Fortschritte der Physik / Progress of Physics: Band 24, Heft 2 1976 [Reprint 2021 ed.] 9783112520383, 9783112520376

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HEFT 2 • 1976 • BAND 24

A K A D E M I E - V E R L A G EVP 10,- M 31728

•

B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an eine Buchhandlung oder an den Akademie-Verlag, D D R - 1 0 8 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle K U N S T UND WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 — in Osterreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 7 0 1 Leipzig, Postfach 160, oder an den Akademie-Verlag, D D R - 1 0 8 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortschritte der Physik" Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur Lösche, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 1 0 8 Berlin, Leipziger Straße 3—4; F e m r u f : 2200441; Telex-Nr. 114420; Postscheckkonto: Berlin 35021; B a n k : Staatsbank der D D R , Berlin, Konto-Nr.: 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 1 0 4 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R - 7 4 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der P h y s i k " erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je B a n d : 144,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je H e f t 12,— M (Preis für die D D R 1 0 , - M). Bestellnummer dieses Heftes: 1027/24/2. © 1976 by Akademie-Verlag Berlin. Printed in the German Democratic Republic.

Fortschritte der Physik 24, 55-83

(1976)

Properties of Equilibrium States on Quasilocal C*-algebras STIG I.

Kungl.

Vetenskapakademien;

ANDERSSON

Institut Mittag-Leffler,

Djursholm,

Sweden

Abstract These notes have grown out of a seminar series and are to a large extent of a review character (this goes especially for section 3). Novel aspects are indicated in connection with the continuous point particle systems and the relations between equilibrium conditions. We concentrate on some less often treated subjects, viz. the decomposition theory and equivalence of the equilibrium conditions. Contents 1. Algebraic Description of Classical and Quantum Systems. 1.1. Preliminary Remarks ' 1.2. Algebraic Formulation 1.3. Classical Lattice and Continuous Systems 1.4. Quantum Spin Systems . 1.5. Continuous Tensor Products

55 55 56 58 60 61

2. Equilibrium States 2.1. Limits of Local Equilibrium States (Gibbs States) 2.2. Global Definitions of Equilibrium States 2.3. Gibbs Random Fields and Markoff Random Fields 2.4. Equivalence of Equilibrium Conditions

67 67 67 70 71

3. Decomposition of Equilibrium States 3.1. Spontaneous Breakdown of Symmetry and Dynamical Instability. Ergodic States 3.2. Specific Decompositions: Extremal, Central and Ergodic Decompositions 3.3. Algebra at Infinity 3.4. Extremal KMS and Extremal Classical Equilibrium States

. .

72 72 73 80 81

1. Algebraic Description of Classical and Quantum Systems 1.1. Preliminary Remarks We will here be dealing with global equilibrium states for some relevant physical systems. More specificly, we will treat decomposition properties of such states as well as the relations between classical and quantum equilibrium conditions. We will limit ourselves to properties of lattice gases and continuous point particle systems in the classical case and to spin systems in the quantum case. More general aspects are indicated here and will be treated in detail in a forthcoming paper A N D E R S S O N [J]. 5

Zeitschrift „Fortschritte der P h y s i k " , Heft 2

56

STIG I . ANDEKSSOU

The relevance of this subject to field theory has been revealed recently by N E L S O N [2, 3] who establishes on a rigorous basis the equivalence of relativistic (Wightman) field theory and a (formal) classical statistical mechanics (SM), in the form of euclidean Marltoff fields. The usefulness of this link has been clearly exhibited by a large number of results recently, (cf. [4, 5] for a comprehensive treatment and references.) 1.2. Algebraic Formulation wo describe the equilibrium states of infinite systems in SM one proceeds along various Tays, e.g. a) One considers the restrictions of the system to each bounded region in space and describe states by either probability measures on the finite configuration space classical case) or by density (traceclass) matrices on a suitable Hilbert space (quantum case). Local equilibrium states are then the Gibbs states. b) Equilibrium states are characterized globally; i.e. by specific conditions directly on the inifinite systems. Of course (a) involves taking the thermodynamic limit and since this might be tricky (non-unique limit in the presence of phase transitions; dependence on the boundary conditions [6]) there are good reasons to pursue (b). Due to the infinite number of degrees of freedom associated with a full description of a system it is expected that algebras will provide as useful a language for SM as it does for relativistic field theory. We therefore adopt this point of view and associate with each system a 0*-algebra, the self-adjoint (s.a.) elements of which correspond to observables (in some sense) and the dual space of which contains the expectation functionals. The special "dress" of a 0*-algebra is of course a technical choice which provides us with a rich but still managable structure for which a large body of mathematical results are known. Given a physical system, to associate with it an algebra is of course not a unique operation and also the mathematical structure associated with a given system is richer than just that of a 0*-algebra. The algebras used for describing systems in SM (and field theory) are typically obtained as follows: To each 0 £ S(R") ( = set of open relatively compact subsets of R") (or $(Z")) we associate a C*-algebra (with identity) 91(0) the local sub-C*-algebra. The set { « ( 0 ) | 0 £ (S(R')} is given the following structure: (1) 0 , cz 02 =$> 21(0,) cz 21 (0 2 ) i.e. we identify 21(0,) with a subalgebra of 2l(0 2 ) by the following normpreserving map 21(0,)

