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Lesmethodes Mathematiques De La Theorie Quantique Des Champs LES METHODES MATHEMATIQUES DE LA THEORIE QUANTIQUE DES CHAMPS COLLOQUES INTERNATIONAUX DU CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE N” 248 LESMETHODES MATHEMATIQUES DE LA THEORIE QUANTIQUE DES CHAMPS MARSEILLE 23-27 JUIN 1975 EDITIONS DU CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE 15, QUAI ANATOLE-FRANCE — 75700 PARIS 1976 Actes du Colloque international n° 248 sur (LES Mé'rH'omas MATHéMAnQUEs DE LA 'I‘HJ’EORIE QUANTIQU'E DES CHAMPS >, organisé dans le cadre des colloques internationaux du Centre National de la Recherche Scientifique, 2‘1 Marseille du 23 au 27 juin 1975, par Monsieur F. GUERRA, Professeur é l’Université de Salerne, Professeur Associé 2‘1 l’Université d’Aix-Mar- seille II - Luminy, Monsieur D.W. ROBINSON; Professeur 2‘1 l’Université d’Aix-Marseille II - Luminy, et Monsieur R. STORA, Directeur de recherche au C.N.R.S. © Centre National de la Recherche Scientifique, Paris, 1976. ISBN 2-222-01925-7 Le Coiloque International sur " 1es Méthodes Mathématiques de 1a Théorie Quantique des Champs" a réuni a Marsei11e 120 participants du 23 au 27 Juin 1975. Ce Colloque a été nntivé par 1a symbioSe croissante entre la Mécanique Statistique et 1e Théorie Quantique des Champs. La plupart des communications ont trait 3 1a description de résuitats pubiiés ici pour la premiere fois. Nous aimerions remercier 1es orateurs et tous 1es participants de la Conférence pour i'atnnsphére scientifique extrémement stimuiante qu'ils ont créée pendant leur séjour a Marseille. Nous remercions également 1e Centre National de la Recherche Scientifique et 1'UER Scientifique de Luminy pour 1eur géné- reuse contribution financiére. Enfin nous aimerions remercier Mesdames S. Guidice11i et N. Jean pour 1eur dévouement et 1a compétence avec laquelie e11es ont organisé ce Colloque. )w‘1’n(\ W W R. Stora a Robinson F. Guerra 0.“. SOMMAIRE S. Albeverio and R. Hoegh—Krohn pages Quasi invariant Measures, Symmetric Diffusion Processes and 11 Quantum Fields. H. Araki Relative Entropy and its Application 61 H.J. Borchers 0n the Equivalence Problem between Wightman and Schwinger 81 Functionals. J.S. Feldman and K. Osterwalder 4 101 The Construction of A43 Quantum Field Models. J. Frfilich Poetic Phenomena in (Two Dimensional) Quantum Field 111 Theory : Non Uniqueness of Vacuum, the Solitons and All that. L. Gross Euclidean Fermion Fields 131 J. Glinm and A. Jaffe Bounds in P(¢ )2 Quantum Field Models. 147 J. Slim and A. Jaffe Critical Problems in Quantum Fields 157 J. Glinln, A. Jaffe and Th. Spencer 4 Existence of Phase Transitions for: W2 Quantum Fields 175 F. Guerra Exponential Bounds in Lattice Field Theory 185 G. Jona_Lasinio Critical Behaviour in Terms of Probabilistic Concepts 207 A. Klein and L. Landau The : 2 : Field : Infinite Volume Limit and Bounds on the Physical Mass 223 O.A. Mc Bryan Convergence of the Vacuum Energy Density, Bounds and Existence of Hightman Functions for the Yakawa2 Model 237 A. Neveu Semi Classical Quantization Methods in Field Theory 253 C.M. Newman Classifying General Ising Models 273 R. Raczka Present Status of Canonical Quantum Field Theory 289 J.E. Roberts Perturbations of Dynamics and Group Cohomology 313 L. Rosen Statistical Mechanics Methods in Quantum Field Theory : Classical Boundary Conditions 325 R. Schra‘der A Possible Constructive Approach to ¢2 347 R. Seiler Massive Thirring Model and Sine-Gordon Theory 363 Th. Spencer The bathe Salpeter Kernel in P(“f)2 375 Colloques Internationaux C.N.R.S. N° 248 — Les méthodes mathématiques de la théorie quantique des champs Quasi Invariant Measures, Symmetric Diffusion Processes and Quantum Fields * by Sergio Albeverio and Raphael H¢egh—Krohn Institute of Mathematics University of Oslo Blindern, Oslo (Norway) RESUME 0n dérive plusieurs propriétés des mesures de probabilité quasi—invariantes, en particulier on étudie Ies propriétés de fermabilité, ergodicité, et des perturbations des systémes associés. Ces résultats sont appliqués aux champs quantiques at on montre que pour les interactions polynomes a deux dimensions, le vide physique restreint aux champ initial est une mesure de la classe supérieure. ABSTRACT We show that for a large class of quasi invariant probability measures on a separable Hilbert space with a nuclear rigging the Dirichlet form V?Ivg dy in L2(dp) is closable and its closure defines a positive self-adjoint operator H in L (d;0, with zero as an eigenvalue to the eigenfunction 1 . The connectiog with the hamiltonian formalism and canonical commutation relations is also studied. We show moreover that H is the infinitesimal generator of a symmetric time homogeneous Markov process on the rigged Hilbert space, with invariant measure [A . For strictly positive )1 this process is ergodic if and only if [A is ergodic, which is the case if and only if zero is a simple eigenvalue of H . Moreover we study perturbations of H and )4 as well as