Talk:Majorana equation

Natural Units

It's not in natural units though is it? If it were c and h-bar wouldn't need to appear.

Fixed. linas (talk) 00:03, 27 October 2008 (UTC)

its own anti-particle

If I remember correctly, the Majorana is its own anti-particle, right? (I mean, that's what "real representation of the Lorentz group" means, under the covers, right? ...there are always subtle points, so I don't want to just wade into this). Thus, the current experiments searching for neutrino-less double beta decay are actually looking for Majorana neutrinos, is that right? Or are there also more complicated spinors that are their own anti-particles? If so, what are they? Just curious.

Anyway, article should mention that no known particles are Majorana, but that perhaps the neutrinos might be, etc. which is why the see-also section is populated with neutrino-related topics. linas (talk) 23:55, 26 October 2008 (UTC)

Errors and attempts to fix them

The August 2019 version of this article stated:

The Majorana equation is

$-i{\partial \!\!\!{\big /}}\psi +m\psi _{c}=0\qquad \qquad (1)$

This relation leads to the alternate expression

$i{\partial \!\!\!{\big /}}\psi _{c}+m\psi =0\qquad \qquad (2)$.

This cannot be right. The reason is easy to see: if these are subtracted one gets

$-i{\partial \!\!\!{\big /}}(\psi +\psi _{c})+\eta m(\psi _{c}-\psi )=0$

Now if "majorana is it's own antiparticle", that implies that $\psi =\psi _{c}$ or maybe $\psi =\eta \psi _{c}$ for some complex number $\eta .$ Now if $\psi =\psi _{c}$ we get an equation describing a massless field. If $\psi =-\psi _{c}$ then the kinetic term vanishes. So adding and subtracting these two leads to nonsense.

Update. Well, eqn 2 is just wrong. It should have been $-i{\partial \!\!\!{\big /}}\psi _{c}+m\psi =0$ which seems to resolve the above issue. 67.198.37.16 (talk) 21:14, 9 December 2020 (UTC)

I am trying to fix all of this now, but this is challenging, because all of the texts available are problematic in various ways. I think they're all correct, but they use inconsistent notation and either use squonky, questionable proof techniques, or make questionable statements without providing proof, or seem to be built on unfounded assumptions. I thought I could whip this article into shape in a day or two, and instead its becoming a real chore. BTW, I am looking at these:

• Andreas Aste, (2010) "A Direct Road to Majorana Fields", Symmetry 2010, 2, pp.1776-1809; doi:10.3390/sym2041776 arXiv:0806.1690 hep-th

• Palash B. Pal (2011) "Dirac, Majorana and Weyl fermions", American Journal of Physics 79, p485. arXiv:1006.1718 hep-ph

• Eckart Marsch (2011) "The Two-Component Majorana Equation-Novel Derivations and Known Symmetries" Journal of Modern Physics, 2011, 2, 1109-1114

• Eckart Marsch (2012) "On the Majorana Equation: Relations between Its Complex Two-Component and Real Four-Component Eigenfunctions", International Scholarly Research Network, ISRN Mathematical Physics, Volume 2012, Article ID 760239, 17 pages doi:10.5402/2012/760239 Hindawi

• Eckart Marsch, (2013) "A New Route to the Majorana Equation", Symmetry vol 5 issue 4, pp.271-286; doi:10.3390/sym5040271. PDF

Anyone who reads this in the next day or two (or next week or two) is invited to help sort out the, uhhh, current mess. I'm trying to get to it, I really am. BTW, this really is all about trying to get to the Elko spinor thing, which cannot be done unless this article is done correctly, first. I'm somewhat dubious that the Elko thing is even real, and not some algebra error, but given the nasty state of affairs with the various explanations and derivations of Majorana that can be found, its quite hard to say. 67.198.37.16 (talk) 18:18, 2 December 2020 (UTC)

Progress report. All of the above references leave a lot to be desired for both clarity and rigor. I have not found any errors in any of the papers, but the poor structure confused the heck out of me; I finally have a clean and clear and crisp derivation of all of the equations and identities. (Edit history of this talk page will reveal my confusion). I will update the article tomorrow. Meanwhile, one more curious reference:

• de Gouvêa, A.; Kayser, B.; Mohapatra, R.N. Manifest CP violation from Majorana phases. Phys. Rev. 2003, D67, 053004.

Phew. This took much more effort than I wanted it to. 67.198.37.16 (talk) 04:31, 11 December 2020 (UTC)

Hermitian operator

I have made a slight edit to the main page, to include the requirement of a Hermitian operator; this immediately tells you that the mass term must be a skew-symmetric (anti-symmetric) matrix. I think the issues you raised above arise because you ignore the condition of a Hermitian operator. (To manifest the Hermiticity, it is more convenient to multiply both sides of the Majorana equation by i, which of course doesn't change the equation, but it highlights that you need a Hermitian operator.) Brienanni (talk) 09:41, 3 December 2020 (UTC)

Thanks. Yes, that change is reasonable. For me, the meta-issue is the so-called "elko spinor". From what I can tell, some authors ignore it. Others seem to rediscover it. And yet others make some dramatic claims regarding it's quantization, as a dark matter candidate. I am not yet comfortable with all the extra signs and conjugations and inner products floating around to be able to finish this article without introducing extra sign errors myself. My goal is write this article clearly enough to make it completely obvious exactly how Elko inter-relates to this, and where all the minus signs are going.

If you are interested, try reading some of the Elko literature. It is annoyingly low-quality and poorly written, which makes all this harder than it should have been. One starting point is this: Draft:Mass dimension one fermions which was recently AfD'ed. In order to revive it, or some semblance of it, I started to write Draft:ELKO Theory but I am not convinced that ... well, that its real and not due to bizarre confusion about sign conventions. 67.198.37.16 (talk) 21:22, 3 December 2020 (UTC)

TODO List

Things the article should cover:

• Explicit CPT operators for 2-spinor variant
• Explicit CPT operators for 4-spinor variant
• Lagrangian
• Quantum field operators

67.198.37.16 (talk) 02:50, 17 December 2020 (UTC)