o^\o1 € 2I(0 2 )

(amplification map).

This property is called isotony. (2) Let G be some invariance group (e.g. translations) and g rg£ *Aut 2t a representation of G in the *-automorphism group of 21 denoted *Aut SC. Then r 9 2i(0) = 2l(0 ff ) (3) 0 , H 0 2 = 0 => [21(0,), 2i(0 2 )j = 0

(covariance) (locality)

(4) The local algebra e> Qf>) where n e is a cyclic representation of 91 in a Hilbert space with cyclic vector Q e 6 such t h a t £>(•) = [Qq, *,(•) Qt). Furthermore, n e is irreducible if and only if q is pure. For classical systems by standard spectral theory, the canonically associated cyclic r e p r e s e n t a t i o n ' s unitarily equivalent to a multiplication operator representation in L 2 (K, (i) (Bochner-Godement theorem) where ¡x corresponds to q.

58

STIG I . ANDERSSON

Briefly : dense embedding

+ L\K, fi)

V

*

where F is a surjection of

°(K) onto jre(9t) Qq by

2Ida

â e g«, °(K)

ti Q(a) Q, e

and F e x t is the extension of V to the closures. Note: (£xa{K) = for K compact. (Soo°(-S0 are the continuous functions vanishing at infinity). We will also need the concept of Gel'fand transform and Gel'fand transformation. Given an abelian 0*-algebra St, we let Q be the character space of 21 (identified to the space of its maximal regular ideals). If 21 is separable Q is metrizable and if 21 has a unit then Q is w*-compact. Given A € 21, the mapping is called the Gel'fand transform of A. The mapping 21 9 A

A{-) € Eoo°(-Q) is called the OeVfand transformation

of A and is defined by A(y)

1.3. Classical Lattice and Continuous Systems In the following we are only interested in describing situations with a variable particle number so we use the grand canonical ensemble of Gibbs (GCE). Let us just briefly review the algebraic description of the two kinds of systems in the title of this subsection. C l a s s i c a l c o n t i n u o u s p o i n t p a r t i c l e s y s t e m . (Infinitesystem in Rv, v = 3 usually). The Ruelle configuration space is the set of functions X: R" —> Z+ (positive integers) such that £ X(x) < oo V K (compact) a R" to guarantee that we don't get infinite densities locally. So configuration space is a subspace of (Z+)Rv restricted so that we have locally finite configurations. Equivalently configurations could be described as follows : a specific configuration could be identified to a sequence in R", {xnj say, which could be finite or infinite. Since, the order in which sequences are enumerated doesn't mat we get the following picture [7] : let 36* = space of locally finite sequences in R'' (i.e. {xn} s.t. 2J X(xn) < oo, V K a Rv, K compact) M nK and let N be the equivalence relation of identifying permuted sequences, then £ = 3î*/N = space of equivalence classes of sequences. One uses these two descriptions interchangeably and calls X* the space of locally finite configuration of labelled particles and 36 the space of locally finite configurations of unlabelled particles.

