Investigating Belinfante Tensor: Relation to Conserved Current

[kent davidge]

I was reading this pdf http://research.physics.illinois.edu/Publications/theses/copies/Bandyopadhyay/Chapter_3.pdf

I can show myself that $\partial_\mu T^{\mu \nu} = 0$ and $\int T^{0 \nu} = \int \Theta^{0 \nu}$ if $T^{\mu \nu} = \Theta^{\mu \nu} + \partial_\alpha B^{\alpha \mu \nu}$. However I'm unable to see how $T^{\mu \nu}$ is related to the actual conserved current (not shown in the PDF) $$\frac{\partial \mathcal L}{\partial (\partial_0 \varphi)} S^{jk} \varphi (x) + (\Theta^{k0}x^j - \Theta^{j0}x^k)$$ Is it hard to show their relation?

 

[samalkhaiat]

Is it hard to show their relation?

No, it is very easy, up to a total divergence (super potential), it is $$M^{0\mu\nu} = x^{\mu}T^{0\nu} - x^{\nu} T^{0\mu} .$$
See how the Eq(2.46) was obtained in the attached PDF.