# Complex structures and quantum representations for scalar QFT in curved spacetimes

## TL;DR

This paper demonstrates the equivalence of Schr"odinger and holomorphic representations in scalar quantum field theory on curved spacetimes, providing explicit formulas and confirming their consistency within the General Boundary Formulation.

## Contribution

It explicitly constructs and compares the Schr"odinger and holomorphic representations for Klein-Gordon theory in curved spacetimes, confirming their equivalence and extending previous results.

## Key findings

- Explicit expressions for vacuum states in both representations
- Confirmed the equivalence of transition amplitudes in GBF
- Aligned complex structure with earlier phase space results

## Abstract

We confirm the equivalence of the Schr\"odinger representation and the holomorphic one, based on previous results of the General Boundary Formulation (GBF) of quantum field theory. 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 Schr\"odinger vacuum (which determines this representation) and for the corresponding complex structure on the space of classical solutions (which determines the holomorphic representation). 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. We also transcribe the complex structure to phase space and show that it agrees with earlier results.