Complex Numbers, Quantum Mechanics and the Beginning of Time

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.