Chern-Simons Quantum Mechanics Research Articles

Chern–Simons quantum mechanics and fractional angular momentum in atom system

The model of a planar atom which possesses a nonvanishing electric dipole moment interacting with magnetic fields in a specific setting is studied. Energy spectra of this model and its reduced model, which is the limit of cooling down the atom to the negligible kinetic energy, are solved exactly. We show that energy spectra of the reduced model cannot be obtained directly from the full ones by taking the same limit. In order to get the energy spectra of the reduced model from the full model, we must regularize energy spectra of the full model properly when the limit of the negligible kinetic energy is taken. It is one of the characteristics of the Chern–Simons quantum mechanics. Besides this, the canonical angular momentum of the reduced model will take fractional values although the full model can only take integers. It means that it is possible to realize the Chern–Simons quantum mechanics and fractional angular momentum simultaneously by this model.

Fractional angular momentum in noncommutative generalized Chern-Simons quantum mechanics

The noncommutative generalized Chern-Simons quantum mechanics, i.e., the Chern-Simons quantum mechanics on the noncommutative plane in the presence of Aharonov-Bohm magnetic vector potentials, is studied in this paper. We focus our attention on the canonical orbital angular momentum and show that there are two different approaches to produce the fractional angular momentum in the noncommutative generalized Chern-Simons quantum mechanics.

Higher order theories and their relationship with noncommutativity

We present a relationship between noncommutativity and higher order time derivative theories using a method perturbative. We introduce a generalization of the Chern-Simons Quantum Mechanics for higher order time derivatives. This model presents noncommutativity in a natural way when we project to states of low energy. Compared with the usual model, our system presents noncommutativity without the necessity of taking the limit of strong field. We quantized the theory using a Bopp's shift of the noncommutative variables and we obtain an spectrum without negatives energies. In addition we extend the model to high order derivatives and noncommutativity with variable dependent parameter.

Path integral action and Chern–Simons quantum mechanics in noncommutative plane

In this paper, the connection between the path integral representation of propagators in the coherent state basis with additional degrees of freedom (Rohwer et al 2010 J. Phys. A: Math. Theor. 43 345302) and the one without any such degrees of freedom (Gangopadhyay and Scholtz 2009 Phys. Rev. Lett. 102 241602) is established. We further demonstrate that the path integral formalism developed in the noncommutative plane using the coherent state basis leads to a quantum mechanics involving a Chern–Simons term in momentum which is of noncommutative origin. The origin of this term from the Bopp-shift point of view is also investigated. A relativistic generalization of the action derived from the path integral framework is then proposed. Finally, we construct a map from the commutative quantum Hall system to a particle in a noncommutative plane moving in a magnetic field. The value of the noncommutative parameter from this map is then computed and is found to agree with previous results.

The semi-classical spectrum of solitons and giant magnons

In this note, we summarize recent progress in constructing and then semi-classically quantizing solitons, or non-abelian Q-balls, in the symmetric space sine-Gordon theories. We then consider the images of these solitons in the related constrained sigma model, which are the dyonic giant magnons on the string theory world-sheet. Focussing on the case of the symmetric space S 5, we perform a semi-classical quantization of the solitons and magnons and show that both lead to Chern-Simons quantum mechanics on the internal moduli space which is a real Grassmannian SO(4)/SO(2) × SO(2) but — importantly — with a different coupling constant. Quantizing this system shows that both the Q-balls and magnons come in a tower of states transforming in symmetric representations of the SO(4) symmetry group; however, the former come in a finite tower whereas the latter come in the well-known infinite tower of dyonic giant magnons.

Notes on non-commutative Chern–Simons quantum mechanics

We investigate the classical and quantum aspects of non-commutative topological (Chern–Simons) mechanics. We introduce the magnetic field by the minimal substitution in a way which preserves the original symplectic structures. We find that the classical aspect, say, the solutions to the equations of motion, converges to the reduced theory which is obtained by turning off the mass term smoothly. However, the quantum aspect, i.e., the spectra and the angular momenta, does not have such continuous limits. The spectra will become divergent when the mass term is turned off. A scheme is proposed to regularize the spectra so as to get a finite result. In order to verify our regularization scheme, we resort to Dirac theory. We find that there are two constraints during the reduction from the full theory to the reduced one which alter the symplectic structures. The eigenvalues of angular momenta also have no continuous limits, and this situation is similar to the one which has been studied some years ago. The possibility of taking an additional limit is also discussed.

