scholarly journals A NOTE ON NONCOMMUTATIVE CHERN–SIMONS MODEL ON MANIFOLDS WITH BOUNDARY

2002 ◽  
Vol 17 (03) ◽  
pp. 141-155 ◽  
Author(s):  
ADRIÁN R. LUGO
Keyword(s):  
Canonical Quantization ◽  
Quantum Hall ◽  
Simons Theory ◽  
Quantum Hall Systems ◽  
Integrable Functions ◽  
Dimensional Boundary ◽  
The One ◽  
Definition Of ◽  
Square Integrable Functions ◽  
Chern Simons

We study field theories defined in regions of the spatial noncommutative (NC) plane with a boundary present delimiting them, concentrating in particular on the U(1) NC Chern–Simons theory on the upper half-plane. We find that classical consistency and gauge invariance lead necessary to the introduction of K0-space of square integrable functions null together with all their derivatives at the origin. Furthermore the requirement of closure of K0 under the *-product leads to the introduction of a novel notion of the *-product itself in regions where a boundary is present, that in turn yields the complexification of the gauge group and to consider chiral waves in one sense or other. The canonical quantization of the theory is sketched identifying the physical states and the physical operators. These last ones include ordinary NC Wilson lines starting and ending on the boundary that yield correlation functions depending on points on the one-dimensional boundary. We finally extend the definition of the *-product to a strip and comment on possible relevance of these results to finite quantum Hall systems.

scholarly journals SINGLE-MODE APPROXIMATION AND EFFECTIVE CHERN–SIMONS THEORIES FOR QUANTUM HALL SYSTEMS

2003 ◽  
Vol 17 (31n32) ◽  
pp. 5875-5891 ◽  
Author(s):  
K. SHIZUYA
Keyword(s):  
Landau Level ◽  
Quantum Hall ◽  
Single Mode ◽  
Collective Excitations ◽  
Effective Theory ◽  
Simons Theory ◽  
Quantum Hall Systems ◽  
Single Mode Approximation ◽  
Chern Simons ◽  
Chern Simons Theory

A unified description of elementary and collective excitations in quantum Hall systems is presented within the single-mode approximation (SMA) framework, with emphasis on revealing an intimate link with Chern–Simons theories. It is shown that for a wide class of quantum Hall systems the SMA in general yields, as an effective theory, a variant of the bosonic Chern–Simons theory. For single-layer systems the effective theory agrees with the standard Chern–Simons theory at long wavelengths whereas substantial deviations arise for collective excitations in bilayer systems. It is suggested, in particular, that Hall-drag experiments would be a good place to detect out-of-phase collective excitations inherent to bilayer systems. It is also shown that the intra-Landau-level modes bear a similarity in structure (though not in scale) to the inter-Landau-level modes, and its implications on the composite-fermion and composite-boson theories are discussed.

scholarly journals Chern-Simons theory of multicomponent quantum Hall systems

2010 ◽  
Vol 81 (19) ◽  
Author(s):  
W. Beugeling ◽  
M. O. Goerbig ◽  
C. Morais Smith
Keyword(s):  
Quantum Hall ◽  
Simons Theory ◽  
Quantum Hall Systems ◽  
Chern Simons ◽  
Chern Simons Theory

THE QUANTUM GROUP REALIZATIONS OF 3-D CHERN–SIMONS THEORIES

1995 ◽  
Vol 10 (01) ◽  
pp. 39-49
Author(s):  
C. RAMÍREZ ◽  
L. F. URRUTIA
Keyword(s):  
Quantum Group ◽  
Deformation Parameter ◽  
Canonical Quantization ◽  
Classical Case ◽  
Quantum Algebra ◽  
Simons Theory ◽  
R Matrix ◽  
The One ◽  
Chern Simons ◽  
Chern Simons Theory

The algebra of the integrated connections and of their traces is considered in the one-genus sector of classical and quantum Chern–Simons theory. In the classical case this algebra is braid-like and although the corresponding Jacobi identities are satisfied, the associated r-matrix does not satisfy the classical Yang–Baxter equations. However, it turns out this algebra originates a "quantum" algebra SU (2)q given by its trace algebra. Canonical quantization of the above algebra is performed and a one-parameter expression for the operator ordering is considered. The same quantum algebra with a modified deformation parameter, nontrivially depending on ħ, is obtained.

