Leonard Gross

Existence and uniqueness of physical ground states

Abstract It is proved that if A is a bounded Hermitian operator on a probability Hilbert algebra which preserves positivity and is continuous from L2 to Lp for some p > 2 then ∥ A ∥ is an eigenvalue of A. A sufficient condition is given for its multiplicity to be one. Applications are given to the proof of existence and nondegeneracy of physical ground states in quantum field theory for physical systems involving Fermions or Bosons.

Logarithmic Sobolev inequalities on loop groups

Let G be a compact Lie group. Denote by m the Brownian bridge measure on the loop group Y ≡ {g ϵ C([0, 1]; G): g(0) = g(1) = e }. The finite energy subgroup of Y determines in a natural way a gradient operation for functions on Y. The following logarithmic Sobolev inequality is proven, ∝ f2, log ¦f¦dm ⩽ ∝ {¦gradf(y)¦2 + V(y) f (y)2} dm + ∥f∥2log∥f∥wherein ∥f∥ denotes the L2(m) norm and V is a potential which is quadratic in the associated Lie algebra valued Brownian motion. The inequality is derived by a method of inheritance from the known inequality for the G valued Brownian motion.

Decay of correlations in classical lattice models at high temperature

In classical statistical mechanical lattice models with many body potentials of finite or infinite range and arbitrary spin it is shown that the truncated pair correlation function decays in the same weighted summability sense as the potential, at high temperature.

Norm invariance of mass-zero equations under the conformal group

It is known that a suitable collection of solutions of the free‐field Maxwells equations is a Hilbert space with respect to an appropriate norm, and that the inhomogeneous Lorentz group acts on this Hilbert space in a unitary and irreducible manner. It is shown that this representation extends to a unitary representation of the conformal group of Minkowski space. Similar results are obtained for other mass‐zero relativistic equations.

Two dimensional Yang-Mills theory via stochastic differential equations

Abstract The equation of parallel transport is given meaning in the quantized two dimensional Euclidean Yang-Mills theory by interpreting it as a stochastic differential equation in the complete axial gauge. The expectation of products of Wilson loops is evaluated using the stochastic differential calculus and a method of Bralic. Euclidean invariance of the theory is proven.

Integration and nonlinear transformations in Hilbert space

The purpose of this paper is to generalize to Hilbert space a well known theorem of Jacobi on the transformation of integrals under a change of coordinates in En. In the process we develop some of the relevant integration theory over Hilbert space. Among the various theories [14; 3; 4; 5; 9; 10 ], of integration over a real Hilbert space 77 we shall utilize the one [14; 3; 4; 5], which seems most relevant to the quantum theory of fields and which is intimately connected with the Wiener integral. The formulation of the theory which we adopt is given in Segal [14, pp. 116-118] and we refer the reader to this paper for details. In outline this formulation consists in associating with 77 in an invariant way a probability space (S, m) and a map/—->/ which assigns a measurable function / on (S, m) to each tame function / on 77, a tame function being one which, roughly speaking, depends only on a finite number of coordinates in 77. The theory of integration over 77 is then the theory of integration over (5, m) with particular emphasis on those questions which arise from the relation of 77 to (S, m). In case H is real P2(0, 1) the probability space may be taken to be Wiener space. We first describe (Theorem 1) a class of continuous functions on 77 which are determined by their continuity properties alone and which correspond in a natural way to measurable functions on (S, m), i.e., functions/ other than tame functions for which/makes sense. The type of continuity involved plays a central role in the remainder of the paper and generalizes ordinary continuity in P„ in the relevant way for the purposes of integration theory. Theorem 2 relates the convergence of sequences of these functions to convergence in probability of the associated measurable functions. Theorem 3 is a density theorem for the group of automorphisms of the algebra La(S, m) and provides a background for the approximation method used in the following theorem. Theorem 4 is a generalization of Jacobis theorem for a transformation of the form T = I+K where 7 is the identity operator and K is nonlinear, small and smooth. In Theorem 5 we avoid the smallness requirement by considering a one-parameter group of nonlinear transformations on 77 on whose generator sufficient conditions are imposed which ensure that the elements of the group satisfy a Jacobi-like theorem. The case of linear transformations has already been investigated by Segal [16].

Hall’s transform and the Segal-Bargmann map

It is shown how Hall’s transform for a compact Lie group can be derived from the infinite dimensional Segal-Bargmann transform by means of stochastic analysis.

A Poincaré lemma for connection forms

Denote by P the space of piecewise smooth curves in Rn beginning at the origin. A path 2-form is a function h on P such that for each element σ in P, h(σ) is a 2-form at the endpoint of σ with values in a Lie algebra G. For example, if A is a smooth G valued connection form on Rn with curvature F and parallel translation operator P(σ) then the equation LA(σ) = P(σ)−1 F(σ(1)) P(σ) defines LA as a path 2-form. A necessary and sufficient condition is given to characterize those path 2-forms which arise in this way. By way of application it is shown that the Birula-Mandelstam generalization of Maxwells equations to nonabelian gauge fields is equivalent to the Yang-Mills equation.

On the formula of Mathews and Salam

Abstract A direct proof of the Mathews-Salam formula in the Euclidean region is given for a Dirac field interacting with a classical time-dependent field and with a scalar Boson field. Cutoffs are not removed. The result is cast in the setting of a probability gage space.

Convergence of U(1)3 lattice gauge theory to its continuum limit

It is shown that in three space-time dimensions the pure U(1) lattice gauge theory with Villain action and fixed coupling constant converges to the free electromagnetic field as the lattice spacing approaches zero. The same holds for the Wilson action on the electric sector.