### The Spinning Motions of All Fermions and Bosons as Implied by Pauli Matrices Containing Complex Conjugates in a Combined Spacetime Four-Manifold

#### Abstract

By substituting *z* = *a + bi* and *1/z* = *a – bi* for *i* and –*i* into one of Pauli matrices and then casting *(x,y) = (0,a+bi)* as *(x,y,z) = (0,a,b)* and *(x,y) = (a-bi,0)* as *(x,y,z) = (a,-b,0)* by the geometry of our previously formulated combined spacetime 4-manifold = *{(t+ti,x+yi,y+zi,z+xi)}*, this paper generalizes the Dirac equation for a free electron into an equation that gives the motion *(t,x(t),y(t),z(t))* for any free fermion or boson to be a uniform circular flow around two semi-circles connected with each other by an angle equal to 0, 30, 60, 90, or 180 degrees depending on the electric charge possessed by the particle. Even purely algebraically, any fermion or boson must correspond to a number on the complex unit circle, since the Dirac equation admits a Pauli matrix of *z* with modulus equal to one but not necessarily equal to *i *and all energies in free space must satisfy this Dirac equation as derived from the universally true equation of “energy-squared minus pc-squared equal to rest-energy-squared.”

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Applied Physics Research ISSN 1916-9639 (Print) ISSN 1916-9647 (Online)

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