โ Scribed by Arthur Jaffe
No coin nor oath required. For personal study only.
We study a heat kernel e &;H defined by a self-adjoint Hamiltonian H acting on a Hilbert space H, and a unitary representation U( g) of a symmetry group G of H, normalized so that the ground vector of H is invariant under U( g). The triple [H, U( g), H] defines a twisted partition function Z g and a twisted Gibbs expectation
We say that [H, U( g), H] has a Feynman Kac representation with a twist U( g), if one can construct a function space and a probability measure d+ g on that space yielding (in the usual sense on products of coordinates) ( } ) g = } d+ g . Bosonic quantum mechanics provides a class of specific examples that we discuss. We also consider a complex bosonic quantum field .(x) defined on a spatial s-torus T s and with a translation-invariant Hamiltonian. This system has an (s+1)-parameter abelian twist group T s _R that is twist positive and that has a Feynman Kac representation. Given { # T s and % # R, the corresponding paths are random fields 8(x, t) that satisfy the twist relation 8(x, t+;)=e i0% 8(x&{, t). We also utilize the twist symmetry to understand some properties of ``zero-mass'' limits, when the twist {, % lies in the complement of a set ( sing of singular twists.
๐ SIMILAR VOLUMES
We express the structure of some positive polynomials in several variables, as squares of rational function with universal denominators, using functional analysis methods.