Quantum evolution across singularities: the case of geometrical resolutions

TL;DR

This paper investigates quantum evolution in singular space-times, focusing on Hamiltonians with multiple operator structures, and finds that scalar field evolution exists in specific parameter regimes with reflection properties.

Contribution

It extends previous minimal subtraction methods to more complex Hamiltonians arising from geometrical space-time resolutions, analyzing scalar field behavior in generalized null-brane geometries.

Findings

Scalar field evolution exists for discrete parameter values.

Coordinates reveal a reflection property in scalar propagation.

A family of pp-wave geometries exhibits particle reflection at singular limits.

Abstract

We continue the study of time-dependent Hamiltonians with an isolated singularity in their time dependence, describing propagation on singular space-times. In previous work, two of us have proposed a "minimal subtraction" prescription for the simplest class of such systems, involving Hamiltonians with only one singular term. On the other hand, Hamiltonians corresponding to geometrical resolutions of space-time tend to involve multiple operator structures (multiple types of dependence on the canonical variables) in an essential way. We consider some of the general properties of such (near-)singular Hamiltonian systems, and further specialize to the case of a free scalar field on a two-parameter generalization of the null-brane space-time. We find that the singular limit of free scalar field evolution exists for a discrete subset of the possible values of the two parameters. The coordinates we introduce reveal a peculiar reflection property of scalar field propagation on the generalized (as well as the original) null-brane. We further present a simple family of pp-wave geometries whose singular limit is a light-like hyperplane (discontinuously) reflecting the positions of particles as they pass through it.