Classical Mechanics And Quantum Field Theory On Noncommutative Spacetime

There are a lot of studies that generalize the ordinary spacetime to a noncommu-tative (NC) spacetime and provide interesting physical models. Because these studies rely only on a formal extension without experiments, there are many possible choices to construct NC spacetimes. A different kind of NC spacetimes usually gives different results that should generally not contradict with the very basic physical principle.The present paper tries to study noncommutative classical mechanics and non-commutative field theory as well. As to noncommutative classical mechanics, we in-vestigate the kinetics of a nonrelativistic particle interacting with a constant external force on a Lie-algebraic noncommutative space. The structure constants of a Lie al-gebra, also called noncommutative parameters, are constrained in general due to some algebraic properties, such as the antisymmetry and Jacobi identity. Through solving the constraint equations the structure constants satisfy, we obtain two new sorts of al-gebraic structures, each of which corresponds to one type of noncommutative spaces. Based on such types of noncommutative spaces as the starting point, we analyze the classical motion of the particle interacting with a constant external force by means of the Hamiltonian formalism on a Poisson manifold. Our results provide new trajecto-ries of motion governed mainly by marvelous extra forces. The extra forces with the unimaginable tχ-,(χχ)-, and (χχ)-dependence besides with the usual t-, χ-, and χ-dependence, originating from a variety of noncommutativity between different spatial coordinates and between spatial coordinates and momenta as well, deform greatly the particle’s ordinary trajectories we are quite familiar with on the Euclidean (commuta-tive) space.As to noncommutative field theory, we construct a deformed Wess-Zumino model on the BFNC superspace where the Bosonic and Fermionic coordinates are no longer commutative with each other. Using the background field method, we calculate the primary one-loop effective action based on the deformed action. By comparing the two actions, we find that the deformed Wess-Zumino model is not renormalizable. To obtain a renormalizable model, we combine the primary one-loop effective action with the deformed action, and then calculate the secondary one-loop effective action based on the combined action. After repeating this process to the third time, we finally give the one-loop renormalizable action by using our specific techniques of calculation.Inspired by the result obtained, we construct a new BFNC Wess-Zumino model with the complex mass and interacting constant, analyze possible divergent terms in its effective action by using the dimensional analysis method, and further prove that the new BFNC Wess-Zumino model is renormalizable to all orders in perturbation theory.