Regularization from Superpositions of Time Evolutions

TL;DR

This paper introduces a novel regularization method using superpositions of time evolutions with Gaussian filters, improving the behavior of short-time kernels in path integrals for singular potentials and quantum field theories.

Contribution

It demonstrates how interference in superpositions of evolutions can produce smooth, removable regulators, offering a new perspective on regularization in quantum mechanics and quantum field theory.

Findings

Gaussian superpositions act as energy filters suppressing high energies.

Short-time kernels become well-behaved for singular potentials.

The method recovers the original dynamics as parameters approach zero.

Abstract

Short-time approximations and path integrals can be dominated by high-energy or large-field contributions, especially in the presence of singular interactions, motivating regulators that are suppressive yet removable. Standard regulators typically impose such suppressions by hand (e.g. cutoffs, higher-derivative terms, heat-kernel smearing, lattice discretizations), while here we show that closely related smooth filters can arise as the conditional map produced by interference in a coherently controlled, postselected superposition of evolutions. A successful postselection implements a single heralded operator that is a coherent linear combination of time-evolution operators. For a Gaussian superposition of time translations in quantum mechanics, the postselected step is $V_{\sigma,\Delta t}=e^{-iH\Delta t}\,e^{-\frac12\sigma^2\Delta t^2H^2}$, i.e.\ the desired unitary step multiplied by a Gaussian energy filter suppressing energies above order $1/(\sigma\Delta t)$. This renders short-time kernels in time-sliced path-integral approximations well behaved for singular potentials, while the target unitary dynamics is recovered as $\sigma\to0$ and (for fixed $\sigma$) also as $\Delta t\to0$ at fixed $t$. In scalar QFT, a local Gaussian smearing of the quartic coupling induces a positive $(\sigma^2/2)\phi^8$ term in the Euclidean action, providing a symmetry-compatible large-field stabilizer; it is naturally viewed as an irrelevant operator whose effects can be renormalized at fixed $\sigma$ (together with a conventional UV regulator) and removed by taking $\sigma\to0$. We give short-time error bounds and analyze multi-step success probabilities.