I've recently learned of a method which can symmetrise the canonical energy-momentum tensor,

$$T_{\mu\nu} = \eta_{\mu\nu} \mathcal{L} - \sum_{a} \frac{\partial \mathcal{L}}{\partial (\partial^\mu \varphi_a)} \partial_\nu \varphi_a,$$

using the $(-,+,+,+)$ signature, where we add an additional total derivative term,

$$T_{\mu\nu} \rightarrow T_{\mu\nu} + \partial^\lambda \chi_{\lambda\mu\nu},$$

$$\chi_{\lambda\mu\nu} = K_{\lambda\mu\nu} + K_{\mu\nu\lambda} + K_{\nu\mu\lambda},\quad K_{\lambda\mu\nu} = -\frac{i}{2} \sum_{a,b} \frac{\partial \mathcal{L}}{\partial (\partial^\lambda \varphi_a) } (J_{\mu\nu})_{ab} \varphi_b,$$

where the $(J_{\mu\nu})_{ab}$ are the representations of the Lorentz algebra under which the fields $\varphi_a$ transform, given here.

Having confirmed this works for the electromagnetic Lagrangian, and understand that $\chi_{\lambda\mu\nu}$ is zero for scalars (which is fine as the original canonical SEM tensor is symmetric), I then attempted to try this on the Dirac Lagrangian,

$$\mathcal{L} = \bar\psi (i\displaystyle{\not}\partial - m) \psi,$$

But I've either made some mistakes, potentially in the handling of the Grassmann-valued fields, or just got a bit stuck at the end.

First, I found the canonical SEM tensor to be

$$T_{\mu\nu} = i \bar\psi \gamma_\mu \partial_\nu \psi,$$

Which is asymmetric in $\mu$ and $\nu$. Now, the correction term:

$$K_{\lambda\mu\nu} = -\frac{i}{2} \left( \frac{\partial \mathcal{L}}{\partial (\partial^\lambda \psi)} S_{\mu\nu} \psi + \frac{\partial \mathcal{L}}{\partial (\partial^\lambda \bar\psi)} S_{\mu\nu} \bar\psi \right),\quad S_{\mu\nu} = \frac{i}{4} \left[ \gamma_\mu, \gamma_\nu \right].$$

The $\frac{\partial \mathcal{L}}{\partial (\partial^\lambda \bar\psi)}$ factor is zero, so the second term disappears.

$$K_{\lambda\mu\nu} = \frac{1}{8} \frac{\partial \mathcal{L}}{\partial (\partial^\lambda \psi)} \left[ \gamma_\mu, \gamma_\nu \right] \psi,\quad \frac{\partial \mathcal{L}}{\partial (\partial^\lambda \psi)} = -i \bar\psi \gamma_\lambda$$

$$\Rightarrow \chi_{\lambda\mu\nu} = -\frac{i}{8} \bar\psi \left( \gamma_\lambda \left[ \gamma_\mu, \gamma_\nu \right] + \gamma_\mu \left[ \gamma_\nu, \gamma_\lambda \right] + \gamma_\nu \left[ \gamma_\mu, \gamma_\lambda \right] \right) \psi$$

$$= -\frac{i}{8} \bar\psi \left( \left\{ \gamma_\mu, \gamma_\nu \right\} \gamma_\lambda - \left\{ \gamma_\lambda, \gamma_\nu \right\} \gamma_\mu - \left[ \gamma_\mu, \gamma_\lambda \right] \gamma_\nu \right) \psi$$

$$= -\frac{i}{4} \bar\psi \left( -\eta_{\mu\nu}\gamma_\lambda + \eta_{\lambda\nu}\gamma_\mu + \eta_{\mu\lambda}\gamma_\nu + \gamma_\lambda \gamma_\mu \gamma_\nu \right) \psi.$$

