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I have been studying complex scalar fields, and in Peskin and Schroeder, An Introduction to Quantum Field Theory, (chapter 2, problem 2, part d— on page 34) they ask you to compute the conserved charges for two equally massive complex scalar fields. So far I understand that the corresponding Lagrangian is invariant under $U(2)$ (which gives four separate conserved charges). But in a note, it says there are actually six. I don't see where the other two could come from. Is it related to the Lagrangian being invariant under a bigger symmetry group that I didn't notice?

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  • $\begingroup$ Where are you getting six charges? My copy not only specifies that there should be four charges, it states exactly what they are. $\endgroup$
    – Buzz
    Dec 29 '20 at 23:27
  • $\begingroup$ At the bottom of the page it says: ''With some additional work you can show that there are actually six conserved charges in the case of two complex fields, and $n(2n-1 )$ in the case of $n$ fields, corresponding to the generators of the rotation group in four and $2 n$ dimensions, respectively. The extra symmetries often do not survive when nonlinear interactions of the fields are included.'' $\endgroup$ Dec 29 '20 at 23:36
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Two complex scalar fields $\phi_{1}$ and $\phi_{2}$ can be rewritten as four real fields, in terms of their real and imaginary parts, $$\Phi=\sqrt{2}\left[\begin{array}{c} \Re\{\phi_{1}\} \\ \Im\{\phi_{1}\} \\ \Re\{\phi_{2}\} \\ \Im\{\phi_{2}\} \end{array}\right].$$ For the free theory, the Lagrange density is actually equal to $${\cal L}=\frac{1}{2}\partial^{\mu}\Phi_{i}\partial_{\mu}\Phi_{i}-\frac{m^{2}}{2}\Phi_{i}\Phi_{i},$$ with the $i=1,\ldots,4$ summed over. This is just the sum of four Lagrange densities for four independent real fields $\Phi_{i}$. This is clearly invariant under real $SO(4)$ rotations, of which there are six.

However, these symmetries do not survive under the natural interactions for complex (i.e. charged) scalar fields, such as the current coupling term $\left[\phi_{j}^{*}(\partial^{\mu}\phi_{j})-(\partial^{\mu}\phi_{j}^{*})\phi_{j}\right]A_{\mu}$, (now summed over $j=1,2$).

(My copy of the book—a relatively early printing—does not have the footnote mentioned in the question. However, the relationship between four real fields and two complex fields is discussed a bit further in problem 4.3.)

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