Q-OSCILLATORS AND THEIR APPLICATIONS

Abstract

The recent focus on deformations of algebras called quantum algebras can be attributed to the fact that they appear to be the basic algebraic structures underlying an amazingly diverse set of physical situations. To date many interesting features of these algebras have been found and they are now known to belong to a class of algebras called Hopf algebras. In this thesis, we begin with a study of q-oscillators which have gained prominence in the literature. Primarily introduced as objects providing realizations to some of the quantum algebras, they have assumed an identity of their own in the general framework. Indeed, it has been shown that for a particular form, the algebra may itself support a Hopf structure. Here we examine the relationship among the various forms of the q-oscillator algebra and show that they are essentially inequivalent. This inequivalence is shown to have some important bearings on the Hopf structure associated with one of them. For the latter, a detailed study of its representations reveals an interesting link with the quantum su(2) algebra. In fact it turns out that the two are equivalent not only algebraiclly but also at the level of Hopf algebras. A major portion of this thesis is devoted to the construction of field theories based on q-deformed algebras. In this respect, two versions are considered. In the first, which is realized through the q-oscillators, we provide a systematic formulation of what we refer to as a q-deformed conformal field theory. Here, q-analogues of the primary and secondary fields are proposed and the operator product expansions between the relevant fields are furnished. We also derive the q-Ward identities which provide useful information about the correlation functions. The existence of a solution for the case of a two-point function is further exhibited. In the second version we furnish a realization of the q-Heisenberg algebra through the q operator product expansion of two current fields. The q-analogue of the Sugawara construction is also considerd in this context. In both of these cases, we provide the higher genus extensions using the formalism of Krichever and Novikov. In this regard we obtain two q-deformed versions of the the KN algebra. Finally, some issues pertaining to the formulation of a q-string theory are examined.