As I understand it, the Dirac equation and its solutions depend on the representation of the $\gamma$ matrices one uses. So if I were to use the Dirac representation I would get different mathematical solutions that if I used the Majorana representation.

In particular, Majorana fermions are their own antiparticles (because the solutions are real), so I am pretty sure that solutions in the Majorana representation are not physically equivalent to the ones in Dirac representation, which are not their own antiparticles.

If the only difference between $\gamma_{dirac}$ and $\gamma_{majorana}$ is a change in representation (meaning a different way to express the same $\gamma$ object), why are the solutions physically not equivalent? Why is the physics of the Dirac equation dependant on the representation of the $\gamma$ matrices?

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    $\begingroup$ Majorana fermions are solutions to the Majorana equation not a solution to the Dirac equation. $\endgroup$ – Y2H May 20 '18 at 22:20
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    $\begingroup$ The basic confusion here is that the Majorana representation does not describe Majorana spinors. It's just that Majorana spinors look simpler in the Majorana representation. $\endgroup$ – knzhou May 20 '18 at 22:32
  • $\begingroup$ Don't confuse the representation of the Clifford algebra, which is as physically meaningless as a choice of basis, with the Lorentz representation of the fields! $\endgroup$ – knzhou May 20 '18 at 22:33

The physics is exactly the same in both basis. The solution appears different to you only because you are using a different basis for the spinors, thus the numbers "in the upper square" and in the "lower square" of the spinors are different (base dependant).

Of course you can write Majorana spinors in chiral/Dirac basis (or whatever spinor basis you want). The only difference is that the Majorana condition is not going to be as simple as

$$\psi^* = \psi$$

The general Majorana condition (in a basis indepent form) is: $$C\psi = \psi ^*$$ where $C$ is the charge conjugation operator. The point is that $C$ has different coordinate (read: entries) in different basis. In particular, it's the $2\times 2$ identity in Majorana basis, hence the utility of this basis.

You can also find how a Majorana spinor looks in Dirac basis by using the fact that $C=-i\gamma^2$ in chiral basis.

If you want a reference I found useful, you can look at David Tong QFT lectures on his page.


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