Example of a Poincare anomaly in relativistic quantum mechanics

TL;DR

This paper investigates the Poincare algebra in 1D classical electrodynamics, demonstrating a coordinate-dependent anomaly in quantization methods and analyzing its physical origin.

Contribution

It reveals a Poincare anomaly in quantization using ordinary coordinates and shows invariance on the light-cone, providing insights into coordinate-dependent quantization issues.

Findings

Canonical quantization is Poincare invariant on the light-cone.

Quantization fails to be Poincare invariant in ordinary coordinates.

The physical origin of the anomaly is analyzed.

Abstract

The Poincare algebra of classical electrodynamics in one spatial dimension is studied using light-cone coordinates and ordinary Minkowski coordinates. We show that it is possible to quantize the theory by a canonical quantization procedure in a Poincare invariant manner on the light-cone. We also show that this is not possible when using ordinary coordinates. The physical reason of this anomaly is analysed.