Positive and negative frequency decompositions in curved spacetime

In this note we derive a formula for the positive and negative frequency parts of a solution in terms of the Feynman propagator. Our arguments are valid in the presence of particle creation. We also derive a formula for an operator J, that gives the particle creation rate. The formalism uses complex structures to capture the notion of positive and negative frequencies and thus avoids using analyticity arguments. The results obtained clarify the relation between approaches to quantum field theory based on the complex structure and approaches in which the propagator is the basic object. We will consider only scalar fields for simplicity