Quaternions, Lorentz Group and the Dirac Theory

TL;DR

This paper explores the quaternionic representation of the Lorentz group and Dirac theory, revealing new algebraic structures and their implications for quantum field theory and spinor representations.

Contribution

It introduces a subgroup of quaternionic matrices isomorphic to the Lorentz group and develops a quaternionic Dirac theory with complex quaternions, linking to standard spinor formalisms.

Findings

Quaternionic subgroup $Spin(2,\mathbb{H})$ isomorphic to Lorentz group

Hermiticity and SU(2) symmetry require an imaginary unit $i$

Recovery of conventional Dirac theory with automatic anti-symmetrization

Abstract

It is shown that a subgroup of $SL(2,{\mathbb H})$, denoted $Spin(2,{\mathbb H})$ in this paper, which is defined by two conditions in addition to unit quaternionic determinant, is locally isomorphic to the restricted Lorentz group, $L_+^\uparrow$. On the basis of the Dirac theory using the spinor group $Spin(2,{\mathbb H})$, 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-G\"ursey 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, ${\mathbb H}^c$, in which a space-time point is represented by a Hermitian quaternion, and that the isomorphism $SL(1,{\mathbb H}^c)/Z_2\cong L_+^\uparrow$ is a direct consequence of the fact $Spin(2,{\mathbb H})/Z_2\cong L_+^\uparrow$. Using $SL(1,{\mathbb H}^c)\cong SL(2,{\mathbb C})$, we make explicit the Weyl spinor indices of the spinor-quaternion, which is the Dirac spinor defined over ${\mathbb H}^c$.