# QFT - rewriting a conserved quantity

center o bass
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.

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.

center o bass
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?

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