On the decomposition of wightman functionals in the euclidean framework

We consider Wightman functionals having the property that the Schwinger functions can be represented by a Euclidean invariant measure on the space of tempered distributions. The strong form of the Osterwalder-Schrader positivity condition is shown to imply that the measure is positive and some restrictions on the Schwinger functions are discussed which guarantee that this condition holds. The ergodic decomposition of the measure leads to a decomposition of the Wightman functional into such with cluster property. We discuss also the role of the positivity condition in connection with a general criterion for the existence of a decomposition.     

From the Euclidean group to the Poincaré group via Osterwalder-Schrader positivity

Given a continuous representation of the Euclidean group inn+1 dimensions, together with a covariant system of subspaces, which satisfies Osterwalder-Schrader positivity, we construct a continuous unitary representation of the orthochronous Poincaré group inn+1 dimensions satisfying the spectral condition. A similar result holds for the covering groups of the Euclidean and Poincaré group.     

On the connetion between Euclidean and Hamiltonian lattice field theories of vortices and anyons

In the (2+1)-dimensional non-compact Abelian lattice Higgs model Euclidean correlation functions of vortices and, after adding the Chern-Simons term to the action, correlation functions of transmuted matter fields (anyons) are set up. These correlation functions satisfy Osterwalder-Schrader positivity. Via the transition to continuous Euclidean time, vortex and anyon operators within a Hamiltonian lattice formulation are obtained, respectively, and their respective dual algebras are displayed.     

The FKG inequality for the Yukawa2 quantum field theory

We establish the FKG correlation inequality for the Euclidean scalar Yukawa₂ quantum field model and, when the Fermi mass is zero, for pseudoscalar Yukawa₂. To do so we approximate the quantum field model by a lattice spin system and show that the FKG inequality for this system follows from a positivity condition on the fundamental solution of the Euclidean Dirac equation with external field. We prove this positivity condition by applying the Vekua-Bers theory of generalized analytic functions.Research partially supported by the National Research Council of Canada.Alfred P. Sloan Foundation Fellow.     

Random fields on Riemannian manifolds: A constructive approach

We extend to Euclidean fields on a wide class of Riemannian manifolds two results which have proven to be crucial in the construction of interacting quantum fields in the flat case, namely local regularity properties of the free covariance in two dimensions and Osterwalder-Schrader positivity.Work supported in part by Ministero della Pubblica Istruzione     

Quantum Field Theory on Curved Backgrounds. I. The Euclidean Functional Integral

We give a mathematical construction of Euclidean quantum field theory on certain curved backgrounds. We focus on generalizing Osterwalder Schrader quantization, as these methods have proved useful to establish estimates for interacting fields on flat space-times. In this picture, a static Killing vector generates translations in Euclidean time, and the role of physical positivity is played by positivity under reflection of Euclidean time. We discuss the quantization of flows which correspond to classical space-time symmetries, and give a general set of conditions which imply that broad classes of operators in the classical picture give rise to well-defined operators on the quantum-field Hilbert space. In particular, Killing fields on spatial sections give rise to unitary groups on the quantum-field Hilbert space, and corresponding densely-defined self-adjoint generators. We construct the Schrödinger representation using a method which involves localizing certain integrals over the full manifold to integrals over a codimension-one submanifold. This method is called sharp-time localization, and implies reflection positivity.     

Long Range Order for Lattice Dipoles

We consider a system of classical Heisenberg spins on a cubic lattice in dimensions three or more, interacting via the dipole-dipole interaction. We prove that at low enough temperature the system displays orientational long range order, as expected by spin wave theory. The proof is based on reflection positivity methods. In particular, we demonstrate a previously unproven conjecture on the dispersion relation of the spin waves, first proposed by Fröhlich and Spencer, which allows one to apply infrared bounds for estimating the long distance behavior of the spin-spin correlation functions.     

