# P(0)2 Euclidean (Quantum) Field Theory

Barry Simon
Pages: 414
https://www.jstor.org/stable/j.ctt13x16st

1. Front Matter
(pp. i-vi)
2. PREFACE
(pp. vii-viii)
BARRY SIMON
(pp. ix-xii)
4. INTRODUCTION
(pp. xiii-2)

These lecture notes are intended to introduce the reader to Euclidean ideas in quantum field theory and then to develop one approach, the “correlation inequality” method, to the simplest model of an interacting quantum field theory, the P($\emptyset$)2model of a self-coupled Bose field in two dimensional space-time. We have tried hard to make them accessible to non-trivial subsets of both the mathematics community and the physics community. We have emphasized the probabilistic Euclidean strategy toward P($\emptyset$)2over the Hamiltonian strategy, which in the hands of Glimm and Jaffe dominated the period from 1964 to 1971 and which has played...

5. CHAPTER I GAUSSIAN RANDOM PROCESSES, Q-SPACE AND FOCK SPACE
(pp. 3-45)

If one considers the free relativistic quantum field, one is led quite naturally to an attempt at simultaneously diagonalizing the time zero fields which form a family of (unbounded) commuting observables. While this idea is implicit in some of the earliest work in quantum field theory (see the historical note in [71]), it was not until the 1950’s that Friedrichs [50] and Segal [158] faced up to the measure theoretic complexities involved in a careful treatment of this diagonalization. In a series of papers [159-162], Segal developed the theory from this point of view and especially emphasized the connection with...

6. CHAPTER II AXIOMS, I
(pp. 46-80)

In this chapter, we will discuss three axiom schemes for relativistic scalar quantum fields: the Garding-Wightman axioms for fields [207], their translation to vacuum expectation values [203] and the Osterwalder-Schrader axioms for Euclidean region “Green’s functions” (or, as we shall call them, Schwinger functions) [143], Unfortunately, the situation for the Osterwalder-Schrader axioms has been complicated by an error in their original paper [143]. The equivalence of the first two and some of the resulting theory has worked its way into the monographs of Jost [110] and Streater-Wightman [189] and for this reason, we intend to be especially sketchy on those...

7. CHAPTER III THE FREE EUCLIDEAN FIELD
(pp. 81-105)

In this chapter, we will consider the field theory associated to the free field Schwinger functions (11.43/45). The possibility of finding a “field theory” whose expectation values are the Schwinger functions is not an automatic consequence of the axiom schemes of the last chapter. We will return to what properties of general Wightman fields imply the existence of Euclidean field theories in the next chapter. Here we will rely on a positivity property first emphasized by Symanzik [192, 193, 194] who used it to develop the free Euclidean field. As we will see in Section IV.3, this positivity property is...

8. CHAPTER IV AXIOMS, II
(pp. 106-130)

The considerations of the last chapter suggest that first it should be useful to consider quantum field theories associated with Euclidean region fields rather than just with Euclidean Green’s functions and secondly that we might expect a connection between such theories and the theory of Markov processes. In fact, our development of the connection between Euclidean-Markov field theories and Wightman field theories with additional properties closely parallels the discussion of the connection between Markov processes and positivity preserving semigroups [32]. In Sections IV. 1,2 we present Nelson’s germinal work [132, 134] on constructing Wightman fields from Euclidean field theories and...

9. CHAPTER V INTERACTIONS AND TRANSFER MATRICES
(pp. 131-178)

Thus far we have described general frameworks and trivial models. We now begin the serious business of constructing nontrivial models. We will try to construct models by local perturbations of the free field model — following thereby the time-honored practice of Lagrangian field theory [10]. It is at this point that the famous infinities of quantum field theory enter. There is a natural hierarchy of formal models going under codenames: P($\emptyset$)2, Y2, (${\emptyset ^4}$)3, Y3, (${\emptyset ^4}$)4· Each successive model has more infinities than the one before and it was Wightman [204] who proposed looking at the models successively — trying to understand each...

