Selecting the complex structure for quantization in the quantum case

Determine a principled prescription for choosing the complex structure that underlies the quantization of field theories, such as the Klein–Gordon theory on curved spacetimes, in the genuinely quantum case beyond static or stationary scenarios, providing criteria that ensure mathematical consistency and physical viability.

Background

The complex structure determines the Kähler geometry used for quantization and fixes the Gaussian measure defining the quantum state space. In the classical setting, the authors adopt a dynamical prescription derived from the Hamiltonian vector field.

For the quantum theory, the authors note that while various choices exist (and coincide in special cases such as static or stationary spacetimes), a general, widely accepted criterion for selecting the complex structure remains unresolved.