Stability of Covariant Relativistic Quantum Theory

In this paper we study the relativistic quantum-mechanical interpretation of the solution of the inhomogeneous Euclidean Bethe-Salpeter equation. Our goal is to determine conditions on the input to the Euclidean Bethe-Salpeter equation so the solution can be used to construct a model Hilbert space and a dynamical unitary representation of the Poincaré group. We prove three theorems that relate the stability of this construction to properties of the kernel and driving term of the Bethe-Salpeter equation. The most interesting result is that the positivity of the Hilbert space norm in the non-interacting theory is not stable with respect to Euclidean covariant perturbations defined by Bethe-Salpeter kernels. The long-term goal of this work is to understand which model Euclidean Green functions preserve the underlying relativistic quantum theory of the original field theory. Understanding the constraints imposed on the Green functions by the existence of an underlying relativistic quantum theory is an important consideration for formulating field-theory motivated relativistic quantum models.This work supported in part by the U.S. Department of Energy, under contract DE-FG02-86ER40286     

Nelson's symmetry and all that in the Yukawa2 and (φ4)3 field theories

We prove Guerra's theorem, φ bounds and Fröhlich bounds in the Y₂ and φ₃⁴ field theories. Among our technical results of interest is a proof that Z ≠ 0 in φ₃⁴ and that the spatially cutoff vacuum in Y₂ has a charge zero component. The two main inputs are Osterwalder-Schrader positivity in the spatial direction as well as in the time direction, and a finite renormalization of the “usual” partition function and Hamiltonian so that Euclidean and Hamiltonian counterterms match exactly.     

Structure of the Space of Ground States in Systems with Non-Amenable Symmetries

We investigate classical spin systems in d ≥  1 dimensions whose transfer operator commutes with the action of a nonamenable unitary representation of a symmetry group, here SO(1,N); these systems may alternatively be interpreted as systems of interacting quantum mechanical particles moving on hyperbolic spaces. In sharp contrast to the analogous situation with a compact symmetry group the following results are found and proven: (i) Spontaneous symmetry breaking already takes place for finite spatial volume/finitely many particles and even in dimensions d = 1,2. The tuning of a coupling/temperature parameter cannot prevent the symmetry breaking. (ii) The systems have infinitely many non-invariant and non-normalizable generalized ground states. (iii) The linear space spanned by these ground states carries a distinguished unitary representation of SO(1, N), the limit of the spherical principal series. (iv) The properties (i)–(iii) hold universally, irrespective of the details of the interaction. Membre du CNRS     

Path space ideas and minimal unitary dilations

Two special features of Minimal Unitary Dilations for kernels on general groups are examined: the existence of a reflection operator and the semigroup property for kernels. The somewhat intermediate property of reflection positivity is analyzed. The relation of these notions to path spaces, Markov path spaces and Osterwalder-Schrader path spaces is determined.     

Time-ordered products and Schwinger functions

It is shown that every system of time-ordered products for a local field theory determines a related system of Schwinger functions possessing an extended form of Osterwalder-Schrader positivity and that the converse is true provided certain growth conditions are satisfied. This is applied to the ₃ ⁴ theory and it is shown that the time-ordered functions andS-matrix elements admit the standard perturbation series as asymptotic expansions.     

Equivalence of Euclidean and Wightman field theories

A new inversion formula for the Laplace transformation of tempered distributions with supports in the closed positive semiaxis is obtained. The inverse Laplace transform of a tempered distribution is defined by means of a limit of a special distribution constructed from this distribution. The weak spectral condition on the Euclidean Green's functions implies that some of the limits needed for the inversion formula exist for any Euclidean Green's function with an even number of variables. We then prove that the initial Osterwalder-Schrader axioms [1] and the weak spectral condition are equivalent with the Wightman axioms.The research described in this publication was made possible in part by Grant No. 93-011-147 from the Russian Foundation for Basic Research     

Euclidean supersymmetry and the many-instanton problem

There is no Hermitean supersymmetry in Euclidean four-space. The simplest supersymmetry has complex four-component spinorial parameters. We give its algebraic structure and the automorphisms of the algebra, as well as a representation in terms of fields and an invariant Lagrangian. The results are relevant to the counting and the construction of the solutions of the many-instanton problem.     

