  It is a commonplace that any theory can be written in any coordinates via
tensor calculus. But it is claimed that spinors as such cannot be represented
in coordinates in a curved space-time. What general covariance means for
theories with fermions is thus unclear. In fact both commonplaces are wrong.
Though it is not widely known, Ogievetsky and Polubarinov (OP) constructed
spinors in coordinates in 1965, helping to spawn nonlinear group
representations. Locally, these spinors resemble the orthonormal basis or
"tetrad" formalism in the symmetric gauge, but they are conceptually
self-sufficient. The tetrad formalism is de-Ockhamized, with 6 extra fields and
6 compensating gauge symmetries. OP spinors, as developed nonperturbatively by
Bilyalov, admit any coordinates at a point, but "time" must be listed first:
the product of the metric components and the matrix diag(-1,1,1,1) must have no
negative eigenvalues to yield a real symmetric square root function of the
metric. Thus the admissible coordinates depend on the types and values of the
fields. Apart from coordinate order and spinorial two-valuedness, OP spinors
form, with the metric, a nonlinear geometric object, with Lie and covariant
derivatives. Such spinors avoid a spurious absolute object in the
Anderson-Friedman analysis of substantive general covariance. They also permit
the gauge-invariant localization of the infinite-component gravitational energy
in GR. Density-weighted spinors exploit the conformal invariance of the
massless Dirac equation to show that the volume element is absent. Thus instead
of a matrix with 16 components, one can use weighted OP spinors coupled to the
9-component symmetric unimodular square root of the conformal metric density.
The surprising mildness of the restrictions on coordinates for the
Schwarzschild solution is exhibited. (edited)
