Discussion with Howard Haber and "world expert" about paper: "The nonequivalence of Weyl andMajorana neutrinos with standardmodel gauge interactions" [ps][pdf] , by R.Plaga
An extract of the discussion is given below.
There seem to be 2 major points of contention. Point 1. The mathematical form of the Lagrangian of a massless Majorana neutrino, that has the identical phenomenology as the observed neutrino. (main criticism of H.Haber)Point 2. The correct normalisation factor in the mathematical definition of the Majorana neutrino. (main criticism of world expert, shared by Haber)In both cases the contention is about a factor sqrt(2).However, this factor sqrt(2) in point 1. and point 2. is not the same, i.e. even using the normalisation factor suggested by the worldexpert and H.Haber in 2., I do not get the mathematical form of the Lagrangian suggested by H.Haber for point 1. 1. The mathematical form of the Lagrangian of a massless Majorana neutrino, that has the identical phenomenology as the observed neutrino. (main criticism of H.Haber)Email of R.Plaga from 11.12.99WHAT I do imply, however, is:  Basic assertion of manuscript (alternative formulation): part 1:
part 2:
part 3:
Answer of H.Haber 24.7.00
You seem to believe that these arguments would somehow be modified if
My eq.(7) and Haber´s eq.(22) (eq. (28) is the same with mass
term and no
email of H.Haber from 14.10.99
My Lagrangian eq.(7) is derived in section 3 of my manuscript, but
Email from 2.8.00 from R.Plaga

Indirectly H.Haber answers to this in his short write up where he
notes after eq.(28):
Haber short write up p.4
The factor 1/2 that appears in eq.(22) in front
of the kinetic energy term
is correct. Just like in scalar field theory,
if one uses real fields, the corresponding
coefficient of the kinetic energy term is 1/2
while for complex field it is 1.
end of quote
I disagree:
email 11.8.00 from R.Plaga
You are of course right that when going from a complex to a
real scalar field, a factor 1/2 appears in the kinetic term
(e.g. Bjorken,Drell Relativistic Quantum fields, chapter 12.5).
However it is not correct to conclude the same happens when going
from Weyl to Majorana fields. As a result of the
factor 1/2 in the kinetic term, the complex and real
scalar field CAN be distinguished on the KINEMATICAL level.
That this must be true can be seen from
the fact that the neutral scalar field has 1 degree of freedom and
the
complex 2, which leads of course to observable differences.
(see Landau,Lifshitz Vol4, § 12 where the factor 1/2 is explained
by noting that real scalar fields have 1/2 as many degrees of freedom).
Contrary to the scalar case Weyl and Majorana fields have the SAME
number of degrees of freedom (namely 2) and [...]
it is clear that their kinetic terms should then be equal.
end if email
\nu_M = \nu_L + C (\bar{\nu_L})^{T}, where the \bar = Dirac conjugation.
One does not need 1/\sqrt{2} as a normalization
factor in the RH side of
this equality.
The kinetic term for a Majorana field is
0.5 \bar{\nu_M} \gamma_{\mu} d/dx_{\mu} \nu_M,
the factor 0.5 in front being very important.
The physical reason for the
factor 0.5 is the same as the reason for which
the same factor appears in
the kinetic term of a REAL scalar field.
As it is not difficult to show, one has (neglecting
a term which
represents a total derivative)
0.5 \bar{\nu_M} \gamma_{\mu} d/dx_{\mu}
\nu_M =
\bar{\nu_L} \gamma_{\mu} d/dx_{\mu}
\nu_L,
and there are no additional factors of \sqrt{2},
which form the basis of
the author's arguments.
In my opinion, the author's article is
just wrong because of this
trivial normalization error that he makes when
he expresses
\nu_M in terms of \nu_L and
C (\bar{\nu_L})^{T}.
end of email
Haber uses the same normalisation as the world
expert in the definition
of the Majorana neutrino in his short write up,
eq.(20).
I quote some references to bolster the derivation
of
the normalisation given in section 2 of my manuscript
Email 2.8.00 from R.Plaga
References confirming the normalisation of the
Majorana
field used by eq.(2) my manuscript.
*******************************************************************

