Holomorphic quantum mechanics with a quadratic Hamiltonian constraint

TL;DR

This paper explores different quantization representations of a finite-dimensional quadratic Hamiltonian system, revealing significant differences in Hilbert space structures and spectral properties, with implications for quantum cosmology.

Contribution

It introduces a unique inner product for holomorphic, antiholomorphic, and mixed representations, and analyzes their impact on the Hilbert space and spectral properties in quadratic constrained systems.

Findings

Holomorphic representation yields spectra consistent with classical expectations.

Antiholomorphic representation produces spectra violating classical expectations.

Path integral approach relates to matrix elements of identity in coherent state bases.

Abstract

A finite dimensional system with a quadratic Hamiltonian constraint is Dirac quantized in holomorphic, antiholomorphic and mixed representations. A unique inner product is found by imposing Hermitian conjugacy relations on an operator algebra. The different representations yield drastically different Hilbert spaces. In particular, all the spaces obtained in the antiholomorphic representation violate classical expectations for the spectra of certain operators, whereas no such violation occurs in the holomorphic representation. A subset of these Hilbert spaces is also recovered in a configuration space representation. A propagation amplitude obtained from an (anti)holomorphic path integral is shown to give the matrix elements of the identity operators in the relevant Hilbert spaces with respect to an overcomplete basis of representation-dependent generalized coherent states. Relation to quantization of spatially homogeneous cosmologies is discussed in view of the no-boundary proposal of Hartle and Hawking and the new variables of Ashtekar.