= Einstein-Cartan theory
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{title2=$(e^a,\omega^{ab})$}
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An independent <vierbein> and metric-compatible <spin connection> allow <spin> density to source a <torsion tensor>. The connection equation from the <Einstein-Hilbert action> is algebraic in torsion. In the absence of connection-dependent matter it fixes the <Levi-Civita connection>. For a minimally coupled <Dirac field>, eliminating the connection generates a local four-fermion interaction. Thus its equivalent torsion-free second-order <action> includes this contact term; simply dropping it defines a different matter theory.
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