Minimal Proper-time in Quantum Field Theory

TL;DR

By integrating Nambu's proper-time idea into a Schrödinger-functional QFT and introducing a physical minimal proper time $τ_{ ext{min}}$, the paper presents a Lorentz-invariant UV regulator that yields controlled unitarity violation and exponential suppression of high-energy modes. This framework reproduces conventional QFT at low energies while predicting a deterministic UV regime with an energy-dependent effective Planck constant, and it suggests an interpretation of renormalization in terms of proper-time evolution and dimensional reduction. A spectral weight function $f(λ)$ governs the soft relaxation of the Nambu constraint, enabling a smooth transition from quantum to deterministic behavior at trans-Planckian scales and offering a consistent path to asymptotic safety. The approach preserves renormalizability, circumvents Haag’s theorem through non-unitary evolution, and opens avenues for gauge theories and open-system phenomenology, situating QFT as an effective, finite theory in the ultraviolet. Overall, the work provides a coherent UV completion perspective with testable implications for high-energy behavior and foundational quantum dynamics.

Abstract

We propose a generalization of quantum field theory within Schrödinger'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 $τ_{\min}$. The introduction of $τ_{\min}$ 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features suggest that a minimal proper-time formulation renders quantum field theory an effective but finite theory, superseded at trans-Planckian energies.