# QFT - rewriting a conserved quantity

## Main Question or Discussion Point

Hey! I'm trying to learn QFT now and I'm currently reading David Tong's online lectures;

http://www.damtp.cam.ac.uk/user/tong/qft/one.pdf.

At page 17 finds the conserved current

$$j^\mu = - \omega^\rho_{\ \nu} T^{\mu}_{\ \rho} x^\nu$$

where i have understood T to be the energy momentum tensor. He further states that it can be rewritten as

$$(J^\mu)^{\sigma \rho} = x^\rho T^{\mu \sigma} - x^\sigma T^{\mu \rho}.$$

I am not that good manipulating tensors yet and my question is how one goes about showing this, step by step.

## Answers and Replies

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Bill_K
Science Advisor
Since ωμν is antisymmetric, ωμν = ½(ωμv - ω). We can therefore write (1.54) as
jμ = - ½(ωρv - ω)Tμρxν.
In the second term, switch the index labels ρ and v:
jμ = ½ωρv(Tμvxρ - Tμρxv)
He then defines (Jμ)ρv to be the quantity in parentheses.

Since ωμν is antisymmetric, ωμν = ½(ωμv - ω). We can therefore write (1.54) as
jμ = - ½(ωρv - ω)Tμρxν.
In the second term, switch the index labels ρ and v:
jμ = ½ωρv(Tμvxρ - Tμρxv)
He then defines (Jμ)ρv to be the quantity in parentheses.
Thanks! And there has also been one applied lowering operator and one raising operator?

What is the motivation behind doing this manipulation btw? To make it look kind of like a cross product?