Representations and physics

[Lester Welch]

To moderator: 1) Sorry about the excessive lines I included in a
previous follow-up. I'll try to fix that problem. (My newsreader is
relatively new to me) 2) I posted this question in that follow-up
but it attracted no answer - I'm surmising because it was buried.
Hope this one makes it.
====

By choosing different representations of the Dirac matrices - all
connected by a unitary transformation - one can derive the equations
of motion for Dirac or Majorana neutrinos. A unitary transformation
isn't supposed to change the physics and yet Dirac and Majorana
neutrinos are different (e.g., Majorana neutrinos are their own anti-
particle). What am I missing? Thanks.

[Hendrik van Hees]

Lester Welch wrote:

> By choosing different representations of the Dirac matrices - all
> connected by a unitary transformation - one can derive the equations
> of motion for Dirac or Majorana neutrinos. A unitary transformation
> isn't supposed to change the physics and yet Dirac and Majorana
> neutrinos are different (e.g., Majorana neutrinos are their own anti-
> particle). What am I missing? Thanks.

Again one has to emphasize that Dirac and Majorana fermions are
different. There is no unitary transformation which can carry a Dirac
spinor into a Majorana spinor. That's most easily seen if you think
about the dimensions of the spinor spaces.

A Dirac spinor is a four-component spinor. The spinor space is a direct
sum of the two Spin-1/2 representations of the proper orthochronous
proper Poincare group (1/2,0) \oplus (0,1/2). It consists of two Weyl
spinors. You can take them as left and right handed spinors. This
provides however an irreducible representation of the orthochonous
Poincare group, i.e., the group generated by the orthochronous proper
Poincaregroup and space reflections. A space reflection interchanges
left and right handed spinors. Further, if the Dirac particle has mass,
the two components mix, and chiral symmetry is broken. The free
Lagrangian reads

L_{Dirac}=\bar{psi}(i \gamma^{mu} \partial_{mu} -m ) \psi,

where gamma^{mu} are any set of Dirac matrices fulfilling

{gamma^{mu},\gamma^{nu}}=g^{mu nu}.

A Majorana spinor is a two-component spinor, i.e., a Weyl spinor, e.g.,
the left handed one. There is no non-trivial representation of space
reflections, and thus parity is not conserved by a theory containing
Majorana spinors. The Lagrangian reads

L_Majorana=i \chi^{dagger} \bar{\sigma}^{mu} \partial_mu \chi
+m/2 \chi^{T} \sigma^2 \chi
-m/2 \chi^{dagger} \sigma^2 \chi^*,

where

\bar{\sigma}^{0}=1, \bar{sigma}^mu=-\sigma^mu for mu \in {1,2,3},

where \sigma^mu for mu \in {1,2,3} are the usual Pauli matrices.

Note that the mass term only is different from 0 in the quantized
version of the theory, i.e., when \chi denotes fermionic field
operators within the operator formulation or Grassmann-number valued
fields within the path-integral formulation.

--
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]

Thanks for your patience and time.

I think I'm getting closer to an understanding. Sanity check. Would
you agree that the following is correct?
===

If one chooses a Majorana representation of the gamma matrices - as is
done in

http://arxiv.org/abs/hep-th/9712113

and solves the equations one does not get Majorana fermions unless one
imposes the condition that

\psi = C \psi that is that the wave function is equal to its charge
conjugate (see e.g., page 227 of Srednicki)

If one doesn't impose that condition then, indeed, one gets a unitary
transform of the solutions obtained by using the Dirac-Pauli
representation of the gamma matrices.
===

I understand that the Lagrangians for Majorana fermions are different
from the Dirac fermions and that the solutions have different
properties. I'm trying to understand where the difference comes from
since both are solutions to the Dirac equation - or where does the
difference in the Lagrangians come from.

Thanks again for your infinite (?) patience.

I worked for four years at RWTH in Aachen on the Gargamelle experiment
more years ago than I wish to admit. The discovery of neutral
currents was a exciting time. In retirement I'm still trying to learn
physics.

Lester Welch