Quantization of singular systems and incomplete motions

TL;DR

This paper develops a rigorous quantization method for singular spaces and incomplete motions, crucial for quantum cosmology, using induced representations of C*-algebras and Schwartz's Hilbert subspaces, enabling concrete calculations of the universe's wave-function.

Contribution

It introduces a novel quantization framework based on C*-algebra representations and Hilbert subspaces, addressing previous limitations in quantum cosmology.

Findings

Provides a basis for generalized eigenfunction expansions.

Enables concrete calculation of the universe's wave-function.

Accounts for freedom in the induction procedure.

Abstract

The need for a mathematically rigorous quantization procedure of singular spaces and incomplete motions is pointed out in connection with quantum cosmology. We put our previous suggestion for such a procedure, based on the theory of induced representations of C*-algebras, in the light of L. Schwartz' theory of Hilbert subspaces. This turns out to account for the freedom in the induction procedure, at the same time providing a basis for generalized eigenfunction expansions pertinent to the needs of quantum cosmology. Reinforcing our previous proposal for the wave-function of the Universe, we are now able to add a concrete prescription for its calculation.