Stochastic quantisation of a gauge field in the spatial axial gauge

The stochastic quantisation method of Nelson can be applied to quantise an Abelian gauge field in the spatial axial gauge.     

Generalized connection forms in the unification of gauge interactions

Within a differential-geometrical framework, the notion of a generalized connection form on a principal fibre bundle is applied to a generalized model for the unification of two (or more) gauge interactions. An example with gauge groups SU(2) and U(1) is considered.     

Generalized measures in gauge theory

LetP M be a principalG-bundle. We construct well-defined analogs of Lebesgue measure on the spaceA of connections onP and Haar measure on the groupG of gauge transformations. More precisely, we define algebras of cylinder functions on the spacesA,G, andA/G, and define generalized measures on these spaces as continuous linear functionals on the corresponding algebras. Borrowing some ideas from lattice gauge theory, we characterize generalized measures onA,G, andA/G in terms of graphs embedded inM. We use this characterization to construct generalized measures onA andG whenG is compact. The uniform generalized measure onA is invariant under the group of automorphisms ofP. It projects down to the generalized measure onA/G considered by Ashtekar and Lewandowski in the caseG = SU(n). The generalized Haar measure onG is right- and left-invariant as well as Aut(P)-invariant. We show that averaging any generalized measure onA against generalized Haar measure gives aG-invariant generalized measure onA.     

Towards a generalized distribution formalism for gauge quantum fields

We prove that the distributions defined on the Gelfand-Shilov spacesS with < 1 and, hence, more singular than hyperfunctions, retain the angular localizability property. Specifically, they have uniquely determined support cones. This result enables one to develop a distribution-theoretic technique suitable for the consistent treatment of quantum fields with arbitrarily singular ultraviolet and infrared behavior. The proof covering the most general and difficult case = 0 is based on the use of the theory of plurisubharmonic functions and Hörmander'sL ²-estimates.This work was supported in part by a Soros Humanitarian Foundation Grant awarded by the American Physical Society.     

Functional integration and gauge ambiguities in generalized abelian gauge theories

We consider the covariant quantization of generalized abelian gauge theories on a closed and compact n$n$-dimensional manifold whose space of gauge invariant fields is the abelian group of Cheeger–Simons differential characters. The space of gauge fields is shown to be a non-trivial bundle over the orbits of the subgroup of smooth Cheeger–Simons differential characters. Furthermore each orbit itself has the structure of a bundle over a multi-dimensional torus. As a consequence there is a topological obstruction to the existence of a global gauge fixing condition. A functional integral measure is proposed on the space of gauge fields which takes this problem into account and provides a regularization of the gauge degrees of freedom. For the generalized p$p$-form Maxwell theory closed expressions for all physical observables are obtained. The Green’s functions are shown to be affected by the non-trivial bundle structure. Finally the vacuum expectation values of circle-valued homomorphisms, including the Wilson operator for singular p$p$-cycles of the manifold, are computed and selection rules are derived.     

Spinor electrodynamics in terms of gauge-invariant quantities

We give a formulation of classical spinor electrodynamics in terms of gauge-invariant quantities. The set of invariants consists of bilinear combinations of spinor fields (currents), a real-valued covector field, and a complex scalar field of modulus one. The presented result is a first step towards formulating quantum electrodynamics in terms of gauge-invariant fields.     

The choice of test functions in gauge quantum field theories

We discuss the problem of the choice of test functions in gauge quantum field theories. Analysis of explicitly soluble models suggests that the test function spaces which are suitable for local and covariant formulation of gauge theories are the Gelfand and Shilov spaces , +>1. We also discuss a possible generalization of the spectral condition.     

On gauge theories of mass

The classical Einstein–Hilbert action in general relativity extends naturally to a blow-up (in the sense of algebraic geometry) of the usual space of pseudo-Riemannian metrics; this presents the metric tensor _gik$g_{ik}$ as a kind of Goldstone boson associated to the real scalar field defined by its determinant. This seems to be quite compatible with the Higgs mechanism in the standard model of particle physics.     

Functional integral of QED in terms of gauge-invariant quantities

We reduce the functional integral of quantum electrodynamics to an integral containing only local gauge invariant quantities. The set of (bosonic) invariants contains bilinear combinations of the spinor field and a real-valued covector field.     

On the stability of gauge fields in higher dimensions

The stability of solutions to gauge field equations in higher dimensions is studied. It is found that in the absence of a topological lower bound, these solutions are unstable.     

A new 2-form for connections on surfaces with boundary

A natural gauge-invariant 2-form is introduced on the space of connections over a compact oriented surface with boundary. It is shown that this 2-form descends to the moduli space of flat connections, under a group of based gauge transformations. An explicit expression (in terms of holonomy variations) is given for the resulting 2-form, and it is related to the symplectic structure of the extended moduli space introduced by L. C. Jeffrey.     

Simplicial gauge theory and quantum gauge theory simulation

We propose a general formulation of simplicial lattice gauge theory inspired by the finite element method. Numerical tests of convergence towards continuum results are performed for several SU(2) gauge fields. Additionally, we perform simplicial Monte Carlo quantum gauge field simulations involving measurements of the action as well as differently sized Wilson loops as functions of β.     

Abelian gauge theories on homogeneous spaces

An algebraic technique of separation of gauge modes in Abelian gauge theories on homogeneous spaces is proposed. An effective potential for the Maxwell-Chern-Simons theory on S ³ is calculated. A generalization of the Chern-Simons action is suggested and analyzed with the example of SU(3)/U(1) X U(1).     

Local currents for the GL(N,C) self-dual Yang-Mills equation

By using a simple Bäcklund-like transformation which linearizes the GL(N, C) self-dual Yang-Mills equation, an infinite number of local conservation laws for this equation are constructed. In the SL(N, C) case, the currents become trivial, which explains why these currents are not found in SU(N) gauge theory.     

Quasi-quantum groups as internal symmetries of topological quantum field theories

We show that every topological quantum field theory (understood as a functor) has an associated quasi-quantum group of internal symmetries.     

The GL q (2) gauge theory

We present some partial results on theq-deformation of the GL(2) Yang-Mills theory. In particular, the irreducible representations needed to describe the complete set of physical states are obtained by a simple procedure.     

C *-algebraic quantization and the origin of topological quantum effects

Concepts from the theory of abstract operator algebras are used to solve the problem of quantizing a particle moving on an arbitrary locally compact homogeneous space. Inequivalent quantizations are identified with inequivalent irreducible representations of the corresponding C ^*-algebra. Topological terms in the action (or Hamiltonian) are found to be representation-dependent, and are automatically induced by the quantization procedure. Known charge quantization conditions turn out to be identically satisfied. Several examples are considered, among them the Dirac monopole and the Aharonov-Bohm effect.     

Generalized holomorphic structures

We investigate generalized holomorphic structures in generalized complex geometry. We find that a generalized holomorphic vector bundle carries a generalized complex structure on its total space if some additional conditions hold. We prove that generalized holomorphicity is equivalent to the integrability of a distribution on the total space, and a family of linear Dirac structures associated with this distribution is a generalized complex structure if a further condition holds. Under the same condition, we also prove that local generalized holomorphic frames exist around a regular point.     

Self-dual monopoles in a seven-dimensional gauge theory

The existence of self-dual or anti-self-dual monopoles of a seven-dimensional generalized Yang-Mills-Higgs theory is proved using the second-order equations of motion. The behavior of solutions can be used to recognize self- or anti-self-duality. Moreover, it is shown that, in the class of the field configurations under discussion, the solutions are, in fact, unique.