The emergent nature of time and the complex numbers in quantum cosmology

TL;DR

This paper explores the fundamental role of complex numbers in quantum mechanics and cosmology, proposing that multiple complex structures may underlie the nature of time and quantum evolution in the universe.

Contribution

It introduces the idea that quantum mechanics may involve multiple complex structures, especially relevant in quantum cosmology and non-time-oriented spacetimes.

Findings

Time in quantum mechanics relates to complex Hilbert spaces.

In non-time-oriented spacetimes, complex quantum mechanics may need to be abandoned.

Multiple complex structures could underpin quantum gravity and the nature of time.

Abstract

The nature of time in quantum mechanics is closely related to the use of a complex, rather than say real, Hilbert space. This becomes particularly clear when considering quantum field theory in time dependent backgrounds, such as in cosmology, when the notion of positive frequency ceases to be well defined. In spacetimes lacking time orientation, i.e without the possibility of defining an arrow of time, one is forced to abandon complex quantum mechanics. One also has to face this problem in quantum cosmology. I use this to argue that this suggests that, at a fundamental level, quantum mechanics may be really real with not one, but a multitude of complex structures. I relate these ideas to other suggestions that in quantum gravity time evolution may not be unitary, possibly implemented by a super-scattering matrix, and the status of CPT.