Structure of gauge theories

Elementary interactions are formulated according to the principle of minimal interaction although paying special attention to symmetries. In fact, we aim at rewriting any field theory on the framework of Lie groups, so that, any basic and fundamental physical theory can be quantized on the grounds of a group approach to quantization. In this way, connection theory, although here presented in detail, can be replaced by “jet-gauge groups” and “jet-diffeomorphism groups.” In other words, objects like vector potentials or vierbeins can be given the character of group parameters in extended gauge groups or diffeomorphism groups. As a natural consequence of vector potentials in electroweak interactions being group variables, a typically experimental parameter like the Weinberg angle $$\vartheta _W$$ ϑ W is algebraically fixed. But more general remarkable examples of success of the present framework could be the possibility of properly quantizing massive Yang–Mills theories, on the basis of a generalized Non-Abelian Stueckelberg formalism where gauge symmetry is preserved, in contrast to the canonical quantization approach, which only provides either renormalizability or unitarity, but not both. It proves also remarkable the actual fixing of the Einstein Lagrangian in the vacuum by generalized symmetry requirements, in contrast to the standard gauge (diffeomorphism) symmetry, which only fixes the arguments of the possible Lagrangians.

GAUGE-INVARIANT REFORMULATION OF THE VECTOR SCHWINGER MODEL WITH A PHOTON MASS TERM AND ITS HAMILTONIAN, PATH INTEGRAL AND BRST FORMULATIONS

Using the Stueckelberg formalism, we construct a gauge-invariant version of the vector Schwinger model (VSM) with a photon mass term studied by one of us recently. This model describes two-dimensional massive electrodynamics with massless fermions, where the left-handed and right-handed fermions are coupled to the electromagnetic field with equal couplings. This model describing the 2D massive electrodynamics becomes gauge-noninvariant (GNI). This is in contrast to the case of the massless VSM which is a gauge-invariant (GI) theory (as a consequence of demanding the regularization for the theory to be GI). In this work we first construct a GI theory corresponding to this model describing the 2D massive electrodynamics, using the Stueckelberg formalism and then we recover the physical contents of the original GNI theory studied earlier, under some special gauge choice. We then study the Hamiltonian, path integral and BRST formulations of this GI theory under appropriate gauge-fixing. The theory presents a new class of models in the 2D quantum electrodynamics with massless fermions but with a photon mass term.

VECTOR SCHWINGER MODEL WITH A PHOTON MASS TERM: GAUGE-INVARIANT REFORMULATION, OPERATOR SOLUTIONS AND HAMILTONIAN AND PATH INTEGRAL FORMULATIONS

We consider the vector Schwinger model (VSM) describing two-dimensional electrodynamics with massless fermions, where the left-handed and right-handed fermions are coupled to the electromagnetic field with equal couplings, with a mass term for the U(1) gauge field and then study its operator solutions and the Hamiltonian and path integral formulations. We emphasize here that although the VSM has been studied in the literature rather widely but only without a photon mass term (which was a consequence of demanding the regularization for the VSM to be gauge-invariant (GI)). The VSM with a photon mass term is seen to be a gauge-noninvariant (GNI) theory. Using the standard Stueckelberg formalism we then construct a GI theory corresponding to the proposed GNI model. From this reformulated GI theory, we further recover the physical contents of the proposed GNI theory under a very special gauge choice. The theory proposed and studied here presents a new class of models in the two-dimensional quantum electrodynamics with massless fermions but with a photon mass term.

4-VECTOR FIELDS AND INDEFINITE METRICS

We generalize the Stueckelberg formalism in the (1/2, 1/2) representation of the Lorentz Group, see Ref. 1–2. We analize the problem of the mass generation and of the indefinite metrics from the modern viewpoints, see Ref. 3–4. Some relations to other modern-physics models are found.

LOCAL GAUGE FIELDS BASED ON A YANG–MILLS THEORY IN LOOP SPACE

We construct a Yang–Mills theory in loop space (the space of all loops in Minkowski space) with the Kac–Moody gauge group in such a way that the theory possesses reparametrization invariance. On the basis of the Yang–Mills theory, we derive the usual Yang–Mills theory and a non-Abelian Stueckelberg formalism extended to local antisymmetric and symmetric tensor fields of the second rank. The local Yang–Mills field and the second-rank tensor fields are regarded as components of a Yang–Mills field on the loop space.

LOCAL GAUGE FIELDS BASED ON A U(1) GAUGE THEORY IN LOOP SPACE

We present a U(1) gauge theory defined in loop space, the space of all loops in Minkowski space. On the basis of the U(1) gauge theory, we derive a local field theory of the second-rank antisymmetric tensor field (Kalb-Ramond field) and the Stueckelberg formalism for a massive vector field; the second-rank antisymmetric tensor field and the massive vector field are regarded as parts of a U(1) gauge field on the loop space. We also consider the quantum theories of the second-rank antisymmetric tensor field and the massive vector field on the basis of a BRST formalism for the U(1) gauge theory in loop space. In addition, reparametrization invariance in the U(1) gauge theory is discussed in detail.

A GENERAL SOLUTION OF THE MASTER EQUATION FOR A CLASS OF FIRST ORDER SYSTEMS

Inspired by the formulation of the Batalin-Vilkovisky method of quantization in terms of “odd time,” we show that for a class of gauge theories which are first order in the derivatives, the kinetic term is bilinear in the fields, and the interaction part satisfies some properties, it is possible to give the solution of the master equation in a very simple way. To clarify the general procedure we discuss its application to Yang-Mills theory, massive (Abelian) theory in the Stueckelberg formalism, relativistic particle and to the self-interacting antisymmetric tensor field.

A SIMPLE MANNER FOR THE QUANTIZATION OF ANOMALOUS GAUGE THEORIES

The chiral Schwinger model is a massive vector theory at the quantum level. We construct the gauge invariant action using Stueckelberg formalism from this. Then the resulting action is exactly the same as the modified action obtained by path-integral formalism. We propose a simple manner for the quantization of anomalous gauge theories.