59

Properties of Equilibrium States on Quasilocal C*-algebras

It turns out that the natural topology to describe convergence on this space is the following (Lanford-topology): given m, A, K (TO positive integer, A cz R" open bounded, K compact cz R" and / l e i ) then a subbasis for the -topology is given by ®ak = {X € 36 / £ X(x) = £ X(x) = ml 1 / zha xac } With this topology the following is known about 36: Let Of be the a-algebra of Borel sets on 36 Proposition

1.1.: (Lanford) if coincides with the c-algebra generated by W„m =

€ 36 j£X(x)

= mj

B runnirg over relatively compact Borel sets in R" andTOover Z+. Proposition

1.2.: (Lanford) 36 is Polish

In the algebraic frame states should r.ow correspond to Borel probability measures on 36. However, in order that the classical algebra E°(36) has a unit (and for some other reasons) one would like 36 to be compact. I t seems there are not way of compactifying 36 so 36 itself is less suitable for such an approach. As for the lattice gas configuration space is K = {0, 1}Z" ^ the space of functions X: Zv {0, 1} and a specific configuration X may also be identified with the set of its occupied lattice sites {x e Z" | X(x) = 1} € ¿P{Z*) ( = set of all subsets of Zv). So we identify the infinite configuration space K = {0,

= &>{Z').

Giving {0, 1} the discrete topology makes it compact and thus by Tychonoff's theorem K is compact in the product topology. States are now probability measures on configuration space K i.e. linear functionals on (£°(K), continuous when K carries its sup-norm topology. Thus we call (£"(K) = 21 the quasilocal observable algebra of the system. The local sub-(7*-algebras %(A) are then the sets of cylinder (tame) functions: A i K{A) & A{X) = A{X C\ A) If x € Z" we have rx € *Aut

V X (p).

-

by

r* A{X) = A{X — x)

• V X € 0>{Z>)

which completes the quasilocal structure. In order to be able to study physically relevant objects we should also introduce interactions. Clearly we cannot pick any potential, but a minimal requirement should be that it produces a thermodynamic behaviour. We leave these requirements till they are needed in specific situations and state here just the formal expressions and definitions (c.f. [ is given by YaX 1.4. Quantum Spin Systems [5] In contrast to the two foregoing situations we now will meet a non-abelian algebra as the quasilocal algebra of observables. Again lattice sites are parametrized according to Z". Let for each point x € Zv, $qx ^ )

(i.e. 0 finite).

For 91(0) we take the algebra of all bounded operators on § ( 0 ) (finite matrix algebra). For 0 , cz 0 2 the natural isomorphism fQ(02) = § ( 0 j ) (x) § ( 0 2 / 0 j ) identifies 91(0,) with a subalgebra of 2l(0 2 ). 21 is then defined as the norm-closure of the "union" of all 21(0), which means the inductive limit in the following sense: for any 0 we have the (amplification) map i0: 21(0) U 31(0) then we define the local algebra by 0

• •

Six = -i^-y 91(0)

w.r.t. i0 and the quasilocal algebra

21 = 91l-

(norm closure).

Lattice translations act in a natural way and we have also O 1 n O 2 = 0 ^ [ 9 l ( O 1 ) , 2 l ( O 2 ) ] = O. Furthermore, we have the asymptotic abelianness lim 11[a, r^lll = 0 |*|-MO

w.r.t. the lattice translations \/ a, b 6 21.

The lattice sites are occupied by particles interacting tjirough fc-body potentials (*) such that: • 4>*(x1,...,xk)a%({xl,...>xk\) and by an interaction we mean a sequence = (k)ks,2- 4> may obviously be considered as an operator-valued set function defined on finite sets in Z" by (X) = 4>k(z1, ...,xk),

X = \xu ..., xk]

61

Properties of Equilibrium States on Quasilocal C*-algebras

and {X) for X aZ", at Z* and 3. \\\\ = ¿ ¡ < oo. With a configuration A we X30 associate the interaction energy, due to , by XcA 1.5. Continuous Tensor Products There is an alternative formulation of a classical point particle system which is close to the foregoing sub-section. For references to the general theory of continuous tensor products used in this section c.f. [9, 10]: As we saw lattice systems, classical and quantum, are naturally described by tensor products, i.e. one considers the situation in each lattice point and set up the proper Hilbert space with observables acting in it for each such lattice site. The complete system is then described by taking infinite (discrete) tensor products. Utilizing the concept of continuous tensor product (OTP), including a trivial form of the Araki-Wood's embedding theorem, we here set up a description of a continuous classical system in complete analogy to the way in which lattice systems are described. Besides this nice analogy, this formulation also offers a particularly direct way for embedding the classical observable algebra into the corresponding quantum algebra. We give here the mathematical structure in a slightly abstract way. Let X be a locally compact, separable and metrizable TVS (topological vector space) the indexing space and let / a(t) A = At the time evolution of the system described by 2i. A state m on 9i is then said to be a KMS state of 91 w.r.t. the automorphism group a at inverse temperature ft if m{BAt) (w(AtB)) could be continued as a holomorphic function in the strip 0 < y < ¡) (—¿3 < y < 0) continuous at the boundaries for all A, B 6 91 and such that the boundary value satisfies w(BAt+if)

=

a>(AtB).

