Twist Positivity

Abstract

We study a heat kernel e−βH 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 Zg and a twisted Gibbs expectation 〈·〉g, Zg=TrH(U(g−1)e−βH) and 〈·〉g=TrH(U(g−1)·e−βH)/TrH(U(g−1)e−βH). We say that {H, U(g), H} is twist positive if Zg>0. 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 dμg on that space yielding (in the usual sense on products of coordinates) 〈·〉g=∫·dμg. Bosonic quantum mechanics provides a class of specific examples that we discuss. We also consider a complex bosonic quantum field ϕ(x) defined on a spatial s-torus Ts and with a translation-invariant Hamiltonian. This system has an (s+1)-parameter abelian twist group Ts×R that is twist positive and that has a Feynman–Kac representation. Given τ∈Ts and θ∈R, the corresponding paths are random fields Φ(x, t) that satisfy the twist relationΦ(x, t+β)=eiΩθΦ(x−τ, t). We also utilize the twist symmetry to understand some properties of “zero-mass” limits, when the twist τ, θ lies in the complement of a set ϒsing of singular twists.