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XIII. ECKS Theory
By the same procedure by which the tetradic Einstein-Cartan form
(12.5) is obtained from (11.6), one may rewrite the action (13.16)
into a functional of the Cartan potentials $\{
{{\mathbf{e}}^a},{{\bs{\omega }}^{ab}}\} $. First, the Lagrangian
(13.12) is redefined like in (12.3):
\[{S_{\text{D}}}[{\mathbf{e}},{\mathbf{\omega }},\psi ] = \int d
{x_4}{\operatorname{I} ^4}{\mathcal{L}_{\text{D}}}(x) : = \int d
{x_4}{L_{\text{D}}}(\psi ,\mathcal{D}\psi ,x)\]
Next, one uses the identities (A.2) and (B.3) for $p=1$ to derive:
\[{{\text{I}}^4}{\gamma
^a}{\mathcal{D}_a} = {{\text{I}}^4}{\gamma ^a}\delta
_a^b{\mathcal{D}_b} = {\bs{\gamma }} \wedge
\mathcal{D}{\quad}{\bs{\gamma }} : = {\gamma ^a}{{\bs{\eta }}_a}\]
where ${{\bs{\eta }}_a}$ is the 3-form (C.3b); the (coordinate
free) gradient coderivative is defined as:
\[\mathcal{D} : =
{{\mathbf{e}}^a}{\mathcal{D}_a} = \nabla + \frac{1}{2}{{\bs{\omega
}}_{bc}}{\sigma ^{bc}} = \nabla + \frac{1}{2}{{\bs{\omega
}}^{bc}}{\sigma _{bc}}\]
The result is the tetradic form of the Dirac Lagrangian in
Riemann-Cartan space:
\[{L_{\text{D}}}(\psi
,{\mathcal{D}}\psi ,x) = \frac{1}{2}i\left( {\bar \psi {\bs{\gamma
}} \wedge {\mathcal{D}} \psi - \bar {\mathcal{D}}\bar \psi \wedge
{\bs{\gamma }}\psi } \right) - {{\eta }}m\bar \psi \psi \]
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 (13.20) the source term of (13.2) . This yields the torsion
equation of the ECD-theory (13.4), with the tensor components of the
spin density at the right-hand side given by
\[\sigma _{ab}^c =\frac{1}{2}
i\bar \psi ({\gamma ^c} \wedge {\sigma _{ab}} + {\sigma _{ab}}
\wedge {\gamma ^c})\psi = \frac{1}{2}i\bar \psi \left\{ {{\gamma
^c}, {\sigma _{ab}}} \right\}\psi \]
The Dirac spin tensor is totally antisymmetric in
its indices; see (13.27).