Then we take the derivative:

$$\partial^\lambda \chi_{\lambda\mu\nu} = -\frac{i}{4} \bar\psi \left( -\eta_{\mu\nu}\displaystyle{\not}\partial + \gamma_\mu \partial_\nu + \gamma_\nu \partial_\mu + \displaystyle{\not}\partial \gamma_\mu \gamma_\nu - \overleftarrow{\displaystyle{\not}\partial} \eta_{\mu\nu} + \overleftarrow{\partial_\nu} \gamma_\mu + \overleftarrow{\partial_\mu} \gamma_\nu + \overleftarrow{\displaystyle{\not}\partial} \gamma_\mu \gamma_\nu \right) \psi$$

$$= -\frac{i}{4} \bar\psi \left( -\eta_{\mu\nu}\displaystyle{\not}\partial + \gamma_\mu \partial_\nu + \gamma_\nu \partial_\mu + \gamma_\mu \gamma_\nu \displaystyle{\not}\partial + \left[ \displaystyle{\not}\partial, \gamma_\mu \gamma_\nu \right] - \overleftarrow{\displaystyle{\not}\partial} \eta_{\mu\nu} + \overleftarrow{\partial_\nu} \gamma_\mu + \overleftarrow{\partial_\mu} \gamma_\nu + \overleftarrow{\displaystyle{\not}\partial} \gamma_\mu \gamma_\nu \right) \psi.$$

We can use the equations of motion to cancel some terms:

$$\displaystyle{\not}\partial \psi = -i m \psi,\quad \bar\psi \overleftarrow{\displaystyle{\not}\partial} = i m \bar\psi$$

$$\Rightarrow \partial^\lambda \chi_{\lambda\mu\nu} = -\frac{i}{4} \bar\psi \left( \gamma_\mu \partial_\nu + \gamma_\nu \partial_\mu + \left[ \displaystyle{\not}\partial, \gamma_\mu \gamma_\nu \right] + \overleftarrow{\partial_\nu} \gamma_\mu + \overleftarrow{\partial_\mu} \gamma_\nu \right) \psi,$$

But at this point I'm not quite sure how to simplify things any more.


1 Answer 1


I found the identity I needed as the fifth miscellaneous identity on the gamma matrices Wikipedia article:

$$\gamma_\sigma \gamma_\mu \gamma_\nu = -\eta_{\sigma\mu}\gamma_\nu - \eta_{\mu\nu}\gamma_\sigma + \eta_{\sigma\nu}\gamma_\mu - i \epsilon_{\rho\sigma\mu\nu} \gamma^\rho \gamma^5,$$

$$\left[ \gamma_\sigma, \gamma_\mu \gamma_\nu \right] = \gamma_\sigma \gamma_\mu \gamma_\nu - \gamma_\mu \gamma_\nu \gamma_\sigma = 2\eta_{\nu\sigma}\gamma_\mu - 2\eta_{\mu\sigma}\gamma_\nu,$$

$$\left[ \displaystyle{\not}\partial, \gamma_\mu \gamma_\nu \right] = 2\gamma_\mu \partial_\nu - 2\gamma_\nu \partial_\mu$$

$$\Rightarrow \partial^\lambda \chi_{\lambda\mu\nu} = -\frac{i}{4} \bar\psi \left( 3\gamma_\mu \partial_\nu - \gamma_\nu \partial_\mu + \overleftarrow{\partial_\nu} \gamma_\mu + \overleftarrow{\partial_\mu} \gamma_\nu \right) \psi.$$

Finally we can get the modified canonical SEM tensor:

$$T_{\mu\nu} + \partial^\lambda \chi_{\lambda\mu\nu} = i \bar\psi \gamma_\mu \partial_\nu \psi - \frac{i}{4} \bar\psi \left( 3\gamma_\mu \partial_\nu - \gamma_\nu \partial_\mu + \overleftarrow{\partial_\nu} \gamma_\mu + \overleftarrow{\partial_\mu} \gamma_\nu \right) \psi$$

$$= \frac{i}{4} \bar\psi \left( \gamma_\mu \partial_\nu + \gamma_\nu \partial_\mu - \overleftarrow{\partial_\nu} \gamma_\mu - \overleftarrow{\partial_\mu} \gamma_\nu \right) \psi.$$

This is symmetric in $\mu$ and $\nu$ as desired and expected.


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