Pitts, J. Brian (2011) The Nontriviality of Trivial General Covariance: How Electrons Restrict 'Time' Coordinates, Spinors (Almost) Fit into Tensor Calculus, and 7/16 of a Tetrad Is Surplus Structure. [Preprint]
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Abstract
It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors _as such_ cannot be represented in coordinates in a curved spacetime. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov 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, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually selfsufficient and more economical. The typical tetrad formalism is deOckhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The OgievetskyPolubarinov formalism, by contrast, is (nearly) Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "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 in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and _values_ of the fields involved.
Apart from coordinate order and the usual spinorial twovaluedness, (densitized) OgievetskyPolubarinov spinors form, with the (conformal part of the) metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the AndersonFriedman analysis of substantive general covariance. They also permit the gaugeinvariant localization of the infinitecomponent gravitational energy in General Relativity. Densityweighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O(1,3)$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use densityweighted OgievetskyPolubarinov spinors coupled to the 9component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors (related to fermions, with halfintegral spin) and tensors and the like (related to bosons, with integral spin) is achieved, such as regarding conservation laws.
Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited.
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Item Type:  Preprint  

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Keywords:  electron, spinor, fermion, general covariance, geometric object, nonlinear group representation, conventionality of simultaneity, Schwarzschild solution  
Subjects:  Specific Sciences > Physics > Fields and Particles Specific Sciences > Mathematics Specific Sciences > Physics > Relativity Theory Specific Sciences > Physics > Symmetries/Invariances 

Depositing User:  Dr. Dr. J. Brian Pitts  
Date Deposited:  10 Nov 2011 13:58  
Last Modified:  10 Nov 2011 13:58  
Item ID:  8895  
URI:  http://philsciarchive.pitt.edu/id/eprint/8895 
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The Nontriviality of Trivial General Covariance: How Electrons Restrict 'Time' Coordinates, Spinors (Almost) Fit into Tensor Calculus, and 7/16 of a Tetrad Is Surplus Structure. (deposited 05 Nov 2011 13:02)
 The Nontriviality of Trivial General Covariance: How Electrons Restrict 'Time' Coordinates, Spinors (Almost) Fit into Tensor Calculus, and 7/16 of a Tetrad Is Surplus Structure. (deposited 10 Nov 2011 13:58) [Currently Displayed]
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