and S are bijective. The last implication follows from the next proposition. u n In the setting of the above proposition we can use ϒ to construct two pairs of complementary projections: Proposition 2.7 Suppose that ϒ is invertible. Then (+) (+) (+) A12 := ||1 ϒ −1 ||2

is the projection onto Z1 alongZ2 ,

(−) := ||2(−) ϒ −1 ||1(−) A21

is the projection onto Z2(−) along Z1(+) ,

(+) (+) (+) A21 := ||2 ϒ −1 ||1

is the projection onto Z2

(−) (−) (−) A12 := ||1 ϒ −1 ||2

is the projection onto Z1

.

(+)

(−)

(+)

along Z1 ,

(−)

(−)

along Z2 .

(+)

In particular, (+) (−) A12 + A21 = 1,

.

(+) (−) A21 + A12 = 1.

(+) Proof First we check that A12 is a projection:

( .

(+) )2

A12

= ||1 ϒ −1 ||2 ||1 ϒ −1 ||2 (+)

(+)

(+)

= ||1 ϒ −1 (||2 ||1 (+)

(+)

(+)

(+)

+ ||1 ||2 )ϒ −1 ||2 (−)

(−)

(+)

(+)

= A12 .

Moreover, (+)

(+)

(+)

A12 = ||1 (||2

.

+ ||1 )ϒ −1 = ϒ −1 (||1 (−)

(+)

(−)

(+)

+ ||2 )||2 .

(+) But (||2(+) + ||1(−) )ϒ −1 and ϒ −1 (||1(+) + ||2(−) ) are invertible. Hence R(A12 )= (+) (+) (+) (−) R(||1 ) and N(A12 ) = N(||2 ) = R(||2 ). This proves the statement of the (+) proposition about A12 . The remaining statements are proven analogously. u n

3 Evolutions on Hilbertizable Spaces In this section we investigate the concept of an evolution (family) in the Hilbertizable setting and without the additional pseudo-unitary structure, which will be added in later sections. Already in the present setting we can try to define

42

J. Derezi´nski and D. Siemssen

abstract versions of “forward/backward”, “Pauli–Jordan”, “particle/antiparticle”, “Feynman/anti-Feynman” propagators and derive their basic properties. The existence of the abstract version of the Feynman propagator (inverse) will depend on a certain property called “asymptotic complementarity”, which in general is not guaranteed to hold. In the next section we will see that this property automatically holds under some natural assumptions typical of QFT. Throughout the section, −∞ ≤ t− < t+ ≤ +∞. For brevity, we write I := ]t− , t+ [. Without limiting the generality, we will always assume that 0 ∈ I . Moreover, we will always assume that either t± are both finite or both infinite. The case where only one of t± is infinite can be deduced from the other cases. In typical situations to define an evolution one first introduces a time-dependent family of operators t |→ −iB(t) that generate R(t, s). The necessary and sufficient conditions for an operator B to generate an autonomous evolution or, what is equivalent, a strongly continuous one-parameter group are well-known and relatively simple. In the non-autonomous case, there exist various relatively complicated theorems describing sufficient conditions. Unfortunately, it seems that a complete theory on this subject is not available. In the first part of this section we will avoid discussing the topic of generators of a non-autonomous evolutions apart from heuristic remarks. We will treat the evolution as given. The Cauchy data operator M := ∂t +iB(t) will be just a heuristic concept, without a rigorous meaning. However, “bisolutions” and “inverses of M” will be rigorously defined. They will be the main topic of this section. In Sect. 3.6 we describe a possible approach to the generation of non-autonomous evolutions based on a theorem of Kato.

3.1 Concept of an Evolution Definition 3.1 Let W be a Banach space. We say that the two-parameter family I × I e (t, s) |→ R(t, s) ∈ B(W)

.

(29)

is a strongly continuous evolution (family) on W if for all r, s, t ∈ I , we have the identities R(t, t) = 1,

.

R(t, s)R(s, r) = R(t, r),

(30)

and the map (29) is strongly continuous. One can also consider evolutions parametrized by the closed interval I cl := [t− , t+ ] instead of I , with the obvious changes in the definition.

An Evolution Equation Approach to Linear Quantum Field Theory

43

Note also that Definition 3.1 involves both forward and backward evolution, since we do not assume t ≥ s in R(t, s). In other words, this definition is a generalization of a one-parameter group instead of a one-parameter semigroup.

3.2 Generators of Evolution Until the end of this section we consider a strongly continuous evolution R(t, s), t, s ∈ I , on a Hilbertizable space W. If R(t, s) = R(t − s, 0) for all t, s, t − s, 0 ∈ I , we say that the evolution is autonomous. An autonomous evolution can always be extended to R × R in the obvious way. Setting R(t) := R(t, 0), we obtain a strongly continuous oneparameter group. As we have already mentioned, we can then write R(t) = e−itB , where B is a certain unique, densely defined, closed operator called the generator of R. For non-autonomous evolutions, the concept of a generator is understood only under some special assumptions. Heuristically, the operator-valued function I e t |→ B(t) is called the (time-dependent) generator of R(t, s) if ( ) B(t) := i∂t R(t, s) R(s, t).

.

Note that the evolution should satisfy in some sense i∂t R(t, s)v = B(t)R(t, s)v,

.

−i∂s R(t, s)v = R(t, s)B(s)v. A possible rigorous meaning of the concept of a time-dependent generator will be discussed in Sect. 3.6.

3.3 Bisolutions and Inverses of the Cauchy Data Operator Introducing the (heuristic) generator B(t), we can consider the (still heuristic) Cauchy data operator M := ∂t + iB(t).

.

Let Cc (I, W) denote continuous compactly supported functions from the open interval I to W. We will say that an operator E is a bisolution resp. an inverse or Green’s operator of M if it is maps Cc (I, W) → C(I, W) and satisfies

44

J. Derezi´nski and D. Siemssen

(heuristically) ( ) ( ) ∂t + iB(t) Ew = 0, E ∂t + iB(t) v = 0, . ( ) ( ) ∂t + iB(t) Ew = w, E ∂t + iB(t) v = v.

.

resp.

(31a) (31b)

for all w ∈ Cc (I, W) and for v in an appropriate domain inside C(I, W). (Note that M is an example of an unbounded operator, hence problems with its domain are not surprising.) A possible rigorous version of (31a) and (31b) will be given in Sect. 3.7. Note that all the definitions of inverses and bisolution that we will give below will be rigorous. Yet, for the time being, they will satisfy the conditions (31a) or (31b) only on a heuristic level. The following definition introduces the two most natural inverses and the most natural bisolution: Definition 3.2 Define the operators E • : Cc (I, W) → C(I, W) (E • w)(t) :=

f

E • (t, s)w(s) ds,

.

• = PJ, ∨, ∧,

(32)

I

by their temporal integral kernels E PJ (t, s) := R(t, s),

.

E ∨ (t, s) := θ (t − s)R(t, s), E ∧ (t, s) := −θ (s − t)R(t, s). E PJ is called the Pauli–Jordan bisolution and E ∨ , E ∧ are called the forward resp. backward inverse. Jointly, we call them classical propagators. Clearly, we have E PJ = E ∨ − E ∧ ,

.

(34)

which is analogous to (2a). If I is finite, then the operators E ∨ , E ∧ , E PJ can be extended to bounded operators on the Hilbertizable space L2 (I, W) = L2 (I ) ⊗ W. If I = R, typically they are not bounded on L2 (R, W). However, if R(t, s) is uniformly bounded (which we will typically assume), then E ∨ , E ∧ , E PJ are bounded as operators −s L2 (I, W) → s L2 (I, W) for s > 12 .

An Evolution Equation Approach to Linear Quantum Field Theory

45

3.4 Bisolutions and Inverses Associated with Involutions Let S+ and S− be two bounded involutions, (−)

(+)

(−)

(||− , ||− ),

.

(+)

(||+ , ||+ )

the corresponding two pairs of complementary projections and (−)

(+)

(−)

(Z− , Z− ),

.

(+)

(Z+ , Z+ ).

the corresponding two pairs of complementary subspaces (see Sect. 2.8). For finite t+ , t− , we set S± (t) = R(t, t± )S± R(t± , t),

.

(+)

(+)

||± (t) := R(t, t± )||± R(t± , t),

(+)

(+)

Z± (t) := R(t, t± )Z± .

||± (t) := R(t, t± )||± R(t± , t), Z± (t) := R(t, t± )Z± ,

(−)

(−)

(−)

(−)

If t± = ±∞, we assume that there exists T such that on ] − ∞, −T [ and ]T , ∞[ the evolution is autonomous, that is, there exist B± such that ±t, ±s > T ⇒ R(t, s) = e−i(t−s)B± .

.

We also assume that e−i(t−s)B± S± = S± e−i(t−s)B± .

(35)

.

We then set S± (t) = R(t, s)S± R(s, t),

.

(+)

(+)

||± (t) := R(t, s)||± R(s, t),

(+)

(+)

Z± (t) := R(t, s)Z± ,

||± (t) := R(t, s)||± R(s, t), Z± (t) := R(t, s)Z± ,

(−)

(−)

(−)

(−)

where ±s > T is arbitrary. Definition 3.3 For any t, s ∈ I , we define (+)

(+)

(+)

E± (t, s) := R(t, s)||± (s) = ||± (t)R(t, s)..

(36a)

(−) (−) (−) E± (t, s) := R(t, s)||± (s) = ||± (t)R(t, s).

(36b)

.

46

J. Derezi´nski and D. Siemssen (+)

(−)

Define the operators E± and E± : Cc (I, W) → C(I, W) by their temporal (+) (+) (−) integral kernels (36). E± are called the ||± -in/out bisolutions and E± are called (−) the ||± -in/out bisolutions. Clearly, we have (+) (−) E PJ = E± + E± ,

(37)

.

which is analogous to (2b). (+) (−) Definition 3.4 We say that asymptotic complementarity holds for (Z+ , Z− ) if for some (and thus for all) t ∈ I ,

( (+) ) (−) Z+ (t), Z− (t)

.

is a pair of complementary subspaces of W. Suppose that asymptotic complemen(+) (−) tarity holds for (Z+ , Z− ). Then we define (+)

(−)

AF(+) (t), the projection onto Z+ (t) along Z− (t), .

(38a)

(−) (+) AF(−) (t), the projection onto Z− (t) along Z+ (t).

(38b)

.

It is the pair of projections associated to the direct sum decomposition W = (+) (−) Z+ (t) ⊕ Z− (t). We also define the operator E F : Cc (I, W) → C(I, W) as in (32) by its temporal integral kernel E F (t, s) := θ (t − s)R(t, s)AF(+) (s) − θ (s − t)R(t, s)AF(−) (s).

.

(+)

(−)

E F is called the Z+ -out Z− -in inverse. The following definition is fully analogous to the previous one: (−) (+) Definition 3.5 We say that asymptotic complementarity holds for (Z+ , Z− ) if for some (and thus for all) t ∈ I ,

( (−) ) (+) Z+ (t), Z− (t)

.

is a pair of complementary subspaces of W. Suppose that asymptotic complemen(−) (+) tarity holds for (Z+ , Z− ). Then we define (−)

(+)

(39a)

(+)

(−)

(39b)

AF(−) (t), the projection onto Z+ (t) along Z− (t), .

.

AF(+) (t), the projection onto Z− (t) along Z+ (t).

An Evolution Equation Approach to Linear Quantum Field Theory

47

It is the pair of projections associated to the direct sum decomposition W = (−) (+) Z+ (t) ⊕ Z− (t). We also define the operator E F : Cc (I, W) → C(I, W) as in (32) by its temporal integral kernel E F (t, s) := θ (t − s)R(t, s)AF(−) (s) − θ (s − t)R(t, s)AF(+) (s).

.

(−)

(+)

E F is called the Z+ -out Z− -in inverse. We clearly have R(t, s)AF(±) (s) = AF(±) (t)R(t, s),

.

R(t, s)AF(±) (s) = AF(±) (t)R(t, s), Heuristically, E F and E F are inverses in the sense of (31b). If I is finite, then they are bounded on L2 (I, W). Asymptotic complementarity is trivially satisfied for the pairs (W, {0}) and ({0}, W). The corresponding inverses are simply E ∨ and E ∧ , defined in Definition 3.2. Remark 3.6 Let us explain the notation that we are using: + and − without parenthesis are related to t+ or “out”, resp. t− or “in”. (+) and (−) inside parenthesis denote the “particle space”, resp. the “antiparticle space”.

3.5 Identities Involving Bisolutions and Inverses We make all assumptions of Sect. 3.4. In addition we suppose that asymptotic complementarity holds for both pairs of subspaces. Using Proposition 2.7, we have the following: Proposition 3.7 The projections defined in (38) and (39) are given explicitly by AF(+) (t) = ||+ (t)ϒ(t)−1 ||− (t), AF(−) (t) = ||− (t)ϒ(t)−1 ||+ (t), (+)

.

and

(+)

(−)

(−)

AF(+) (t) = ||− (t)ϒ(t)−1 ||+ (t), AF(−) (t) = ||+ (t)ϒ(t)−1 ||− (t), (+)

(+)

(−)

where ϒ(t) is the invertible operator defined by ϒ(t) :=

.

) 1( 2 + S− (t)S+ (t) + S+ (t)S− (t) . 4

(−)

48

J. Derezi´nski and D. Siemssen

Observe that ϒ(t) has the properties ϒ(t) = R(t, s)ϒ(s)R(s, t),

.

ϒ(t)S± (t) = S± (t)ϒ(t). Thus we have defined the inverses E ∨ , E ∧ , E F , E F and the bisolutions E PJ , (−) (+) (−) E− , E+ , E+ . They satisfy the relations (34) and (37). As described in the introduction, in the setting of the Minkowski space and, more generally, of a stationary spacetime, they satisfy several other identities. These identities do not hold in general. Instead, we have: (+) E− ,

Proposition 3.8 The following three identities hold: (+)

(−)

(+)

(−)

(E+ − E+ − E− + E− )(t, s) = R(t, s)(S+ − S− )(s); .

.

(E F + E F − E ∨ − E ∧ )(t, s) =

(40)

1 R(t, s)ϒ(s)−1 [S− (s), S+ (s)]; 4

1 (+) (−) (+) (−) (E F − E F )(t, s) − (E+ − E+ + E− − E− )(t, s) 2 . ] [ 1 = R(t, s)ϒ(s)−1 S+ (s) − S− (s), [S+ (s), S− (s)] . 8

(41)

(42)

Proof Equation (40) is straightforward. Obviously, the difference of two inverses is a bisolution. The temporal kernels of the following bisolutions have very simple forms: .

(E F − E ∧ )(t, s) = R(t, s)AF(+) (s), .

(43)

(E F − E ∨ )(t, s) = −R(t, s)AF(−) (s), .

(44)

(E F − E ∧ )(t, s) = R(t, s)AF(−) (s), .

(45)

(E F − E ∨ )(t, s) = −R(t, s)AF(+) (s).

(46)

Taking the sum of (43) and (46) we obtain (E F + E F − E ∨ − E ∧ )(t, s) = R(t, s)(AF(+) − AF(−) )(s)

.

= R(t, s)ϒ(s)−1 (||+ ||− − ||− ||+ )(s), (+)

(+)

(+)

(+)

An Evolution Equation Approach to Linear Quantum Field Theory

49

which yields (41). Taking the difference of (43) and (45) we obtain the following identities: E F (t, s) − E F (t, s) = R(t, s)(AF(+) − AF(−) )(s)

.

( ) 1 R(t, s)ϒ(s)−1 (1 + S+ )(1 + S− ) − (1 − S+ )(1 − S− ) (s) 4 1 = R(t, s)ϒ(s)−1 (S+ + S− )(s) 2 (1 1 = R(t, s) (S+ + S− )(s) + ϒ(s)−1 8 2 ) × (S+ + S− − S− S+ S− − S+ S− S+ )(s) ,

=

u n

This yields (42).

Note that for the standard choice of propagators in a stationary QFT the right hand sides of (40), (42) and (41) vanish. It is easy to see that they do not have to vanish in general. The identities (41) and (42) simplify in some important situations: Proposition 3.9 Assume that asymptotic complementarity holds. Further, suppose that for any (and hence for all) t ∈ I S− (t)S+ (t) = S+ (t)S− (t).

.

(47)

Then EF + EF = E∨ + E∧, .

.

EF − EF =

1 (+) (−) (+) (−) (E − E+ + E− − E− ). 2 +

(48a) (48b)

Equation (47) is satisfied in a number of interesting situations. In particular, if the evolution is autonomous, it is natural to assume that S+ = S− =: S• , requiring that (+) (−) it commutes with the generator B. Then E± and E± collapse to two bisolutions: (+)

(+)

(−)

(−)

E+ = E− =: E (+) ,

.

E+ = E− =: E (−) . Thus, in the autonomous case, the identity (48a) holds and (48b) can be rewritten as E F − E F = E (+) − E (−) .

.

50

J. Derezi´nski and D. Siemssen

3.6 Almost Unitary Evolutions on Hilbertizable Spaces So far we discussed generators of an evolution only in a heuristic way. In this subsection we will describe a setting that allows us to make this concept precise. In view of our applications, we will introduce generators of evolutions on Hilbertizable spaces that one might call almost unitary evolutions. For the remainder of this section, we consider the scale of Hilbertizable spaces Wα , α ∈ [0, 1], as in Sect. 2.4. Theorem 3.10 (cf. Thm. C.10 of [9]) Let {B(t)}t∈I be a family of densely defined, closed operators on W0 . Suppose that the following conditions are satisfied: ( ) (a) W1 ⊂ D B(t) so that B(t) ∈ B(W1 , W0 ) and I e t |→ B(t) ∈ B(W1 , W0 ) is norm continuous. (b) For every t ∈ I , scalar products (· | ·)0,t and (· | ·)1,t compatible with W0 resp. W1 have been chosen. Denote the corresponding Hilbert spaces W0,t and W1,t . ˜ (c) B(t) is self-adjoint in the sense of W0,t and the part B(t) of B(t) in W1 is self-adjoint in the sense of W1,t . (d) For a positive C ∈ L1loc (I ) and all s, t ∈ I , f ||w||0,t ≤ ||w||0,s exp| f ||v||1,t ≤ ||v||1,s exp|

t

C(r) dr|,

.

s t

C(r) dr|. s

Then there exists a unique family of bounded operators {R(t, s)}s,t∈I , on W0 with the following properties: (i) For all r, s, t ∈ I , we have the identities (30). (ii) R(t, s) is W0 -strongly continuous. It preserves W1 and is W1 -strongly continuous. Hence it preserves Wα , 0 ≤ α ≤ 1, and is Wα -continuous. Moreover, f ||R(t, s)||α,t ≤ exp|

t

2C(r) dr|,

.

s, t ∈ I.

s

(iii) For all w ∈ W1 and s, t ∈ I , i∂t R(t, s)w = B(t)R(t, s)w,

.

−i∂s R(t, s)w = R(t, s)B(s)w, where the derivatives are in the strong topology of W0 . We call {R(t, s)}s,t∈I the evolution generated by B(t).

An Evolution Equation Approach to Linear Quantum Field Theory

51

Note that, if t± are finite, the above theorem remains true if we everywhere replace I with I cl = [t− , t+ ], and L1loc ]t− , t+ [ with L1 [t− , t+ ], provided that we consider only the right/left-sided derivatives at t− /t+ . Sometimes it is convenient to use an easy generalization of Theorem 3.10, where the generator is perturbed by a bounded operator. It is also proven in [9]. Theorem 3.11 Suppose that {B0 (t)}t∈I satisfies all the assumptions of Theorem 3.10 and I e t |→ V (t) ∈ B(W0 ) is a norm continuous family of operators. Let B(t) := B0 (t) + V (t). Then there exists a unique family of bounded operators {R(t, s)}s,t∈I , on W0 satisfying all properties of Theorem 3.10. If we want to pass to the real case, we use the following obvious fact: Proposition 3.12 If W0 has a conjugation which preserves W1 , then R(t, s) is real for t, s ∈ I if and only if its generator B(t) is anti-real for all t ∈ I .

3.7 Rigorous Concept of a Bisolution and Inverse Under the assumptions of Theorem 3.10 it is possible to propose a rigorous version of a concept of a (left) bisolution and a (left) inverse, and check that they are satisfied by the E • that we have constructed. Proposition 3.13 1. Let v ∈ Cc (I, W1 ) ∩ Cc1 (I, W0 ). Then for E • = E PJ , E− , E− , E+ , E+ we have (+)

) ( E • ∂t + iB(t) v = 0,

.

and for E • = E ∧ , E ∨ , E F , E F we have ) ( E • ∂t + iB(t) v = v.

.

2. Let w ∈ Cc (I, W1 ). Then for E • = E PJ we have ( ) ∂t + iB(t) E • w = 0,

.

and for E • = E ∧ , E ∨ we have ( .

) ∂t + iB(t) E • w = w.

(−)

(+)

(−)

52

J. Derezi´nski and D. Siemssen

Proof Let us prove (1) for E F : f t ( ( ) ) ( ) F ∂s + iB(s) v (t) = . E AF(+) (t)R(t, s) ∂s + iB(s) v(s) ds t−

f

t+

−

( ) AF(−) (t)R(t, s) ∂s + iB(s) v(s) ds

t

f

t

=

( ) AF(+) (t)∂s R(t, s)v(s) ds

t−

f

t+

−

( ) AF(−) (t)∂s R(t, s)v(s) ds

t

=A

F(+)

(t)v(t) + AF(−) (t)v(t)

= v(t). Let us prove (2) for E ∨ : (( .

) ) ( ) ∂t + iB(t) E ∨ w (t) = ∂t + iB(t)

f

t

R(t, s)w(s) ds t−

f

= R(t, t)w(t) +

t

( ) ∂t + iB(t) R(t, s)w(s) ds

t−

= w(t). u n

4 Evolutions on Pseudo-Unitary Spaces Pre-pseudo-unitary spaces are Hilbertizable spaces with a distinguished bounded Hermitian form. They can be viewed as complexifications of pre-symplectic spaces—real spaces with a distinguished bounded antisymmetric form. In practice, one usually assumes that the Hermitian or pre-symplectic form is non-degenerate. Then these spaces are called pseudo-unitary, resp. symplectic. A transformation preserving the structure of a pseudo-unitary, resp. symplectic space is called pseudo-unitary, resp. symplectic. Krein spaces constitute an especially well-behaved class of pseudo-unitary spaces. The Krein structure adds interesting new features to bisolutions and inverses of M := ∂t + iB(t). The most interesting new fact is the automatic validity of asymptotic complementarity if the “in particle space” is maximally positive and the “out antiparticle space” is maximally negative. This implies the existence of the Feynman and anti-Feynman inverses.

An Evolution Equation Approach to Linear Quantum Field Theory

53

We will also discuss generators of 1-parameter groups preserving the pseudounitary structure, called pseudo-unitary generators. In particular, we will introduce the so-called stable pseudo-unitary generators, which possess positive Hamiltonians. They are distinguished both on physical and mathematical grounds. Especially good properties have strongly stable pseudo-unitary generators, whose positive Hamiltonians are bounded away from zero. We discuss two constructions of a pseudo-unitary evolution R(t, s). starting from a time-depenent generator I e t |→ B(t). The first construction uses a nested pair of Hilbertizable spaces W1 ⊂ W0 , where W0 is equipped with a Hermitian form Q, and B(t) : W1 → W0 . The second construction uses a nested pair of Hilbertizable spaces W 1 ⊂ W− 1 equipped with a pairing Q and B(t) : W 1 → W− 1 . The 2 2 2 2 pseudo-unitary space W0 is obtained by interpolation. Later on we will use both constructions.

4.1 Symplectic and Pseudo-Unitary Spaces Definition 4.1 A pre-symplectic space is a real vector space Y equipped with an antisymmetric form ω, called a pre-symplectic form Y × Y e (v, w) |→ ∈ R.

.

If ω is non-degenerate, then Y is called a symplectic space. If the dimension of Y is infinite, we assume that Y is Hilbertizable and ω is bounded. Definition 4.2 We will say that a bounded invertible operator R on a pre-symplectic space (Y, ω) preserves ω if = .

.

If in addition ω is non-degenerate, we will say that R is symplectic. Definition 4.3 A pre-pseudo-unitary space is a complex vector space W equipped with a Hermitian form Q W × W e (v, w) |→ (v | Qw) ∈ C.

.

If Q is non-degenerate, then W is called a pseudo-unitary space. If the dimension of W is infinite, we assume that W is Hilbertizable and Q is bounded. Definition 4.4 We will say that a bounded invertible operator R on (W, Q) preserves Q if (Rv | QRw) = (v | Qw).

.

(49)

54

J. Derezi´nski and D. Siemssen

If in addition Q is non-degenerate, we will say that R is pseudo-unitary. Note that even if one starts from a real (pre-)symplectic space, it is useful to consider its complexification. In Sect. 2.6 we described how to pass from the real to complex formalism. In this section we will treat the complex formalism as the standard one. In the context of a pre-pseudo-unitary space treated as the complexification of a pre-symplectic space it is natural to consider conjugations that anti-preserve (and not preserve) Q, that is (v | Qw) = −(v | Qw), see (26). Definition 4.5 An antilinear involution v |→ v on a pseudo-unitary space (W, Q) which anti-preserves Q and such that there exists a compatible scalar product (· | ·)• satisfying (v | w)• = (v | w)• ,

.

will be called a conjugation on (W, Q). As in (24), given a conjugation we can define the real subspace WR of W. The restriction of Q to WR is clearly a pre-symplectic space.

4.2 Admissible Involutions and Krein Spaces Let (W, Q) be a pre-pseudo-unitary space. Definition 4.6 A (bounded) involution S• on W will be called admissible if it preserves Q and the scalar product (v | w)• := (v | QS• w) = (S• v | Qw)

.

(50)

is compatible with the Hilbertizable structure of W. Sometimes we will write W• to denote the space W equipped with the scalar product (50). Definition 4.7 A pre-pseudo-unitary space is called a Krein space if it possesses an admissible involution. Clearly, every Krein space is pseudo-unitary. Proposition 4.8 If S• is an admissible involution on (W, Q), then S• is self-adjoint and unitary on W• . For any admissible involution S• , we define the corresponding particle projection ||•(+) and particle space Z•(+) , as well as the antiparticle projection ||•(−) and

An Evolution Equation Approach to Linear Quantum Field Theory

55

antiparticle space Z(−) • , as in (27). Note the following relations: (+) (−) (−) (v | w)• = (||(+) • v||• w)• + (||• v | ||• w)• ,

.

(+) (−) (−) (v | Qw) = (||(+) • v | ||• w)• − (||• v | ||• w)• .

Let us make an additional comment on Krein spaces with conjugations. Proposition 4.9 Suppose that (W, Q) is a Krein space with conjugation. If S• is an admissible anti-real involution, then iS• is real and we have (+)

||•

.

= ||(−) • ,

(−) Z(+) • = Z• ,

(+) so that W = Z(+) • ⊕ Z• .

4.3 Basic Constructions in Krein Spaces Let (W, Q) be a Krein space. Definition 4.10 Let Z ⊂ W. We define its Q-orthogonal complement as follows: Z⊥Q := {w ∈ W | (w | Qv) = 0, v ∈ Z}.

.

If (· | ·)• is a scalar product, we also have the •-orthogonal complement Z⊥• := {w ∈ W | (w | v)• = 0, v ∈ Z}.

.

Proposition 4.11 1. If Z is a closed subspace, then so is Z⊥Q , and (Z⊥Q )⊥Q = Z. ⊥Q ⊥Q 2. If Z1 , Z2 are complementary in W, then so are Z1 , Z2 . 3. If (v | Qw) = (v | S• w)• (equivalently, if S• is admissible), then Z⊥Q = S• Z⊥• . Definition 4.12 Let A ∈ B(W). We define its Q-adjoint as follows: (A∗Q v | Qw) = (v | QAw),

.

v, w ∈ W.

If (· | ·)• is a scalar product, we also have the •-adjoint of A (A∗• v | w)• = (v | Aw)• ,

.

v, w ∈ W.

Proposition 4.13 1. If (v | Qw) = (v | S• w)• (equivalently, if S• is admissible), then A∗Q = S• A∗• S• .

56

J. Derezi´nski and D. Siemssen

2. Let (||(+) , ||(−) ) be a pair of complementary projections. Then (||(+)∗Q , ||(−)∗Q ) is also a pair of complementary projections and R(||(±)∗Q ) = N(||(∓)∗Q ) = R(||(∓) )⊥Q = N(||(±) )⊥Q .

.

Proposition 4.14 Let Z•(+) be a closed subspace of W. Set Z•(−) := Z•(+)⊥Q . The following conditions are equivalent: 1. Z•(+) satisfies

and

.

v ∈ Z•(+) ⇒ (v | Qv) ≥ 0, .

(51)

v ∈ Z•(−) ⇒ (v | Qv) ≤ 0.

(52)

2. Z•(+) is a maximal closed subspace of W with the property (51). 3. The spaces Z•(+) and Z•(−) are complementary, and if (||•(+) , ||•(−) ) is the corresponding pair of projections, then S• := ||•(+) − ||•(−) is an admissible involution. Definition 4.15 If Z•(+) satisfies the conditions of Proposition 4.14, then it is called a maximally positive subspace. Analogously we define maximally negative subspaces.

4.4 Pairs of Admissible Involutions Let S1 , S2 be a pair of admissible involutions on a Krein space (W, Q). We will describe some structural properties of such a pair. Let ||i(+) , ||i(−) , Zi(+) , Zi(−) , i = 1, 2, be defined as in (27). Set K := S2 S1 ,

.

(+) 1 − K

c := ||1

(−)

where c is interpreted as an operator from Z1

1+K

(−)

||1 ,

(+)

to Z1 .

Proposition 4.16 K is pseudo-unitary and invertible. K is positive and ||c|| < 1 with respect to (· | ·)1 and (· | ·)2 . We have S1 KS1 = S2 KS2 = K −1 , .

.

S1

1−K 1−K 1−K S1 = S2 S2 = − . 1+K 1+K 1+K

(53) (54)

An Evolution Equation Approach to Linear Quantum Field Theory

57

Proof K is pseudo-unitary as the product of two pseudo-unitary transformations. The inequality (v | Kv)1 = (S1 v | QS2 S1 v) = (S1 v | S1 v)2 ≥ a(S1 v | S1 v)1 = a(v | v)1

.

with a > 0 shows the positivity of K with respect to (· | ·)1 and its invertibility. This 1−K || < 1. Hence ||c|| < 1. implies || 1 +K The identities (53) and (54) are direct consequences of the definition of K and S12 = S22 = 1. u n Proposition 4.17 Using the decomposition W = Z1(+) ⊕ Z1(−) we have .

[ ] 1−K 0 c = ∗ ,. (55a) c 0 1+K ] [ (1 + cc∗ )(1 − cc∗ )−1 −2c(1 − c∗ c)−1 , K =. (55b) −2c∗ (1 − cc∗ )−1 (1 + c∗ c)(1 − c∗ c)−1 ] [ ] [ c(1 − c∗ c)−1 10 (1 − cc∗ )−1 (+) (+) ,. ||1 = (55c) , ||2 = −c∗ (1 − cc∗ )−1 −c∗ c(1 − c∗ c)−1 00 ] [ ] [ 00 −cc∗ (1 − cc∗ )−1 −c(1 − c∗ c)−1 (−) (−) ,. ||1 = (55d) , ||2 = c∗ (1 − cc∗ )−1 (1 − c∗ c)−1 01 ] [ [ ] 2c(1 − c∗ c)−1 1 0 (1 + cc∗ )(1 − cc∗ )−1 . S1 = , S2 = −2c∗ (1 − cc∗ )−1 −(1 + c∗ c)(1 − c∗ c)−1 0 −1 (55e)

Moreover, if S1 is an admissible involution and ||c|| < 1, then S2 given as in (55e) is an admissible involution. Proof Equation (54) implies (55a). From the definition of c (or (55a)) we obtain [

1 .K = −c∗ [ 1 = −c∗

]−1 1 c c∗ 1 ] ][ (1 − cc∗ )−1 −c(1 − c∗ c)−1 −c . 1 −c∗ (1 − cc∗ )−1 (1 − c∗ c)−1

−c 1

][

This yields (55b). From S2 = KS1 we obtain (55c), (55d) and (55e).

u n

The involutions S1 and S2 correspond to the pairs of complementary subspaces (+) (−) (+) (−) (Z1 , Z1 ), resp. (Z2 , Z2 ). The following proposition implies the existence of two other direct sum decompositions. This fact plays an important role in the construction of the (in-out) Feynman inverse.

58

J. Derezi´nski and D. Siemssen (+)

(−)

(+)

(−)

Proposition 4.18 The pairs of subspaces (Z1 , Z2 ) and (Z2 , Z1 ) are complementary. Here are the corresponding projections: (+)

[ ] 1c (+) (+) = ||1 ϒ −1 ||2 00 [ ] 0 −c (−) (−) = = ||2 ϒ −1 ||1 0 1 [ ] 1 0 (+) (+) = = ||2 ϒ −1 ||1 −c∗ 0 ] [ 0 0 (−) (−) = ||1 ϒ −1 ||2 = ∗ c 1

(+)

along Z2 ,

(−)

along Z1 ,

(+)

along Z1 ,

(−)

along Z2 ,

A12 =

projects onto Z1

(−)

projects onto Z2

.

A21

(+)

A21

(−)

A12

projects onto Z2 projects onto Z1

(−)

(+)

(−)

(+)

where ϒ

.

−1

] 4 4 0 1 − cc∗ = . = = 0 1 − c∗ c (2 + S2 S1 + S1 S2 ) (1 + K)(1 + K −1 ) [

u n

Proof We apply Proposition 2.6. We can reformulate Proposition 4.18 as follows.

Proposition 4.19 Let Z1 be an maximally positive subspace and Z2 an maximally negative space. Then they are complementary. Proof By Proposition 4.14 there exist admissible involutions S1 and S2 such that u n Z1 = Z1(+) and Z2 = Z2(−) . Hence, it suffices to apply Proposition 4.18.

4.5 Pseudo-Unitary Generators Let (W, Q) be a pre-pseudo-unitary space. Definition 4.20 We say that a densely defined operator B on W infinitesimally preserves Q if B is the generator of a one-parameter group e−itB on W and (v | QBw) = (Bv | Qw),

.

v, w ∈ D(B).

(56)

If in addition Q is non-degenerate, then we will say that B is a pseudo-unitary generator. The quadratic form defined by (56) will be called the energy or Hamiltonian quadratic form of B on D(B). Proposition 4.21 Let B be a generator of a one-parameter group on W. Then e−itB , t ∈ R, preserves Q if and only if B infinitesimally preserves Q.

An Evolution Equation Approach to Linear Quantum Field Theory

59

Proof Let us show ⇐. Assume first that v, w ∈ D(B). Then .

d −itB (e v | Qe−itB w) = i(Be−itB v | Qe−itB w) − i(e−itB v | QBe−itB w) = 0. dt

Hence (e−itB v | Qe−itB w) = (v | Qw).

.

(57)

By the density of D(B) and the boundedness of Q and e−itB , (57) extends to the whole W. In the proof of ⇒ we use the above arguments in the reverse order (with the exception of the density argument, which is not needed). u n The following proposition describes a large class of pseudo-unitary transformations and pseudo-unitary generators on Krein spaces. Proposition 4.22 Suppose that (W, Q) is a Krein space and S• is an admissible involution with the corresponding scalar product (· | ·)• . If B is a densely defined operator on W, self-adjoint in the sense of W• and commuting with S• , then B is a pseudo-unitary generator on (W, Q) in the sense of Definition 4.20. Proof Clearly, e−itB is a unitary operator on W• commuting with S• . Therefore, it u n is a pseudo-unitary transformation (see Definition 4.4). Definition 4.23 A densely defined operator B on a Krein space (W, Q) is called a stable pseudo-unitary generator if it is similar to self-adjoint, N(B) = {0}, and sgn(B) is an admissible involution. B is called a strongly stable pseudo-unitary generator if in addition it is invertible. In other words, a stable pseudo-unitary generator has a positive Hamiltonian and a strongly stable generator has a positive Hamiltonian bounded away from zero.

4.6 Bisolutions and Inverses Let (W, Q) be a Krein space. Then naturally L2 (I, W) is also a Krein space with the Hermitian form f ( ) .(v | Qw) := v(t) | Qw(t) dt, I

and compatible scalar products f (v | w)• :=

(

.

I

) v(t) | w(t) • dt.

60

J. Derezi´nski and D. Siemssen

Let {R(t, s)}t,s∈I be a strongly continuous pseudo-unitary evolution on (W, Q). We denote by B(t) the (heuristic) generator of R(t, s). Recall that in Sect. 3.3 we considered the (heuristic) Cauchy data operator M = ∂t + iB(t). Note that M = ∂t + iB(t) is (heuristically) anti-Q-Hermitian on L2 (I, W ). We will give a rigorous version of this statement a little later, in (59). In Sect. 3.3 we introduced various inverses and bisolutions of M, considered as operators Cc (I, W) → C(I, W). In this subsection we add the Krein structure to the picture. First, as in Sect. 3.3, we define the Pauli–Jordan bisolution E PJ , and the forward and backward inverses E ∨ , resp. E ∧ . ∨ Proposition 4.24 E PJ is Q-Hermitian ( and ) the Q-adjoint of E is contained in ∧ −E . More precisely, for v, w ∈ Cc I, W we have

(v | QE PJ w) = (E PJ v | Qw),

.

(v | QE ∧ w) = −(E ∨ v | Qw). Consider now non-classical bisolutions and inverses. If t± are finite, let us select two arbitrary admissible involutions S+ , S− . If t± = ±∞, recall that we assumed that for large ±t, ±s we have R(t, s) = e−i(t−s)B± . We assume that B± are stable pseudo-unitary generators and we set ( ) S± := sgn B± .

(58)

.

Note that (58) implies that (35) is satisfied and S± are admissible. (+) (−) (+) (−) (+) (−) Define ||± , ||± , Z± , Z± , as well as E± , E± , as in Definition 3.3. Note (+) (+) (−) (−) that, for any t ∈ I , Z+ (t) and Z− (t) are maximally positive and Z+ and Z− are maximally negative. This immediately implies (+)

(−)

(+)

(−)

Proposition 4.25 E± and E± are Q-Hermitian. E± is Q-positive and E± is Q-negative. More precisely, for v, w ∈ Cc (I, W) we have (+)

(+)

(v | QE± v) ≥ 0,

(−)

(−)

(v | QE± v) ≤ 0.

(E± w | Qv) = (w | QE± v),

.

(E± w | Qv) = (w | QE± v),

(+) (−)

Proof Consider for definiteness the case of finite t± . It holds ( (+) (+) (+) ) (v | QE± v) = ||± w | Q||± w , ( (−) (−) (−) ) (v | QE± v) = ||± w | Q||± w ,

.

where w =

f I

R(t± , t)v(t) dt.

u n

An Evolution Equation Approach to Linear Quantum Field Theory

61

What is more remarkable, under the present assumptions by Proposition 4.11 ( (+) (−) ) (2) asymptotic complementarity holds automatically for both Z+ , Z− and ( (−) (+) ) F F Z+ , Z− . Therefore, we can define the inverses E and E , as in Definition 3.4 and 3.5. F F Proposition ( 4.26) The Q-adjoint of E is contained in −E . More precisely, for v, w ∈ Cc I, W we have

(E F w | Qv) = −(w | QE F v).

.

Proof By (49) we have R(s, t)∗Q = R(t, s). Clearly, Z± (t)⊥Q = Z± (t), and hence by Proposition 4.11 (+)

AF(+) (t)∗Q = AF(+) (t),

.

