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

Pitts, J. Brian (2013) Time and Fermions: General Covariance vs. Ockham's Razor for Spinors. Proceedings of the 4th Conference on Time and Matter - TAM 2013. pp. 185-198. ISSN 978-961-6311-79-3

 Preview
PDF (Contribution to conference proceedings)
TAM2013TimeandFermionsConfProc.pdf - Published Version

## Abstract

It is a commonplace in the foundations of physics, attributed to Kretschmann, that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics and mathematics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. 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, enhancing the unity of physics and helping to spawn particle physicists' concept of 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 typical tetrad formalism is thus de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. As developed nonperturbatively by Bilyalov, OP spinors admit any coordinates at a point, but time' must be listed first; time' is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag(-1,1,1,1)$, the product of which must have no negative eigenvalues. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved.

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. Important results on Lie and covariant differentiation are recalled and applied. The rather mild consequences of the coordinate order restriction are explored in two examples: the question of the conventionality of simultaneity in Special Relativity, and the Schwarzschild solution in General Relativity.

 Export/Citation: EndNote | BibTeX | Dublin Core | ASCII/Text Citation (Chicago) | HTML Citation | OpenURL
 Social Networking:

Item Type: Published Article or Volume
Creators:
CreatorsEmailORCID
Pitts, J. Brianjbp25@cam.ac.uk
Additional Information: The proceedings editor put his paper at the physics arxiv here: http://arxiv.org/abs/1312.4407 I take that to indicate permission for me to act similarly.
Keywords: spinor, fermion, general covariance, geometric object, nonlinear group realization, conventionality of simultaneity, Schwarzschild solution
Subjects: Specific Sciences > Physics > Fields and Particles
Specific Sciences > Mathematics
Specific Sciences > Physics > Quantum Gravity
Specific Sciences > Physics > Quantum Field Theory
Specific Sciences > Physics > Relativity Theory
Specific Sciences > Physics > Symmetries/Invariances
Depositing User: Dr. Dr. J. Brian Pitts
Date Deposited: 25 Jul 2014 12:50
Item ID: 10900
Journal or Publication Title: Proceedings of the 4th Conference on Time and Matter - TAM 2013
Publisher: University of Nova Gorica Press
Subjects: Specific Sciences > Physics > Fields and Particles
Specific Sciences > Mathematics
Specific Sciences > Physics > Quantum Gravity
Specific Sciences > Physics > Quantum Field Theory
Specific Sciences > Physics > Relativity Theory
Specific Sciences > Physics > Symmetries/Invariances
Date: 2013
Page Range: pp. 185-198
ISSN: 978-961-6311-79-3
URI: http://philsci-archive.pitt.edu/id/eprint/10900