文档介绍：Foundations of Physics, Vol. 16, No. 8, 1986
Relativistic Algebraic Spinors and
Quantum Motions in Phase Space
P. R. Holland I
Received February 15, 1985; revised July 23, 1985
Following suggestions of SchOnberg and Bohm, we study the tensorial phase space
representation of the Dirac and Feynman-Gell-Mann equations in terms of the
complex Dirac algebra C4, a Jordan-Wigner algebra G4, and Wigner transfor-
mations. To do this we solve the problem of the conditions under which elements in
C 4 generate minimal ideals, and extend this to G4. This yields the linear theory of
Dirac spin spaces and tensor representations of Dirac spinors, and the spin -1 wave
equations are represented through fermionic state vectors in a higher space as a set
of interconnected tensor relations.
1. INTRODUCTION
In a series of papers, Sch6nberg (1) has brought out the connections which
exist between the algebras of quantum field theory, Clifford algebras and
geometry. Recently, Bohm (2) has used some of Sch6nberg's results to derive
a Liouville equation for the massless Dirac-like equation (Dirac operator
applied to a general Dirac number). In this approach one takes the density
matrix (or a non-Hermitian extension of this, the characteristic matrix) as
a basic descriptive element from which the wave function is an abstraction.
By treating the wave equation as a relation in a higher vector space, the
application of a Wigner transformation (3) leads to equations in phase space
which may pared with analogous classical equations. A particular
point here is to treat the Wigner transformed characteristic matrix not as
directly related to a phase space probability distribution function but
rather as a field which defines constants of the motion. C4)
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