Set Of Dirac Matrices Research Articles

A Gauge and Lorentz covariant approximation for the quark propagator in an arbitrary gluon field

We decompose the quark propagator in the presence of an arbitrary gluon field with respect to a set of Dirac matrices. The four-dimensional integrals which arise in first order perturbation theory are rewritten as line-integrals along certain field lines, together with a weighted integration over the various field lines. It is then easy to transform the propagator into a form involving path ordered exponentials. The resulting expression is non-perturbative and has the correct behavior under Lorentz transformations, gauge transformations and charge conjugation. Furthermore it coincides with the exact propagator in first order of the coupling g. No expansion with respect to the inverse quark mass is involved, the expression can even be used for vanishing mass. For large mass the field lines concentrate near the straight line connection and simple results can be obtained immediately.

On the calculus of four-spinors

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 x A x B 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 [ A û B ] 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 .

Spinor fields in general relativity

A unitary field theory is developed with two fundamental field variables, a spinor and a bispinor-vector (set of Dirac matrices). The theory is required to be invariant under general co-ordinate transformations and similarity (spinor) transformations, and it is shown, under a minimum of restrictive hypotheses, that Einstein’s concept of teleparallelism is implied. Dirac’s equation is formulated in this unitary theory, and the electromagnetic field potentials are identified. An action density is proposed for the field which appears to exhibit gravitational and electromagnetic field properties consistent with experiment. There are indications that the charge of the elementary particles is quantized. Most of the paper is devoted to a specialization in which the ‘natural’ derivative of the Dirac matrices is made to vanish. The generalization of this theory is briefly considered at the end, and is found to introduce other fields, including a pseudo-scalar field which may account for nuclear forces.