Regularization from Superpositions of Time Evolutions
Eliahu Cohen, Tomer Shushi
TL;DR
This work addresses the problem of ill-behaved short-time propagators in the presence of singular interactions by introducing an interference-based regulator arising from postselected, coherently controlled superpositions of time evolutions. The main approach yields a Gaussian energy filter in quantum mechanics via the postselected step $V_{\sigma,\Delta t}=e^{-iH\Delta t} e^{- rac12\sigma^2\Delta t^2 H^2}$, which suppresses high-energy components while recovering the target unitary dynamics in the limit $\sigma\to0$, and a local Gaussian averaging of couplings in scalar QFT that induces a positive, symmetry-preserving $\frac{\sigma^2}{2}\phi^8$ term in the Euclidean action. Key contributions include explicit short-time expansions and error bounds for the postselected Kraus map, a detailed kernel analysis for singular QM settings (e.g., Coulomb, spike potentials, and outside-negative wells), and a renormalization-aware framework for the QFT regulator at fixed $\sigma$. The work provides a physically motivated, operational interpretation for smooth regulators and suggests practical avenues for lattice implementations and exploration of non-unitary extensions constrained by complete positivity. The proposed regulators are removable, preserving the standard unitary dynamics in the appropriate limits, and offer a principled link between interference, regularization, and renormalization in quantum theories.
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 $V_{σ,Δt}=e^{-iHΔt}\,e^{-\frac12σ^2Δt^2H^2}$, i.e.\ the desired unitary step multiplied by a Gaussian energy filter suppressing energies above order $1/(σΔt)$. This renders short-time kernels in time-sliced path-integral approximations well behaved for singular potentials, while the target unitary dynamics is recovered as $σ\to0$ and (for fixed $σ$) also as $Δt\to0$ at fixed $t$. In scalar QFT, a local Gaussian smearing of the quartic coupling induces a positive $(σ^2/2)φ^8$ 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 $σ$ (together with a conventional UV regulator) and removed by taking $σ\to0$. We give short-time error bounds and analyze multi-step success probabilities.