(−)

AF(−) (t)∗Q = AF(−) (t).

Now, f

(

(E F w | Qv) =

.

) w(t) | QE F (s, t)∗Q v(s) dt ds

I

and E F (s, t)∗Q = θ (s − t)AF(+) (t)∗Q R(s, t)∗Q − θ (t − s)AF(−) (t)∗Q R(s, t)∗Q

.

= θ (s − t)AF(+) (t)R(t, s) − θ (t − s)AF(−) (t)R(t, s) = θ (s − t)R(t, s)AF(+) (s) − θ (t − s)R(t, s)AF(−) (s) = −E F (t, s). u n (+) With the choice (58), the bisolutions E± are called the in/out positive frequency (−) bisolutions, the bisolutions E± are called the in/out negative frequency bisolutions, and the inverses E F , resp. E F are called the Feynman, resp. the anti-Feynman inverse.

4.7 The Cauchy Data Operator in the Krein Setting The Cauchy data operator M is the sum of two unbounded operators: ∂t and iB(t). Therefore, it is not easy to choose its domain. We will discuss two possible approaches to this question. In this subsection we will describe

62

J. Derezi´nski and D. Siemssen

the “operator approach”. Sects. 4.8 and 4.9 will discuss the “quadratic form approach”. Suppose that (W0 , Q) is a pre-pseudo-unitary space. Theorem 4.27 Suppose that I e t |→ B(t) is an operator on a Krein space W0 . Suppose that W1 is a Hilbertizable space densely and continuously embedded in W0 and all the assumptions of Theorem 3.10 are satisfied. In addition, assume that for all t ∈ I the operators B(t) infinitesimally preserve Q. Then the evolution R(t, s) on W0 preserves Q. Proof Recall that among the assumptions of Theorem 3.11 there is W1 ⊂ D(B(t)). Besides, for any w ∈ W1 this theorem implies that i∂t R(t, s)w = B(t)R(t, s)w.

.

Hence the above theorem follows by repeating the proof of Proposition 4.21, where we use W1 instead of D(B). u n Suppose that (W0 , Q) is a Krein space and I e t |→ B(t) is a family of pseudounitary generators satisfying the assumptions of Theorem 4.27. We can treat the Cauchy data operator M = ∂t + iB(t)

.

as a densely defined operator on the Krein space L2 (I, W0 ) with the domain Cc (I, W1 ) ∩ Cc1 (I, W0 ). Now we can give a rigorous meaning to its anti-QHermiticity: for v, w ∈ Cc (I, W1 ) ∩ Cc1 (I, W0 ), ( (( ) ( ) ) ) w | Q ∂t + iB(t) v = − ∂t + iB(t) w | Qv .

.

(59)

For the remaining part of this subsection we assume that I is finite and two admissible involutions S+ , S− have been chosen. Proposition 4.28 E F and E F extend to bounded operators on L2 (I, W0 ), their ranges are dense and their nullspaces are {0}. Proof The boundedness is obvious. By Proposition 3.13 (1) for any v Cc (I, W1 ) ∩ Cc1 (I, W0 ) we have

∈

( ) E F ∂t + iB(t) v = v.

.

Hence R(E F ) contains Cc (I, W1 ) ∩ Cc1 (I, W0 ), which is dense in L2 (I, W0 ). The same argument shows that R(E F ) is dense in L2 (I, W0 ). Now N(E F ) = R(E F∗Q )⊥Q = R(E F )⊥Q = {0}.

.

u n

An Evolution Equation Approach to Linear Quantum Field Theory

63

Thus by Proposition 4.28, for finite I we can define operators with dense domains M F := (E F )−1 ,

.

M F := (E F )−1 .

They satisfy (M F )∗Q = M F

.

and 0 belongs to their resolvent set.

4.8 Nested Pre-Pseudo-Unitary Pairs In this and the following subsection we describe the “quadratic form approach” to dynamics on pre-pseudo-unitary spaces. Such an approach usually requires weaker assumptions. In this approach the starting point is a nested pair of Hilbertizable spaces equipped with a Hermitian pairing. The pre-pseudo-unitary space is then obtained by interpolation. Let us describe this simple construction in detail. In the next subsection we will describe evolutions on such nested pairs. Definition 4.29 Let λ > 0. A nested pre-pseudo-unitary pair (W−λ , Wλ , Q) consists of a pair of Hilbertizable spaces W−λ , Wλ , where Wλ is densely and continuously embedded in W−λ and a Hermitian pairing, that is a sesquilinear form Wλ × W−λ e (v, w) |→ (v | Qw) ∈ C,

.

which is Hermitian on Wλ , i.e., (v | Qw) = (w | Qv),

v, w ∈ Wλ ,

.

(60)

and bounded, i.e., for some (hence all) compatible norms || · ||λ,• and || · ||−λ,• on Wλ , resp. W−λ there exists C• such that |(v | Qw)| ≤ C• ||v||λ,• ||w||−λ,• .

.

(61)

In what follows, let (W− 1 , W 1 , Q) be a nested pre-pseudo-unitary pair. (The 2

2

parameter 12 can be changed to any positive number, it is chosen here in view of our future applications.) Let Wλ , λ ∈ [− 12 , 21 ], be the Hilbertizable spaces obtained by interpolation from the nested pair (W 1 , W− 1 ). 2

2

64

J. Derezi´nski and D. Siemssen

Proposition 4.30 For λ ∈ [− 12 , 21 ], there exists a unique family of Hermitian pairings Qλ Wλ × W−λ e (v, w) |→ (v | Qλ w) ∈ C.

.

such that Q 1 = Q and if − 12 ≤ λ1 ≤ λ2 ≤ 12 , v ∈ W−λ1 ⊂ W−λ2 , w ∈ Wλ2 ⊂ 2 Wλ1 , then (v | Qλ1 w) = (v | Qλ2 w).

(62)

.

Proof By (61), Q can be viewed as a bounded map from W− 1 to the antidual of 2 W 1 . The antidual of W 1 coincides with W− 1 . Hence, Q ∈ B(W− 1 ). 2 2 2 2 The restriction of Q to W 1 , by (61) and (60) is a bounded map to the antidual 2 of W− 1 , which is W 1 . Hence Q ∈ B(W 1 ). 2

2

2

By interpolation, that is, Proposition 2.1 (2), for λ ∈ [− 12 , 12 ], the restriction of Q to Wλ is bounded. u n In what follows we drop the subscript λ from Qλ , which is allowed because of (62). In particular, for λ = 0, we obtain a bounded Hermitian form on W0 : W0 × W0 e (v, w) |→ (v | Qw) ∈ C.

.

4.9 Evolutions on Nested Pre-Pseudo-Unitary Pairs Recall that in Theorem 4.27 we constructed a pre-pseudo-unitary dynamics starting from a pre-pseudo-unitary space W0 and its subspace W1 . In this subsection we give a slightly different construction of such a dynamics which starts from a nested pre-pseudo-unitary pair (W 1 , W− 1 , Q). 2

2

Definition 4.31 Let (W 1 , W− 1 , Q) be a nested pre-pseudo-unitary pair. If a 2 2 bounded operator R on W− 1 , restricts to a bounded operator on W 1 , and 2

2

(Rv | QRw) = (v | Qw),

v ∈ W− 1 ,

.

2

w ∈ W1 , 2

then we say that R preserves (W 1 , W− 1 , Q). 2

2

Applying complex interpolation we obtain Proposition 4.32 Suppose that R preserves (W 1 , W− 1 , Q). Then for 0 ≤ λ ≤ 12 2 2 it restricts to an operator preserving (Wλ , W−λ , Q). In particular, R preserves Q on W0 in the usual sense.

An Evolution Equation Approach to Linear Quantum Field Theory

65

Definition 4.33 Suppose that B is an operator on W− 1 with domain containing 2 W 1 . We say that B infinitesimally preserves (W 1 , W− 1 , Q) if B is a generator of 2 2 2 a group on W− 1 , its part B˜ in W 1 is a generator of a group on W 1 , and 2

2

2

(Bv | Qw) = (v | QBw),

v, w ∈ W 1 .

.

(63)

2

The quadratic form defined by (63) is called the energy or Hamiltonian quadratic form of B on W 1 . 2

Proposition 4.34 Suppose that B is an operator on W− 1 that infinitesimally 2

preserves (W− 1 , W 1 , Q). Then e−itB , t ∈ R, preserves (W− 1 , W 1 , Q). 2

2

2

2

Proof First we check that (e−itB v | Qe−itB w) = (v | Qw),

.

v, w ∈ W 1 . 2

Then, by continuity, we extend (64) to v ∈ W− 1 . 2

(64) u n

The following theorem can be viewed as an alternative to Theorem 4.27: Theorem 4.35 Suppose that all assumptions of Theorem 3.11 are satisfied where W0 is replaced with W− 1 and W1 is replaced with W 1 . In addition, assume that 2 2 for all t ∈ I the operators B(t) infinitesimally preserve (W− 1 , W 1 , Q). Then the 2 2 evolution R(t, s) preserves (W− 1 , W 1 , Q). In particular, it is pre-pseudo-unitary 2 2 on W0 . Let us compare the constructions of Theorem 4.27 and of Theorem 4.35. In both cases we obtain a (pre-)pseudo-unitary evolution on a (pre-)pseudo-unitary space. However, in the former case we have a fixed space W1 contained in D(B(t)) for all t. In the latter case, we do not have information about the domain of the generator of the evolution on W0 , that is, of the part of B(t) in W0 . On the other hand, in practice the assumptions of Theorem 4.35 can be weaker.

5 Abstract Klein–Gordon Operator The usual Klein–Gordon operator acts on, say, Cc∞ (M), where M is a Lorentzian manifold, and is given by the expression (11). K can be interpreted as a Hermitian operator in the sense of the Hilbert space L2 (M). (One of the main ideas of our paper is the usefulness of this interpretation.) After the identification of M with I × E, where I corresponds to the time variable and E describes the spatial variables, we can identify L2 (M) with L2 (I, K), where K = L2 (E), see (105) for more details.

66

J. Derezi´nski and D. Siemssen

Then, formally, the operator K is given by the expression ( ) 1 ( ) Dt + W (t) − L(t), K := Dt + W ∗ (t) 2 α (t)

.

(65)

where α(t) involves the metric tensor, W (t) consists mostly of the 0th component of the potential and L(t) is a magnetic Schrödinger operator on E. We will describe this identification in more detail in Sect. 7.3. In this section we study (65) in an abstract setting. We are interested in various inverses and bisolutions of K. We treat K as an abstract Hilbert space and L(t), α(t), W (t) as given abstract operators. The results of this section will be applied to the usual Klein–Gordon operator in Sect. 7. In order to study propagators associated with the abstract Klein–Gordon operator, we first introduce a certain scale of Hilbertizable spaces Wλ . Each member of this scale is the direct sum of two Sobolev-type spaces based on K describing the “configurations” and the “momenta”. The space W0 has the structure of a Krein space and will play the central role in quantization. The Cauchy data for K on W0 undergo a certain pseudo-unitary evolution whose generator B(t) is made out of L(t), W (t), α 2 (t). After imposing boundary conditions we can define various propagators, first associated with the Cauchy data operator M = ∂t + iB(t), and then associated with the operator K itself. Note that the formula (65) does not give a rigorous definition of a unique closed operator. Actually, the analysis of possible closed realizations of K and the corresponding inverses is quite subtle and depends strongly on whether I is finite or not. In the case of a finite I one first needs to impose appropriate boundary conditions at the initial time t− and the final time t+ . These conditions lead to a construction of E F and E F , which are bounded inverses of M. They are then used to define GF and GF , which are bounded inverses of K. Inverting them we obtain a well-posed realization of the Klein–Gordon operator K. The situation is different and less understood if I = R. The Feynman and antiFeynman inverse can be constructed, however they are not bounded on L2 (R, K). We conjecture that there exists a distinguished self-adjoint realization K s.a. of K such that these inverses are the boundary values of the resolvent of K s.a. from above and below at zero. The conjecture can be easily shown in some special cases, e.g., if K is stationary. We describe some arguments in favor of the conjecture, notably, we sketch a possible construction of the resolvent. There exist recent papers that show this conjecture if K corresponds to the Klein–Gordon operator on asymptotically Minkowskian spaces satisfying a non-trapping condition. Throughout the section we need to overcome a number of technical issues. First, it is convenient to assume that the Hamiltonian used in the construction of the phase space is bounded away from zero, or in physical terms, that the mass is strictly positive. However, physical systems may have a zero mass. This is solved by assuming that the phase space is constructed not directly from the Hamiltonian H (t), but from H0 (t) which differs by a constant b.

An Evolution Equation Approach to Linear Quantum Field Theory

67

Another problem is the regularity. Assumption 5.36 allows us only to perform the basic construction. We introduce the additional Assumption 5.6(ρ), which for ρ = 0 coincides with Assumption 5.36 and for ρ > 0 guarantees additional regularity. With strengthened hypotheses we are able to show some desired properties of the propagators and of the Klein–Gordon operator. Note that the above issue essentially disappears if we assume that everything is smooth, therefore it can be considered as purely academic, of interest only to specialists in operator theory. Nevertheless, we try to give an honest (if not optimal) treatment of this question.

5.1 Basic Assumptions on the Abstract Klein–Gordon Quadratic Form Throughout this section we assume that K is a Hilbert space and for t ∈ I cl we are given the following operators on K: 1. self-adjoint L(t) for which there exists b ∈ R such that L0 (t) := L(t) + b are positive invertible, 2. bounded invertible self-adjoint α(t), 3. operator W (t). We will say that an operator-valued function I e t |→ A(t) ∈ B(K) is absolutely norm continuous if there exists c ∈ L1 (I ) such that c ≥ 0 and f

s

||A(t) − A(s)|| ≤

c(τ ) dτ,

.

t ≤ s,

t, s ∈ I.

t

Here is the basic assumption that we will use in this section. Assumption 5.36 1. For any t ∈ I there exist 0 < c1 ≤ c2 such that c1 L0 (0) ≤ L0 (t) ≤ c2 L0 (0)

. 1

1

(66)

and I cl e t |→ L0 (0)− 2 L0 (t)L0 (0)− 2 ∈ B(K) is absolutely norm continuous. 2. I cl e t |→ α 2 (t) ∈ B(K) is absolutely norm continuous. 1 3. I cl e t |→ W (t)L0 (t)− 2 ∈ B(K) is absolutely norm continuous and there exists a < 1 such that 1

||α(t)−1 W (t)L0 (t)− 2 || ≤ a.

.

68

J. Derezi´nski and D. Siemssen

For β ∈ R, t ∈ I cl , define the scales of Hilbert spaces β

Kβ,t := L0 (t)− 2 K.

(67)

.

Then by (66) and Sect. 2.4, for β ∈ [−1, 1], Kβ,t is compatible with Kβ,0 . Thus we obtain the scale of Hilbertizable spaces Kβ ,

.

β ∈ [−1, 1].

Assumption 5.36 seems insufficient to define the Klein–Gordon operator. However, we can define the Klein–Gordon quadratic form by setting (f1 | Kf2 ) =

.

f ( ( ) Dt + W (t) f1 (t) | I

f

−

) ) 1 ( Dt + W (t) f2 (t) dt

α 2 (t)

( ) 1 1 L0 (t) 2 f1 (t) | L0 (t) 2 f2 (t) dt + b

f

I

(

) f1 (t) | f2 (t) dt,

(68)

I

f1 , f2 ∈ Cc (I, K1 ) ∩ Cc1 (I, K0 ). Note that K is a Hermitian form in the sense of the Hilbert space L2 (I ) ⊗ K = L2 (I, K). Formally, it corresponds to the operator given by the expression (65). Unfortunately, K is not a semibounded form, hence the usual theory of quadratic forms does not apply. Therefore, it is not easy to interpret K as a closed operator on L2 (I, K). We will come back to this question in Sect. 5.9 under more restrictive assumptions.

5.2 Pseudo-Unitary Evolution on the Space of Cauchy Data The following analysis is essentially an adaptation of [8, 9] to the abstract setting. We consider the scale of Hilbertizable spaces Wλ = Kλ+ 1 ⊕ Kλ− 1 ,

.

2

2

] [ 1 1 . λ∈ − , 2 2

(69)

Of special importance are the energy space

.

the dynamical space and the dual energy space

Wen := W 1 = K1 ⊕ K0 , 2

Wdyn := W0 = K 1 ⊕ K− 1 , 2

W∗en

2

:= W− 1 = K0 ⊕ K−1 . 2

An Evolution Equation Approach to Linear Quantum Field Theory

69

[ ] [ ] u1 v ∈ W−λ and 1 ∈ Wλ with |λ| ≤ 12 we introduce the pairing defined For u2 v2 [ ] 01 by the charge operator Q = . In other words, 10 ([u ]

.

1

u2

|Q

[ ]) v1 = (u1 | v2 ) + (u2 | v1 ). v2

Note that (W− 1 , W 1 , Q) is a nested pseudo-unitary pair, see Definition 4.29. 2 2 Moreover, (W0 , Q) is a Krein space. Indeed, [

1

0 L0 (t)− 2 .St := 1 L0 (t) 2 0

]

is a bounded self-adjoint involution on the Hilbert space 1

1

W0,t = L0 (t)− 4 K ⊕ L0 (t) 4 K.

(70)

.

Equation (70) is compatible with W0 . Hence St is an admissible involution. Introduce the Hamiltonians ] ] [ [ L(t) W ∗ (t) L0 (t) W ∗ (t) , H . .H (t) = (t) = 0 W (t) α 2 (t) W (t) α 2 (t) Proposition 5.2 Suppose Assumption 5.36 holds. There exist 0 < c1 ≤ c2 such that on K ⊕ K we have [ [ ] ] L0 (t) 0 L (t) 0 ≤H0 (t) ≤ c2 0 , 0 1 0 1 [ [ ] ] 1 0 1 0 ≤QH0 (t)Q ≤ c2 . c1 0 L0 (t) 0 L0 (t)

c1

.

Therefore, 1

W 1 ,t := H0 (t)− 2 (K ⊕ K)

.

2

1 2

W− 1 ,t := (QH0 (t)Q) (K ⊕ K) 2

is compatible with W 1 , 2

is compatible with W− 1 . 2

In particular, H0 (t) is a positive operator on K ⊕ K with the form domain W 1 . 2

70

J. Derezi´nski and D. Siemssen

Let Wλ,t , λ ∈ R, denote the scale of Hilbert spaces defined by interpolation from the nested pair W 1 ,t , W− 1 ,t . Clearly, Wλ,t are compatible with the Hilbertizable 2

2

spaces Wλ for λ ∈ [− 12 , 21 ]. Introduce the generators [ .B(t) := QH (t) =

[

] W (t) α 2 (t) , L(t) W ∗ (t)

B0 (t) := QH0 (t) =

] W (t) α 2 (t) . L0 (t) W ∗ (t)

Proposition 5.3 Suppose Assumption 5.36 holds. Then for any λ ∈ R, B0 (t) is a unitary operator from W 1 +λ,t to W− 1 +λ,t . Besides, it is a self-adjoint operator in 2 2 the sense of W− 1 +λ,t with the domain W 1 +λ,t . Therefore, 2

2

1

1

Wλ,t = |B0 (t)|−λ+ 2 W 1 ,t = |B0 (t)|−λ− 2 W− 1 ,t .

(71)

.

2

2

Proof We drop 0 and (t) from H0 (t), B0 (t). First note that H is bounded from K 1 ⊕ K0 to K− 1 ⊕ K0 . Hence B is bounded from W 1 = K 1 ⊕ K0 to W− 1 = 2 2 2 2 2 K0 ⊕ K− 1 . Now, 2

(Bu | Bv)− 1 ,t = (QH u | (QH Q)−1 QH v) = (u | H v) = (u | v) 1 ,t .

.

2

2

This proves the unitarity of B from W 1 ,t to W− 1 ,t . 2 2 Let u, v ∈ W 1 . Then 2

(u | Bv)− 1 ,t = (u | (QH Q)−1 QH v) = (QH u | (QH Q)−1 v) = (Bu | v)− 1 ,t

.

2

2

proves the Hermiticity in the sense of W− 1 ,t with the domain W 1 ,t . Clearly, an 2 2 invertible Hermitian operator is self-adjoint. Now we obtain W− 1 ,t = |B|W 1 ,t ,

.

2

2

from which (71) and all the remaining statements of the proposition follow.

u n

Proposition 5.4 Suppose Assumption 5.36 holds. Then I e t |→ B(t) : W 1 → 2 W− 1 satisfies the assumptions of Theorem 4.35. Therefore, it defines an evolution 2

R(t, s) on Wλ , − 12 ≤ λ ≤ 12 , pseudo-unitary in the sense of (W−λ , Wλ , Q). In particular, the evolution R(t, s) is pseudo-unitary on (W0 , Q). ) ( Proof First we check that t |→ B0 (t) ∈ B W 1 , W− 1 is norm continuous. 2 2 Besides, B0 (t) is self-adjoint in the sense of the the scalar products (· | ·)− 1 ,t and 2 (· | ·) 1 ,t . 2

An Evolution Equation Approach to Linear Quantum Field Theory

71

There exists c ∈ L1 (I ) such that for t, s ∈ I cl , f t || || ( ) || − 12 − 12 || H0 (t) − H0 (s) H0 (s) || ≤ 2 . ||H0 (s) c(u) du. s

Therefore, .

|| || ( f t ) 1 1 || || c(u) du , ||H0 (s)− 2 H0 (t)H0 (s)− 2 || ≤ exp 2 s

|| || (f s ) 1 1 || || hence ||v|| 1 ,t = ||H0 (t) 2 H0 (s)− 2 || ||v|| 1 ,s ≤ exp t c(τ ) dτ ||v|| 1 ,s . 2

2

Similarly, there exists c ∈ L1 (I ) such that for t, s ∈ I cl , .

2

f t || || ) 1 || 1( || c(u) du. ||(QH0 (s)Q)− 2 QH0 (t)Q − QH0 (s)Q (QH0 (s)Q)− 2 || ≤ 2 s

Therefore, || || 1 1 || || −1 . ||(QH0 (t)Q) 2 (QH0 (s)Q) (QH0 (t)Q) 2 || || || ( f t ) || − 21 − 21 || = ||(QH0 (s)Q) QH0 (t)Q(QH0 (s)Q) || ≤ exp 2 c(u) du , s

|| || (f s ) 1 || 1 || hence ||v||− 1 ,s = ||H0 (s)− 2 H0 (t) 2 || ||v||− 1 ,t ≤ exp t c(τ ) dτ ||v||− 1 ,t . 2 2 2 Thus the assumptions of Theorem 3.10 are satisfied and B0 (t) defines a dynamics on Wλ for − 12 ≤ λ ≤ 12 . The perturbation B(t) − B0 (t) is bounded. Therefore, the assumptions of Theorem 3.11 are satisfied, and B(t) also defines a dynamics. Finally, both B0 (t) and B(t) infinitesimally preserve the pseudo-unitary nested pair (W 1 , W− 1 , Q). Hence the assumptions of Theorem 4.35 hold and R(t, s) is 2 2 pseudo-unitary in the sense of (W 1 , W− 1 , Q). u n 2

2

5.3 Propagators on a Finite Interval Assume that I = ]t− , t+ [ is finite. Suppose Assumption 5.36 holds. Let R(t, s) be the corresponding pseudo-unitary evolution on the Krein space W0 , whose existence is guaranteed by Proposition 5.4. Thus we are now in the setting of Sect. 4.6. First we define the propagators for the Cauchy data. The classical propagators E PJ , E ∨ , E ∧ are introduced as in Definition 3.2. Then we choose two admissible (+) involutions S+ , S− on W0 . We then define the non-classical propagators E± ,

72

J. Derezi´nski and D. Siemssen

(+)

E± , E F , E F as in Definitions 3.3, 3.4 and 3.5. (Recall that the asymptotic complementarity is automatically satisfied.) All the propagators for the Cauchy data are bounded operators on L2 (I, W0 ). Clearly, we can write E• =

[

.

] • E• E11 12 . • E• E21 22

(72)

We define the propagators for the abstract Klein–Gordon operator by selecting the upper right element of the matrix of (72) and possibly by multiplying it by a conventional factor: • G• := iE12 ,

• = PJ, ∨, ∧, F, F; .

(73a)

(+) (+) G± := E±,12 ,

(−) (−) G± := −E±,12 .

(73b)

.

Theorem 5.5 Equation (73) are bounded operators on L2 (I, K). They satisfy GPJ∗ = −GPJ ;

.

G∨∗ = G∧ ; GF∗ = GF ; (+)∗

= G± ≥ 0,

(−)∗

= G± ≥ 0.

G± G±

(+) (−)

We expect that typically G• with • = ∨, ∧, F, F have a zero nullspace and a dense range. (This will be proven below under some additional assumptions.) If this is the case, we can define K • := G•−1 ,

.

• = ∨, ∧, F, F.

(75)

Note that K • can be viewed as well-posed realizations of the Klein–Gordon operator on a slab with appropriate (non-self-adjoint) boundary conditions. Clearly, K ∨∗ = K ∧ ;

.

K F∗ = K F .

An Evolution Equation Approach to Linear Quantum Field Theory

73

5.4 Propagators on the Real Line Let I = R. We assume that for ±t > T the operators L(t), α(t) and W (t) do not depend on t. Therefore, the generators B(t) for ±t > T do not depend on t, so that they can be denoted B± . We also assume that B± are stable and set S± := sgn(B± ). We define the propagators E • undestood as operators from L2c (R, W0 ) to L2loc (R, W0 ). Then we define the operators G• just as in (73), interpreted as operators L2c (R, K) to L2loc (R, K). Obvious analogs of Propositions 5.13 and 5.14 hold for I = R. The above propagators play a central role in Quantum Field Theory on the spacetime M. In fact, they correspond to the positive energy Fock representations of incoming and outgoing quantum fields, as will be sketched in Sect. 7. We actually believe that this choice is also distinguished for very different reasons by purely mathematical arguments. It probably corresponds to a distinguished (maybe unique) self-adjoint realization of the Klein–Gordon operator. We formulate our expectation in the following conjecture. Conjecture 5.6 For a large class of asymptotically stationary and stable abstract Klein–Gordon operators the following holds: 1. There exists a distinguished self-adjoint operator K s.a. on L2 (R, K) such that on D(K s.a. ) the quadratic form (68) coincides with (f1 | K s.a. f2 ). 2. For s > 12 the following statements hold in the sense of −s L2 (R, K) → s L2 (R, K): .

s-lim(K s.a. − ie)−1 = GF , e-0

s-lim(K s.a. + ie)−1 = GF . e-0

We will discuss arguments in favor of this conjecture in Sect. 5.7.

5.5 Perturbation of the Evolution by the Spectral Parameter Now we are going to compute the resolvent of well-posed realizations of the Klein– Gordon operator. To this end in this subsection we introduce the perturbed evolution Rz (t, s).

74

J. Derezi´nski and D. Siemssen

Let z ∈ C. We define [

] 00 , 10 [ ] W (t) α 2 (t) = B(t) − zZ. Bz (t) := L(t) − z W ∗ (t) Z :=

.

Note that Z is a bounded operator on each Wλ : 1

1

1

1

1

1

||Zv||λ = ||L(t) 2 (λ− 2 ) v1 || ≤ ||L(t) 2 (λ+ 2 ) v1 || + ||L(t) 2 (λ− 2 ) v2 || = ||v||λ .

.

Proposition 5.7 Suppose Assumption 5.36 holds. Then t |→ Bz (t) generates an evolution Rz (t, s) on Wλ , λ ∈ [− 12 , 21 ]. We have Rz (t, s)∗ QRz (t, s) = Q,

Rz (t, s)∗ = QRz (s, t)Q.

.

(77)

In particular, if z ∈ R, then Rz (t, s) is pseudo-unitary on W0 . Formally the equation (z + K)f = 0

.

is equivalent to [ ] [ ] u1 u = Bz (t) 1 , .i∂t u2 u2 u1 = f,

u2 =

) 1 ( D + W (t) f. t α 2 (t)

Therefore, the evolution Rz (t, s) can be used to construct the resolvent of realizations of K.

5.6 Resolvent for Finite Intervals Assume again that I is finite. We are back in the setting of Sect. 5.3. Recall that we impose Assumption 5.36 and choose admissible involutions S+ , S− . Let (+) (−) (+) (−) (Z+ , Z+ ) and (Z− , Z− ) be the corresponding particle/antiparticle spaces. For z ∈ C and t ∈ I , set (+)

(+)

Z±,z (t) := Rz (t, t± )Z+ ,

.

(−)

(−)

Z±,z (t) := Rz (t, t± )Z± .

(78)

An Evolution Equation Approach to Linear Quantum Field Theory

75

Let | { RS := z ∈ C | for some (hence all)t ∈ I the pair of subspaces ( (+) ) } (−) Z+,z (t), Z−,z (t) is complementary .

.

By Proposition 4.11 (2), | { RS := z ∈ C | for some (hence all) t ∈ I the pair of subspaces ( (−) ) } (+) Z+,z (t), Z−,z (t) is complementary .

.

For z ∈ RS we define (+)

(−)

(−)

(+)

(−)

(+)

(+)

(−)

.

AF(+) (t), z

the projection onto Z+,z (t) along Z−,z (t),

AF(−) (t), z

the projection onto Z−,z (t) along Z+,z (t);

AF(−) (t), z

the projection onto Z+,z (t) along Z−,z (t),

AF(+) (t), z

the projection onto Z−,z (t) along Z+,z (t).

Set EzF (t, s) := θ (t − s)Rz (t, s)AF(+) (s) − θ (s − t)Rz (t, s)AF(−) (s), z z

.

EzF (t, s) := θ (t − s)Rz (t, s)AzF(−) (s) − θ (s − t)Rz (t, s)AzF(−) (s); F GFz (t, s) := iEz,12 (t, s), F (t, s). GFz (t, s) := iEz,12

Proposition 5.8 For z ∈ RS the operators GFz , GzF are bounded and satisfy the resolvent equation: .

GFz − GFw = (z − w)GFz GFw , .

(79)

GFz − GFw = (z − w)GFz GFw .

(80)

(QEzF )∗ = −QEzF , .

(81)

Besides, .

(GFz )∗ = GzF .

(82)

76

J. Derezi´nski and D. Siemssen

Proof The boundedness is obvious. Let us prove (79). A straightforward computation yields ) ( F (t, s) = (z − w) EzF ZEw

f

t+

F EzF (t, τ )ZEw (τ, s) dτ

.

t−

f

=

t

θ (t − s)(z − w) s s

f + θ (s − t)(z − w) f

t s

− θ (t − s)(z − w) f

t− t

− θ (s − t)(z − w) f

t−

AF(−) (t)Rz (t, τ )ZRw (τ, s)AF(−) z w (s) dτ AF(+) (t)Rz (t, τ )ZRw (τ, s)AF(−) z w (s) dτ AF(+) (t)Rz (t, τ )ZRw (τ, s)AF(−) z w (s) dτ

t+

− θ (s − t)(z − w) f

AF(+) (t)Rz (t, τ )ZRw (τ, s)AF(+) z w (s) dτ

s t+

− θ (t − s)(z − w) t

AF(−) (t)Rz (t, τ )ZRw (τ, s)AF(+) z w (s) dτ AF(−) (t)Rz (t, τ )ZRw (τ, s)AF(+) z w (s) dτ.

By the fundamental theorem of calculus this equals .

=

( ) θ (t − s)AF(+) (t) Rz (t, s) − Rw (t, s) AF(+) z w (s) ( ) (t) Rw (t, s) − Rz (t, s) AF(−) + θ (s − t)AF(−) z w (s) ( ) F(+) − θ (t − s)Az (t) Rz (t, t− )Rw (t− , s) − Rz (t, s) AF(−) w (s) ( ) F(−) F(+) − θ (s − t)Az (t) Rz (t, t− )Rw (t− , s) − Rw (t, s) Aw (s) ( ) (t) − Rz (t, t+ )Rw (t+ , s) + Rz (t, s) AF(+) − θ (s − t)AF(−) z w (s) ( ) (t) − Rz (t, t+ )Rw (t+ , s) + Rw (t, s) AF(+) − θ (t − s)AF(−) z w (s).

We rearrange this, obtaining .

=

( ) F(−) θ (t − s)AF(+) (t)Rz (t, s) AF(+) z w (s) + Aw (s) ) ( (t) + AF(+) (t) Rw (t, s)AF(+) − θ (t − s) AF(−) z z w (s) ( ) F(−) (t)Rz (t, s) AF(+) − θ (s − t)AF(−) z w (s) + Aw (s) ) ( (t) + AF(+) (t) Rw (t, s)AF(−) + θ (s − t) AF(−) z z w (s)

An Evolution Equation Approach to Linear Quantum Field Theory

.

77

− Rz (t, t− )AF(+) (t− )AF(−) z w (t− )Rw (t− , s).

(83a)

+ Rz (t, t+ )AF(−) (t+ )AF(+) z w (t+ )Rw (t+ , s)

(83b)

which simplifies to .

=

θ (t − s)AF(+) (t)Rz (t, s) − θ (t − s)Rw (t, s)AF(+) z w (s) (t)Rz (t, s) + θ (s − t)Rw (t, s)AF(−) − θ (s − t)AF(−) z w (s)

F = EzF (t, s) − Ew (t, s).

since, for any z, w, ( ) ( F(+) ) (−) R AF(−) (t− ) w (t− ) = Z− = N Az ) ( F(−) ) ( (+) (t+ ) . R AF(+) w (t+ ) = Z+ = N Az

.

Thus we have proven that ) ( F F (t, s) = EzF (t, s) − Ew (z − w) EzF ZEw (t, s).

.

Taking the 1, 2 component of (84) we obtain (79). To prove (81) we use (77). This implies (82).

(84) u n

Clearly, if we can define K F , K F by (75) as operators with dense domains, then we have GFz = (z + K F )−1 ,

.

GFz = (z + K F )−1

and resolvent set of K F = resolvent set of K F ⊂ RS.

.

5.7 Resolvent for I = R In this subsection we will give some arguments supporting Conjecture 5.6. We will sketch a construction of a family of operators which we expect to be the resolvent of the (putative) distinguished self-adjoint realization of the abstract Klein–Gordon operator for I = R. We impose the assumptions of Sect. 5.4 (but it is likely that more assumptions are needed). Our analysis will not be complete. First note that we cannot repeat the constructions of Sect. 5.6 without major changes. In fact, for I = R in the definitions (78) one should take t → ±∞.

78

J. Derezi´nski and D. Siemssen

However, for z /∈ R the evolution Rz (t, s) blows up on a part of W0 and decays on another part as t → ±∞. We need to define the projections onto the (distorted) positive/negative part of the spectrum of B±,z (defined analogously to B± in Sect. 5.4). This is straightforward for B± , because they are self-adjoint. However, B±,z in general are not self-adjoint. For simplicity, we will assume that B± are strongly stable, so that their spectrum has a gap around 0. Then for small Re(z) the operators B±,z are “bisectorial”, which is sufficient for the construction of these projections. This is described in Proposition 7.2 of [8], which implies the following proposition: Proposition 5.9 There exists ζ0 > 0 such that the strip {ζ ∈ C | −ζ0 ≤ Re(ζ ) ≤ ζ0 } is contained in the resolvent set of B±,z . Moreover, the operators 1( 1+ τ →∞ 2 1( := lim 1− τ →∞ 2

(+) ||±,z := lim

.

(−)

||±,z

1 πi 1 πi

f

iτ

−iτ

f

iτ

−iτ

) (B±,z − ζ )−1 dζ , (B±,z − ζ )−1 dζ

)

constitute a pair of complementary projections commuting with B±,z such that ( ) ( (+) ) σ B±,z ||±,z = σ B±,z ∩ {w ∈ C | Re(w) ≥ 0}, ( ( ) (−) ) σ B±,z ||±,z = σ B±,z ∩ {w ∈ C | Re(w) ≤ 0}.

.

Set ( ) (±) (±) Z±,z (t) := R lim Rz (t, τ )||±,z Rz (τ, t) ,

Im(z) ≥ 0

( ) (∓) (∓) Z±,z (t) := R lim Rz (t, τ )||±,z Rz (τ, t) ,

Im(z) ≤ 0.

.

τ →±∞ τ →±∞

We can now complement Conjecture 5.6 with an additional conjecture about the resolvent of K s.a. : Conjecture 5.10 We expect that for Im(z) ≥ 0 for some (hence all) t ∈ R the pair ( (+) ) ( ) (−) of subspaces Z+,z (t), Z−,z (t) is complementary. Let AF(+) (t), AF(−) (t) be the z z pair of projections corresponding to this pair of spaces. Equivalently, we expect that for Im(z) ≤ 0 for some (hence all) t ∈ R the pair ( (−) ) ( F(−) ) (+) F(+) of subspaces Z+,z (t), Z−,z (t) is complementary. Let Az (t), Az (t) be the pair of projections corresponding to this pair of spaces. We also introduce Ez (t, s) := θ (t − s)Rz (t, s)AF(+) (s) − θ (s − t)Rz (t, s)AF(−) (s), z z

.

Im(z) > 0;

An Evolution Equation Approach to Linear Quantum Field Theory

79

Ez (t, s) := θ (t − s)Rz (t, s)AzF(−) (s) − θ (s − t)Rz (t, s)AzF(+) (s),

Im(z) < 0;

Gz (t, s) := iEz,12 (t, s),

Im(z) /= 0.

Then we conjecture that Gz defines for z ∈ C\R a bounded operator on L2 (R, K) with a dense range and a trivial nullspace such that Gz − Gw = (z − w)Gz Gw , .

.

G∗z

= Gz .

(85) (86)

Thus Gz is the resolvent of a self-adjoint operator, which we can call K s.a and treat as the distinguished self-adjoint realization of K. Let us sketch some arguments in favor of the above conjecture. First of all, in the stationary case, that is, if B(t) does not depend on t, the conjecture is true (with minor additional assumptions) following the arguments of [8] and [9]. The conjecture is also true if K = C. In fact, the operator K is then essentially the well-known 1-dimensional magnetic Schrödinger operator. It should not be difficult to generalize this to the case of a finite dimensional K, or bounded L(t) and W (t). Unfortunately, for a generic spacetime, L(t) is unbounded and non-stationary. We know in this case that asymptotic complementarity holds for real z, so we can expect it to hold in a neighborhood of R. Ez are well defined as quadratic forms on, say, Cc (R, W0 ) and (86) is easily checked. However, we do not know how to control the norm of Ez (t, s) for large t, s, and hence to show the boundedness of Ez . We expect that .

lim e∓itB±,z ||±,z = 0,

Im(z) ≥ 0; .

(87)

lim e∓itB±,z ||±,z = 0,

Im(z) ≤ 0.

(88)

(±)

t→+∞

(∓)

t→−∞

(This follows under a slightly stronger assumption from Proposition 7.2 of [8].) Now the resolvent Eq. (85) should follow by the same calculation as in the proof of Proposition 5.8, except that the terms (83a) and (83b) should be zero by (87) and (88).

5.8 Additional Regularity In order to have better properties of the Klein–Gordon form, and in particular, to guarantee that it defines an operator, it will be useful to introduce a family of assumptions more restrictive than Assumption 5.36. This family will depend on a parameter ρ ≥ 0.

80

J. Derezi´nski and D. Siemssen

Assumption 5.6 (ρ) 1. For any t ∈ I there exist 0 < c1 ≤ c2 such that c1 L0 (0)1+ρ ≤ L0 (t)1+ρ ≤ c2 L0 (0)1+ρ

.

