Noncommutative Time in Quantum Field Theory

TL;DR

This paper investigates the implications of noncommutative time in quantum field theory, revealing fundamental inconsistencies especially related to causality, and aligns these findings with string theory constraints.

Contribution

It provides a comprehensive analysis of noncommutative time in quantum field theory across multiple formalisms, demonstrating inherent inconsistencies and clarifying their relation to string theory.

Findings

Noncommutative time leads to inconsistency in quantum field theories.

Causality issues are central to the problems caused by noncommutative time.

String theory restricts noncommutative time, except in light-like cases.

Abstract

We analyze, starting from first principles, the quantization of field theories, in order to find out to which problems a noncommutative time would possibly lead. We examine the problem in the interaction picture (Tomonaga-Schwinger equation), the Heisenberg picture (Yang-Feldman-K\"all\'{e}n equation) and the path integral approach. They all indicate inconsistency when time is taken as a noncommutative coordinate. The causality issue appears as the key aspect, while the unitarity problem is subsidiary. These results are consistent with string theory, which does not admit a time-space noncommutative quantum field theory as its low-energy limit, with the exception of light-like noncommutativity.