weak limits of quasi invariant measures and their associates Markov processes. Finally we apply our results to quantum fiels. In particular we show that for polynomial interactions in two space-time dimensions the physical vacuum restricted to the 6 -algebra generated by the time zero fields is a measure )A in the above class of quasi invariant measures and the physical Hamiltonian coincides on a dense domain of L (H) with the generator of the Markov process given by the Dirichlet form determined by )4 . ’ Work supported by the Norwegian Research Council for Science and the Humanities. -11— 1. Introduction Within the general theory of Markov stochastic processes with continuous time parameter and finite dimensional state space the class of diffusion processes is of special importance due to its connection with second order partial differential equations. Since moreover every such Markov process is the solution of a stochastic differential equation, one has a beautiful interplay of the theory of partial differential equa— tions, diffusion processes and stochastic differential equations. For this we refer to [1], [21, [3]. This paper introduces our study of the extension of these subjects, and in particular of the theory of Markov diffusion processes, to the infinite dimen— sional case. For a more detailed account and further results see [4]. We first mention shortly some previous work. A whole direc— tion of early studies, mainly connected with the names Friedrichs, Gelfand and Segal, arose in connection with problems of quantum fields and in particular of the representations of canonical commutation relations, see e.g. [5]. A related stimulating influence came from Feynman's path integral formulation of quan— tum dynamics, see the references in [6]. Some studies dealing with differential and stochastic differential equations in in- finite dimensional spaces are in [71—[11] and references therein. Results from constructive field theory which are most related to our subject will be mentioned below. Let us now summarize the content of our paper. In section 2 we start by assembling some facts about Gelfand's representation of Weyl's canonical commutation relations by means of probability measures on N' , quasi invariant with respect to translations by elements in N , where NcKcN'. is a real separ- able Hilbert space with a nuclear rigging. References to previous work on this representation are [5] and [12]. We then isolate a class of quasi invariant measures, which we call measures with first order regular derivatives and which in the finite dimensi— onal case correspond to the density functions having L2 deri- vatives. This class is suitable for the construction of the self- _12_ adjoint positive operator H associated with the Dirichlet form Ivf-vg do and acting in the representation space L2(du) for the canonical commutation relations. The relation of Dirichlet forms * with the canonical forma- lism has been discussed, modulo some domain questions, by Araki, in his algebraic approach to the Hamiltonian fOrmalism and canoni- cal commutation relations [13]. Some of our results in this sec— tion can be looked upon as providing analytic versions of alge— braic derivations of Araki. The self-adjoint operator H mentioned above is the Friedrichs operator given by the closure of the Dirichlet form, first defined on a dense set F2 of finitely based 02 functions. H is non negative and has the eigenvalue zero with the eigenfunction identically equal to 1 in L2(du). e-tH is a Markov semigroup so that H is the infinitesimal generator of a time homogeneous Markov process on N' with invariant measure u . Accordingly we call H the diffusion operator associated with u . A (possibly strict) self adjoint extension of H is the Operator fi = v*v with domain equal to the domain of the closed gradient operator v in L2(du) . We have H = M on F2 . e-tH is also a Markov semigroup and in this case we have both time ergodic and N—ergodic decompositions of u, L2(du), fl and the representation of canonical commutation relations,which coincide for a class of measures u containing the so called strictly positive measures. For such measures zero is a simple eigen— value of fi if and only if u is ergodic, which in turn is equivalent with the representation of the canonical commutation relations given by u being irreducible. In section 3 we study perturbations of quasi invariant measures a with regular first order derivatives and of the associated diffusion operators. we also find sufficient con— ditions for the stability under weak limits of the correspondence between quasi invariant measures with regular first order deri— *Dirichlet forms have also been considered in related contexts in [31] and [32]. -13_ vatives and the associated diffusion processes. In section 4 we apply the general results of the preceding sectionsto the case of quantum fields. Stochastic methods in constructive quantum field theory are of course not new.
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