New Gauge Invariant Formulation of the Chern-Simons Gauge Theory: Classical and Quantal Analysis

TL;DR

This paper presents a new gauge invariant formulation of Chern-Simons gauge theory, analyzing classical and quantum aspects, and clarifies the role of gauge invariance, physical states, and gauge fixing in the theory.

Contribution

It introduces a gauge invariant formulation consistent with gauge fixing, constructs gauge invariant generators, and clarifies the physical state structure in Chern-Simons theory.

Findings

Gauge invariant generators obey Poincaré algebra.

Physical states require gauge invariant longitudinal modes.

Clarification of Dirac's dressing function effects.

Abstract

Recently proposed new gauge invariant formulation of the Chern-Simons gauge theory is considered in detail. This formulation is consistent with the gauge fixed formulation. Furthermore it is found that the canonical (Noether) Poincar\'e generators are not gauge invariant even on the constraints surface and do not satisfy the Poincar\'e algebra contrast to usual case. It is the improved generators, constructed from the symmetric energy-momentum tensor, which are (manifestly) gauge invariant and obey the quantum as well as classical Poincar\'e algebra. The physical states are constructed and it is found in the Schr\"odinger picture that unusual gauge invariant longitudinal mode of the gauge field is crucial for constructing the physical wavefunctional which is genuine to (pure) Chern-Simons theory. In matching to the gauge fixed formulation, we consider three typical gauges, Coulomb, axial and Weyl gauges as explicit examples. Furthermore, recent several confusions about the effect of Dirac's dressing function and the gauge fixings are clarified. The analysis according to old gauge independent formulation a' la Dirac is summarized in an appendix.