TRANSLUMINAL ENERGY QUANTUM MODELS OF THE PHOTON AND THE ELECTRON

Richard Gauthier

Santa Rosa Junior College, 1501 Mendocino Ave., Santa Rosa CA 95401, U.S.A. E-mail: richgauthier@gmail.com www.superluminalquantum.org

A photon is modeled by an uncharged transluminal energy quantum (TEQ) moving at 2c along an open 45-degree helical trajectory with radius (where is the helical pitch or wavelength of the photon). A transluminal spatial model of an electron is

composed of a charged pointlike transluminal energy quantum circulating at an extremely high frequency of hz in a closed, double-looped helical trajectory whose helical pitch or wavelength is one Compton wavelength . The transluminal energy quantum has energy and momentum but not rest mass, so its speed is not limited by c. The TEQ’s speed is superluminal 57% of the time and subluminal 43% of the time, passing through c twice in each trajectory cycle. The TEQ’s maximum speed in the electron’s rest frame is 2.516c and its minimum speed is c / 2 or .707c . The electron model’s helical trajectory parameters are selected to produce the electron’s spin ! / 2 and approximate (without small QED corrections) magnetic moment e! / 2m (the Bohr magneton) as well as its Dirac equation-related “jittery motion” (zitterbewegung) angular frequency 2mc

2 / ! , amplitude ! / 2mc and internal speed c. The two possible helicities of the electron model correspond to the electron and the positron. With these models, an electron is like a closed circulating photon. The electron’s inertia is proposed to be related to the electron model’s circulating internal momentum mc.

Keywords: Transluminal; Superluminal; Photon; Electron; Zitterbewegung; Inertia; Model  1. Introduction

Dirac’s1,2 theory of the relativistic electron did not include a model of the electron itself, and assumed the electron was a point-like particle. Schrödinger3 analyzed the results of the Dirac equation for a free electron, and described an extremely high-frequency zitterbewegung or jittery motion, which appears to be due to the interference between positive and negative energy terms in the solution. Barut and Bracken4 analyzed Schrödinger’s Zitterbewegung results and proposed a spatial description of the electron where the zitterbewegung would produce the electron’s spin as the orbital angular momentum of the electron’s internal system, while the electron’s rest mass would be the electron’s internal energy in its rest frame. Barut and Thacker5 generalized Barut and Bracken’s analysis of the internal geometry of the Dirac electron to a proper-time relativistic formalism. Hestenes6,7,8,9 reformulated the Dirac equation through a mathematical approach (Clifford algebra) that brings out a geometrical trajectory approach to understanding zitterbewegung and to modeling the electron, such as identifying the phase of the Dirac spinor with the spatial angular momentum of the electron. A trajectory approach to the Dirac theory has also been utilized by Bohm and Hiley10, who describe the electron’s spin angular momentum and its magnetic moment as due to the

circulatory motion of a point-like electron. However, none of the above work in modeling the electron’s jittery motion has a superluminal aspect.

The photon has previously been modeled geometrically with several approaches, with results quantitatively similar to those in the present superluminal quantum model of the photon. Ashworth11 used a classical model of the photon to obtain a radius

for the photon and a superluminal internal speed of 2 c, the same quantitative results for the photon as in the present paper. Kobe12 obtained the same quantitative result for the photon radius as in the present paper, based on a helical approach and quantum mechanical considerations. Sivasubramanian et al.13, using a model of the photon that is helical and explicitly internally superluminal, independently arrived at the same radius ! / 2" for a photon as in the present paper.

The objectives of the present paper are: to 1) present a simply derived transluminal helical quantum model for the photon, 2) present a related transluminal closed helical quantum model of the electron having experimental and theoretical features of the Dirac equation’s electron, 3) relate the electron and photon models to the Heisenberg uncertainty relations, 4) propose a new approach to understanding the electron’s inertia, and 5) suggest approaches for testing the proposed models of the transluminal energy

/ 2R λ π= λ202.5 10×

/h mc

/ 2λ π

quantum model of the photon and the transluminal energy quantum model of the electron.

