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Hestenes' Tetrad and Spin Connections

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Abstract

Defining a spin connection is necessary for formulating Dirac's bispinor equation in a curved space-time. Hestenes has shown that a bispinor field is equivalent to an orthonormal tetrad of vector fields together with a complex scalar field. In this paper, we show that using Hestenes' tetrad for the spin connection in a Riemannian space-time leads to a Yang-Mills formulation of the Dirac Lagrangian in which the bispinor field Ψ is mapped to a set of SL(2,R)× U(1) gauge potentials F α K and a complex scalar field ρ. This result was previously proved for a Minkowski space-time using Fierz identities. As an application we derive several different non-Riemannian spin connections found in the literature directly from an arbitrary linear connection acting on the tensor fields (F α K, ρ). We also derive spin connections for which Dirac's bispinor equation is form invariant. Previous work has not considered form invariance of the Dirac equation as a criterion for defining a general spin connection.

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Authors and Affiliations

  1. Lockheed Martin Corporation, Naval Electronic and Surveillance Systems (137-205), 199 Borton Landing Road, Moorestown, New Jersey

    Frank Reifler & Randall Morris

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  1. Frank Reifler
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  2. Randall Morris
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Correspondence to Frank Reifler.

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Reifler, F., Morris, R. Hestenes' Tetrad and Spin Connections. Int J Theor Phys 44, 1307–1324 (2005). https://doi.org/10.1007/s10773-005-4688-8

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  • DOI: https://doi.org/10.1007/s10773-005-4688-8

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