[Paper Review] Embedding Second Class Systems via Symplectic Gauge-invariant Formalism

Jorge Ananias Neto, Mendes, A. C. R.+3 more|ArXiv.org|Sep 11, 2001

Numerical methods for differential equations2 references3 citations

TL;DR

This paper introduces a novel symplectic gauge-invariant formalism to embed second-class constrained systems into first-class gauge theories without the ambiguity issues plaguing BFFT and iterative methods. By extending the phase space with Wess-Zumino (WZ) variables and modifying the symplectic matrix to render it non-singular, the method successfully restores gauge symmetry in both Abelian and non-Abelian models—demonstrated via the Proca, nonlinear sigma, and chiral Schwinger models—while preserving dynamical equivalence to the original theories.

ABSTRACT

In this paper we reformulate Abelian and non-Abelian noninvariant systems as gauge invariant theories using a new constraint conversion scheme, developed on the symplectic framework. This conversion method is not plagued by the ambiguity problem that torments the BFFT and iterative methods and also it seems more powerful since it does not require special modifications to handle with non-Abelian systems.

Motivation & Objective

• To resolve the ambiguity problem in constraint conversion methods like BFFT and iterative schemes when introducing Wess-Zumino (WZ) variables.
• To develop a unified formalism that handles both Abelian and non-Abelian second-class systems without requiring algebra-specific modifications.
• To reveal hidden gauge symmetries in original phase space, rather than only in extended WZ space, as seen in prior approaches.
• To demonstrate the method’s effectiveness on key models in high-energy physics: the nonlinear sigma model and the chiral Schwinger model.
• To show that a single WZ field suffices for gauge restoration, avoiding the need for multiple fields that exacerbate ambiguity.

Proposed method

• The method is built on the symplectic framework, where the symplectic matrix's singularity signals constraints; the formalism renders it non-singular by introducing arbitrary functions of original and WZ variables into the first-order Lagrangian.
• It reformulates second-class systems as gauge-invariant theories by extending the phase space with WZ fields, ensuring the resulting constraints are first-class.
• The procedure avoids the ambiguity of prior methods by not relying on iterative or BFFT-style reconstruction of the symplectic structure.
• The formalism is applied to Abelian models (Proca, nonlinear sigma model, chiral Schwinger model) and then to the non-Abelian Proca model, showing no need for algebra-specific adjustments.
• Gauge symmetry is verified via infinitesimal transformations, and unitary gauge fixing restores the original Dirac brackets, confirming dynamical equivalence.
• The Dirac bracket structure is computed in both original and extended phase spaces to confirm consistency and equivalence.

Experimental results

Research questions

• RQ1 Can a constraint conversion method be developed that avoids the ambiguity problem inherent in BFFT and iterative approaches when introducing Wess-Zumino fields?
• RQ2 Does the symplectic gauge-invariant formalism preserve dynamical equivalence between the original second-class system and the resulting first-class gauge theory?
• RQ3 Can this method reveal hidden gauge symmetries in the original phase space, rather than only in the extended WZ space?
• RQ4 Is the formalism universally applicable to both Abelian and non-Abelian models without requiring modifications based on the algebraic structure?
• RQ5 Can a single Wess-Zumino field suffice to restore gauge symmetry, reducing the complexity and ambiguity of prior methods?

Key findings

• The symplectic gauge-invariant formalism successfully converts second-class constraints into first-class ones without ambiguity, unlike BFFT and iterative methods.
• In the nonlinear sigma model, a hidden gauge symmetry is revealed in the original phase space, contrasting with previous approaches that localize symmetry only in the extended WZ space.
• For the chiral Schwinger model, the formalism cancels the chiral anomaly and restores gauge symmetry using only one WZ field, avoiding the need for multiple fields that increase ambiguity.
• The non-Abelian Proca model is successfully embedded into a gauge-invariant theory without algebra-specific modifications, proving the method's robustness for non-Abelian systems.
• After unitary gauge fixing, the Dirac brackets in the extended phase space exactly reproduce the original Dirac brackets, confirming dynamical equivalence.
• The method achieves gauge invariance with minimal extension—only one WZ field per constraint—making it more efficient and less ambiguous than prior schemes.