Conservation Law: Still Alive!?
V. Dvoeglazov firstname.lastname@example.org
Titular C of the Escuela de Fisica of the UAZ
This is a brief note for “pedestrians,” which is devoted to the explanation of recent ideas of G. Ziino, A. Barut, D. V. Ahluwalia, and myself in the theory of neutral particles.
A number of articles
discussing the parity violation effect is enormous. In yet another good popular article of the series “Fisica y
Historia” Dr. Felix Valdez (Guanajuato, Mexico) presented a wide panorama for
a general auditorium of the questions connected with the concept of the parity
and its non-conservation, expressing the ideas that have prevailed in the brains
of physicists in the last thirty years. However, another alternative already
exists to the conclusion that the author of  defends on the basis of the
results of the experiment of Yang and Lee . I am convinced that the
interpretation of that experiment, by way of physical laws is not being
invariant with respect to the parity operation, can be misleading. Yet we
don’t know much about “the world through the looking glass.”
Based on articles that have
been published over the last few years in scientific journals by Profs. Ziino
, Barut , Ahluwalia  and myself [6,
7, 8], I would like to present a framework that rejects the interpretation
previously presented. Taking into
account the goals of the journals (“Journal of Theoretics” and “Revista de
Fisica — UNISON, Mexico) where I submitted two versions of the paper (English
and Spanish), the propagation of physics for the young generation and making
progress in the unification of physics through theoretical ideas, I will try to
make the presentation as accessible as I am able, (on the B. Sc. and M. Sc.
level) but mathematically rigorous.
Beforehand, let us remind ourselves of the Dirac equation, which describes charged particles in the relativistic frameworks. It has been known for every physicist:
In this formulae gm
are 4 x 4 matrices that give the most
simple representation of hypercomplex numbers from the mathematical viewpoint; m
is the mass of the particles with the intrinsic angular momentum ħ/2,
the electron and the positron; ψ(x)
is the field (wave) function, i.e. we use the well-accustomed notation of
practically all of the books on quantum theories. The Einstein rule is implied, namely, the summation in the
indices which are repeated (m = 0 . . . 3). From the above equation after one introduces the
interaction with a 4-vector potential in it, the quantum-electrodynamic
processes (i. e. the processes with elementary particles) can be calculated on
using the perturbation theory.
Since long ago Ziino for the
first time considered (1989, ref. ) an equation, which is similar to the one
that Dirac deduced, but with the opposite sign in the mass term. The latter was
proposed by Prof. M. Markov , the author of the well-known idea of friedmons.
If the equations are considered separately, the difference in the sign
doesn’t have any physical significance, which is the consequence of the CPT
theorem in the theories for charged particles, the theorem dealing with discrete
symmetries of space inversion, charge conjugation, and time reversal. This also
can be seen in a simple way, by means of verification of the invariance of the
fermion current in the quantum electrodynamics, with respect to transformation
with the g5
However, those two equations
can provide a theoretical construction of neutral particles, such as the
neutrino when they are considered jointly.
Ziino and Barut used the matrix g5
as the matrix for charge conjugation. This means that the conjugation for all of
the system, which consists of a fermion and a 4-vector potential. The Dirac
function and its conjugate function (of an antifermion) obey the Dirac equations
with the opposite signs for the mass term. Then, the self/anti-self charge
conjugate states are automatically chiral states
Nevertheless, they continue
being massive. Ziino and
Barut discovered that in the framework of this formulation:
· The problem of
the missing right-handed neutrino (anti-clockwise if one sees a particle that is
moving towards the observer) is absent. The image of the β
→ p +
, has to be identified with the process that indeed occurs
+ ve .
· The parity
operation is divided naturally to the external parity corresponding to the
and the internal parity (which are defined in accordance with the selection of
the Lorentz group representation and the dynamic properties of the particle).
· The Lagrangian
in the theory with massive neutrinos is invariant with respect to both parts of
the parity operation. This shows a great divergence from the model of Glashow,
Salam, and Weinberg (GSW). However,
in a particular case, one could describe the V
electroweak phenomenology. It is a consequence of this non-invariance
of the weak processes with respect to the external parity, that one cannot be
considered as the true space-inversion operation for the (
, 0) ⊕ (0,
) Lorentz group representation.
· In the massless
limit, one recovers the Weyl theory for two-component neutrinos.
In refs. [5, 6, 7, 8] other
types of bispinors were used, in fact the Majorana spinors , which differ
from (although they are connected with) the Dirac ones.
Authors of those works arrived at the same conclusion as their
predecessors and found other interesting theoretical features which were missed
in refs. [3, 4]. For example, it was demonstrated that one can build the field
operator which would contain four Majorana-like states and would preserve the parity as the
theory which follows from the Barut-Ziino Lagrangian.I
New equations (cf. with ) in the coordinate space are:
= 0 ,
= 0 ,
= 0 ,
= 0 ,
where the momentum-space
self (S) /anti-self (A)
charge conjugate bispinors (r
and λ) are defined from the beginning.
