A derivation of Maxwell equations in quaternion space

At the root of the present paper is the purely mathematical fact that the Lorentz group (or, more precisely, its Lie algebra) can be represented in a number of isomorphic ways. In addition to the 4 times 4 real matrices of {rm SO}(1,3) that come from the defining representation on Minkowski space, one also has the 2 times 2 complex matrices of {rm SL}(2,Bbb C) that relate to Dirac spinors (more precisely, bispinors) and the 3 times 3 complex matrices of {rm SO}(3,Bbb C) that relate to the representation on the space of 3-spinors, which also are used in the Majorana-Oppenheimer representation of relativistic wave mechanics. par It is in relation to the latter representation of Lorentz transformations that one can introduce complex quaternions, as well. That is because the set of all unit real quaternions inherits a multiplication from the quaternion multiplication rules that makes it a group that is isomorphic to the simply-connected covering group {rm SU}(2) of the Euclidean rotation group {rm SO}(3,Bbb R). Interestingly, {rm SU}(2), despite the fact that its defining representation involves complex matrices, is not a complex Lie group, but a real one. Under complexification, {rm SU}(2) becomes {rm SL}(2,Bbb C), {rm SO}(3,Bbb R) becomes {rm SO}(3,Bbb C), and the real quaternions become complex quaternions. Thus, the unit complex quaternions are a group that is isomorphic to {rm SL}(2,Bbb C) or {rm SO}(3,Bbb C), and locally isomorphic to the Lorentz group. par One can then define a generalized mechanics in which the group of motions no longer has to be the group of Euclidean rigid motions, but possibly the Poincaré group, in its various representations, and one finds that one still has to deal with fictitious accelerations that arise from differentiating the group action twice. (There is a consistent error in the present paper regarding the representation of the contribution from the angular acceleration.) This was the basis for the way that Feynman once showed how one could derive the Maxwell equations as a consequence of the Lorentz force equation in a noninertial frame, when one included the fictitious accelerations. Basically, the bold E field was due to the linear, angular, and centripetal accelerations, while the bold B field was due to the Coriolis acceleration. This is also the basis for Larmor's theorem that the motion of an electric charge in a constant, uniform, magnetic field is mechanically equivalent to the motion of the particle in a rotating frame in which the angular velocity of rotation is proportional to the magnetic field strength. par One can similarly represent the Dirac equation, which is closely related to the Maxwell equations, in various ways, depending upon the way that one is representing the Lorentz group, and the authors briefly discuss the way that it gets represented in terms of complex quaternions.

Chrisitianto Vic , Smarandache Florentin
Delphenich David H.
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