Definition 2.3.: We identify the KMS-states with the equilibrium states of the system. The physical reasons supporting this definition are among others the following: 1. For lattice systems it has been proved that any translation invariant state maximizing the difference of entropy per lattice site and energy per lattice site is KMS. 2. Thermodynamic limits of local equilibrium states (Gibbs' states) are KMS (for a wide class of lattice interactions and where thermodynamic limit means in the van Hove sense for example). 3. The KMS condition implies the Greenberg/Robinson equations which generalizes the Kirkwood-Salsburg equations to quantum systems. 4. Under various decompositions the extremal KMS states have a nice and direct physical interpretation (cf. section 3). 5. "Comparing" the classical und quantum systems in a sense to be made precise in subsection 2.4. the KMS and DLR conditions "match" in a very natural manner. Remark 2.3.: The most interesting question whether the KMS condition can really be used to characterize equilibrium states has not been answered in the sense that it is not known if all KMS states come from translation invariant states maximizing the difference of entropy and energy per lattice site (in the lattice case). For general properties of KMS states we refer the reader to the previously mentioned Hugenholtz' lectures (and to subsection 2.4). 2.3. Gibbs' "Random Fields and Markoff Random Fields There is another very intersting ^spect of the local equilibrium states which may have some implications for the study of DLR-states and the relations to field theory.

Properties of Equilibrium States on Quasilocal C*-algebras

71

Theorem 2.1.: (AVERINTSEV, SHERMAN, SPITZER, SULLIVAN [15—19]). For lattice systems any local Gibbs' random field with a fc-body translation invariant finite range potential is a k: th order translation invariant positive local Markoff random field and vice versa. This result was obtained gradually by SPITZER, AVERINTSEV, SHERMAN and SULLIVAN who all based their works on ideas due to DOBRUSHIN [18]. On the lattice Zv for v = 1 the above result remains valid also globally, whereas for v 2 very interesting non-uniqueness results have been found by DOBRUSHIN and collaborators, results which are related to the phenomena of phase transitions (cf. subsection 4.1). So even if we knew nothing of statistical mechanics and Gibbs' states but were interested solely in Markoff random fields, we would nevertheless meet with the local equilibrium state-prescription of Gibbs'. From a heuristic point of view the above theorem is natural since we certainly associate a situation of equilibrium with some kind of smeared-out property, where the system roughly speaking "looks the same everywhere". So upon measuring something inside a bounded region we expect influence to matter only if it comes from the nearest neighbour region. (Markoff property). 2.4. Equivalence of Equilibrium Conditions^ Having discussed at some length the two seemingly very different equilibrium conditions, KMS and DLR, used for quantum and classical systems respectively, the natural question as for their relationship arises. From a physical point of view a connection between them should exist to the same extent as to which there exists a link between classical and quantum physics. Various other equilibrium conditions (not described here) are identified in a more or less trivial way, a situation which is certainly not true for the DLR and KMS conditions. The results proven so far could be briefly summarized as follows: BRASCAMP [20], in order to compare the two conditions, (the KMS automorphism has of course no meaning on the classical system) embeds the classical observable algebra for a lattice gas into the center of the corresponding quantum algebra. For a large class of potentials the corresponding induced quantum evolution and KMS automorphism is studied. I t is shown that the KMS condition, in terms of the correlation functions of the classical system, "coincides" with the D L R condition. ARAKI and ION [21] on the other hand (again for lattice systems and a large class of potentials) construct a non-abelian version of (2.3), called the Gibbs' condition, the equivalence of which with KMS is direct. More formally one could state the results as follows: Let Jt be the quantum algebra of a lattice ga,s,^V the image of the classical algebra in Ji i.e. Ji c= center Jt and ¿V = L^IK, /i) {Jt and Ji viewed as W*-algebras) ARAKI and ION work in the quantum region ("outside" Ji in a specific s3nse) by constructing the Gibbs' condition there and identify it to KMS. They then show that the Gibbs' condition is a genuine extension of D L R by establishing their equivalence on Ji. Brascamp works in the classical region by translating KMS into a condition on the probability measures on K (classical configuration space). This condition is then identified to DLR. KMS (4) Gibbs' cond. Jt

on Jt

on Jt

(1)|

|(3)

