QFT - rewriting a conserved quantity

  • #1

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

[tex] j^\mu = - \omega^\rho_{\ \nu} T^{\mu}_{\ \rho} x^\nu[/tex]

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

[tex](J^\mu)^{\sigma \rho} = x^\rho T^{\mu \sigma} - x^\sigma T^{\mu \rho}.[/tex]

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

Answers and Replies

  • #2
Bill_K
Science Advisor
Insights Author
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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.
 
  • #3
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?
 
  • #4
What is the motivation behind doing this manipulation btw? To make it look kind of like a cross product?
 

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