By Sweetser D.B.
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Extra resources for Doing physics with quaternions
Sample text
A 57 This combination of differential quaternion operator, quaternion potential and quaternion 4−velocity generates the covariant form of the Lorentz operator in the Lorenz gauge, minus a factor of the charge e which operates as a scalar multiplier. Implications By writing the covariant form of the Lorentz force as an operator acting on a potential, it may be possible to create other laws like the Lorentz force. For point sources in the classical limit, these new laws must have the form of Coulomb’s law, F = k e e’/r^2.
I call this the "2 zero" rule: if there are two zeros in the generator of a law in physics, the law is classical. Newton’s 2nd Law for an Inertial Reference Frame in Cartesian Coordinates Define a position quaternion. R t, R Operate on this once with the differential operator to get the velocity quaternion. V d ,0 dt t, R 1, R Operate on the velocity to get the classical inertial acceleration quaternion. A d ,0 dt 1, R 0, R This is the standard form for acceleration in Newton’s second law in an inertial reference frame.
They are worked out for quaternions here in detail to solidify the connection between the machinery of quantum mechanics and quaternions. The conjugate of the square of the norm equals the square of the norm of the two terms reversed. X, t X Xt X t XxX X xX These are identical, because the terms involving the cross produce will flip signs when their order changes. For products of squares of norms in quantum mechanics, Φ Φ Φ Φ This is also the case for quaternions. X , 0 2 2 t, X t, X t,X t,X t,X t,X , 0 ^2 2 If the signs of each pair of component are the same, the two sides will be equal.