(abelian) KMS (1) — ( 2 ) = c

BRASCAMP

¡¿j (3) — ( 4 ) =

Zeitschrift „Fortschritte der Physik", Heft 2

DLR ION/ARAKI

72

STIG I . A N D E R S S O N

We do not enter a detailed account for these theorems but refer the reader to the original articles. In this connection we announce that the commutative diagram above is true in a more abstract setting, depending upon properties of the KMS modular automorphism group and the existence of Radon-Nikodym derivatives on general von Neumann algebras. This is the subject for a forthcoming paper [1], The deep point is that the KMS condition plays an intrinsic role in non-commutative integration theory and in operator theory; even if one did not know of statistical mechanics one would nevertheless encounter the KMS condition in e.g. the search for non-commutative analogues of the Radon-Nikodym theorem. Very loosely and somewhat speculatively speaking, by the results of the foregoing subsection, the KMS condition (being non-abelian Gibbs' prescription) is also related (at least for lattice systems in one dimension) to the Markoff property. We do not enter any account of this here. 3. Decomposition of Equilibrium States

3.1. Spontaneous Breakdown of Symmetry and Dynamical Instability. Interpretation of Ergodic States [5, 22—24] Partially the decomposition theory was initially inspired by a desire to understand better the concept of spontaneous breakdown of symmetry. Before we elaborate on the relevance of this concept for the decomposition theory, however, we need to write down some general definitions which will occur. We will follow the given references [ yl(•) 6 E r ° A 2 ( - ) 6 S = convex cone of continuous convex functions. The convexity of the squares of the elements in (£r°(E) is easily seen. We next give Definition

3.4.: We introduce the following ordering (BISHOP — DE LEETJW) & < fi2

J'(4>) MM) E

< f A2() MM) E

• V A 6 21s.

76

STIG I. ANDERSSON

Theorem 3.3.: (CARTIER, F E L L , M E Y E R ) If E is metrizable and ^ < ¡i2 then there exists a family {fia}aiE of probability measures on E s.t. 1. fi„ has resultant a 2. a -> fi„(4>) is a Borel function for all £ E°(2?) and = f

PM)

E

Rewriting /x2 = f /¿„/¿¡(da) we see that if lu1 < ju2, the decomposition associated with fi2 is finer in the sense that it may be accomplished in two steps, one of which is the decomposition accociated with /¿j. Theorem 3.4.: Let ¡¡Sy and be two abelian W,"*-algebras in n(%)' and let ¡jlx and fi2 be measures the two 38x- and ^ V of Q resp. Then C

^./I!
)2 ni(d) = (Q, Pxn(A) P^n(A) P^) rg € *Aut 91. We let EG be the G-invariant states. As a general remark we would also like to remind you about the fact that for E the property of being a simplex is connected to uniqueness of the maximal measure the property of being metrizable is connected to the fact that ¡j, is carried by S'G. Metrizability of E is in turn connected to separability assumptions on 91 e.g. the weak separability assumption of Ruelle: ConditionS: For k = 1, ...,n there are countable families (9i«1...c(„) and (Sc satisfying the general (mild) conditions we described earlier.) Theorem 3.17.: a) is convex, compact and a Choquet simplex. b) A state g € is a factor state (i.e. has s.r.c.) iff 0 is an extremal KMS-state w.r.t. cj> i.e. extremal in r 9 . Theorem 3.18.: Lst g 6 r 9 and let ¡u be the measure giving the central decomposition of q ( = decomposition at infinity). Then a) supp ¡x a JT^ b) /x is the unique measure on r 9 carried by the extremal points of i.e. by the states of s.r.c. ( = factor states). There is one paper [25] which has nice physical results supporting the interpretation we gave earlier of the extremal KMS states. This paper studies a class of Weiss-Ising