2. A Transluminal Energy Quantum Approach to Modeling the Electron and the Photon

The present approach is a unified approach to modeling both the electron and the photon with transluminal helical trajectories. The electron model has several features of the Dirac electron’s zitterbewegung. Point-like entities are postulated called transluminal energy quanta (TEQs), which are distinct from electrons and photons themselves and which compose an electron or a photon. These TEQs (which may be subluminal, luminal or superluminal) have an energy E, with its associated frequency ! related by E = h! , an angular frequency ! = 2"# , an instantaneous momentum

!P

with its associated wavelength , and an electric charge (in the case of the electron). One TEQ forms a photon or an electron. An electron’s TEQ is found to alternate between subluminality and superluminality passing through the speed of light, while a photon’s TEQ is always superluminal. TEQs move in helical trajectories, which may be open (for a photon) or closed (for an electron). Movement of one of these TEQs along its trajectory produces an electron or a photon. The type of helical trajectory and the associated charge or lack of charge determines which particle is produced. More details about the models presented below are provided in Gauthier14,15.

3. The Transluminal Energy Quantum Model of the Photon

A photon is modeled as a TEQ traveling along an open helical trajectory of radius R and pitch (wavelength) . The trajectory makes an angle with the forward direction. In this helical trajectory, R, ! and ! are related geometrically by . By incorporating into the model the photon’s experimentally known linear momentum and the photon’s experimentally known angular momentum (spin) s = ! , a second relationship is found:

. Combining these two relationships containing gives . This result, combined with the photon’s experimentally known energy relationship where is the photon quantum’s frequency, leads to the photon model.

The photon model has the following properties:

1) The forward angle ! of the photon TEQ’s helical trajectory is 45o .

2) The radius of the photon TEQ’s helical trajectory is

3) The speed of the photon’s TEQ is c 2 = 1.414.. c along its helical trajectory.

Using these results, for a right-handed photon traveling in the +z direction, the equations for the trajectory of the transluminal quantum (neglecting a possible phase factor) are:

( ) cos( ) ,2

( ) sin( ) ,2

( ) ,

x t t

y t t

z t ct

λ ωπλ ωπ

=

=

=

(1)

where is the angular frequency of the photon, f is the photon’s frequency in cycles per second and ! is the photon’s wavelength. In the transluminal photon model, ! is the pitch of the helix, i.e. the distance along the helical axis corresponding to one rotation of the superluminal quantum along its helical trajectory.

Similarly, for this right-handed photon, the equations for the components of the momentum of the superluminal quantum along its trajectory are:

(2)

The x and y components of momentum are 90 degrees out of phase with the x and y position values.

4. The Transluminal Energy Quantum Model of the Electron

If the open helical trajectory of the photon model is converted into a closed, double-looped helical trajectory, the TEQ is given an electric charge , and several helical parameters corresponding to an electron’s experimental and theoretical properties are set, we get the transluminal energy quantum model of the electron. Besides having the electron’s experimental spin value and the magnetic moment of the Dirac electron, the transluminal energy quantum model of the

λ

λθ

tan 2 /Rθ π λ=

/p h λ=

tan / 2 Rθ λ π=tanθ 2 Rλ π=

E hν= ν

/ 2R λ π=

! = 2" f = 2"c / #

( ) sin( ) ,

( ) cos( ) .

( ) .

x

y

z

hp t t

hp t t

hp t

ωλ

ωλ

λ

= −

=

=

e−

electron, described below, quantitatively embodies the electron’s zitterbewegung. Zitterbewegung refers to the Dirac equation’s predicted rapid oscillatory motion of a free electron that adds to its center-of-mass motion. No size or spatial structure of the electron has so far been observed experimentally. High-energy electron scattering experiments by Bender et al.16 have put an upper value on the electron’s size at about m. Yet the Schrödinger3 zitterbewegung results suggest that the electron’s rapid oscillatory motion has an amplitude length of Rzitt = ! / 2mc = 1.9 !10

"13m and an angular frequency of ! zitt = 2mc

2 / ! = 1.6 "1021 / sec , twice the angular frequency ! o = mc

2 / ! of a photon whose energy is that contained within the rest mass of an electron. Furthermore, in the Dirac equation solution the electron’s instantaneous speed (its eigenvalue solution) is c, although experimentally observed electron speeds are always less than c. An acceptable spatio-temporal model of the electron would presumably contain these zitterbewegung properties.

In the TEQ model of the electron, the electron is composed of a negatively charged point-like TEQ moving along a closed, double-looped helical trajectory in the electron model’s rest frame, that is, the frame where the TEQ’s trajectory closes on itself. The TEQ electron model structurally resembles a closed, double-looped circulating TEQ photon model having angular frequency ! o = mc

2 / ! and wavelength !C = h /mc (the Compton wavelength). The electron model’s TEQ moves in a closed double-looped helical trajectory having a circular axis of circumference !C / 2 . The radius of the TEQ trajectory’s closed double-looped circular axis is set to be R0 =

12 ! / mc = 1.9!10"13m .

