I was trying to solve Problem 3.2 ( problem 3 in chapter 2) in Quantum Field Theory and the standard model for Matthew D. Schwartz. The problem says :

Calculate the conserved currents Kμνα associated with (global) Lorentz transformations Xμ → Λμν Xν.

I just want to make sure whether I start right in my solving. I'm not asking for a full solution for the problem. I just need some help in the beginning of solving it. This is how I begun: enter image description here


No, your index structure is messed up. The argument in $\Phi(x_\mu)$ means to say that the field depends on all four spacetime coordinates, $\mu$ is not a fixed index. In particular, you can't pull the Lorentz matrix out of the field's argument because then your left hand side has no free indices while the right hand side has two free indices $\mu$ and $\nu$.

  • $\begingroup$ Ah you're right. Thank you. Can you help me for a good start? $\endgroup$ – user65035 Jul 18 '18 at 8:29
  • $\begingroup$ Try to consider the action $S$ of the theory and then expand the Lagrangian density and the integration measure in $\delta x$ up to first order, where $\delta x = x'-x$ and $x'=\Lambda x$. $\endgroup$ – Photon Jul 18 '18 at 8:33
  • $\begingroup$ You mean that the lagrangian will be in terms of the vector X and not in terms of fields? $\endgroup$ – user65035 Jul 18 '18 at 8:59
  • $\begingroup$ No, it will be in terms of the fields but they themselves depend on $x$, so you need to expand them in $\delta x$ as well. $\endgroup$ – Photon Jul 18 '18 at 9:02

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