# Neutrinos, quasiparticles and Majorana fermions

In connection with topological quantum computing I encountered term Majorana fermions. According to Wikipedia these are:

A Majorana fermion, also referred to as a Majorana particle, is a fermion that is its own antiparticle.

The Wiki also states

With the exception of the neutrino, all of the Standard Model fermions are known to behave as Dirac fermions at low energy (after electroweak symmetry breaking), and none are Majorana fermions. The nature of the neutrinos is not settled - they may be either Dirac or Majorana fermions.

I am a little bit confused by this. Since neutrinos are electrically neutral, I would expected that they are their own antiparticles. At the same time, I know that there is also a lepton number which is negative for antiparticles. So neutrino and anti-neutrion can still differ in value of the lepton number.

Does this all mean that Majorana fermions and anti-fermions have positive lepton number?

In connection to quantum topological computing where so-called anyons quasiparticles are used, the Wiki states

In superconducting materials, a Majorana fermion can emerge as a (non-fundamental) quasiparticle, more commonly referred to as a Bogoliubov quasiparticle in condensed matter physics. This becomes possible because a quasiparticle in a superconductor is its own antiparticle.

So, why in case of quasiparticles, they are their own antiparticles and in case of neutrinos they are not?

I would appreaciate a more or less laymen explanation, if possible, as I am not well trained in mathematical background of particle physiscs.

• Merely being electrically neutral doesnt stop a particle from having a different antiparticle. Eg, the neutron is electrically neutral but it's not its own antiparticle, and a neutron annihilates with an antineutron in a reaction that's similar to proton + antiproton annihilation. But of course a neutron is a composite particle, not a fundamental particle like a neutrino is. Apr 6, 2020 at 13:37

Since neutrinos are electrically neutral, I would expected that they are their own antiparticles.

Not everything that is electrically neutral is its own antiparticle. The neutron and the antineutron are different particles (one has quark composition $$udd$$, the other has quark composition $$\overline{u}\overline{d}\overline{d}$$).

At the same time, I know that there is also a lepton number which is negative for antiparticles. So neutrino and anti-neutrion can still differ in value of the lepton number.

If the neutrino turns out to be its own antiparticle, then lepton number is not a conserved quantity.

Does this all mean that Majorana fermions and anti-fermions have positive lepton number?

Because lepton number is no longer conserved in that case, I don't think the lepton number of a Majorana lepton is well-defined.

So, why in case of quasiparticles, they are their own antiparticles and in case of neutrinos they are not?

Quasiparticles aren't always their own antiparticles. There are quasiparticles that behave as Dirac fermions as well, with distinct antiparticles: for example, in graphite (https://www.nature.com/articles/nphys393). And we don't know that neutrinos aren't their own antiparticles - we aren't currently able to measure precisely enough to find out (mainly because neutrinos barely ever interact with anything in the first place, and the process that would tell us that it's a Majorana fermion, namely, neutrinoless double-beta decay, is projected to be extremely rare even among neutrino interactions).

So ultimately, we don't know of anything preventing neutrinos from being their own antiparticles. And we have observed both types of fermions as quasiparticles.