1+ρ

1+ρ

and I cl e t |→ L0 (0)− 2 L0 (t)1+ρ L0 (0)− 2 ∈ B(K) is absolutely norm continuous. ρ ρ 2. I cl e t |→ L0 (t) 2 α 2 (t)L0 (t)− 2 ∈ B(K) is absolutely norm continuous. ρ −1−ρ ρ −1+ρ ∈ B(K) are 3. I cl e t |→ L0 (t) 2 W (t)L0 (t) 2 , L0 (t)− 2 W (t)L0 (t) 2 absolutely norm continuous and there exists a < 1 such that ρ

−1+ρ 2

ρ

−1−ρ 2

||L(t)− 2 α(t)−1 W (t)L0 (t)

.

||L(t) 2 α(t)−1 W (t)L0 (t)

|| ≤ a, || ≤ a.

Note that Assumption 5.36 coincides with Assumption 5.6(0). If 0 ≤ ρ ' ≤ ρ, then Assumption 5.6(ρ) implies Assumption 5.6(ρ ' ). Proposition 5.12 Suppose Assumption 5.6(ρ) holds. Then the following is true: 1. For any t ∈ I cl and for |β| ≤ 1 + ρ, the Hilbert spaces Kβ,t and Kβ,0 are compatible. Hence we can extend the scale of Hilbertizable spaces Kβ to |β| ≤ 1+ρ 2. We can extend the scale of Hilbertizable spaces Wλ to |λ| ≤ 12 + ρ. Besides, for any t ∈ I cl and such λ the Hilbert spaces Wλ,t are compatible with Wλ . 3. For any −ρ− 12 ≤ λ ≤ ρ− 12 the function I e t |→ B(t) : Wλ+1 → Wλ satisfies the assumptions of Theorem 3.11, and hence defines an evolution R(t, s) on Wλ , |λ| ≤ 12 + ρ. For −ρ − 12 ≤ λ ≤ ρ − 12 this evolution satisfies ( ) ∂t + iB(t) R(t, s)u = 0,

.

u ∈ Wλ+1 .

(89)

Proof Assumption 5.6(ρ) implies immediately that the Hilbertizable spaces K±(1+ρ),t and K±(1+ρ),0 coincide. Therefore (1) follows by the Kato–Heinz inequality. Let us drop (t). We have ⎡ .

θ 2

⎤

[

α2

]

⎡

−1−θ 2

0 ⎦ W ⎣L0 −1+θ ⎣L 0 ∗ L W 2 0 0 L0 0

⎤

⎡

θ 2

−1−θ 2

0 ⎦ ⎣L 0 W L 0 = 1

−θ L0 2

θ 2

θ

⎤

− L0 α 2 L0 2 ⎦ . −1+θ −θ L0 2 W ∗ L0 2

(90) Equation (90) is bounded for θ = ρ and θ = −ρ. Hence by interpolation it is bounded for −ρ ≤ θ ≤ ρ. Hence B0 is bounded from Wθ+ 1 to Wθ− 1 . 2

2

An Evolution Equation Approach to Linear Quantum Field Theory

81

Let −ρ ≤ θ ≤ ρ. Equation (90) can be represented as [ .

−θ L0 αL0 2 θ 2

0

]⎡

⎤[

−1−θ 2

θ

2 −1 0 ⎣L 0 α W L 0 1 1

−1+θ 2

L0

1 − θ2

W ∗ α −1 L0

⎦

1

]

0 − θ2

θ

0 L02 αL0

The two extreme terms are invertible. Besides, −1−θ 2

θ

||L02 α −1 W L0

.

|| ≤ a < 1.

Hence the middle term is also invertible. This proves the invertibility of B0 (t) as a map from Wθ+ 1 to Wθ− 1 . 2

2

Let −ρ − 12 ≤ θ ≤ ρ + 12 . Then for some θ0 ∈ [− 21 , 12 ] and n ∈ Z we have θ = n + θ0 . Then Wθ,t = B0 (t)n Wθ0 ,t . But as Hilbertizable spaces Wθ0 ,t = Wθ0 . We have just proved that B0 (t)n are bounded invertible in the sense Wθ0 → Wθ0 +n for θ0 + n ≤ ρ + 12 . Hence Wθ0 +n,t = Wθ0 +n . Hence (2) is true. (3) is proven in a similar way as Proposition 5.4. n u Note that Assumption 5.6(ρ) is especially important for ρ = 12 . Then (89) holds with λ = 0, so that B(t) can be interpreted as an (unbounded) pseudo-unitary generator on the Krein space W0 with the domain W1 .

5.9 The Abstract Klein–Gordon Operator Analysis of differential operators with variable coefficients of low regularity is a rather technical and complicated subject, even if these coefficients are scalar. In our case these coefficients have values in unbounded operarators, hence it is not surprising that defining Klein–Gordon operators with low regularity conditions is difficult and messy. For completeness, in the following theorems we give conditions which allow us to define abstract Klein–Gordon operators. Note that we do not attempt to be optimal. Proposition 5.13 Suppose Assumption 5.6( 12 ) holds. Let • = ∨, ∧, F, F. 1. Let α −2 (−i∂t + W )f ∈ Cc1 (I, K− 1 ).

f ∈ Cc (I, K 3 ) ∩ Cc1 (I, K 1 ),

.

2

2

2

(91)

Then G• Kf = f.

.

(92)

82

J. Derezi´nski and D. Siemssen

2. If the space of f satisfying (91) is dense in K, then N(G• ) = {0} and R(G• ) is dense. ) ) 1 1( 1( 3 3. If in addition I e t |→ L(t)− 4 ∂t α(t)−2 L(t)− 4 , L(t)− 4 ∂t W (t) L(t)− 4 ∈ B(K) are continuous, then (91) is equivalent to f ∈ Cc (I, K 3 ) ∩ Cc1 (I, K 1 ) ∩ Cc2 (I, K− 1 ).

.

2

2

2

Proof (1): By Proposition 3.13 (1) we know that if u ∈ Cc (I, W1 ) ∩ Cc1 (I, W0 ), then ) ( E • ∂t + iB(t) u = u.

(93)

.

f ∈ Cc (I, K 3 ) ∩ Cc1 (I, K 1 ) implies that α −2 (−i∂t + W )f ∈ Cc (I, K 1 ). Therefore, 2

2

[ u :=

.

2

]

f ∈ Cc (I, W1 ) ∩ Cc1 (I, W0 ). −α −2 (−i∂t + W )f

(94)

Hence we can apply (93) to u. We have ] L −i∂t + W ∗ .Q(−i∂t + B) = −i∂t + W α2 ][ [ ][ ] 1 0 1 (−i∂t + W ∗ )α −2 −K 0 = . 0 1 0 α 2 α −2 (−i∂t + W ) 1 [

For u given by (94) we have Q(−i∂t + B)u =

.

[ ] −Kf . 0

We obtain ] [ • Kf ) −E12 .u = E ∂t + iB(t) u = • Kf , −E22 •

(

which yields (92). By (92) the range of G• contains (91) contains. The same argument shows that the range of G•∗ contains (91). If the space of (91) is dense, we have N(G• ) = R(G•∗ )⊥ = {0}.

.

This proves (2). (3) follows by checking that ∂t α −2 ∂t f and ∂t α −2 Wf are in Cc (I, K− 1 ). 2

u n

An Evolution Equation Approach to Linear Quantum Field Theory

83

Thus, if I is finite, under the assumptions of Proposition 5.13 (3) we can define an unbounded closed operator K • in the sense of L2 (I, K) with 0 belonging to its resolvent set. It corresonds to the quadratic form (68) with the appropriate boundary conditions. For instance, the boundary conditions of K F are ] f (t∓ ) (∓) ∈ Z∓ , −α −2 (−i∂t + W )f (t∓ )

[ f ∈ D(K F ) ⇒

.

] f (t∓ ) (±) ∈ Z∓ . −α −2 (−i∂t + W )f (t∓ )

[ f ∈ D(K ) ⇒ F

Clearly, the above construction is indirect. It does not mean that the expression (65) is well defined in the sense of the Hilbert space L2 (I, K). Under more stringent conditions, such as those described in the following proposition, one can directly interpret (65) as an operator: Proposition 5.14 Suppose that Assumption 5.6(1) holds. In addition, assume that ) ( ) 1 ( I cl e t |→ ∂t α −2 (t) L0 (t)− 2 , ∂t W (t) L0 (t)−1 ∈ B(K) are norm continuous families. Then K, as defined in (65), maps Cc (I, K2 ) ∩ Cc1 (I, K1 ) ∩ Cc2 (I, K0 )

.

(95)

into Cc (I, K0 ). Hence (95) can serve as a dense domain of the operator K. Proof We rewrite K as ( i ) 1 1 1 2 ∗ D D W (t)D Dt + W (t) + + ∂ t t t α 2 (t) t α 2 (t) α 2 (t) α 2 (t) ( i i ) 1 W (t) − 2 ∂t W (t) − L(t). + W ∗ (t) 2 W (t) + ∂t 2 α (t) α (t) α (t)

K=

.

u n

Now in the I = R case we can strengthen Conjecture 5.10: Conjecture 5.15 Impose the assumptions of Conjecture 5.10 and Proposition 5.14, Then the operator K with the domain, say, (95) is essentially self-adjoint and its closure K s.a. satisfies Conjecture 5.10.

6 Bosonic Quantization In this section we describe the basics of quantization used in bosonic QFT. It involves two steps. First, we select a classical phase space, and we associate with it an algebra of Canonical Commutation Relations. Second, we choose a representation of this algebra.

84

J. Derezi´nski and D. Siemssen

We will describe four formalisms used in Step 1 as summarized in the table in Sect. 1.11. We also describe how to define Fock representations, which is the most common realization of Step 2. Note that, unlike in the introduction and the next section, in this section we do not put “hats” on quantized operators to reduced the notational burden.

6.1 Real (or Neutral) Formalism 6.1.1

Canonical Commutation Relations

Suppose that Y is a real vector space equipped with an antisymmetric form ω, i.e., (Y, ω) is a pre-symplectic space. Let CCR(Y) denote the complex unital ∗-algebra generated by o(w), w ∈ Y, satisfying 1. o(w)∗ = o(w), 2. the map Y e w |→ o(w) is linear, 3. the canonical commutation relations hold, [ ] o(v), o(w) = i,

.

v, w ∈ Y.

Let W := CY be the complexification of Y and Q the corresponding Hermitian form, as described in (25): (v | Qw) := i,

.

v, w ∈ W.

We extend o to W, so that it is complex antilinear: o(wR + iwI ) := o(wR ) − io(wI ),

.

wR , wI ∈ Y.

Then we have, for all v, w ∈ W, o∗ (w) := o(w)∗ = o(w), [ ] o(v), o∗ (w) = (v | Qw).

.

6.1.2

Fock Representation

Assume in addition that W is Krein. Let S• be an admissible anti-real involution on W, see Sect. 4.2. Let ||•(+) , Z•(±) = ||•(+) W be the corresponding particle projection and space, see Sect. 2.8.

An Evolution Equation Approach to Linear Quantum Field Theory

85

It is well-known that the two-point function ( ) ω• o(v)o∗ (w) = (||•(+) v | Q||•(+) w),

.

v, w ∈ W,

uniquely determines a centered pure quasi-free state on the algebra CCR(Y). To describe the corresponding GNS representation first note that the charge Q defines on Z•(+) a positive definite scalar product (z1 | z2 ) := (z1 | Qz2 ),

.

z1 , z2 ∈ Z•(+) .

(96)

Hence for z ∈ Z•(+) we can introduce the standard annihilation, resp. creation operators a• (z) and a•∗ (z) acting on the bosonic Fock space rs (Z•(+) ), see e.g. [7]. The state ω• is given by the vacuum o• ∈ rs (Z•(+) ): ω• ( · ) = (o• | · o• ),

.

and the representation is given by o• (w) := a• (||•(+) w) + a•∗ (||•(+) w),

.

o∗• (w) := a•∗ (||•(+) w) + a• (||•(+) w),

w ∈ W.

Note that if z ∈ Z•(+) , then o• (z) = o∗• (z) = a• (z),

.

o• (z) = o∗• (z) = a•∗ (z). 6.1.3

Two-Component Representation

Let X be a real Hilbertizable space and X∗ is its dual. The pairing of X and X∗ will be denoted . We equip X∗ ⊕ X with the symplectic form (u2 , v2 )ω(u1 , v1 ) = − ,

.

(u1 , v1 ), (u2 , v2 ) ∈ X∗ ⊕ X.

Then Y := X∗ ⊕ X is a symplectic space and CY is a Krein space. Consider the ∗-algebra generated by φ(u), u ∈ X∗ , π(v), v ∈ X satisfying 1. φ(u)∗ = φ(u), π(v)∗ = π(v); 2. the maps X∗ e u |→ φ(u), X e v |→ π(v) are linear; 3. the CCR in the two-component form hold: [ .

] [ ] φ(u1 ), φ(u2 ) = π(v1 ), π(v2 ) = 0; u1 , u2 ∈ X∗ , v1 , v2 ∈ X. [ ] φ(u), π(v) = i, u ∈ X∗ , v ∈ X.

86

J. Derezi´nski and D. Siemssen

We can pass to the formalism of Sect. 6.1.1 by setting u ∈ X∗ , v ∈ X.

o(v, u) := π(v) + φ(u),

.

Indeed, [ ] ( ) o(v2 , u2 ), o(v1 , u1 ) = i − .

.

We can extend φ, π to CX∗ e u |→ φ(u) and CX e v |→ π(v) by antilinearity. Then we can replace the symplectic form by the Hermitian form ( ) (iv2 , u2 ) | Q(iv1 , u1 ) = + ( ) = i − .

.

Every anti-real admisssible involution S• on CX∗ ⊕ CX determines a Fock state ω• . Let [ ] (+) (+) ||•11 ||•12 (+) .||• = (+) (+) . ||•21 ||•22 be the corresponding “particle projection”. The conditions ||•(+)∗ Q = Q||•(+) and (+)

||•(+) + ||•

= 1 yield (+)∗

(+)

||•22 = ||•11 ,

.

(+)

(+)

(+)

(+)∗

(+)

||•22 + ||•22 = ||•11 + ||•11 = 1,

.

(+)

||•12 = ||•12 , (+)

(+)∗

(+)

||•21 = ||•21 , (+)

(+)

(+)

||•12 + ||•12 = ||•21 + ||•21 = 0.

The two-point correlation functions of the state ω• are given by ( ) < (+) > ω• π(v)π(v ' ) = v | ||•21 v ' , . ( ) < (+) ' > ω• φ(u)φ(u' ) = u | ||•12 u ,. ( ) < (+) > ω• φ(u)π(v) = i u | ||•11 v , .

.

( ) < (+) > (+) ω• π(v)φ(u) = −i v | ||•22 u = −i.

(97a) (97b) (97c) (97d)

An Evolution Equation Approach to Linear Quantum Field Theory

87

6.2 Complex (or Charged) Formalism 6.2.1

Charged Canonical Commutation Relations

Suppose that W is a complex vector space equipped with a Hermitian form (v | Qw),

.

v, w ∈ W.

Let CCR(W) denote the complex unital ∗-algebra generated by w(w) and w ∗ (w), w ∈ W, such that 1. w ∗ (w) = w(w)∗ , 2. the map W e w |→ w ∗ (w) is linear, 3. the canonical commutation relations in the complex form hold: [ ] w(v), w ∗ (w) = (v | Qw),

.

[ ∗ ] w (v), w ∗ (w) = 0,

v, w ∈ W.

We can pass from the complex to real formalism as follows. Set ) 1 ( oR (w) := √ w(w) + w ∗ (w) , 2

.

) 1 ( oI (w) := √ w(w) − w ∗ (w) , i 2

Then [ .

] oR (w2 ), oR (w1 ) = i Im(w2 | Qw1 ), [ ] oR (w2 ), oI (w1 ) = 0, [ ] oI (w2 ), oI (w1 ) = i Im(w2 | Qw1 ).

Therefore, according to the real formalism of Sect. 6.1.1, the phase space is W⊕W considered as a real space, and on each W we put the symplectic form Im(· | Q·).

6.2.2

Fock Representations

Assume, in addition, that (W, Q) is Krein. Let S• be an admissible involution on W and introduce ||•(±) , Z•(±) := ||•(±) W as in Sect. 2.8. Then we have a unique centered pure quasi-free state on CCR(W) defined by ( ) ω• w(v)w ∗ (w) = (v | Q||•(+) w), ( ) ω• w ∗ (v)w(w) = −(w | Q||•(−) v), ( ( ) ) ω• w ∗ (v)w ∗ (w) = ω• w(v)w(w) = 0. .

88

J. Derezi´nski and D. Siemssen

Let us describe explicitly the GNS representation of ω• . The space Z•(+) is a Hilbert space with the scalar product (96). The space Z•(−) has the scalar product (z1 | z2 ) := −(z2 | Qz1 ) = −(z1 | Qz2 ),

.

z1 , z2 ∈ Z•(−) ,

(98)

and is also a Hilbert space. The state ω• is represented by the Fock vacuum (o · o) in the doubled Fock space ) ( rs Z•(+) ⊕ Z•(−) = rs (Z•(+) ) ⊗ rs (Z•(−) ).

.

Denote the creation and annihilation operators by a•∗ and a• . The fields w in the GNS representation given by ω• will be denoted by w• . We have w• (w) := a• (||•(+) w) + a•∗ (||•(−) w),

.

w•∗ (w) := a•∗ (||•(+) w) + a• (||•(−) w). The operator dr(S• ) plays the role of a charge: Proposition 6.1 We have a U (1) group of symmetries eisdr(S• ) w• (w)e−isdr(S• ) = e−is w• (w),

.

eisdr(S• ) w•∗ (w)e−isdr(S• ) = eis w•∗ (w).

6.2.3

Two-Component Representations

Suppose V is a complex Hilbertizable space and V∗ is its antidual. The pairing of V and V∗ is denoted (· | ·). We equip V∗ ⊕ V with the Hermitian form ( (v2 , u2 ) | Q(v1 , u1 )) = (u2 | v1 ) + (v2 | u1 ).

.

Then V∗ ⊕ V is a Krein space. Consider the ∗-algebra generated by ψ(u), ψ ∗ (u), u ∈ V∗ , η(v), η∗ (v), v ∈ V satisfying 1. ψ(u)∗ = ψ ∗ (u), η(v)∗ = η∗ (v); 2. the maps V∗ e u |→ ψ(u), V e v |→ η(v) are antilinear; 3. the CCR in the complex version of the two-componenet form hold (we write only non-zero commutators): ] [ ψ(u), η∗ (v) = i(u | v), [ ∗ ] ψ (u), η(v) = i(v | u),

.

An Evolution Equation Approach to Linear Quantum Field Theory

89

for u ∈ V∗ , v ∈ V. To pass to the formalism of Sect. 6.2.1, we set w(v, u) := iη(v) + ψ(u),

.

w ∗ (v, u) := −iη∗ (v) + ψ ∗ (u),

u ∈ V∗ , v ∈ V.

Indeed, [ ] w(v2 , u2 ), w ∗ (v1 , u1 ) = (u2 | v1 ) + (v2 | u1 ).

.

Every admisssible involution S• on V∗ ⊕ V determines a Fock state ω• . Let (±) .||•

[ ] (±) (±) ||•11 ||•12 = (±) (±) . ||•21 ||•22

be the corresponding “particle and antiparticle projection”. The conditions ||•(±)∗ Q = Q||•(±) and ||•(+) + ||•(−) = 1 yield (±)∗ (±) ||•22 = ||•11 ,

.

(±)∗ (±) ||•12 = ||•12 ,

(+) (−) (+) (−) ||•22 + ||•22 = ||•11 + ||•11 = 1,

.

(±)∗ (±) ||•21 = ||•21 ,

(+) (−) (+) (−) ||•12 + ||•12 = ||•21 + ||•21 = 0.

The two-point correlation functions of the state ω• are given by ( ( ) ) ( (+) ' ) ω• η(v)η∗ (v ' ) = ω• η∗ (v ' )η(v) = v | ||•21 v ,. (99a) ( ) ( ) ( (+) ) ω• ψ(u)ψ ∗ (u' ) = ω• ψ ∗ (u' )ψ(u) = u | ||•12 u' , . (99b) ( ( ) ( ) ) ( ) (+) (−) ω• ψ(u)η∗ (v) = i u | ||•11 v , ω• η∗ (v)ψ(u) = −i u | ||•11 v , . (99c) ( ( ) ( ) ) ( ) (+) (−) ω• η(v)ψ ∗ (u) = −i v | ||•22 u , ω• ψ ∗ (u)η(v) = i v | ||•22 u , (99d) .

7 Klein–Gordon Equation and Quantum Field Theory on Curved Space-Times In this section we formulate the main results of this paper in the setting of QFT on curved spacetimes, more precisely, on a globally hyperbolic manifolds equipped with electromagnetic and scalar potentials. We describe the role various propagators play in the theory of quantum fields satisfying the Klein–Gordon equation.

90

J. Derezi´nski and D. Siemssen

The usual presentations of this topic make the assumption that all coefficients in the Klein–Gordon equation are smooth. The results that we have obtained in the previous sections allow us to consider systems with much lower regularity.

7.1 Half-Densities on a Pseudo-Riemannian Manifold Let M be a manifold. A half-density on M is an assignment of a complex function to every coordinate patch satisfying the following condition: if x |→ f (x) and x ' |→ f ' (x ' ) are two such assignments, then | ∂x ' | 1 |2 ' ' | | f (x ) = f (x), | ∂x

.

| '| | | where | ∂x ∂x | denotes the Jacobian of the change of coordinates. The space of square integrable half-densities will be denoted L2 (M). Thus if we choose coordinates x and the support of a function f is contained in the corresponding coordinate patch, then f f1 (x)f2 (x) dx, (100) .(f1 | f2 ) = where dx = dx 1 · · · dx d is the Lebesgue measure. Note that the scalar product (100) is independent of coordinates. Suppose that M is a pseudo-Riemannian manifold with a metric tensor, which μ in coordinates | | x = [x ] is given by the matrix g(x) = [gμν (x)]. Let |g|(x) := | det[gμν (x)]|. M has a distinguished density, which in the coordinates x is given √ by of √ scalar functions, square integrable with respect to √ |g|(x) dx. The space |g|(x) dx, is denoted L2 (M, |g|). Obviously / 1 | |g| 4 f ∈ L2 (M) L2 (M, |g|) e f →

.

is a unitary map.

7.2 Klein–Gordon Equation on Spacetime and the Conserved Current Suppose a pseudo-Riemannian manifold is equipped with a vector field [Aμ (x)], called the electromagnetic potential, and the scalar potential Y (x).

An Evolution Equation Approach to Linear Quantum Field Theory

91

In this and the following subsection we develop the basic formalism thinking of A, Y as smooth functions. In Sect. 7.4 we translate this fomalism to the setting of Sect. 5, which alows for low regularity. The Klein–Gordon operator, written first in the scalar and then in the half-density formalism in any local coordinates is 1

1

1

1

K := −|g|− 2 (Dμ − Aμ )|g| 2 g μν (Dν − Aν ) − Y,

.

1

K := −|g|− 4 (Dμ − Aμ )|g| 2 g μν (Dν − Aν )|g|− 4 − Y. For functions u, v ∈ C ∞ (M) introduce the current, which again we write first in the scalar, then in the half-density formalism: ( ) 1 j μ (x; u, v) := − u(x)g μν (x)|g| 2 (x) Dν − Aν (x) v(x) ( ) 1 − Dν − Aν (x) u(x)g μν (x)|g| 2 (x)v(x), ( ) 1 1 j μ (x; u, v) := − u(x)g μν (x)|g| 4 (x) Dν − Aν (x) |g|− 4 (x)v(x)

.

( ) 1 1 − Dν − Aν (x) |g|− 4 (x)u(x)g μν (x)|g| 4 (x)v(x). We check that if u, v solve the Klein–Gordon equation, that is Ku = Kv = 0,

.

then the current j μ (x; u, v) is conserved, that is ∂μ j μ (x; u, v) = 0.

.

Let M be globally hyperbolic, see e.g. [2]. If o ⊂ M, then J ∨ (o) denotes the future shadow, and J ∧ (o) the past shadow of o, that is, the set of all points in M that can be reached from o by future/past directed causal paths. A set o ⊂ M is called space compact if there exists a compact o ⊂ M such that o ⊂ J ∨ (o) ∪ J ∧ (o). Csc (M) denotes the set of continuous functions on M with a space compact support. Let Wsc denote the set of smooth space compact solutions to the Klein–Gordon equation. For u, v ∈ Wsc , f (u | Qv) :=

j μ (x; u, v) dsμ (x)

.

(102)

E

does not depend on the choice of a Cauchy surface E, where dsμ (x) denotes the natural measure on S times the normal vector. Equation (102) defines a Hermitian form on Wsc called the charge.

92

J. Derezi´nski and D. Siemssen

7.3 Foliating the Spacetime Let us fix a diffeomorphism I × E → M, where I is, as usual, [t− , t+ ] or R, and E is a manifold. In other words, equip M with a time function t = x 0 ∈ I such that all the leaves of the foliation Et = {t} × E are identified with a fixed manifold E. The generic notation for a point of E will be x-. The restriction of g to the tangent space of Et defines a time-dependent family of metrics on E, denoted h(t) = h = [hij ]. We make the assumption that all h are Riemannian, or, what is equivalent, that the covector dt is always timelike. We set |h| = det h. In coordinates, the metric can be written as gμν dx μ dx ν = −α 2 dt 2 + hij (dx i + β i dt)(dx j + β j dt),

.

g μν ∂μ ∂ν = −

1 (∂t − β i ∂i )2 + hij ∂i ∂j . α2

for some α(x) > 0 and [β i (x)]. We have |g| = α 2 |h|. The Klein–Gordon operator in the half-density formalism can now be written 1

K = |g|

.

− 14

1 |g| 2 (D0 − β Di − A0 + β Ai ) 2 (D0 − β i Di − A0 + β i Ai )|g|− 4 α

i

1

i

1

1

− |h|− 4 (Di − Ai )|h| 2 hij (Dj − Aj )|h|− 4 − Y ( ) 1 ( ) Dt + W (t) − L(t), = Dt + W ∗ (t) 2 α (t)

(103)

where 1

1

1

L(t) := |g|− 4 (Di − Ai )|g| 2 hij (Dj − Aj )|g|− 4 + Y . (104a) ) 1 ( ) ( 1 1 i i α,i − Ai |h| 2 hij Dj + α,i − Aj |h|− 4 + Y . = |h|− 4 Di − 2α 2α (104b)

.

W (t) := β i Di − A0 + β i Ai +

i i i |g|,0 − β |g|,i 4|g| 4|g|

(104c)

The temporal component of the current, again in the half-density formalism, is j 0 (x; u, v) = u(x)

.

+

1 α 2 (x) 1

α 2 (x)

(

( ) D0 − W (x) v(x)

) D0 − W (x) u(x)v(x).

An Evolution Equation Approach to Linear Quantum Field Theory

93

We use the half-density formalism to define the spaces L2 (M) and L2 (E). We define L2 (I ) using the Lebesgue measure. We have L2 (M) = L2 (I, E) = L2 (I ) ⊗ L2 (E),

(105)

.

We treat E as equipped with the metric h(t). The operator L(t) is a Hermitian operator on Cc∞ (E) in the sense of the Hilbert space L2 (E). We have written it in two ways: (104a) looks simpler, but the expression (104b) is manifestly covariant with respect to a change of coordinates on E. For brevity, we will write K = L2 (E). Note that K in (103) has the form of an abstract Klein–Gordon operator considered in Sect. 5. Choosing Et for the Cauchy surface we can rewrite (102) as f (u | Qv) =

u(t, x-)

.

E

f

1 α 2 (t, x-)

( ) D0 + W (t, x-) v(t, x-) dx

( ) D0 + W (t, x-) u(t, x-)

+ E

1 α 2 (t, x-)

v(t, x-) dx,

u, v ∈ K.

7.4 Classical Propagators After identifying M = I × E, (105) and (103) show that we are in the setting of Sect. 5 devoted to the abstract Klein Gordon operator. From now on we impose Assumption 5.36. We introduce the formalism of Sect. 5, such as the Hilbertizable spaces Kβ , β ∈ [−1, 1] and Wλ , λ ∈ [− 21 , 12 ], as in (67) and (69), the generator of the evolution B(t) and the evolution itself R(t, s). In particular, the space W0 is a Krein space equipped with the form Q. It is easy to construct the classical propagators in this setting. First we define G• , • = ∨, ∧, PJ as operators Cc (I, K) → C(I, K) as in (73a). By the Schwartz kernel theorem they possess distributional kernels, which we denote G• (x, y) = G• (t, x-; s, y-), It is obvious that .

supp G∧ ⊂ {(t, x-; s, y-) ∈ M × M | t ≤ s}, supp G∨ ⊂ {(t, x-; s, y-) ∈ M × M | t ≥ s}.

One can expect their support to be even smaller, more precisely, that they are causal: .

supp G∧ ⊂ {(x, y) ∈ M × M | x ∈ J ∧ (x)}, supp G∨ ⊂ {(x, y) ∈ M × M | x ∈ J ∨ (x)}, supp GPJ ⊂ {(x, y) ∈ M × M | x ∈ J (x)}.

94

J. Derezi´nski and D. Siemssen

If g, A, Y are smooth, this is very well-known, proven in numerous sources. Under Assumption 5.36 this is presumably also true. Under slightly more restrictive assumptions it follows from Theorem E1 of [9], see also [23] for a different approach.

7.5 Non-Classical Propagators If I is finite, we choose two admissible involutions S± on W0 . If I = R, we assume that the spacetime ( is) stationary for large times and assume that B± are stable. We set S± := sgn B± , which are automatically admissible involutions. With help of these admissible involutions, we define the non-classical propaga(+) (−) tors G± , G± , GF , GF . If I is finite, the non-classical propagators can be interpreted as bounded operators on L2 (M). If GF , GF have zero nullspaces, then we define K F := GF−1 ,

.

K F := GF−1 ,

which can be treated as well-posed realizations of the Klein–Gordon operator satisfying K F∗ = K F . If I = R, then the non-classical propagators can be understood as, say, operators Cc (I, L2 (E)) → C(I, L2 (E)). Suppose we impose the assumption of Proposition 5.14. It is then clear that the operator K is Hermitian (or, as it is often termed, symmetric) on Cc∞ (M). One can ask about the existence of its self-adjoint extensions. This is the subject of following conjecture, which is essentially a spacetime version of Conjectures 5.6 and 5.15. Conjecture 7.1 For a large class of asymptotically stationary and stable Klein– Gordon operators the following holds: 1. The operator K with the domain Cc∞ (M) is essentially self-adjoint in the sense of L2 (M). Denote its unique self-adjoint extension by K s.a. . 2. In the sense of −s L2 (M) → s L2 (M), for s > 12 , .

s-lim(K s.a. − ie)−1 = GF , e-0

s-lim(K s.a. + ie)−1 = GF . e-0

Note that Conjecture 7.1 is true in the stable stationary case, see [8] and [9]. As proven by Vasy [33] and Nakamura–Taira [27–29], it is also true for some classes of asymptotically Minkowskian spacetimes. Kami´nski described a counterexample to a certain strong version of this conjecture [24].

An Evolution Equation Approach to Linear Quantum Field Theory

95

7.6 Charged Fields Let us now describe the formalism of classical and quantum field theory in our setting. Note that the formalism described in the introduction used pointlike fields. In this section pointlike fields may be ill-defined because of insufficient smoothness. Therefore we prefer to use smeared fields. Recall that elements of W0 can be written as two component vectors: [ ] w1 . .w = w2 The space W0 is preserved by the dynamics R(t, s). We treat time t = 0 as the “reference time”. For any t ∈ I , u ∈ K− 1 , v ∈ K 1 we define the following functionals on W0 : 2

2

f =

.

=

f

( ) u(x ) R(t, 0)w 1 (x ) dx, ( ) x ) dx, u(x ) R(t, 0)w 1 (f

( ) v(x ) R(t, 0)w 2 (x ) dx,

= −i

f =i

( ) x ) dx. v(x ) R(t, 0)w 2 (-

From the symplectic structure associated with the charge form Q we derive the Poisson brackets between ψt , ψt∗ , ηt , ηt∗ . Below we present only the non-zero cases: { .

{

} ψt (u), ηt∗ (v) =

ψt∗ (u), ηt (v)

}

f u(x )v(x ) dx, f

=

x ) dx. u(x )v(-

The first step of quantization is the replacement of the Poisson bracket by i times the commutator. Thus we obtain the commutation relations f [ ] ˆ t (u), ηˆ t∗ (v) = i u(. ψ x )v(x ) dx, [ ∗ ] ψˆ t (u), ηˆ t (v) = i

f u(x )v(x ) dx.

Then one chooses the in and the out Fock state. Recall that they are determined by two admissible involutions S± . From S± we obtain two pairs of complementary (+) (−) (+) (−) projections ||± , ||± . Following the recipe (99), ||± and ||± are used to define

96

J. Derezi´nski and D. Siemssen

the Fock states ω± = (o± | · o± ). In the slab geometry case this is done using the fields ψˆ t± , ψˆ t∗± , ηˆ t± , ηˆ t∗± . In the unrestricted time case it is done using ψˆ t , ψˆ t∗ , ηˆ t , ηˆ t∗ with ±t > T . We can also smear the fields with spacetime functions. Suppose that, say, f ∈ Cc (I, K− 1 ). Set 2

f ψ[f ] :=

.

( ) ψt f (t, · ) dt,

ψ ∗ [f ] :=

f

( ) ψt∗ f (t, · ) dt.

The Poisson bracket of the fields is known then as the Peierls bracket: ff { } ∗ f1 (x)GPJ (x, y)f2 (y) dx dy. . ψ[f1 ], ψ [f2 ] = −

(107)

Similarly, we can smear the quantum fields: ˆ ] := ψ[f

.

f

( ) ψˆ t f (t, ·) dt,

ψˆ ∗ [f ] :=

f

( ) ψˆ t∗ f (t, ·) dt.

(108)

The commutator of fields is expressed by the Peierls bracket. [ ] ˆ 1 ], ψˆ ∗ [f2 ] = −i ψ[f

ff f1 (x)GPJ (x, y)f2 (y) dx dy.

.

(109a)

The vacuum expectation values of the products of fields are expressed by the positive/negative frequency bisolutions: ( ) ˆ 1 ]ψˆ ∗ [f2 ]o± = o± | ψ[f

ff

( ) ˆ 1 ]o± = o± | ψˆ ∗ [f2 ]ψ[f

(+)

(109b)

(−)

(109c)

f1 (x)G± (x, y)f2 (y) dx dy, .

.

ff

f1 (x)G± (x, y)f2 (y) dx dy.

Let T{} denote the time-ordered product and and T{} the reverse time-ordered product. Assume in addition the Shale condition for ω+ and ω− , so that o+ and o− can be treated as vectors in the same representation. Then the vacuum expectation values of the time-ordered and reverse time-order products divided by the overlap of the vacua is expressed by the Feynman, resp. anti-Feynman inverses: ( ) ff ˆ 1 ]ψˆ ∗ [f2 ]}o− o+ | T{ψ[f = −i . f1 (x)GF (t, s)f2 (y) dx dy, . (o+ o− ) ( ) ff ˆ 1 ]ψˆ ∗ [f2 ]}o+ o− T{ψ[f =i f1 (x)GF (t, s)f2 (y) dx dy. (o− | o+ )

(110a) (110b)

An Evolution Equation Approach to Linear Quantum Field Theory

97

Note, however, that as already mentioned in the introduction the RHS of (110) is well-defined also if the Shale condition is not satisfied. Here are identities that can be used to prove the above formulas: ( ) ( ) ψt (u) = ψs R(s, t)22 u + iηs R(s, t)12 u , . ( ( ) ) ψt∗ (u) = ψs∗ R(s, t)22 u − iηs∗ R(s, t)12 u . .

(111a) (111b)

They follow the definition of ψt and ψt∗ and from the pseudounitarity of R(s, t). Commuting ψs (v) with (111b) we obtain {ψs (v), ψt (u)} = −i(v | R(s, t)12 u) f x )GPJ (s, x-; t, y-)u(y ) dx dy, = − v(-

.

from which the formula for the Peierls bracket (107) follows. Clearly, the quantized version of (111) with all fields decorated with hats is also true. It implies (109a) Remark 7.2 In most physics literature one uses pointlike fields, denoted typically ˆ x ∈ M, (not to be confused with the spatially smeared fields ψˆ ∗ (x), ψ(x), ψˆ t∗ (u), ψˆ t (u)). Formally, the smeared-out fields are given by ˆ ] := ψ[f

f

.

ψˆ ∗ [f ] :=

f

ˆ f (x)ψ(x) dx, f (x)ψˆ ∗ (x) dx.

Smeared-out fields are more typical for the mathematics literature, since they can be interpreted as closed densely defined operators (at least in a linear QFT). Nevertheless, pointlike fields are convenient. We used them in the introduction. Note that identities (109a), (109b), (109c), (110a) and (110b) are equivalent to identities (4a), (18a), (18b), (19a) and (19b).

7.7 Neutral Fields Suppose that the electromagnetic potential [Aμ ] is absent. Then the Klein–Gordon operator 1

1

1

K := |g|− 4 ∂μ |g| 2 g μν ∂ν |g|− 4 − Y.

.

is real. In particular, the spaces Kβ , Wλ can be equipped with the usual complex conjugation and their real subspaces Kβ,R , Wλ,R can be defined. Note that W0,R

98

J. Derezi´nski and D. Siemssen

is a real Krein space equipped with the symplectic form := Im(v | Qw).

.

Also note that the evolution satisfies R(t, s) = R(t, s) and thus it can be restricted to W0,R . For any t ∈ I , u ∈ K− 1 ,R , v ∈ K 1 ,R we define the following functionals on 2 2 W0,R : f =

.

( ) u(x ) R(t, 0)w 1 (x ) dx, f

= −i

( ) v(x ) R(t, 0)w 2 (x ) dx.

From the symplectic structure given by the form ω we derive the Poisson brackets between φt , πt . Below we present the only non-zero relation: } { φt (u), πt (v) =

.

f u(x )v(x ) dx.

The first step of quantization is the replacement of the Poisson bracket by i times the commutator. Thus we obtain the commutation relations f [ ] ˆ . φt (u), π ˆ t (v) = i u(x )v(x ) dx. Then one chooses the in and the out Fock state—as in the charged case. In addition, in the slab geometry case, we demand that the two admissible involutions S± on W0 are anti-real. (In the unrestricted time case this is automatic.) Thus the (+) (−) two pairs of complementary projections ||± , ||± obtained from S± are real. We can also smear the fields with space-time functions. Suppose that, say, f ∈ Cc (I, K− 1 ,R ). Set 2

ˆ ] := φ[f

f

.

( ) φˆ t f (t, ·) dt.

Now we have the following identities: [ ] ˆ 1 ], φ[f ˆ 2 ] = −i . φ[f ( ) ˆ 1 ]φ[f ˆ 2 ]o± = o± | φ[f

ff

ff f1 (x)GPJ (x, y)f2 (y) dx dy, . (+) f1 (x)G± (x, y)f2 (y) dx dy, .

