Zeta-regularized vacuum expectation values

TL;DR

This paper demonstrates that $ta$-regularized vacuum expectation values can be understood as continuum limits of discretization schemes, with applications shown in quantum systems and gauge theories, connecting mathematical regularization with physical interpretations.

Contribution

It establishes the physical meaning of $ta$-regularized vacuum expectations as continuum limits and analyzes their convergence in quantum and gauge theory contexts.

Findings

Discretization schemes converge to $ta$-regularized vacuum expectations.

Application to a 1D hydrogen atom evaluated on quantum hardware.

Computation of vacuum expectations in gauge theories.

Abstract

It has recently been shown that vacuum expectation values and Feynman path integrals can be regularized using Fourier integral operator $\zeta$-function, yet the physical meaning of these $\zeta$-regularized objects was unknown. Here we show that $\zeta$-regularized vacuum expectations appear as continuum limits using a certain discretization scheme. Furthermore, we study the rate of convergence for the discretization scheme using the example of a one-dimensional hydrogen atom in $(-\pi,\pi)$ which we evaluate classically, using the Rigetti Quantum Virtual Machine, and on the Rigetti 8Q quantum chip "Agave" device. We also provide the free radiation field as an example for the computation of $\zeta$-regularized vacuum expectation values in a gauge theory.