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

Field quantization in 5D flat and warped space-times with Z2-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 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 theories are formulated by the Dirac's bra-ket 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