(112a) (112b)

An Evolution Equation Approach to Linear Quantum Field Theory

( ) ff ˆ 1 ]φ[f ˆ 2 ]}o− o+ | T{φ[f = −i f1 (x)GF (t, s)f2 (y) dx dy, . (o+ | o− ) ( ) ff ˆ 1 ]φ[f ˆ 2 ]}o+ o− | T{φ[f =i f1 (x)GF (t, s)f2 (y) dx dy. (o− | o+ )

99

(112c) (112d)

As in the charged case, also here the Shale conditions is required for the LHS of (112c) and (112d) to be well-defined. Remark 7.3 In most physics literature one uses pointlike fields, denoted typically ˆ φ(x), x ∈ M. Formally, the smeared-out fields are given by ˆ ] := .φ[f

f

ˆ f (x)φ(x) dx.

for f ∈ Cc∞ (M, R), Acknowledgments J.D. acknowledges the support of National Science Center (Poland) under the grant UMO-2019/35/B/ST1/01651. He also benefited from the support of Istituto Nazionale di Alta Matematica “F. Severi”, through the Intensive Period “INdAM Quantum Meetings (IQM22)”. Moreover, he is grateful to C. Gérard, M. Wrochna and W. Kami´nski for useful discussions.

References 1. Bär, C., Fredenhagen, K. (eds.): Quantum Field Theory on Curved Spacetimes. Lecture Notes in Physics, vol. 786. Springer, Berlin (2009). https://doi.org/10.1007/978-3-642-02780-2 2. Bär, C., Ginoux, N., Pfäffle, F.: Wave Equations on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematical Physics. European Mathematical Society, Helsinki (2007). https://doi.org/10.4171/037 3. Birrell, N.D., Davies, P.C.W.: Quantum Fields in Curved Space. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1984) 4. Björken, J.D., Drell, S.D.: Relativistic Quantum Fields. McGraw-Hill, New York (1965) 5. Bogoliubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields, 3 edn. John Wiley & Sons, Hoboken (1980) 6. Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds. Commun. Math. Phys. 208(3), 623–661 (2000). https://doi.org/10.1007/s002200050004 7. Derezi´nski, J., Gérard, C.: Mathematics of Quantization and Quantum Fields. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2013) 8. Derezi´nski, J., Siemssen, D.: Feynman propagators on static spacetimes. Rev. Math. Phys. 30(3), 1850006 (2018) 9. Derezi´nski, J., Siemssen, D.: An evolution equation approach to the Klein–Gordon operator on curved spacetime. Pure Appl. Anal. 1(2), 215–261 (2019). https://doi.org/10.2140/paa.2019.1. 215 10. DeWitt, B.S.: Quantum field theory in curved spacetime. Phys. Rep. 19(6), 295–357 (1975). https://doi.org/10.1016/0370-1573(75)90051-4

100

J. Derezi´nski and D. Siemssen

11. Duistermaat, J.J., Hörmander, L.: Fourier integral operators. II. Acta Math. 128(1), 183–269 (1972). https://doi.org/10.1007/BF02392165 12. Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford University Press, Oxford (2018) 13. Fewster, C.J., Verch, R.: Dynamical locality and covariance: what makes a physical theory the same in all spacetimes? Ann. Henri Poincaré 13(7), 1613–1674 (2012). https://doi.org/10. 1007/s00023-012-0165-0 14. Fewster, C.J., Verch, R.: The necessity of the Hadamard condition. Classical Quantum Gravity 30(23), 235027 (2013). https://doi.org/10.1088/0264-9381/30/23/235027 15. Friedlander, F.G.: The Wave Equation on a Curved Space-Time. Cambridge University Press, Cambridge (1975) 16. Fukuma, M., Sugishita, S., Sakatani, Y.: Propagators in de Sitter space. Phys. Rev. D 88, 024041 (2013) 17. Fulling, S.A.: Aspects of Quantum Field Theory in Curved Space-Time. London Mathematical Society Student Texts, vol. 17. Cambridge University Press, Cambridge (1989) 18. Fulling, S.A., Narcowich, F.J., Wald, R.M.: Singularity structure of the two-point function in quantum field theory in curved spacetime, II. Ann. Phys. 136(2), 243–272 (1981). https://doi. org/10.1016/0003-4916(81)90098-1 19. Gérard, C., Wrochna, M.: Hadamard property of the in and out states for Klein-Gordon fields on asymptotically static spacetimes. Ann. Henri Poincaré 18(8), 2715–2756 (2017). https://doi. org/10.1007/s00023-017-0573-2 20. Gérard, C., Wrochna, M.: The massive Feynman propagator on asymptotically Minkowski spacetimes. Am. J. Math. 141(6), 1501–1546 (2019) 21. Hollands, S.: Renormalized quantum Yang–Mills fields in curved spacetime. Rev. Math. Phys. 20(9), 1033–1172 (2008). https://doi.org/10.1142/S0129055X08003420 22. Hollands, S., Wald, R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223(2), 289–326 (2001). https://doi.org/10. 1007/s002200100540 23. Hörmann, G., Sanchez Sanchez, Y., Spreitzer, C., Vickers, J.A.: Green operators in low regularity spacetimes and quantum field theory. Classical Quantum Gravity 37(17), 175009 (2020). https://doi.org/10.1088/1361-6382/ab839a 24. Kami´nski, W.: Non-self-adjointness of the Klein-Gordon operator on a globally hyperbolic and geodesically complete manifold: an example. Ann. Henri Poincaré 23, 4409–4427 (2022) 25. Kato, T.: Linear evolution equations of “hyperbolic” type. J. Fac. Sci. Univ. Tokyo Sect. I 17, 241–258 (1970) 26. Leray, J.: Hyperbolic Differential Equations. Unpublished Lecture Notes. The Institute for Advanced Study, Princeton (1953) 27. Nakamura, S., Taira, K.: Essential self-adjointness of real principal type operators. Ann. Henri Lebesgue 4, 1035–1059 (2021). https://doi.org/10.5802/ahl.96 28. Nakamura, S., Taira, K.: Essential self-adjointness for the Klein-Gordon type operators on asymptotically static spacetime. Commun. Math. Phys. 398(3), 1153–1169 (2022) 29. Nakamura, S., Taira, K.: A remark on the essential self-adjointness for Klein-Gordon type operators. Ann. Henri Poincaré (2022). https://doi.org/10.1007/s00023-023-01277-2 30. Parker, L.E., Toms, D.J.: Quantum Field Theory in Curved Spacetime. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2009)

An Evolution Equation Approach to Linear Quantum Field Theory

101

31. Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179(3), 529–553 (1996). https://doi.org/10.1007/ BF02100096 32. Rumpf, H., Urbantke, H.K.: Covariant “in–out” formalism for creation by external fields. Ann. Phys. 114, 332–355 (1978). https://doi.org/10.1016/0003-4916(78)90273-7 33. Vasy, A.: Essential self-adjointness of the wave operator and the limiting absorption principle on Lorentzian scattering spaces. J. Spectral Theory 10(2), 439–461 (2020) 34. Wald, R.M.: The back reaction effect in particle creation in curved spacetime. Commun. Math. Phys. 54(1), 1–19 (1977). https://doi.org/10.1007/BF01609833 35. Wald, R.M.: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago Lectures in Physics. University of Chicago Press, Chicago (1994)

Renormalization of Spin–Boson Interactions Mediated by Singular Form Factors Davide Lonigro

1 Introduction The description of physical systems obeying the laws of quantum mechanics usually involves a distinction between a singled out subsystem under experimental control, usually finite-dimensional and simply referred to as an “open quantum system”, and an external “environment” (or bath) which is typically infinite-dimensional and affects the system in a nontrivial way, yielding diverse effects like population decay, noise, or loss of quantum coherence [1–6]. Two possible, and complementary, approaches to the study of such systems are available: one either focuses ab initio on the open system alone, whose dynamics will be described by a completely positive and trace-preserving map on the Hilbert space of the open system (e.g. a Gorini–Kossakowski–Lindblad–Sudarshan semigroup [7, 8]), or tackles the problem directly at the “system+environment” level, which encompasses the choice of a legitimate self-adjoint operator—the Hamiltonian—generating the joint unitary dynamics of the system and the environment; the evolution of the open system alone is thus recovered by tracing out the degrees of freedom of the environment. Spin–boson models and their many generalizations, modeling an open system (e.g. an atom) interacting with a structured boson field via annihilation and creation of field excitations, are typical examples of such models. They provide a treatable, yet usually realistic, description of many quantum phenomena of both fundamental and technological interest, and are widely used in diverse fields like quantum information, quantum optics and quantum simulation, to name a few [9, 10]. From

D. Lonigro (o) Dipartimento di Fisica and MECENAS, Università di Bari, and INFN, Sezione di Bari, Bari, Italy Dipartimento di Matematica, Università di Bari, Bari, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Correggi, M. Falconi (eds.), Quantum Mathematics II, Springer INdAM Series 58, https://doi.org/10.1007/978-981-99-5884-9_3

103

104

D. Lonigro

the mathematical point of view, a lot of effort has been devoted to the investigation of the properties of spin–boson models [11–18] and generalized spin–boson (GSB) models [19–24]. The prototypical example of such models is the following one. Let .H = L2μ (X) be the space of measurable complex-valued functions on a measure space .(X, μ) with finite quadratic norm, that is, f ||f ||2 :=

.

|f (k)|2 dμ(k) < ∞,

(1)

where, here and in the following, all integrals are understood to be extended on the full space X. We denote as .F(H) ≡ F the symmetric (or bosonic) Fock space with single-particle space .H, defined as usual: F=

O

.

Sn H (n) ,

H (n) =

n∈N

n O

H,

(2)

j =1

with .Sn being the symmetrization operator on .H (n) . This is the Hilbert space to be associated with the boson environment, with the Hilbert space of the atom simply being .h = C2 . On their tensor product .H = C2 ⊗F, we define the following operator: ( H0 =

.

ωe 0 0 ωg

) ⊗ I + I ⊗ dr(ω),

(3)

with .ωe , ωg ∈ R, and .dr(ω) being the second quantization of the multiplication operator associated with a real-valued, measurable function .ω : X → R; the latter is self-adjoint on its maximal domain .D(dr(ω)), see e.g. [25, 26], and so is .H0 on .D(H0 ) = C2 ⊗ D(dr(ω)). The self-adjoint operator .H0 represents the free (or decoupled) Hamiltonian of the theory; the function .ω is the dispersion relation of the field. We shall restrict our attention to boson fields with strictly positive mass, that is, we shall suppose m := inf ω(k) > 0.

(4)

( ) Hf = H0 + λ σ + ⊗ a(f ) + σ − ⊗ a † (f ) ,

(5)

.

k∈X

We then define .

with .λ ∈ R being a coupling constant, .σ ± being the usual ladder operators on .C2 : σ+ =

.

(

01 00

)

= (σ − )† ,

(6)

Renormalization of Spin–Boson Interactions Mediated by Singular Form Factors

105

and .a(f ) , a † (f ) being the annihilation and creation operators associated with some function .f ∈ H, which we denote as the form factor of the model. This Hamiltonian, which is self-adjoint on .D(Hf ) = D(H0 ), corresponds to a variation of the standard spin–boson model in which counter-rotating terms—those not preserving the total number of excitations—are discarded, a procedure often denoted as the rotating-wave approximation (RWA) [27], and finds applications in a wide range of situations; in particular, it reduces to the well-known Jaynes–Cummings model [28– 30] in the case of a monochromatic field, simply obtained by taking .μ as an atomic (Dirac) measure. A limitation of the model is the following: in order for the expressions above to correctly define a self-adjoint operator on .H, the form factor f must be a squareintegrable function, .f ∈ H. Without this constraint, the annihilation operator .a(f ), while still densely defined, fails to admit a densely defined adjoint [31]. Nevertheless, a rigorous extension to singular (i.e. non-normalizable, .f ∈ / H) form factors is both desirable for physical purposes, since singular form factors are often found in the physical literature, as well as expected on the basis of the comparison with similar models such as rank-one singular perturbations of differential operators [32–34] and Friedrichs (or Friedrichs–Lee) models [35–37]. It is also worth noting that, taking into account Eq. (4), this problem essentially corresponds to describing GSB models whose form factors exhibit ultraviolet (UV) divergences; cf. [38–42] for a detailed discussion of the same problem for other models of matter–field interaction. An extension of the model above accommodating possibly singular form factors, together with analogous (but perturbative) results for all GSB models, was obtained in [24] by exploiting the formalism of Hilbert scales. Given .s ∈ R, define .Hs as the space of all measurable functions satisfying the constraint f ||ωs/2 f ||2 =

.

ω(k)s |f (k)|2 dμ < ∞;

(7)

as a direct consequence of Eq. (4), this condition is either stronger (if .s > 0) or weaker (if .s < 0) than the normalization condition (1), whence a scale of Hilbert spaces, .

. . . ⊃ H−2 ⊃ H−1 ⊃ H0 ≡ H ⊃ H1 ⊃ H2 ⊃ . . . ,

(8)

each naturally endowed with the norm .||f ||s := ||ωs/2 f ||, is obtained. In the same way, defining .Fs as the space of all sequences .w = {w (n) }n∈N (with .w (n) being a completely symmetric n-particle function) satisfying || || ||(dr(ω) + 1)s/2 w ||2 < ∞, F

.

(9)

106

D. Lonigro

with .||·||F denoting the norm on the Fock space .F, a scale of Fock spaces is obtained: .

. . . ⊃ F−2 ⊃ F−1 ⊃ F0 ≡ F ⊃ F1 ⊃ F2 ⊃ . . . ,

(10)

each space being naturally endowed with the norm .||w||Fs := ||(dr(ω)+1)s/2 w||F . All inclusions in Eqs. (8) and (10) are dense. With these definitions, the basic idea at the root of [24] is the following: given a singular form factor .f ∈ H−s for .s > 0, a legitimate self-adjoint operator to be associated with the (otherwise just formal) expression (5) can be constructed by interpreting .a(f ) and .a † (f ), instead as unbounded operators on the Fock space .F, as continuous (bounded) maps on the scale of Fock spaces. The construction of such a “singular” spin–boson model was performed in the case .f ∈ H−1 by means of a careful choice of the self-adjointness domain, and then further generalized to the case .f ∈ H−2 ; remarkably, in the latter case one needs to “trade” the excitation energy .ωe of the two-level system in Eq. (5) with a new parameter representing its dressed, or renormalized, excitation energy. This is reminiscent of the renormalization procedures usually encountered in quantum field theories. The “price to pay” for this extension will be a modification of the original operator domain and, in the case .f ∈ H−2 \ H−1 , even of its form domain. In this work we shall summarize the aforementioned results for the spin–boson model with a rotating-wave approximation, with particular emphasis on the ideas at the ground of the renormalization procedure, and then discuss the extension of the same techniques to other instances of generalized spin–boson (GSB) models whose interaction term has an analogous rotating-wave structure, thus allowing for an explicit calculation of the resolvent instead of relying on perturbative methods. Namely, we shall investigate: • an extension of the rotating-wave spin–boson model in which the two-level system .h = C2 is replaced by a system of arbitrary (but finite) dimension decomposed into the direct sum of an “excited” and a “ground” sector, .h = he ⊕ h g ; • the multi-atom generalization of the rotating-wave spin–boson model, living on N the Hilbert space .H = C2 ⊗ F, with N being the number of two-level systems. The work is organized as follows. In Sect. 2 we briefly revise the formalism of singular annihilation and creation operators; in Sect. 3 we summarize the main results for the rotating-wave spin–boson model, first with form factor .f ∈ H−1 and then with form factor up to .f ∈ H−2 ; in Sects. 4–6 we investigate the aforementioned generalizations of said model, presenting preliminary results and discussing future developments; concluding remarks are provided in Sect. 7.

Renormalization of Spin–Boson Interactions Mediated by Singular Form Factors

107

2 Singular Annihilation and Creation Operators For convenience, we shall start by briefly recalling the construction of singular annihilation and creation operators presented in [24], which enables us to model atom–field interactions mediated by singular form factors. Given a dispersion relation .ω ≥ m and the corresponding free field operator .dr(ω), let .{Hs }s∈R , .{Fs }s∈R be the corresponding scales of Hilbert spaces as defined in the introduction. Recall that, in particular, for all .r, s ∈ R the operator r .(dr(ω) + 1) can be continuously extended to an isometry between the spaces .Fs and .Fs−2r [32, 33, 43]. Consequently, for all .s > 0 the triple .(Fs , F, F−s ) is a Gelfand triple [44–46], with the spaces .F±s being mutually dual with respect to the duality pairing (w, o) ∈ F−s × F+s |→ (w, o)F−s ,Fs / \ := (dr(ω) + 1)−s/2 w, (dr(ω) + 1)+s/2 o , F

.

(11)

with .F being the scalar product on the Fock space .F. Proposition 1 ([24], Props. 3.4, 3.5 and 3.7) Let .f ∈ H−s for some .s ≥ 1. Then the following statements hold true: (i) the operator .a(f ) : Fs → F acting as .a(f ) o = 0 and, for all .n ≥ 1, as .

f ( ) √ a(f ) w (n) (k1 , . . . , kn−1 ) = n f (kn )w (n) (k1 , . . . , kn−1 , kn ) dμ(kn ), (12)

is well-defined and bounded; (ii) its adjoint .a † (f ) := a(f )∗ : F → F−s with respect to the pairing (11) is likewise well-defined and bounded, and, for .w ∈ F+1 , ( .

) a † (f ) w (n) (k1 , . . . , kn , kn+1 ) =√

(E n

1 n+1

+w

(n)

j th

w

(n)

'''' (k1 , . . . , kn+1 , . . . , kn )f (kj )

j =1

) (k1 , . . . , kn )f (kn+1 ) ;

(13)

108

D. Lonigro

(iii) finally, there exists a sequence of regular form factors .{f i }i∈N ⊂ H such that .

|| || || || lim ||a(f ) − a(f i )|| = 0, i→∞ B(Fs ,F)

|| || || || lim ||a † (f ) − a † (f i )|| = 0, i→∞ B(F,F−s ) (14)

and this happens if and only if .||f − f i ||−s → 0. Above, .o is the vacuum state of the boson field, i.e. the single normalized element (up to an irrelevant phase term) of .H(0) , while .B(·, ·) stands for the linear space of continuous maps between two Banach (in this case Hilbert) spaces. The proof of (i)–(ii) is basically an application of standard estimates on the norm of .a(f ) w (cf. [31, Eq. (5)] or [47]) translated within the language of Fock scales, while (iii) follows from the fact that .H is densely embedded into .H−s . Clearly, when .f ∈ H, the action of both operators above coincide with that of the “standard” annihilation and creation operators (defined as unbounded operators on .F) whenever the latter are defined, thus justifying our notation and nomenclature; when .f ∈ H−s \ H, i.e. in the case of non-normalizable form factors, the singular operators can be approximated by the regular ones in a suitable topology.

3 Renormalization of the Rotating-Wave Spin–Boson Model Let us start by considering the spin–boson model in Eq. (5) with a regular form factor .f ∈ H; without loss of generality, we shall set the ground energy of the atom to zero, .ωg = 0. By using the obvious isomorphism .H = C2 ⊗ F = F ⊕ F, the model can be conveniently written in a formal matrix fashion: ( Hf,ωe =

.

) ωe + dr(ω) λa(f ) , λa † (f ) dr(ω)

(15)

) ( with domain .D Hf,ωe = C2 ⊗ D(dr(ω)) = D(dr(ω)) ⊕ D(dr(ω)). For future convenience, in the remainder of this section we will explicitly indicate the dependence on .ωe . The domain includes, in particular, the state ( w0 =

.

) o , 0

(16)

representing the configuration in which the atom is in its excited state and the field is in its vacuum state .o ∈ F. Notice that, in particular, the mean value and the variance

Renormalization of Spin–Boson Interactions Mediated by Singular Form Factors

109

of the total energy of the system in this state read .

/

2 Hf,ω e

\ w0

< > Hf,ωe w = ωe , . 0

< >2 − Hf,ωe w = λ2 ||f ||2 . 0

(17) (18)

If .f ∈ / H, Eq. (15) does not define a legitimate operator on .F with the domain prescription chosen above: even by interpreting .a(f ) , a † (f ) in the sense of Proposition 1, the latter operator has values outside the Fock space .F when applied on .D(dr(ω)) ⊕ D(dr(ω)). A different domain prescription, which will turn out to depend both on the choice of form factor f as well as the coupling constant .λ, is therefore required. We shall present the main results hereafter; a summary of our findings can be found in Table 1 at the end of this section.

3.1 Case f ∈ H−1 Let us start by addressing the case .f ∈ H−1 . First, to this purpose it is useful to compute the resolvent of the regular model above. The latter reads as follows [24]: for all .we , wg ∈ F, and .z ∈ C \ R, 1

(

we . Hf,ωe − z wg ⎛

)

) ⎞ ( 1 G−1 (z) w − λa(f w ) e g f,ωe dr(ω)−z )⎠ ( =⎝ 1 1 1 † (f ) G−1 (z) w − λa(f ) w −λ a e f,ωe dr(ω)−z g dr(ω)−z dr(ω)−z wg (19)

where .G−1 f,ωe (z), the propagator of the model, is defined as the bounded inverse [24, Lemma 5.3] of the operator with domain .D(dr(ω)) Gf,ωe (z) = ωe − z + dr(ω) − λ2 Sf (z),

.

(20)

where Sf (z) = a(f )

.

1 a † (f ) . dr(ω) − z

(21)

Now, a close scrutiny to Eqs. (19)–(21) shows that, provided that one interprets a(f ) and .a † (f ) as continuous maps on the scale of Fock spaces constructed before (see Proposition 1), the latter expression is indeed well-defined even when, more

.

110

D. Lonigro

generally, .f ∈ H−1 ; indeed, (dr(ω)−z)−1

a †(f )

a(f )

F −−−→ F−1 −−−−−−−→ F+1 −−→ F,

(22)

.

whence the map .Sf (z) in Eq. (21) is bounded on .F. This simple observation is at the root of the following theorem: Theorem 1 ([24], Theorem 5.5) Let .f ∈ H−1 , .ωe , λ ∈ R, and let .Hf,ωe be the operator on .F ⊕ F with domain {( D(Hf,ωe ) =

.

}

)

oe 1 a † (f ) oe og − λ dr(ω)+1

: oe , og ∈ D(dr(ω)) ,

(23)

acting as ( Hf,ωe

.

oe

)

1 a † (f ) oe og − λ dr(ω)+1

(( ) ) ωe + dr(ω) − λ2 Sf (−1) oe + λ a(f ) og = . 1 a † (f ) oe dr(ω)og + λ dr(ω)+1

(24)

Then: (i) for .f ∈ H, the operator coincides with the spin–boson Hamiltonian in Eq. (15); (ii) for .f ∈ H−1 \ H, the operator is self-adjoint, its resolvent reads as in Eq. (19), and there exists a sequence .{f i }i∈N ⊂ H such that .Hf i ,ωe → Hf,ωe in the norm resolvent sense. In particular, this happens iff ||f i − f ||−1 → 0.

.

(25)

Points (i) and the first part of (ii) are essentially a direct consequence of the mapping properties of .a(f ) and .a † (f ), while the last part of (ii) follows from Proposition 1(iii). It is worth discussing the structure of the operator domain (23), which involves the presence of an additional term .λ(dr(ω) + 1)−1 a † (f ) in the second component of the states. If .f ∈ H, the latter term is in .D(dr(ω)) and can be simply reabsorbed in the definition of .og , so that the excited and ground state wavefunctions can be chosen independently from each other. Instead, if .f ∈ H−1 \ H, the additional term does not belong to .D(dr(ω)) and cannot be reabsorbed, thus behaving as a “singular” potential effectively coupling the two components. Physically, in order for the variance of the total energy of the system in a state to be finite, its free boson energy must diverge. In this sense, in the limit .Hf i ,ωe → Hf,ωe , the operator domain

Renormalization of Spin–Boson Interactions Mediated by Singular Form Factors

111

of the model exhibits a highly “irregular” behavior—it is constant for all .i ∈ N, and changes abruptly in the limit—while the resolvent behaves regularly. Finally, notice that the operator domain in the case .f ∈ H−1 \H does not include .w0 , the excited state of the atom interacting with the boson vacuum as defined in Eq. (16); however, .w0 is still in the form domain. This is in full accordance with Eq. (17): the variance of the total energy must diverge, while its average value does not change.

3.2 Case f ∈ H−2 Without any further modification, the result above cannot be extended to a larger class of form factors. Namely, if .f ∈ H−s \ H−1 for some .s > 1, then the operator .Sf (z) in Eq. (21) (and thus the propagator itself) fails to be well-defined. Indeed, again by the properties of .a(f ) and .a † (f ) collected in Proposition 1, we have a †(f )

(dr(ω)−z)−1

F −−−→ F−s −−−−−−−→ F−s+2 ⊃ F+s ,

.

(26)

the latter inclusion being proper since .s > 1, whence .a(f ) cannot be applied. However, as long as .s ≤ 2, the difference between the values of .Sf (z) in two distinct points outside the spectrum of .dr(ω), say .z ∈ C \ R and .z0 = −1, is in fact well-defined. Indeed, because of the first resolvent identity, the difference −1 − (dr(ω) + 1)−1 is “more regular” than each of the two terms alone, .(dr(ω) − z) i.e. it satisfies (dr(ω) − z)−1 − (dr(ω) + 1)−1 : F−s → F−s+4 ,

.

(27)

whence a †(f )

(dr(ω)−z)−1 −(dr(ω)+1)−1

a(f )

F −−−→ F−s −−−−−−−−−−−−−−−−→ F−s+4 ⊂ F+s −−→ F,

.

(28)

the latter inclusion holding provided that .s ≤ 2. This observation suggests that it may be possible to define a renormalized version of the operator-valued map .Sf (z) accommodating a form factor .f ∈ H−s \ H−1 up to .s = 2. To this purpose, however, a slightly stronger assumption will be needed. Given .s ∈ (1, 2] and .r ∈ [s − 1, 1], we define r .H−s

{ f := f ∈ H−s :

} |f (k)|2 s−r dμ(k) = O(n ) , [ω(k) + (n − 1)m]s

(29)

that is, the subspace of functions in .H−s such that, in addition, the sequence of integrals in Eq. (29) decays “sufficiently quickly” as .n → ∞. As shown in [24, Lemma 6.2 and Lemma 6.5], this technical request, which is typically satisfied in

112

D. Lonigro

realistic models, allows one to define a renormalized propagator in the following ˜ f (z) such that sense: there exists an (unbounded) operator-valued map .S ˜ f (z) = Sf (z) + ||f ||2 ; • if .f ∈ H−1 , then .S −1 r ˜ f (z) is still a well• if .f ∈ H−s for some .s ∈ (1, 2] and .r ∈ [s − 1, 1], then .S defined operator which, in addition, is relatively bounded with respect to .dr(ω) (and even infinitesimally relatively bounded if .r < 1). Consequently, looking at the definition (20) of the operator .Gf,ωe (z), the following holds: given .ω˜ e ∈ R and the operator on .F defined via ˜ f (z), ˜ f,ω˜ (z) = ω˜ e − z + dr(ω) − λ2 S G e

(30)

.

we can conclude that ˜ f,ω˜ (z) = Gf,ω (z) with the latter as defined in Eq. (20), • if .f ∈ H−1 , then .G e e provided that ω˜ e = ωe + λ2 ||f ||2−1 ;

(31)

.

r ˜ f,ω˜ (z) is still a • if .f ∈ H−s for some .s ∈ (1, 2] and .r ∈ [s − 1, 1], then .G e well-defined operator with domain .D(dr(ω)).

/ Heuristically, this can be interpreted as follows: the divergence of .Sf (z) for .f ∈ H−1 is “cured” by adding an “infinite constant” to it; correspondingly, the same constant must be added to the excitation energy .ωe of the atom. This construction allows us to prove the following theorem, which further extends the results of Theorem 1: Theorem 2 ([24], Theorem 6.6) Let .f ∈ Hr−s for some .s ∈ [1, 2] and .r ∈ [s − 1, 1], .ω˜ e , λ ∈ R, and let .H˜ f,ω˜ e be the operator on .F ⊕ F with domain as in Eq. (23), acting as ( H˜ f,ω˜ e

.

=

oe

)

1 og − λ dr(ω)+1 a † (f ) oe ) (( ) ˜ f (−1) oe + λ a(f ) og ω˜ e + dr(ω) − λ2 S 1 dr(ω)og + λ dr(ω)+1 a † (f ) oe

.

(32)

Then (i) if .s = 1, the operator coincides with the spin–boson Hamiltonian in Eq. (15), provided that .ωe is as given by Eq. (31); (ii) if .s ∈ (1, 2] and .r ∈ [s − 1, 1), then, for all .λ ∈ R, the operator is selfadjoint, and there exist sequences .{f i }i∈N ⊂ H, .{ωei }i∈N ⊂ R such that ˜ f,ω˜ in the strong resolvent sense; in particular, this happens iff .Hf i ,ωi → H e e

Renormalization of Spin–Boson Interactions Mediated by Singular Form Factors

113

Table 1 Classification of the main properties of the regular spon–boson model and the singular one for all values .1 ≤ s ≤ 2 and .s − 1 ≤ r ≤ 1, as given by Theorems 1–2. In order: type of approximation by regular models, admissible values of the coupling constant, average value and variance of the total energy of the system in the state .w0 . Notice that the finiteness (or lack thereof) of the latter two quantities reflects whether the form domain and operator domain, respectively, of the corresponding singular model are dependent or independent of the coupling Coupling class

Approxim. Norm resolv. Strong resolv.

Coupling const. Arbitrary Arbitrary Arbitrary

Strong resolv.

Small

.H .H−1

\H \ H−1 , .s > 1, r < 1 r .H−s \ H−1 , .s > 1, r = 1 r

.H−s

||f i − f ||−s → 0

.

and

Mean energy .w0

.∞

Variance energy .w0 2 ||f ||2 .∞ .∞

.∞

.∞

.ωe .ωe

.λ

ωei + λ2 ||f i ||2−1 → ω˜ e ;

(33)

(iii) if .s ∈ (1, 2] and .r = 1, then the same statements as in point (ii) hold for sufficiently small .λ. The proof follows similar arguments as the one of Theorem 1. Equation (33) accounts for the interpretation of the parameter .ω˜ e as the renormalized (or dressed) excitation energy of the atom: since .f ∈ / H−1 , necessarily .||f i ||−1 diverges and thus, in order for the latter to be finite, and whence the model to be well-defined, the bare one must diverge as well. This is a simple example of a renormalization procedure. It is not required in the case .f ∈ H−1 , where both energies are finite and only differ by a shift. We point out that, while in the case .f ∈ H−1 Theorem 1 ensures the possibility to approximate the singular model via regular one in the norm resolvent sense, in r the more general case .f ∈ H−s Theorem 2 only provides strong resolvent sense. Mathematically, this is essentially a consequence of the fact that, differently from ˜ f (z) is unbounded. Whether what happens in the case .f ∈ H−1 , here the operator .S this result may be improved, at least under additional assumptions, is a currently open question. Finally, as a further difference from the case .f ∈ H−1 , notice that now the state .w0 does not even belong to the form domain of the renormalized operator, again in full accordance with Eq. (17): since the bare excitation energy “is infinite” (in the sense discussed before), necessarily the average value of the total energy in that state must diverge as well. A summary of all possible cases, as given by Theorems 1–2, is reported in Table 1.

114

D. Lonigro

4 Renormalization of Higher-Dimensional Models The construction presented above is not exclusive of the qubit scenario: similar results as in the two-level case may be obtained, mutatis mutandis, to some classes of generalized spin–boson (GSB) models whose interaction exhibits a similar rotating-wave structure. We shall hereafter examine some preliminary results for a higher-dimensional generalization of the model above. We shall replace the two-level atom of Sect. 3 with a multi-level atom, associated with a Hilbert space .h which can be decomposed into the direct sum of two sectors, .h = he ⊕ hg , effectively behaving like the two levels of a spin–boson model, in the following sense. Choosing the decoupled Hamiltonian .H0 on the Hilbert space .H = h ⊗ F as ( H0 =

.

Ee 0 0 Eg

) ⊗ I + I ⊗ dr(ω),

(34)

with .Ee ∈ B(he ) and .Eg ∈ B(hg ) symmetric and nonnegative, we set, for some integer r, H = H0 + λ

.

r ( E ( ) ( )) Ej+ ⊗ a fj + Ej− ⊗ a † fj ,

(35)

j =1

where .Ej+ ∈ B(hg , he ), .Ej− = (Ej− )† ∈ B(he , hg ), and .f1 , . . . , fr ∈ H are form factors. The physical interpretation of this model is straightforward. The system behaves like a multilevel atom whose energy levels can be grouped into two sectors, .he and .hg ; the interaction with the boson field causes r possible transitions between levels belonging to separate sectors, while transitions between levels of the same sector are not allowed; the j th allowed transition from .he to .hg (resp. from .hg to .he ) involves the creation (resp. annihilation) of a boson with wavefunction .fj . In the case .hg = C (i.e. a single ground level), this model was first studied in [48], and the properties of the corresponding reduced dynamics were investigated in [49, 50]. Following a similar strategy as in the spin–boson case, we can exploit the isomorphism .H = h ⊗ F = He ⊕ Hg , with .Hs := hs ⊗ F for .s ∈ {e, g}, and write the model above in a formal matrix fashion: ( ) he λAf .Hf = (36) , λA†f hg where we are using the shorthands .f = (f1 , . . . , fr ), hs = Es ⊗ I + I ⊗ dr(ω),

.

s ∈ {e, g}

(37)

Renormalization of Spin–Boson Interactions Mediated by Singular Form Factors

115

and Af =

r E

.

( ) Ej+ ⊗ a fj .

(38)

j =1

Following the clear analogies with the spin–boson model analyzed in Sect. 3, the extension of this model to the case .f1 , . . . , fr ∈ H−1 can be performed by following a similar route as in Theorem 1. Indeed, in analogy with Eq. (20), define Gf (z) = he − z − λ2 Sf (z),

(39)

.

where r E ( ) 1 1 † E − ⊗ a † (fl ) , A = Ej+ ⊗ a fj .Sf (z) = Af hg − z j hg − z f

(40)

j,l=1

the latter being still a well-defined, bounded operator on .He as long as .f1 , . . . , fr ∈ H−1 , because of the usual mapping properties of singular annihilation and creation operators. Remarkably, in the particular case .hg = C (i.e. the ground sector being one-dimensional), in which necessarily .Ej+ = |ej > and .Ej− = 1, since the propagator fails to be defined in such a case, cf. Eq. (26); however, again as a consequence of the first resolvent identity, the difference .Sf (z) − Sf (−1) is indeed well-defined, cf. Eq. (28), thus suggesting the possibility of pursuing a renormalization procedure analogous to (but mathematically more intricate than) the one for the spin–boson model. We shall explore this possibility elsewhere.

5 Perspective: Many-Body Spin–Boson Model Let us now consider the many-body generalization of the spin–boson model. The N corresponding Hamiltonian, to be defined on the Hilbert space .H = C2 ⊗ F, can be obtained as follows. The decoupled Hamiltonian reads H0 = K ⊗ I + I ⊗ dr(ω),

(43)

.

where K=

.

(

N E ( ) I ⊗ · · · ⊗ Kj ⊗ · · · ⊗ I ,

Kj =

j =1

ωe,j 0 0 ωg,j

) ,

(44)

with .ωe,j , .ωg,j ∈ R. Given .f1 , . . . , fN ∈ H, we set Hf = H0 + λ

.

N ( E ( )) ( ) σj+ ⊗ a fj + σj− ⊗ a † fj ,

(45)

j =1

where j th

σj±

.

N '''' O ) ( = I ⊗ ··· ⊗ σ± ⊗··· ⊗ I .

(46)

j =1

Leaving to a future work a detailed discussion about the implementation of singular form factors in this model, we shall show that, as a starting point, this Hamiltonian can be indeed recast in a matrix fashion akin to the one in Eq. (15) for the spin– boson model, or Eq. (36) for its multilevel generalization, thus allowing for similar calculations.

Renormalization of Spin–Boson Interactions Mediated by Singular Form Factors

117

To this purpose, since the dimension of the spin subsystem grows exponentially with N , it will be largely convenient to introduce a clever representation by packing together all states in which an equal number of atoms are, say, in their excited state: H=

N O

.

H(j ) =

H(j ) ,

j =0

nj O

F,

nj =

l=1

( ) N j

(47)

with .H(j ) including the boson wavefunctions corresponding to the .nj states in which j atoms are excited and .N − j atoms are in their ground state. Notably, the number of such sectors grows linearly with N. Similarly, for all .s ∈ R we will write Hs =

N O

.

(j ) Hs ,

(j ) Hs

j =0

=

nj O

Fs .

(48)

l=1

We shall briefly examine explicitly the case .N = 2. In this case, the decomposition (47) reads .H = H(0) ⊕ H(1) ⊕ H(2) , with .H(0) = H(2) = F and .H(1) = F ⊕ F. Correspondingly, the Hamiltonian .Hf1 ,f2 can be written as ⎞ λa(f2 ) λa(f1 ) 0 hee ⎜ λa † (f2 ) heg λa(f1 ) ⎟ 0 ⎟, =⎜ ⎝ λa † (f1 ) 0 hge λa(f2 ) ⎠ λa † (f1 ) λa † (f2 ) hgg 0 ⎛

Hf1 ,f2

.

(49)

with .hxx ' = ωx,1 + ωx ' ,2 + dr(ω) for .x, x ' ∈ {e, g}, with the various terms of the Hamiltonian being grouped accordingly. More compactly, ⎞ h2 λA1,2 0 ⎟ ⎜ † = ⎝ λA1,2 h1 λA0,1 ⎠ , 0 λA†0,1 h0 ⎛

Hf1 ,f2

.

(50)

all terms to be found with direct inspection with Eq. (49), bringing about a simple block tridiagonal structure; in particular, note that (

)

A1,2 = a(f2 ) a(f1 ) ,

.

( A0,1 =

) a(f1 ) , a(f2 )

(51)

We are now in a situation similar to that of the rotating-wave spin–boson model. When .f1 , f2 ∈ H, the Hamiltonian is self-adjoint by construction, with domain 4 (0) ⊕ D(1) ⊕ D(2) , where .D(0) = D(dr(ω)) = D(2) .D(Hf1 ,f2 ) = C ⊗ F = D and .D(1) = D(dr(ω)) ⊕ D(dr(ω)). When .f1 , f2 ∈ H−1 \ H, by interpreting

118

D. Lonigro

( ) ( ) a fj , a † fj in the sense of Proposition 1 we have

.

(2) A1,2 : H(1) +1 → H ,

A†1,2 : H(2) → H(1) −1 ; .

(0)

A†0,1 : H(1) → H−1 ,

.

A0,1 : H+1 → H(1) ,

(0)

(52) (53)

suggesting that, by a suitable choice of domain generalizing Eq. (23), it is possible to define a singular 2-atom spin–boson Hamiltonian with form factors .f1 , f2 ∈ H−1 , that may be approximated in the norm resolvent sense by sequences of regular models. Similar considerations can be done for higher values of N , for which again a block tridiagonal structure analogous to Eq. (50) emerges. Finally, we point out that the scheme outlined above is sufficiently flexible to account for a more general choice of the operator K in Eq. (44)—that is, to include a suitable class of spin–spin interactions. Indeed, the procedure can be reproduced without substantial effort by just requiring K to be block-diagonal with respect to the decomposition (47) of the Hilbert space, that is, KH(j ) ⊆ H(j )

.

(54)

for all .j = 0, . . . , N . In words, spin–spin interactions that preserve the total number of atoms in their excited state can be included in this scheme as well. As a concrete example, in the case .N = 2 outlined above we have the freedom to add an interaction term which flips the states of the two spins whenever they are different: such a term will merely add off-diagonal terms to the central block of the Hamiltonian .Hf1 ,f2 in Eq. (49), but its block tridiagonal structure represented in Eq. (50), up to a proper redefinition of .h1 , will not be disrupted.

