# A Majorana Lagrangian and Majorana/Dirac matrices

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1. Aug 11, 2017

### mbond

In Lancaster & Burnell book, "QFT for the gifted amateur", chapter 48, it is explained that, with a special set of $\gamma$ matrices, the Majorana ones, the Dirac equation may describe a fermion which is its own antiparticle.

Then, a Majorana Lagrangian is considered:
$\mathcal{L}=\bar{\nu}i\gamma^\mu\partial_{\mu}\nu-$mass terms
where $\nu$ is for the Majorana fields. This Lagrangian is developed, using the usual Dirac $\gamma$ matrices and not the Majorana ones, and good looking Dirac equations are obtained.

My question is: why using the Dirac matrices to develop the Lagrangian instead of the Majorana ones? If I try the calculation with the Majorana $\gamma$ I obtain odd looking equations that don't look right.

Thank you for any help.

2. Aug 11, 2017

### George Jones

Staff Emeritus
Look at exercise (36.4).

3. Aug 12, 2017

### mbond

>
Thank you. But is there a unitary transformation between the Majorana $\bar\gamma$ matrices and the Dirac $\gamma$ matrices? I don't think so, for example $U\bar\gamma^0=\gamma^0U => U=0$. Actually the physics is different with the two sets of matrices, with antiparticle in one case and no antiparticle in the other.

4. Aug 12, 2017

### George Jones

Staff Emeritus
What do you mean by "usual Dirac $\gamma$ matrices"?

Which "representation" of the (Dirac) gamma matrices? Dirac? Weyl/chiral (as on page 324)?

See problem 2 of
http://users.physik.fu-berlin.de/~jizba/FU-petr/FU-ubungen.pdf

I think that the first $U$ in the problem should have a $-\sigma^2$ at the bottom left.