[Paper Review] 1D Diffeomorphism Invariant Model of a Free Scalar Relativistic Particle: Supervariable and BRST Approaches

B. Chauhan, A. K. Rao+2 more|arXiv (Cornell University)|Dec 30, 2019

Algebraic structures and combinatorial models2 references4 citations

TL;DR

This paper develops a 1D diffeomorphism-invariant model of a free scalar relativistic particle using the supervariable and BRST approaches, deriving off-shell nilpotent and absolutely anticommutative (anti-)BRST symmetries. It introduces a novel modified Bonora-Tonin supervariable approach and proves the existence of the Curci-Ferrari-type restriction through symmetry considerations and the (anti-)chiral supervariable approach, ensuring consistency of the quantum theory.

ABSTRACT

We apply the supervariable approach to derive the proper quantum Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetries for the 1D diffeomorphism invariant model of a free scalar relativistic particle by exploiting the infinitesimal classical reparameterization (i.e. 1D diffeomorphism) symmetry of this theory. We derive the conserved and off-shell nilpotent (anti-)BRST charges and prove their absolute anticommutativity property by using the virtues of Curci-Ferrari (CF)-type restriction of our present theory. We establish the sanctity of the existence of CF-type restriction (i) by considering the (anti-)BRST symmetry transformations of the coupled (but equivalent) Lagrangians, and (ii) by proving the symmetry invariance of the Lagrangians within the framework of supervariable approach. We capture the nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges within the framework of (anti-)chiral supervariable approach (ACSA) to BRST formalism. One of the novel observations of our present endeavor is the derivation of CF-type restriction by using the modified Bonora-Tonin (BT) supervariable approach (while deriving the (anti-)BRST symmetries for the target spacetime and/or momenta variables) and by symmetry considerations of the Lagrangians of the theory. The rest of the (anti-)BRST symmetries, for the other variables, are derived by using the newly proposed ACSA. We also demonstrate the existence of CF-type restriction in the proof of absolute anticommutativity of the (anti-)BRST charges.

Motivation & Objective

• To derive off-shell nilpotent and absolutely anticommutative (anti-)BRST symmetries for a 1D diffeomorphism-invariant free scalar relativistic particle.
• To establish the existence of the Curci-Ferrari-type restriction through symmetry invariance and supervariable formalism.
• To generalize the (anti-)chiral supervariable approach (ACSA) to include matter and gauge fields in diffeomorphism-invariant theories.
• To demonstrate that the (anti-)BRST charges are conserved and their anticommutativity is protected by the CF-type restriction.

Proposed method

• Employing the modified Bonora-Tonin (BT) supervariable approach to derive (anti-)BRST symmetries for target spacetime and momentum variables.
• Applying the (anti-)chiral supervariable approach (ACSA) to systematically derive (anti-)BRST symmetries for all other variables in the theory.
• Imposing the Curci-Ferrari (CF)-type restriction to ensure off-shell nilpotency and absolute anticommutativity of the (anti-)BRST charges.
• Using symmetry considerations of the coupled Lagrangians to justify the CF-type restriction as a physical constraint.
• Verifying the consistency of the BRST and anti-BRST symmetries through explicit transformation laws and invariance of the action.
• Proving the absolute anticommutativity of the (anti-)BRST charges by showing their BRST/anti-BRST transformations vanish only when the CF-type restriction holds.

Experimental results

Research questions

• RQ1 How can the (anti-)BRST symmetries be consistently derived for a 1D diffeomorphism-invariant free scalar relativistic particle using supervariable techniques?
• RQ2 What is the role of the Curci-Ferrari-type restriction in ensuring the nilpotency and absolute anticommutativity of the (anti-)BRST charges in this model?
• RQ3 Can the modified Bonora-Tonin supervariable approach be extended to derive (anti-)BRST symmetries for matter and gauge fields in diffeomorphism-invariant theories?
• RQ4 How does the (anti-)chiral supervariable approach (ACSA) provide a geometric framework for deriving the (anti-)BRST symmetries in this context?
• RQ5 Under what conditions is the CF-type restriction physically valid, and how is it derived from symmetry principles rather than ad hoc assumptions?

Key findings

• The (anti-)BRST charges are conserved and off-shell nilpotent, with their nilpotency and absolute anticommutativity proven via the (anti-)chiral supervariable approach.
• The Curci-Ferrari-type restriction emerges naturally from symmetry considerations of the Lagrangian and is shown to be invariant under (anti-)BRST transformations.
• The modified Bonora-Tonin supervariable approach successfully derives (anti-)BRST symmetries for target spacetime and momentum variables, extending the formalism to diffeomorphism-invariant systems.
• The absolute anticommutativity of the (anti-)BRST charges is rigorously established only when the CF-type restriction is imposed, confirming its physical necessity.
• The theory remains invariant under (anti-)BRST transformations only on the submanifold defined by the CF-type restriction, validating its role as a physical constraint.
• The derivation of the CF-type restriction via symmetry and supervariable methods provides a novel, geometrically consistent foundation for BRST quantization in 1D diffeomorphism-invariant models.