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.     

The quantum Sturm-Liouville equation,quantum resolvent,quantum integrals,and quantum KdV: The fast decrease case

We construct quantum operators solving the quantum versions of the Sturm-Liouville equation and the resolvent equation, and show the existence of conserved currents. The construction depends on the following input data: the basic quantum field O(k) and the regularization.     

Classical solutions to a quadratically nonlinear gauge theory

We present some classical solutions to a gauge theory based on quadratically nonlinear Lie algebras without a central term. We observe that instanton-like and meron-like solutions force the internal fields to behave like solitons.     

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.     

A note on gauge symmetry breaking

We show that the breaking of Abelian gauge symmetry implies the existence of dipole singularities in the correlation functions of the (Abelian) Higgs model. We also show that the noninvariance of the Wightman functions does not preclude the implementability of the global gauge symmetry. An explicit example of gauge symmetry breaking (Ferrari's model) is discussed.     

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.     

Loop Group-Valued CS and WZNW models,integrable systems,and self-dual Yang-Mills theory

Affine CS and WZNW theories with values in infinite-dimensional (loop) groups are proposed. It appears that the affine CS theory naturally introduces a spectral parameter into a CS theory. The Sinh-Gordon, KdV, and nonlinear Schrödinger equations are obtained, via Hamiltonian reductions, from the affine WZNW. It is shown that the self-dual Yang-Mills (SDYM) equation is related to the equation of motion of the affine WZNW and, thus, symmetry algebra underlying the SDYM can be identified with the affine two-loop Kac-Moody algebra of the affine WZNW.K. C. Wong Research Award Winner, address after 28 October 1992: Department of Mathematics, University of Queensland, Brisbane, Queensland 4072, Australia.     

Convergence of Logarithmic Quantum Mechanics to the Linear One

We study the nonlinear logarithmic Schrödinger equation in three dimensions. We establish the existence of the solutions of general quasi-linear Schrödinger equations. Finally, we show the convergence of the logarithmic quantum mechanics to the linear regime.     

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.     

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.     

Operator ideals and the statistical independence in quantum field theory

We consider analytic properties of a class of dynamical systems which are defined by the action of certain homomorphism groups on von Neumann algebras if restricted to subalgebras. In particular, the analyticity of nuclear maps in the nuclear norm is shown. Furthermore, the statistical independence will be derived from nuclearity conditions. These results give new insight in the statistical independence of commuting algebras.This paper is a result of a collaboration with H. J. Borchers.     

Generalized nuclearity conditions and the split property in quantum field theory

Generalized nuclearity conditions that are applicable in arbitrary superselection sectors of a quantum field theory and to theories with a maximal temperature are discussed. They are shown to imply the (distal) split property and to impose specific restrictions on the spectral properties of modular operators associated with local algebras and vectors of compact energy support.     

A generalized self-dual Chern-Simons Higgs theory

In the recently discovered Chern-Simons model, the reduction to a Bogomol'nyi bound or self-duality depends crucially on the specific form of the Higgs potential energy function, which is characterized by a ⁶ type self-interaction. The purpose of this paper is to show that a much wider class of Higgs self-interaction may be allowed to achieve self-duality provided that the kinetic energy term of the Higgs scalar is suitably modified. The existence of topological multivortex solutions is also established. Furthermore, it is remarked that the Meissner effect may occur in the model.     

Modular inclusion,the Hawking temperature,and quantum field theory in curved spacetime

A recent result by Borchers connecting geometric modular action, modular inclusion and spectrum condition, is applied in quantum field theory on spacetimes with a bifurcate Killing horizon (these are generalizations of black-hole spacetimes, comprising the familiar black-hole spacetime models). Within this framework, we give sufficient, model-independent conditions ensuring that the temperature of thermal equilibrium quantum states is the Hawking temperature.     

Many boson realizations of universal nonlinearW ∞-algebras,modified KP hierarchies,and graded lie algebras

An infinite number of free field realizations of the universal nonlinear ^(N) ( ₁₊ ^(N) ) algebras, which are identical to the KP Hamiltonian structures, are obtained in terms ofp plusq scalars of different signatures withp –q =N. They are generalizations of the Miura transformation, and naturally give rise to the modified KP hierarchies via corresponding realizations of the latter. Their characteristic Liealgebraic origin is shown to be the graded SL(p, q).     

Feynman integral and complex classical trajectories

Let _t be an analytic solution of the Schrödinger equation with the initial condition . Let _t be the solution of the Schrödinger equation with the initial condition =, where is an analytic function. When 0, then _t (x) _t (x)¹ ( _t (x)), where _t (x) trajectory starting from x. We relate this result to Feynman's sum over trajectories and complex stochastic differential equations.     

Self-dual Vortices in a Maxwell–Chern–Simons Model with Non-minimal Coupling

We study self-dual vortex solutions in a Maxwell – Chern – Simons model with anomalous magnetic moment. We establish the existence of multivortex solutions and obtain the quantized energy and the magnetic flux. We also prove the uniqueness of solutions when there is only one vortex point.      

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.