6 Perspective: Interior-Boundary Conditions for Spin–Boson Models The results of [24], together with the partial results presented in this paper, represent a first attempt towards a more systematic theory of GSB models with singular form factors. In this regard, it is useful to point out that such form factors—typically, exhibiting ultraviolet (UV) divergences—have been long encountered in various other models of nonrelativistic quantum mechanical matter interacting with boson fields. In these cases, one typically finds a scenario analogous to the one for GSB models: for sufficiently mild divergences one seeks a form-theoretical approach; when such techniques are not feasible, a renormalization procedure analogous to the one considered by Nelson for the eponymous model [31] may be attempted— again, without generally retaining information on the domain. In the latter case, the limiting Hamiltonian, say .H∞ , is obtained as the norm resolvent limit of .HA + EA

Renormalization of Spin–Boson Interactions Mediated by Singular Form Factors

119

as .A → +∞, where .HA is a properly regularized Hamiltonian (e.g. via a cutoff procedure) and .EA a diverging energy shift. This procedure shows similarities with the renormalization of the spin–boson model for .f ∈ H−2 presented in Sect. 3: however, roughly speaking, in that case the excitation energy of the qubit “takes charge” of the diverging energy shift instead of the Hamiltonian itself. A novel approach to UV divergences in quantum field theories which, differently from the techniques briefly summarized above, can be adapted to different scenarios and accounts for an explicit determination of the operator domain, was recently proposed by Teufel and Tumulka [38, 39] and has been successfully applied to diverse other particle–field Hamiltonians, see e.g. [40, 41] (also cf. [42] for the abstract setting). This approach is based on a systematic use of interior-boundary conditions (IBC), that is, abstract boundary conditions relating sectors with distinct number of field excitations. The basic idea at the core of this approach is to employ a restriction–extension procedure analogous to the one used to define singular perturbations of possibly infinite rank [51] (also cf. [52]). Without entering into details, the procedure can be roughly summarized as follows: • starting from the decoupled (self-adjoint) Hamiltonian .H0 , one considers the restriction of .H0 to a properly chosen dense subspace of the Hilbert space, thus obtaining a “minimal” operator admitting infinitely many self-adjoint extensions; • such self-adjoint extensions are then characterized via suitable abstract boundary conditions on the domain of the adjoint of the minimal operator; • one finally searches for the boundary conditions corresponding to the formal, UV-divergent operator—i.e. such that, by applying the formal expression to the states satisfying said boundary conditions, all divergent terms cancel out. In the process, the domain of the adjoint is shown to comprise vectors with a “regular” component in the domain of the decoupled Hamiltonian .H0 , plus an additional “singular” part. As such, the procedure sketched above clearly shows important similarities with the renormalization procedure of the models discussed in Sects. 3–5. In both cases, the key idea is to properly redefine the domain of the decoupled Hamiltonian by means of a singular term chosen so that the divergences produced by the formal Hamiltonian cancel out. While leaving a more detailed discussion to future works, the IBC method can be already predicted to be consistent with the results of [24]. Indeed, applying the same steps followed e.g. in [41, Section 1], the self-adjointness domain of the singular spin–boson model of Sect. 3 ends up being the space of vectors .w = we ⊕ wg such that the vector ( ) ( ) 1 0 we † σ− ⊗ a (f ) w = +λ .w + (55) 1 † wg H0 + 1 dr(ω)+1 a (f ) we

120

D. Lonigro

(with .σ− as in Eq. (6)) is in .D(H0 ) = D(dr(ω)) ⊕ D(dr(ω)), which is achieved by taking .we ≡ oe ∈ D(dr(ω)) and .wg = og − λ(dr(ω) + 1)−1 a † (f ) oe for some .og ∈ D(dr(ω)), thus obtaining the very same expression as in Eq. (23). This argument strongly supports the IBC method as a natural candidate for a general theory of GSB models with singular form factors—even those not sharing the rotating-wave structure of the Hamiltonians discussed in Sects. 3–5. Of course, while the general recipe to construct such models will be the same, the results will crucially depend on the particular choice of the operators mediating the matter–field interaction: for instance, replacing the operator .σ− in Eq. (55) with another one will generally cause both components of the wavefunctions in the domain to acquire a singular part, thus preventing an explicit evaluation of the resolvent.

7 Concluding Remarks In this work we have reviewed the application of the formalism of singular annihilation and creation operators, defined as continuous maps on a scale of Fock spaces, to the nonperturbative study of simple models describing a rotating-wave interaction, mediated by non-normalizable form factors, between quantum systems with a finite number of degrees of freedom (e.g. spins) with a structured boson field. To this extent, we have summarized known results for the simplest instance of such systems, i.e. the rotating-wave spin–boson model, and discussed the application of similar techniques to more complicated systems. Future works shall be devoted to a more detailed developments of the ideas presented in the last sections. As a common thread, for such Hamiltonians the accommodation of form factors in the class .H−1 seems to only involve a careful choice of the operator domain, compatibly with existing perturbative results for all models in the class of generalized spin–boson models; the case .H−2 is instead much less trivial, already in the case of the spin–boson model, where a nontrivial renormalization procedure is required. Acknowledgments We acknowledge support by MIUR via PRIN 2017 (Progetto di Ricerca di Interesse Nazionale), project QUSHIP (2017SRNBRK). This work is partially supported by Istituto Nazionale di Fisica Nucleare (INFN) through the project “QUANTUM” and by the Italian National Group of Mathematical Physics (GNFM-INdAM). We also thank Paolo Facchi for many fruitful discussions.

References 1. Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press on Demand (2002) 2. Ingold, G.-L.: Path integrals and their application to dissipative quantum systems. In: Coherent Evolution in Noisy Environments, pp. 1–53. Springer, Berlin (2002)

Renormalization of Spin–Boson Interactions Mediated by Singular Form Factors

121

3. Grifoni, M., Paladino, E., Weiss, U.: Dissipation, decoherence and preparation effects in the spin-boson system. Eur. Phys. J. B 10, 719–729 (1999) 4. Thorwart, M., Paladino, E., Grifoni, M.: Dynamics of the spin-boson model with a structured environment. Chem. Phys. 296, 333–344 (2004) 5. Clos, G., Breuer, H.-P.: Quantification of memory effects in the spin-boson model. Phys. Rev. A 86, 012115 (2012) 6. Costi, T.A., McKenzie, R.H.: Entanglement between a qubit and the environment in the spinboson model. Phys. Rev. A 68, 034301 (2003) 7. Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17, 821–825 (1976) 8. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976) 9. Weiss, U.: Quantum Dissipative Systems, vol. 13. World Scientific, Singapore (2012) 10. Leggett, A.J., Chakravarty, S., Dorsey, A.T., Fisher, M.P.A., Garg, A., Zwerger, W.: Dynamics of the dissipative two-state system. Rev. Mod. Phys. 59, 1 (1987) 11. Hübner, M., Spohn, H.: Spectral properties of the spin-boson Hamiltonian. Ann. I. H. P. Phys. Théorique 62, 289–323 (1995) 12. Hirokawa, M.: An expression of the ground state energy of the Spin-Boson model. J. Funct. Anal. 162, 178–218 (1999) 13. Hirokawa, M.: Remarks on the ground state energy of the spin-boson model: an application of the Wigner-Weisskopf model. Rev. Math. Phys. 13, 221–251 (2001) 14. Amann, A.: Ground states of a spin-boson model. Ann. Phys. 208, 414–448 (1991) 15. Davies, E.B.: Symmetry breaking for molecular open systems. Ann. I. H.P. Phys. Théorique 35, 149–171 (1981) 16. Fannes, M., Nachtergaele, B., Verbeure, A.: The equilibrium states of the spin-boson model. Commun. Math. Phys. 114, 537–548 (1988) 17. Hübner, M., Spohn, H.: Radiative decay: nonperturbative approaches. Rev. Math. Phys. 7, 363– 387 (1995) 18. Reker, J.: Existence of resonances for the spin-boson model with critical coupling function. J. Math. Anal. Appl. 483, 123628 (2020) 19. Arai, A., Hirokawa, M.: On the existence and uniqueness of ground states of a generalized spin-boson model. J. Funct. Anal. 151, 455–503 (1997) 20. Arai, A.: Essential spectrum of a self-adjoint operator on an abstract Hilbert space of Fock type and applications to quantum field Hamiltonians. J. Math. Anal. Appl. 246, 189–216 (2000) 21. Takaesu, T.: On generalized spin-boson models with singular perturbations. Hokkaido Math. J. 39, 317–349 (2010) 22. Falconi, M.: Self-adjointness criterion for operators in Fock spaces. Math. Phys. Anal. Geom. 18, 2 (2015) 23. Teranishi, N.: Self-adjointness of the generalized spin-boson Hamiltonian with a quadratic boson interaction. Hokkaido Math. J. 44, 409–423 (2015) 24. Lonigro, D.: Generalized spin-boson models with non-normalizable form factors. J. Math. Phys. 63, 072105 (2022) 25. Reed, M., Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, SelfAdjointness, vol. 2. Elsevier, Amsterdam (1975) 26. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics 1: C*- and W*-algebras. Symmetry groups. Decomposition of states. In: Operator Algebras and Quantum Statistical Mechanics. Springer, Berlin (1987) 27. Agarwal, G.S.: Rotating-wave approximation and spontaneous emission. Phys. Rev. A 7, 1195 (1973) 28. Shore, B.W., Knight, P.L.: The Jaynes-Cummings model. J. Mod. Opt. 40, 1195–1238 (1993) 29. Phoenix, S.J.D., Knight, P.L.: Establishment of an entangled atom-field state in the JaynesCummings model. Phys. Rev. A 44, 6023 (1991) 30. Vogel, W., de Matos Filho, R.L.: Nonlinear Jaynes-Cummings dynamics of a trapped ion. Phys. Rev. A 52, 4214 (1995)

122

D. Lonigro

31. Nelson, E.: Interaction of nonrelativistic particles with a quantized scalar field. J. Math. Phys. 5, 1190–1197 (1964) 32. Albeverio, S., Kurasov, P.: Singular Perturbations of Differential Operators: Solvable Schrödinger-Type Operators, vol. 271. Cambridge University Press, Cambridge (2000) 33. Simon, B.: Spectral analysis of rank one perturbations and applications. CRM Proceedings and Lecture Notes, 8, 109–149 (1995) 34. Posilicano, A.: A Krein-like formula for singular perturbations of self-adjoint operators and applications. J. Funct. Anal. 183, 109–147 (2001) 35. Dereziñski, J., Früboes, R.: Renormalization of the Friedrichs hamiltonian. Rep. Math. Phys. 50, 433–438 (2002) 36. Facchi, P., Ligabo, M., Lonigro, D.: Spectral properties of the singular Friedrichs–Lee Hamiltonian. J. Math. Phys. 62, 032102 (2021) 37. Lonigro, D.: The self-energy of Friedrichs–Lee models and its applications to bound states and resonances. Eur. Phys. J. Plus 137, 492 (2022) 38. Teufel, S., Tumulka, R.: Hamiltonians without ultraviolet divergence for quantum field theories. Quantum Stud.: Math. Found. 8, 17–35 (2021) 39. Teufel, S., Tumulka, R.: Avoiding ultraviolet divergence by means of interior–boundary conditions. In: Finster, F., Kleiner, J., Röken, C., Tolksdorf, J. (eds.) Quantum Mathematical Physics: A Bridge Between Mathematics and Physics, pp. 293–311. Springer International Publishing, Cham (2016) 40. Lampart, J., Schmidt, J., Teufel, S., Tumulka, R.: Particle creation at a point source by means of interior-boundary conditions. Math. Phys. Anal. Geom. 21, 1–37 (2018) 41. Lampart, J., Schmidt, J.: On Nelson-type Hamiltonians and abstract boundary conditions. Commun. Math. Phys. 367, 629–663 (2019) 42. Binz, T., Lampart, J.: An abstract framework for interior-boundary conditions. arXiv:2103.17124 (2021) 43. Albeverio, S., Kuzhel, S., Nizhnik, L.: Singularly perturbed self-adjoint operators in scales of Hilbert spaces. Ukr. Math. J. 59, 787–810 (2007) 44. Bohm, A.: Rigged Hilbert space and quantum mechanics. Tech. rep. Texas Univ., Austin. Center for Particle Theory (1974) 45. De La Madrid, R.: The role of the rigged Hilbert space in quantum mechanics. Eur. J. Phys. 26, 287 (2005) 46. Bohm, A., Gadella, M.: Dirac Kets, Gamow Vectors and Gel’fand Triplets: The Rigged Hilbert Space Formulation of Quantum Mechanics. Lectures in Mathematical Physics at the University of Texas at Austin. Springer, Berlin (1989) 47. Ginibre, J., Nironi, F., Velo, G.: Partially classical limit of the Nelson model. Ann. Henri Poincare 7, 21–43 (2006) 48. Garraway, B.: Nonperturbative decay of an atomic system in a cavity. Phys. Rev. A 55, 2290 (1997) 49. Lonigro, D., Chru´sci´nski, D.: Quantum regression beyond the Born-Markov approximation for generalized spin-boson models. Phys. Rev. A 105, 052435 (2022) 50. Chru´sci´nski, D., Hesabi, S., Lonigro, D.: On Markovianity and classicality in multilevel spinboson models. Scientific Reports 13, 1518 (2023) 51. Posilicano, A.: Self-adjoint extensions of restrictions. Oper. Matrices 2, 483–506 (2008) 52. Posilicano, A.: On the self-adjointness of H + A ∗ +A. Math. Phys. Anal. Geom. 23, 1–31 (2020)

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom Plus Wall Case Marco Olivieri

1 Introduction The intermolecular and interatomic interactions are at the basis of several important phenomena which occur in our world [2, 4, 39]. If we consider two neutral atoms, it is a well known fact [35] that the fluctuations of the charge distribution of one atom create an instantaneous dipole which polarizes the other atom, and this triggers the emergence of multipole moments which influence back the dipole of the first atom. This process gives rise to an attractive interaction which is known as the van der Waals interaction. Van der Waals forces have universal decaying behaviors with respect to the distances between the interacting interfaces, and depend only on the geometry of the interfaces. There are two paradigmatic simple examples where this is evident: the interaction between two hydrogen atoms and the interaction between an hydrogen atom and an infinite, plane surface, perfect conductor (called, from now on, “wall”). Introducing the fine structure constant .α, whose approximate value is α=

.

1 0 in a quantum mechanical description, i.e., .

M. Olivieri (O) Department of Mathematics, Aarhus University, Aarhus C, Denmark e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Correggi, M. Falconi (eds.), Quantum Mathematics II, Springer INdAM Series 58, https://doi.org/10.1007/978-981-99-5884-9_4

123

124

M. Olivieri

where the interaction is considered instantaneous and originated only by the static Coulomb potentials, its decay for the aforementioned examples is

qm

WL

.

⎧ C1 (α) ⎪ ⎪ ⎨− 6 , L = ⎪ C ⎪ ⎩− 2 (α) , L3

for two hydrogen atoms, (2) for a hydrogen atom and a wall,

for suitable values of .C1 (α), C2 (α) > 0 in the chosen units system, see [3, 5, 27]. This description neglects, however, the retardation effects given by the interference with the quantum fluctuations of the field. If we take in consideration the fact that the electromagnetic field propagates at the speed of light (which is finite) the interaction is retarded. The behavior in (2) holds, indeed, up to a distance of approximately 10 Bohr radii. At this distance, the information about the first atom’s electron motion reaches the second interface in a time that is comparable with the average circulation time of the electron. This breaks the correlation between the two objects and weakens the interaction [26]. This effect was studied and formalized in 1948 by Casimir and Polder [15], from which the phenomenon took its name. By perturbation theory techniques they showed how, for the cases of the two atoms and the atom plus the wall, the behavior of the interaction energy with quantum fields, QFT now denoted by .WL , is

QFT

WL

.

⎧ D1 (α) ⎪ ⎪ , ⎨− L7 = ⎪ ⎪ ⎩− D2 (α) , L4

for two hydrogen atoms, (3) for a hydrogen atom and a wall,

for suitable .D1 (α), D2 (α) > 0 in the chosen units system and a distance .L > 1 large enough. Despite being a remarkable result of quantum field theory, the theoretical work of Casimir and Polder is not mathematically rigorous, mainly because they calculated only the first terms of the perturbative expansions of the interaction energy. Aim of the present work is to give a rigorous mathematical proof, with precise estimates, of the calculation of the interaction energy for the case of the atom plus the wall. The quantum nature of the van der Waals forces was first studied by London [29]. The first mathematical rigorous result is due to Lieb and Thirring in [28], where they derived an upper bound for the interaction energy between molecules, obtaining the universal .L−6 decay. The analysis was completed in [3] for the case of several atoms, where the correct leading order expression was derived. The literature about van der Waals interaction is extensive and includes results about further order expansions [9] and about interactions between various types of interfaces [37]. Casimir and Polder studied the retardation in the interaction in a non relativistic quantum electrodynamics description via a fourth (second) order expansion for the energy for two atoms (atom plus wall). At the best of our knowledge, there are

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

125

many results in the physics literature continuing the line of research of Casimir and Polder (see [31] for an extensive bibliographic collection and [12, 20, 21, 36] for monographs about the Casimir-Polder effect and van der Waals forces), but few ones with a theoretical, mathematical rigorous approach. In [33, 34] the two atoms case is studied by the authors, who derived the decay .L−7 using a path integral formulation, making however the strong assumption that the cumulants over the second order give smaller contributions in terms of the inverse of the distance. In [32] one of the two authors obtained again the .L−7 decay using similar techniques and estimating the higher order cumulants too, but assuming a dipole approximation and strong binding of the electrons to the nuclei (harmonic traps approximating the Coulomb attraction). Nevertheless, the cancellation of the van der Waals term of order .L−6 is not obtained by the contribution of the radiation, which is a fundamental mechanism to explain the retardation effects, as it is clear from [15]. The cancellation of the van der Waals term is recovered in Koppen’s PhD thesis [26]: the author considers a quantum electrodynamics model introducing an infrared cutoff in the Hamiltonian and studying the fourth order perturbative expansion of the energy in dipole approximation. To take the infrared limit is, however, known to be a very difficult problem and the result is affected by the same problem of considering a truncated perturbative expansion. Other rigorous results concern only the Casimir effect [14] where the interaction with the matter is encoded in the boundary conditions and the radiation, described via a scalar field, is influenced only by the geometry of the classical interfaces: the vacuum energy is calculated in [10, 11, 13, 22–24]. In [16] the authors apply the same techniques as [33] to the case of the atom and the wall reobtaining the behaviors (2) and (3), but still lacking full mathematical rigor. The rigorous proof of the Casimir-Polder effect for the general setting is, thus, still an open problem. In this paper we study the Casimir-Polder effect for the case of the atom interacting with the wall. In [5] the van der Waals interaction energy for the electrostatic setting is rigorously computed and is coherent with the decay (2): qm

WL = −

.

( α2 ) α2 , + O L3 L5

(4)

where L is the distance between the atom’s nucleus and the wall and .α is the fine structure constant. It is a common strategy in quantum field theory to consider this as a small parameter and study expansions of the physical quantities w.r.t. .α, see [6–8], and we are going to follow the same approach providing expansions of the interaction energy in terms of .α. In order to prove the appearance of the retardation effects and relative faster decay of the interaction after a suitable distance for the quantum fields, we consider the Pauli-Fierz model. The Pauli-Fierz model has been widely used to solve problems in non relativistic quantum electrodynamics [17–19]. We make the following assumptions:

126

M. Olivieri

(A1) the dipole approximation; (A2) reduction of the action of the Hamiltonian to the interaction with 0 and 1 photons between matter and field. Our approach relies on the use of precise estimates for the ground state energies of the interaction system and free systems inspired by the perturbation theory, like in [8], and on the calculation of line integrals on the complex plane inspired by [15]. This last step allows us to obtain the important cancellation of the van der Waals term in (4) generated by the Coulomb contribution and to derive the new leading term, as stated in the main result in Theorem 3: QFT

WL

.

=−

α ℵα,L , L4

(5)

which, for short distances (less than 10 Bohr radii), gives again the decay in (4) because .ℵα,L = αL, while for large distances (bigger than 100 Bohr radii) gives the −4 behavior predicted in (3) because .ℵ .L α,L = const., and in the intermediate region expresses the transition between the two values. The techniques used let us enlighten how the retardation effects are originated from the exchanges of one photon with the matter and the interaction with the vacuum flactuations (see the calculations in Sect. 3.3). At the best of our knowledge, our result is the first one where the Casimir-Polder effect for the model of the atom plus the wall is proven with rigorous estimates and without recurring to infrared cut-off or to perturbative expansions. The result is, nevertheless, unsatisfactory in some aspects: one would like to drop assumptions .(A1) and .(A2) and obtain the result for the full model. Furthermore, as explained in Sect. 4, the result gives the decay behavior of the interaction energy discussed above, but the error produced in the calculations is smaller than the leading term only up to approximately .82.5 Bohr radii. After that distance the expression (5) ceases to be the leading term because, for technical difficulties, parts of the error term are uniform on the distance. In a future paper we would like to give the result for the full, non approximated model with an error suitably dependent on the distance. The structure of the paper is the following: • in Sect. 2 we introduce the Pauli-Fierz model, a quantum electrodynamics model describing the joint system of the hydrogen atom interacting with the radiation and we recall a result from [8], adapted to our approximated model, for the estimate of the ground state energy of this free system; • in Sect. 3 we introduce a modified version of the Pauli-Fierz model to describe the interaction between the hydrogen atom and radiation with the wall, whose construction is justified in Appendix 3, and we state our main result in Theorem 3. The proof is given in the following parts: in Sects. 3.1 and 3.2 we prove upper and lower bounds, respectively, for the ground state energy of the interaction system. Then, in Sect. 3.3, we calculate the difference between the ground state energies of the interaction and free systems with the technical line integral calculations postponed in Appendix 2.

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

127

• In Sect. 4 we discuss the relation between the term in (5) and the error terms to identify the leading term in the different regimes of distance. y • In Appendix 1 we collected some useful estimates on the one-photon vector .o# useful through all the paper. We make a comment here about the notation that is going to be used in the paper: .C > 0 is going to denote a positive constant which is independent of the parameters of interest .α and L and which can vary from line to line. The notation .O(·) has to be intended in the usual sense, but we remark that we did not take track of the dependence on the ultra-violet cut-off .A, meaning that we assume .A to be fixed, independent of .α and L and we are not interested in studying the ultraviolet problem. For the quadratic forms of operators we use sometimes the notation .w := . Furthermore, for a vector .v ∈ H in a Hilbert space . H, and two operators .A, B acting on . H, where B admits an inverse, we use the following fractional notation to be interpreted as the order of operations below: .

A Av = v := B −1 Av. B B

2 Free Hydrogen Atom with Radiation: Pauli-Fierz Model We consider a non-relativistic, quantum, spinless electron in a hydrogen atom model, therefore interacting via an electrostatic Coulomb potential with a fixed nucleus. We study the joint system of the electron and a quantum electromagnetic field with their mutual interaction. We fix the nucleus of the atom in position .0 ∈ R3 and define the position variable of the electron to be .x = (x1 , x2 , x3 ) ∈ R3 , so that the Hilbert space associated to the hydrogen atom model is .L2 (R3 ; dx). The radiation is described in a Fock space representation rs (h) =

∞ O

.

h⊗s n ,

(6)

n=0

where the n-th sector is associated to n photons, and the one photon space is h := L2 (R3 ; C2 ; dk),

.

(7)

of square integrable functions with two components in the complex numbers associated to the two perpendicular polarization directions of the electromagnetic field. In our notation, we are going to denote by superscripts the components of the vectors in the sectors of the Fock space: w ∈ rs (h),

.

w = (w (0) , w (1) , w (2) , . . .),

and denote by .o := (1, 0, 0, . . .) the vacuum vector.

(8)

128

M. Olivieri

The Hilbert space for the full system is .

H = L2 (R3 ; dx) ⊗ rs (h).

(9)

We can define operator-valued distribution for the creation and annihilation operators {ay† (k), ay (k)}y =1,2 ,

.

which create and destroy a photon, respectively, with frequency .k ∈ R3 for each direction of polarization .y and have the following canonical commutation relations, for .β, y ∈ {1, 2} and .k, h ∈ R3 , [aβ (k), ay (h)] = 0 = [aβ† (k), ay† (h)],

[aβ (k), ay† (h)] = δβ,y δ(k −h).

.

(10)

The associated field operators are then, for any .λ ∈ h, a(λ) =

E f

a † (λ) =

dk λy (k)ay (k),

.

E f

dk λy (k)ay† (k).

(11)

y =1,2

y =1,2

If the wall is at infinite distance, it does not affect the system composed by the hydrogen atom and the radiation. The dynamics is generated by the so-called PauliPF and is formally defined by the following Fierz Hamiltonian, that we denote by .H∞ sum PF H∞ = (P ⊗ 1 − α 1/2 A∞ (x))2 + 1 ⊗ Hf −

.

α ⊗ 1, |x|

(12)

where .α plays the role both of the square of the charge and of the coupling between matter and field. Here .P = i∇x is the momentum operator for the electron and E f

Hf = dr(ω) =

.

dk ω(k) ay† (k)ay (k),

y =1,2

is the free energy operator for the field with the usual dispersion relation for the massless photons ω(k) = |k|.

.

(13)

The vector field potential .A∞ (x) describes the interaction between electron and field. It can be expressed as the sum − A∞ (x) = A+ ∞ (x) + A∞ (x),

.

(14)

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

129

where † A+ ∞ (x) = a (λ∞ (x)),

.

A− ∞ (x) = a(λ∞ (x)),

(15)

− and .A+ ∞ (x) and .A∞ (x) create and annihilate a photon with state .λ∞ (x), respectively, from the interaction with an electron with position variable x. The expression of the form factor .λ∞ (x) = (λ∞,y (x))y =1,2 , with .λ∞,y ∈ L∞ (R3 ; h3 ), is given by

λ∞,y (x) =

.

χA (k) ey (k)eikx , 2π |k|1/2

y = 1, 2,

(16)

with .χA being defined, for a fixed, finite .A ≥ 1, as {

( |k| ) , .χA (k) = χ A

χ (r) =

1, if

r < 1/2,

0, if

r > 1,

χ ∈ [0, 1],

(17)

where .χ ∈ C0∞ (R+ ). In this way .χA is a cut-off function for frequencies of the photons over .|k| < A. The .(ey )y =1,2 are the two polarization vectors which form k with .kˆ = |k| an orthonormal basis for .R3 . The vector field can be rewritten in a formal but useful way by means of the operator-valued distributions E f

A∞ (x) =

.

y =1,2

R3

dk

χA (k) ey (k)(ay (k)eikx + ay† (k)e−ikx ). 2π |k|1/2

(18)

Since .|α| ≤ 1 and .λ∞,y , ω−1/2 λ∞,y ∈ L∞ (R3 ; h3 ), y = 1, 2, by [38, Theorem 13.3] and Kato-Rellich Theorem, the Pauli-Fierz Hamiltonian is self-adjoint on the domain . D(H∞ ) = H 2 (R3 ) ⊗ D(dr(ω)) (for more general conditions see [25]). Assuming to work in Coulomb gauge, expressed by the condition .∇x · A(x) = 0, the Pauli-Fierz Hamiltonian can be rewritten, calculating the square, in the following way: H∞ = hα + Hf − 2α 1/2 Re(P A∞ (x)) + αA2∞ (x),

.

(19)

where, from now on, we drop the tensor products with the identity in order to ease the notation. The .hα is the hydrogen atom Hamiltonian hα = −Ax −

.

α , |x|

|x| α 3/2 uα (x) = √ e−α 2 , 8π

eα = −

α2 , 4

(20)

with .uα and .eα being the ground state and ground state energy, respectively. When needed, we are going to use as well the notation .h1 := hα=1 , e1 := eα=1 , u1 := uα=1 .

130

M. Olivieri

As we anticipated, we are going to work with the dipole approximated Hamiltonian whose action is restricted to 0 and 1 photons. In order to do that we introduce the projector on the first sectors of the Fock space || : rs (h) −→ C ⊕ h

.

w− | → (w (0) , w (1) ), whose action on pure tensors is ||(f ⊗ w) := f ⊗ ||w = f ⊗ (w (0) , w (1) ),

f ∈ L2 (R3 ; dx), w ∈ rs (h). (21)

.

We apply the substitution below on the new Hamiltonian .||H∞ ||† , called dipole approximation, A± ∞ (x)

.

|−→

± A± ∞ (0) =: A∞

(22)

which makes the argument of the creation and annihilation operators to be λ∞ = (λ∞,y )y =1,2 ,

λ∞,y := λ∞,y (0) =

.

χA (k) ey (k), 2π |k|1/2

(23)

obtaining the new approximated, free Hamiltonian − H∞ := hα + Hf − 2α 1/2 ReP A∞ + α||λ∞ ||2 + 2αA+ ∞ A∞ ,

.

(24)

acting on the space .

H∞ := || H = L2 (R3 ; dx) ⊗ (C ⊕ h).

(25)

We observe that the third term in (24) is the only one which changes the number of photons. Let us further denote by E∞ := inf σ (H∞ ),

.

(26)

the ground state energy of the approximated, free Hamiltonian. We are now ready to state an adaptation of the result [8, Theorem 2.1] in our setting with at most one photon. Let us introduce the following scalar products on . H∞ , # := ,

.

∗ := ,

(27)

and the vectors 1/2 o∞ (hα − eα + Hf )−1 P uα ⊗ A+ ∞ o, # := 2α

.

1/2 o∞ P uα ⊗ Hf−1 A+ ∗ := 2α ∞ o,

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

131

where the second one is not a vector belonging to the Hilbert space, but it is going to appear only in expressions which make sense. We also define the following vector −1 + + o∗1 = Hf−1 Pf A− ∞ Hf A∞ A∞ o,

.

where .Pf := dr(k) is the momentum operator for the field. Theorem 1 The following estimate holds for the energy of the free Hamiltonian, 2 3 ∗ 2 4 −1 E∞ = eα + α||λ∞ ||2 − ||o∞ # ||# − 4α ||o1 ||∗ + O(α log(α )).

.

(28)

Proof From [8, Theorem 5.1] we get the upper bound, adapted for the Hamiltonian with 0 and 1 photons, 2 3 ∗ 2 4 −1 E∞ ≤ eα + α||λ∞ ||2 − ||o∞ # ||# − 4α ||o1 ||∗ + O(α log(α )),

.

(29)

by choosing a suitable trial function. The term .α||λ∞ ||2 appears because we do not consider the normal ordered Hamiltonian like in the cited paper. We observe that instead of having an error of order .O(α 4 ) we get a .O(α 4 log(α −1 )) term because of 2 the appearance of the additional term .α||λ∞ ||2 ||o∞ # || in the calculations compared to the original version, which is treated in a similar way as its analogous in the interaction model (see formula (66)). By [8, Theorem 5.2] we obtain the lower bound 2 3 ∗ 2 4 −1 E∞ ≥ eα + α||λ∞ ||2 − ||o∞ ∗ ||∗ − 4α ||o1 ||∗ + O(α log(α )).

.

(30)

∞ 2 2 The substitution of .||o∞ ∗ ||∗ with .||o# ||# produces the error term, thanks to [8, Lemma C.5], 2 ∞ 2 5 −1 ||o∞ ∗ ||∗ − ||o# ||# = O(α log(α )),

.

which is reabsorbed in the error term .O(α 4 log(α −1 )).

(31) u n

3 Interaction Model: Atom and Wall We are now ready to define the interaction Hamiltonian, which shares, except for the presence of the Coulomb potential with the wall, the same structure with the free Hamiltonian, but in the vector potential it is clear how the presence of the wall influences the energy. Without loss of generality we can consider the wall to be parallel to the plane .E0 = {(0, x2 , x3 ) | x2 , x3 ∈ R}, translated in the .x1 direction by a distance .y > 0 in the positive semi-line, so that the conductor wall is described by .Ey = {(y, x2 , x3 ) | x2 , x3 ∈ R}.

132

M. Olivieri

˜

2

0

Fig. 1 Interaction described by the image charge method

The space for the particle is set to be L2 (R3y ; dx)

R3y = {x = (x1 , x2 , x3 ) ∈ R3 | x1 < y}.

where,

.

(32)

We express the distance as a multiple of the Bohr radius, given by the inverse of the fine structure constant .α to make it homogeneous with the physical quantities we are going to introduce in the following, so that y = Lα −1 ,

.

L > 1.

(33)

By an abuse of notation, we denote by y both the length (33) and the vector y = (Lα −1 , 0, 0),

(34)

.

the choice being clear from the context (Fig. 1). The Coulomb interaction with the wall is equivalent, thanks to the well known image charge method, to the interaction with a mirror atom with inverted charges: ) ( 1 1 1 1 1 + + − , − .Vy (x) = 2|y| |x˜y | |x − 2y| |x˜y − x| 2

x˜y := (2y − x1 , x2 , x3 ).

By [5, Lemma 2.1] we know that .Vy ≤ 0. For future purposes, we make the following split of the potential Vy> := −

.

1 , 2|x˜y − x|

Vy< := Vy − Vy> ,

(35)

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

133

and observe that, in .R3y , the following bounds hold: there exists a .C > 0 such that |Vy< (x)| ≤

.

C , y

for any x ∈ R3y , .

f

(36)

f dx (Vy> (x))2 |u(x)|2 ≤ C

R3y

R3y

dx |P u(x)|2 ,

for any u ∈ H01 (R3y ), (37)

the second one being a Hardy-type inequality proven in [5, Lemma 3.1]. The electromagnetic field is described by the Fock space with photons with positive frequencies in the direction normal to the wall: rs (h+ ) =

∞ O

.

⊗s n h+ ,

h+ := L2 (R+ × R2 ; C2 ; dk),

(38)

n=0

where the two polarization directions of the photons are taken into account. The full Hilbert space is, in this case, .

Hy = L2 (R3y ) ⊗ rs (h+ ),

(39)

and the Hamiltonian generating the dynamics is formally given by the expression HyPF := hα + Hf+ − 2α 1/2 Re(P Ay (x)) + αA2y (x) + αVy (x),

.

(40)

where the free field energy is E f

Hf+ :=

.

+ 2 y =1,2 R ×R

dk ω(k)ay† (k)ay (k).

(41)

Here again we can split .Ay in creation and annihilation parts − Ay (x) = A+ y (x) + Ay (x),

.

(42)

where † A+ y (x) = a (λy (x)),

.

A− y (x) = a(λy (x)),

(43)

134

M. Olivieri

the form factor .λy (x) = (λy,y (x))y =1,2 , with .λy,y ∈ L∞ (R3y ; h3+ ), this time being ⎞ e(1) y (k)2 cos(k1 (x1 − y)) χA (k) i(k2 x2 +k3 x3 ) ⎜ (2) ⎟ e .λy,y (x) = ⎝ ey (k)2i sin(k1 (x1 − y)) ⎠ . 2π |k|1/2 (3) ey (k)2i sin(k1 (x1 − y)) ⎛

(44)

(j )

Here by .ey is the j -th component of the .y -th polarization vector. In Appendix 3 we give a justification of the definition of this Hamiltonian as the right one to describe the model of the atom interacting with the wall. Theorem 5.7 in [30] ensures the self-adjointness of the Hamiltonian, provided that the following conditions are satisfied: following the notation of the mentioned paper, we choose α 2 PF .M = R+ × R ; .H = Hy ; .ω(k) = |k|; .λ = λy ; .V = − |x| + αVy (x). In particular, recalling that .x˜y = (2y − x1 , x2 , x3 ), ∇ · λy,y =

.

χA (k) (1) {e (k)∂x1 (ei(k2 x2 +k3 x3 ) cos(k1 (x1 − y))) π |k|1/2 y i(k2 x2 +k3 x3 ) i sin(k1 (x1 − y))) + e(2) y (k)∂x2 (e i(k2 x2 +k3 x3 ) + e(3) i sin(k1 (x1 − y)))} y (k)∂x3 (e

=−

χA (k) k · ey (k) ei(k2 x2 +k3 x3 ) sin(k1 (x1 − y)) = 0, π |k|1/2

y = 1, 2.

Therefore [30, Theorem 5.7] can be applied and .HyPF is self-adjoint on .

D(HyPF ) = D((−AD ) ⊗ 1) ∩ D(1 ⊗ dr(|k|)),

(45)

where .−AD is the Dirichlet Laplacian. As for the free model, we reduce the action of this Hamiltonian to the 0 and 1 photons space. By an abuse of notation, we denote again by .|| the projector over the 0-th and 1-st Fock sectors of .rs (h+ ) and apply an PF † analogous dipole approximation as (22) for the .A± y in .||Hy || to obtain − Hy := hα + Hf+ + αVy − 2α 1/2 ReP Ay (x) + α||λy ||2 + 2αA+ y Ay ,

.

(46)

where now the argument of the creation and annihilation operators .A± y has the form ⎛

λy = (λy,y )y =1,2 ,

.

λy,y

⎞ e(1) y (k)2 cos(k1 y) χA (k) ⎜ (2) ⎟ := λy,y (0) = ⎝ −ey (k)2i sin(k1 y) ⎠ . 2π |k|1/2 (3) −ey (k)2i sin(k1 y) (47)

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

135

Introducing the ground state energy Ey := inf σ (Hy ),

(48)

.

we estimate it in the next theorem. Theorem 2 For any .L > 1, we have ( α2 ) α2 2 # 2 + α||λ || − ||o || + O y y # L3 L5 ) ( + O(α 4 log(α −1 )) + O α 2 Le−L/2 .

Ey = eα −

.

(49)

The proof consists in giving upper and lower bounds, which is the content of Sects. 3.1 and 3.2, respectively. We use the estimates for the free energy .E∞ , the estimates for the energy of the interaction system .Ey from Theorems 1 and 2, respectively, and the important estimates from Proposition 1 in Sect. 3.3, to prove the main theorem of the paper. Theorem 3 For any .L > 1, we have α + O(α 4 log(α −1 )) L4 ( α3 ( α2 ) ) ) ( + O α 2 Le−L/2 + O 5 + O 2 log(α −1 ) , L L

WyQF T = Ey − E∞ = −ℵα,L

.

(50)

where ℵα,L :=

.

/ ( )\ 1 1 αL arctan , 6π αL(h1 − e1 ) xu1

(51)

and we have the following behaviors depending on the chosen regime for the distance:

ℵα,L

.

⎧ ⎪ ⎪ ⎪αL, ⎪ ⎪ ⎪ ⎨ 16 η = α , ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 ||(h − e )−1/2 xu ||2 , 1 1 1 6π

if 1 < L ≤

16 , 3

16 −1+η α , η ∈ (0, 1), 3 16 −1 α . if L ≥ 3

if L =

(52)

The interpretation of the result is going to be studied in Sect. 4. Remark 1 In the error terms in Theorem 3 it is not stated explicitly the dependence on the ultraviolet cut-off .A. Some of the bounds depend, indeed, on the choice of .A that anyway we keep fixed and independent on .α and L, and therefore they should be

136

M. Olivieri

improved if one wanted to study the removal of the cut-off. Therefore, the estimates hold considering .α as a small parameter compared to the other quantities and .A as a fixed constant value. Since we are not interested in studying the ultra-violet problem and since the key quantities come from the interactions with photons with low momenta, we may set for simplicity .A = 1 to keep it independent on the numerical value of .α. Remark 2 The cancellation of the van der Waals term and the leading term in the model are obtained considering only interactions with zero and one photons. It is therefore reasonable that the interactions with a higher number of photons and without considering the dipole approximation would contribute only by error terms. The technical difficulties in doing so consist in a heavier computational cost (the number of terms to bound is really higher), in dealing with the position-dependence on the field operators and on the fact these bounds require the derivation of number estimates for the states of minimal energy of the interaction model. This is what we aim to prove in a future paper.

