The Exact Similarity between the Positron and the Electron Equations in a P and the T Violations ()
1. An Invariant Form of the Dirac Equation
1.1. The Hestenes Form of the Dirac Equation
D. Hestenes has established in [1] a form of the Dirac equation of the electron in a galilean frame
, by using the Clifford algebra 
associated with the Minkowski space
, or Space-Time Algebra (STA) [2] (see Note below). This form eliminates the use of the complex Dirac matrices and spinor and replaces them in STA by a biquaternion
, element of 
even sub-algebra of
, which expresses the wave function of the electron.
The form given to 
is the following
(1)
where:
expresses the invariant probability density;
;
corresponds to the Lorentz rotation such that 
and 
are the velocity and the invariant probability current;
is the Yvon “angle” [3] (independently rediscovered by Hestenes [1]).
The vectors
, are such that the bivectors 
or 
define, multiplied by
, the angular momentum of the electron, in spin “up” or “down” respectively.
Let us consider the Dirac equation
(2)
and the equivalences [1]
(in “up”), 
(in “down”),
.
Denoting
, one obtains the Hestenes form of the Dirac equation
(3)
in “up”, 
in “down”
1.2. An Invariant Form of the Hestenes Equation
Multiplying Equation (2) on the right by 
(
), we obtain [4]
(4)
in “up”, 
in “down”, where the bivectors 
define the infinitesimal rotation of the moving frame
, or proper frame of the electron.
This equation is independent with respect to all galilean frame. The dimension of each term is an energy. In this form the velocity v is associated with 
and gives the equation of the free electron, considered when the potential A is null. So this form appears as the most appropriated among the other invariant forms.
Note: The Clifford algebra 
associated with an euclidean space 
is a real associative algebra, generated by 
and the vectors of
, whose elements are identified to the ones of the Grassmann algebra
. Furthermore this algebra implies the use of the inner products in 
denoted
, 
for products of a p-vector 
by a vector a of 
which correspond to the operation so-called (by the physicists) “contraction on the indices”. The product 
(
) defines the signature 
of
. Its principal properties are:
1) The Clifford product of two elements 
of 
is denoted AB and verifies the fundamental relation
.
2) All element A of is the sum of terms in the form
,
.
3)
.
4) If p vectors 
are orthogonal their Clifford product verifies
.
2. The P and T Violations
2.1. The CPT Transforms
We recall the usual presentation of the passage from the equation of the electron to the one of the positron by means of the CPT transforms that we apply to Equation (4).
1) C (Charge) changes 
into
.
2) P (Parity) changes 
into 
and so
, 
into
,
.
3) T (Time reversion) changes 
into 
and so 
in
.
Equation (4) becomes, after the change of sign of 
and v, associated with the C change,
(5)
However the T transformation seems to imply that the positrons come from the future, contrarily to experiments in particular in the LEP, and these kinds of transformations are to be imperatively interpreted in a different way.
2.2. Theoretical Justification of a P and the T Violations
Note that the changes of sign of 
and 
are made to put the terms containing the spin and the velocity in agreement with the change of sign of the term containing the charge.
So one has not to change by P 
into 
but, instead, to keep 
unchanged and to change the sign of the term of the equation containing the spin, and so not to change the spin.
In the same way one has not to change by the T transform v into 
but, instead, to keep v unchanged and to change the sign of the term of the equation containing v, and so not to change the velocity v.
So the positron velocity may be clearly considered in direction to the future and the strange interpretation of T by Stückelberg and Feymann is to be abandoned.
Equation (5) is changed into
(6)
that is
(7)
This equation is exactly the same as the equation of the electron, with the condition that, the charge 
being changed into
, one has to change the electromagnetic potential A acting on the electron in 
acting on the positron:
.
2.3. The CP-Takabayasi Transform
In [5], Equation (10.3b), T. Takabayasi avoids the change of v into 
by the following transform:
3)' The angle 
is changed into 
which gives
, v remaining unchanged.
A difference with what precedes is the fact the sign minus is placed directly in front of the term containing the mass and so that T may be removed from CPT to be replaced by the above transform on the “angle”
.
About the spin the addition of 
to the “angle”
, which in fact concerns not the vectors of M but the bivectors (see for example [6], p. 37), may change a bivector 
into
. This addition of 
to 
is coherent with P which nevertheless needs to be violated as it is explained above.
Note that Takabayasi had assigned the sign minus obtained by his transform to the mass, not to the term containing the mass, and deduced a wrong hypothesis of a negative mass for the positron (that we have unluckily repeated in [6], p. 37).
3. Conclusions
About the behaviour of the positron in the LEP experiments we have sent to Michel Spiro, President of Scientific Board of the LEP, then of the LHC, the following message:
“The form of the positron equation is the same as the electron equation, except that the charge is positive. So one can expect that, in similar experiences, the behaviour of the positron ought to be the same as the one of the electron to the condition that the sign of the electromagnetic potential A is changed.”
We have received the following answer (which does not engage its author on the validity of the above calculations) from Dr. Rolf Landua, Head of Education and Public Outreach (CERN):
“I can confirm that this statement is correct. It has also undergone many experimental tests, e.g. by testing the equality of the magnetic moment of the electron and the positron. This measurement is hitherto the most precise comparison of a physical property of particle and antiparticle.”