Quantization of first-class constraints with structure functions

As part of a program to desuperize BRST symmetries, we show how to translate the BRST construction completely in terms of standard symplectic geometry in the case where the constraints are derived from a foliation on a configuration space. The consequences of this approach on quantization are investigated. As a corollary, we solve (in a restricted setting) the problem of how to deal in quantization with (globally independent) first-class constraints with structure functions rather than structure constants.     

Coordinate-free quantization of second-class constraints

The conversion of second-class constraints into first-class constraints is used to extend the coordinate-free path-integral quantization, achieved by a flat-space Brownian motion regularization of the coherent-state path-integral measure, to systems with second-class constraints.     

Canonical formalism of gauge theories with reducible constraints

A procedure for the canonical quantization of gauge theories with reducible constraints (that is, linearly dependent) is proposed. The procedure consists of extending the initial phase space and filling out the initial system of constraints to an aggregate of linearly indepndent constraints. The equivalence is shown between the proposed quantization scheme and canonical quantization when only the linearly independent constraints are chosen from the initial system of constraints.Translated from Izvestiya Vysshikh.Uchebnykh Zavedenii, Fizika, No. 7, pp. 64–68, July, 1985.     

The effective action for superfield Lagrangian quantization in reducible hypergauges

The rules of local superfield Lagrangian quantization in reducible non-Abelian hypergauge functions are formulated for an arbitrary gauge theory. The generating functionals of standard and vertex Greens functions which depend on the Grassmann variable via super(anti)fields and sources are constructed. The difference between the local quantum and the gauge fixing action determines an almost Hamiltonian system such that translations with respect to along the solutions of this system define the superfield BRST transformations. The Ward identities are derived and the gauge independence of the S-matrix is proved.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 5966, October, 2004.     

Canonical quantization with quadratic constraints

Based on an analysis first suggested by Bryce S. DeWitt, we have found that a special case of the general classical theory involving quadratic constraints can be quantized canonically, in the sense that the quantum constraints are consistent. In particular, this special case contains all known physical theories of bosons,including Einstein'sGeneral Theory of Relativity. The quantum constraints for this theory are given explicitly in an appendix.     

On the BFT-BFV quantization of gauge invariant systems with linear second class constraints

We study the Hamiltonian path integral formulation for generic systems with first class and linear second class constraints.ift01001@ufrj     

Geometric quantization of symplectic manifolds with respect to reducible non-negative polarizations

The leafwise complex of a reducible non-negative polarization with values in the prequantum bundle on a prequantizable symplectic manifold is studied. The cohomology groups of this complex is shown to vanish in rank less than the rank of the real part of the non-negative polarization. The Bohr-Sommerfeld set for a reducible non-negative polarization is defined. A factorization theorem is proved for these reducible non-negative polarizations. For compact symplectic manifolds, it is shown that the above complex has finite dimensional cohomology groups, more-over a Lefschetz fixed point theorem and an index theorem for these non-elliptic complexes is proved. As a corollary of the index theorem, we deduce that the cardinality of the Bohr-Sommerfeld set for any reducible real polarization on a compact symplectic manifold is determined by the volume and the dimension of the manifold. Supported in part by NSF grant DMS-93-09653, while the author was visiting University of California Berkeley.     

Geometric quantization of the multidimensional Kepler problem

The geometric quantization scheme of Czyz and Hess is applied to the (n - 1)-dimensional quadric in complex projective space. As the quadratic is the orbit manifold of the n-dimensional Kepler problem and the geodesic flow on the n-dimensional euclidean sphere, we thus obtain the quantum energy levels and their multiplicities for these Hamiltonian systems.     

Geometric quantization of the five-dimensional Kepler problem

An extension of the Hurwitz transformation to a canonical transformation between phase spaces allows conversion of the five-dimensional Kepler problem into that of a constrained harmonic oscillator problem in eight dimensions. Thus a new regularization of the Kepler problem is established. Then, following Dirac, we quantize the extended phase space, imposing constraint conditions as superselection rules. In that way the interchangeability of the reduction and the quantization procedures is proved.     

Geometric quantization and constraints in field theory

The main objective of this series of lectures is a discussion of the problem of quantization of systems with constraints, first studied by P.A.M. Dirac. I want to reinterprete Dirac's approach to quantization of constraints in the framework of geometric quantization, and then use it to discuss some aspects of quantized Yang-Mills fields. We begin with a review of geometric quantization and the implied relationship between the co-adjoint orbits and the irreducible unitary representations of Lie groups. Next, we discuss an intrinsic Hamiltonian formulation of a class of field theories which includes gauge theories and general relativity. Quantization of this class of field theories is discussed. Dirac's approach to quantization of constraints is reinterpreted in the framework of geometric quantization.     

Periodic orbit quantization of the anisotropic Kepler problem

The periodic orbit quantization on the anisotropic Kepler problem is tested. By computing the stability and action of some 2000 of the shortest periodic orbits, the eigenvalue spectrum of the anisotropic Kepler problem is calculated. The aim is to test the following claims for calculating the quantum spectrum of classically chaotic systems: (1) Curvature expansions of quantum mechanical zeta functions offer the best semiclassical estimates; (2) the real part of the cycle expansions of quantum mechanical zeta functions cut at appropriate cycle length offer the best estimates; (3) cycle expansions are superfluous; and (4) only a small subset of cycles (irreducible cycles) suffices for good estimates for the eigenvalues. No evidence is found to support any of the four claims.     

On the quantization of gravity

We analyse the construction of quantum states for the case of gravity and matter using the solution of the 2-submanifold boundary value problem and “sum over paths” quantisation. This leads to a specification of such states in terms of a complete commuting set of operators. The “sum over topologies” definition is obtained only by a very ad hoc assumption which is made precise. The problem of the arbitrariness of the background metric discussed and resolved by analogy with QED.     

Algebraic quantization on a group and nonabelian constraints

A generalization of a previous group manifold quantization formalism is proposed. In the new version the differential structure is circumvented, so that discrete transformations in the group are allowed, and a nonabelian group replaces the ordinary (central)U(1) subgroup of the Heisenberg-Weyl-like quantum group. As an example of the former we obtain the wave functions associated with the system of two identical particles, and the latter modification is used to account for the Virasoro constraints in string theory.Research partially supported by the Conselleria de Cultura de la Generalitat Valenciana, the Plan de Formacion del Personal Investigador, and the Comision Asesora de Investigacion Cientifica y Tecnica (CAICYT)On leave of absence from the IFIC, Centro Mixto Universidad de Valencia—C.S.I.C. and the Departamento de Fisica Teorica de la Universidad de Valencia     

Remarks on the geometric quantization of the Kepler problem

The geometric quantization of the (three-dimensional) Kepler problem is readily obtained from the one of the harmonic oscillator using a Segre map. The physical meaning of the latter is discussed.