The Feynman-Kac formula in deformation quantization

TL;DR

This work establishes a Feynman–Kac–type relation within deformation quantization by linking the star exponential of a Hamiltonian to the ground-state energy via a Wick-rotated phase-space integral. The authors derive a Fourier–Dirichlet expansion for the star exponential, show that the long-time (imaginary) limit isolates $E_0$, and provide explicit expressions for several 1D systems, including the free particle, harmonic oscillator, linear potential, general quadratic forms, and a damped oscillator. The resulting formula offers an efficient, operator-free route to compute ground-state energies and strengthens the conceptual link between deformation quantization and path-integral methods. The work suggests natural extensions to quantum field and string theories, promoting a unified view of quantum dynamics across formalisms.

Abstract

We introduce the Feynman-Kac formula within the deformation quantization program. Constructing on previous work it is shown that, upon a Wick rotation, the ground state energy of any prescribed physical system can be obtained from the asymptotic limit of the phase space integration of the star exponential of the Hamiltonian operator. Some examples of this correspondence are provided showing a novel and efficient way of computing the ground state energy for some physical models.