Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:hep-th/9912216 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Canonical approach to 2D WZNW model, non-Abelian bosonization, and anomalies

BRANISLAV SAZDOVIĆ

Published Jul 20, 2000• 6 citations
arXivHTMLPDF

Abstract

The gauged WZNW model has been derived as an effective action, whose Poisson brackets algebra of the constraints is isomorphic to the commutator algebra of operators in quantized fermionic theory. As a consequence, the Hamiltonian as well as the usual Lagrangian non-Abelian bosonization rules have been obtained for the chiral currents and chiral densities. The expression for the anomaly has been obtained as a function of the Schwinger term, using canonical methods.

DOI
10.1103/physrevd.62.045011

Related papers

A Quantum Gauge Group Approach to the 2D SU(n) WZNW Model

Paolo Furlan, Ludmil K. Hadjiivanov, Ivan T. Todorov +1 more

The canonical quantization of the WZNW model provides a complete set of exchange relations in the enlarged chiral state spaces that include the Gauss components $M_\\pm$, ${\\bar M}_\\pm$ of the monodromy matrices $M$, $\\bar M$. Regarded as new dynamical variables, the elements of $M$ and $\\bar M$ cannot be identified -- they satisfy different exchange relations. Accordingly, the two dimensional theory expressed in terms of the left and right movers' fields does not automatically respect monodromy invariance. Continuing our recent work on the subject, we show that the definition of the appropriate physical subspace of the $SU(n)$ WZNW model involves an extended quantum group symmetry in a finite dimensional Fock space. In the simplest ($n=2$) case we explicitly construct the subspace of $U_q(sl(2))\\otimes U_q(sl(2))$ invariant vectors. Monodromy invariance is also restored in a weak sense.

•Jan 1, 1996•13 citations

A quantum model with one bosonic degree of freedom

Ning Wu, Tunan Ruan

Conventionally, when we construct a quantum model, we must first know the corresponding classical model. Applying the correspondence between the classical Poisson brackets and the canonical commutator, we can find the canonical quantization condition. Through the example given in this paper, we will find that it is not necessary to do this. In the example, a Lagrangian operator is operator gauge invariant. After localization, in order to keep the operator gauge invariance of the operator action, we must introduce a gauge potential. The Euler - Lagrange equation of motion of q gives the usual operator equation of motion, and the gauge potential gives a constraint. This constraint is just the usual canonical quantization condition.

Journal of Physics A Mathematical and General•Dec 7, 1996•0 citations

New quantum mechanical model

Wu Ning

A quantum mechanical model with one bosonic degree of freedom is discussed in detail. Conventionally, when a quantum mechanical model is constructed, one must know the corresponding classical model. And by applying the correspondence between the classical Poisson brackets and the canonical commutator, the canonical quantization condition can be obtained. In the quantum model, study of the corresponding classical model is needed first. In this model, the Lagrangian is an operator gauge invariant. After localization, in order to keep gauge invariance, the operator gauge potential must be introduced. The Eular-Lagrange equation of motion of the dynamical argument gives the usual operator equation of motion. And the operator gauge potential just gjves a constraint. This constraint is just the usual canonical quantization condition.

Science China Mathematics•Jan 1, 1996•0 citations

Trace dynamics: the classical Lagrangian and Hamiltonian dynamics of matrix models

Stephen L. Adler

In this chapter we set up a classical Lagrangian and Hamiltonian dynamics for matrix models. The fundamental idea is to set up an analog of classical dynamics in which the phase space variables are non-commutative, and the basic tool that allows one to accomplish this is cyclic invariance under a trace. Since no assumptions about commutativity of the phase space variables (such as canonical commutators/anticommutators) are made at this stage, the dynamics that we set up is not the same as standard quantum mechanics. Quantum mechanical behavior will be seen to emerge only when, in Chapters 4 and 5, we study the statistical mechanics of the classical matrix dynamics formulated here. In Section 1.1, we introduce our basic notation for bosonic and fermionic matrices, and give the cyclic identities that will be used repeatedly throughout the book. In Section 1.2, we define the derivative of a trace quantity with respect to an operator, and give the basic properties of this definition. In Section 1.3, we use the operator derivative to formulate a Lagrangian and Hamiltonian dynamics for matrix models. In Section 1.4, we introduce a generalized Poisson bracket appropriate to trace dynamics, constructed from the operator derivative defined in Section 1.2, and give its properties and some applications. Finally, in Section 1.5 we discuss the relation between the trace dynamics time evolution equations, and the usual unitary Heisenberg picture equations of motion obtained when one assumes standard canonical commutators/anticommutators. Bosonic and fermionic matrices and the cyclic trace identities We shall assume finite-dimensional matrices, although ultimately an extension to the infinite-dimensional case may be needed. The matrix elements of these matrices will be constructed from ordinary complex numbers, and from complex anticommuting Grassmann numbers.

