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Applied Physics and Mathematics

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Quaternion algebra and the description of spin
M.A. Mikaelyan

In non-relativistic quantum mechanics the particle with spin
½ is associated with the evolution of four real functions, which
combine with each other in two complex functions that form a
spinor. With the help of quaternion algebra an alternative variant
of combining in one quaternion function is considered [1].
Ultimately a description of spin particle appears to be identical
to the description of spinless particle; in all formulas instead of
the usual imaginary unit the quaternion imaginary unit figures.
At the same time, unlike standard description, «switching» of
magnetic field by the use of well-known substitution in momentum
operator leads to the correct value of intrinsic magnetic
moment of the particle. Relativistic generalization of the
description of spin particle is formulated and corresponds to
the usage of biquaternion algebra (biquaternion is a quaternion
with complex components). As in non-relativistic case the wave
function is a one-component quantity. It means an absence of
matrices in the formalism; this fact considerably simplifies the
consideration. In the language of biquaternion algebra the Dirac
and Klein-Gordon equations are formulated. Simultaneously
the quaternion formulation of Lorentz transformations and
Maxwell equations are represented.
Keywords: quantum mechanics, spin, quaternion, biquaternion,
Pauli equation, Dirac equation.

Contacts: E-mail: mikhail@bk.ru

Pp. 76-84.

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