1. Dirac Equation with Real Wave Functions
The relativistic Dirac Equation, describing a free Fermion of mass m is given by [1] [2]:
(1)
One may separate the Dirac operator
and the complex wave function
into their real and imaginary parts [3] [4]
With
all real components.
As will be shown, these 4 components represent two fermions and two anti-fermions. Each pair is the source for two opposing spin states.
After some work and boosting to a system moving with the particle along the +x axis (
),
the Dirac equations take the form:
(2)
(3)
(4)
(5)
This shows, that,
is coupled with
and
is coupled with
. As will be shown later, linear combinations of these represent spin-up and spin-down fermion and anti-fermion.
2. Solution with 8 Real Components
Each
is a 2-vector with real components. Thus, the Dirac Equation is actually 8 equations of real components with coupled pairs
, and
.
Applying a time derivative to the first equation of each pair and using the second component of each pair, leads to:
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
These 8 real components equations demonstrate the existence of coupled pairs: (
), (
), (
) and (
).
These equations show, that every fermion is composed of 4 real fields which are coupled in a yet to be explored manner.
The solutions are described in the following:
Here
, where
is the x component of the momentum.
In order to better understand the nature of the Equations (6)-(13), one can boost the coordinates to the fermion’s rest frame along a random axis, say the x-axis.
When boosted to the fermion’s rest frame, where
, one can omit all
terms (
).
Equations (2)-(5) then read (for all four wave functions):
(14)
And likewise for all 8 components
and
:
(15)
(16)
Solving this equation by setting
or
shows that all components of the fermion at its rest frame, are oscillating at a rate given by
.
Inserting the values in MKS units one obtains, for all fermions
.
For an electron
MeV/c2 and thus
GHz. (=0.512 MeV). For a proton,
MeV/c2 and
GHz and for the τ quark
GeV/c2 and
GHz (Table 1).
Table 1. Masses and equivalent electromagnetic range Properties of electron, proton and tau-quark.
particle |
|
EM range |
Mass |
electron |
7.77 × 1011 GHz |
Soft X-ray |
0.512 MeV |
Proton |
14.28 × 1014 GHz |
Gamma |
939 MeV |
τ Quark |
2.63 × 1017 GHz |
Gamma |
173 GeV |
3. A Model for the Fermion Internal Structure
Previously [3] it was shown that the description of Quantum Mechanics can be done by use of real fields and operators. The Schrodinger equation was then shown to be describing a double string model with particle exchange between the two strings. It was also shown, that all known quantum mechanical unique phenomena, such as interference and entanglement, can be explained based on real fields approach [3] [4].
The same argumentation was applied to Dirac equation, demonstrating a double string coupling.
Dirac equation shows us, that the internal frequency of each component of the fermion equals the rest mass of that fermion.
Even though a fermion is assumed to be made of 2 components, U and D, each one inherits its oscillation frequency
from the same total fermion mass m, and not from U's or D's individual masses (if any).
As shown earlier, each
is a 2-vector with real components.
We will assume next that for a reference frame near static with respect to the fermion
(17)
(18)
even in the non-boosted system. Same is true for
(19)
(20)
Therefore,
(21)
(22)
(23)
(24)
This makes the components
coupled to
and same for
to
.
One may write
(25)
(26)
(27)
(28)
where the integration in x is over the negligible size of the fermion, and where
represents the average amplitude over that extent.
This allows us to neglect the extra terms on the r.h.s of the equations and obtain, even in a non-stationary frame of reference (relative to the fermion):
(29)
(30)
(31)
(32)
These equations indicate to a coupling by exchange mechanism between
and
. When looking at the change in amplitude of
we get:
(33)
Internal fluctuations of
and
at rate
indicate an energy exchange between the two components. Same argument is valid for the internal fluctuations of
and
So, during the exchange,
gains
in amplitude per second if
is negative, and loses amplitude if
is positive.
Recall that they oscillate at a rate of 1.24 × 1011 GHz for an electron. This is a picture of two adjacent strings, oscillating in anti-phase, at said frequency (see Figure 1). This ant-phase is a must, in order to keep the particle's momentum zero in the non-x (perpendicular) direction.
We assume that at most, the whole mass participates in the kinetic energy transfer mechanism. This assumption puts an upper limit on
, the kinetic energy transfer.
An upper limit estimate of the kinetic energy transfer between
and
in the electron (change in amplitude per second is velocity) will then be given by:
If
then the time average of
is
.
So, the average kinetic energy transfer between the coupled strings in the electron,
is less than 2.73 × 10−25 eV/sec. For a proton the upper limit is 9.2 × 10−19 eV/sec and 3.12 × 10−14 eV/sec for the tau quark. All cases have values far below our detection capabilities.
Same arguments hold for the kinetic energy transfer between
and
.
Figure 1. Energy transfer model based on anti-phase oscillations between the two components
and
of a fermion.
The interaction may be due to some yet unknown particles, where possible candidates may well be those suggested by Harrari [5] and Hubsch [6].
4. Conclusions
All fermions are made of pairs of coupled fields (strings) with an internal tension related to mutual attraction forces, affected by Planck’s constant. The solution to Dirac equation gives rise to four, real, 2-vector fields
,
,
, and
where (
) are coupled via linear combinations to yield spin-up and spin-down fermions. Likewise, (
) are coupled via linear combinations to represent spin-up and spin-down anti-fermions.
An investigation of the free fermion internal frequency is discussed, hinting to an exchange interaction between the two components of which a fermion is made of. An upper limit estimate is given to the strength of this interaction.