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The axial current is defined as jμ5=ˉψγμγ5ψ. 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 jμ5 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 jμ5's?
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  • as for 3), may I ask what is the meaning of a time-ordered correlation function of jμ'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 jμjν, only jμ. If jμjν doesn't have much use, why would jμ5jν5? [I'm truly curious because I know barely nothing about these matters...] Commented Apr 7, 2016 at 22:22
  • or put it another way: if jμjν doesn't have a precise meaning, why would jμ5jν5? Commented Apr 7, 2016 at 22:25
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    @AccidentalFourierTransform The use of these quantities is that we can calculate them, and see if current conservation μjμ holds. Since μjμ5jνjρ0, the axial current is not conserved.
    – knzhou
    Commented Apr 7, 2016 at 22:26
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    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.
    – knzhou
    Commented Apr 7, 2016 at 22:29
  • I see, nice :-) 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 ψ(x)eiαγ5ψ(x),αR of a massless fermionic theory with the usual Dirac action SD[ψ]=ˉψ(x)iDμγμψ(x)ddx 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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