The Non-commutative Electromagnetic Field Theory

We deal with the electromagnetic field in the non-commutative space-time by path integral. In this process, we think of three aspects: mapping between A^ in the non-commutative space-time and A^ in the commutative space-time; reducing Lagrangian by replacing star product with ordinary product; corresponding relation of integral between the non-commutative space and commutative space. Then we obtain 5-matrix which are represented by A^. We calculate the element Feynman diagrams in the first order.