Abstract:
Among various approaches in proving gauge independence, models containing an explicit gauge dependence are convenient. The well-known example is the gauge parameter in the covariant gauge fixing which is of course most suitable for the perturbation theory but a negative metric prevents us from imaging a dynamical picture. Noncovariant gauge such as the Coulomb gauge is on the contrary used for many physical situations. Therefore it is desirable to include both cases. More than ten years ago, Steinmann introduced a function (distribution) which can play this role in his attempt on discussing quantum electrodynamics (QED) in terms of the gauge invariant fields solely. The method is, however, broken down in the covariant case: the invariant operators are ill-defined because of 1/p^2 singularity in the Minkowski space. In this paper, we apply his function to the path integral: utilizing the arbitrariness of the function we first restrict it to be able to have a well- defined operator, and then a Hamiltonian with which we can build up the (Euclidean) path integral formula. Although the formula is far from covariant, a full covariant expression is recovered by reviving the components which have been discarded under the construction of the Hamiltonian. There is no pathological defects contrary to the operator formalism. With the aid of the path integral formula, the gauge independence of the free energy as well as the S-matrix is proved. Moreover the reason is clarified why it is so simple and straightforward to argue gauge transformations in the path integral. Discussions on the quark confinement is also presented.

Abstract:
In this paper we investigated the problem of the existence of invariant meaures on the local gauge group. We prove that it is impossible to define a {\it finite} translationally invariant measure on the local gauge group $C^{\infty}({\bf R}^n,G)$(where $G$ is an arbitrary matrix Lie group).

Abstract:
We introduce a new path integral representation for slave bosons in the radial gauge which is valid beyond the conventional fluctuation corrections to a mean-field solution. For electronic lattice models, defined on the constrained Fock space with no double occupancy, all phase fluctuations of the slave particles can be gauged away if the Lagrange multipliers which enforce the constraint on each lattice site are promoted to time-dependent fields. Consequently, only the amplitude (radial part) of the slave boson fields survives. It has the special property that it is equal to its square in the physical subspace. This renders the functional integral for the radial field Gaussian, even when non-local Coulomb-type interactions are included. We propose i) a continuum integral representation for the set-up of further approximation schemes, and ii) a discrete representation with an Ising-like radial variable, valid for long-ranged interactions as well. The latter scheme can be taken as a starting point for numerical evaluations.

Abstract:
We propose a three-fold covering of the group ${U}(2)$ as a gauge group for the electroweak interactions for the purpose of describing fields with integer and fractional electric charges with respect to the residual electromagnetic gauge group after a spontaneous breaking of the gauge symmetry. In a more general scheme we construct a three-fold covering of ${U} (n)$ and consider for the case $n=2$ several representations which are used in the construction of a model of the electroweak interactions in a subsequent paper.

Abstract:
We show that in the functional integral formalism of U(1) gauge field theory some formal manipulation such as interchange of order of integration can yield erroneous results. The example studied is analysed by Fubini theorem.

Abstract:
We compute the gauge field functional integral giving the scalar product of the SU(2) Chern-Simons theory states on a Riemann surface of genus > 1. The result allows to express the higher genera partition functions of the SU(2) WZNW conformal field theory by explicit finite dimensional integrals. Our calculation may also shed new light on the functional integral of the Liouville theory.

Abstract:
We find that sometimes the usual definition of functional integration over the gauge group through limiting process may have internal difficulties.

Abstract:
We consider the discrete analogue of a fractional integral operator on the Heisenberg group, for which we are able to prove nearly sharp results by means of a simple argument of a combinatorial nature.

Abstract:
We present a refinement of a recently found gauge-gravity relation between one-loop effective actions: on the gauge side, for a massive charged scalar in 2d dimensions in a constant maximally symmetric electromagnetic field; on the gravity side, for a massive spinor in d-dimensional (Euclidean) anti-de Sitter space. The inclusion of the dimensionally regularized volume of AdS leads to complete mapping within dimensional regularization. In even-dimensional AdS, we get a small correction to the original proposal; whereas in odd-dimensional AdS, the mapping is totally new and subtle, with the `holographic trace anomaly' playing a crucial role.

Abstract:
We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaat-Heckman theorem, and the Weyl integral formula respectively. In each case, we first introduce the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups), and describe the finite dimensional integration formulae. We then discuss some applications to path integrals and give an overview of the relevant literature. The applications we deal with include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two-dimensional Yang-Mills theory.