Gauge Invariant Quantization Research Articles

Magnetic quantization over Riemannian manifolds

We demonstrate that Weyl's pioneering idea (1918) to intertwine metric and magnetic fields into a single joint connection can be naturally realized, on the phase space level, by the gauge-invariant quantization of the cotangent bundle with magnetic symplectic form. Quantization, for systems over a noncompact Riemannian configuration manifold, may be achieved by the introduction of a magneto-metric analog of the Stratonovich quantizer — a family of invertible, selfadjoint operators representing quantum δ functions. Based on the quantizer, we construct a generalized Wigner transform that maps Hilbert–Schmidt operators into L2 phase-space functions. The algebraic properties of the quantizer allow one to extract a family of symplectic reflections, which are then used to (i) derive a simple, explicit, and geometrically invariant formula for the noncommutative product of functions on phase space, and (ii) construct a magneto-metric connection on phase space. The classical limit of this product is given by the usual multiplication of functions (zeroth-order term), the magnetic Poisson bracket (first-order term), and by the magneto-metric connection (second-order term).PACS Nos.: 02.40.-k, 11.10.Nx

Symplectic areas, quantization, and dynamics in electromagnetic fields

A gauge invariant quantization in a closed integral form is developed over a linear phase space endowed with an inhomogeneous Faraday electromagnetic tensor. An analog of the Groenewold product formula (corresponding to Weyl ordering) is obtained via a membrane magnetic area, and extended to the product of N symbols. The problem of ordering in quantization is related to different configurations of membranes: A choice of configuration determines a phase factor that fixes the ordering and controls a symplectic groupoid structure on the secondary phase space. A gauge invariant solution of the quantum evolution problem for a charged particle in an electromagnetic field is represented in an exact continual form and in the semiclassical approximation via the area of dynamical membranes.

Quantization of anomalous gauge field theory and BRST-invariant models of two-dimensional quantum gravity

We analyze the problems with the so called gauge invariant quantization of the anomalous gauge field theories originary due to Faddeev and Shatashvili (FS). Our analysis bring to a generalization of FS method which allows to construct a series of classically equivalent theories which are non equivalent at quantum level. We prove that these classical theories are all consistent with the BRST invariance of the original gauge symmetry with suitably augmented field content. As an example of such a scenario, we discuss the class of physically distinct models of two dimensional induced gravity which are a generalization of the David-Distler-Kawai model.

Explicit gauge-invariant quantization of the Schwinger model on a circle in the functional Schrödinger representation.

We solve the Schwinger model on a circle by first finding the explicit ground state functional(s). Having done this, we give the structure of the Hilbert space and derive bosonization formulas in this formalism.

BFV quantization on hermitian symmetric spaces

Gauge-invariant BFV approach to geometric quantization is applied to the case of hermitian symmetric spaces G/ H. In particular, gauge invariant quantization on the Lobachevski plane and sphere is carried out. Due to the presence of symmetry, master equations for the first-class constraints, quantum observables and physical quantum states are exactly solvable. BFV-BRST operator defines a flat G-connection in the Fock bundle over G/ H. Physical quantum states are covariantly constant sections with respect to this connection and are shown to coincide with the generalized coherent states for the group G. Vacuum expectation values of the quantum observables commuting with the quantum first-class constraints reduce to the covariant symbols of Berezin. The gauge-invariant approach to quantization on symplectic manifolds synthesizes geometric, deformation and Berezin quantization approaches.

Gauge invariant quantization and induced braid statistics in 2+1 dimensional U(1) theories

We use the Euler-Lagrange equations of motion in order to solve the constraints for various types of two dimensional U(1) gauge theories. It is shown that, using this method, one obtains a Lagrangian which is gauge independent. In particular, we discuss the anyonic behavior of theories which contain fermions coupled to the Chern-Simons term, with and without the Maxwell kinetic energy term. The existence of braid statistics for the free fermions in the adiabatic limit is discussed, as part of a consistent perturbative expansion around this limit.