This gives the TEQ model the electron’s spin

s = R0 ! pC = (! / 2mc)!mc = ! / 2 , where pC = mc is the linear Compton momentum of the circulating TEQ photon-like object composing the TEQ electron model. The TEQ trajectory’s helical radius is set to be 2R0 . This gives the TEQ electron model a magnetic moment whose z-component is equal in magnitude to the Bohr magneton (See Appendix). After following its helical trajectory around this circular axis once, the electron TEQ’s trajectory is 180o out of phase and doesn’t close on itself. But after the TEQ traverses its helical trajectory around the circular axis a second time, the TEQ’s trajectory is back in phase with itself and closes upon itself. The total longitudinal distance along its

circular axis that the circulating TEQ has traveled before its trajectory closes is !C .

In its rest frame, the electron’s TEQ carries energy

E = !! o = mc2 , while the TEQ carries the electron’s

negative charge –e. The coordinates for the closed, double-looped

helical spatial trajectory for the TEQ in the electron model can be expressed in rectangular coordinates by:

(3)

where Ro = ! / 2mc and ! o = mc2 / ! . These equations

correspond to a left-handed photon-like object of wavelength !C , circulating counterclockwise (as seen above from the +z axis) in a closed double loop. An image of the electron TEQ’s trajectory described by equation (3) and the trajectory of a mirror-image TEQ positron are shown in Figure 1.

(a) Trajectory of the TEQ electron model.

              (b) Trajectory of the TEQ positron model Fig. 1. Trajectories of the transluminal energy quantum of (a) the electron model and (b) the positron model, (the latter showing a cut-away mathematical spindle-torus surface on which the two trajectories are located. (The circle in the x-y plane in (a) is the axis of the TEQ’s closed double-looped helical trajectory.)

1810−

0 0 0

0 0 0

0 0

( ) (1 2 cos( ))cos(2 ) ,

( ) (1 2 cos( ))sin(2 ) ,

( ) 2 sin( ) ,

x t R t t

y t R t t

z t R t

ω ω

ω ω

ω

= +

= +

=

The velocity components of the electron model’s transluminal energy quantum are obtained by differentiating the position coordinates of the TEQ in equation (3) with respect to time, giving:

 

 

 

From equation (4) it is found that the maximum speed of the electron’s TEQ is 2.516c, while its minimum speed is c / 2 = .707c. A graph of the speed of the electron’s TEQ versus the TEQ’s angle of rotation in the x-y plane as the electron’s TEQ circulates in its closed double-looped helical trajectory is shown in Figure 2. The TEQ completes each trajectory cycle in 12.56 or 4! radians.

Fig. 2. The speed of the electron model’s TEQ along its double-looped helical trajectory as a function of the angle of rotation of the TEQ in the x-y plane.

From equation (4) the circulating transluminal energy quantum spends approximately 57% (more precisely 56.64%) of its time traveling superluminally along its trajectory and 43% (more precisely 43.36%) of its time traveling subluminally. The TEQ passes twice through the speed of light c while completing one closed helical trajectory. This passage of the TEQ from superluminal speeds through c to subluminal speeds and back again to superluminal speeds is not a problem

from a relativistic perspective. This is because it is the TEQ’s electric charge -e that is moving at these speeds and not the average center of mass/energy of the TEQ electron model, which remains at rest in the electron model’s rest frame. The TEQ electron model as a whole would move at a speed less than c.

5. Similarities Between the Dirac Equation’s Free Electron Solution and the Transluminal Energy Quantum Model of the Electron

The transluminal energy quantum model of the electron shares a number of quantitative properties with the Dirac equation’s electron with zitterbewegung: 1) The zitterbewegung internal frequency of

! zitt = 2mc2 / ! = 2! o .

2) The zitterbewegung radius R0 = ! / 2mc = Rzitt . Using the equations (3) for x(t), y(t) and z(t) of the TEQ electron model, the x, y and z rms (root mean square) values, which are the values of in the Heisenberg uncertainty relation, are all calculated to be

R0 = ! / 2mc , where is the radius of the closed helical axis of the TEQ electron model. This is also the value of the amplitude of the Dirac electron’s jittery motion found by Schrödinger3. These rms results for the electron model are only obtained when the radius of the TEQ’s double-looped helix is as in equation

