Time and Fermions: General Covariance vs. Ockham's Razor for Spinors

TL;DR

This paper revisits the representation of spinors in curved spacetime, demonstrating that they can be formulated in coordinates without tetrads, challenging traditional views and exploring implications for covariance and simultaneity.

Contribution

It introduces a coordinate-based formulation of spinors in curved spacetime, showing they are self-sufficient and more economical than tetrad formalism, with implications for covariance.

Findings

OP spinors resemble tetrads but are conceptually self-sufficient

Admissible coordinates depend on field values, not just form

Coordinate order affects the interpretation of simultaneity

Abstract

It is a commonplace attributed to Kretschmann that any local physical theory can be represented in arbitrary coordinates using tensor calculus. But the literature also claims that spinors _as such_ cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent, so what is general covariance for fermions? In fact both commonplaces are wrong. Ogievetsky and Polubarinov (OP) constructed spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn nonlinear group representations. Roughly and locally, OP spinors resemble the orthonormal basis or tetrad formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The tetrad formalism is thus de-Ockhamized, with six extra components and six compensating gauge symmetries. As developed nonperturbatively by Bilyalov, OP spinors admit any coordinates at a point, but 'time' must be listed first; 'time' is defined by an eigenvalue problem involving the metric and diag(-1,1,1,1), the product of which must have no negative eigenvalues. Thus even formal general covariance requires reconsideration; admissible coordinates should be sensitive to the types and _values_ of the fields. Apart from coordinate order and the usual spinorial two-valuedness, (densitized) Ogievetsky-Polubarinov spinors form, with the (conformal part of the) metric, a nonlinear geometric object, with Lie and covariant derivatives. The mild consequences of coordinate order are explored for the conventionality of simultaneity in Special Relativity and the Schwarzschild solution in General Relativity.