1. B.Kayser, The Physics of Massive neutrinos,
World Scientific,Singapore,1989 p.89:
\nu´=fM_D/M_R F
f= 1/sqrt(2) (\Psi_L+(\Psi_L)^c)
\nu´ is identified on p.90 l.4 as Majorana
neutrino, in the massless
limit (here:M_R > \inf) eq.(6) my manuscript
occurs


2. M.Fukugita,A.Suzuki (eds.)
Physics and Astrophysics of Neutrinos, Springer,1991
Article: "Physics of Neutrinos" by M.Fukugita,T.Yanagida
section 7, eq(8):
"\chi_{+} = 1/{sqrt(2)}(\psi + \psi^c)
where \psi^c is the charge conjugation of \psi"
(my eq(2))


3. T.D.Lee, Particle Physics and Introduction
to Field Theory,
Harwood,Chur,1981,p.54 (Excercise (ii) of section
3):
"Hint for (ii):Define the Majorana field operator
\Psi_M = 1/sqrt(2) (\Psi_L + \Psi_L^c) " (my
eq(6))


4. P.Roman,Theory of Elementary Particles,North
Holland,1960
Chapter 4g, Majorana Theory,p.307. eq 4.37
"\chi = 1/sqrt(2) (\Psi + \Psi^c)"
as definition for Majorana field.
4.39b gives the ususal anticommutation relations
for \chi.(my eq(2))


5. W.Pauli, Relativistic Field Theories, Rev.Mod.Phys.13,203(1941)
p.225,eq.100
v = 1/sqrt(2) (u+C^*u*)
(where u is the usual Dirac spinor and the second
term charge
conjugation in Pauli´s notation).(my eq(2))

end of email
The world expert gave the following references
to bolster
his normalisation factor.
Email 12.7.00 from world expert
For further details see Rev. Mod. Phys. 59 (1987) 671 and the
discussion of the Majorana mass term (subsection C) on pages
681682.
In particular, please note footnote 12 on page 682.
Another useful article might be that of P. Mannheim in Int. J. Theor.
Phys. 23 (1984) 643. I have read it many years ago  all the necessary
normalization factors are rigorously derived, if I remember correctly,
in
this article.
end of email
Email 2.8.00 from R.Plaga
He then cites
>P. Mannheim in Int. J. Theor.
>Phys. 23 (1984) 643 in which
as basic reference.
This reference contains the above definition
of the Majorana
neutrino \nu_M AS WRITTEN BY ME *WITH* THE FACTOR
1/sqrt{2}!!!
equation (19):
\Psi_M= 1/sqrt{2} (\Psi_L^W + (\Psi_L^W)^C)
is identical to equation (6) my manuscript.
Rev. Mod. Phys. 59 (1987) 671 by Bilenky,Petcov
indeed gives the same normalisation as the referee,
but quotes Mannheim as more BASIC Majorana reference
(beginning of Appendix A of Bilenky,Petcov).
end of email
To be absolutely sure that there is no misunderstanding
I asked
P.Mannheim personally, if my normalisation is
wrong.
Email 15.11.00 from R.Plaga
Using standard textbook normalisations and notations
(to be absolutely definite: the ones of BjorkenDrell
vol.2)
everywhere, is the following decomposition
of a Majorana spinor field "\chi" into a
Weyl spinor field \Psi^W_L and its own charge
conjugate (\psi^W_L)^C WRONG?
\chi=[\psi^W_L+(\psi^W_L)^C]/\sqrt 2$)
( ) yes, the decomposition is WRONG
( ) no, this decomposition is not WRONG
end of email
Email 17.11.00 from P.Mannheim
As regards your question, the
relation $\chi=[\psi^W_L+(\psi^W_L)^C]/\sqrt
2$ as given in my Majorana
paper as Eq. (49) is certainly correct.
end of email