10. CHAPTER VI NELSON’S SYMMETRY AND ITS APPLICATION
(pp. 179-212)

The FKN formula tells us that the Euclidean field is a path integral over the time zero Minkowski field. This path integral is manifestly Euclidean convariant, invariance under rotations by π/2 yields:

THEOREM VI.1 (Nelson’s Symmetry).

$\langle {\Omega _o},{e^{ - t{H_\ell }}}{\Omega _o}\rangle = \langle {\Omega _o},{e^{ - \ell {H_t}}}{\Omega _o}\rangle$. (VI.1)

Proof.By the FKN formula, we need only prove that:

$\int {d{\mu _o}} \exp \left( { - \int_o^t {ds\int_{ - \ell /2}^{\ell /2} {dx:P\left( {\emptyset \left( {x,s} \right)} \right):} } } \right) = \int {d{\mu _o}\exp \left( { - \int_o^\ell {ds\int_{ - t/2}^{t/2} {dx:P\left( {\emptyset \left( {x,s} \right)} \right):} } } \right)}$

and this follows from the invariance of μ0under Euclidean motions and the covariance of$\emptyset$. ■

At first sight (VI.1) is striking looking although once one understands that it is an expression of Lorentz invariance, it is not quite so mysterious. What is perhaps more surprising is how powerful it...

11. CHAPTER VII DIRICHLET BOUNDARY CONDITIONS
(pp. 213-254)

In the theory of statistical mechanical systems such as the Ising model, the clever use of one or more kinds of boundary conditions plays a major role [80,154, 47], It is therefore not surprizing that a similar situation occurs in the P($\emptyset$)2model, especially when statistical mechanical methods are employed. In this chapter we discuss in detail one type of boundary condition, the Dirichlet boundary conditions. An analysis of more general kinds of boundary conditions is possible (see [90] for the one-dimensional case, and [91] for P($\emptyset$)2) and it is our expectation that other kinds of boundary conditions will eventually...

12. CHAPTER VIII THE LATTICE APPROXIMATION AND ITS CONSEQUENCES
(pp. 255-314)

We now turn towards controlling the infinite volume limit of the cutoff Schwinger functions. Since there is an analogy between statistical mechanics and Euclidean field theory, it is natural to ask what methods are available for controlling the infinite volume limit of the correlation functions there. There are three [154]:

(a)The transfer matrix method.This is restricted to one-dimensional systems or at least to going to infinity in only one direction. We have already seen in Section V. 4 how to extend the method to P($\emptyset$)2-models; in particular, this method completely solves the problems for one-dimensional theories (anharmonic oscillators)....

13. IX. THE CLASSICAL ISING APPROXIMATION AND ITS APPLICATIONS
(pp. 315-358)

There is a large amount of elegant machinery which has been developed in the statistical mechanical theory of Ising ferromagnets. One set of results holds with general kinds of (even) single spin distributions, i.e., for what we have called general ferromagnets. These results also generally extend to systems with suitable (ferromagnetic) many body interactions. While we have not used these many body interactions in our field theory analogies, we will see that the possibility of many body interactions is not unrelated to the possibility of arbitrary single spin distributions. This first set of results includes GKS and FKG inequalities; the...

14. CHAPTER X ADDITIONAL RESULTS AND TECHNIQUES: A BRIEF INTRODUCTION
(pp. 359-378)

Thus far we have presented the general Euclidean philosophy towards the P($\emptyset$)2model and described in detail the correlation inequality — Lee-Yang approach towards the control of the infinite volume theory. In this final chapter, we want to say something about two other approaches of controlling the infinite volume limit (Sections 1,2, Section 5) and about some generalities concerning boundary conditions and the definition of equilibrium state (Sections 3,4). We will make no attempt at a comprehensive review but intend mainly to emphasize to the reader the many additional and powerful ideas which complement the methods of Chapters VIII and...

15. REFERENCES
(pp. 379-392)
16. Back Matter
(pp. 393-393)