Noncommutative geometry framework and the Feynman's proof of Maxwell equations

The main focus of the present work is to study the Feynman's proof of the Maxwell equations using the NC geometry framework. To accomplish this task, we consider two kinds of noncommutativity formulations going along the same lines as Feynman's approach. This allows us to go beyond the standard case and discover nontrivial results. In fact, while the first formulation gives rise to the static Maxwell equations, the second formulation is based on the following assumption m[x(j),x(k)]=delta(jk)+imtheta(jk)f. The results extracted from the second formulation are more significant since they are associated to a nontrivial theta-extension of the Bianchi-set of Maxwell equations. We find div(theta) B=eta(theta) and (partial derivativeB(s)/partial derivativet)+epsilon(kjs)(partial derivativeE(j)/partial derivativex(k))=A(1)(d(2)f/dt(2))+A(2)(df/dt)+A(3), where eta(theta), A(1), A(2), and A(3) are local functions depending on the NC theta-parameter. The novelty of this proof in the NC space is revealed notably at the level of the corrections brought to the previous Maxwell equations. These corrections correspond essentially to the possibility of existence of magnetic charge sources that we can associate to the magnetic monopole since div(theta) B=eta(theta) is not vanishing in general. (C) 2003 American Institute of Physics.