, cf. [65, 66]). In the following we shall deal only with relativistic unified field theories.
If we define , with , then transforms like a tangent vector under point transformations of the , and as a covariant vector under homogeneous transformations of the . The may be used to relate covariant vectors and by Thus, the metric tensor in the space of homogeneous coordinates and the metric tensor of are related by with The inverse relationship is given by with The covariant derivative for tensor fields in the space of homogeneous coordinates is defined as before (cf. Section 2.1.2):
In this section, we briefly present Cartan’s one-form formalism in order to make understandable part of the literature. Cartan introduces one-forms () by The reciprocal basis in tangent space is given by . Thus, . The metric is then given by . The covariant derivative of a tangent vector with bein-components is defined via Cartan’s first structure equations,34. The covariant derivative of a tangent vector with bein-components then is
By further external derivation35 on we arrive at the second structure relation of Cartan,
A relative tensor of type and of weight at a point on the manifold transforms liketensor density. We can form a tensor from by introducing , where is a Lorentz-metric. Note that The dual to a 2-form (skew-symmetric tensor) then is defined by
In connection with conformal transformations , the concept of the gauge-weight of a tensor is introduced. A tensor is said to be of gauge weight if it transforms by Equation (56) as
Objects that transform as in Equation (67) but with respect to a subgroup, e.g., the linear group, affine group , orthonormal group , or the Lorentz group , are tensors in a restricted sense; sometimes they are named affine or Cartesian tensors. All the subgroups mentioned are Lie-groups, i.e., continuous groups with a finite number of parameters. In general relativity, both the “group” of general coordinate transformations and the Lorentz group are present. The concept of tensors used in Special Relativity is restricted to a representation of the Lorentz group; however, as soon as the theory is to be given a coordinate-independent (“generally covariant”) form, then the full tensor concept comes into play.
Then, by a transformation from ,38. Moreover, which, according to Equation (70), expresses the invariance of the space-time distance to the origin:
Now, contravariant 2-spinors () are the elements of a complex linear space, spinor space, on which the matrices are acting39. The spinor is called elementary if it transforms under a Lorentz-transformation ascontravariant dotted spinors are those transforming with the complex-conjugate matrix : inverse matrices,
Higher-order spinors with dotted and undotted indices transform correspondingly. For the raising and lowering of indices now a real, antisymmetric -matrix with components is needed, such that
Next to a spinor, bispinors of the form , etc. are the simplest quantities (spinors of 2nd order). A vector can be represented by a bispinor ,
In order to write down spinorial field equations, we need a spinorial derivative,Weyl equation:
Dirac- or 4-spinors with 4 components , , may be constructed from 2-spinors as a direct sum of contravariant undotted and covariant dotted spinors and : For , we enter and ; for , we enter and In connection with Dirac spinors, instead of the Pauli-matrices the Dirac -matrices (-matrices) appear; they satisfy[53, 54]:
The generally-covariant formulation of spinor equations necessitates the use of -beins , whose internal “rotation” group, operating on the “hatted” indices, is the Lorentz group. The group of coordinate transformations acts on the Latin indices. In Cartan’s one-form formalism (cf. Section 2.1.4), the covariant derivative of a 4-spinor is defined by
Equation (89) is a special case of the general formula for the covariant derivative of a tensorial form , i.e., a vector in some vector space , whose components are differential forms,40.
A Riemannian space is called (locally) stationary if it admits a timelike Killing vector;