{title2=$\gamma^\mu$}
{wiki=Gamma_matrices}

Gamma matrices give a matrix representation of the <Clifford algebra> of spacetime.

Gamma matrix

After using the <Clifford algebra>, integrations by parts and the <Dirac equation>, every allowed Lorentz-covariant symmetric rank-two expression with one derivative reduces, modulo terms vanishing <on shell> and identically conserved improvements, to an overall multiple of
$$
\bar\psi\gamma^{(\mu}\overleftrightarrow{\partial}^{\nu)}\psi.
$$
Thus the general nontrivial tensor in the stated class is
$$
\widehat T^{\mu\nu}
=C\left[
\bar\psi\gamma^\mu\partial^\nu\psi
+\bar\psi\gamma^\nu\partial^\mu\psi
-(\partial^\nu\bar\psi)\gamma^\mu\psi
-(\partial^\mu\bar\psi)\gamma^\nu\psi
\right].
$$
It is manifestly symmetric. Differentiating it, commuting partial derivatives and using
$$
i\gamma^\rho\partial_\rho\psi=m\psi,
\qquad
i(\partial_\rho\bar\psi)\gamma^\rho=-m\bar\psi,
$$
makes the terms cancel pairwise, proving $\partial_\mu\widehat T^{\mu\nu}=0$ on shell. The normalization $C$ remains free at this stage.

Solved by gpt-5.6-sol high.

Solution

Paper 301 / 1 / b / Solution

Choose $C=i/4$. Expanding
$$
S^{\lambda\mu\nu}=\frac12\bar\psi[\gamma^\lambda,\gamma^\mu]\gamma^\nu\psi
$$
with the <Clifford algebra>, applying the product rule and then using the <Dirac equation> and its adjoint gives
$$
\widehat T^{\mu\nu}
=T^{\mu\nu}+\frac i4\left[
\partial_\lambda S^{\lambda\mu\nu}
-\partial^\nu(\bar\psi\gamma^\mu\psi)
\right].
$$
This is the <Belinfante-Rosenfeld stress-energy tensor> written as an improvement of the canonical tensor.

The translation charges are
$$
P^\nu=\int d^3x\,T^{0\nu},
\qquad
\widehat P^\nu=\int d^3x\,\widehat T^{0\nu}.
$$
For spatial $\nu$, their difference is the integral of $\partial_iS^{i0\nu}-\partial^\nu(\bar\psi\gamma^0\psi)$. For $\nu=0$, use conservation of the <Dirac current> to replace $\partial^0(\bar\psi\gamma^0\psi)$ by a spatial divergence; also $S^{00\nu}=0$. Hence $\widehat P^\nu-P^\nu$ is always a spatial boundary integral. Under the usual decay boundary condition it vanishes, so both currents generate the same four-momentum.

Solved by gpt-5.6-sol high.

Paper 301 / 1 / c / Solution

{c}
{wiki}

For a vector space with quadratic form $Q$, its Clifford algebra is generated by vectors subject to $v^2=Q(v)1$. In a basis this is equivalently $\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2\eta^{\mu\nu}1$.


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