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In chiral perturbation theory, we introduce the parametrization of the Goldstone manifold $$ U = \exp\left( i \frac{1}{f_\pi} \pi_a \sigma_a\right), $$ where $\sigma_a$ are the Pauli matrices. Most sources (Weinberg, quantum theory of fields 2, chapter 19 and Stefan Scherer, introduction to Chiral Perturbation Theory, for example) identify these parameters to the charged pion by $$ \pi_a \sigma_a = \begin{pmatrix} \pi_0 & \sqrt 2 \pi_+ \\ \sqrt 2 \pi_- & -\pi_0 \end{pmatrix} = \begin{pmatrix} \pi_3 & \pi_1 - i \pi_2 \\ \pi_1 + i \pi_2 & -\pi_3 \end{pmatrix}. $$

However, in M. Schwartz' Quantum field theory and the standard model, the identification is $\sqrt 2 \pi_+ = \pi_1 + i\pi_2$. This makes more sense to me. The positively charged pion has positive isospin, and this relates the pion with positive charge to the raising operator $\sqrt 2 \sigma_+ = \sigma_1 + i \sigma_2$. Is this the correct identification? Is there a source where anyone discusses this in more depth, as all these sources just writes this down.

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  • $\begingroup$ Make sure they all use the same convention: does the $\pi^+$ field destroy or create a $\pi^+$ state in all sources? $\endgroup$ Dec 14, 2021 at 18:38
  • $\begingroup$ I count on the fact that $\pi^+$ always corresponds to the creation of a positively charged pion, but non of the sources talk much about it. And I noticed the clash afterward, it seems all to be rather confusing. He never explicitly states the form of the generators, so I do not know. $\endgroup$ Dec 14, 2021 at 20:09

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The dyslexia purgatory in the rabbit hole you've slipped into is "fields-versus-states". Weinberg, Scherer, and lots of purists don't discuss matrix elements, and hence fields and states; but, rather, neutral interaction terms in lagrangians. Your comment leaves what a "positively charged pion" is ambiguous: a field or a state?

In any case, the majority of users, like Weinberg, cf the definition of charged Ws here, use the convention you start with. As a result, the corresponding neutral interaction term of the σ model, i terms of fields, is $$ \propto \bar p \pi^+ n \propto \bar \psi \pi^+ \begin{pmatrix} 0&1\\0 &0\end{pmatrix} \psi , $$ visibly neutral, albeit counterintuitive, as you complained.

Few people will discuss the fact that it assumes the non- vanishing matrix element entering PCAC is $\langle 0| \pi^+|\pi^+\rangle$, i.e. $\pi^+|0\rangle \propto |\pi^-\rangle$, which would drive you (and me, and half of Schwartz, in case you noticed his inconsistency between (22.17) and (30.91)!) crazy...

They would rather take your (presumed favorite) $$ \pi^+|0\rangle \propto |\pi^+\rangle, $$ which, however, hides a hermitean conjugation in the interest of sanity. In this matrix element convention, all the +s and the -s add up to 0, so it is the matrix element
$$ \langle p| \pi^+|n\rangle \propto \langle p|p\rangle $$ which is neutral.

Much of the literature follows the leading, legit, convention, but lots of people stick to the second, and some just shrug the torpedoes off and assume the well-meaning reader will chase the signs... and wink at consistency...

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  • $\begingroup$ Ouuuf, i've got some work to do before handing in my thesis... Thank you for the clarification $\endgroup$ Dec 14, 2021 at 21:21
  • $\begingroup$ Is it correct that, no matter the choice of convention, the $ \pi_1 + i\pi_2$-operator is the creation operator for a state with positive electric and third-isospin-component charge? (I hope so, for the sake of my crumbling sanity) $\endgroup$ Dec 14, 2021 at 21:27
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    $\begingroup$ Yes. Sanity reclaimed. $\endgroup$ Dec 14, 2021 at 21:31

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