82

STIG I . A N D B B S S O N

models (for which the time-evolution is not an automorphism on the quasi-local observable algebra) and it compares the characterizations of pure thermodynamical phases as extremal KMS states and as extremal time- or space-invariant states. The conclusion is that the correct characterization of the pure thermodynamical phases is in terms of extremal KMS states. B. Classical L a t t i c e Gas Given an interaction (j>, we denote by A,p the set of equilibrium states for (i.e. solutions of the DLR-equations). Lemma: 3.4. A,f is convex and a>*-compact. It is also a Choquet simplex. Here and in the following cj> is assumed to satisfy just the general conditions described in connection with the observable algebra of this system. Theorem 3.19.: A state Q € Av has s.r.c. iff it is an extremal point of A,p. Proof: Idea of proof is to show Q non-extremal Q doesn't have s.r.c. K compacts 91 = ©"(if) = &oo(K) which is weakly dense in L°° (K, //) provided supp /x = K. Let {$Qe, ji e , a>e) be the representation canonically associated with q and fi the corresponding probability measure on K (Riesz-Markov) so that (#), 1) o non-extremal o 3 A € Av s.t. X Si Q. (A not multiple of p.) This means ([26] prop. 2.5.1) 3 T , dY) & h(X, Y) fiA(X, dY) = fA(X, Y) h(4>, Y) pA(, dT) given any A cz Z" and ¡x 0) region of the variable t correspond to different physical processes. In elastic scattering (see Fig. 6 a) one has t si 0, for the energy is not changed q = (0, p' — p). The momentum transfer becomes time-like in the case of production processes, as for instance e+ + e~ + n~ (see Fig. 6b). Here t = (Ep< + Ep)2 is the square of the total energy in the center-of-momentum system. Obviously the production of a pion pair is possible only for values t ig (27rthreshold). It can be shown that the

a)

b)

Fig. 6. a) Elastic scattering

b) Inclnstie process

pion from factor is an analytic function in the complex t plane, except a cut along the real positive axis starting from t = 4mn2. On the real negative axis (space-like virtual photons) Fn is a real function. Both regions of t are separated by an unphysical domain (see Fig. 7). These properties are the starting-point for a variety of models [49—51], based on the use of dispersion relations and the existence of a nn resonance with spin J = 1 and isospin T = 1, the p° meson 9 ). However, the exporessions for F„ obtained in these models are, because of their complicated structure, not suitable for our estimate of AEF. 3m t unphysical region

e-IT- scattering s p a c e - l i k e photons

AmI2T

16rrr—

production time-like

Ami k

Ret

processes photons

Fig. 7. The complex i-plane

On the experimental side the most reliable data so far are those obtained from the storage rings at Novosibirsk [-54] and Orsay [55], It is the measurement of the process e+ + e~ 7i+ + Tt~ that allows a direct determination of ^„(i)) in the time-like region. The result (following Ref. [55]) is displayed in Fig. 8, showing \F„|2 as a function of the energy E =

]/s/2.

The p° resonance can be seen directly, thus providing an impressive confirmation of the hypothesis of p meson dominance in the measured region. The experimental results obtained so far in the space-like region are compatible with this hypothesis at least for low momentum transfer. 9

) It is interesting to observe that the existence of vector mesons has first been postulated in order to understand the behaviour of form factors [49, 52], Afterwards they were then discovered in experiments [53].

94'

U . E.SCHRÖDER

I n the case of the bound pion one needs information on in the space-like region. Starting from the knowledge of F„ for time-like momenta one might think of an analytic continuation of the form factor into the space-like region. This method has been applied by several authors [56]. One should however remark that by this method, from an uncomplete, error affected knowledge of the input data, as obtained from experiments, considerable errors may arise in the result [57]. Therefore, in order to obtain a practicable expression for Fn in the region i ^ Owe shall apply the hypothesis of p° dominance, confirmed in the time-like region, directly on the process of pion scattering by an external potential.

