[Paper Review] Path Integrals, and Classical and Quantum Constraints

John R. Klauder|ArXiv.org|Jul 22, 2005

Photonic and Optical Devices4 references3 citations

TL;DR

This paper presents a path integral formulation for quantum systems with classical and quantum constraints using coherent state path integrals and projection operators, avoiding traditional issues like gauge fixing, Faddeev-Popov determinants, and Dirac brackets. The method directly imposes regularized quantum constraints and ensures consistency with the abstract operator formalism, offering a unified treatment of first- and second-class constraints without auxiliary variables or indefinite metrics.

ABSTRACT

Systems with constraints pose problems when they are quantized. Moreover, the Dirac procedure of quantization prior to reduction is preferred. The projection operator method of quantization, which can be most conveniently described by coherent state path integrals, enables one to directly impose a regularized form of the quantum constraints. This procedure also overcomes conventional difficulties with normalization and second class constraints that invalidate conventional Dirac quantization procedures.

Motivation & Objective

• To develop a path integral formulation that consistently quantizes systems with classical and quantum constraints.
• To overcome limitations of the standard Dirac quantization procedure, especially for second-class constraints and normalization issues.
• To provide a formulation that directly incorporates quantum constraints via projection operators within a coherent state path integral framework.
• To eliminate the need for auxiliary variables, ghosts, or indefinite metrics in constrained quantization.
• To ensure compatibility with the fundamental operator formulation of quantum mechanics through coherent state representations.

Proposed method

• Uses coherent state path integrals to represent the time evolution of quantum systems with constraints.
• Introduces a projection operator IÊ to restrict dynamics to the physical Hilbert space defined by constraints.
• Employs a double discretization: time steps ε and refinement steps M to properly regularize constraint terms up to second order in ε.
• Constructs a path integral measure involving δ-functions and Gaussian integrals over Lagrange multipliers λα to enforce constraints.
• Expands the constraint term Φα to second order in ε to maintain consistency in the continuum limit.
• Uses a normalization constant c = −4πiM/ε to ensure correct normalization in the path integral expression.

Experimental results

Research questions

• RQ1 How can quantum constraints be consistently imposed in a path integral formulation without relying on gauge fixing?
• RQ2 What is the role of coherent states in ensuring the correct correspondence between classical and quantum Hamiltonians in constrained systems?
• RQ3 How can second-class constraints be treated without introducing Dirac brackets or auxiliary variables?
• RQ4 Can a path integral formulation be constructed that directly projects onto the physical Hilbert space while preserving unitarity and normalization?
• RQ5 What modifications are required in the path integral measure to correctly handle both first- and second-class constraints?

Key findings

• The proposed path integral formulation successfully enforces quantum constraints via a projection operator IÊ, ensuring dynamics occur only in the physical Hilbert space.
• The method avoids gauge fixing, Faddeev-Popov determinants, Gribov ambiguities, and auxiliary fields, resolving long-standing issues in constrained quantization.
• Second-order regularization of constraint terms in ε ensures consistency and convergence in the continuum limit.
• The use of coherent states ensures that the classical Hamiltonian in Cartesian coordinates matches the quantum Hamiltonian up to O(ħ) corrections, preserving physical correspondence.
• The final path integral expression matches the abstract operator evolution e^{-i(ÊHÊ)T}, confirming consistency with standard quantum mechanics.
• The formulation is free of indefinite metrics and does not require Dirac brackets, providing a manifestly consistent treatment of both first- and second-class constraints.