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140413 - The self-coupled Einstein-Cartan-Dirac equations in terms of Dirac bilinears.pdf (317.13 kB)
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The self-coupled Einstein-Cartan-Dirac equations in terms of Dirac bilinears

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journal contribution
posted on 2023-05-20, 16:57 authored by Shaun Inglis, Peter JarvisPeter Jarvis
In this article we present the algebraic rearrangement, or matrix inversion of the Dirac equation in a curved Riemann–Cartan spacetime with torsion; the presence of non-vanishing torsion is implied by the intrinsic spin-1/2 of the Dirac field. We then demonstrate how the inversion leads to a reformulation of the fully non-linear and self-interactive Einstein–Cartan–Dirac field equations in terms of Dirac bilinears. It has been known for some decades that the Dirac equation for charged fermions interacting with an electromagnetic field can be algebraically inverted, so as to obtain an explicit rational expression of the four-vector potential of the gauge field in terms of the spinors. Substitution of this expression into Maxwell's equations yields the bilinear form of the self-interactive Maxwell–Dirac equations. In the present (purely gravitational) case, the inversion process yields two rational four-vector expressions in terms of Dirac bilinears, which act as gravitational analogues of the electromagnetic vector potential. These 'potentials' also appear as irreducible summand components of the connection, along with a traceless residual term of mixed symmetry. When taking the torsion field equation into account, the residual term can be written as a function of the object of anholonomity. Using the local tetrad frame associated with observers co-moving with the Dirac matter, a generic vierbein frame can described in terms of four Dirac bilinear vector fields, normalized by a scalar and pseudoscalar field. A corollary of this is that in regions where the Dirac field is non-vanishing, the self-coupled Einstein–Cartan–Dirac equations can in principle be expressed in terms of Dirac bilinears only.


Publication title

Journal of Physics A: Mathematical and Theoretical





Article number









School of Natural Sciences


Institute of Physics Publishing Ltd.

Place of publication

United Kingdom

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© 2019 IOP Publishing Ltd. © 2018 IOP Publishing Ltd. This is the Accepted Manuscript version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at:

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Expanding knowledge in the mathematical sciences; Expanding knowledge in the physical sciences