Gamma matrices representations

[Lester Welch]

I understand that there are different representations of the Dirac
Gamma matrices - Pauli-Dirac, Weyl, Majorana - all related by a
unitary transformation.

However I read about a "Majorana mass" etc. Since mass is a scalar
and should not vary under a unitary transformation why are there
diferent "masses?"

I hope my confusion does make my question unintelligible. Thanks.

[Hendrik van Hees]

Lester Welch wrote:

> I understand that there are different representations of the Dirac
> Gamma matrices - Pauli-Dirac, Weyl, Majorana - all related by a
> unitary transformation.

That's correct. All representations are of course equivalent. For almost
all calculations in perturbation theory, the representation doesn't
matter at all, and you can live with the Clifford-algebra property

{\gamma^{\mu},\gamma^{\nu}}=\eta^{\mu \nu},

where \eta^{\mu \nu}=diag(1,-1,-1,-1) denote the components of the
metric tensor (in about half of the literature the other convention
with diag(-1,1,1,1) is used, but that's also physically equivalent, one
must only be careful not to mix one convention with another in one
calculation).

For some purposes, a special representation can have advantages. For
Dirac particles in the massless limit the chiral representation is
advantegeous. If you like to consider the non-relativistic limit of the
Dirac particle in an external em. field in Coulomb gauge, the Dirac
representation is the most convenient starting point.
>
> However I read about a "Majorana mass" etc. Since mass is a scalar
> and should not vary under a unitary transformation why are there
> diferent "masses?"

The one thing has nothing to do with the other. A Majorana particle is
described by another kind of field, namely a Weyl spinor and has a mass
term different from the Dirac-mass term (see, e.g., Peskin/Schroeder,
An Introduction to Quantum Field Theory).

--
Hendrik van Hees Institut für Theoretische Physik
Phone: +49 641 99-33342 Justus-Liebig-Universität Gießen
Fax: +49 641 99-33309 D-35392 Gießen
http://theory.gsi.de/~vanhees/faq/

[Lester Welch]

On Jan 5, 3:09Â*pm, Hendrik van Hees
<Hendrik.vanH...@theo.physik.uni-giessen.de> wrote:
[[Mod. note -- Many lines of excess quoted text snipped. -- jt]]
> > However I read about a "Majorana mass" etc. Â*Since mass is a scalar
> > and should not vary under a unitary transformation why are there
> > diferent "masses?"
>
> The one thing has nothing to do with the other. A Majorana particle is
> described by another kind of field, namely a Weyl spinor and has a mass
> term different from the Dirac-mass term (see, e.g., Peskin/Schroeder,
> An Introduction to Quantum Field Theory).

[[Mod. note -- Quoted signature snipped. -- jt]]

Thanks for the answer. Let me ask a couple related questions. I read
that the (massless) neutrinos in the Majorana representation are their
own anti-particle but not in the Pauli-Dirac representation. If
unitary transformations don't change the physics - how can this be?
Are the fields obtained by the canonical quantization of the different
representations of the Dirac equation different? If so, isn't that a
change of the physics?

[Mihai Cartoaje]

The present articles on the Dirac equation in curved space-time have
sketchy proofs that are difficult to follow. So I am working on an
article about a Dirac equation in curved space-time that has all the
calculations in detail:
http://geocities.com/repstsb/dirac.html