Soliton quantization in lattice field theories

Quantization of solitons in terms of Euclidean region functional integrals is developed, and Osterwalder-Schrader reconstruction is extended to theories with topological solitons. The quantization method is applied to several lattice field theories with solitons, and the particle structure in the soliton sectors of such theories is analyzed. A construction of magnetic monopoles in the four-dimensional, compactU(1)-model, in the QED phase, is indicated as well.     

A new proof of the existence and nontriviality of the continuum ϕ 2 4 and ϕ 3 4 quantum field theories

We use Schwinger-Dyson equations combined with rigorous “perturbation-theoretic” correlation inequalities to give a new and extremely simple proof of the existence and nontriviality of the weakly-coupled continuum ? ₂ ⁴ and ? ₃ ⁴ quantum field theories, constructed as subsequence limits of lattice theories. We prove an asymptotic expansion to order λ or λ² for the correlation functions and for the mass gap. All Osterwalder-Schrader axioms are satisfied except perhaps Euclidean (rotation) invariance.     

Functional Integral Construction of the Massive Thirring model: Verification of Axioms and Massless Limit

We present a complete construction of a Quantum Field Theory for the Massive Thirring model by following a functional integral approach. This is done by introducing an ultraviolet and an infrared cutoff and by proving that, if the “bare” parameters are suitably chosen, the Schwinger functions have a well defined limit satisfying the Osterwalder-Schrader axioms, when the cutoffs are removed. Our results, which are restricted to weak coupling, are uniform in the value of the mass. The control of the effective coupling (which is the main ingredient of the proof) is achieved by using the Ward Identities of the massless model, in the approximated form they take in the presence of the cutoffs. As a byproduct, we show that, when the cutoffs are removed, theWard Identities have anomalies which are not linear in the bare coupling. Moreover, we find for the interacting propagator of the massless theory a closed equation which is different from that usually stated in the physical literature.     

A completely solvable model of the nonlinear Boltzmann equation

In one space and one time dimension, a model of the nonlinear Boltzmann equation is presented that is exactly solvable for all initial conditions. Furthermore, this model has the following desirable properties: (i) conservation of the number of particles; (ii) energy conservation; (iii) nonlinearity; (iv) positivity of distribution functions; and (v) unique equilibrium state (for any given density) which is approached as t → ∞ for most physically interesting initial conditions. Some of the simple properties of this model are studied.     

Remarks on some new models of interacting quantum fields with indefinite metric

We study quantum field models in indefinite metric. We introduce the modified Wightman axioms of Morchio and Strocchi as a general framework of indefinite metric quantum field theory (QFT) and present concrete interacting relativistic models obtained by analytical continuation from some stochastic processes with Euclidean invariance. As a first step towards scattering theory in indefinite metric QFT, we give a proof of the spectral condition on the translation group for the relativistic models.     

The IR stability of de Sitter QFT: physical initial conditions

This work uses Lorentz-signature in-in perturbation theory to analyze the late-time behavior of correlators in time-dependent interacting massive scalar field theory in de Sitter space. We study a scenario recently considered by Krotov and Polyakov in which the coupling g turns on smoothly at finite time, starting from g = 0 in the far past where the state is taken to be the (free) Bunch–Davies vacuum. Our main result is that the resulting correlators (which we compute at the one-loop level) approach those of the interacting Hartle–Hawking state at late times. We argue that similar results should hold for other physically-motivated choices of initial conditions. This behavior is to be expected from recent quantum “no hair” theorems for interacting massive scalar field theory in de Sitter space which established similar results to all orders in perturbation theory for a dense set of states in the Hilbert space. Our current work (1) indicates that physically motivated initial conditions lie in this dense set, (2) provides a Lorentz-signature counter-part to the Euclidean techniques used to prove such theorems, and (3) provides an explicit example of the relevant renormalization techniques.