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I'm studying particle physics with an experimental approach. I have still few theory lectures including QFT. However, I'm lost about calculating Noether current. I saw this formula for example in my lecture: $$ J_\mu=-\frac{\partial L}{\partial\partial_\mu \Phi} \tau \Phi $$ with tau an $n x n$ matrix for a $n$ components field (if I understand correctly). Yet, given a Lagrangian and a field, I can't find the right answer. How to determine tau for example?

Example here: in a past exam, I had to determine Noether current for a charged particle. Lagrangian is involving complex field. I understand that I can express this field as the sum of 2 real fields, but one of them multiplied by i. However how am I supposed to find tau?

$$L=\partial_\mu\Phi^\dagger\partial^\mu\Phi-m^2\Phi^\dagger\Phi$$ $$\Phi=\frac{1}{\sqrt{2}}(\Phi_1+i\Phi_2) \; from \; correction$$ $$\tau=\begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix} \; from \; correction$$ $$J_\mu=\frac{1}{2i}(\partial_\mu \Phi\Phi^\dagger-\partial_\mu\Phi^\dagger\Phi)\; from \; correction$$

What are the tools to calculate this current without mistakes?

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