Minimal Proper-time in Quantum Field Theory

Abstract

We propose a generalization of quantum field theory within Schrodinger's functional representation, inspired by Nambu's proper-time formulation of quantum mechanics. The key motivation for this generalization is to incorporate a fundamental, Lorentz-invariant minimum scale, which in this formulation is played by a minimal proper time $\tau_{\min}$. The introduction of $\tau_{\min}$ leads to several significant effects at very high energies: it modifies the Heisenberg uncertainty principle, induces a controlled violation of unitarity, and suppresses high-energy modes. This minimal scale renders the theory asymptotically safe through a mechanism akin to dimensional reduction, while reproducing all the standard results at low energies, where quantum field theory emerges. Remarkably, the same framework can accommodate a deterministic regime at energies approaching the Planck scale. These features suggest that a minimal proper-time formulation renders the quantum field theory an effective but finite theory, superseded at trans-Planckian energies.