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.

 

[Bill_K]

Since ω_μν is antisymmetric, ω_μν = ½(ω_μv - ω_vμ). We can therefore write (1.54) as
j^μ = - ½(ω_ρv - ω_vρ)T^μρx^ν.
In the second term, switch the index labels ρ and v:
j^μ = ½ω_ρv(T^μvx^ρ - T^μρx^v)
He then defines (J^μ)^ρv to be the quantity in parentheses.

 

[center o bass]

Since ω_μν is antisymmetric, ω_μν = ½(ω_μv - ω_vμ). We can therefore write (1.54) as
j^μ = - ½(ω_ρv - ω_vρ)T^μρx^ν.
In the second term, switch the index labels ρ and v:
j^μ = ½ω_ρv(T^μvx^ρ - T^μρx^v)
He then defines (J^μ)^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?