Prog. Theor. Phys. Vol. 117 No. 3 (2007) pp. 501-532
Quaternions, Lorentz Group and the Dirac Theory
Department of Physics, Nagoya University, Nagoya 464-4602, Japan
(Received January 5, 2007)
It is shown that a subgroup of SL(2,H),
denoted Spin(2,H) in this paper,
which is defined by two conditions in addition to unit quaternionic
determinant, is locally isomorphic to
the restricted Lorentz group, L+↑.
On the basis of the Dirac theory using the spinor group
Spin(2,H), in which the charge conjugation transformation
becomes linear in the quaternionic Dirac spinor,
it is shown that the Hermiticity requirement
of the Dirac Lagrangian, together with the persistent
presence of the Pauli-Gürsey SU(2) group, requires
an additional imaginary unit (taken to be the ordinary one, i)
that commutes with Hamilton's units, in the theory.
A second quantization is performed with this i incorporated into the theory,
and we recover the conventional Dirac theory with an automatic
`anti-symmetrization' of the field operators.
It is also pointed out that we are naturally led to the scheme of
complex quaternions, Hc, in which a space-time point
is represented by a Hermitian quaternion, and that the isomorphism
is a direct consequence of the fact
we make explicit the Weyl spinor indices of the spinor-quaternion,
which is the Dirac spinor defined over Hc.
DOI : 10.1143/PTP.117.501
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Citing Article(s) :
Progress of Theoretical Physics Vol. 126 No. 5 (2011) pp. 903-914
Note on a New Quaternionic Approach to Relativity and the Dirac Theory