The algebra of observables in Gau{\ss}ian normal spacetime coordinates

TL;DR

This paper investigates the canonical structure of Gau{ ext}ian normal spacetime coordinates, revealing that a local algebra of observables cannot be constructed due to non-commuting gauge conditions, leading to inherently non-local Dirac brackets.

Contribution

It demonstrates that the spacetime version of the radial gauge does not admit a local algebra of observables, unlike the spatial version, due to non-commuting gauge conditions.

Findings

Local observables cannot be constructed in the spacetime radial gauge.

The Dirac bracket becomes inherently non-local.

A complete set of local observables with local Dirac brackets does not exist.

Abstract

We discuss the canonical structure of a spacetime version of the radial gauge, i.e. Gau{\ss}ian normal spacetime coordinates. While it was found for the spatial version of the radial gauge that a "local" algebra of observables can be constructed, it turns out that this is not possible for the spacetime version. The technical reason for this observation is that the new gauge condition needed to upgrade the spatial to a spacetime radial gauge does not Poisson-commute with the previous gauge conditions. It follows that the involved Dirac bracket is inherently non-local in the sense that no complete set of observables can be found which is constructed locally and at the same time has local Dirac brackets. A locally constructed observable here is defined as a finite polynomial of the canonical variables at a given physical point specified by the Gau{\ss}ian normal spacetime coordinates.