The Chern–Simons term induced at high temperature and the quantization of its coefficient

By perturbative calculations of the high-temperature ground-state axial vector current of fermion fields coupled to gauge fields, an anomalous Chern–Simons topological mass term is induced in the three-dimensional effective action. The anomaly in three dimensions appears just in the ground-state current rather than in the divergence of ground-state current. In the Abelian case, the contribution comes only from the vacuum polarization graph, whereas in the non-Abelian case, contributions come from the vacuum polarization graph and the two triangle graphs. The relation between the quantization of the Chern–Simons coefficient and the Dirac quantization condition of magnetic charge is also obtained. It implies that in a (2+1)-dimensional QED with the Chern–Simons topological mass term and a magnetic monopole with magnetic charge g present, the Chern–Simons coefficient must be also quantized, just as in the non-Abelian case. Received: 7 April 1999 / Published online: 3 November 1999     

Quantum Field Formalism for the Electromagnetic Interaction of Composite Particles in a Nonrelativistic Gauge Model II

By generalizing a model previously proposed, a classical nonrelativistic U(1)×U(1) gauge field model for the electromagnetic interaction of composite particles in (2+1) dimensions is constructed. The model contains a Chern–Simons U(1) field and the electromagnetic U(1) field, and it describes both a composite boson system or a composite fermion one. The second case is considered explicitly. The model includes a topological mass term for the electromagnetic field and interaction terms between the gauge fields. By following the Dirac Hamiltonian formalism for constrained systems, the canonical quantization for the model is realized. By developing the path integral quantization method through the Faddeev–Senjanovic algorithm, the Feynman rules of the model are established and its diagrammatic structure is discussed. The Becchi–Rouet–Stora–Tyutin formalism is applied to the model. The obtained results are compared with the ones corresponding to the previous model.     

Deformation Quantization and the Baum–Connes Conjecture

 Alternative titles of this paper would have been ``Index theory without index' or ``The Baum–Connes conjecture without Baum.' In 1989, Rieffel introduced an analytic version of deformation quantization based on the use of continuous fields of C ^* -algebras. We review how a wide variety of examples of such quantizations can be understood on the basis of a single lemma involving amenable groupoids. These include Weyl–Moyal quantization on manifolds, C ^* -algebras of Lie groups and Lie groupoids, and the E-theoretic version of the Baum–Connes conjecture for smooth groupoids as described by Connes in his book Noncommutative Geometry. Concerning the latter, we use a different semidirect product construction from Connes. This enables one to formulate the Baum–Connes conjecture in terms of twisted Weyl–Moyal quantization. The underlying mechanical system is a noncommutative desingularization of a stratified Poisson space, and the Baum–Connes Conjecture actually suggests a strategy for quantizing such singular spaces. Received: 30 April 2002 / Accepted: 2 October 2002 Published online: 17 April 2003 RID="⋆" ID="⋆" Supported by a Fellowship from the Royal Netherlands Academy of Arts and Sciences (KNAW). Communicated by H. Araki, D. Buchholz and K. Fredenhagen     

Covariant canonical quantization

We present a manifestly covariant quantization procedure based on the de Donder–Weyl Hamiltonian formulation of classical field theory. This procedure agrees with conventional canonical quantization only if the parameter space is d=1 dimensional time. In d>1 quantization requires a fundamental length scale, and any bosonic field generates a spinorial wave function, leading to the purely quantum-theoretical emergence of spinors as a byproduct. We provide a probabilistic interpretation of the wave functions for the fields, and we apply the formalism to a number of simple examples. These show that covariant canonical quantization produces both the Klein–Gordon and the Dirac equation, while also predicting the existence of discrete towers of identically charged fermions with different masses. Covariant canonical quantization can thus be understood as a “first” or pre-quantization within the framework of conventional QFT. PACS 04.62.+v; 11.10.Ef; 12.10.Kt     

Renormalization of an abelian gauge theory in stochastic quantization

The renormalization of an abelian gauge field coupled to a complex scalar field is disccused in the stochastic quantization method. The supper space formulation of the stochastic quantization method is used to derived the Ward Takahashi identities assocoated with supersymmetry. These Ward Takahashi identities together with previously derived Ward Takahshi identities associated with gauge invariance are shown to be sufficient to fix all the renormalization constant in temrs of scaling of the fields and of the parameters appearing in the stochastic theory.     

