Twist Positivity

We study a heat kernel e 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) e&;H) and ( } ) g=TrH (U(g) } e) TrH (U(g) e). 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 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 and with a translation-invariant Hamiltonian. This system has an (s+1)-parameter abelian twist group T_R that is twist positive and that has a Feynman Kac representation. Given { # T and % # R, the corresponding paths are random fields 8(x, t) that satisfy the twist relation 8(x, t+;)=e8(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. 1999 Academic Press