Relativistic quantum mechanics on non-commutative spaces

A brief review of non-relativistic and relativistic quantum mechanics is carried out, with particular attention to concepts that are modi ed in the noncommutative setting. The Hilbert-Schmidt operator formulation of non-relativistic, non-commutative quantum mechanics is generalized to the relativistic setting of 4-dimensional non-commutative space-time. The generators of Lorentz transformations are derived in this formalism and compared with the commutative case. It is shown that the non-interacting, non-commutative Dirac equation is Lorentz invariant. Lorentz invariance can be maintained when electromagnetic interactions are included provided that the gauge and Dirac elds are composed through an appropriate star product. The appropriate gauge transformations for the non-commutative Dirac equation are found. The non-commutative C, P, T symmetries of the interacting Dirac equation are investigated. The free Dirac equation and the interacting Dirac equation for the case of a constant background magnetic eld are studied on the operator level. Systems con ned to a speci c space-time volume, and the associated boundary conditions, are studied using appropriate projection operators. As a speci c example the Dirac equation in an in nitely long cylinder is considered. An operator valued action, which yields the interacting Dirac equation as the equation of motion, is derived and evaluated in a coherent state basis. The constant background magnetic eld is revisited in the coherent state basis. This establishes the link to the standard star product formulation of non-commutative quantum eld theories. Useful properties of the star product are derived in an appendix.National Research Foundatio