canonical Energy momentum tensor

[Maria Isabel Iguti]

Dear friends, I try to derive Energy momentum Tensor, for Rarita Swinger  lagragian  in D=3+1 dimensions.  How can I variate the lagrangian with respect  to the derivative of the field?

[Leo Stein]

Dear Maria,

One easy way you can do this is with Implode[] and Explode[], which take expressions like CD[-a][field[b]] and turn them into tensors e.g. CDfield[b,-a]. Explode undoes this. You can then use VarD to take variational derivatives with respect to the derivative of said field.

I tried to demonstrate this in the attached notebook, using the Spinors package to represent Rarita-Schwinger spinors (I tried to agree with Penrose and Rindler Eq. (5.10.35)). Please only consider this as a code example — I did not try to check any factors of +-1, i, or other conventions. Also this can surely be made more automatic and general, but I just wanted to show how to use Implode[] and Explode[].

Notice that inLagrangian-variation-xPert-VarD.nbfrom the xAct-contrib examples, there are examples that show that you get the true curved-space symmetric stress-energy tensor by varying the appropriately densitized Lagrangian with respect to the metric. However I never did an example getting the stress-energy tensor with spinors.

Please let me know if you find some error, track down the conventions, generalize the code, or succeed in getting the Rarita-Schwinger energy-momentum tensor by varying w.r.t. the metric!

Best

Leo

On Thu, Jul 18, 2024 at 5:22 PM Maria Isabel Iguti <m.i...@unesp.br> wrote:

Dear friends, I try to derive Energy momentum Tensor, for Rarita Swinger  lagragian  in D=3+1 dimensions.  How can I variate the lagrangian with respect  to the derivative of the field?

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