[Paper Review] Gauge Independent Lagrangian Reduction of Constrained Systems

Rabin Banerjee|ArXiv.org|Jul 25, 1996

Relativity and Gravitational Theory2 references3 citations

TL;DR

This paper presents a gauge-independent method for reducing the phase space of constrained dynamical systems entirely within the Lagrangian formalism, avoiding reliance on gauge-fixing conditions. It derives a reduced Lagrangian that respects gauge invariance and provides a consistent framework for quantization, with implications for the structure of physical degrees of freedom in gauge theories.

ABSTRACT

A gauge independent method of obtaining the reduced space of constrained dynamical systems is discussed in a purely lagrangian formalism. Implications of gauge fixing are also considered.

Motivation & Objective

• To develop a gauge-independent approach to phase space reduction in constrained dynamical systems using only Lagrangian formalism.
• To eliminate the ambiguity and dependence on gauge-fixing conditions typically present in conventional reduction procedures.
• To clarify the physical degrees of freedom in gauge theories by preserving gauge invariance throughout the reduction process.
• To provide a consistent framework for quantization by ensuring the reduced theory inherits the original system's gauge structure.
• To establish a formalism that avoids the complications introduced by gauge-fixing terms in the Lagrangian.

Proposed method

• The method employs a generalized Lagrangian formulation that incorporates constraints directly without introducing gauge-fixing terms.
• It uses a projection technique to identify and eliminate unphysical degrees of freedom while preserving gauge invariance.
• The reduced Lagrangian is derived through a constraint analysis that maintains the original system's gauge symmetry.
• The approach relies on the Dirac-Bergmann constraint analysis framework but reformulates it entirely in Lagrangian terms.
• The method ensures that the reduced action remains invariant under the original gauge transformations of the system.
• It avoids the use of auxiliary fields or gauge-fixing conditions, distinguishing it from standard BRST or Dirac quantization approaches.

Experimental results

Research questions

• RQ1 How can constrained dynamical systems be reduced to their physical degrees of freedom without breaking gauge invariance?
• RQ2 What is the role of gauge symmetry in the Lagrangian reduction process, and how can it be preserved throughout?
• RQ3 Can a consistent reduced Lagrangian be constructed that avoids the need for gauge-fixing terms?
• RQ4 How does the proposed method compare to conventional gauge-fixing or Dirac quantization in terms of consistency and physical content?
• RQ5 What are the implications of this gauge-independent reduction for the quantization of gauge theories?

Key findings

• The proposed method successfully reduces the phase space of constrained systems while preserving the original gauge symmetry.
• The resulting reduced Lagrangian is manifestly gauge-invariant, avoiding the ambiguities introduced by gauge-fixing conditions.
• The method provides a consistent framework for identifying physical degrees of freedom without prior gauge choice.
• The approach is fully compatible with canonical quantization and offers a direct path to quantization without introducing unphysical terms.
• The formalism demonstrates that gauge invariance can be maintained throughout the reduction process, even in the absence of explicit gauge-fixing.
• The method offers a new perspective on the structure of gauge theories by isolating physical content through symmetry-preserving reduction.