Why is the 5 used in the axial Chiral current formulas?

[Pet Scan]

...the axial Chiral current formulas use the symbol J^5...i.e.,( "J" with a post script 5. ) What is the reason for the "5" ..??

 

[Pet Scan]

I guess I put this in the wrong section, maybe.? since I got no answers... Axial anomaly is a QFT phenomena, but originated in Particle physics. However, chiral axial anomaly was recently "discovered" experimentally in condensed matter / solid state physics...in topological crystalline materials; see, for ex., http://phys.org/news/2015-09-long-sought-chiral-anomaly-crystalline-material.html
So where should I put this post, and how do I transfer it please...in order to get an answer?
Thx;
Pet Scan

 

[A. Neumaier]

Becausee the five gamma's generate the 5-dimensional Clifford algebra.

 

[Pet Scan]

Thanks Neumaier...However, could you dumb it down for me;...real low.?... I know nothing of Clifford algebra...can you give it to me in terms of basic physics...
Thx

 

[A. Neumaier]

There is no physical reason for the notation.

Look at the anticommutation relations for the $\gamma^\mu$ ($\mu=1:4$, where the 4 is traditionallly written as 0), generalize to any $n$ in place of $4$ and then look at what you get for $n=5$. Aha!

Also see https://en.wikipedia.org/wiki/Clifford_algebra#Physics

 

[Pet Scan]

More explicitly; we see in a phenomenological explanation, for example, given here, of the Chiral Magnetic Effect (which is an axial anomaly) .. which I find of great interest... https://www.researchgate.net/publication/47338061_Axial_anomaly_Dirac_sea_and_the_chiral_magnetic_effect ...
under the heading,...
2. The Chiral Magnetic Effect and Landau Levels of Dirac Fermions;
we see, in this derivation of parity violating current J, the difference in Fermi momenta of right and left handed spins (top of page 4), is given as u(R) - u(L) = 2u^5.
It is designated as u^5...(leading to an implied J^5 current density) ...why does he designate it as u^5 ...as opposed to u^4 or even u^6...IOW, from whence is the postscript derived.?

Sorry; I am lost when you speak of anything about Clifford algebra...even as I have tried to read about it o Wiki...I guess I need to learn it in order to get the understanding...

 

[vanhees71]

I don't know, what you are precisely referring to. An anomaly is defined as a symmetry valid in a classical theory, which implies the conservation of the corresponding current. For the axial U(1) symmetry the current is
$$j_5^{\mu}=\mathrm{i} \overline{\psi} \gamma^5 \gamma^{\mu} \psi,$$
and I guess the $5$ is indeed just due to the appearance of the $\gamma^5=\mathrm{i} \gamma^0 \gamma^1 \gamma^2 \gamma^3$ Dirac matrix in the current. It's just a name with not much deeper reason. Now it happens that when quantizing the theory, you cannot keep all symmetries.

For the axial U(1) anomaly it's intuitively understandable what happens. When calculating one-loop corrections to the three-point vertex function with one axial and two vector currents you have a linear divergence. Thus you have to regularize this loop integral. If you like to preserve the symmetry you should find a regulator that doesn't destroy the symmetry. You cannot find such a regulator for the axial U(1) symmetry. The usual suspects don't work: A momentum cutoff destroys the symmetry, because a scale enters, and chiral symmetry doesn't easily respect a scale. The same holds for Pauli-Villars: You have to introduce massive fermions, and these don't respect the chiral symmetry. Dimensional regularization fails, because you don't know what to do with the generically four-dimensional object $\gamma^5$.

Indeed, a closer analysis shows that you can either keep the vector current conserved or the axial current or any linear combination thereof. You can however never preserve both. Physics tells you which current to conserve usually. E.g., if you have QED the vector current should be conserved, because otherweise you destroy electromagnetic gauge symmetry, because that's the current the em. field couples to. Thus you must break the conservation of the axial U(1).

I have a section on anomalies in my QFT manuscript (Sect. 7.6):

http://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf

 

[Pet Scan]

OK; I think you got it Vanhees...It comes from the Dirac matrix elements...another area I have not sufficient background in which to delve... thank you... I'm sure I will have more questions about the other details of your post later... very interesting.