scholarly journals NON-ABELIAN FRACTIONAL QUANTUM HALL STATES AND CHIRAL COSET CONFORMAL FIELD THEORIES

2000 ◽  
Vol 15 (30) ◽  
pp. 4857-4870 ◽  
Author(s):  
D. C. CABRA ◽  
E. FRADKIN ◽  
G. L. ROSSINI ◽  
F. A. SCHAPOSNIK
Keyword(s):  
Quantum Hall ◽  
Field Theories ◽  
Simons Theory ◽  
Conformal Field Theories ◽  
Fractional Quantum ◽  
Fractional Quantum Hall ◽  
Conformal Field ◽  
Quantum Hall States ◽  
Effective Lagrangian ◽  
Chern Simons

We propose an effective Lagrangian for the low energy theory of the Pfaffian states of the fractional quantum Hall effect in the bulk in terms of non-Abelian Chern–Simons (CS) actions. Our approach exploits the connection between the topological Chern–Simons theory and chiral conformal field theories. This construction can be used to describe a large class of non-Abelian FQH states.

scholarly journals ON CANONICAL QUANTIZATION OF THE GAUGED WZW MODEL WITH PERMUTATION BRANES

2011 ◽  
Vol 26 (26) ◽  
pp. 4647-4660
Author(s):  
GOR SARKISSIAN
Keyword(s):  
Riemann Surface ◽  
Phase Space ◽  
Boundary Conditions ◽  
Canonical Quantization ◽  
Simons Theory ◽  
Gauge Fields ◽  
Time Line ◽  
Chern Simons ◽  
Special Case ◽  
Chern Simons Theory

In this paper we perform canonical quantization of the product of the gauged WZW models on a strip with boundary conditions specified by permutation branes. We show that the phase space of the N-fold product of the gauged WZW model G/H on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of the double Chern–Simons theory on a sphere with N holes times the time-line with G and H gauge fields both coupled to two Wilson lines. For the special case of the topological coset G/G we arrive at the conclusion that the phase space of the N-fold product of the topological coset G/G on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of Chern–Simons theory on a Riemann surface of the genus N-1 times the time-line with four Wilson lines.

scholarly journals Hermite Functions and Fourier Series

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 853
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano del Olmo
Keyword(s):  
Fourier Transform ◽  
Hermite Functions ◽  
Quantum Oscillator ◽  
Linear Combinations ◽  
Ladder Operators ◽  
Orthonormal Bases ◽  
Integrable Functions ◽  
The Fourier Transform ◽  
The One ◽  
Square Integrable Functions

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.

scholarly journals THE EDGE STATES OF THE BF SYSTEM AND THE LONDON EQUATIONS

1993 ◽  
Vol 08 (04) ◽  
pp. 723-752 ◽  
Author(s):  
A.P. BALACHANDRAN ◽  
P. TEOTONIO-SOBRINHO
Keyword(s):  
Scalar Field ◽  
Scaling Limit ◽  
Quantum Hall ◽  
Simons Theory ◽  
Massless Scalar ◽  
Electromagnetic Current ◽  
Massless Scalar Field ◽  
Edge States ◽  
Chern Simons ◽  
London Equations

It is known that the 3D Chern–Simons interaction describes the scaling limit of a quantum Hall system and predicts edge currents in a sample with boundary, the currents generating a chiral U(1) Kac-Moody algebra. It is no doubt also recognized that, in a somewhat similar way, the 4D BF interaction (with B a two-form, dB the dual *j of the electromagnetic current, and F the electromagnetic field form) describes the scaling limit of a superconductor. We show in this paper that there are edge excitations in this model as well for manifolds with boundaries. They are the modes of a scalar field with invariance under the group of diffeomorphisms (diffeos) of the bounding spatial two-manifold. Not all diffeos of this group seem implementable by operators in quantum theory, the implementable group being a subgroup of volume-preserving diffeos. The BF system in this manner can lead to the w1+∞ algebra and its variants. Lagrangians for fields on the bounding manifold which account for the edge observables on quantization are also presented. They are the analogs of the (1+1)-dimensional massless scalar field Lagrangian describing the edge modes of an Abelian Chern-Simons theory with a disk as the spatial manifold. We argue that the addition of “Maxwell” terms constructed from F∧*F and dB∧*dB does not affect the edge states, and that the augmented Lagrangian has an infinite number of conserved charges—the aforementioned scalar field modes—localized at the edges. This Lagrangian is known to describe London equations and a massive vector field. A (3+1)-dimensional generalization of the Hall effect involving vortices coupled to B is also proposed.

scholarly journals Dualities in Quantum Hall System and Noncommutative Chern-Simons Theory

2002 ◽  
Vol 2002 (01) ◽  
pp. 002-002 ◽  
Author(s):  
Alexander Gorsky ◽  
Ian I Kogan ◽  
Chris Korthals-Altes
Keyword(s):  
Quantum Hall ◽  
Simons Theory ◽  
Quantum Hall System ◽  
Chern Simons ◽  
Chern Simons Theory
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