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Working through David Tong's sheet here https://www.damtp.cam.ac.uk/user/tong/qft/oh1.pdf and can't follow how to get the Levi-Cevita symbol out the front? Its equation 15. I was looking at trying to use an identity with the Kronecker Deltas in the energy momentum tensor but I am really stuck. Below is the conserved charge, where does the $\epsilon_{ijk}$ come from?

$$Q_i = \epsilon_{ijk} \int d^3x (x^jT^{0k} - x^kT^{0j})$$

I can follow up to the conserved current and get the correct value for charge but I am missing the Levi-Cevita

Qmechanic
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milkcookie
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1 Answers1

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The conserved quantity that arises due to spatial rotational invariance is $$ L^{jk} = Q^{jk} = \int \mathrm{d}^3 x\ (x^j T^{0k} - x^k T^{0j}) $$

Unlike the classical angular momentum pseudovector $\mathbf{x \times p}$, this is expressed as a matrix:

$$ L^{ij} = \begin{pmatrix} 0 & L_{xy} & L_{xz} \\ -L_{xy} & 0 & L_{yz} \\ -L_{xz} & -L_{yz} & 0 \end{pmatrix} $$

whose components may look familiar - they are similar to the magnetic field components in the electromagnetic field tensor $F^{jk}$. Again, similar to the electromagnetic case, to convert $L^{jk}$ back into the classical vector form, you have to use the Levi-Civita symbol: $$ L_i = \epsilon_{ijk}L^{jk} $$

However, as Luboš Motl remarks here, it is usually more efficient to leave it tensor form.

Nihar Karve
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