GAUGE INVARIANCE AND BROKEN SYMMETRIES IN ANYON SUPERFLUIDS

We review aspects of broken symmetry and the nature of long range order in theories of anyons starting with bosons with a statistical interaction. We introduce a novel gauge invariant quantization scheme that allows the identification of local and gauge invariant order parameters. The connection between spin and statistics is reviewed and the consequences of broken symmetries in the anyon representation are discussed. An anyon gas is studied in the Bogoliubov approximation, it is determined that the ground state is a condensate of charge-flux composites with “quasi-long-range order” at zero temperature, a “weak” gap in the spectrum and finite helicity modulus. The system is disordered at nonzero temperatures. The disorder is not caused by Goldstone bosons but by the strong infrared behavior arising from the Coulomb interaction induced by the long-range statistical interaction. The properties of topological vortices in nonrelativistic and in relativistic Landau-Ginzburg theories are studied in detail. We study the physics of the mean-field ansatz and quasi-long range order in a simple exactly soluble relativistic model. This model exhibits a novel phenomenon of charge redistribution to the boundaries and restoration of translational invariance in the infinite volume limit. It also illuminates the physics of quasi-long-range order with a gap in the spectrum, statistical charge polarization by external magnetic fields and the role of “large” gauge transformations.

Gauge Invariant Quantization on Riemannian Manifolds

Gauge Invariant Quantization on Riemannian Manifolds

Gauge invariant quantization on Riemannian manifolds

For every pointwise polynomial function on each fiber of the cotangent bundle of a Riemannian manifold M M , a family of differential operators is given, which acts on the space of smooth sections of a vector bundle on M M . Such a correspondence may be considered as a rule to quantize classical systems moving in a Riemannian manifold or in a gauge field. Some applications of our construction are also given in this paper.

Lattice mechanism for gauge boson mass generation in chiral theories

It is shown that in lattice gauge theories with a general form of the lattice fermion action, masses of the gauge bosons coupled to the chiral fermion currents are inevitably generated after gauge invariant quantization. The origin of this mechanism is considered.

Commutators of Gauss-law operators in chiral gauge theories

A field theoretic derivation, by nonperturbative functional methods in even space-time dimensions, of the equal-time commutators of the Gauss-law operators is presented. For the anomalous formulation of chiral gauge theories, the Schwinger term in the cummutators is expressed in a closed form in terms of the consistent current divergence anomaly. However, in the gauge-invariant quantization, with the addition of the Wess-Zumino lagrangian, the commutators are free of the Schwinger term.

Gauge-Invariant Quantization of Abelian and Non-Abelian Theories

A review is given for the gauge-invariant quantization method of Abelian and non-Abelian theories based on the explicit solution of the constraint equation (Gauss equation). Considered are the problems whose solution in the Dirac method depends on the gauge. They include: the calculation of the one-particle Green function, the problem of gauge non-uniqueness, quantization of the theories with interaction containing time derivatives of fields. Within this quantization method the perturbation theory is formulated for description of bound states in QED and QCD.

Quantum vortices

We make the contribution of vortices to the partition function of the abelian Higgs model explicit without resorting to approximation. The formulation is a gauge-invariant quantization that generalizes to non-abelian gauge theories. We also calculate an effective potential that is of physical significance because it is guaranteed gauge invariant.

Isoscalar giant vibrations in the quantized time-dependent deformed oscillator model

The gauge-invariant quantization prescription is applied to the solutions of the time-dependent deformed-oscillator model (TDDOM) for the isoscalar giant multipole resonances in light atomic nuclei. The quantized TDDOM results are in good agreement with those obtained in the generator-coordinate method. The problems found in the application of the gauge-invariant quantization method to the non-separable Hamilton systems are presented.

On confinement in pure Yang-Mills theories

By using a gauge-invariant quantization procedure in the loop-space previously proposed by the authors, a simple realization of 't Hooft's algebra is obtained which suggests that the dynamics of the system singles out the permanent confinement region in 't Hooft's phase diagram.

The gauge-invariant quantization method for multidimensional Hamilton systems

The gauge-invariant quantization method is extended for multidimensional hamilton systems which are separable or nearly separable in the sense of the Kolmogorov-Arnold-Moser theorem. The method reduces to the coherent gauge-invariant quantization prescription for time-periodic gauge-invariant functions.

The generator coordinate method, path integrals, and quantized time-dependent mean field motion

The two parameter generator coordinate (GCM) method is used to derive the equation of motion for time-dependent mean fields associated to large amplitude vibrations. To achieve this a semclassical assumption and the gaussian overlap approximation is introduced into the GCM before the variational principle is applied. The result consists in time-dependent Hartree-Fock equations with periodic boundary conditions and a semiclassical quantization rule to select those TDHF orbits which correspond to the stationary states. The prescription turns out to be identical to the one obtained recently by path integral techniques and gauge invariant quantization.