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

Abstract: This paper describes the canonical quantization of the U(1) gauge field across all four regions in the Rindler coordinates in the Lorentz-covariant gauge. Concretely, in the four regions (future, past, left and right Rindler-wedges) in the Rindler coordinates, the gauge-fixed Lagrangian in the Lorentz-covariant gauge is obtained, which is composed of the U(1) gauge field, the B-field and ghost fields. Since the U(1) gauge and B-fields are decoupled from the ghost fields by the property of the U(1) gauge theory, the U(1) gauge field and the B-field are examined in this study. Then, by solving the equations of motion obtained from the gauge-fixed Lagrangian, the solutions of each mode of the U(1) gauge field and the B-field can be obtained. Following this, with the Klein-Gordon inner-product defined in the Rindler coordinates, the normalization constants of each of those mode-solutions are determined. Subsequently, formulating the canonical commutation relations of the U(1) gauge field and its canonical conjugate momentum, the equal-time commutation relations of the coefficient of each mode-solution in each direction of the U(1) gauge field in each region of the Rindler coordinates are obtained. From these, it can be seen that those coefficients have physical meaning as creation/annihilation operators. The polarization vectors in each region of the Rindler coordinates are also given in this study.