Quantum evolution across singularities

TL;DR

This paper proposes a novel method, analogous to renormalization, to define unitary quantum evolution across space-time singularities, demonstrated on free fields in the compactified Milne universe.

Contribution

It introduces a new approach to handle quantum evolution through singularities, extending the mathematical framework for such problematic space-time regions.

Findings

Successfully constructs unitary evolution across singularities

Applies the method explicitly to free fields in Milne universe

Shows consistency with previous matching conditions

Abstract

Attempts to consider evolution across space-time singularities often lead to quantum systems with time-dependent Hamiltonians developing an isolated singularity as a function of time. Examples include matrix theory in certain singular time-dependent backgounds and free quantum fields on the two-dimensional compactified Milne universe. Due to the presence of the singularities in the time dependence, the conventional quantum-mechanical evolution is not well-defined for such systems. We propose a natural way, mathematically analogous to renormalization in conventional quantum field theory, to construct unitary quantum evolution across the singularity. We carry out this procedure explicitly for free fields on the compactified Milne universe and compare our results with the matching conditions considered in earlier work (which were based on the covering Minkowski space).