[Paper Review] New Measures for the Quantization of Systems with Constraints

John R. Klauder|ArXiv.org|Jul 22, 1996

Quantum optics and atomic interactions3 references3 citations

TL;DR

This paper introduces new measures for quantizing constrained systems using coherent-state phase-space path integrals, extending the formalism to include reparameterization-invariant Hamiltonians and first-class constraints with spectral gaps. The key contribution is a refined quantization procedure that ensures consistency with quantum constraint subspaces, particularly for systems with non-degenerate constraint spectra.

ABSTRACT

Based on the results of a recent reexamination of the quantization of systems with first-class and second-class constraints from the point of view of coherent-state phase-space path integration, we give additional examples of the quantization procedure for reparameterization invariant Hamiltonians, for systems for which the original set of Lagrange multipliers are elevated to dynamical variables, as well as extend the formalism to include cases of first-class constraints the operator form of which have a spectral gap about the value zero that characterizes the quantum constraint subspace.

Motivation & Objective

• To develop a consistent quantization framework for dynamical systems with first-class and second-class constraints.
• To extend the phase-space path integral approach using coherent states to handle reparameterization-invariant Hamiltonians.
• To incorporate systems where first-class constraints have a spectral gap around zero, characterizing the quantum constraint subspace.
• To ensure the quantization procedure respects the physical subspace defined by constraints, especially in cases with non-degenerate spectra.
• To generalize existing quantization methods to include Lagrange multipliers as dynamical variables.

Proposed method

• Utilizes coherent-state phase-space path integrals as the foundational quantization technique.
• Elevates original Lagrange multipliers to dynamical variables in the path integral formulation.
• Applies the formalism to reparameterization-invariant systems, preserving diffeomorphism invariance.
• Introduces new measures in the path integral that account for the spectral gap in first-class constraint operators.
• Constructs the quantum constraint subspace by projecting onto states where the constraint operator's spectrum is separated from zero.
• Employs operator ordering and measure modifications to maintain unitarity and consistency in the path integral.

Experimental results

Research questions

• RQ1 How can the coherent-state path integral be adapted to consistently quantize systems with first-class constraints?
• RQ2 What modifications to the path integral measure are required for reparameterization-invariant Hamiltonians?
• RQ3 How can spectral gaps in first-class constraint operators be used to define the physical quantum subspace?
• RQ4 In what way do elevated Lagrange multipliers affect the quantization of constrained systems?
• RQ5 Can the formalism be extended to systems with non-degenerate constraint spectra while preserving physical consistency?

Key findings

• The path integral formulation successfully incorporates reparameterization-invariant Hamiltonians through dynamical Lagrange multipliers.
• The method ensures that the quantum constraint subspace is precisely defined by the spectral gap of the constraint operator about zero.
• New measures in the path integral are derived that maintain consistency with the physical subspace for first-class constraints.
• The formalism generalizes previous approaches by including systems where constraint operators have discrete, non-zero spectral gaps.
• The approach preserves unitarity and gauge invariance in the quantized theory, even when constraints are non-trivial.
• The results demonstrate a consistent quantization procedure applicable to a broader class of constrained systems than previously addressed.