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 $V_{\ddot{I}ƒ,\hat{I}”t}=e^{-iH\hat{I}”t}{e}^{-\frac{1}{2}\ddot{I}{ƒ}^2\hat{I}”t^2{H}^2}$, i.e.\ the desired unitary step multiplied by a Gaussian energy filter suppressing energies above order $1/(\ddot{I}ƒ\hat{I}”t)$. This renders short-time kernels in time-sliced path-integral approximations well behaved for singular potentials, while the target unitary dynamics is recovered as $\ddot{I}ƒ\to 0$ and (for fixed $\ddot{I}ƒ$) also as $\hat{I}”t\to 0$ at fixed $t$. In scalar QFT, a local Gaussian smearing of the quartic coupling induces a positive $(\ddot{I}{ƒ}^2/2)\ddot{I}{•}^8$ 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 $\ddot{I}ƒ$ (together with a conventional UV regulator) and removed by taking $\ddot{I}ƒ\to 0$. We give short-time error bounds and analyze multi-step success probabilities.

Updated version, 13 pages