Comparison with group-averaged physical inner products

Determine whether the physical inner product induced by the spectral embedding of reduced Schutz-clock states is equal to the physical product obtained from an independently specified group-averaging or refined-algebraic-quantization construction.

Background

The reconstructed extended states are distributional in the noncompact clock sector and are not trace-class with respect to the kinematical clock Hilbert space. The paper therefore defines a positive sesquilinear distribution and an induced physical inner product through the generalized spectral embedding of the reduced Hilbert space.

The authors distinguish this construction from refined algebraic quantization and phase-space group averaging. They state that equality between the induced product and an independently constructed group-averaged product cannot be assumed without specifying the averaging domain and normalization. Establishing that equality remains unresolved.