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XII. Einstein-Cartan Theory
In algebraic manipulations, gamma-matrices may be treated like
vectors in GA, e.g.
\[\begin{gathered}
{{\mathbf{\gamma }}^c}({{\mathbf{\gamma }}^a} \wedge
{{\mathbf{\gamma }}^b}) = {{\mathbf{\gamma }}^c} \cdot
({{\mathbf{\gamma }}^a} \wedge {{\mathbf{\gamma }}^b}) +
{{\mathbf{\gamma }}^c} \wedge {{\mathbf{\gamma }}^a} \wedge
{{\mathbf{\gamma }}^b}\quad \hfill \\ \qquad \qquad \; = {\eta
^{ca}}{{\mathbf{\gamma }}^b} - {\eta ^{cb}}{{\mathbf{\gamma }}^a} +
{{\mathbf{\gamma }}^a} \wedge {{\mathbf{\gamma }}^b} \wedge
{{\mathbf{\gamma }}^c} \hfill \\ \end{gathered} \]
With definition (D.2) of the spin matrices, it is then easily
shown that the spin-density tensor of the Dirac field (12.27) can be
written in the form:
\[{\sigma ^{abc}} =
\frac{1}{2}i\bar \psi \left\{ {{\sigma ^{ab}},{\gamma ^c}}
\right\}\psi = \frac{1}{2}i\bar \psi \left( {{\gamma ^a} \wedge
{\gamma ^b} \wedge {\gamma ^c}} \right)\psi \]
which makes it evident that the spin-density tensor is totally
antisymmetric in its indices.
A relation with the antisymmetric part of the
energy-momentum tensor (12.31) is established through the
divergence
\[{{\mathcal{D}}_c}{\sigma
^{cab}} := \frac{1}{2}i\left[ {\left( {{{\bar {\mathcal{D}}}_c}\bar
\psi } \right){{\mathbf{\gamma }}^a} \wedge {{\mathbf{\gamma }}^b}
\wedge {{\mathbf{\gamma }}^c}\psi + \bar \psi {{\mathbf{\gamma }}^a}
\wedge {{\mathbf{\gamma }}^b} \wedge {{\mathbf{\gamma }}^c}\left(
{{{\mathcal{D}}_c}\psi } \right)} \right]\]
A straightforward calculation with the help of the identity
(12.32) and use of the Dirac equations (12.30) gives the on-shell
result:
\[{\mathcal{D}_c}{\sigma
^{cab}} = {\mathcal{D}_c}{\sigma ^{abc}} = {\tau ^{ab}} - {\tau
^{ba}} = 2{\tau ^{[ab]}}\]
So the divergence of the spin current (12.33) is directly related to
the anti-symmetric part of the canonical energy momentum
tensor (12.31). This relation is the basis for the
Belinfante-Rosenfeld (BR) construction of the symmetric
energy-momentum tensor
\[\tau _{{\text{BR}}}^{ab} : =
{\tau ^{ab}} - \frac{1}{2}{\mathcal{D}_c}{\sigma ^{cab}} = {\tau
^{(ab)}}\]
In the case of the Dirac field the BR construction amounts to
simply symmetrizing the canonical energy-momentum tensor ${\tau
^{ab}}$. It works out this way because in this particular case the
spin-density tensor is totally antisymmetric. [
Wikipedia: Belinfante–Rosenfeld stress–energy tensor]