Global Geometric Deformations of the Virasoro Algebra,Current and Affine Algebras by Krichever–Novikov Type Algebras

In two earlier articles we constructed algebraic-geometric families of genus one (i.e. elliptic) Lie algebras of Krichever–Novikov type. The considered algebras are vector fields, current and affine Lie algebras. These families deform the Witt algebra, the Virasoro algebra, the classical current, and the affine Kac–Moody Lie algebras respectively. The constructed families are not equivalent (not even locally) to the trivial families, despite the fact that the classical algebras are formally rigid. This effect is due to the fact that the algebras are infinite dimensional. In this article the results are reviewed and developed further. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics. The algebras are of relevance in the global operator approach to the Wess–Zumino–Witten–Novikov models appearing in the quantization of Conformal Field Theory.     

Landau analog levels for dipoles in non-commutative space and phase space

We investigate the analog of Landau quantization, for a neutral polarized particle in the presence of homogeneous electric and magnetic external fields, in the context of non-commutative quantum mechanics. This particle, possessing electric and magnetic dipole moments, interacts with the fields via the Aharonov–Casher and He–McKellar–Wilkens effects. For this model we obtain the Landau energy spectrum and the radial eigenfunctions of the non-commutative space coordinates and non-commutative phase space coordinates. Also we show that the case of non-commutative phase space can be treated as a special case of the usual non-commutative space coordinates.     

Quantum Dynamical Yang–Baxter Equation¶Over a Nonabelian Base

In this paper we consider dynamical r-matrices over a nonabelian base. There are two main results. First, corresponding to a fat reductive decomposition of a Lie algebra ?=?⊕?, we construct geometrically a non-degenerate triangular dynamical r-matrix using symplectic fibrations. Second, we prove that a triangular dynamical r-matrix naturally corresponds to a Poisson manifold ?^⋆×G. A special type of quantization of this Poisson manifold, called compatible star products in this paper, yields a generalized version of the quantum dynamical Yang–Baxter equation (or Gervais–Neveu–Felder equation). As a result, the quantization problem of a general dynamical r-matrix is proposed. Received: 19 May 2001 / Accepted: 19 November 2001     

An Explicit Formula for the Natural and Conformally Invariant Quantization

Lecomte (Prog Theor Phys Suppl 144:125–132, 2001) conjectured the existence of a natural and conformally invariant quantization. In Mathonet and Radoux (Existence of natural and conformally invariant quantizations of arbitrary symbols, math.DG 0811.3710), we gave a proof of this theorem thanks to the theory of Cartan connections. In this paper, we give an explicit formula for the natural and conformally invariant quantization of trace-free symbols thanks to the method used in Mathonet and Radoux and to tools already used in Radoux [Lett Math Phys 78(2):173–188, 2006] in the projective setting. This formula is extremely similar to the one giving the natural and projectively invariant quantization in Radoux.     

Magnetothermal factor of intersubband relaxation of 2D electrons from a highly doped heterotransition in AlGaAs (Si)/GaAs

Characteristics of the Shubnikov-de Haas transverse magnetoresistance oscillations of 2D electrons in highly dopedAlGaAs(Si)/GaAs heterostructures are investigated in the present paper. Anomalies caused by the occupation of two quantization subbands are revealed for samples with 2D-electron density n_s>7·10¹¹ cm⁻² at T=1.7–16 K and magnetic field induction B up to 7.4 T. The dependences of the normalized oscillation amplitude on the magnetic field show bends that typically displace toward weaker magnetic fields with decreasing temperature and electron density n_s. A nonmonotonic (oscillating) dependence of the Dingle temperature on the experimental temperature is found. These anomalies are interpreted for a model of the occupation of two quantization subbands with electrons. They are caused by the competitive character, of intersubband 2D-electron scattering. Small-angle relaxation times are estimated for 2D electrons of the zero and first quantization subbands. S. A. Esenin Ryazan' State Pedagogical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 52–57, January, 2000.     

