## Abstract

This paper begins with a heuristic derivation of the explicit and covariant law of multiplication of the Dirac algebra, regarded as a matrix algebra. To achieve this the 'space-time’ *R*_{4} of signature —2 is embedded in a six-dimensional flat space* R*_{6} of signature 0, —2, or — 4, as the case may be. The metric tensor need not be diagonal. The 15 elements of the algebra, excluding the unit element, then transform as a bi-vector in *R*_{6}. Just as two-spinors are associated with four-dimensional Lorentz transformations, so four-spinors, i. e. the vectors and tensors of a four-dimensional complex linear vector space *S*_{4}, naturally associate themselves with Lorentz transformations in six dimensions. After some incidental work the group of proper orthogonal transformations in *R*_{6} is considered, together with the group of spin transformations *S* it induces in *S*_{4}. One then arrives at once at the known local isomorphisms *SO*(5, 1) ≈ *SU**(4), *SO*(4, 2) ≈ SU(2, 2), *SO*(3, 3) ≈ SL(4, *R*); and the generic form of *S* in each case is explicitly set down. The second part of the work starts with an approach to the law of multiplication from a deductive point of view. It is based upon the observation that transformations of the form *SMŚ* of any skew-symmetric 4 x 4 matrix by an arbitrary unimodular matrix *S* leave its skew-symmetry and its determinant ∆ invariant; at the same time ∆ is a perfect square, so that the six linearly independent elements *M*_{uv} of *M* can be brought into linear one-one correspondence with six real variables *x*^{A} in such a way that ∆ is the square of the metric ground form *gAB ^{xAxB}* of an

*R*

_{є}. This correspondence between

*x*

^{A}and

*M*

_{uv}leads to a basis of six skew-symmetric vector spinors

*u*

_{Auv}and their duals

*û*

_{A}

^{uv}. The skew-symmetrized matrix product

*e*

_{AB}=

*u*

_{[AûB]}is deductively shown to obey the law of multiplication of the Dirac algebra. The spinors

*H, C, T*and a vector

*q*

_{A}associated with Hermitian conjugation, complex conjugation and transposition of the

*e*

_{AB}are examined from a (spin-) tensorial point of view.

*H*

^{-1}, now written as

*g*

_{u̇v}is introduced as a metric in

*S*4. Then

*T*

^{uv}and

*C*

^{uv}are identical, and further

*C*

^{uv}] is self-dual. The theory is now made fully covariant (in the sense that the restriction det

*S*= 1 is dropped) by attaching suitable spin weights and anti-weights to the various spinors which occur. A new vector-spinor

*v*

_{A}is defined which is a kind of six-dimensional analogue of the set of Dirac matrices; and with its help a formal six-dimensional generalization of Dirac’s equation is written down. The last part of this paper deals with the generalization of the appropriate parts of the preceding theory to the situation in which

*g*

_{AB}becomes the metric tensor of a Riemann space

*V*

_{6}.

## Footnotes

This text was harvested from a scanned image of the original document using optical character recognition (OCR) software. As such, it may contain errors. Please contact the Royal Society if you find an error you would like to see corrected. Mathematical notations produced through Infty OCR.

- Received July 13, 1967.

- Scanned images copyright © 2017, Royal Society

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