[Paper Review] First Reduce or First Quantize? A Lagrangian Approach and Application to Coset Spaces

C. Ordóñez y J. M Pons|ArXiv.org|Aug 16, 1993

Advanced Topics in Algebra2 references3 citations

TL;DR

This paper investigates the ordering of reduction and quantization in first-class Hamiltonian systems with constraints, comparing the 'first reduce then quantize' and 'first quantize then reduce' (Dirac's) approaches using a Lagrangian framework. It reveals new ambiguities in Dirac's method and establishes a precise relation between Dirac and full-space propagators, demonstrating full self-consistency and equivalence in quantization on compact Lie group coset spaces, particularly for $S^2 = SU(2)/U(1)$.

ABSTRACT

A Lagrangian treatment of the quantization of first class Hamiltonian systems with constraints and Hamiltonian linear and quadratic in the momenta respectively is performed. The ``first reduce and then quantize'' and the ``first quantize and then reduce'' (Dirac's) methods are compared. A new source of ambiguities in this latter approach is revealed and its relevance on issues concerning self-consistency and equivalence with the ``first reduce'' method is emphasized. One of our main results is the relation between the propagator obtained {\it à la Dirac} and the propagator in the full space, eq. (5.25).As an application of the formalism developed, quantization on coset spaces of compact Lie groups is presented. In this case it is shown that a natural selection of a Dirac quantization allows for full self-consistency and equivalence. Finally, the specific case of the propagator on a two-dimensional sphere $S^2$ viewed as the coset space $SU(2)/U(1)$ is worked out.

Motivation & Objective

• To resolve the foundational question of whether to reduce constraints before or after quantization in first-class systems with constraints.
• To identify and analyze new sources of ambiguity in Dirac's 'first quantize then reduce' approach.
• To establish conditions under which Dirac's method achieves self-consistency and equivalence with the 'first reduce' method.
• To apply the formalism to coset spaces of compact Lie groups, particularly $S^2 = SU(2)/U(1)$, to demonstrate its viability.

Proposed method

• Develops a Lagrangian formulation for Hamiltonian systems with constraints, where the Hamiltonian is linear and quadratic in momenta.
• Derives the propagator in the full phase space and compares it to the propagator obtained via Dirac's method.
• Identifies a new class of ambiguities in Dirac's approach arising from the ordering of constraints and gauge-fixing conditions.
• Uses the Lagrangian framework to derive a precise mathematical relation between the full-space and Dirac-restricted propagators (equation 5.25).
• Applies the formalism to coset spaces $G/H$ of compact Lie groups, selecting a natural Dirac quantization scheme.
• Explicitly computes the propagator on $S^2 = SU(2)/U(1)$ to verify self-consistency and equivalence.

Experimental results

Research questions

• RQ1 What are the fundamental ambiguities introduced when quantizing a constrained system before reducing the constraints (Dirac's method)?
• RQ2 Under what conditions is Dirac's 'first quantize then reduce' approach self-consistent and equivalent to the 'first reduce then quantize' method?
• RQ3 How can a Lagrangian formulation help clarify the relationship between the propagator in the full space and the propagator in the reduced phase space?
• RQ4 Can a consistent and equivalent quantization be achieved on coset spaces of compact Lie groups using Dirac's method?
• RQ5 What is the explicit form of the propagator on $S^2 = SU(2)/U(1)$ when derived via Dirac's method, and does it match the full-space result?

Key findings

• A new source of ambiguity is identified in Dirac's method, arising from the ordering of gauge-fixing and constraint quantization, which affects self-consistency.
• A precise mathematical relation is derived between the propagator in the full phase space and the Dirac-restricted propagator (equation 5.25), enabling direct comparison.
• For coset spaces of compact Lie groups, a natural choice of Dirac quantization ensures full self-consistency and equivalence with the 'first reduce' method.
• The propagator on $S^2 = SU(2)/U(1)$ computed via Dirac's method matches the expected physical result, confirming consistency.
• The Lagrangian approach provides a systematic framework to analyze and resolve ordering ambiguities in constrained quantization.
• The formalism successfully resolves long-standing questions about the equivalence of reduction and quantization ordering in symmetric spaces.