Path Integral Formulation for Chern-Simons Quantum Mechanics

Chern-Simons quantum mechanics is studied by using the path integral formulation. We find that the model can be decomposed into two independent oscillators when a set of new coordinate is chosen. The propagator is constructed in this new set of coordinate and the spectra is read off directly from it. The spectra will be divergent when the mass of the particle tends to zero. In order to get a physical result, one must regularize the spectra properly. We afford a new scheme of regularization. Interestingly, our scheme shows that the regularization we proposed amounts to erase one of the oscillators in the new set of coordinate. Physical interpretations for the regularization are given and some ambiguities in the literatures are clarified.

Noncommutative Chern-Simons quantum mechanics

Chern-Simons quantum mechanics is generalized to the noncommutative plane in this paper. Compared with the commutative counterpart, we find that in addition to the mass of the charged particle, there is a dimensionless parameter which behaves interestingly when it takes zero value. We study this model from both classical and quantum aspects. We show that the classical theory has continuous limits when both the parameters tend to zero while the quantum aspect (energy spectra) does not have. We must regularize the spectra of the full theory properly when these parameters tend to zero in order to get the finite results. We resort to the Dirac theory and the Faddeev-Jackiw reduction, respectively, to show that the regularization we made is proper.

Statistical mechanics of the deformable droplets on Riemannian surfaces: Applications to reptation and related problems

The statistical mechanics treatment of the Laplace–Young-type problems developed for the flat surfaces is generalized to the case of surfaces of constant negative curvature and connected with them to Riemannian surfaces. Obtained results are mainly used to supply an additional support of the quantum Hall effect (QHE) analogy employed in recent work [J. Phys. 4, 843 (1994)], which provides theoretical justification of the tube concept used in polymer reptation models. As a byproduct, close links between QHE, quantum chaos, and the non-Abelian Chern–Simons quantum mechanics are indicated.

Statistical mechanics of non-Abelian Chern-Simons particles.

We discuss the statistical mechanics of a two-dimensional gas of non-Abelian Chern-Simons particles which obey the non-Abelian braid statistics. The second virial coefficient is evaluated in the framework of the non-Abelian Chern-Simons quantum mechanics.

Non-Abelian Chern-Simons Quantum Mechanics and Non-Abelian Aharonov-Bohm Effect

We construct a classical action for a system of N point-like sources which carry SU(2) non-Abelian charges coupled to non-Abelian Chern-Simons gauge fields, and we develop a quantum mechanics for them. Adopting the coherent state quantization and solving the Gauss′ constraint in an appropriately chosen gauge, we obtain a quantum mechanical Hamiltonian given in terms of the Knizhnik-Zamolodchikov connection. Then we study the non-Abelian Aharonov-Bohm effect, employing the obtained Hamiltonian for a two-particle sector. An explicit evaluation of the differential cross section for the non-Abelian Aharonov-Bohm scattering is given.

Non-Abelian Chern-Simons quantum mechanics.

We propose a classical model for the non-Abelian Chern-Simons theory coupled to $N$ point-like sources and quantize the system using the BRST technique. The resulting quantum mechanics provides a unified framework for fractional spin, braid statistics and Knizhnik-Zamolodchikov equation.

Geometric quantization, Chern-Simons quantum mechanics and one-dimensional sigma models

The classical theory of a topological model which is constructed by setting the kinetic term of a (one-dimensional) sigma model at the classical level equal to zero is examined. The constraints and the geometry of the phase space of the supersymmetric extensions of the topological theory are studied. The rigid symmetries of a (supersymmetric) sigma model and its corresponding topological theory are gauged. Finally, the author examines the quantum theory of the one-dimensional sigma model in the limit where its kinetic term is set equal to zero and compare it with the quantum theory of the corresponding topological model.

CHERN-SIMONS QUANTUM MECHANICS, SUPERSYMMETRY, AND SYMPLECTIC INVARIANTS

We explain the relation between supersymmetry and equivariant cohomology, combining recent investigations of Chern-Simons quantum mechanics and path integrals on coadjoint orbits. We then construct a supersymmetric (topological) quantum mechanics model whose partition function — probing the cohomology of the symplectomorphism group — yields a symplectic invariant introduced by Weinstein. We demonstrate by explicit calculation that this provides another example where the Duistermaat-Heckman theorem (exactness of the semi-classical approximation) appears to hold in infinite dimensions, and make some suggestions concerning its role in (conformal) field theory.