Dirac quantization of a supersymmetric field theory

The superfield of one-space dimensional field theory is quantized using Dirac's method of quantization of systems with constraints. The quantization is shown to be consisted with that of the component fields.     

Topological quantization of mass in extended gauge theories with a monopole

Alvarez's treatment of topological charge quantization is generalized to include extended objects like the Dirac string in the presence of a magnetic pole. We rederive the topological mass quantization of the Abelian gauge field in (2+1)-dimensional spacetime previously derived by Henneaux and Teitelboim. A plausible argument is given for the general 2p + 1 cases in which the present method works.     

Cosmic origin of quantization

The origin of quantization is attributed - via the mechanism of “stochastic quantization” - to the universal interaction of every particle with the background gravitational force due to all other particles of the Universe. A formula for Planck's action constant h, obtained on the basis of this idea, yields the correct order of magnitude for h when implemented with current cosmological data.     

Periodic orbit quantization of chaotic maps by harmonic inversion

A method for the semiclassical quantization of chaotic maps is proposed, which is based on harmonic inversion. The power of the technique is demonstrated for the baker's map as a prototype example of a chaotic map.     

Proper Dirac quantization of a free particle on a D-dimensional sphere

We show that an unambiguous and correct quantization of the second-class constrained system of a free particle on a sphere in D dimensions is possible only by converting the constraints to Abelian gauge constraints, which are of first class in Dirac's classification scheme. The energy spectrum is equal to that of a pure Laplace-Beltrami operator with no additional constant arising from the curvature of the sphere. A quantization of Dirac's modified Poisson brackets for second-class constraints is also possible and unique, but must be rejected since the resulting energy spectrum is physically incorrect.     

Exact solutions for the early Friedmann-Robertson-Walker universe in general scalar-tensor theories

Homogeneous and isotropic cosmologies with trace-free energy-momentum tensors are studied in general scalar-tensor theories. A method is presented which allows one to construct exact solutions for theories with arbitrary coupling functionω(φ). Particular attention is paid to Schwinger's theory.     

Connes' tangent groupoid and strict quantization

We address one of the open problems in quantization theory recently listed by Rieffel. By developing in detail Connes' tangent groupoid principle and using previous work by Landsman, we show how to construct a strict flabby quantization, which is moreover an asymptotic morphism and satisfies the reality and traciality constraints, on any oriented Riemannian manifold. That construction generalizes the standard Moyal rule. The paper can be considered as an introduction to quantization theory from Connes' point of view.     

Magnetic charge quantization and generalized imprimitivity systems

The quantum mechanical concept of an active translation operation in an external magnetic field is discussed, and an integral version of the kinetic momentum components' commutation relations in terms of a generalized imprimitivity system is formulated. Magnetic charge quantization then follows from a cocyclelike identity in complete analogy with Dirac's original derivation. A generalized system of imprimitivity for the Dirac monopole is explicitly constructed with no strings attached.     

Elementary algebra of the Euclidean group,with application to magnetic charge quantization

A simple treatment of the algebra of the Euclidean group E3 is based on the introduction of a second group of rotations. Dirac's quantization of magnetic charge appears as the quantization of the generator of rotations about the axis connecting the electric and magnetic charges.     

On the Fedosov deformation quantization beyond the regular Poisson manifolds

A simple iterative procedure is suggested for the deformation quantization of (irregular) Poisson brackets associated to the classical Yang–Baxter equation. The construction is shown to admit a pure algebraic reformulation giving the Universal Deformation Formula (UDF) for any triangular Lie bialgebra. A simple proof of classification theorem for inequivalent UDF's is given. As an example the explicit quantization formula is presented for the quasi-homogeneous Poisson brackets on two-plane.     

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.     

Electrical engineering implications of the quantized sine-Gordon model

Using the Coleman mapping, for a particular value of the coupling constant, the forced sine-Gordon theory is transformed into massive 1+1 dimensional quantum electrodynamics. For the latter, Schwinger's paper time method is employed to calculate the exact transition rate and the resulting V-I characteristic for a model Josephson junction.     

Path integral quantization and the ground-state wave functional for multiplier scalar-vector field systems

With the help of path-integral quantization and Fradkin's approach, we obtain a new representation in the Schrödinger picture of the multiplier scalar-vector fields and the ground-state functional. We show that the model is equivalent to free scalar fields with the same mass.     

Deformation quantization, superintegrability and Nambu mechanics

Phase Space is the framework best suited for quantizing superintegrable systems—systems with more conserved quantities than degrees of freedom. In this quantization method, the symmetry algebras of the hamiltonian invariants are preserved most naturally. We illustrate the power and simplicity of the method through new applications to nonlinear σ-models, specifically for Chiral Models and de Sitter N-spheres, where the symmetric quantum hamiltonians amount to compact and elegant expressions, in accord with the Groenewold-van Hove theorem. Additional power and elegance is provided by the use of Nambu Brackets (linked to Dirac Brackets) involving the extra invariants of superintegrable models. The quantization of Nambu Brackets is then successfully compared to that of Moyal, validating Nambu’s original proposal, while invalidating other proposals.