Canonical Quantization of the U(1) Gauge Field in the right Rindler-wedge in the Rindler Coordinates

TL;DR

This paper performs canonical quantization of the U(1) gauge field in the right Rindler-wedge, deriving mode solutions, commutation relations, and showing the Unruh effect for the gauge field in accelerated frames.

Contribution

It provides a detailed canonical quantization framework for the U(1) gauge field in Rindler coordinates, including mode solutions and the Unruh temperature effect.

Findings

Derived mode solutions for the gauge and B fields.

Established canonical commutation relations in Rindler coordinates.

Demonstrated the Unruh temperature for the gauge field in accelerated frames.

Abstract

In this study, the canonical quantization of the U(1) gauge field in the Lorentz-covariant gauge in the right Rindler-wedge (RRW) of the four-dimensional Rindler coordinates is performed. Specifically, we obtain the gauge-fixed Lagrangian by the Lorentz-covariant gauge in the RRW of the Rindler coordinates, which is composed of the U(1) gauge field and B-field. Then, we obtain the mode-solutions of the U(1) gauge field and B-field by solving the equations of motion obtained from that gauge-fixed Lagrangian. Subsequently, defining the Klein-Gordon inner-product in the RRW of the Rindler coordinates, we determine the normalization constants of all directions of the mode-solutions of the U(1) gauge field and B-field. Then, for the U(1) gauge field given by those normalized mode-expanded solutions, we obtain the commutation relations of the creation and annihilation operators defined in the RRW of the Rindler coordinates by formulating the canonical commutation relations. In addition, we provide a polarization vector for the annihilation operators obtained in this way. Using these result, we show that the Minkowski ground state can be expressed as the outer-product of the left and right Rindler-wedges state on which those creation and annihilation operators act. Then, tracing out the left Rindler states of that Minkowski ground state, we obtain the density matrix of the U(1) gauge field in the RRW. From this, we show that the U(1) gauge field in a constant accelerated system will feel the Unruh temperature as well.