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DeWitt, C. (ed.); Stora, R. (ed.)
[Glimm, J.; Jaffe, A.; Lanford, O. E. III; Ruelle, D.; Griffiths, R. B.; Lieb, Elliott H.; Ginibre, J.; Hepp, K.; Epstein, H.; Glaser, V.]

Statistical mechanics and quantum field theory. Mécanique statistique et théorie quantique des champs. Université de Grenoble-Summer School of Theoretical Physics Les Houches 1970. (English) Zbl 0242.46002

New York-London-Paris: Gordon and Breach Science Publishers. xvi, 552 p. (1971).
This volume is devoted to mathematical studies in the theory of quantum systems with an “infinite number of degrees of freedom”. The subject is obviously too broad and cannot be exhausted, even during an eight week session. This book does not duplicate the proceedings of two other summer schools, whose programs covered the same area; Varenna in 1968, Cargese in 1969. [R. Jost (ed.), Rendiconti della scuola internazionale di fisica “Enrico Fermi” – Local quantum theory (1969); D. Kastler (ed.), Cargese lectures in physics. Vol. 4 (New York 1970)].
“The lecture notes are presented in an order very close to the order of the courses. Constructive , expounded by J. Glimm and A. Jaffe, provides at present the best support in favour of the systems of axioms proposed by A. S. Wightman, H. Araki, R. Haag, D. Kastler, H. Lehmann, K. Symanzik, W. Zimmermann, showing that they are likely to be realized in a non-trivial manner. Some of the problems under consideration, in particular that of the infinite volume limit, are similar to problems occurring in . The course of D. Ruelle on the foundations of equilibrium statistica mechanics, brings forth these analogies. However, whereas the mathematics used in quantum field theory consists mostly of the of and has been treated by Glimm and Jaffe themselves, statistical mechanics primarily requires the use of functional analysis of bounded operators, and this has been treated by O. E. Lanford.
These introductory courses have been followed by those of K. Hepp and H. Epstein on renormalization theory, of R. Griffiths on phase transitions, of E. Lieb on soluble models in statistical mechanics, and of J. Ginibre on applications of functional integration in statistical mechanics, thus treating some of the most interesting current problems.
Shortage of time has prevented us from treating such important topics as axiomatic field theory, non-relativistic scattering, ergodic systems, transport phenomena, approach to equilibrium to mention only a few examples. This book is intended primarily for physicists working in equilibrium statistical mechanics or quantum field theory, and for those wishing to size up the extent to which current conjectures are being confirmed.” So runs the editors’ preface.
As the preface says this tome contains the lecture notes from a summer school. The Les Houches schools used to instruct young tyros in the field they covered, ’synthesizing present knowledge of rapidly developing areas of physics and helping the student to enter the world of research’. A look at the list of ’students’ at this one reveals this wasn’t so here; it was a gathering largely of experts or of mathematicians of standing. The editors state the book is intended primarily for physicists, but it should be of interest to mathematicians concerned with functional analysis among other things. F. J. Dyson [Bull. Am. Math. Soc. 78, 635–652 (1972; Zbl 0271.01005)] has remarked the present divorce between mathematics and physics. The subject of this book is one where they are much closer together again and mathematicians recently entering the field have conspicuously contributed to its successes. The terminology from physics may be unfamiliar to many but at least in these accounts much of it is defined.
To start with the pure mathematics course. O. E. Lanford (Selected topics in functional analysis – 106 pp.) expounded on algebras of bounded operators with a digression on Choquet theory. These are tools most important for the ‘local algebra’ formulations of quantum field theory, after Haag and Kastler, and particularly of statistical mechanics. Lanford’s notes provide a rather slick introduction to what is required, with neat proofs and informative prose. Contents: Preliminaries; Integral representations on compact convex sets – existence and uniqueness; \(C^*\)-algebras – spectrum and resolvent, commutative \(C^*\)-algebras, the , positive and the Gel’fand-Segal construction, and irreducible representations; von Neumann algebras – von Neumann and Kaplansky density theorems, projections, their comparison and disjointness, tensor product decompositions, classification of factors, dimension theory of \(\text{II}_1\) factors.


J. Glimm and A. Jaffe (Quantum field theory models – 108 pp.) to build rigorous models of relativistically invariant quantum field theories require more of unbounded operators. They have developed much of the necessary theory relating to these themselves. This is not the place to attempt a potted outline of their programme and reference to some of their main articles [Phys. Rev., II. Ser. 176 (1968), 1945–1951 (1969; Zbl 0177.28203); Ann. Math. (2) 91, 362–401 (1970; Zbl 0191.27005); Acta Math. 125, 203–267 (1970); Commun. Pure Appl. Math. 22, 401–414 (1969; Zbl 0167.42804)], to a review by J. Glimm [Adv. Math. 3, 101–125 (1969)] and to their talks at Varenna must suffice. Glimm and Jaffe have now given three courses on constructive quantum field theory – Varenna 1968, Les Houches 1970, London 1971 [Boson quantum field models, Proc. Lond. Math. Soc., Instructional Conf. on the Mathematics of contemporary Physics (1973)]. This one supersedes that in Varenna for great advances had been made and the mathematical techniques had been refined. It is however more condensed. That in London shows a different emphasis not that of the original series of paper. The complicated nature of the subject has meant many sketch proofs but the course is surprisingly self contained. It does of course include the improvements made on the first proofs of several points. Contents: The \(\varphi^{2n}_2\) model – , annihilation-creation forms; Q-space (Fock space as \(L^2\) on a underlying a Gaussian stochastic process), \(:\varphi^n:\) as a multiplication operator, \(H_0\) as a Hermite operator; the Hamiltonian \(H(g)\), positivity, essential self-adjointness domains, spectrum; removing the space cut-off; Lorentz covariance and the Haag-Kastler axioms. The Yukawa\(_2\) model – the \(Y_2\) Hamiltonian, limits of ; positivity for \(Y_2\) Hamiltonian; resolvent convergence, graph convergence and self-adjoint-ness; the Heisenberg picture finite propagation speed, the field operators.


