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摘要
En 中文
We study a heat kernel e(-beta H) defined by a self-adjoint Hamiltonian H acting on a Hilbert space s, 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), s} defines a twisted partition function 3(g) and a twisted Gibbs expectation [.]g;3(g) = Tr-s(U(g(-1)) e(-beta H)) and [.](g) = Tr-s(U(g(-1)) . e(-beta H))/Tr-s( U(g(-1)) e(-beta H)). We say that {H, U(g), s} is twist positive if 3(g)>0. We say that {H, U(g), s} has a Feynman-Kac representation with a twist U(g), if one can construct a Function space and a probability measure d mu(g) oil that space yielding (in the usual sense on products of coordinates) [.](g)=integral . d mu(g). Bosonic quantum mechanics provides a class of specific examples that we discuss. We also consider a complex bosonic quantum field phi(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 x R that is twist positive and that has a Feynman-Kac representation. Given tau is an element of T-s and 0 is an element of R, the corresponding paths are random fields Phi(x, t) that satisfy the twist relation Phi(x, t + beta) = e(\Omega 0)Phi(x - tau, t). We also utilize the twist symmetry to understand some properties of zero-mass limits, when the twist tau, 0 lies in the complement of a set Y-sing of singular twists. (C) 1999 Academic Press.
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