A rigorous path integral for quantum spin using flat-space Wiener regularization

TL;DR

This paper develops a rigorous continuum path-integral formulation for quantum spin systems by relating spin-coherent states to Schrödinger semigroups on Euclidean space, using Wiener regularization.

Contribution

It introduces a novel, mathematically rigorous path-integral approach for quantum spins based on flat-space Wiener regularization, extending previous ideas to a continuum setting.

Findings

Derivation of a rigorous path-integral formula for spin Hamiltonians.

Connection between spin-coherent states and Schrödinger semigroups on R^2.

Application of Feynman-Kac-Itô formula in the context of quantum spins.

Abstract

Adapting ideas of Daubechies and Klauder [J. Math. Phys. {\bf 26} (1985) 2239] we derive a rigorous continuum path-integral formula for the semigroup generated by a spin Hamiltonian. More precisely, we use spin-coherent vectors parametrized by complex numbers to relate the coherent representation of this semigroup to a suitable Schr\"odinger semigroup on the Hilbert space $L^2(R^2)$ of Lebesgue square-integrable functions on the Euclidean plane $R^2$. The path-integral formula emerges from the standard Feynman-Kac-It\^o formula for the Schr\"odinger semigroup in the ultra-diffusive limit of the underlying Brownian bridge on $R^2$. In a similar vein, a path-integral formula can be constructed for the coherent representation of the unitary time evolution generated by the spin Hamiltonian.