Path Integral for Space-time Noncommutative Field Theory

TL;DR

This paper formulates a path integral approach for space-time noncommutative field theories, analyzing their quantization, unitarity, and energy conditions, revealing fundamental conflicts in their consistent quantization.

Contribution

It introduces a path integral formulation based on Schwinger's action principle for space-time noncommutative theories and examines their canonical structure and quantization issues.

Findings

Path integral formulation aligns with Yang-Feldman approach

Naive interaction picture quantization is justified for space-time noncommutative theories

Unitarity and positive energy conditions cannot be simultaneously satisfied

Abstract

The path integral for space-time noncommutative theory is formulated by means of Schwinger's action principle which is based on the equations of motion and a suitable ansatz of asymptotic conditions. The resulting path integral has essentially the same physical basis as the Yang-Feldman formulation. It is first shown that higher derivative theories are neatly dealt with by the path integral formulation, and the underlying canonical structure is recovered by the Bjorken-Johnson-Low (BJL) prescription from correlation functions defined by the path integral. A simple theory which is non-local in time is then analyzed for an illustration of the complications related to quantization, unitarity and positive energy conditions. From the view point of BJL prescription, the naive quantization in the interaction picture is justified for space-time noncommutative theory but not for the simple theory non-local in time. We finally show that the perturbative unitarity and the positive energy condition, in the sense that only the positive energy flows in the positive time direction for any fixed time-slice in space-time, are not simultaneously satisfied for space-time noncommutative theory by the known methods of quantization.