Algebrodynamics in complex space-time and the complex-quaternionic origin of Minkowski geometry

This short well-written paper is suitable for students and researchers with basic knowledge of special relativity, hypercomplex algebra and differential geometry. par First, biquaternions Bbb{H} otimes Bbb{C} represented in M(2,Bbb{C}) are embedded in Bbb{C}^4 with complex coordinates z_{mu} in Bbb{C}, 0leq mu leq 3. The internal 3-complex-parameter transformations {rm SO}(3,Bbb{C}) preserve biquaternion multiplication. The trace z_0 remains invariant, z_1,z_2 and z_3 undergo complex rotations, and z_0^2-sigma and sigma = z_1^2+z_2^2+z_3^2 are two complex invariants. The real (metric Minkowski space) interval S^2=sigmasigma^* = T^2 - |X|^2geq 0 is reparametrized by two pure quaternions bold{p}, bold{q} (real and imaginary parts of z_1, z_2, z_3) to (positive definite) physical time T=|bold{p}|^2+|bold{q}|^2, and a physical Euclidean radius vector X=2bold{p}timesbold{q}. {rm SO}(3,Bbb{C}) transformations now describe proper Lorentz transformations. par Next, increments delta X and delta T are discussed in terms of increments deltabold{p} and deltabold{q}. delta T appears irreversible, any closed point particle path bold{p}(lambda), bold{q}(lambda), with variable lambda in Bbb{C}, has nonzero physical time T, perhaps related to time delay. Then the restoration of the "hidden" complex deltabold{p}, deltabold{q} structure from the Minkowski space-time delta X, delta T is discussed, mapping the complex null cone delta sigma=0 to the real light cone delta S^2=0. Only a common 1-parameter rotation with axis delta X remains. In general there is a "hidden" geometric phase alpha(deltabold{p}, deltabold{q}) invariant under Lorentz transformations, with alpha being possibly related to quantum interference. If alpha is fixed the angle theta between deltabold{p} and deltabold{q} can be computed, and depends on the velocity v and on alpha with extremes theta=0 and theta=pi, which might be related to quantum spin. par Finally, "hidden" algebraic dynamics in complex-quaternionic space and its image in Minkowski space-time is discussed using techniques of differential geometry. It is found that the real Minkowski interval is the modulus of the complex proper time of an elementary observer. In order to restore time order the introduction of an evolution curve is suggested. Variations of the alpha phase could be related to quantum uncertainty and interference.

Kassandrov Vladimir V.
Hitzer Eckhard M.S.
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Английский, Русский
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