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The axial current is defined as $$j^\mu_5 = \bar{\psi} \gamma^\mu \gamma_5 \psi.$$ This quantity is important when studying anomalies. Explicitly working out components, the axial current is just the current due to left-handed fermions minus the current due to right-handed ones, just like the usual current is their sum.

The above is true just by definition, but it leaves me unsure what the axial current and the associated charge are.

  • Is there a nice, physical interpretation of $j^\mu_5$ besides the one I gave above?
  • How does one measure the axial current/charge experimentally?
  • What is the physical meaning of a time-ordered correlation function of $j^\mu_5$'s?
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  • $\begingroup$ as for 3), may I ask what is the meaning of a time-ordered correlation function of $j^\mu$'s? that is, what is the meaning when you use vector currents instead of axial currents? I'm asking because , in my very limited experience with QFT, I've never seen $\langle j^\mu j^\nu\cdots\rangle$, only $\langle j^\mu\rangle$. If $\langle j^\mu j^\nu\cdots\rangle$ doesn't have much use, why would $\langle j^\mu_5 j^\nu_5\cdots\rangle$? [I'm truly curious because I know barely nothing about these matters...] $\endgroup$
    – AccidentalFourierTransform
    Commented Apr 7, 2016 at 22:22
  • $\begingroup$ or put it another way: if $\langle j^\mu j^\nu\cdots\rangle$ doesn't have a precise meaning, why would $\langle j^\mu_5 j^\nu_5\cdots\rangle$? $\endgroup$
    – AccidentalFourierTransform
    Commented Apr 7, 2016 at 22:25
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    $\begingroup$ @AccidentalFourierTransform The use of these quantities is that we can calculate them, and see if current conservation $\partial_\mu \langle j^\mu \cdots \rangle$ holds. Since $\partial_\mu \langle j_5^\mu j^\nu j^\rho \rangle \neq 0$, the axial current is not conserved. $\endgroup$
    – knzhou
    Commented Apr 7, 2016 at 22:26
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    $\begingroup$ I really want to assign an interpretation to this quantity, because physical meaning is really, really slippery in QFT, to the point that some purists tell me nothing has any meaning besides S-matrix elements. This makes it hard to tell what I'm actually doing in a calculation. $\endgroup$
    – knzhou
    Commented Apr 7, 2016 at 22:29
  • $\begingroup$ I see, nice :-) $\endgroup$
    – AccidentalFourierTransform
    Commented Apr 7, 2016 at 22:31

1 Answer 1

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  1. The axial (also called "chiral") current is the Noether current for the axial transformation $$ \psi(x) \mapsto \mathrm{e}^{\mathrm{i}\alpha \gamma^5}\psi(x),\alpha\in\mathbb{R}$$ of a massless fermionic theory with the usual Dirac action $$ S_D[\psi] = \int \bar\psi(x)\mathrm{i}D_\mu\gamma^\mu\psi(x)\mathrm{d}^d x$$ so it is classically non-conserved only by the transformation of the mass term in a massive Dirac action.

  2. One experimental significance of the non-conservation of the axial current is neutral pion decay, see this question and references therein. Heuristically, it's the axial current and not the usual current that plays a role there because the pion is a pseudoscalar, not a scalar, and hence must couple to another pseudo-object (which the axial current is) to give a scalar amplitude. By explicit calculation, you find that the chiral anomaly is directly related to the decay width of the pion.

  3. No idea. :)
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