The quantum states in this formulation are not the eigenstates of the
parity operator as we have the property λS
respect to this operation, nor are they the eigenstates of the helicity
operator (as opposed to the Dirac bispinors which describe an electron and a
positron)II . They are bi-ortonormal ones from the
mathematical point of view. Also,
considering the gauge interactionsIII one could see that
although the neutrino (which can be massive) is a neutral particle with
respect to the electric charge operator, it is not a neutral particle with
respect to another operator of the charge, the so-called chiral charge Qch.IV
These two operators of the charge anticommute what was mentioned by Barut
and Ziino. Thus we are induced to
accept that the electron and the neutrino are but different states of the same
object; when in the free state they cannot interfere each other (compare with
the model of CSW where also a left-handed electron and a left-handed neutrino
form a doublet).
From the group-theoretical
viewpoint this construct represents a type of the Lorentz-invariant model (of
the Wigner type). Wigner
discovered that the inversion sub-group could be formed in several ways, as
either a 4-group, a dihedral group, or as a C2
x C4 , and the discrete symmetry operators in the Fock
quantum space could commute or anticommute.
These ideas are, of course,
are related to the concept of the so-called “mirror” particles. According to
R. Volkas, R. Foot et al., Z.
Berezhiani and R. Mohapatra, and Z. Silagadze  there may exist a mirror
world “with the same microphysics as our own one but with opposite P-
From a philosophical
viewpoint, this construct is as functional (it allows us to explain the old
enigmas of the standard model) as beautiful (mathematically), as well as
rational since it was built on the basis of the use of a minimal number of
ingredients. That coincides the great Russian encyclopedist, I. A. Efremov’s
view (he was also the science fiction writer and had a second doctorateIV
in the geological sciences): “what is beautiful is functional, what is
functional is beautiful” (I. A. Efremov,”Leaf of Razor”).
conclusion: the idea of a parity
violation in the electroweak interactions, perhaps will be left only as a
historical memory in the same manner as flageston idea; an idea which the great
Russian scientist M. V. Lomonosov denied and experimentally disproved in the
XVIII century. Likewise the idea of
a violation of the energy conservation as expressed by Dirac! Was displaced by
W. Pauli’s proposal of a new particle: the
neutrino. Who knows what new points of view will help in
resolving the apparent contradictions between the Copenhagen School and those
such as L. de Broglie, A. Einstein, N. Rosen, B. Podol’sky, M. Sachs, and
others who have criticized it.
Although in a theoretical
sense, I believe that the idea of invariance of the operators that corresponds
to the dynamic variables with respect to the Parity operation is a very
attractive theoretical idea in the sense that the parity violation is
“fictitious” as we observe only a part of the relevant processes.
But in philosophiae naturalis, experimental validation will always have
the last word.
below, please, remember that the field operators contain two sets of quantum
states, for a particle and its antiparticle.
helicity operator is connected with the concept of the spin, a discrete
phaseless variable that is defined for the field functions in each
representation of the Lorentz group. The physical sense of that operator is that
it defines projections of the intrinsic angular momentum to the direction of the
The reader can find many articles where the gauge invariance is explained
in detail on the accessible level, e.g., J. Enrique Barradas, Revista de Fisica
— UNISON, No. 20 (1995) p. 9.
learned recently that a similar idea was expressed by Prof. R. Marahak some time
In Russia there are two doctorate degrees.
Generally, the second degree is awarded after the defense of one’s thesis if
there is a proposed new direction of research.
 J. F. Valdez, Revista de Fisica—UNISON,
No. 20, Sept. 1995, p. 32.
 T. F. Lee and C. N. Yang, Phys.
Rev. 104 (1956) 254.
 G. Ziino, Ann. Fond. L. de Brogue
14 (1989) 427; ibid 16 (1991) 343; Int. J.
Mod. Phys. A, in press; also see E. Recami and G. Ziino, Nuovo Cim. 33A
 G. Ziino and A. Barut, Mod.
Phys. Lett. A8 (1993) 1011.
 D. V. Ahluwalia, M. B. Johnson,
T. Goldman, Mod. Phys. Lett. A9 (1994)
439; D. V. Ahiuwalia, Int. J. Mod. Phys. A11 (1996) 1855.
 V. V. Dvoeglazov, Rev.
Mex. Fis. Suppl. (Proc. XVII Simposium of Nuclear Physics. Oaxtepec, Mexico,
Jan. 4-7, 1995) 41 Suppl. 1. (1995) 159.
 V. V. Dvoeglazov, Int.
J. Theor. Phys. 34 (1995) 2467.
18] V. V. Dvoeglazov, Nuovo
Cimento 108A (1995) 1467.
 M. A. Markov, ZhETF 7 (1937) 579,
603; JINR Communications D1345, Dubna, 1963.
 E. Majorana, Nuovo Cim. 14 (1937) 171.
 E. P. Wigner, in “Group Theoretical
Concepts and Methods in Elementary Particle Physics” (Lectures of the
Istanbul Summer School of Theoretical Physics, 1962, Ed. F.Gursey), Gordon &
Breach, 1965, p.37.
 Z. Silagadze, Phys. Atom. Nucl. 55 (1992) 392; ibid 60 (1997) 272; hep-ph /9908208
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