Tesis Magíster
An extensión of the landau problem to quantum field theory
Autor
Loewe, Marcelo
Gamboa, Jorge
Pontificia Universidad Católica de Chile
Institución
Resumen
The noncommutativity of spacetime is an old idea, which dates back to the 1930's and isattributed to Heisenberg. The idea was first developed by Snyder, who published his analysisin 1947 [1]. It has been revived in recent years within the context of string theory and sincethen, noncommutative field theories have attracted much attention [2].In the first part of this thesis we begin to study the Landau problem [3] as a motivation tononcommutativity. In this scenario the momentum space of this system is noncommutativeby considering the kinetic momentum. On the other hand, the noncommutative coordinatespace arises when considering a strong magnetic field. La ter, we study an harmonic oscillatorin a magnetic field, solving both systems using ladder operators.Within noncommutative quantum mechanics (NCQM) the Moyal correspondence is introduced,in which all products are replaced by the star product [4] in order to obtain thenoncommutative version of quantum mechanics. In addition, we show the equivalence withthe Bopp shift, a transformation connecting the noncommutative to commutative variables.Using the generalized Bopp shift we transform sorne two dimensional systems defined in anoncommutative phase space into a commutative one: the isotropic harmonic oscillator, theisotropic harmonic oscillator in a magnetic field and the central field. These are solved usingour previous analysis of ladder operators.In the second part of the thesis we want to extend the idea of noncommutative variablesto field theory. For this we use the most simple way to find noncommutativity, the Landauproblem, i.e. we follow the idea of noncommutativity of kinetic momenta to realize theidea. The idea of noncommutative fields has been considered for example, in [26] through amodification of the commutation relations between fields and momenta.In this work we postulate a Lagrangian with certain extra terms accompanied with coefficientsthat violate Lorentz and CPT symmetry. This aspect is similar to the Standard ModelExtension (SME) [39], but we want to emphasize that this theory does not ha ve the samemotivation as SME.