3. Vector Meson Dominance and the Additional Potential So we shall assume that the pion form factor may be described by the dominating contribution of the p° meson (see Fig. 9). In lowest order of the coupling constant the scattering amplitude for Mott scattering with p° dominance is thus obtained to be

j/d)

(p.* — (2tc)3 pPo

p'f) pPa'

>" ' M g 2

g

/p"71

T

I

TTT" ^py-^-'Ì?) > M2

V - V-

(10)

I Fl

605040302010

T"

600

700

—I 800 800

T

9 00

E [MeV]

Fig. 8. Pion form factor as a function of energy

Here A'{q) is the Fourier transform of the electromagnetic potential, fpnn, Fpr = li p 2 /2y p are coupling constants and {—g^ -f- q^M2)j{q2 — Mp2) is the propagator of the p° meson (Mp = 765 MeV). For the Coulomb field considered here one has AM

Ze = in —— 2nd(p0' - p0) gM0.

(11)

95

Structure Effects of H a d r o n s in Hadronic Atoms

Because of current conservation the second term in the p° propagator does not contribute. The calculation leads to the following result for the differential cross section 1

dQ

4 p V sin 4 6/2 P

q2

+

1

M

(12)

2

2

\ir~

R

x W

/

I/ pirnr1 f

«

}U

v

u

IT

'p Fig. 9. Teynman diagram illustrating p dominance

But the first factor is the well-known cross section for a point-like scalar particle, and thus one obtains the form factor of the pion Fn(q»)

= /p^-Tpy

•

q 2

(13)

From the normalization ^ ( 0 ) = 1 one gets /pn 7t

= 1 10)

(14)

and hence the expression M*

Remembering the interpretation of the form factor as the Fourier transform of the charge distribution, the corresponding potential is now easily obtained. From the Poisson equation in momentum space q W

p

( q ) = ^ e ( q ) ,

(q)

=

e

F(q)

(16)

one immediately reads off the potential (see also Eq. (15)) (17)

The Fourier transform of before,

yields, taking into account the factor

Ve(q)

7,P%

pM

F

=

—Ze2

suppressed

F-MGR

~ — + Z e *

—

.

(18>

Therewith the contribution exceeding the Coulomb potential, that is the additional potential determined by the form factor, is found to be -Mpr

e

V'(r)

10

=

+Ze2

.

) This relation can be tested for t h e p7r+7i~ vertex. F r o m t h e decays p -»- 7r+ TZ~ a n d p finds /pTtn/iyp2 = 1.03 ± 0.16 [5S].

(19)

e+e~ one

96

U . E.SCHRODER

In Eq. (10) the p° meson has been treated as a stable particle. This however is not correct and, strictly speaking, one has to include the decay width r p s» 125 MeV in the propagator of the p° meson. The calculation including a decay width F p different from zero leads to the following additional terms [59] V"(r) =

(20)

e-W - Zar*rexp{-Mer).

The contribution of these terms to the energy shift however amount to only a few percent of the main contribution originating from V'{r) (19). Therefore, in the following these small corrections to V will be neglected. 4. Calculation of the Energy Shift With a knowledge of V we can proceed to calculate the energy level shifts due to the pion form factor. It is sufficient to consider V as a perturbation and to use the first-order approximation11) (21).

AEr = fd3r y>*(r) V'(r) v{r).

For the spherical symmetric perturbation V'(r) the energy shift is obtained in firstorder perturbation theory by (22)

AEnJ = jdr • r\Rnil{r)f V'(r) where Rn.i = —

2 \3 (n - I - 1)11/2 naJ 2n[{n -f Z)!]3e x P {

na\ ( n a ) ^ n + t ( m ) '

a

fiZe2

(23)

is the non-relativistic Coulomb wave function12) corresponding to the bound state characterized by the quantum numbers n = 1, 2, . . n — 1 I = 0, 1, . . . The associated Laguerre polynomials are defined as * £„»(*) =

(-1)»

¿n-m n\ ezz~ e~'zu (n r— m) ! dzn~v

(24)

Now the most intense electric dipole transitions are observed between circular orbits n,l — n — 1 [1 0 by a negative contribution. One obtains in first-order approximation (Ze)*

(r~*)nih

(37)

where (j—4)m,/ denotes the mean value of r~4 in the state ipn ¡. In this equation one of the factors depends only on the atomic states »/)„,(> while the other, the structure constant xB1, is characteristic for the hadron. The potential (36) depends quadratically on the external field and therefore varies like r~l. Thus it is a potential of long range compared to the strong interaction. The influence of strong interaction and that of the extended charge distribution of the nucleus decreases rapidly for higher-lying states. This is not so in the case of the long-range potential (36). Consequently, in the states of higher angular momentum the effect of polarizability will dominate that of the strong interaction and that of nuclear charge distribution as well. Hence a separation of these effects will be possible. In order to calculate the energy shift (37) one has to know the polarizability of the hadron 17

) For systems with spherical symmetry which are considered here, the polarizability is a scalar.

