Complex Numbers, Quantum Mechanics and the Beginning of Time

TL;DR

This paper explores how complex structures in classical solutions relate to quantum field quantization in curved spacetime, using real tunneling geometries and Euclidean field theory concepts.

Contribution

It introduces a novel connection between real tunneling geometries and the complex structures used in quantizing fields in curved space.

Findings

Link established between tunneling geometries and complex structures

Relation to Osterwalder-Schrader Euclidean approach clarified

Implications for field quantization in curved spacetime

Abstract

A basic problem in quantizing a field in curved space is the decomposition of the classical modes in positive and negative frequency. The decomposition is equivalent to a choice of a complex structure in the space of classical solutions. In our construction the real tunneling geometries provide the link between the this complex structure and analytic properties of the classical solutions in a Riemannian section of space. This is related to the Osterwalder- Schrader approach to Euclidean field theory.