Investigating Belinfante Tensor: Relation to Conserved Current

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The discussion centers on the relationship between the Belinfante tensor \( T^{\mu \nu} \) and the conserved current in the context of field theory. It establishes that \( \partial_\mu T^{\mu \nu} = 0 \) and \( \int T^{0 \nu} = \int \Theta^{0 \nu} \) when \( T^{\mu \nu} = \Theta^{\mu \nu} + \partial_\alpha B^{\alpha \mu \nu} \). The relationship to the conserved current is clarified as straightforward, involving the expression \( M^{0\mu\nu} = x^{\mu}T^{0\nu} - x^{\nu} T^{0\mu} \), which is derived from the provided PDF reference.

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kent davidge
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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?
 
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kent davidge said:
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.
 

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