Classical theory of the interaction between a spinor field and the gravitational field: First-order field equations
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The classical theory of the interaction of a neutral spin-1/2 Dirac field and the gravitational field is studied. For the purely gravitational part of the Lagrangian, written in terms of a vierbein and the local connection coefficient ω aμb , (regarded as independent field variables), the usual first-order form is adopted. For the Dirac part, however, a different choice is made, in which the covariant derivative ofψ is built with the aid of the vierbein instead of with ω aμb . This still yields a first-order formalism, but one in which ω aμb is related to the vierbein in the same way as it would be in the absence ofψ. This ensures that the global connection Г μν α remains symmetric inμ andv in the presence ofψ. The way in which the vierbein field equation leads to a familiar Einstein equation with a symmetric and conserved stress tensor on its right side is also analyzed.
KeywordsField Equation Classical Theory Gravitational Field Einstein Equation Spinor Field
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