The foundations of statistical mechanics have recently been developed by people strongly influenced by the ‘local algebra’ formulation of quantum field theory. It is perhaps not surprising that some of the problems of the two fields are similar. D. Ruelle’s course (Equilibrium statistical mechanics of infinite systems – 26 pp.) is a review of equilibrium statistical mechanics and it should be, unifying classical and quantum, lattice and continuum cases. Since this happy state has in fact not yet been attained, there appear quasi-theorems, that is results that should hold in the unifying framework hoped for; they may also be interpreted as proven mathematical fact in some cases. They are certainly suggestive and there are questions and exercises, too. The lectures presented were really rather short but they do supplement D. Ruelle’s book [Statistical mechanics. Rigorous results (1969; this Zbl 0177.57301)]. Contents: Finite subsystems, quasilocal structure, , Gibbs states, simplex structure of global equilibrium states, equilibrium conditions, KMS condition, role of symmetries, entropy of a translation invariant state, variational principle for invariant equilibrium states, Gibbs’ phase rule, symmetry breakdown; Appendix – states as probability measures, states on \(C^*\)-algebras, locally normal states, equilibrium conditions for continuous systems, ground states.


R. B. Griffiths (Phase transitions – 40 pp.) sets out more of the phenomena that statistical mechanics would like to explain and is more discursive in his approach. Though he is a lucid expositor this article and the following are probably the least accessible to people unfamiliar to the jargon. Contents: Phase transitions in nature and in physical models; Ising model; Peierls’ argument for nearest neighbour ferromagnets; for other systems; Griffiths-Kelly-Sherman inequalities; one-dimensional systems – Dysons’s argument: Mermin-Wagner arguments; problems.


E. Lieb (Models in Statistical Mechanics – 46 pp.) goes over nine models, and discusses very clearly the solutions known and some of their properties. The exposition overlaps with that of his Boulder 1969 lectures. There is of course a great deal of mathematical activity in the solution of statistical mechanical models and there are notable outstanding problems. Contents: Monomer-dimer problem, Takahashi model, Coulomb problem in one dimension, transfer matrix formalism, the dimer problem, two-dimensional Ising problem, ferroelectric problems.


J. Ginibre’s lectures (Some applications of functional integration in statistical mechanics – 102 pp.) treat methods introduced into quantum mechanics by R. P. Feynman. However there is notorious difficulty in rigorously defining the integrals written down by Feynman for interactions more complicated than the physically essentially trivial quadratic. Indeed J. F. D y s o n (loc. cit.) proposes the mathematical well founding of the Feynman history integral method as one of his two main challenges. Ginibre undertakes statistical mechanics with the Feynman approach and here the method reduces to use of the well-known Wiener integral. In thorough notes he develops the mathematics and applies it, ending with the polaron model which is a sort of simplified quantum field theory. Contents: and the Feynman-Kac formula; thermodynamic functions and reduced ; infinite volume limit and fugacity expansions; Kirkwood-Salzburg equations, algebraic and cluster properties of reduced density matrices; integration of Gaussian functionals and applications.


K. Hepp (Renormalization theory – 72 pp.) undertook to give a unified presentation of the renormalization of Feynman amplitudes. His lectures contain a unifying review of the rigorous results in mathematical style setting out the conditions but often proving by reference. The article will probably prove to be a standard reference itself. But the subject is not one of wide interest; perhaps it should be for a lot of mathematics has gone into it. Contents: Hamiltonian formalism; axiomatic quantum field theory; the axioms of renormalization; analytic, additive and P-space renormalizations; ; renormalization of the Hamiltonian.


H. Epstein and V. Glaser (Le rôle de la localité dans la renormalisation perturbative en théorie quantique des champs - 36 pp.) summarize an article of theirs which is to appear. Contents: Géneralités sur le decoupage des distributions, “comptage des puissances”, le point de vue de la soustraction des divergences par des contre termes infinis, equivalence avec le formalisme habituelle, problèmes de la limite adiabatique.
The ties between the quantum field theory and statistical mechanics suggested in this book have recently become stronger yet, reinforced by work of E. Nelson [Construction of quantum fields from Markoff fields, J. Funct. Anal. 12, 97–112 (1973; Zbl 0252.60053)] and much further progress in the programme of constructive field theory has been made.
The book is well printed (in East Germany) but expensive. There are quite a few misprints, ranging up to ’Haid Square Laltice Gap’ on p. 257, but most of these errors I think are faithfully copied from the preprints. This volume is likely to be viewed as a specialist reference work and won’t receive the attention it deserves from mathematicians. That’s a pity.
Reviewer: Patrick D. F. Ion
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MSC:

81-06 Proceedings, conferences, collections, etc. pertaining to quantum theory
82-06 Proceedings, conferences, collections, etc. pertaining to statistical mechanics
00B25 Proceedings of conferences of miscellaneous specific interest
46-06 Proceedings, conferences, collections, etc. pertaining to functional analysis
47-06 Proceedings, conferences, collections, etc. pertaining to operator theory

Citations:

Zbl 0271.01005; Zbl 0177.28203; Zbl 0191.27005; Zbl 0167.42804; Zbl 0177.57301; Zbl 0252.60053
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