Noncommutative Solvable Models in Quantum and Field Theories

Abstract

The dissertation work substantially bears our contribution to the development of noncom mutative (NC) theories. It mainly provides a comparative study of ordinary complex scalar 4 D and complex Grosse-Wulkenhaar (GW) model. In this context, relevant physical quan tities such as energy momentum tensors (EMTs) are explicitly computed and improved to satisfy known physics based properties in line with the Wilson and Jackiw techniques. In all these theories, the dilatation symmetry is broken and the breaking terms are discussed. As expected, all computed physical quantities for ordinary complex 4 D noncommutative eld theory (NCFT) are easily recovered from the results obtained for the complex GW NCFT by setting = 0. A generalization of the Hamiltonian formulation developed by Gomis et al is perfomed and analyzed for the renormalizable Grosse-Wulkenhaar 4 D model. The dynamical noncommutativity introduced by Aschieri et al [3] is implemented and discussed in the case of a new class of renormalizable NC eld theories (RNCFT) built on the GW 4 scalar eld model de ned in Euclidean space. Our investigations show that the twisted GW action is not invariant under global translation. Such an undesirable feature has been got round by imposing a constraint on the Lagrangian action, which is nothing but the equation of motion governing the GW harmonic term. Contrarily to pevious works, both ordinary GW and twisted GWmodels provide nonlocally conserved and nonsymmetric EMT, angular momentum tensor (AMT) and dilatation current (DC) due to the presence of the harmonic term. Fortunately, all these physical quantities can be subjected to well known Jackiw and Wilson regularization procedures to acquire the local conservation property. We de ne the twisted connections in NC spaces and discuss NC gauge transformations. Then, the Yang-Mills (YM) action, twisted in the dynamical Moyal space, is proved to be invariant under U (1) local gauge transformation with the parameter = 0 + x, where is an innitesimal parameter and 0 a constant. The NC gauge invariant currents are explicitly computed. These currents are locally conserved. Besides, the main properties of the harmonic oscillator in the framework of a dynam ical noncommutativity realized through a twisted Moyal product are discussed. Working in the NC con guration space, explicit spectrums of harmonic oscillator with non-vanished momentum-momentumbracketarederived and the spectrums computed. It should be pointed out that, in order to maintain the Bose-Einstein statistics, the model parameters and must satisfy the relation 2 2 =0. Therefore, the parameters and re ect the intrin sic noncommutativity between positions and momenta, respectively, (as a Planck constant encodes the noncommutativity of position and momentum).