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XII. Einstein-Cartan Theory
To obtain the source term ${{\bs{\tau }}_a}$ at the
right-hand side of (12.13), the Dirac Lagrangian density (12.26) is
varied with respect to the potentials $\{ {{\mathbf{e}}^a}\} $. The
definitions (C.3) imply: $\delta {\mathbf{\eta }} = -\delta
{{\mathbf{e}}^a} \wedge {{\bs{\eta }}_a}$ and $\delta {{\bs{\eta
}}_b} = -\delta {{\mathbf{e}}^a} \wedge {{\bs{\eta }}_{ab}}$, which
then directly leads to the source term:
\[{{\bs{\tau }}_a}
= \frac{1}{2}i\left( {\bar {\mathcal{D}}\bar \psi
\wedge {\gamma ^b}{{\bs{\eta }}_{ab}}\psi
- \bar \psi {\gamma ^b}{{\bs{\eta }}_{ab}}
\wedge {\mathcal{D}}\psi } \right) +
{{\bs{\eta }}_a}m\bar \psi \psi \]
With the identity ${{\bs{\eta }}_{ab}} \wedge {{\mathbf{e}}^c} =
\delta _b^c{{\bs{\eta }}_a} - \delta _a^c{{\bs{\eta }}_b}$ and the
definition of the coderivative (12.25) this may be reworked into the
sum of two contributions
\[\begin{gathered}{{\bs{\tau
}}_a} = \frac{1}{2}\left[ {\left( {i{{\bar {\mathcal{D}}}_b}\bar
\psi {\gamma ^b}\psi + m\bar \psi \psi } \right) - \left( {i\bar
\psi {\gamma ^b}{{\mathcal{D}}_b}\psi - m\bar \psi \psi } \right)}
\right]{{\bs{\eta }}_a} \hfill \\ {+}\frac{1}{2}i\left( {\bar \psi
{\gamma ^b}{{\mathcal{D}}_a}\psi - {{\bar {\mathcal{D}}}_a}\bar \psi
{\gamma ^b}\psi } \right){{\bs{\eta }}_b} \quad \qquad \qquad
\qquad \hfill \\ \end{gathered} \]
The terms in the first line vanish if the spinor fields are
required to satisfy the Dirac equations
\[i{\gamma
^b}{{\mathcal{D}}_b}\psi - m\psi = 0{\quad}i{\bar
{\mathcal{D}}_b}\bar \psi {\gamma ^b} + m\bar \psi = 0\]
This leaves the last term of (12.29) with components of the on shell
(canonical) Dirac energy-momentum tensor ${{\bs{\tau }}_a}: = {\tau
_a}^b{{\bs{\eta }}_b}$ given by:
\[{\tau _a}^b =
\frac{1}{2}i\left( { {\bar \psi {\gamma ^b}{{\mathcal{D}}_a}\psi -
{\bar {\mathcal{D}}}_a}\bar \psi {\gamma ^b}\psi } \right) \]
This expression is obviously not symmetric in its indices.
However, as it turns out, the existence of a
spin current demands the canonical energy-momentum tensor to have an
anti-symmetric part so as to fulfill the requirement of local
angular momentum conservation; see:
Wikipedia: Belinfante–Rosenfeld stress–energy tensor.