102

U . E.SCHRODER

xEl. On the other hand, from a measurement of AEm the structure constant xEl could be determined. We next consider the expression for the polarizability following from nonrelativistic perturbation theory. The interaction of the induced dipole with the nuclear Coulomb field is described by VE1 = - Z e * d - ^ j .

(38)

The definition of the dipole operator

d = jd 3r'e(ry

r'

(39)

contains the actual charge distribution which is, in general, not known explicitly. In the following we shall assume r > r', that is the hadron stays outside the nucleus. For the atomic states with higher quantum numbers n, I, which are of interest here, this condition is fulfilled. Accordingly the nuclear charge can be considered as the source of the external electric field acting on the hadron. The expression for AEE1 then reads in second-order perturbation theory | < W . H 4 I AE B 1 = - e * Z * Z — =r-, an',l' (-&« — & 0 ) + K&n'.l' — tin.l)

(40)

where aH refers to the internal states of the hadron. A further approximation is now introduced by neglecting the atomic excitation energies in the denominator of Eq. (40) which are small compared to the excitation energies of the hadron. This shall be called the static approximation, for because of the relative small atomic frequencies the inner motion of the hadron may adjust to the momentary atomic state. B y using the closure relation for the atomic states ifn',i' one obtains with r \\z (4D This relation is of the same structure as Eq. (37) and from comparison the following expression for the polarizability of the hadron results XE1

~

26

f

E*

- E

•

{42)

A direct evaluation of this sum is however difficult. Later on, in the quark model for mesons we shall circumvent the problem of summing over the intermediate states by use of the variational method. I n another method one expresses the sum of dipole matrix elements in Eq. (42) by an integral over the total photoabsorption cross section for E 1 transitions. For the proton and many nuclei this cross section is known from experiment, and thus xE1 can be determined by this sum rule. Bsfore continuing the discussion of Kfa we next consider the other polarization shifts. 2. Other Multipole Moments a) Polarization of the nucleus First of all, as already mentioned, the energy shift (41) due to the polarizability of the hadron has to be completed by the effect of nuclear polarizability. The total interaction of the electric dipoles with the corresponding Coulomb fields is then described by the

Structure Effects of Hadrons in H'adronic Atoms

,103

potential '

VE1 = —e2(Z) - Zd) • - L

'

(43)

where D denotes the dipole operator for the nucleus. As before from this one obtains in the static approximation18) the following expression for the total polarizability B1~

£

Ef* - E0" + E*

- E0*

+

144)

where Xg1 is the polarizability of the nucleus p

~ ^«

In order to calculate x^ it is suitable to use the connection to the total y absorption cross section for E1 radiation in the long wave-length limit (see e.g. Ref. [7S\) alm\E) = 4jPE^e* e

Dt |^N)\2 1) smaller values than in the dipole case. Further the energy shifts in Eq. (60) become small for higher values of X because of their peculiar dependence on r . This is easily verified for the states with high quantum numbers I considered here (X < I ) , where the singularity at r = 0 does not matter. We shall discuss this later in section V. 3. First Information on xEl An order of magnitude estimate of the polarizability can be obtained by writing Eq. (42) approximately V - l , "

-

3

6

E

'

(

' '

While the smallest possible mean excitation energy E is determined by the pion mass, the matrix element (0Hl d? |, )

(73) '

where the trial function 4> = (I AW) 4>a contains the unperturbed ground state 4>0, the perturbation W, and the variational parameter A. Because of the spherical symmetry of {r) the following expectation values in Eq. (73) vanish: W = W+ = II W = WH — W+WW = 0 25 ). On thus obtains the considerable simpler expression 0

0

E E(X)

0

+

2XWW

+

PWH0W

'

= 1

+

+

0

(74)

A2WW

Since we are interested in perturbative contributions proportional to F2 only, and W itself contains F linearly, we may expand the denominator, obtaining26) E (A)

~

E o +

2AWW

+

A2[W,

H0]

W .

(75)

Looking for the minimum value of E(A), the polarizability now readily can be determined by comparison with Eq. (36). This calculation is simplified by the assumption F\\z and yields / ?r