Combinatorial quantization of the Hamiltonian Chern-Simons theory I

Motivated by a recent paper of Fock and Rosly [6] we describe a mathematically precise quantization of the Hamiltonian Chern-Simons theory. We introduce the Chern-Simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly. The lattice model enjoys the symmetry with respect to a quantum gauge group. Using this fact we construct the algebra of observables of the Hamiltonian Chern-Simons theory equipped with a *- operation and a positive inner product.Supported by Swedish Natural Science Research Council (NFR) under the contract F-FU 06821-304 and by the Federal Ministry of Science and Research, AustriaPart of project P8916-PHY of the Fonds zur Förderung der wissenschaftlichen Forschung in ÖsterreichSupported in part by DOE Grant No DE-FG02-88ER25065;     

Explicit quantization of the Chern-Simons action

We quantize the three-dimensional Chern-Simons action explicitly. We found that the geometric quantization of the action strongly depends on the topology of the (fixed-time) Riemann surface. On the disk the phase space and the symplectic structure are the same as those of the (chiral) Wess-Zumino-Witten model. On the torus the Hilbert space is the vector space of characters of Kac-Moody algebras. The fusion rules of the primary fields are derived from theclassical matching condition of the holonomy. In general case, the wave-functional of the theory is the generating function of the current insertion in Wess-Zumino-Witten model.     

Supermembrane with non-Abelian gauging and Chern-Simons quantization

We present the non-Abelian gaugings of supermembranes for general isometries for compactifications from eleven-dimensions, starting with an Abelian case as a guide. We introduce a super Killing vector in eleven-dimensional superspace for a non-Abelian group G associated with the compact space B for a general compactification, and couple it to a non-Abelian gauge field on the world-volume. As a technical tool, we use teleparallel superspace with no manifest local Lorentz covariance. Interestingly, the coupling constant is quantized for the non-Abelian group G, due to its non-trivial mapping . Received: 15 September 2004, Revised: 12 November 2004, Published online: 14 January 2005 PACS: 11.25.Mj, 11.25.Tq, 04.50. + h, 04.65. + e     

Hamiltonian BRST quantization of the conformal string

We present a new formulation of the tensionless string (T = 0) where the space-time conformal symmetry is manifest. Using a Hamiltonian BRST scheme we quantize this Conformal String and find that it has critical dimension D = 2. This is in keeping with our classical result that the model describes massless particles in this dimension. It is also consistent with our previous results which indicate that quantized conformally symmetric tensionless strings describe a topological phase away from D = 2.

We reach our result by demanding nilpotency of the BRST charge and consistency with the Jacobi identities. The derivation is presented in two different ways: in operator language and using mode expansions.

Careful attention is paid to regularization, a crucial ingredient in our calculations.     

Becchi-Rouet-Stora-Tyutin quantization and Hamiltonian formalism

An introductory review of BRST hamiltonian formalism is presented. The method of quantization of gauge and string theories is discussed. A few simple examples are presented to illustrate the BRST techniques.     

Relativistic Hamiltonian dynamics. II. Momentum-dependent interactions, confinement and quantization

The previously developed covariant classical relativistic N-particle dynamics is extended to momentum-dependent interactions and generalized to a gauge-independent constraint reduction. This reduction is made via center-of-momentum variables as well as via the more conventional individual particle variables. A canonical quantization is then carried out. The two-body problem is discussed in detail for the case of momentum-dependent interactions. It is demonstrated that such interactions can give rise to dynamical confinement both classically and quantum mechanically. The prototype interaction −β²(ξ · π)² has a harmonic oscillator type spectrum and shows a linear dependence of the binding energy on the angular momentum for small particle rest masses (m₁, m₂ β).     

Hamiltonian quantization of the non-anomalous generalized Schwinger model

We quantize a generalized version of the Schwinger model, where the two chiral sectors couples with different strengths to theU(1) gauge field. Starting from a theory which includes a generalized Wess-Zumino term, we obtain the equal time commutation relation for physical fields, both the singular and non-singular cases are considered. The photon propagators are also computed in their gauge dependent and invariant versions.     

The quantization of the Hamiltonian in curved space

The construction of the quantum-mechanical Hamiltonian by canonical quantization is examined. The results are used to enlighten examples taken from slow nuclear collective motion. Hamiltonians, obtained by a thoroughly quantal method (generator-coordinate method) and by the canonical quantization of the semiclassical Hamiltonian, are compared. The resulting simplicity in the physics of a system constrained to lie in a curved space by the introduction of local Riemannian coordinates is emphasized. In conclusion, a parallel is established between the result for various coordinates and a proposed procedure for quantizing the semiclassical Hamiltonian for a single coordinate.Partially supported by Fundação Calouste Gulbenkian, Lisboa.     

Hamiltonian N=2 superfield quantization

We present a superfield construction of Hamiltonian quantization with N=2 supersymmetry generated by two fermionic charges Q^a. As a byproduct of the analysis we also derive a classically localized path integral from two fermionic objects Σ^a that can be viewed as “square roots” of the classical bosonic action under the product of a functional Poisson bracket.     

Discrete Hamiltonian symmetries and semiclassical quantization

Most quantum Hamiltonian systems exhibit discrete symmetries. Allowing for these is crucial when properly calculating the fluctuation properties of the quantal spectrum. These properties are then employed to distinguish between classically chaotic or non-chaotic quantum systems. In general, semiclassical quantization procedures do not take into account irreducible representations of the Hamiltonian. A procedure is presented to take these into account in semiclassical quantization schemes and calculate some of the energy eigenvalues belonging to a specific irreducible representation.