On spinless null propagation in five-dimensional space-times with approximate space-like Killing symmetry

Abstract

Five-dimensional (5D) space-time symmetry greatly facilitates how a 4D observer perceives the propagation of a single spinless particle in a 5D space-time. In particular, if the 5D geometry is independent of the fifth coordinate then the 5D physics may be interpreted as 4D quantum mechanics. In this work we address the case where the symmetry is approximate, focusing on the case where the 5D geometry depends weakly on the fifth coordinate. We show that concepts developed for the case of exact symmetry approximately hold when other concepts such as decaying quantum states, resonant quantum scattering, and Stokes drag are adopted, as well. We briefly comment on the optical model of the nuclear interactions and Millikan's oil drop experiment.

The 4D Dirac Equation in Five Dimensions

“…This completes our description of 5D null propagation, as an alternative to the null path-integral formulation [15][16][17]. Dirac operators over curved manifolds are commonly introduced using standard Clifford algebras and spin connections [21].…”

“…In this work, we define a special 5D Clifford algebra using the O (1,4) symmetry and propose a constraint equation for 5D null propagation. We then use a recent interpretation of the 5D geometry [15][16][17] where a 5D null constraint yields a 4D on-shell constraint. In other words, the norm of the 5-momentum of a 5D photon in flat space-time η AB p A p B = 0 [A, B = 0, 1, 2, 3, 5, and η AB = diag(−1, 1, 1, 1, 1)] yields the 4D on-shell constraint for a particle of mass m, η αβ p α p β = −m 2 c 2 (α, β = 0, 1, 2, 3), where we used p 5 ≡ mc.…”

“…If the 5D geometry is independent of the fifth coordinate, then the 5D physics can be interpreted as 4D quantum mechanics. If the 5D geometry is independent of time (i.e., the zeroth coordinate), then the 5D physics can be interpreted as 4D statistical mechanics [15][16][17]. Both mechanics and statistics pictures apply when the geometry is independent on both time and fifth coordinate.…”

Electromagnetism from 5D gravity: beyond the Maxwell equations

“…Therefore, the hierarchical physical interpretation is not covariant. Reference [16] demonstrates how particle propagation in a 5D space-time with approximate x 5 -translation symmetry can be given a physical picture. In the first order, we use 4D mechanics [15].…”

“…Here, we impose two requirements, which make the 5D field parameterization unique. First, the parameterization submits to the geometric picture of the foliation of the 5D metric along x 5 and, second, it provides an anomaly-free theory for single-particle propagation in 4D space-time, according to the principles of general relativity [13][14][15][16].…”