A new method of quantization of classical solutions

The stochastic quantization method of Parisi and Wu is used to derive exact equations for the correlators of quantum fluctuations around the classical solution in the massless φ ⁴ theory. The equations obtained are then solved in the lowest orders of perturbation theory, and the first correction to the free propagator of a quantum fluctuation is calculated. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 5, 381–386 (10 March 1996) Publsihed in English in the original Russian journal. Edited by Steve Torstveit.     

Hamilton—Jacobi quantization of constrained systems

The path integral quantization of contrained systems is analysed using Hamilton-Jacobi formalism. The integrability conditions are investigated and the results are in agreement with those obtained by Dirac’s method. Presented at the 10th Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June, 2001.     

The Complete Solution of the Classical SL(2,ℝ/U(1) Gauged WZNW Field Theory

We prove that any gauged WZNW model has a Lax pair representation, and give explicitly the general solution of the classical equations of motion of the SL(2,ℝ/U(1) theory. We calculate the symplectic structure of this solution by solving a differential equation of the Gelfand–Dikii type with initial state conditions at infinity, and transform the canonical physical fields non-locally onto canonical free fields. The results will, finally, be collected in a local B?cklund transformation. These calculations prepare the theory for an exact canonical quantization. Received: 9 June 1998 / Accepted: 7 March 1999     

Simulation of the resistive state of superconducting films in a magnetic field on the basis of the nonstationary ginzburg-landau equation

By numerical simulation of the temporal two-dimensional Ginzburg-Landau equation, we study the resistive state in superconducting bridges with dimensions ξ ≪ d ≪ λ. It is found that the basis of the resistive state here is, as for d ≫ λ, the vortical structures (vortices) whose motion defines the resistive state. It is shown that the motion of vortices is stochastic in a certain range of currents and magnetic fields. We give a classification of possible dynamic and stochastic modes and examine the transitions from the current flow mode, which is observed for large magnetic fields and small transport currents, to the mode of fast phase slippage. The symmetry breaking effect of the resistive state, which results in cross tension with a quadrupole structure, has been detected. Institute of Physics of Microstructures, Russian Academy of Sciences, Nizhny Novgorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenü, Radiofizika, Vol. 40, Nos. 1–2, pp. 213–231, January–February, 1997.     

The Fokker–Planck Equation, and Stationary Densities

The most general local Markovian stochastic model is investigated, for which it is known that the evolution equation is the Fokker–Planck equation. Special cases are investigated where uncorrelated initial states remain uncorrelated. Finally, stochastic one-dimensional fields with local interactions are studied that have kink-solutions.     

Stochastic quantization of the nonlinear sigma model and the background field method

The background field method is a useful scheme for calculation of the effective action in conventional quantum field theory. In stochastic quantization this approach is introduced by using auxiliary fields, as suggested by Okano. In this work, we implement the background field method, using the normal coordinate expansion, for the nonlinear sigma model on a general Riemannian manifold in the context of stochastic quantization. We also calculate, making use of this novel formulation, the action necessary for investigation of the divergences, at least at the one-loop level.     

Strings, world-sheet covariant quantization and Bohmian mechanics

The covariant canonical method of quantization based on the De Donder–Weyl covariant canonical formalism is used to formulate a world-sheet covariant quantization of bosonic strings. To provide the consistency with the standard non-covariant canonical quantization, it is necessary to adopt a Bohmian deterministic hidden-variable equation of motion. In this way, string theory suggests a solution to the problem of measurement in quantum mechanics. PACS 11.25.-w; 04.60.Ds; 03.65.Ta     

Spectroscopy of the Einstein–Maxwell–Dilaton–Axion black hole

The entropy spectrum of a spherically symmetric black hole was derived via the Bohr–Sommerfeld quantization rule in Majhi and Vagenas’s work. Extending this work to charged and rotating black holes, we quantize the horizon area and the entropy of an Einstein–Maxwell–Dilaton–Axion black hole via the Bohr–Sommerfeld quantization rule and the adiabatic invariance. The result shows the area spectrum and the entropy spectrum are respectively equally spaced and independent on the parameters of the black hole.