3.1 Upper Bound In this subsection we are going to prove, in the theorem below, an upper bound for .Ey providing in this way the first step of the proof for Theorem 2. We use the convention, for .f, g ∈ L2 (R3y ) and .w, o ∈ h+ , =

E f

.

3 y =1,2 Ry

f dx

R3+

dk f (x)g(x)wy (k)oy (k).

(53)

Theorem 4 There exists a function .ϕy ∈ D(Hy ) such that, for any .L > 1, .

α2 ≤ eα − 3 + α||λy ||2 − ||o#y ||2#

L ( α2 ) + 4α 3 ||o1∗ ||2∗ + O 5 + O(α 4 log(α −1 )) + O(α 2 Le−L/2 ). L (54)

In order to prove the theorem we construct the trial function .ϕy in the following y way: we define the vector .o# in an analogous way as we did for the relative free version: o# := 2α 1/2 (hα − eα + Hf+ )−1 P uα ⊗ A+ y o.

.

y

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

137

We then introduce the trial function ˜ 1∗ ) + o , ϕy := uα ⊗ (o + 2α 3/2 o # y

.

√ ˜ 1∗ := 2o1∗ | 3 is the restriction of .o1∗ to the positive .k1 where the vector .o k∈R+ frequencies. We calculate the norm of the trial function. Lemma 1 The trial function .ϕy has the following norm ||ϕy ||2 = 1 + O(α 3 log(α −1 )).

.

(55)

Proof Since the vacuum vector and the last addend composing the trial function y live in two different Fock sectors, since .uα ⊥o# and .uα ⊗ o has norm 1, we can write ˜ 1∗ ||2 + ||o ||2 , ||ϕy ||2 = 1 + 4α 3 ||o # y

.

we can conclude by applying Lemma 3 to the last term.

u n

In the following we are going to make use of the exponential decay of the ground state of the hydrogen atom reformulated in the next lemma to exploit the dependence on the parameters for our setting. We introduce .ζy ∈ C0∞ (R3 ), a smooth, radial characteristic function, with { ζy (x) =

1,

for

|x| ≤ 14 y,

0,

for

|x| ≥ 31 y,

.

(56)

which localizes the electron in a neighborhood of the origin strictly smaller than the distance from the wall. Lemma 2 There exists .C > 0 such that the following holds, for any .L > 1, ||uα (1 − ζy )||2 ≤ CL2 e−L .

.

(57)

The proof is a straightforward direct calculation of the norm. Localizing in a neighborhood of zero we can consider the Taylor expansion of the potential .αVy : Vy (x) = −

.

( |x|4 ) (x · y) ˆ 2 + |x|2 fodd (x) , + O + 8y 4 y5 8y 3

for any x ∈ BR (0), R > 0, (58)

138

M. Olivieri

where .fodd is an odd function in x such that .|fodd (x)| ≤ C|x|3 . A direct consequence is that, for the Coulomb potential of interaction with the wall we have the estimate, recalling that .y = Lα −1 , = +

.

( α2 ) ( C) α2 > + α||(1 − ζ + O )u || ||V u || + y α α y y L3 L5 ) ( 2 2 ( ) α α = − 3 + O 5 + O α 2 Le−L/2 , L L ≤−

(59)

where for the localized part we used (58), while for the complementary part we used Lemma 2, (36) and (36), the last one giving ||Vy> uα || ≤ C||P uα || ≤ Cα.

(60)

.

We are now ready to prove Theorem 4. Proof (of Theorem 4) Let us calculate the quadratic form of the Hamiltonian on the function .ϕy ϕy = uα ⊗o + uα ⊗2α 3/2 o˜ 1 + oy

.

∗

+ 2Re o

˜ 1∗ | Hy o >. + 2Re + 2Re = O(α 4 ), 2Re| ≤ Cα 5/2 ||o ˜ 1∗ ||||Vy uα ||||o || = O(α 4 ), 4α 5/2 |Re 1, α2 + α||λy ||2 − ||o#y ||2# + 4α 3 ||o1∗ ||2∗ L3 ( α2 ) ( ) + O 5 + O(α 4 log(α −1 )) + O α 2 Le−L/2 . L

Ey ≥ eα −

.

(72)

Proof Let .wy denote the normalized ground state of .Hy so that Ey = .

.

(73)

We decompose .wy in the following way: wy = uα ⊗ oy + Ry ,

.

(74)

where • .oy := L2 (R3y ) and we further decompose 3/2 ˜ 1 o∗ + R∗ , oy = o(0) y + 2ηα

.

(75)

(0)

where .oy is the component of .oy in the zero-th Fock sector and the conditions R∗(0) = 0,

.

define .R∗ and .η.

˜ 1∗ | R∗(1) >∗ = 0, uα ⊗oy + < Hy >Ry + 2Re.

.

(79)

We analyze each term separately. Let us start from the quadratic form in .uα ⊗ oy : < Hy >uα ⊗oy = u

.

(0) α ⊗oy

+ uα ⊗2α 3/2 ηo1∗ + uα ⊗R∗

3/2 + 2Re y | Hy uα ⊗ (2α

+ 2Re + 2Re,

(80)

where u

.

(0)

α ⊗oy

( α2 ) ( ( )) α2 2 = eα + α||λy ||2 − 3 + O 5 + O α 2 Le−L/2 |o(0) y | , L L

(81)

thanks to (59), and ˜ 1∗ ||2∗ + O(α 4 ) = 4α 3 |η|2 ||o1∗ ||2∗ + O(α 4 ), uα ⊗2α 3/2 ηo1∗ = 4α 3 |η|2 ||o

.

(82)

due to (36), (36) and by symmetries in the .∗-norm. The quadratic term for .R∗ gives 2 uα ⊗R∗ = (eα + α||λy ||2 )||R∗ ||2 + ||R∗ ||2∗ + αuα ||R∗ ||2 + α||A− y R∗ ||

.

≥ ||R∗ ||2∗ + C1 α||R∗ ||2 ,

(83)

2 where we bounded from below .||A− y R∗ || by zero, used (36), (36) with (60) and chose .0 < C1 ≤ ||λy ||2 + α −1 eα − Lα − CαLe−L/2 . For the cross terms we have 3/2 ˜ 1 2Re = 0, y | Hy (uα ⊗ 2α

.

(84)

142

M. Olivieri

because .Pj uα ⊥uα , j = 1, 2, 3, by mismatch of Fock sectors, and 2Re = 0,

(85)

.

(0)

because .R∗ = 0. We use again this last property together with (76) in ˜ 1∗ )> 2Re = 4(eα + α||λy ||2 + αuα )Re η. + 2αRe 0 to be chosen later and the standard Fock estimate, for any .w ∈ rs (h+ ), 1/2 ||A− w||, y w|| ≤ ||λy ||L∞ (R3y ;h+ ) ||N

(87)

.

where .N is the number operator, to obtain ˜ 1∗ ||2 (86) ≥ − Cα(α −1 eα + α||λy ||2 + α 2 Le−L/2 )ε1 ||R∗ ||2 − Cα 4 ε1−1 |η|2 ||o

.

≥ − C2 αε1 ||R∗ ||2 + O(α 4 ),

(88)

where we used also (36), (36) and chose .0 < C2 ≤ C(||λy ||2 +α −1 eα +CαLe−L/2 ). This implies that, using (81)–(85) and (88) in (80), we get ( α2 ) 2 3 2 1 2 2 uα ⊗oy ≥ eα + α||λy ||2 − 3 |o(0) y | + 4α |η| ||o∗ ||∗ + ||R∗ ||∗ L ( α2 ) ( ) + α(C1 − C2 ε1 )||R∗ ||2 + O 5 + O α 2 Le−L/2 + O(α 4 ). L (89)

.

For the quadratic form of the remainder, y

Ry = κoy + Ry# + 2Re,

.

(90)

#

we have ( y ) y y 2 κoy = |κ|2 ||o# ||2# + (eα + α||λy ||2 )||o# ||2 + α||A− y o# || + αoy

.

#

#

= |κ|

2

y ||o# ||2#

+ O(α log(α 4

−1

)),

(91)

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

143

where we used (136), (137) and (139). For the quadratic form in .Ry# we have # 2 Ry# = ||Ry# ||2# + (eα + α||λy ||2 )||Ry# ||2 + α||A− y Ry || + αRy# .

.

(92)

# 2 We estimate .||A− y Ry || by zero for a lower bound, and we observe that

||Ry# ||2# = ||P Ry# ||2 −

.

/α \ − eα ||Ry# ||2 + ||Ry# ||2∗ |x| Ry#

≥ (1 − ε2−1 α)||P Ry# ||2 − (ε2 α + eα )||Ry# ||2 + ||Ry# ||2∗ ,

(93)

where we used a Hardy-type inequality for the Coulomb potential, a CauchySchwarz weighted with a parameter .ε2 > 0, and α2 # 2 ||R || + >Ry# L y ) ( α2 + ε3 α ||Ry# ||2 − ε3−1 α||P Ry# ||2 , ≥− C L

Ry# ≥ − C

.

(94)

thanks again to a Cauchy-Schwarz inequality, this time weighted with .ε3 > 0, and used (36). We conclude that Ry# ≥ (1 − (ε2−1 + ε3−1 )α)||P Ry# ||2 + (C3 − ε2 − ε3 )α||Ry# ||2 + ||Ry# ||2∗ ,

.

(95)

where we chose .0 < C3 ≤ ||λy ||2 + α −1 eα − C Lα . y For the cross term, using the orthogonality of .Ry# and .o# in the .#−scalar product, y

y

2Re = (eα + α||λy ||2 ) 2Re

.

− # # + 2αRe + 2αRe, y

y

(96)

and we observe that, thanks to a Cauchy-Schwarz inequality weighted with a parameter .ε4 > 0, −1 − # − # 2 2 − 2 2αRe ≥ − Cαε4 ||Ay Ry || − Cε4 α|κ| ||Ay o# || , y

.

y

≥ − Cαε4 ||Ry# ||2 + O(α 4 ),

(97)

where we used Lemma 3 and (87). By a Cauchy-Schwarz inequality, (36), (37) and Lemma 3 we have 2αRe ≥ − Cαε5−1 |κ|2 ||Vy o# ||2 − Cαε5 ||Ry# ||2

.

y

y

≥ O(α 4 ) − Cαε5 ||Ry# ||2 .

(98)

144

M. Olivieri

Again by Lemma 3 and by a Cauchy-Schwarz weighted with a parameter .ε6 > 0, we have, since .α −1 eα + ||λy ||2 is positive, (eα + α||λy ||2 ) 2Re ≥ −Cε6 α||Ry# ||2 − Cε6−1 α|κ|2 ||o# ||2 y

.

y

= −Cε6 α||Ry# ||2 + O(α 4 log(α −1 )).

(99)

Collecting (97)–(99) and plugging them into (96) we get 2Re ≥ −C(ε4 + ε5 + ε6 )α||Ry# ||2 + O(α 4 log(α −1 )), y

.

(100)

and using this last estimate, (91) and (95) we get 6 ) ( E y Ry ≥ |κ|2 ||o# ||2# + ||Ry# ||2∗ + α C3 − C εj ||Ry# ||2

.

j =2

+ (1 − Cα)||P Ry# ||2

+ O(α log(α −1 )). 4

(101)

Now we analyze the last term in (79) 3/2 2Re = 2Re y | Hy Ry > + 2Re 2Re = 2Re, + 2Re ≥ −C|η|2 α 4 ||o ˜ 1∗ ||2 − Cα|κ|2 ||Vy o ||2 = O(α 4 ). 2Re 0, (36), (37) to get ˜ 1∗ | αVy Ry# > 2Re 0 and Lemma 3 to get 2αRe ≥ −Cαε8 ||R∗ ||2 − Cαε8−1 |κ|2 ||Vy o# ||2 y

.

y

= −Cαε8 ||R∗ ||2 + O(α 4 ).

(108)

For the second addend we apply a Cauchy-Schwarz inequality and use (60) and (36) to get 2αRe ≥ − Cα||Vy uα ||||R∗ ||||Ry# ||

.

≥ − Cα 2 ||R∗ ||2 − Cα 2 ||Ry# ||2 .

(109)

Collecting then (103), (105), (106), (108) and (109) and plugging them into (102) we get (0)

y

2Re = − 2Re(oy κ)||o# ||2# − Cα(ε7 + α)||Ry# ||2

.

− Cα(ε8 + α)||R∗ ||2 − Cαε7 ||P Ry# ||2 + O(α 4 ). (110)

146

M. Olivieri

We finally collect the inequalities (89), (101) and (110) and plug them into (79) to obtain the following lower bound for the quadratic form of the Hamiltonian

.

( ( 2 α2 ) (0) ) y 2 3 2 1 2 2 ≥ eα + α||λy ||2 − 3 |o(0) y | + |κ| − 2Re(oy κ) ||o# ||# + 4α |η| ||o∗ ||∗ L 7 ) ( E + ||R∗ ||2∗ + α(C1 − C2 ε1 − Cε8 − Cα)||R∗ ||2 + α C3 − α − εj ||Ry# ||2 j =2

+ (1 − Cα(1 + ε7 ))||P Ry# ||2 + O

( α2 ) L5

( ) + O α 2 Le−L/2 + O(α 4 log(α −1 )).

Choosing the .εj , .j = 1, . . . , 8 such that C1 − C2 ε1 − Cε8 − Cα > 0,

.

C3 − α −

7 E

εj > 0,

1 − Cα − Cαε7 > 0.

j =2

we can bound from below the positive terms involving .||R∗ ||2 , .||Ry# ||2 , .||P Ry# ||2 and (0)

||R∗ ||2∗ . Using that .|oy | ≤ 1, we complete the square and bound

.

(0)

2 (0) 2 |κ|2 − 2Re(oy κ) = |κ − o(0) y | − |oy | ≥ −1,

.

(111)

finally making us obtain ( α2 ) y 2 2 3 2 1 2 ≥ eα + α||λy ||2 − 3 |o(0) y | − ||o# ||# + 4α |η| ||o∗ ||∗ L ( α2 ) ( ) + O 5 + O α 2 Le−L/2 + O(α 4 log(α −1 )). L

.

Comparing the result with the upper bound obtained in Theorem 4 let us bound (0) |oy |2 and .|η|2 by 1 plus terms which, multiplied with the rest, can be reabsorbed u n in the error terms, concluding the proof of the desired lower bound.

.

3.3 Evaluation of the Norms Joining together the upper and lower bounds for .Ey obtained in Theorems 4 and 5 and subtracting the estimate of .E∞ by Theorem 1, we get α2 y 2 2 + α(||λy ||2 − ||λ∞ ||2 ) + ||o∞ # ||# − ||o# ||#. L3 ( α2 ) ( ) + O 5 + O α 2 Le−L/2 + O(α 4 log(α −1 )). L

WyQF T = Ey − E∞ = −

.

(112) (113)

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

147

Introducing the quantity .

y 2 2 E := α(||λy ||2 − ||λ∞ ||2 ) + ||o∞ # ||# − ||o# ||# ,

(114)

our goal in this section is to prove an estimate for . E by the following proposition. Proposition 1 For any .L > 1, .

E=

( α3 ) α α2 −1 − ℵ +O log(α ) +O(α 4 log(α −1 ))+O(α 3 L2 e−L ). α L3 L4 L2

(115)

Proof We split the proof in three parts: we evaluate the norms involving the .λ terms, then the ones involving the .o terms and finally we sum the results and give the estimate above by studying some complex line integrals. Let us recall the definitions (23) and (47) of .λy and .λ∞ , respectively, for reader’s convenience. Then, f ||λy ||2 =

dk

.

R3+

2 (k) E { χA ey(1) 2 (k) cos2 (k1 y) π 2 |k| y =1,2

} +(ey(2) 2 (k) + ey(3) 2 (k)) sin2 (k1 y) ,

ˆ e1 (k), e2 (k)} is an where we denoted by .R3+ := R+ × R2 . We now use that .{k, 3 orthonormal basis, for a.e. .k ∈ R , to have E )2 . e(j = 1 − kˆj2 , j = 1, 2, 3, (116) y y =1,2

and plug in the previous expression, using some goniometric formulas, to get f ||λy ||2 =

.

R3+

dk

2 (k) { χA (1 − kˆ12 )(1 + cos(2k1 y)) 2π 2 |k|

} +(2 − kˆ22 − kˆ32 )(1 − cos(2k1 y)) .

Since we have only even integrands in the .k1 variable, we can turn the integration on the whole .R3 getting a factor .1/2, and we separate also the integer from the oscillatory parts: f .

=

dk R3

2 (k) χA (3 − kˆ12 − kˆ22 − kˆ32 ) 4π 2 |k| f χ 2 (k) + dk A2 (−1 − kˆ12 + kˆ22 + kˆ32 ) cos(2k1 y). 4π |k| R3

148

M. Olivieri

ˆ 2 = 2, we recognize the first term to be the Using that .3 − kˆ12 − kˆ22 − kˆ32 = 3 − |k| expression of the norm for .λ∞ and therefore we obtain f 2 2 .α(||λy || − ||λ∞ || ) = α dk fy (k), (117) R3

where we denoted by fy (k) :=

.

2 (k) 1 χA (−1 − kˆ12 + kˆ22 + kˆ32 ) cos(2k1 y). 2 4π |k|

(118)

We turn now the attention to the .o terms. Let us calculate 2 ||o# ||2# = 4α||(hα − eα + Hf+ )−1/2 P uα ⊗ A+ y o|| |2 | f | | P uα | λy (k)|| + O(α 3 L2 e−L ). = 4α dk dx | 1/2 (hα − eα + |k|) R3+ ×R3

.

y

where the last error term was obtained completing the domain of integration in the position variable and using that f C 3 2 −L α L e . (119) . dx |(hα − eα + |k|)−1/2 P uα (x)|2 ≤ |k| {x1 ≥y} Expanding the square, we obtain, by similar calculations to the .λ terms and recalling the expression of .fy (k) in (118), y 2 .||o ||# #

|2 | | | P uα | + | λ (k) = 8α dkdx | ∞ | (hα − eα + |k|)1/2 R3+ ×R3 |2 | f | | P uα | fy (k) + O(α 3 L2 e−L ). dkdx || + 8α | 1/2 3 3 (h − e + |k|) α α R+ ×R f

We observe that both the integrals can be extended to the whole .R3 dropping a factor 2 2 thanks to the even integrand, recovering the expression for .||o∞ # ||# and therefore write, 2 2 ||o∞ # ||# − ||o# ||# = −4α y

.

||2 || || || P uα || fy (k) + O(α 3 L2 e−L ). dk || || 2 || 1/2 3 (h − e + |k|) α α R L (120)

f

We use the relation .2iP uα = (hα − eα )xuα to write 2 2 ||o∞ # ||# − ||o# ||# = −α

.

y

|| || || (hα − eα )xuα ||2 3 2 −L || dk || || (h − e + |k|)1/2 || 2 fy (k) + O(α L e ). 3 α α R L (121)

f

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

149

Calling || || || (hα − eα )xuα ||2 || || .G(k) := || (h − e + |k|)1/2 || 2 , α

α

(122)

L

we can finally write 2 2 ||o∞ # ||# − ||o# ||# = −α y

.

f R3

dk G(k) fy (k) + O(α 3 L2 e−L ).

(123)

Let us calculate now the difference between the norms of the .λ and .o terms. In order to do so, we observe that, by explicit calculations, = 3.

.

(124)

Therefore, by (124), (117) and (123), f . E =α dk fy (k) (1 − G(k)) + O(α 3 L2 e−L ) = R3

| | \ / | (hα − eα ) (hα − eα )2 || 3 2 −L xu − dk fy (k) xuα || | α + O(α L e ) = 3 (h − e + |k|) 3 α α R | | \ (/ f | (hα − eα )|k| | | | xuα dk fy (k) xuα | =α 3(hα − eα + |k|) | R3 | | \) / | 2(hα − eα )2 | | xuα + O(α 3 L2 e−L ), | − xuα | 3(hα − eα + |k|) | f

=α

where we used the second resolvent formula to reduce to a common denominator and perform the calculation above. The estimate of the oscillatory integrals are proven in Appendix 2. We show in Lemma 4 how the second term in the expression above produces an error of order α4 .O( L ), while in Proposition 2 we show how the first integral is responsible for the cancellation of the van der Waals term coming from the Coulomb interaction and produces the new leading term. This concludes the proof of Proposition 1. u n This concludes the proof of the main Theorem 3: by Theorem 4, Theorem 5 3 2 −L and ) 1 we get, observing that the error .O(α L e ) can be absorbed in ( Proposition 2 −L/2 .O α Le , QFT

WL

.

( α2 ) ( ) α2 + O(α 4 log(α −1 )) + O α 2 Le−L/2 = + O 5 3 L L ( α3 ( α2 ) ) 2 α = − 4 ℵα,L + O 5 + O 2 log(α −1 ) L L L ( 2 −L/2 ) 4 −1 , + O(α log(α )) + O α Le

= E−

with .ℵα,L defined in (52).

150

M. Olivieri

4 Discussion of the Result In this section we analyze the result of Theorem 3 to get information about the leading term of the interaction energy in the different regimes for the distance. Let us recall the expression of .ℵα,L obtained in the calculations for the proof of Proposition 2: ℵα,L

.

)\ / ( 1 1 αL arctan , = αL(h1 − e1 ) xu1 6π QFT

and the expression of the interaction energy .WL reader’s convenience

obtained in Theorem 3 for the

α + O(α 4 log(α −1 )) L4 ( α3 ) ( α2 ) ) ( + O α 2 Le−L/2 + O 5 + O 2 log(α −1 ) . L L

QFT

= Ey − E∞ = − ℵα,L

WL

.

(125)

(126)

We want to study this expression and compare the first term with the behavior of the error terms. The spectral gap value for the spectrum of the hydrogen atom Hamiltonian .h1 is 3 −1 ≤ 16 . . , 3 16 so that .(h1 − e1 ) −1 Let us consider the regime .L ≥ 16 3 α : in this case, the argument of the .arctan in .ℵα,L is smaller than 1 and by a Taylor expansion and functional calculus it can be approximated by ( 1 ) 1 ||(h1 − e1 )−1/2 xu1 ||2 + O 2 2 , 6π α L

ℵα,L =

.

which, plugged in (126), it gives that the first two order terms are, since .L ≥ QFT

WL

.

=−

α ||(h1 − e1 )−1/2 xu1 ||2 + O(α 4 log(α −1 )), 6π L4

(127) 16 −1 3 α ,

(128)

where the second term is dominant and expresses an error bigger than the first term. 16 −1 For the regime . 16 3 < L < 3 α , we introduce the parameter .η > 0 such that the interval can be described as L=

.

16 −1+η α , 3

η ∈ (0, 1).

(129)

For these distances, by a Taylor expansion and recalling that .||xu1 ||2 = 12, we have ℵα,L =

.

( )\ 3α −η 16 η 8α η / arctan α + O(α 2η ), = 9π 16(h1 − e1 ) xu1 3

(130)

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

151

which, plugged in (126), gives that the relevant terms in the interaction energy are ( α 1+2η ) 16 α 1+η + O(α 4 log(α −1 )) + O(α 5−2η ). + O (131) 3 L4 L4 ( ) The first term is the leading term for .η ∈ 13 , 1 , otherwise the leading term is of order .O(α 4 log(α −1 )). For the remaining regime, .1 < L ≤ 16 3 , we recover the expression of the van der Waals term, because, again by a Taylor expansion, we have QFT

WL

.

=−

ℵα,L =

.

αL ||xu1 ||2 + O(α 2 L) = αL + O(α 2 L), 12

(132)

and then the leading term reads QFT

WL

.

=−

α2 . L3

(133)

Finally, we can collect here below the expressions of the leading terms of the energy and the associated values of .ℵα,L depending on the distance:

ℵα,L

.

⎧ ⎪ αL ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 16 η = α ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 ||(h − e )−1/2 xu ||2 1 1 1 6π

if 1 < L ≤

16 , 3

16 −1+η α , η ∈ (0, 1), 3 16 −1 α , if L ≥ 3 if L =

and

QFT

WL

.

⎧ α2 ⎪ ⎪ − ⎪ ⎪ ⎪ L3 ⎪ ⎪ ⎨ 1+η = − 16 α ⎪ ⎪ 3 L4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩O(α 4 log(α −1 ))

if 1 < L ≤

16 , 3

(1 ) 16 −1+η α ,1 , , η∈ 3 3 16 −1+ 1 3. α if L ≥ 3

if L =

As a remark, we underline the fact that the expression of the leading term we would QFT −1 is have liked to obtain for .WL in the regime .L > 16 3 α QFT

WL

.

=−

α ||(h1 − e1 )−1/2 xu1 ||2 , 6π L4

(134)

but the precision used in the calculation does not allow to produce an error small enough to make the term above to appear as leading term. The problem seems to

152

M. Olivieri

be, anyway, just of technical nature. Furthermore, it is really intrinsic in the method used that some terms of the error obtained are uniform in L. Therefore, whatever the degree of precision of the error in .α, one can always find a distance large enough such that (134) is no longer the leading term. In the right units, .α −1 corresponds to the value of a half of Bohr radius, and expressing the distance .y = Lα −1 , L represents half of the number of Bohr radii. In conclusion, plugging the numerical values of the parameters .

16 = 5.3, 3

16 −1+ 1 3 = 165, α 3

(135)

we see that our result proves the Casimir-Polder effect for all the distances up to approximately .82.5 Bohr radii.

Appendix 1: Technical Inequalities y

Lemma 3 The following estimates hold for the vector o# ||o# ||2 = O(α 3 log(α −1 )), .

.

y

2 3 ||A− y o# || = O(α ), . y

(136) (137)

||P o# ||2 = O(α 5 log(α −1 )), .

(138)

oy = O(α 5 log(α −1 )), .

(139)

||Vy o# ||2 = O(α 5 log(α −1 )).

(140)

y

#

y

Proof Let us start by proving (136): 2 ||o# ||2 = 4α||(hα − eα + Hf+ )−1 P uα ⊗ A+ y o|| |2 2 (k) || E f f | χA P uα | | ≤ Cα dx dk | |k| (hα − eα + |k|) | R3y R3+

.

y

y =1,2

( ) × ey(1)2 cos2 (k1 y) + (ey(2)2 + ey(3)2 ) sin2 (k1 y) f 2 (k) χA ≤ Cα 3 = O(α 3 log(α −1 )), dk 3 2 2 α ) |k|(|k| + 16 R3 where we used that the integrand is an even function of k1 to extend the integral to 3 2 the whole space and that the spectral gap of the hydrogen atom is 16 α .

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

153

For inequality (137), we find, by similar calculations and use of symmetries, that − y .||Ay o || #

f ≤ Cα ×

(f dx

R3y

R3+

dk

2 (k) χA P uα |k| (hα − eα + |k|)

E ( ))2 ey(1)2 cos2 (k1 y) + (ey(2)2 + ey(3)2 ) sin2 (k1 y) y =1,2

f

≤ Cα

R3

dx((hα − eα )xuα (x))2

(f dk R3

χA (k)2 )2 = O(α 3 ). |k|2

In order to prove (138) we observe that P uα is odd. On the subspace of antisymmetric functions the infimum of the spectrum of hα is strictly bigger than eα . This, and the fact that we can choose a y0 > 0 small enough such that hα + y0 (−A) α is a perturbation of hα , gives us the inequality −(1−y0 )A− |x| ≥ eα , which implies P 2 < y0−1 (hα − eα ). We apply this to y 2 .||P o || #

| |f | 2 (k)(h − e )1/2 P u |2 χA α α α| | ≤ dk | dx | | R3 |k|(hα − eα + |k|)2 | R3 | |f f | 2 (k)(h − e )1/2 P u |2 χA 1 1 1| | −1 5 ≤ y0 Cα dk | dx | = O(α 5 log(α −1 )), | R3 |k|( 3 α 2 + |k|)2 | R3 y0−1 Cα

f

16

where we used similar calculations as in the proof of (136). We prove now (139): by (36), (37), (136) and (138) we have α y |αoy | ≤ C ||o# ||2 + >oy # # y

.

≤C

α5 y y log(α −1 ) + α||o# ||||P o# || = O(α 5 log(α −1 )). L

(141)

For inequality (140) we use again (36), (37), (136) and (138): ||Vy o2# ||2 ≤ C

.

α4 y log(α −1 ) + C||P o# ||2 = O(α 5 log(α −1 )). L2

(142)

Appendix 2: Estimates of Oscillatory Integrals In this technical Appendix we collected the lemmas which prove the estimates of the oscillatory integrals needed to evaluate the quantity . E in Sect. 3.3.

154

M. Olivieri

Lemma 4 For any .L > 1, | / f | .α dk fy (k) xuα ||

| \ ( α4 ) (hα − eα )2 || xu . = O α (hα − eα + |k|) | L

R3

Proof Let us first bound the quantity f .J :=

dk

R3

fy (k) , α 2 + |k|

(143)

(144)

where we recall the expression 2 (k) 1 χA (−1 − kˆ12 + kˆ22 + kˆ32 ) cos(2k1 y). 4π 2 |k|

fy (k) =

.

(145)

Let us change to spherical coordinates .(ρ, ϕ, θ ) ∈ (0, +∞) × (0, 2π ) × (0, π ) so that .ρ = |k|, .k1 = ρ cos(θ ) and .(−1 − kˆ12 + kˆ22 + kˆ32 ) = −2 cos2 θ and then J gives .

−

1 2π 2

f

f

+∞ 0

=−

1 π

f

2π

dρ

π

dϕ 0

f

+∞ 0

0

2 dθ ρ sin θ χA (ρ)

2 dρ χA (ρ)

ρ 2 (α + ρ)

f

π

(α 2

1 (cos θ )2 cos(2ρy cos θ ) + ρ)

dθ sin θ (cos θ )2 cos(2ρy cos θ ).

0

A further change of variables .τ = cos θ gives .

−

1 π

f

+∞

dρ 0

=− {

1 π

f

2 (ρ)ρ χA (α 2 + ρ)

+∞

dρ 0

f

1

−1

dτ τ 2 cos(2ρyτ )

2 (ρ)ρ χA (α 2 + ρ)

} −4 sin(2ρy) + 2(2ρy)2 sin(2ρy) + 8yρ cos(2ρy) . (2ρy)3

Changing again variables .σ = ρy we have, recalling that .y = Lα −1 , ( ) 2 σ σ ( ) f +∞ (α) χ A y cos(2σ ) sin(2σ ) 1 − . =O sin(2σ ) + dσ .J = − 2 2 πy 0 σ L (yα + σ ) 2σ From this, using that | | | | / \ / \ | | (hα − eα )2 | | (h1 − e1 )2 2 | | | | . xuα | (h − e + |k|) | xuα = α xu1 | (α 2 (h − e ) + |k|) | xu1 , α α 1 1 and the spectral theorem we conclude the proof.

(146) u n

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

155

In the next lemma we show by complex line integration techniques, inspired by the original work of Casimir and Polder [15], the main integral term gives the fundamental cancellation of the van der Waals term and produces the new leading term. Proposition 2 For any .L > 1, | | \ / f | (hα − eα )|k| | | | xuα .α dk fy (k) xuα | 3(hα − eα + |k|) | R3 ( α3 ) α α2 = 3 − ℵα,L 4 + O 2 log(α −1 ) , L L L

(147)

where .ℵα,L is defined in (153). Proof We write explicitly the expression of the integral and denote it by I , f χ 2 (k) α dk A (−1 − kˆ12 + kˆ22 + kˆ32 ) cos(2k1 y) .I := 2 |k| 4π R3 | | \ / | (hα − eα )|k| | | xuα . | × xuα | 3(h − e + |k|) | α

α

Let us pass to spherical coordinates .(ρ, ϕ, θ ) ∈ (0, +∞) × (0, 2π ) × (0, π ) so that ρ = |k|, .k1 = ρ cos(θ ) and .(−1 − kˆ12 + kˆ22 + kˆ32 ) = −2 cos2 θ ,

.

α .− 2π 2

f

+∞ f 2π 0

0

f

π

0

2 dρ dϕ dθ ρ 2 sin θ χA (ρ)

| | \ / | (hα − eα ) | | xuα . × cos2 θ cos(2ρy cos θ ) xuα || 3(hα − eα + ρ) |

By an explicit calculation, the integration in .ϕ gives only a .2π factor and the one in the .θ variable: f π sin(2ρy) cos(2ρy) sin(2ρy) + − . dθ sin θ cos2 θ cos(2ρy cos θ ) = ρy ρ2y2 2ρ 3 y 3 0 ) ( 1 −i i + 2 2 + 3 3 e2iρy + h.c., = 2ρy 2ρ y 4ρ y where in the last line we used Euler formulas for sine and cosine. Plugging in the original calculation and making explicit the dependence on .α we have α .I = 6π

f

+∞ 0

||2 || || || (h1 − e1 )1/2 || || dρ ρ || (α 2 (h − e ) + ρ)1/2 xu1 || 1 1 ) } {( i i 1 × − 2 2 − 3 3 e2iρy + h.c. . ρy ρ y 2ρ y 2

2 χA (ρ)

156

M. Olivieri

+ ext

+

+ int + − − int

− ext

−

Fig. 2 Complex line integrals

Let us define the class of integrals below, for a measurable set .B ⊆ C: f IB :=

dz g(z),

.

B

||2 ( || ) || || (h1 − e1 )1/2 i 1 α 2 2 i || || e2izy . − z χA (z) || 2 g(z) := xu − 1 || zy 6π (α (h1 − e1 ) + z)1/2 z2 y 2 2z3 y 3 Here .χA (z) ≡ χA (|z|), z ∈ C, denotes the complex extension of the frequencies cut-off defined only for real arguments. Thanks to this notation, we can rewrite the integral over the half-line as a limit introducing a parameter .ε ∈ (0, 1): I = lim (I(ε,ε−1 ) + I(ε,ε−1 ) ).

.

ε→0

(148)

For both the integrals, their integrands are analytic in .{z ∈ C | Rez > 0} \ Bε (0). Then, integrating over any closed path in that domain gives zero as result. Let us interpret the interval .(ε, ε−1 ) as part of two closed paths (Fig. 2): + + −1 + yε,ε ) ∪ yext ∪ i(ε−1 , ε) −1 := −yint ∪ (ε, ε

.

− − −1 − yε,ε ) ∪ −yext ∪ −i(ε−1 , ε) −1 := yint ∪ (ε, ε

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

157

where

| | ( π )} ( π )} { { | | + + , yext , yint = z = εeiθ | θ ∈ 0, = z = ε−1 eiθ | θ ∈ 0, 2 2 | | ( π )} ( π )} { { | | − − yext yint = z = εeiθ | θ ∈ − , 0 , = z = ε−1 eiθ | θ ∈ − , 0 . 2 2

.

From the picture we see that [f

f I(ε,ε−1 ) =

.

(ε,ε −1 )

dz g(z) = − [f

f I(ε,ε−1 ) =

(ε,ε −1 )

dz g(z) = −

f + −yint

+

i(ε−1 ,ε)

+

f − yint

+

]

f + yext

f −i(ε−1 ,ε)

+

− −yext

dz g(z), ] dz g(z).

We can immediately observe that the ext terms disappear in the limit of .ε → 0. Indeed .g(ε−1 eiθ ) = 0, for sufficiently large .ε−1 (bigger than .A), thanks to the presence of the cut-off .χA : f . lim dz g(z)# = 0, (149) ε→0 y ± ext

with .g # covering both the cases g and .g. ¯ So it remains to analyze the int and imaginary terms. Changing variables to .z = + , we have εeiθ in the integral over .yint f .

α dz g(z) = + 6π yint

f

π 2

iθ

2 2iθ

iεe dθ ε e

0

( ×

2 χA (ε)

i εeiθ y

−

||2 || || || (h1 − e1 )1/2 || || || (α 2 (h − e ) + εeiθ )1/2 xu1 || 1

1 ε2 e2iθ y 2

1

−

i 3 2ε e3iθ y 3

)

e2iεe

iθ y

.

By the bounded convergence theorem we can switch the limit with the integral and obtain, having in mind that .||xu1 ||2 = 12 and recalling that .y = Lα −1 , f f π 2 1 α −1 . − lim dz g(z) = e3iθ dθ ||xu1 ||2 3iθ 3 + ε→0 −y 6π 0 2e y int =

α2 α −1 π 12 = . 3 6π 2 2y 2L3

− A totally analogous calculation yields the same result for .yint , so that } {f f α2 . − lim dz g(z) + dz g(z) = 3 . + − ε→0 L −yint yint

(150)

158

M. Olivieri

We show how the term of order . Lα4 comes from the integration on the imaginary −1 axis. For the ) change variable setting .w = −iz and denoting by ( path in .i(ε, ε ) we 1 1 1 −2wy : .by (w) := wy + w2 y 2 + 2w3 y 3 e f .

dz g(z)

i(ε−1 ,ε)

α 6π

=

α 6π

=

f

ε−1

||2 || || || (h1 − e1 )1/2 2 || idw w 2 χA (w) || xu || (α 2 (h − e ) + iw)1/2 1 || by (w) 1

ε

f

ε−1

1

| | \ | | (h1 − e1 ) 2 2 | | xu1 by (w), dw w χA (w) xu1 | (−iα 2 (h − e ) + w) | /

1

ε

1

while for the path .i(ε, ε−1 ) we change variable setting .w = iz, obtaining f .

−i(ε−1 ,ε)

α 6π

=

f

dz g(z)

ε−1

ε

| | \ / | | (h1 − e1 ) 2 | xu1 by (w). dw w 2 χA (w) xu1 || (iα 2 (h1 − e1 ) + w) |

Using that / .

| | xu1 ||

| \ | (h1 − e1 ) (h1 − e1 ) | xu1 + (−iα 2 (h1 − e1 ) + w) (iα 2 (h1 − e1 ) + w) | | | \ / | | 2w(h1 − e1 ) | xu1 | = xu1 | 4 (α (h1 − e1 )2 + w 2 ) |

(151)

we sum the two contributions to finally obtain f .

−

f

i(ε−1 ,ε)

=−

dz g(z) − α 6πy 3

f

ε−1 ε

−i(ε−1 ,ε)

dz g(z)

|| || (h1 − e1 )1/2 xu1 2 dw χA (w) || || (α 4 (h − e )2 + w 2 )1/2 1

1

||2 || || 2w 3 y 3 by (w). ||

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

159

By a further change of variable .v = wy = wLα −1 , f .

−

f

i(ε−1 ,ε)

=−

dz g(z) −

α5 6π L4

f

−i(ε−1 ,ε)

( αv ) || || (h1 − e1 )1/2 xu1 || || 4 L (α (h1 − e1 )2 + L−2 α 2 v 2 )1/2 ( ) × 2v 2 + 2v + 1 e−2v

yε−1

2 dv χA

yε

α3 −−→ − ε→0 6π L4

dz g(z) =

f

+∞

0

||2 || || ||

||2 || || ( αv ) || 1/2 xu (h − e ) 1 1 1 || || 2 dv χA || || 2 L || (α 2 (h1 − e1 )2 + v 2 )1/2 || L ( ) × 2v 2 + 2v + 1 e−2v .

Let us split the region of integration in two parts in order to estimate the last integral: (0, 1) ∪ (1, +∞).

.

• .v ∈ (1, +∞): the following estimates holds |f ( ) α 3 || +∞ 2 αv dv χ . A L 6π L4 | 1

|| || || ||

(h1 − e1 )1/2 xu1 (α 2 (h1 − e1 )2 +

v 2 1/2 ) L2

||2 ( | ) || | || 2v 2 + 2v + 1 e−2v | || |

||2 f +∞ ( α3 ) α 3 || || || 1/2 ≤ 2 ||(h1 − e1 ) xu1 || dv e−2v = O 2 . L L 1 2 (2v 2 + 2v + 1)e−2v by 1 thanks to a Taylor • .v ∈ (0, 1): the approximation of .χA expansion produces the following error

| ( αv ) | (α ) | 2 | (2v 2 + 2v + 1)e−2v − 1| ≤ C ||∇χA ||∞ + 1 |v|, | χA L L

.

