[Paper Review] Spinning particle and null-string on $AdS_d$: projective-space approach

Д. В. Уваров|arXiv (Cornell University)|Jul 18, 2017

Black Holes and Theoretical Physics56 references3 citations

TL;DR

This paper proposes spinning particle and tensionless string models on $AdS_d$ using a projective-space Hamiltonian formulation, deriving classical Lagrangians and establishing a Lax-type representation of equations of motion. It demonstrates that a quantum BRST generator is nilpotent for any $d$ under specific operator ordering, while conformal reparametrizations and space-time dilatations are anomalous in Fourier mode ordering.

ABSTRACT

Spinning particle and tensionless string models on $AdS_d$ background in the projective-space realization are proposed as constrained Hamiltonian systems. Various forms of particle and string Lagrangians are derived and classical mechanics is studied including the Lax-type representation of the equations of motion. After that transition to the quantum theory is discussed. Analysis of potential anomalies in the tensionless string model necessitates introduction of ghosts and BRST generator. It is shown that quantum BRST generator is nilpotent for any $d$ if coordinate-momentum ordering for the phase-space bosonic variables, Weyl ordering for the fermions and $cb$ ($\gamma\beta$) ordering for ghosts is chosen, while conformal reparametrizations and space-time dilatations turn out to be anomalous for the ordering in terms of positive and negative Fourier modes of the phase-space variables and ghosts.

Motivation & Objective

• To formulate spinning particle and tensionless string models on $AdS_d$ as constrained Hamiltonian systems using projective-space realization.
• To derive classical Lagrangians and analyze their dynamics, including a Lax-type representation of equations of motion.
• To investigate the transition from classical to quantum theory, particularly focusing on anomalies in the tensionless string model.
• To determine conditions under which the quantum BRST generator remains nilpotent and identify anomalous symmetries in the quantum theory.

Proposed method

• Formulate the models as constrained Hamiltonian systems in projective space, encoding $AdS_d$ geometry through phase-space constraints.
• Derive various forms of particle and string Lagrangians from the Hamiltonian structure, preserving gauge symmetries.
• Construct a Lax-type representation of the equations of motion, enabling integrability analysis.
• Apply operator ordering prescriptions: coordinate-momentum for bosons, Weyl for fermions, and $cb$ ($\gamma\beta$) for ghost fields.
• Analyze quantum anomalies by examining the closure of the BRST algebra under different ordering schemes.
• Study the behavior of conformal reparametrizations and space-time dilatations in terms of positive and negative Fourier modes of phase-space variables and ghosts.

Experimental results

Research questions

• RQ1 Under what conditions is the quantum BRST generator nilpotent in the tensionless string model on $AdS_d$?
• RQ2 How do different operator ordering prescriptions affect the nilpotency of the BRST generator and the consistency of the quantum theory?
• RQ3 Which symmetries—conformal reparametrizations and space-time dilatations—become anomalous in the quantum regime?
• RQ4 Can a Lax-type representation be constructed for the equations of motion of spinning particles and null strings on $AdS_d$?
• RQ5 What role does projective-space realization play in unifying the description of spinning particles and tensionless strings in $AdS_d$?

Key findings

• The quantum BRST generator is nilpotent for any $d$ when using coordinate-momentum ordering for bosons, Weyl ordering for fermions, and $cb$ ($\gamma\beta$) ordering for ghosts.
• Conformal reparametrizations are anomalous in the quantum theory when expressed in terms of positive and negative Fourier modes of phase-space variables.
• Space-time dilatations are also anomalous under the same Fourier mode ordering, indicating a breakdown of these symmetries in the quantum regime.
• The classical dynamics of the spinning particle and null string admit a Lax-type representation, suggesting integrability properties.
• The projective-space formulation successfully unifies the description of spinning particles and tensionless strings on $AdS_d$ within a single constrained Hamiltonian framework.
• The analysis reveals that anomalies in the quantum theory are sensitive to the choice of operator ordering, particularly in the Fourier mode decomposition of the phase-space variables.