Hi there, i'm having a problem calculating the energy momentum tensor for the dirac spinor [tex]\psi (x) =\left(\begin{align}\psi_{L1}\\ \psi_{L2}\\\psi_{R1}\\ \psi_{R2}\end{align}\right)[/tex](free theory).(adsbygoogle = window.adsbygoogle || []).push({});

So, with the dirac lagrangian [tex]\mathcal{L}=i\bar{\psi}\gamma^\mu\partial_\mu\psi-m\bar{\psi}\psi[/tex]in hand i should be able to figure out the energy momentum tensor by using the formula[tex]T^\mu{}_\nu=\frac{\delta\mathcal{L}}{\delta \partial u_A}\partial_\nu u_A-\delta^\mu_\nu\mathcal{L}[/tex]

and since we're assuming the equations of motion to be true we can forget of the latter term in the above equation, and focus on the former.

Here come the probelms.

First of all, i'm not sure on which values does the [itex]A[/itex] index in the [itex]T^\mu{}_\nu[/itex] forumula run: are they 1L, 2L, 1R, 2R?

If so, how can you tell me explicitly how to do the functional derivation [itex]\frac{\delta\mathcal{L}}{\delta\partial u_A}[/itex]?

By writing explicitly the lagrangian (forgetting about the mass term) i get to

[tex]\mathcal{L}=i(\psi^*_{L1},\psi^*_{L2},\psi^*_{1R}, \psi^*_{2R})\gamma^0\partial_0\left(\begin{align} \psi_{L1}\\ \psi_{L2}\\ \psi_{R1}\\ \psi_{R2}\end{align}\right) +i(\psi^*_{L1},\psi^*_{L2},\psi^*_{1R},\psi^*_{2R})\gamma^1\partial_1\left(\begin{align}\psi_{L1}\\ \psi_{L2}\\\psi_{R1}\\ \psi_{R2}\end{align}\right)+i(\psi^*_{L1},\psi^*_{L2}, \psi^*_{1R},\psi^*_{2R})\gamma^2\partial_2\left (\begin{align}\psi_{L1}\\\psi_{L2}\\\psi_{R1}\\ \psi_{R2}\end{align}\right)+ i(\psi^*_{L1},\psi^*_{L2},\psi^*_{1R},\psi^*_{2R})\gamma^3\partial_3\left( \begin{align}\psi_{L1}\\\psi_{L2}\\\psi_{R1}\\ \psi_{R2}\end{align}\right)[/tex]and I stop here becaouse those gamma matrices makes the calculation ridiculously complicated, which bring me nowhere.

Can you folks help me??

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# Energy-momentum tensor for the Dirac spinor

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