(152)

which, by the functional calculus, implies α3 . 6π L4

f 0

1

|| || (h1 − e1 )1/2 xu1 || dv || || (α 2 (h1 − e1 )2 + v 2 )1/2 L2

||2 || | ( αv ) ( | ) || | 2 | 2v 2 + 2v + 1 e−2v − 1| || |χA || L =O

( α3 L2

) log(α −1 ) .

160

M. Olivieri

Therefore we pass to estimate the integral below, where, again by the functional calculus, we can write α3 .− 6π L4

f 0

1

||2 || || || (h1 − e1 )1/2 xu1 || || dv || || || (α 2 (h1 − e1 )2 + v 2 )1/2 || 2 L / ( )\ 1 α αL arctan . =− αL(h1 − e1 ) xu1 6π L4

Introducing the quantity ℵα,L :=

.

/ ( )\ 1 1 αL arctan , 6π αL(h1 − e1 ) xu1

(153)

we can finally state that .

{ f lim −

ε→0

=−

f i(ε−1 ,ε)

dz g(z) −

−i(ε−1 ,ε)

} dz g(z)

( α3 ) α −1 ℵ + O log(α ) . α,L L4 L2

(154)

Collecting the estimates (149), (150) and (154), we conclude the proof of Proposition 2. u n

Appendix 3: Derivation of the Model: Quantization on Half Space For simplicity we consider the half space R3+ := {x = (x1 , x2 , x3 ) ∈ R3 | x1 > 0},

.

with the surface of the conductor being .E0 = {(0, x2 , x3 ) ∈ R3 }, and we obtain the general result by translation and reflection. We denote by .E(x, t) = (E(j ) (x, t))3j =1 , and .B(x, t) = (B(j ) (x, t))3j =1 , .x ∈ R3+ , the components of the classical electric and magnetic fields .E, B ∈ R3 , respectively. The standard boundary conditions for .(E, B) in the presence of a grounded, perfect conductor wall which can be found, for example, in formula (13.106) from [38], are: n(x) ˆ × E(x) = 0,

.

n(x) ˆ · B(x) = 0,

for any x ∈ E0 ,

(155)

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

161

where .nˆ denotes the outward normal versor to the surface of the wall, in our case n(x) ˆ = (1, 0, 0). This implies that the conditions can be rewritten as

.

E(2) (0, x2 , x3 ) = 0 = E(3) (0, x2 , x3 ),

B(1) (0, x2 , x3 ) = 0,

.

(x2 , x3 ) ∈ R2 . (156)

We start from observing that, in the classical setting, the electric field function has to be a solution of the wave equation in the half space with constraints given by the aforementioned boundary conditions for the conductor surface: { .

∂t2 E(j ) (x, t) = −Ax E(j ) (x, t),

x1 > 0,

E

j = 2, 3.

(j )

(0, x2 , x3 , t) = 0,

(157)

We introduce a new electric field on the full space by an odd reflection { =(j ) E

.

(x, t) :=

E(j ) (x, t), −E

(j )

if x1 ≥ 0,

(−x1 , x2 , x3 , t), if x1 < 0.

(158)

The field is assumed to be real and its expansion in Fourier modes as solution of the wave equation has the standard expression (j ) = E (x, t) =

.

f 1 (j ) (j ) dk (β+ (k)ei(kx−ωt) + β− (k)e−i(kx−ωt) ), . (2π )3/2 R3

β+ (k) := F[= E (·, 0)](k), (j )

(j )

(j )

(j )

β− (k) = β+ (k),

(159) (160)

(j ) E is odd in .x1 , where we denoted by . F the Fourier transform in .R3 . Since .= its Fourier transform is odd in .k1 and this implies the following relations for the coefficients: (j )

(j )

β± (−k1 , k2 , k3 ) = −β± (k),

.

j = 2, 3,

(161)

which gives back, using an odd reflection, an expansion for the electric field (159) in terms of sines in the .x1 direction: for .j = 2, 3, (j ) = E (x, t)

.

f 2 (j ) (j ) = dk sin(k1 x1 )(iβ+ (k)ei(k2 x2 +k3 x3 −ωt) − iβ− (k)e−i(k2 x2 +k3 x3 −ωt) ) (2π )3/2 R3+ E f 2 ) i(k2 x2 +k3 x3 −ωt) dk sin(k1 x1 )e(j + h.c.), = y (k)(iβ+,y (k)e (2π )3/2 R3+ y =1,2

162

M. Olivieri

where we projected the Fourier coefficients on the polarization vectors .{ey (k)}y =1,2 β ± (k) =

E

.

β±,y (k)ey (k),

y =1,2

(1)

(2)

(3)

β ± (k) = (β± , β± , β± ),

(162)

and assumed to work in Coulomb gauge. Recalling the Maxwell equations in the vacuum .

∇ ·= E = 0,

E, ∇ ×= B = ∂t =

∇ ·= B = 0,

B, ∇ ×= E = −∂t =

(1) B and of .= E . we can recover the expressions of the components of .= = on We introduce the classical vector potential .A on .R3+ and its extension .A 3 .R , even for the first component and odd for the remaining ones. By the equation = we recover the expression of .A = as well. Collecting the previous formulas B = ∇×A .= for the expansions we finally obtain, for all the fields,

= E(x, t) =

.

E f 1 dk β+,y (k) b(k) + h.c., . (2π )3/2 R3+

(163)

y =1,2

= B(x, t) =

E f 1 dk β+,y (k) k × b(k) + h.c., . (2π )3/2 R3+

(164)

y =1,2

E f dk 1 β+,y (k) b(k) + h.c., 3/2 3 (2π ) R+ ω(k)

(165)

⎞ (1) 2 cos(k1 x1 )ey (k)ei(k2 x2 +k3 x3 −ωt) ⎟ ⎜ (2) .b(k) := ⎝ 2i sin(k1 x1 )ey (k)e i(k2 x2 +k3 x3 −ωt) ⎠ . i(k2 x2 +k3 x3 −ωt) 2i sin(k1 x1 )e(3) y (k)e

(166)

= t) = A(x,

y =1,2

where ⎛

We introduce the rescaled Fourier coefficients .{α±,y }y =1,2 by β±,y (k) = (2π )3/2

.

ω1/2 (k) α±,y (k). 2π

(167)

We further introduce a cut-off .χA ∈ C0∞ (R3 ) for the momenta (see the construction of the Abraham model [38, Chapter 2.4]) and we derive the expansion expression for the original fields .(E, B, A), for .x ∈ R3+ , E(x, t) = = E(x, t)|x1 >0 ,

.

B(x, t) = = B(x, t)|x1 >0 ,

= t)|x1 >0 , A(x, t) = A(x, (168)

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

163

where = t) = A(x,

E f

.

3 y =1,2 R+

dk

χA (k) α+,y (k) 2π |k|1/2 ⎛

⎞ (1) 2 cos(k1 x1 )ey (k)ei(k2 x2 +k3 x3 −ωt) ⎜ i(k2 x2 +k3 x3 −ωt) ⎟ × ⎝ 2i sin(k1 x1 )e(2) ⎠ + h.c. y (k)e (3) i(k x +k x −ωt) 2 2 3 3 2i sin(k1 x1 )ey (k)e

(169)

We want to derive the Fourier modes expansion for the electromagnetic energy too. By the usual definition, this time adapted to the half space, f 1 dx (|E(x, t)|2 + |B(x, t)|2 ) 8π R3+ 3 f 1 E = dx (|E(j ) (x, t)|2 + |B(j ) (x, t)|2 ). 3 8π R+

hf :=

.

j =1

Comparing the integral with the odd extensions for .j = 2, 3 and with the even extension for .j = 1, we can write 3 f 1 E (j ) (j ) dx (|= E (x, t)|2 + |= B (x, t)|2 ) .hf = 3 16π j =1 R f E f 1 E dk |k|α−,y (k)α+,y (k) = dk |k|α−,y (k)α+,y (k), = 2 R3 R3+ y =1,2

y =1,2

where for the second equality we used the usual expression for the electromagnetic energy in the full space and in the third equality we used (167) and the symmetry properties of the .β’s to change the domain of integration. By Wick quantization techniques for polynomial symbols (see [1] for details) we can define .(E, B, A) being the associated quantum field versions of the electromagnetic operators .(E, B, A), respectively. The theory results into the intuitive quantization rules α+,y (k)

.

→

ay† (k),

α−,y (k)

→

ay (k),

(170)

substitution that for a polynomial symbol .p(α+ , α− ) we denote as .(p(α+ , α− ))Wick . In this way we can write A(x) := (A(x, 0))Wick ,

.

Hf+ := (hf )Wick ,

x ∈ R3+

(171)

164

M. Olivieri

which gives the same expression for the operators .Ay (x), Hf given in Sect. 3 by a translation and a reflection in the .x1 variable for .A(x). Acknowledgments This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Project-ID 258734477 - SFB 1173. I thank D. Hundertmark and S. Vugalter for the intensive discussions about the topic and important suggestions about the scaling involved in the problem during my permanence in Karlsruhe Institute of Technology. I thank I. Anapolitanos for the discussions about the van der Waals forces involved in the model studied. I further thank M. Correggi, M. Falconi and L. Morin for suggestions about the presentation of the result, and the anonymous referee, whose accurate comments really helped to improve the presentation and the rigor of the paper.

References 1. Ammari, Z., Nier, F.: Mean field limit for bosons and infinite dimensional phase-space analysis. Ann. Henri Poincaré 9(8), 1503–1574 (2008) 2. Anapolitanos, I., Lewin, M.: Compactness of molecular reaction paths in quantum mechanics. Arch. Ration. Mech. Anal. 236(2), 505–576 (2020) 3. Anapolitanos, I., Sigal, I.M.: Long-range behavior of the van der Waals force. Commun. Pure Appl. Math. 70(9), 1633–1671 (2017) 4. Anapolitanos, I., Lewin, M., Roth, M.: Differentiability of the van der Waals interaction between two atoms. Hal-02144016 (2019) 5. Anapolitanos, I., Badalyan, M., Hundertmark, D.: On the van der Waals interaction between a molecule and a half-infinite plate (2020) ArXiv:2004.04771 6. Bach, V., Fröhlich, J., Pizzo, A.: Infrared-finite algorithms in QED II. The expansion of the groundstate of an atom interacting with the quantized radiation field. Adv. Math. 220, 1023– 1074 (2009) 7. Barbaroux, J.-M., Chen, T., Vougalter, V., Vugalter, S.: On the ground state energy of the translation invariant Pauli-Fierz model. Proc. Am. Math. Soc. 136, 1057–1064 (2007) 8. Barbaroux, J.-M., Chen, T., Vougalter, V., Vugalter, S.: Quantitative estimates on the binding energy for hydrogen in non-relativistic QED. Ann. Henri Poincaré 11(8), 1487–1544 (2010) 9. Barbaroux, J.-M., Hartig, M., Hundertmark, D., Vugalter, S.: Van der Waals-London interaction of atoms with pseudo-relativistic kinetic energy (February 2019) Hal-02043282 10. Bernasconi, F., Graf, G.M., Hasler, D.: The heat kernel expansion for the electromagnetic field in a cavity. Ann. Henri Poincaré 4(5), 1001–1013 (2003) 11. Buenzli, P.R., Martin, P.A.: The Casimir effect. Acta Phys. Pol. B 37(9), 2503–2559 (2006) 12. Buhmann, S.Y.: Dispersion Forces I: Macroscopic Quantum Electrodynamics and GroundState Casimir, Casimir–Polder and van der Waals Forces. Springer, Berlin (2012) 13. Cacciapuoti, C., Fermi, D., Posilicano, A.: Relative-zeta and Casimir energy for a semitransparent hyperplane selecting transverse modes. In: Michelangeli, A., Dell’Antonio, G. (eds.) Advances in Quantum Mechanics: Contemporary Trends and Open Problems, pp. 71– 97. Springer International Publishing, New York City (2017) 14. Casimir, H.B.G.: On the attraction between two perfectly conducting plates. Indag. Math. 10, 261–263 (1948) 15. Casimir, H.B.G., Polder, D.: The influence of retardation on the London-van der Waals forces. Phys. Rev. 73, 360–372 (1948) 16. Cornu, F., Martin, Ph.A.: Atom–wall dispersive forces: a microscopic approach. J. Phys. A Math. Theor. 42(49), 495001 (2009) 17. Correggi, M., Falconi, M., Olivieri, M.: Magnetic Schrödinger operators as the quasi-classical limit of Pauli–Fierz-type models. J. Spectr. Theory 9(4), 1287–1325 (2019)

The Casimir-Polder Effect for an Approximate Pauli-Fierz Model: The Atom. . .

165

18. Correggi, M., Falconi, M., Olivieri, M.: Ground State Properties in the Quasi-Classical Regime (2020). ArXiv:2007.09442 19. Correggi, M., Falconi, M., Olivieri, M.: Quasi-classical dynamics. J. Eur. Math. Soc. Published online first (2022) 20. Dalvit, D., Milonni, P., Roberts, D., da Rosa, F.: Casimir Physics. Springer, Berlin (2011) 21. Dzyaloshinskii, I.E., Lifshitz, E.M., Pitaevskii, L.P.: The general theory of van der Waals forces. Adv. Phys. 10(38), 165–209 (1961) 22. Fermi, D.: Vacuum polarization with zero-range potentials on a hyperplane. Universe 7(4), 92 (2021) 23. Fermi, D., Pizzocchero, L.: Local zeta regularization and the Casimir effect. Prog. Theor. Phys. 126(3), 419–434 (2011) 24. Fermi, D., Pizzocchero, L.: Local Casimir effect for a scalar field in presence of a point impurity. Symmetry 10(2), 38 (2018) 25. Hiroshima, F.: Self-adjointness of the Pauli-Fierz Hamiltonian for arbitrary values of coupling constants. Ann. Henri Poincaré 3(1), 171–201 (2002) 26. Koppen, M.J.: Van der Waals forces in the context of non-relativistic quantum electrodynamics PhD dissertation, Technischen Universität München, (2011). http://mediatum.ub.tum.de/node? id=1071141 27. Lennard-Jones, J.E.: Processes of adsorption and diffusion on solid surfaces. Trans. Faraday Soc. 28, 333–359 (1932) 28. Lieb, E.H., Thirring, W.E.: Universal nature of van der Waals forces for Coulomb systems. Phys. Rev. A 34, 40–46 (1986) 29. London, F.: Zur Theorie und Systematik der Molekularkräfte. Z. Phys. 63(3), 245–279 (1930) 30. Matte, O.: Pauli-Fierz type operators with singular electromagnetic potentials on general domains. Math. Phys. Anal. Geom. 20 (2017) 31. Milton, K.A.: Resource letter VWCPF-1: van der Waals and Casimir–Polder forces. Am. J. Phys. 79(7), 697–711 (2011) 32. Miyao, T.: Note on the retarded van der Waals potential within the dipole approximation. Symmetry Integrability Geom. Methods Appl. 16, 36–70 (2020) 33. Miyao, T., Spohn, H.: The retarded van der Waals potential: Revisited. J. Math. Phys. 50(7), 072103 (2009) 34. Miyao, T., Spohn, H.: Scale dependence of the retarded van der Waals potential. J. Math. Phys. 53(9), 095215 (2012) 35. Pauling, L., Wilson, E.B.: Introduction to Quantum Mechanics: With Applications to Chemistry. Dover Publications, New York (1985) 36. Salam, A.: Non-relativistic QED Theory of the van der Waals Dispersion Interaction. Springer, Cham (2016) 37. Sernelius, B.E.: Fundamentals of van der Waals and Casimir Interactions. Springer, Cham (2018) 38. Spohn, H.: Dynamics of Charged Particles and their Radiation Field. Cambridge University Press, Cambridge (2004) 39. Yu, C.-J., Ri, G.-C., Jong, U.-G., Choe, Y.-G., Cha, S.-J.: Refined phase coexistence line between graphite and diamond from density-functional theory and van der Waals correction. Phys. B: Condens. Matter 434, 185–193 (2014)

Part II

Open Quantum Systems

Asymptotic Dynamics of Open Quantum Systems and Modular Theory Daniele Amato, Paolo Facchi, and Arturo Konderak

1 Motivation The study of the dynamics of open quantum systems has received a strong boost by the recent developments in quantum technologies [57]. In particular, understanding and controlling the decoherence effects, arising from nontrivial interactions of the system under interest with its surroundings, are needed for achieving optimal performance in quantum computers [62]. Moreover, the asymptotic evolution of the system plays a crucial role in reservoir engineering [48, 56], e.g. in the preparation of a target quantum state by relaxation of a system properly coupled to the bath. In this context, as well as for information protection and processing tasks [60, 61], a one-dimensional attractor subspace, namely a unique steady state towards which the dynamics converges, is of little use for practical purposes. This has led many studies to focus on the asymptotic discrete-time evolution of open systems in a general setting, both for finitedimensional [2, 7, 18, 45, 46, 58, 59] and infinite-dimensional systems [11, 12, 30]. Also, the non-unitary continuous-time dynamics at large times was investigated in the literature of the last twenty years, see e.g. [1, 3, 6, 8, 26], with a particular attention to quantum dynamical semigroups [51], completely characterized in the two seminal papers [31, 40] from 1976. However, the strong interest in the study of the stationary states of quantum dynamical semigroups goes back to the seventies [27, 28, 51, 52], mainly motivated by the problem of irreversibility in quantum statistical mechanics [24, 37].

D. Amato · P. Facchi (O) · A. Konderak Dipartimento di Fisica, Università di Bari, Bari, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Bari, Italy e-mail: [email protected]; [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Correggi, M. Falconi (eds.), Quantum Mathematics II, Springer INdAM Series 58, https://doi.org/10.1007/978-981-99-5884-9_5

169

170

D. Amato et al.

This Article is organized as follows. After recalling some basic concepts about quantum channels (Sect. 2.1) and Tomita-Takesaki modular theory (Sect. 2.2), we will discuss a structure theorem (Theorem 1) by Perez-Garcia and Wolf [58, 59] for the attractor subspace and the action of the channel on this subspace. Then, in Sect. 3, we will present sufficient conditions (Theorem 2) and a characterization (Theorem 3) under which the asymptotic dynamics is unitary. Finally, in Sect. 4, we will study the connection between the structure theorem and modular theory, recently already explored in the continuous-time setting by Longo [41].

2 Preliminaries 2.1 Quantum Channels In this Section we set up the notation and recall some known concepts about the dynamics of finite-dimensional open quantum systems. The state of an open quantum system is given by a density operator .ρ, i.e. a positive semidefinite operator of unit trace on .H, the system Hilbert space with dimension d. In the discrete-time point of view, usually adopted in quantum information theory [44], the evolution in the Schrödinger picture of an open quantum system in the unit time is described by a quantum channel .o, namely a completely positive trace-preserving map on .B(H) [35]. We will denote with .B(H, K) the space of bounded operators from the Hilbert space .H to the Hilbert space .K and, in particular, .B(H) = B(H, H). The adjoint map .o† of the channel .o is defined with respect to the HilbertSchmidt scalar product .HS as .

HS = HS ,

A, B ∈ B(H).

(1)

o† is a completely positive unital map on .B(H) and describes the system dynamics in the Heisenberg picture, i.e. the evolution of the system observables. Coming back to the Schrödinger picture, if the system is prepared in the initial state .ρ(0), the evolved state .ρ(n) at time .t = n ∈ N will be given by .ρ(n) = on (ρ(0)). Consequently the system dynamics is described by a sequence of states .ρ(0), ρ(1), . . . , ρ(n), called a quantum Markov chain [45]. The spectrum .spect(o) of a quantum channel .o has the following three major features [58] .

• .1 ∈ spect(o), • .λ ∈ spect(o) ⇒ λ¯ ∈ spect(o), • .spect(o) ⊆ {λ ∈ C | |λ| ≤ 1}.

Asymptotic Dynamics of Open Quantum Systems and Modular Theory

171

The asymptotic dynamics is obtained in the large n-limit and it ends inside the asymptotic, peripheral or attractor subspace of .o, defined as .

O

Attr(o) =

Ker(o − λ1),

(2)

λ∈spectP (o)

i.e. the direct sum of the eigenspaces belonging to the peripheral eigenvalues λ ∈ spectP (o) = {λ ∈ spect(o) | |λ| = 1}.

.

(3)

In particular, if .λ is a primitive M-th root of unity, then the corresponding eigenoperator satisfies oM (Y ) = Y,

.

(4)

and describes a limit cycle of length M. Clearly, the fixed points of .o, i.e. o(Y ) = Y

.

(5)

are (trivial) examples of limit cycles. On the other hand, the orbit of an eigenoperator Y associated with an eigenvalue .λ which is not a root of unity does not close and is almost periodic. Importantly, according to Proposition 6.9 of [58], if .P denotes the eigenprojection onto the fixed point space .Fix(o) of .o, then X ∈ Fix(o) ⇒ supp(X), Ran(X) ⊆ supp(P(I)) ≡ H0 ,

.

(6)

where .supp(A), .Ran(A) stand for the support and the range of the operator A. Obviously, .P(I) is a maximum-rank fixed point of .o. ˜ : B(H0 ) |→ B(H0 ) as Therefore, let us define .o ˜ o(X) = V † o(V XV † )V .

.

(7)

Here .V : H0 |→ H is an isometry satisfying .V † V = IH0 and .V V † = Q, the ˜ can be shown to be a faithful quantum channel (Lemma 6.4 projection onto .H0 . .o of [58]), namely having a full rank fixed point. We have .

˜ Fix(o) = 0 ⊕ Fix(o),

(8)

with the zero block acting on the orthogonal complement .H⊥ 0 of .H0 and, analogously [2], .

˜ Attr(o) = 0 ⊕ Attr(o).

(9)

172

D. Amato et al.

2.2 Tomita-Takesaki Modular Theory The modular theory by Tomita [55] and Takesaki [53, 54] was originally introduced to generalize integration and measure theory to non-Abelian algebras, but it turned out to be a fundamental and general tool in the algebraic description of quantum systems. For instance, it is used for .σ -finite von Neumann algebras where it plays an important role in the classification of factors [19, 20, 47]. Also, it is crucial in quantum statistical mechanics, as it associates with a faithful normal state a (modular) group of automorphisms over the algebra of observables. This establishes a connection with the Kubo-Martin-Schwinger (KMS) condition on equilibrium states [32, 39, 43]. In recent years, left Hilbert algebras and modular theory were employed to obtain the algebra generated by a groupoid in the Schwinger’s picture of quantum mechanics [15–17]. Incidentally, modular theory was also generalized to Jordan algebras, see [33]. We will give here a quick review of modular theory for finite-dimensional algebras. Let .M be a finite-dimensional .∗-algebra over a Hilbert space .H. By a well-known structure theorem [23, p. 74], the Hilbert space .H can be decomposed as M O

H = H⊥ 0 ⊕

.

Hk,1 ⊗ Hk,2 ,

(10)

Mdk ⊗ Ik,2 .

(11)

k=1

so that the algebra .M decomposes as M=0⊕

M O

.

k=1

Here, .H⊥ 0 represents the degeneracy of the algebra, .Mdk is the algebra of matrices over .Hk,1 , with .dk = dim Hk,1 , .Ik,2 is the identity over the Hilbert space .Hk,2 and represents the multiplicity of the algebra .Mdk [25]. By Riesz lemma, every state .σ over the algebra .M [10] can be represented as a density matrix belonging to the algebra: σ = EM

(

1

.

k=1 tr(σk )

0⊕

M O k=1

Ik,2 σk ⊗ mk

) ∈ M,

(12)

with .mk = dim Hk,2 , so that .σ (A) = HS = tr(σ A). For faithful states .σk > 0, and an inner product can be defined on .M as .

σ = tr(σ A† B),

(13)

which makes .(M, < | >σ ) a Hilbert space. In particular, on this Hilbert space it is possible to obtain a cyclic representation of the algebra .M [10, 29, 50].

Asymptotic Dynamics of Open Quantum Systems and Modular Theory

173

Starting from a faithful state .σ , Tomita-Takesaki modular theory allows to obtain a group of unitary automorphisms on the algebra A |→ A−it AAit ,

.

t ∈ R,

A ∈ M.

(14)

The operator .A : M → M is self-adjoint with respect to the inner product .< | >σ , and it is obtained via the polar decomposition of the map .S : A |→ A† . In the non-degenerate case (.H⊥ 0 = 0), the modular group takes the simple form A |→ σ −it Aσ it ,

.

(15)

with .σ > 0 (this can be easily generalized to the degenerate case by taking the maximum-rank state .σ = 0 ⊕ σ˜ , with .σ˜ > 0). Note that the unitary operator .σ it is in the algebra .M. Conversely, starting from a one-parameter group of unitary automorphisms on the algebra .M, one can obtain the equilibrium thermal state via a KMS condition [39, 43]. In particular, the equilibrium thermal state corresponding to the dynamics (15) depends on a parameter .β ∈ R and it is in the form σβ =

.

1 −βH 1 = e σβ. Z tr(σ β )

(16)

Here, .H = − log σ is the generator of the unitary evolution .σ it . Obviously, the faithful state .σ in (12) corresponds to the choice .β = 1.

2.3 Structure Theorem The finer structure for .Attr(o) and the action of a quantum channel .o on such subspace, i.e. of the peripheral map .oP = o|Attr(o) is given in the following theorem by Perez-Garcia and Wolf (Theorem 8 of [59]). Theorem 1 Let .o be a quantum channel and .P the eigenprojection onto its fixed point space .Fix(o). 1. There exists a decomposition Cd = H⊥ 0 ⊕

M O

.

k=1

Hk,1 ⊗ Hk,2 ,

(17)

174

D. Amato et al.

with .H0 = supp(P(I)) and some Hilbert spaces .Hk,i with .k = 1, . . . , M and i = 1, 2, and positive definite density matrices .ρk on .Hk,2 such that

.

.

Attr(o) = 0 ⊕

M O

Mdk ⊗ ρk ,

(18)

k=1

with .dk = dim(Hk,1 ), namely .X ∈ Attr(o) may be represented as X =0⊕

M O

.

xk ⊗ ρk ,

(19)

k=1

for some matrices .xk ∈ Mdk ; 2. There exist unitary matrices .Uk on .Hk,1 and a permutation .π acting on .{1, . . . , M} such that o(X) = 0 ⊕

M O

.

Uk xπ(k) Uk† ⊗ ρk , X ∈ Attr(o).

(20)

k=1

Remark 4 Each cycle of the permutation .π must act on matrices .xk of the same order, consistently with the fact that .Attr(o) is an invariant subspace for .o (Proposition 6.12 of [58]), i.e. o(Attr(o)) = Attr(o).

.

(21)

Remark 5 In the faithful case we have that .dim(H⊥ 0 ) = 0 and the zero factor in (18) disappears. Furthermore, it turns out that .Attr(o† ) is a unital .∗-algebra (Theorem 1 of [11]) with the following structure [23]

.

Attr(o ) = †

M O

Mdk ⊗ Ik,2 ,

(22)

k=1

i.e. (11) without the zero first block. In the language of [9], .Attr(o) is a distorted algebra, in the sense that it is an algebra under the star product A * B := APP (I)−1 B ∈ Attr(o),

.

A, B ∈ Attr(o),

(23)

where .PP denotes the spectral projection onto .Attr(o). See also [4, 14] for the contracted algebra of a dissipative quantum system.

Asymptotic Dynamics of Open Quantum Systems and Modular Theory

175

Remark 6 More generally, Theorem 1 is valid for Schwarz maps [58, 59], namely positive trace-preserving maps obeying to the operator inequality o† (A† )o† (A) < o† (A† A),

.

A ∈ B(H).

(24)

3 Unitary asymptotic dynamics One of the most intriguing features of Theorem 1 is the occurrence of permutations in the asymptotic action (20) of .o, which makes the peripheral map .oP , i.e. the asymptotic dynamics, not generally unitary. Thus one may ask about the conditions on .o under which the map .oP is unitary. Two sufficient conditions are provided in the following Theorem [5]. Theorem 2 Given a quantum channel .o with corresponding peripheral map .oP , then 1. If .o = eL , with .L being a GKLS generator [31, 40] (Markovian channel), then oP (X) = U XU † , with U unitary, 2. If .o2 = o (idempotent channel), then .oP (X) = U XU † , with U unitary. .

The proof, which may be found in [5], is based on a characterization of the absence of permutations in the structure of the asymptotic map. Remark 7 The first sufficient condition is in line with the asymptotics of quantum dynamical semigroups [3]. Remark 8 Note that both conditions are not necessary, since a Markovian channel is invertible (as a linear map), while an idempotent channel, different from the identity, is not. Trivially, the peripheral channel .oP is unitary when there are no permutations in the action (20). Indeed, as we will show in Sect. 4, in such a case the dynamics is obtained from the modular group corresponding to a faithful fixed state. Now, we will provide a characterization for the unitarity of the asymptotic dynamics .oP . Theorem 3 Let .o be a quantum channel with attractor subspace .Attr(o) of the form (18) and peripheral map .oP . Then oP (X) = U XU † ,

.

with X ∈ Attr(o),

(25)

176

D. Amato et al.

for some unitary operator U on .H, iff .ρk ∼ ρπ(k) , namely ρk = Vk ρπ(k) Vk† ,

.

(26)

for some unitaries .Vk : Hπ(k),2 → Hk,2 . Proof Besides being necessary, one can easily show that this condition is also sufficient. Indeed, if .o is faithful, then we can define the operator .U : H0 → H0 as U:

M O

.

φk ⊗ ψk |→

k=1

M O

Uk φπ(k) ⊗ Vk ψπ(k) ,

(27)

k=1

and its extension to .H0 follows by linearity. It is easy to see that this is indeed unitary on .H0 , and it can be extended on the whole .H as .I0 ⊕ U . Moreover, it satisfies the condition stated in Theorem 3. u n Observe that condition (26) holds for unital quantum channels, i.e. .o(I) = I, Indeed, a unital quantum channel satisfies conditions .ρk = Ik,2 /mk and .mk = mπ(k) for all .k = 1, . . . , M. Moreover, unitality also ensures Hilbert-Schmidt unitarity, that is ||o(X)||HS = ||X||HS ,

.

X ∈ Attr(o).

(28)

As a final remark, it is possible to show [5] that every map defined on a set in the form (18), and acting as (20) can be extended to a quantum channel over .B(H). This can be interpreted as the inverse of structure theorem 1.

4 Asymptotic Dynamics and Modular Theory In this Section, we are going to study the relation between the asymptotic dynamics of a quantum channel and Tomita-Takesaki modular theory. In particular, we want to find the conditions under which the asymptotic dynamics .oP can be obtained from the modular evolution group (15) associated with an equilibrium state. In doing this, we will use the structure theorem 1, and we will write the modular group corresponding to fixed states. Let us assume that the quantum channel .o is faithful. We could drop this ˜ defined in Eq. (7). requirement simply by considering the reduced channel .o We start by considering a particular situation, in which the permutation in Eq. (20) has only one cycle, i.e., up to a relabeling, it is in the form π(j ) = j + 1

.

mod M.

(29)

Asymptotic Dynamics of Open Quantum Systems and Modular Theory

177

A state .σ in the attractor space .Attr(o) in (18) will be a fixed point whenever its components .σj satisfy the equation σj = Uj σj +1 Uj†

.

1 < j < M.

mod M,

(30)

By setting .V1 = U1 U2 . . . UM , Eq. (30) reads σ1 = V1 σ1 V1† ,

.

σj = Uj†−1 · · · U1† σ1 U1 · · · Uj −1 ,

2 < j < M,

(31)

so that the fixed points have .σ1 in the commutant {V1 }' = {A ∈ Md1 | [A, V1 ] = 0},

(32)

.

of .V1 , where .[·, ·] stands for the commutator. Writing the spectral decomposition of V1

.

V1 =

n E

.

eiθj Pj(1) ,

θj ∈ R,

(33)

j =1

E we have .σ1 = nj=1 Pj(1) σ1 Pj(1) . We make another requirement, namely .σ1 being in the bicommutant .{V1 }'' ⊂ {V1 }' . In this way, it will be a function of .V1 in the form σ1 =

n E

.

λj Pj(1) .

(34)

j =1

The other components .σk will have a similar structure † σk = Uk−1 · · · U1† σ1 U1 · · · Uk−1 =

n E

.

(k)

λj Pj ,

(35)

j =1 (k)

(1)

† with .Pj = Uk−1 · · · U1† Pj U1 · · · Uk−1 . Note that we can also apply Tomita-Takesaki modular theory to the space .Attr(o), being it a von Neumann algebra with respect to a modified product (see Remark 5). In the Schrödinger picture, the unitary evolution (15) generated by .σ via the modular theory is

X |→ σ it Xσ −it ,

.

X ∈ Attr(o)

(36)

178

D. Amato et al.

with σ it =

M O

.

⎛ ⎞ M n O E (k) ⎝ σkit ⊗ ρkit = eit log λj Pj ⎠ ⊗ ρkit .

k=1

k=1

(37)

j =1

We see that the permutation disappears, and the evolution turns out to be unitary. By setting .λj = eθj , we see that the evolution for .t = 1 is just σ i Xσ −i = oV (X),

.

X ∈ Attr(o)

(38)

where .oV is the map oV (X) =

M O

.

Vk Xk Vk† ⊗ ρk ,

X ∈ Attr(o)

(39)

k=1

with † Vk = Uk Uk+1 · · · UM U1 · · · Uk−1 = Uk−1 · · · U1† V1 U1 · · · Uk−1 .

.

(40)

Note that the map .oV is just .oM P , the peripheral evolution repeated over a complete cycle. The generalization to a permutation with more cycles can be obtained in a similar way. Indeed, the corresponding Tomita-Takesaki evolution turns out to be the M-th power of the asymptotic map .oP , with M being the least common multiple of the lengths of the cycles of the permutation. In particular, this corresponds to the asymptotic evolution if and only if there are no permutations.

5 Conclusions and Outlooks In this Article, we gave further insights into the asymptotics of finite-dimensional open quantum systems, whose study has been strongly motivated by quantum information protection and processing [9, 60, 61]. Theorem 1, giving the structure of the attractor subspace and the action of the channel on it, reveals two important facts. First, up to a zero block in the non-faithful case, the attractor subspace is a .∗algebra with respect to the modified product (5). Second, in the structure of the asymptotic map, partial permutations between the factors in the decomposition (18) may occur according to (20), making the dynamics at large times no longer unitary. The occurrence of permutations also explains how the asymptotic evolution cannot always be regarded as the modular dynamics on the attractor subspace associated with a fixed state, as shown in Sect. 4. More precisely, Tomita-Takesaki

Asymptotic Dynamics of Open Quantum Systems and Modular Theory

179

evolution is the (unitary) peripheral map of the M-times iterated dynamics, with M denoting the least common multiple of the lengths of the cycles of the permutation. This unveils a connection between permutations and the divisibility of the quantum channel which generates the dynamics under consideration, deserving further studies in the future [5]. Also, as already illustrated in the faithful case by Novotný, Maryška, and Jex [46], the asymptotic evolution may not be a Hilbert-Schmidt unitary but it is still unitary with respect to a modified scalar product. In other words, the peripheral map is not generally unitary because of permutations, but it can be shown to be unitary in a weaker sense. Another open problem could be the connection between eventually and asymptotically entanglement breaking maps [38, 49] and the (non-)unitarity of the asymptotic map [34]. Entanglement breaking maps [36] have turned out to be a central topic in the theory of quantum information, specifically in entanglement transferring [21, 22]. In particular, many efforts have been devoted to the proof of the PPT square conjecture [13], recently achieved in the finite-dimensional case [42]. Acknowledgments This work was partially supported by Istituto Nazionale di Fisica Nucleare (INFN) through the project “QUANTUM”, by Regione Puglia and QuantERA ERA-NET Cofund in Quantum Technologies (Grant No. 731473), project PACE-IN, and by the Italian National Group of Mathematical Physics (GNFM-INdAM).

