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XIII. ECKS Theory
To obtain the source term ${{\bs{\tau }}_a}$ at the right-hand side
of (13.1), the Dirac Lagrangian density (13.20) 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 (E.3b) and the definition of the coderivative
(13.19) 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 (13.23) with components of the on-shell
(canonical) Dirac energy-momentum tensor ${{\bs{\tau }}_a}: = {\tau
^b}_a{{\bs{\eta }}_b}$ given by:
\[{\tau^b}_a =
\frac{1}{2}i\left( { {\bar \psi {\gamma ^b}{{\mathcal{D}}_a}\psi -
{\bar {\mathcal{D}}}_a}\bar \psi {\gamma ^b}\psi } \right) \]
This expression is obviously not symmetric in its indices. 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:
Conservation Laws.