Quantum field theory

The new formalism for quantization of gauge systems based on the concept of the
dynamical Hamiltonian recently introduced as a basis for the canonical theory of
quantum gravity was considered in the context of general gauge theories. This and
other Hamiltonian methods, that include Dirac's theory of extended Hamiltonian
and the Hamiltonian reduction formalism were critically examined. It was established
that the classical theories of constrained gauge systems formulated within the
framework of either of the approaches are equivalent. The central to the proof of
equivalence was the fact that the gauge symmetries resuIt in the constraints of the
first class in Dirac's terminology that Iead to redundancy of equations of motion
for some of the canonica variables. Nevertheless, analysis of the quantum theories
showed that in general, the quantum theory of the dynamical Hamiltonian is inequivalent
to those of the extended Hamiltonian and the Hamiltonian reduction. The
new method of quantization was applied to a number of gauge systems, including
the theory of relativistic particle, the Bianchi type IX cosmological model and spinor electrodynamics along side with the traditional methods of quantization. In all of the
cases considered the quantum theory of the dynamical Hamiltonian was found to be
well-defined and to possess the appropriate classical limit. In particular, the quantization
procedure for the Bianchi type IX cosmological spacetime did not run into
any of the known problems with quantizing the theory of General Relativity. On the
other hand, in the case of the quantum electrodynamics the dynamical Hamiltonian
approach led to the quantum theory with the modified self-interaction in the matter
sector. The possible consequence of this for the quantization of the full theory of
General Relativity including the matter fields are discussed.

Working with the creation and annihilation operators of
the Cook formalism we develop methods to analyze the permutation
symmetry of the (n-1)-particle system that results
when the annihilation operator is applied to a system of n
indistinguishable particles having a particular initial
permutation symmetry. These methods are then applied to
the Wigner supermultiplet model.

A discrete formalism for General Relativity was introduced in 1961 by Tulio Regge in the form of a piecewise-linear manifold as an approximation to (pseudo-)Riemannian manifolds. This formalism, known as Regge Calculus, has primarily been used to study vacuum spacetimes as both an approximation for classical General Relativity and as a framework for quantum gravity. However, there has been no consistent effort to include arbitrary non-gravitational sources into Regge Calculus or examine the structural details of how this is done. This manuscript explores the underlying framework of Regge Calculus in an effort elucidate the structural properties of the lattice geometry most useful for incorporating particles and fields. Correspondingly, we first derive the contracted Bianchi identity as a guide towards understanding how particles and fields can be coupled to the lattice so as to automatically ensure conservation of source. In doing so, we derive a Kirchhoff-like conservation principle that identifies the flow of energy and momentum as a flux through the circumcentric dual boundaries. This circuit construction arises naturally from the topological structure suggested by the contracted Bianchi identity. Using the results of the contracted Bianchi identity we explore the generic properties of the local topology in Regge Calculus for arbitrary triangulations and suggest a first-principles definition that is consistent with the inclusion of source. This prescription for extending vacuum Regge Calculus is sufficiently general to be applicable to other approaches to discrete quantum gravity. We discuss how these findings bear on a quantized theory of gravity in which the coupling to source provides a physical interpretation for the approximate invariance principles of the discrete theory.