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Tetrads in General Relativity

XII. Einstein-Cartan Theory

ECD Gravity

   In the ECKS-theory, torsion is sourced by the spin density of matter; see (12.14). In this context, a much studied model is the free Dirac fermionic field of spin $½$ and mass $m$ described by a Dirac spinor $\psi$ minimally coupled to torsion. The matter Lagrangian density of this Einstein-Cartan-Dirac (ECD) theory is given by:

12.18

\[{{\mathcal{L}}_{\text{D}}} : = \frac{1}{2}\left[ {\bar \psi (i {\gamma ^a}{{\mathcal{D}}_a}\psi - m\psi ) - (i {{\bar {\mathcal{D}}}_a}\bar \psi {\gamma ^a} + m\bar \psi )\psi } \right]\]

where $\bar \psi : = {\psi ^\dagger }{\gamma ^0}$ is the adjoint spinor, the dagger indicating the Hermitian conjugate. The Dirac matrices $\left\{ {{\gamma ^a};a = 0,1,2,3} \right\}$ have their special relativistic values and are considered to be constant; see Dirac Algebra.

   The Lagrangian is Hermitian and of first order, i.e. only a first derivative of the fields enters, which in this case is the Fock-Ivanenko coderivative:

12.19

\[{{\mathcal{D}}_a}\psi : = {\partial _a}\psi + {\Gamma _a}\psi {\quad}{\bar {\mathcal{D}_a}}\bar {\psi} : = {({\mathcal{D}_a}\psi )^\dagger } {\gamma ^0}={\partial _a}\bar {\psi} - \bar {\psi}\, {\Gamma _a}\]

12.20

\[{\mathcal{D}_a} := {e^\mu }_a\left( {{\partial _\mu } + \frac{1}{2}{\Sigma _{\mu bc}}{\sigma ^{bc}}} \right)\]

and the generators of the spinor algebra:

12.21

\[{\sigma ^{bc}} : = \frac{1}{2}{\gamma ^b} \wedge {\gamma ^c}{\text{ = }}\frac{1}{4}{\text{[}} {\gamma ^b},{\gamma ^c}{\text{]}} \quad({\sigma ^{bc}}{)^\dagger } = - {\gamma ^0}{\sigma ^{bc}}{\gamma ^0}\]

   The corresponding Hermitian matter action in ECD gravity is

12.22

\[{S_{\text{M}}}[e, \Sigma,\psi] = \int {{d^4}} x\sqrt {\left| g \right|} {\mathcal{L}_{\text{D}}}(x)\]

By construction it is a functional of the Dirac spinors and the vierbein gravitational potentials  $\left\{ {{e_\mu }^a,{\Sigma _\mu }^{ab}} \right\}$. Varying the action with respect to $\bar \psi $, or $\psi $, and setting the result equal to zero, one obtains the Dirac equation (6.19) or its adjoint.