Regularization from Superpositions of Time Evolutions

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.