Complex Structures and Quantum Representations for Scalar QFT in Curved Spacetimes

We confirm the equivalence of the Schrödinger representation and the holomorphic representation, based on previous results of the General Boundary Formulation (GBF) of Quantum Field Theory (QFT). On a wide class of curved spacetimes, we consider real Klein-Gordon theory in two types of regions: interval regions (consisting e.g. of a time interval times all of space), and rod regions (a solid ball of space extended over all of time). Using mode expansions, we provide explicit expressions for the Schrödinger vacua (choosing a vacuum determines a Schrödinger representation) and for the corresponding complex structures on the space of classical solutions (choosing a complex structure determines a holomorphic representation). That is, we parametrize the space of representations through vacua respectively complex structures. We also transcribe the complex structure to phase space and show that it agrees with earlier results. We explicitly construct the map which determines the isomorphism between the two representations. For both representations we give the corresponding coherent states and calculate the generalized free transition amplitudes of the GBF, which coincide and hence confirm the equivalence of the two representations.