Noncommutative Extension of Minkowski Spacetime and Its Primary Application

We propose a noncommutative extension of the Minkowski spacetime by introducing a well-defined proper time from the $\kappa$-deformed Minkowski spacetime related to the standard basis. The extended Minkowski spacetime is commutative, {\em i.e.} it is based on the standard Heisenberg commutation relations, but some information of noncommutativity is encoded through the proper time to it. Within this framework, by simply considering the Lorentz invariance we can construct field theory models that comprise noncommutative effects naturally. In particular, we find a kind of temporal fuzziness related to noncommutativity in the noncommutative extension of the Minkowski Spacetime. As a primary application, we investigate three types of formulations of chiral bosons, deduce the lagrangian theories of noncommutative chiral bosons and quantize them consistently in accordance with Dirac's method, and further analyze the self-duality of the lagrangian theories in terms of the parent action approach.