  In gravitation theory, the realistic fermion matter is described by spinor
bundles associated with the cotangent bundle of a world manifold $X$. In this
case, the Dirac operator can be introduced. There is the 1:1 correspondence
between these spinor bundles and the tetrad gravitational fields represented by
sections of the quotient $\Si$ of the linear frame bundle over $X$ by the
Lorentz group. The key point lies in the fact that different tetrad fields
imply nonequivalent representations of cotangent vectors to $X$ by the Dirac's
matrices. It follows that a fermion field must be regarded only in a pair with
a certain tetrad field. These pairs can be represented by sections of the
composite spinor bundle $S\to\Si\to X$ where values of tetrad fields play the
role of parameter coordinates, besides the familiar world coordinates.