Cambridge University Press eBooks•Aug 26, 2004•0 citations

Classical Fermionic Dynamics

Luther Rinehart

The formulation of classical mechanics applicable to fermionic degrees of freedom is presented in mathematically rigorous terms, including a description of how the mathematical structure relates to the quantization of the theory. Canonical transformations and the algebra of observables are defined and studied. A formula is given for the analog of the Poisson bracket. The quantization of the theory proceeds according to deformation quantization.

arXiv:1505.00237•May 1, 2015•0 citations

A Quantum Gauge Group Approach to the 2D SU(n) WZNW Model

Paolo Furlan, Ludmil K. Hadjiivanov, Ivan T. Todorov

The cannonical quantization of the WZNW model provides a complete set of exchange relation in the enlarged chiral state spaces that include the Gauss components M±, [Formula: see text] of the monodromy matrices M, [Formula: see text]. Regarded as new dynamical variables, the elements of M and [Formula: see text] cannot be identified — they satisfy different exchange relations. Accordingly, the two dimensional theory expressed in terms of the left and right movers' fields does not automatically respect monodromy invariance. Continuing our recent work on the subject we show that the definition of the appropriate physical subspace of the SU(n) WZNW model involves an extended quantum group symmetry in a finite dimensional Fock space. In the simplest (n = 2) case we explicitly construct the subspace of Uq(sl(2)) ⊗ Uq(sl(2)) invariant vectors. Monodromy invariance is also restored in a weak sense.

arXiv:hep-th/9610202•International Journal of Modern Physics A•Jan 10, 1997•13 citations
The gauged WZNW model has been derived as an effective action, whose Poisson brackets algebra of the constraints is isomorphic to the commutator algebra of operators in quantized fermionic theory. As a consequence, the Hamiltonian as well as the usual Lagrangian non-Abelian bosonization rules have been obtained for the chiral currents and chiral densities. The expression for the anomaly has been obtained as a function of the Schwinger term, using canonical methods.
The canonical quantization of the WZNW model provides a complete set of exchange relations in the enlarged chiral state spaces that include the Gauss components $M_\\pm$, ${\\bar M}_\\pm$ of the monodromy matrices MMM, barM\\bar MbarM. Regarded as new dynamical variables, the elements of MMM and barM\\bar MbarM cannot be identified -- they satisfy different exchange relations. Accordingly, the two dimensional theory expressed in terms of the left and right movers' fields does not automatically respect monodromy invariance. Continuing our recent work on the subject, we show that the definition of the appropriate physical subspace of the SU(n)SU(n)SU(n) WZNW model involves an extended quantum group symmetry in a finite dimensional Fock space. In the simplest (n=2n=2n=2) case we explicitly construct the subspace of Uq(sl(2))otimesUq(sl(2))U_q(sl(2))\\otimes U_q(sl(2))Uq​(sl(2))otimesUq​(sl(2)) invariant vectors. Monodromy invariance is also restored in a weak sense.
Conventionally, when we construct a quantum model, we must first know the corresponding classical model. Applying the correspondence between the classical Poisson brackets and the canonical commutator, we can find the canonical quantization condition. Through the example given in this paper, we will find that it is not necessary to do this. In the example, a Lagrangian operator is operator gauge invariant. After localization, in order to keep the operator gauge invariance of the operator action, we must introduce a gauge potential. The Euler - Lagrange equation of motion of q gives the usual operator equation of motion, and the gauge potential gives a constraint. This constraint is just the usual canonical quantization condition.
A quantum mechanical model with one bosonic degree of freedom is discussed in detail. Conventionally, when a quantum mechanical model is constructed, one must know the corresponding classical model. And by applying the correspondence between the classical Poisson brackets and the canonical commutator, the canonical quantization condition can be obtained. In the quantum model, study of the corresponding classical model is needed first. In this model, the Lagrangian is an operator gauge invariant. After localization, in order to keep gauge invariance, the operator gauge potential must be introduced. The Eular-Lagrange equation of motion of the dynamical argument gives the usual operator equation of motion. And the operator gauge potential just gjves a constraint. This constraint is just the usual canonical quantization condition.
In this chapter we set up a classical Lagrangian and Hamiltonian dynamics for matrix models. The fundamental idea is to set up an analog of classical dynamics in which the phase space variables are non-commutative, and the basic tool that allows one to accomplish this is cyclic invariance under a trace. Since no assumptions about commutativity of the phase space variables (such as canonical commutators/anticommutators) are made at this stage, the dynamics that we set up is not the same as standard quantum mechanics. Quantum mechanical behavior will be seen to emerge only when, in Chapters 4 and 5, we study the statistical mechanics of the classical matrix dynamics formulated here. In Section 1.1, we introduce our basic notation for bosonic and fermionic matrices, and give the cyclic identities that will be used repeatedly throughout the book. In Section 1.2, we define the derivative of a trace quantity with respect to an operator, and give the basic properties of this definition. In Section 1.3, we use the operator derivative to formulate a Lagrangian and Hamiltonian dynamics for matrix models. In Section 1.4, we introduce a generalized Poisson bracket appropriate to trace dynamics, constructed from the operator derivative defined in Section 1.2, and give its properties and some applications. Finally, in Section 1.5 we discuss the relation between the trace dynamics time evolution equations, and the usual unitary Heisenberg picture equations of motion obtained when one assumes standard canonical commutators/anticommutators. Bosonic and fermionic matrices and the cyclic trace identities We shall assume finite-dimensional matrices, although ultimately an extension to the infinite-dimensional case may be needed. The matrix elements of these matrices will be constructed from ordinary complex numbers, and from complex anticommuting Grassmann numbers.
The cannonical quantization of the WZNW model provides a complete set of exchange relation in the enlarged chiral state spaces that include the Gauss components M±, [Formula: see text] of the monodromy matrices M, [Formula: see text]. Regarded as new dynamical variables, the elements of M and [Formula: see text] cannot be identified — they satisfy different exchange relations. Accordingly, the two dimensional theory expressed in terms of the left and right movers' fields does not automatically respect monodromy invariance. Continuing our recent work on the subject we show that the definition of the appropriate physical subspace of the SU(n) WZNW model involves an extended quantum group symmetry in a finite dimensional Fock space. In the simplest (n = 2) case we explicitly construct the subspace of Uq(sl(2)) ⊗ Uq(sl(2)) invariant vectors. Monodromy invariance is also restored in a weak sense.