Field Quantization in 5D Space-Time with Z$_2$-parity and Position/Momentum Propagator

TL;DR

This paper develops a comprehensive framework for quantizing fields in 5D space-times with Z$_2$-parity, deriving position/momentum propagators, analyzing their behaviors, and clarifying the relation between different extra-dimensional models without relying on KK-expansion.

Contribution

It introduces a novel method for deriving 5D propagators that accounts for all KK-modes and compares flat and warped models using Dirac's formalism, avoiding KK-expansion.

Findings

Derived 5D position/momentum propagators with boundary conditions

Clarified relation between P/M propagator and KK-expansion series

Resolved the $ ext{} ext{delta}(0)$ problem without KK-expansion

Abstract

Field quantization in 5D flat and warped space-times with Z$_2$-parity is comparatively examined. We carefully and closely derive 5D position/momentum(P/M) propagators. Their characteristic behaviours depend on the 4D (real world) momentum in relation to the boundary parameter ($l$) and the bulk curvature ($\om$). They also depend on whether the 4D momentum is space-like or time-like. Their behaviours are graphically presented and the Z$_2$ symmetry, the "brane" formation and the singularities are examined. It is shown that the use of absolute functions is important for properly treating the singular behaviour. The extra coordinate appears as a {\it directed} one like the temperature. The $\delta(0)$ problem, which is an important consistency check of the bulk-boundary system, is solved {\it without} the use of KK-expansion. The relation between P/M propagator (a closed expression which takes into account {\it all} KK-modes) and the KK-expansion-series propagator is clarified. In this process of comparison, two views on the extra space naturally come up: orbifold picture and interval (boundary) picture. Sturm-Liouville expansion (a generalized Fourier expansion) is essential there. Both 5D flat and warped quantum systems are formulated by the Dirac's bra and ket vector formalism, which shows the warped model can be regarded as a {\it deformation} of the flat one with the {\it deformation parameter} $\om$. We examine the meaning of the position-dependent cut-off proposed by Randall-Schwartz.