References 1. Agredo, J., Fagnola, F., Rebolledo, R.: Decoherence free subspaces of a quantum Markov semigroup. J. Math. Phys. 55, 112201 (2014) 2. Albert, V.V.: Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states. Quantum 3, 151 (2019) 3. Albert, V.V., Jiang, L.: Symmetries and conserved quantities in Lindblad master equations. Phys. Rev. A 89, 022118 (2014) 4. Alipour, S., Chruscinski, D., Facchi, P., Marmo, G., Pascazio, S., Rezakhani, A.T.: Dynamical algebra of observables in dissipative quantum systems. J. Phys. A: Math. Theor. 49, 065301 (2017) 5. Amato, D., Facchi, P., Konderak, A.: Asymptotics of quantum channels. J. Phys. A: Math. Theor. 56, 265304 (2023) 6. Baumgartner, B., Narnhofer, H.: Analysis of quantum semigroups with GKS–Lindblad generators: II. General. J. Phys. A: Math. Theor. 41, 395303 (2008) 7. Baumgartner, B., Narnhofer, H.: The structures of state space concerning quantum dynamical semigroups. Rev. Math. Phys. 24, 1250001 (2012) 8. Baumgartner, B., Narnhofer, H., Thirring, W.: Analysis of quantum semigroups with GKS– Lindblad generators: I. Simple generators. J. Phys. A: Math. Theor. 41, 065201 (2008) 9. Blume-Kohout, R., Ng, H.K., Poulin, D., Viola, L.: Information-preserving structures: a general framework for quantum zero-error information. Phys. Rev. A 82, 062306 (2010) 10. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics: Volume 1: C*- and W*-Algebras. Symmetry Groups. Decomposition of States. Springer, New York (2002)

180

D. Amato et al.

11. Carbone, R., Jenˇcová, A.: On period, cycles and fixed points of a quantum channel. Ann. Henri Poincaré 21, 155–188 (2020) 12. Carbone, R., Pautrat, Y.: Irreducible decompositions and stationary states of quantum channels. Rep. Math. Phys. 77, 293–313 (2016) 13. Christandl, M.: PPT square conjecture. In: Banff International Research Station Workshop: Operator Structures in Quantum Information Theory (2012) 14. Chruscinski, D., Facchi, P., Marmo, G., Pascazio, S.: The observables of a dissipative quantum system. Open Syst. Inf. Dynam. 19, 1250002 (2012) 15. Ciaglia, F.M., Ibort, A., Marmo, G.: Schwinger’s picture of quantum mechanics I: groupoids. Int. J. Geom. Methods Mod. Phys. 16, 1950119 (2019) 16. Ciaglia, F.M., Ibort, A., Marmo, G.: Schwinger’s picture of quantum mechanics II: algebras and observables. Int. J. Geom. Methods Mod. Phys. 16, 1950136 (2019) 17. Ciaglia, F.M., Di Cosmo, F., Ibort, A., Marmo, G., Schiavone, L., Zampini, A.: Feynman’s propagator in Schwinger’s picture of quantum mechanics. Mod. Phys. Lett. A 36, 2150187 (2021) 18. Cirillo, G.I., Ticozzi, F.: Decompositions of Hilbert spaces, stability analysis and convergence probabilities for discrete-time quantum dynamical semigroups. J. Phys. A: Math. Theor. 48, 085302 (2015) 19. Connes, A.: Une classification des facteurs de type III. Ann. Sci. de l’Ecole Norm. Superieure 6, 133–252 (1973) 20. Connes, A.: Noncommutative Geometry. Elsevier, New York (1995) 21. Cuevas, Á., De Pasquale, A., Mari, A., Orieux, A., Duranti, S., Massaro, M., Di Carli, A., Roccia, E., Ferraz, J., Sciarrino, F., Mataloni, P., Giovannetti, V.: Amending entanglementbreaking channels via intermediate unitary operations. Phys. Rev. A 96, 022322 (2017) 22. Cuevas, Á., Mari, A., De Pasquale, A., Orieux, A., Massaro, M., Sciarrino, F., Mataloni, P., Giovannetti, V.: Cut-and-paste restoration of entanglement transmission. Phys. Rev. A 96, 012314 (2017) 23. Davidson, K.R.: C*-Algebras by Example. American Mathematical Society, Rhode Island (1996) 24. Evans, D.E., Lewis, J.T.: Dilations of Irreversible Evolutions in Algebraic Quantum Theory. Dublin Institute for Advanced Studies, Dublin (1977) 25. Facchi, P., Gramegna, G., Konderak, A.: Entropy of quantum states. Entropy 23, 645 (2021) 26. Fagnola, F., Rebolledo, R.: On the existence of stationary states for quantum dynamical semigroups. J. Math. Phys. 42, 1296–1308 (2001) 27. Frigerio, A.: Stationary states of quantum dynamical semigroups. Commun. Math. Phys. 63, 269–276 (1978) 28. Frigerio, A., Verri, M.: Long-time asymptotic properties of dynamical semigroups on W*algebras. Math Z. 180, 275–286 (1982) 29. Gelfand, I., Neumark, M: On the imbedding of normed rings into the ring of operators in Hilbert space. Mat. Sb. 12, 197–217 (1943) 30. Girotti, F.: Absorption in invariant domains for quantum Markov evolutions. Ph.D. Thesis, Università degli Studi di Pavia (2022) 31. Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17, 821–825 (1976) 32. Haag, R., Hugenholtz, N.M., Winnink, M.: On the equilibrium states in quantum statistical mechanics. Commun. Math. Phys. 5, 215–236 (1967) 33. Haagerup, U., Hanche-Olsen, H.: Tomita-Takesaki theory for Jordan algebras. J. Oper. Theory 11, 343–364 (1984) 34. Hanson, E.P., Rouzé, C., Stilck França, D.: Eventually Entanglement Breaking Markovian Dynamics: Structure and Characteristic Times. Ann. Henri Poincaré 21, 1517–1571 (2020) 35. Heinosaari, T., Ziman, M.: The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement. Cambridge University Press, Cambridge (2011) 36. Horodecki, M., Shor, P.W., Ruskai, M.B.: Entanglement breaking channels. Rev. Math. Phys. 15, 629–641 (2003)

Asymptotic Dynamics of Open Quantum Systems and Modular Theory

181

37. Ingarden, R.S., Kossakowski, A.: On the connection of nonequilibrium information thermodynamics with non-Hamiltonian quantum mechanics of open systems. Ann. Phys. 89, 451–485 (1975) 38. Kennedy, M., Manor, M., Paulsen, V.I.: Composition of PPT maps. Quant. Inf. Comput. 18, 0472–0480 (2018) 39. Kubo, R.: Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Japan 12, 570–586 (1957) 40. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976) 41. Longo, R.: The emergence of time. Expositiones Math. 38, 240–258 (2020) 42. Majewski, W.A.: On PPT square conjecture (2021). arXiv:2108.01588 [math-ph] 43. Martin, P.C., Schwinger, J.: Theory of many-particle systems. I. Phys. Rev. 115, 1342–1373 (1959) 44. Nielsen, M.A., Chuang, I.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2002) 45. Novotný, J., Alber, G., Jex, I.: Asymptotic properties of quantum Markov chains. J. Phys. A: Math. Theor. 45, 485301 (2012) 46. Novotný, J., Maryška, J., Jex, I.: Quantum Markov processes: from attractor structure to explicit forms of asymptotic states. Eur. Phys. J. Plus 133, 310–326 (2018) 47. Powers, R.T.: Representations of uniformly hyperfinite algebras and their associated von Neumannn rings. Bull. Am. Math. Soc. 73, 572–575 (1967) 48. Poyatos, J.F., Cirac, J.I., Zoller, P.: Quantum Reservoir Engineering with Laser Cooled Trapped Ions. Phys. Rev. Lett. 77, 4728–4731 (1996) 49. Rahaman, M., Jaques, S., Paulsen, V.I.: Eventually entanglement breaking maps. J. Math. Phys. 59, 062201 (2018) 50. Segal, I.E.: Irreducible representations of operator algebras. Bull. Am. Math. Soc. 53, 73–88 (1947) 51. Spohn, H.: An algebraic condition for the approach to equilibrium of an open N-level system. Lett. Math. Phys. 2, 33–38 (1977) 52. Spohn, H.: Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 52, 569–615 (1980) 53. Takesaki, M.: Tomita’s Theory of Modular Hilbert Algebras and its Applications. SpringerVerlag, Berlin (1970) 54. Takesaki, M.: Theory of Operator Algebras II. Springer, Berlin (2003) 55. Tomita, M.: On canonical forms of von Neumann algebras (Japanese). In: Fifth Functional Analysis Symposium, pp. 101–102, Tôhoku University, Sendai (1967) 56. Verstraete, F., Wolf, M.M., Cirac, J.I.: Quantum computation and quantum-state engineering driven by dissipation. Nat. Phys. 5, 633–636 (2009) 57. Wei, S.-J., Xin, T., Long, G.-L.: Efficient universal quantum channel simulation in IBM’s cloud quantum computer. Sci. China Phys. Mech. Astron. 61, 070311 (2018) 58. Wolf, M.M.: Quantum Channels & Operations: Guided Tour. Lecture Notes (2012). https:// www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichealWolf/QChannelLecture.pdf 59. Wolf, M.M., Perez-Garcia, D.: The inverse eigenvalue problem for quantum channels (2010). arXiv:1005.4545 [quant-ph] 60. Zanardi, P., Rasetti, M.: Noiseless quantum codes. Phys. Rev. Lett. 79, 3306–3309 (1997) 61. Zanardi, P., Venuti, L.C.: Coherent quantum dynamics in steady-state manifolds of strongly dissipative systems. Phys. Rev. Lett. 113, 240406 (2014) 62. Zurek, W.H.: Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715–775 (2003)

Boson Quadratic GKLS Generators Franco Fagnola

1 Introduction Quantum Markov semigroups (QMS), or dynamical semigroups in the Physics terminology, describe the evolution of open quantum systems subject to noise because of the interaction with the surrounding environment under certain markovian approximations. In the case of quasi-free bosons open quantum systems, a class QMS that can be handled in detail are the so-called Gaussian QMS. The structure of their generators was investigated in the seventies (see [13, 31]), shortly after the celebrated GKLS theorem. However, not much progress has been made since then because of various questions related to the unboundedness of bosonic creation and annihilation operators and, consequently, unboundedness of their generator. Tools for the construction and analysis of QMS with unbounded generators [9, 12, 17] have been developed in the meantime. Generators of Gaussian QMS generalize bosonic quadratic Hamiltonians [14, 19, 22, 23, 30] because their formal GKLS representation has a quadratic Hamiltonian in bosonic annihilation and creation operators and Kraus operators (also called noise operators) that are first order polynomials in annihilation and creation operators. As the open system counterpart of boson quadratic Hamiltonians, they present a rich mathematical theory, yet to be explored in infinite dimension. Moreover, the special case of quantum Ornstein-Uhlenbeck processes has been studied in detail revealing several deep properties (see [10] and, more recently, [8]). In this note, based on joint works with J. Agredo and D. Poletti, we outline the construction of the semigroup and some properties of the Markovian dynamics

F. Fagnola (O) Dipartimento di Matematica, Politecnico di Milano, Milano, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Correggi, M. Falconi (eds.), Quantum Mathematics II, Springer INdAM Series 58, https://doi.org/10.1007/978-981-99-5884-9_6

183

184

F. Fagnola

such as irreducibility, ergodicity, existence and uniqueness of invariant states, and the structure of the decoherence-free subalgebra in which the reduced evolution is homomorphic. Our basic idea is to try to formulate the properties of the QMS on an infinite dimensional algebra of in terms matrices of coefficient matrices of operators in a GKLS representation of the generator.

2 Gaussian QMSs Throughout the paper .h denotes the Fock space .h = r(Cd ) with canonical orthonormal basis .(e(n1 , . . . , nd ))n1 ,...,nd ≥0 , isometrically isomorphic to .r(C) ⊗ · · · ⊗ r(C) via the unitary linear map .e(n1 , . . . , nd ) |→ en1 ⊗ . . . ⊗ end . We will use the multi-index notation .n = (n1 , . . . , nd ) in which .|n| = n1 + · · · + nd and denote by D the linear manifold of finite linear combinations of vectors of the canonical orthonormal basis. Let .aj , aj† be the creation and annihilation operator E E defined on the domain of vectors . n∈Nd ξn e(n) ∈ h (. n∈Nd |ξn |2 < +∞) such E that . n∈Nd |n| · |ξn |2 < +∞ by aj e(n1 , . . . , nd ) =

.

aj† e(n1 , . . . , nd ) =

√ /

nj e(n1 , . . . , nj −1 , nj − 1, . . . , nd ),

nj + 1 e(n1 , . . . , nj −1 , nj + 1, . . . , nd ).

Note that D is an essential domain for both .aj and .aj† . The Canonical Commutation Relations (CCR) are written as .[aj , ak† ]ξ = δj k ξ , for all .ξ ∈ D, where .[·, ·] denotes the commutator. For any .g ∈ Cd , define the exponential vector .eg associated with g by eg =

.

E g n1 · · · g nd d e(n1 , . . . , nd ) √1 n ! · · · n 1 d! d n∈N

Creation and annihilation operators with test vector .v ∈ Cd can also be defined on the total set of exponential vectors (see [26]) by a(v)eg = eg ,

.

a † (v)eg =

d eg+εu |ε=0 dε

for all .u ∈ Cd . The above unitary linear map .r(Cd ) |→ r(C) ⊗ · · · ⊗ r(C) eg |→

E

.

n1 ≥0,...,nd

g n1 . . . gdnd en1 ⊗ . . . ⊗ end √1 n1 ! . . . nd ! ≥0

Boson Quadratic GKLS Generators

185

allows one to establish the identities a(v) =

d E

.

v j aj ,

a † (u) =

j =1

d E

uj aj†

j =1

for all .uT = [u1 , . . . , ud ], v T = [v1 , . . . , vd ] ∈ Cd . Note that the linear manifold D turns out to be an essential domain for all the operators considered so far. This also happens for quadrature (or field) operators ( ) √ q(u) = a(u) + a † (u) / 2

.

u ∈ Cd

(1)

that are symmetric and essentially self-adjoint on the domain D by Nelson’s theorem on analytic vectors [29, Th. X.39 p. 202]. The linear span of exponential vectors also is an essential domain for operators .q(u) for the same reason. If the vector u has real (resp. purely imaginary) components one finds position (resp. momentum) operators and the commutation relation .[q(u), q(v)] ⊆ i Im1 (where .Im and .Re denote the imaginary and real part of any complex number). Another set of operators playing an important role are Weyl operators, defined on exponential vectors as follows W (z)eg = e−||z||

.

2 /2−

ez+g

z, g ∈ Cd .

By this definition . = for all .f, g ∈ Cd , therefore .W (z) extends uniquely to a unitary operator on .h. Weyl operators satisfy the CCR in the exponential form, namely, for every .z, z' ∈ Cd , W (z)W (z' ) = e−i Im W (z + z' ). '

.

(2)

√ Moreover .W√(z) is the exponential of the anti self-adjoint operator .−i 2 q(iz) i.e. −i 2 q(iz) = eza † −za . .W (z) = e A QMS .T = (Tt )t≥0 is a weakly.∗ -continuous semigroup of completely positive, identity preserving, weakly.∗ -continuous maps on .B(h). The predual semigroup .T∗ = (T∗t )t≥0 on the predual space of trace class operators on .h is a strongly continuous contraction semigroup. Gaussian QMSs can be introduced starting either by their explicit action on Weyl operators (see formula (7)) or by their candidate generator on a dense subspace. This will be called pre-generator, or form generator, and be represented in a GKLS form (see [13, 31] and also [27, Theorems 5.1, 5.2]) that will be called generalized since operators .Ll , H below are unbounded ) 1 E( ∗ Ll Ll x − 2L∗l xLl + x L∗l Ll . 2 m

£(x) = i [H, x] −

.

l=1

(3)

186

F. Fagnola

where .1 ≤ m ≤ 2d, and ) E ) d ( d ( E κj k † † κj k ζj † ζ¯j oj k aj† ak + aj ak + aj ak + aj + aj , . 2 2 2 2

H =

.

d ( ) E vlk ak + ulk ak† = a(vl• ) + a † (ul• ),

Ll =

(4)

j =1

j,k=1

(5)

k=1

o := (oj k )1≤j,k≤d = o∗ and .κ := (κj k )1≤j,k≤d = κ T ∈ Md (C), are .d × d complex matrices with .o Hermitian and .κ symmetric, .V = (vlk )1≤l≤m,1≤k≤d , U = (ulk )1≤l≤m,1≤k≤d ∈ Mm×d (C) are .m × d matrices and .ζ = (ζj )1≤j ≤d ∈ Cd . The notation .vl• and .ul• refers to the .l-th row of V and U , thought as a vector in .Cd . For an open quantum system of Bosons, the number m typically represents the number of external noises and .Cm is also called noise multiplicity space of .£. We suppose that at least one .Ll is non-zero, namely one among matrices .V , U is non-zero otherwise the pre-generator .£ reduces to .i times the commutator with the quadratic Hamiltonian H . Another application of Nelson’s theorem on analytic vectors [29, Th. X.39 p. 202] shows that H , as an operator with domain D, is essentially selfadjoint. Note that operators .Ll are closable, therefore we will identify them with their closure. It is worth noticing that the above generalized GKLS form is the most general with operators .Ll (resp. the self-adjoint H ) which are first (resp. second) order polynomials in .aj and .ak† . Indeed, in the case where .Ll are as above plus a multiple of the identity operator, exploiting non uniqueness of GKLS representations (see [26, section 30]) one can reduce himself to the previous case. The number of operators .Ll (also called Kraus or noise operators), the parameter m, is chosen to be minimum.

.

Definition 1 A GKLS representation of .£ is mimimal if the number m in (3) is minimum. is It is not difficult to see (Proposition 2.2 in [2]) that a GKLS representation ( ) T ∗ minimal if and only if the intersection of kernels .ker (V ) ∩ ker U , that are subspaces of the noise multiplicity space .Cm , is reduced to .{0}. This condition will be assumed throughout the paper. The pre-generator defined by means of operators .H, Ll determines a unique QMS that can be constructed by the minimal semigroup method (see [2, Appendix A]). For all .x ∈ B(h) consider the quadratic form with domain .D × D < > < > £(x)[ξ ' , ξ ] = i H ξ ' , xξ − i ξ ' , xH ξ

.

> < > < >) 1 E (< ' ξ , xL∗l Ll ξ − 2 Ll ξ ' xLl ξ + L∗l Ll ξ ' xξ 2 m

−

l=1

(6)

Boson Quadratic GKLS Generators

187

Theorem 1 There exists a unique < QMS, .T > = (Tt )t≥0 such that, for all .x ∈ B(h) and .ξ, ξ ' ∈ D, the function .t |→ ξ ' , Tt (x)ξ is differentiable and .

> d < ' ξ , Tt (x) ξ = £(Tt (x))[ξ ' , ξ ] dt

∀ t ≥ 0.

The domain of the generator consists of .x ∈ B(h) for which the quadratic form .£(x) is represented by a bounded operator. Since the operators .H, Ll are unbounded, the domain of the generator .L is not the whole of .B(h). Weyl operators typically do not belong to the domain of .L because a straightforward computation (see [2, Appendix A]) shows that the quadratic form .£(x) is unbounded. In spite of this we have the following explicit formula (see [13, 31]) Theorem 2 Let .(Tt )t≥0 be the quantum Markov semigroup with generalized GKLS generator associated with .H, Ll as above. For all Weyl operator .W (z) we have ( f t / f \ \ ) ( ) 1 t / sZ sZ sZ Re e z, Ce z ds + i Re ζ, e z ds W etZ z .Tt (W (z)) = exp − 2 0 0 (7) where the real linear operators .Z, C on .Cd are ) ) ( T U V − V TU U ∗U − V ∗V + iκ z. + io z + .Zz = 2 2 ( ) ) ( Cz = U ∗ U + V ∗ V z + U T V + V T U z (

(8) (9)

We refer to [2, subsection 4.5] for the proof. An alternative construction of a Gaussian QMS can be done starting from formula (7) and extending the definition of operators .Tt to the Weyl algebra (see [13, 31]). It can be shown (see [28, Theorems 5.1]) that a QMS .T is Gaussian if maps .T∗t of the predual semigroup .T∗ preserve Gaussian states or, still in an equivalent way, Weyl operators are eigenvectors for .Tt . More precisely, Theorem 3 For a QMS .T = (Tt )t≥0 on .B(h) the following are equivalent: 1. Maps .Tt act on Weyl operators as (7) for some .ζ ∈ C and some complex .d × d matrices .C, Z, 2. Maps .T∗t of the predual semigroup transform Gaussian densities into Gaussian densities, 3. The form generator can be represented in a generalized GKLS form by means of operators .H, Ll as in (4),(5).

188

F. Fagnola

3 Irreducibility and Long Time Behaviour In this section we discuss the asymptotic behaviour of a Gaussian QMS with a special attention to existence, uniqueness and convergence to invariant densities. Irreducibility plays a key role because it immediately implies faithfulness and uniqueness of invariant densities. Throughout the paper we consider only normal states on .B(h) identified with their density matrix, a positive operator with unit trace. A density matrix is called faithful if its kernel is reduced to the 0 vector. Definition 2 A density matrix .ρ is invariant if .T∗t (ρ) = ρ for all .t ≥ 0, namely tr (ρTt (x)) = tr (ρx) for all .t ≥ 0 and all x.

.

Keeping in mind the characteristic function of a Gaussian density matrix .ρ with mean .ω and covariance matrix S ) ( 1 .ρ(z) ˆ := tr(ρW (z)) = exp −i Re − Re , 2 a quick inspection at formula (7) shows that it can be invariant only if ( f t / f \ \ ) 1 t / sZ sZ sZ Re e z, Ce z ds + i Re ζ, e z ds tr . tr (ρTt (x)) = exp − 2 0 0 ) ( × ρW (etZ z) ( / \ 1f t / \ ) 1 tZ tZ sZ sZ = exp − Re e z, Se z − Re e z, Ce z ds 2 2 0 ( f t / \ \ ) / tZ sZ · exp −i Re ω, e z + i Re ζ, e z ds 0


< > is constant in time t. In particular, since .Re etZ z, SetZ z and .Re esZ z, CesZ z are non-negative, differentiating the argument of the first exponential, we get .

/ \ / \ Re esZ z, (Z ∗ S + SZ)esZ z + Re esZ z, CesZ z = 0

for all .s ≥ 0. As a result, a Gaussian density matrix is invariant if and only if the covariance matrix S solves the equation .Z ∗ S + SZ + C = 0. This equation is well-known in matrix analysis (see e.g. [4]). In particular, if Z has eigenvalues with stricly negative real parts (i.e. it is stable), a unique solution exists. Moreover, a similar computation on the purely imaginary part of the exponential shows that the mean .ω satisfies .Zω + ζ = 0. Proposition 1 If all eigenvalues of Z have strictly negative real parts then there exists a unique Gaussian invariant density matrix whose covariance matrix S and

Boson Quadratic GKLS Generators

189

mean .ω are the unique solution of Z ∗ S + SZ + C = 0,

.

Zω + ζ = 0.

We refer to [1, Sections 5 and 6], for a detailed discussion of the one-dimensional (.d = 1) case. In this case one can also show that the above condition is necessary for existence of Gaussian invariant densities. In particular, if Z has an eigenvalue with strictly positive real part, then there exists .z ∈ C such that .Tt (W (z)) weakly.∗ converges to 0 as t goes to .+∞. The picture becomes more complicated if .d > 1 because, roughly speaking, the behaviour in the various coordinates can be different. A notion of irreducibility is useful for complexity reduction and, more generally, plays an important role because: 1. An irreducible QMS cannot be decomposed into simpler QMSs, 2. If there exists a normal faithful invariant density matrix irreducibility implies uniqueness and other properties of the dynamics ([20, Lemma 1]), 3. It is a necessary condition for controllability, namely, roughly speaking, the possibility to drive any initial state of the system near a prescribed state. Definition 3 A QMS .T on .B(h) is irreducible if the only projections p such that Tt (p) ≥ p for all .t ≥ 0 are .p = 0 and .p = 1.

.

One can easily show ([17, Theorem II.1]) that condition .Tt (p) ≥ p for all .t ≥ 0, also called subharmonic or increasing projections, 1. Is equivalent to .Tt (p⊥ B(h)p⊥ ) ⊆ p⊥ B(h)p⊥ for all .t ≥ 0, 2. For all initial density matrix .ρ with support in p, i.e. such that .ρ = pρp, the evolved density matrix .T∗t (ρ) is also supported in p, i.e. .T∗t (ρ) = pT∗t (ρ)p. Subharmonic projections of a QMS with unbounded generator in a generalized GKLS, form are characterized in terms of operators .H, Ll by the following result (see [17, Theorem III.1]). More precisely, E in ∗our framework, one can show that the closure of the operator .−iH − (1/2) m l=1 Ll Ll defined on the domain D, denoted by G, generates a strongly continuous contraction semigroup .(Pt )t≥0 on .h and, denoting .Rng(p) the range of a projection p, Theorem 4 A p projection is .T-subharmonic if and only if, for all .t ≥ 0, .l = 1, . . . , m pPt = pPt p

.

and

Ll (Rng(p) ∩ Dom(G)) ⊆ Rng(p) .

Note that, by general results in semigroup theory .Rng(p) ∩ Dom(G) is dense in Rng(p) and the restriction of maps .Pt to .Rng(p) defines a strongly continuous contraction semigroup on this subspace. Intuitively, Theorem 4 says that subharmonic projections are determined by subspaces of .h which are invariant for the operators G and .Ll . In this way, we guess

.

190

F. Fagnola

that, if there are enough linearly independent operator .Ll , then candidate invariant subspace must be invariant for all operators .aj and .ak† and so .T is irreducible. In addition, disregarding for the moment domain problems, if p is a subharmonic projection, then .Rng(p) is also invariant for the commutator .[G, Ll ] and iterated commutators .[G, . . . , [G, Ll ] . . . ] which are first order polynomials in .aj and .ak† . Clearly, we can associate with each such polynomial for which .Rng(p) has to be invariant the vector in .C2d of coefficients as follows [ ] d ( (0) ) E v l• † v lj aj + ulj aj .Ll = (0) , ul• j =1 d ( ) E (1) (1) v lj aj + ulj aj† [G, Ll ] = j =1

[

(1)

]

v l• (1) ul•

] [ (k) (k) T with .k ≤ n Let .Vn be the subspace of .C2d generated by vectors . v l• , ul• determined by iterated commutators up to the order n. A simple dimensionality argument shows that there exists .n∗ ≤ 2d − m for which the dimension of .Vn is maximum for all .n ≥ n∗ . If the dimension of .Vn∗ = V2d−m is equal to 2d, then .Rng(p) is invariant for all creation and annihilation operators and the conclusion follows. Algebraic computations of iterated commutators can be made rigorous by considering .ξ ∈ Dom(Gn ) ∩ Rng(p), for some n, if we know that .Ll Pt ξ belongs to .Dom(G) since | | .[G, Ll ]ξ = [G, Ll ]Pt ξ |

t=0

[

d Ps Ll Pt−s ξ = ds

]

| | |

s=0+ t=0

In [16] we showed that this happens if the domain of the operator G, Ewhich is contained in the domain of the number operator N (the closure of . dj =1 aj† aj defined on D), actually coincides with the domain of N. Indeed, with this assumption, it turns out that .Dom(Gn ) = Dom(N n ) for all .n ≥ 1 [16, Lemma 5] and .Ll (Dom(Gn ) ∩ Rng(p)) ⊆ Dom(Gn−1 ) ∩ Rng(p). In this way we can prove the following Theorem 5 Suppose that .Dom(G) = Dom(N ). If dim.(V2d−m ) = 2d, then the gaussian QMS is irreducible. We refer to [16, Theorem 9] for the proof. Moreover, as a partial converse, (n) denoting .δH (Ll ) = [H, Ll ] and .δH (Ll ) the n-iterated commutator, we can show [16, Theorem 11] that Proposition 2 Suppose that .Dom(G) = Dom(N ), let .dr =dim.(V2d−m ) and (n) assume that the sets .Lin { δH (Ll ) | 1 ≤ l ≤ m, 0 ≤ n ≤ 2d − m } and

Boson Quadratic GKLS Generators

191

Lin { aj , aj† | 1 ≤ j ≤ dr } coincide up to conjugation with a unitary transformation that preserves the CCR. If the gaussian QMS is irreducible, then dim.(V2d−m ) = 2d.

.

Irreducibility allows one to get immediately the following Theorem 6 Let .T be an irreducibile QMS with an invariant density matrix .ρ. Then ρ is faithful and unique. In particular, if an irreducible Gaussian QMS admits a Gaussian invariant density matrix .ρ, then it is faithful, unique among all densities and f f 1 t 1 t ∗ T∗s (ρ0 ) ds = ρ (10) Ts (x) ds = tr(ρx) 1, w− lim . w − lim t→+∞ t 0 t→+∞ t 0

.

for all .x ∈ B(h) and all initial density matrix .ρ0 . Proof The invariant density is faithful because its support projection is subharmonic projection [17, Theorem II.1] and so it coincides with .1. Once proved that it is faithful, it is unique by [20, Lemma 1]. Convergence properties also follow from known general results on QMS [21, Theorems 1.1, 2.1]. We end this section by an illustrative example with .d = 2 (just to simplify computations). Consider the Gaussian QMS√with generator determined by Kraus √ √ operators .L1 = 2μa1 , .L2 = 2λa1† , .L3 = 2a2 (.0 < λ < μ are two parameters) and Hamiltonian ) ( † † − 1 < r < 1. .H = N + r a a2 + a1 a 1 2 In this case, since .κ = 0, .U T V = V T U = 0, the matrices Z and C are also complex linear and, putting .γ 2 = μ2 − λ2 , [ o=

.

] 1r , r1

[ Z=

] −γ 2 + i ir , ir −1 + i

[ C=2

μ2 + λ2 0 0 1

]

The eigenvalues of Z have real part smaller than 0, therefore Z is stable and there exists an invariant Gaussian density matrix by Proposition 1. Vectors in .C4 determined by Kraus operators are L1 -

.

√

2[μ, 0, 0, 0]T ,

L2 -

√

2[0, 0, λ, 0]T ,

L3 -

√ 2[0, 1, 0, 0]T

and are linearly independent. Computing the commutator .[G, L2 ] = (μ2 + λ2 + ir)a1† )1, we find the vector √ [G, L2 ] - − 2λ[0, 0, (μ2 + λ2 + i), ir]T ,

.

√

2λ(−ira2† −

192

F. Fagnola

which is linearly independent of vectors determined by .L1 , L2 , L3 if and only if r /= 0. Therefore, for .r /= 0, the dimension of .V2d−m = V1 is equal to 4 and we can apply Theorems 5, 6 to conclude the Gaussian QMS is irreducible. As a consequence, the invariant density matrix is unique and Gaussian with zero mean and covariance S matrix that can be explicitly computed solving .ZS ∗ +SZ +C = 0. In the next section, applying known results by Frigerio and Verri [21, Theorem 3.3] one can show that, in addition, .T∗t (ρ0 ) converges towards the limit Gaussian density matrix as .t → +∞. For .r = 0, the vector .λ[0, 0, 0, 1]T does not belong to .V1 and the creation operator .a2† does not arise from the commutator .[G, Ll ] or from iterated commutators. Intuitively, this means that the number of particles in the second component can only decrease. More precisely, if we denote .p[0, n2 ] the spectral projection of .N2 , the closure of .a2† a2 defined on D, corresponding to the integer interval .[0, n2 ], a straightforward computation yields

.

) ( £(p[0, n2 ]) = 2 (N2 − 1)+ − N2 p[0, n2 ]

.

where .(m − 1)+ denotes .m − 1 if .m > 1 and 0 otherwise. The right-hand side is a bounded operator and is negative, therefore .p[0, n2 ] belongs to the domain of .L and .Tt (p[0, n2 ]) ≤ p[0, n2 ] for all .t ≥ 0. It follows that the Gaussian QMS is not irreducible.

4 The Decoherence-Free Subalgebra As mentioned in the previous section, often convergence towards invariant densities (10) can be improved considering the decoherence-free subalgebra [5, 7, 15, 25] of .T, defined as { } N(T) = x ∈ B(h) | Tt (x ∗ x) = Tt (x ∗ )Tt (x), Tt (xx ∗ ) = Tt (x)Tt (x ∗ ), ∀t ≥ 0 .

.

This is the biggest sub von Neumann algebra of .B(h) [15, Proposition 2.3, see also [18]] on which maps .Tt act as .∗ -homorphisms. If one considers a GKLS generator with bounded operators .H, Ll instead of (5), (4) it is not hard to show [15, Proposition 2.3] that .N(T) is the commutator n (L ), δ n (L∗ ) with .l = 1, . . . , m, n ≥ 0 where .δ (x) = of the set of operators .δH l H H l [H, x]. In the case of Gaussian QMSs with unbounded operators .Ll , H it can be characterized in a similar way considering generalized commutators. We recall that the generalized commutant of an unbounded operator L is the set of bounded operators x for which .xL ⊆ Lx, namely Lx is an extension of xL. Applying H. Araki’s Theorem 4 p. 1358 in [3], sometimes referred to as duality for Bose fields, (or [24, Theorem 1.3.2 ] (iv) for a proof with our notation, up to a constant in the symplectic from) we can also describe the structure of .N(T).

Boson Quadratic GKLS Generators

193

Theorem 7 The decoherence-free subalgebra of a Gaussian QMS with generator in a generalized GKLS form associated with operators .H, Ll as in (4), (5) is the generalized commutant of the following linear combinations of creation and annihilation operators n δH (L∗l )

n δH (Ll ),

.

l = 1, . . . , m, 0 ≤ n ≤ 2d − m

where .δH (x) = [H, x] denotes the generalized commutator. Moreover .Tt (x) = eitH x e−itH for all .t ≥ 0 and .x ∈ N(T) and, up to unitary equivalence, N(T) = L∞ (Rdc ; C) ⊗ B(r(Cdf )),

.

where .⊗ denotes the tensor product in the sense of .W ∗ -algebras, for a pair of natural numbers .dc , df ≤ d. We refer to [2, Theorems 3.2, 3.7 and Appendix B] for the proof (the bound .2d − 1 there is immediately replaced by .2d − m). By a known result of Frigerio and Verri [21, Theorem 3.3], given a QMS with a faithful invariant density matrix, if .N(T) coincides with the algebra of fixed points, then .T∗t (ρ0 ) converges as t goes to .+∞ for any initial density matrix. It is not difficult to see that the decoherencefree algebra of the Gaussian QMS considered as an example in Sect. 3 is trivial. The fixed point algebra is also trivial by irreducibility, therefore .T∗t (ρ0 ) converges towards the unique invariant density matrix as t goes to .+∞.

5 Conclusions and Outlook We have presented some recent results on Gaussian QMSs on states invariant densities and convergence towards invariant densities. We concentrated on ergodic theorems because of their interest in quantum probability (see e.g. [11]) but many problems are still open. It would be useful to extend Theorem 5 removing the assumption .Dom(G) = Dom(N ) in view of applications, for example to a chain of bosons. Moreover, we can show that convergence towards a unique invariant density matrix is exponentially fast, in a space of Hilbert-Schmidt operators determined by the invariant space (see e.g. [6]), and that the spectral gap of the generator is determined by the spectrum of Z. The solution of these problems will appear in forthcoming papers. The case of an infinite dimensional Hilbert space .h also seems very interesting. The decoherence-free algebra .N(T), in particular, may not be of the type described in Theorem 7. Acknowledgments The author is a member of GNAMPA INdAM. The financial support of GNAMPA 2022 Project “Algebre di Operatori e Probabilità Quantistica” CUP_E55F22000270001 is gratefully acknowledged.

194

F. Fagnola

References 1. Agredo, J., Fagnola, F., Poletti, D.: Gaussian quantum Markov semigroups on a one-mode fock space: irreducibility and normal invariant states. Open Sys. Inf. Dyn. 28(1), 2150001 (2021). https://doi.org/10.1142/S1230161221500013 2. Agredo, J., Fagnola, F., Poletti, D.: The decoherence-free subalgebra of Gaussian quantum markov semigroups. Milan J. Math. 90, 257–289 (2022). https://doi.org/10.1007/s00032-02200355-0 3. Araki, H.: A lattice of von Neumann algebras associated with the quantum theory of a free bose field. J. Math. Phys. 4, 1343–1362 (1963). https://doi.org/10.1063/1.1703912 4. Bellman, R.: Introduction to Matrix Analysis. Classics in Applied Mathematics SIAM. The RAND Corporation, Santa Monica (1960) 5. Blanchard, P., Olkiewicz, R.: Decoherence induced transition from quantum to classical dynamics. Rev. Math. Phys. 15(3), 217–243 (2003). https://doi.org/10.1142/ S0129055X03001631 6. Carbone, R., Fagnola, F.: Exponential L2 -convergence of quantum markov semigroups on B(h). Math. Not. 68, 452–463 (2000). https://doi.org/10.1007/BF02676724 7. Carbone, R., Sasso, E., Umanità, V.: Environment induced decoherence for Markovian evolutions. J. Math. Phys. 56, 092704 (2015). https://doi.org/10.1063/1.4931479 8. Carlen, E.A., Maas, J.: Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems. J. Stat. Phys. 178(2), 319–378 (2020). https://doi.org/10. 1007/s10955-019-02434-w 9. Chebotarev, A.M., Fagnola, F.: Sufficient conditions for conservativity of minimal quantum dynamical semigroups. J. Funct. Anal. 153(2), 382–404 (1998). https://doi.org/10.1006/jfan. 1997.3189 10. Cipriani, F., Fagnola, F., Lindsay, J.M.: Spectral analysis and feller property for quantum Ornstein–Uhlenbeck Semigroups. Commun. Math. Phys. 210, 85–105 (2000). https://doi.org/ 10.1007/s002200050773 11. Crismale, V., Fidaleo, F., Lu, Y.G.: Ergodic theorems in quantum probability: an application to monotone stochastic processes. Ann. Sc. Norm. Super. Pisa Cl. Sci. XVII, 113–141 (2017). https://doi.org/10.2422/2036-2145.201506_009 12. Davies, E.B.: Quantum dynamical semigroups and the neutron diffusion equation. Rep. Math. Phys. 11(2), 169–188 (1977). https://doi.org/10.1016/0034-4877(77)90059-3 13. Demoen, B., Vanheuverzwijn, P., Verbeure, A.: Completely positive maps on the CCR-algebra. Lett. Math. Phys. 2, 161–166 (1977). https://doi.org/10.1007/BF00398582 14. Derezi´nski, J.: Bosonic quadratic Hamiltonians. J. Math. Phys. 58, 121101 (2017). https://doi. org/10.1063/1.5017931 15. Deschamps, J., Fagnola, F., Sasso, E., Umanità, V.: Structure of uniformly continuous quantum Markov semigroups. Rev. Math. Phys. 28(1), 1650003 (2016). https://doi.org/10. 1142/S1230161217400054 16. Fagnola, F., Poletti, D.: On irreducibility of Gaussian quantum Markov semigroups. Infin. Dimens. Anal. Quant. Probab. Relat. Top. 24, 2240001 (2022) 17. Fagnola, F., Rebolledo, R.: Subharmonic projections for a quantum Markov semigroup. J. Math. Phys. 43, 1074–1082 (2002). https://doi.org/10.1063/1.1424475 18. Fagnola, F., Sasso, E., Umanità, V.: The role of the atomic decoherence-free subalgebra in the study of quantum Markov semigroups. J. Math. Phys. 60, 072703 (2019). https://doi.org/10. 1063/1.5030954 19. Flynn, V.P., Cobanera, E., Viola, L.: Deconstructing effective non-Hermitian dynamics in quadratic bosonic Hamiltonians. New J. Phys. 22, 083004 (2020). https://iopscience.iop.org/ article/10.1088/1367-2630/ab9e87 20. Frigerio, A.: Quantum dynamical semigroups and approach to equilibrium. Lett. Math. Phys. 2, 79–87 (1977). https://doi.org/10.1007/BF00398571

Boson Quadratic GKLS Generators

195

21. Frigerio, A., Verri, M.: Long-time asymptotics properties of dynamical semigroups on W ∗ algebras. Math. Z. 180, 275–286 (1982). https://doi.org/10.1007/BF01318911 22. Gough, J.E., James, M.R., Nurdin, H.I.: Squeezing components in linear quantum feedback networks. Phys. Rev. A 81, 023804 (2010). https://doi.org/10.1103/PhysRevA.81.023804 23. Isar, A.: Decoherence and asymptotic entanglement in open quantum dynamics. J. Russ. Laser Res. 28(5), 439–452 (2007). https://link.springer.com/content/pdf/10.1007/s10946-007-00334.pdf 24. Leyland, P., Roberts, J., Testard, D.: Duality for free quantum Fields. Centre de Physique Théorique Marseille, Report CPT-78/P-1016 (1978). https://inspirehep.net/literature/132161 25. Lidar, D.A.: Review of decoherence free subspaces, noiseless subsystems, and dynamical decoupling. Adv. Chem. Phys 154, 295–354 (2014). https://doi.org/10.1002/9781118742631. ch11 26. Parthasarathy, K.R.: An Introduction to Quantum Stochastic Calculus. Birkhauser, Basel (1992) 27. Poletti, D.: Characterization of Gaussian quantum Markov semigroups. In: Accardi, L., Mukhamedov, F., Al Rawashdeh, A. (eds.) Infinite Dimensional Analysis, Quantum Probability and Applications QP41 Conference, Al Ain, UAE, March 28–April 1 (2021). https://link. springer.com/book/9783031061691 28. Poletti, D.: Characterization of Gaussian quantum Markov semigroups. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 25, 2250014 (2022). https://doi.org/10.1142/ S021902572250014X 29. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II: Fourier Analysis SelfAdjointness. Academic Press, San Diego (1975) 30. Teretenkov, A.E.: Irreversible quantum evolution with quadratic generator. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 22(4), 1930001 (2019). https://doi.org/10.1142/ S0219025719300019 31. Vanheuverzwijn, P.: Generators for quasi-free completely positive semigroups. Ann. Inst. H. Poincaré Sect. A 29, 123–138 (1978). https://eudml.org/doc/75994

Part III

Many-Body Quantum Mechanics

Energy Expansions for Dilute Bose Gases from Local Condensation Results: A Review of Known Results Giulia Basti, Cristina Caraci, and Serena Cenatiempo

1 Introduction We consider N bosons in the d-dimensional torus .AT = [−T /2, T /2]d , .d = 2, 3, interacting via a two body non negative, radial and compactly supported potential V with scattering length .a (see [71, App.C] for a definition of the scattering length). Note that with a slight abuse of notation we denote by .a both the three and two dimensional scattering lengths. In the units where the particle mass is set to .m = 1/2 and .h¯ = 1, the Hamilton operator has the form HT = −

N E

.

Ax i +

i=1

E

V (xi − xj )

(1)

1≤i