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Tetrads in General Relativity

XII. Einstein-Cartan Theory

Tetradic Dirac Action

   By the same procedure by which the tetradic Einstein-Cartan form (12.5) is obtained from (11.6), one may rewrite the action (12.22) into a functional of the Cartan potentials $\{ {{\mathbf{e}}^a},{{\bs{\omega }}^{ab}}\} $. First, the Lagrangian (12.18) is redefined like in (12.3):

12.23

\[{S_{\text{D}}}[{\mathbf{e}},{\mathbf{\omega }},\psi ] = \int d {x_4}{\operatorname{I} ^4}{\mathcal{L}_{\text{D}}}(x) : = \int d {x_4}{L_{\text{D}}}(\psi ,\mathcal{D}\psi ,x)\]

   Next, one uses the identities (A.2) and (B.3) for $p=1$ to derive:

12.24

\[{{\text{I}}^4}{\gamma ^a}{\mathcal{D}_a} = {{\text{I}}^4}{\gamma ^a}\delta _a^b{\mathcal{D}_b} = {\bs{\gamma }} \wedge \mathcal{D}{\quad}{\bs{\gamma }} : = {\gamma ^a}{{\bs{\eta }}_a}\]

where ${{\bs{\eta }}_a}$ is the Trautman form (C.3b); the (coordinate free) gradient coderivative is defined as:

12.25

\[\mathcal{D} : = {{\mathbf{e}}^a}{\mathcal{D}_a} = \nabla + \frac{1}{2}{{\bs{\omega }}_{bc}}{\sigma ^{bc}} = \nabla + \frac{1}{2}{{\bs{\omega }}^{bc}}{\sigma _{bc}}\]

   The result of these manipulations is the tetradic form of the Dirac Lagrangian in Riemann-Cartan space:

12.26

\[{L_{\text{D}}}(\psi ,{\mathcal{D}}\psi ,x) = \frac{1}{2}i\left( {\bar \psi {\bs{\gamma }} \wedge {\mathcal{D}} \psi - \bar {\mathcal{D}}\bar \psi \wedge {\bs{\gamma }}\psi } \right) - {{\eta }}m\bar \psi \psi \]

which is minimally coupled to the Riemann-Cartan spacetime via the gauge potentials ${{\mathbf{e}}^a}$ (contained in $\bs{\gamma} $ and ${\eta} ={\operatorname{I} ^4}$) and ${{\bs{\omega }}^{ab}}$ (contained in $\mathcal{D}$). The potentials enter  the Lagrangian linearly.

   Because of this linearity, it is now an easy exercise to calculate from (12.26) the source term of (12.14) . This yields the torsion equation of the ECD-theory (12.16), with the components of the spin-density tensor at the right-hand side given by

12.27

\[\sigma _{ab}^c =\frac{1}{2} i\bar \psi ({\gamma ^c} \wedge {\sigma _{ab}} + {\sigma _{ab}} \wedge {\gamma ^c})\psi = \frac{1}{2}i\bar \psi \left\{ {{\gamma ^c}, {\sigma _{ab}}} \right\}\psi \]

The Dirac spin-density tensor is totally antisymmetric in its indices; see (12.33).