(3). This value is the helical radius required to give the electron model’s z-component of its magnetic moment a magnitude equal to the Dirac equation’s electron magnetic moment magnitude of one Bohr magneton MB . (See Appendix.) 3) The Dirac electron’s speed-of-light result is also contained in the TEQ electron model. The longitudinal speed of the TEQ along the circular helical axis is c, although the TEQ’s actual speed is transluminal. 4) The prediction of the electron’s antiparticle the positron, having opposite helicity to the electron model’s helicity. 5) The calculated spin of the electron, which comes out correctly in the TEQ model due to the double-looping of the TEQ’s helical trajectory and its circular axis. 6) The calculated Dirac magnetic moment of the electron Mz = !MB ( MB is the Bohr magneton). In the TEQ electron model, and My = !.25MB ,

which differs from the Dirac result. But it is the z-component of an electron’s magnetic moment that is experimentally measured.

vx (t) = !c[(1+ 2 cos" ot)sin2" ot +22cos2" ot sin" ot],

vy (t) = c[(1+ 2 cos! ot)cos2! ot "22sin2! ot sin! ot],

vz (t) = c2

2cos! ot . (4)

, and x y zΔ Δ Δ

0R

2R02R0

0xM =

7) The electron’s motion is the sum of its center-of-mass motion and its zitterbewegung . 8) The non-conservation of linear momentum in the zitterbewegung of a free electron, a result first pointed out for the Dirac electron by de Broglie17.

6. The Heisenberg Uncertainty Principle and the Transluminal Energy Quantum Models of the Photon and the Electron

With the transluminal energy quantum model of the photon, the TEQ would be the particle that is actually detected when a single photon is detected in an experiment. Suppose a TEQ photon is traveling in the +z or longitudinal direction. Because of the TEQ photon’s varying transverse x and y position and momentum components as it moves along its open helical trajectory, a range of values of these x and y components would be detected when various photons traveling in the +z direction are measured successively.

A remarkable aspect of the TEQ model of the photon is that the TEQ’s transverse position and momentum variabilities are found to quantitatively match those in the Heisenberg uncertainty relation. This relation says that there is a fundamental limitation on the accuracy of simultaneously measuring two related physical properties, such as the corresponding position and momentum components of an elementary particle or other physical object. Greater accuracy in measuring one of the two properties entails a corresponding lesser accuracy in measuring the corresponding property. The Heisenberg uncertainty relationship for the x coordinate of a particle is stated precisely as !x!px " h / 4# , where is the standard error or rms value (the square root of the statistical variance) in measuring the position of the particle along the x direction, is the standard error or rms value in measuring the particle’s momentum along the same x dimension, and h is Planck’s constant. How does the Heisenberg uncertainty relation apply to detecting a photon in the TEQ photon model? Using equation (1) for transverse x-position of the TEQ and equation (2) for the transverse x-momentum of the TEQ in the photon model, the rms value or for is found to be:

(4)

while the rms value for is found to be:

(5)

Multiplying these rms values and for the TEQ model of the photon gives:

(6)

Comparing this result with the Heisenberg uncertainty relation:

(7)

we see that the uncertainty product of the transverse or x components of position and momentum for the superluminal quantum in the photon model is exactly the minimum value allowed by the Heisenberg uncertainly relation. The same quantitative results are found for and . For the TEQ photon model:

(8)

Any real photon will have a finite value of uncertainty in the measured coordinates of both its position and its momentum. In quantum mechanics, a photon, until it is detected, is described by a mathematical superposition of position states or their corresponding momentum states, each corresponding to a particular wave function with a particular amplitude, frequency and phase. This total quantum wave function describing the photon is then related to the probability of detecting the photon in the regions where the total wave function is non-zero. The transluminal energy quantum model of the photon seems to be consistent with the quantum mechanical interpretation of the photon and the Heisenberg uncertainty principle.

Similarly, the electron is modeled as a helical photon-like object moving in a double-looped circle at the zitterbewegung angular frequency ! zitt = 2mc

2 / ! with a forward velocity c and Compton momentum pC = h / !C = mc (where is the Compton

wavelength h /mc of the photon-like object composing the electron.) This circle (see Figure 1) is the circular axis (of radius Rzitt = ! / 2mc ) of the TEQ electron model’s helix. The rms values for position in the electron model in the x, y and z directions all give

Rrms = ! / 2mc as mentioned earlier. Combining this rms position result with the calculated rms value mc / 2 for the circulating momentum mc in the x and y directions gives:

!x!px = (! / 2mc)(mc / 2 ) = .707(! / 2) = .707 h

4" (9)

xΔ

xpΔ

xΔ ( )x t1 ,

22x λ

πΔ =

xpΔ ( )xp t1 .2xhpλ

Δ =

xΔ xpΔ

1 1( )( ) .2 42 2x

h hx p λπ λ π

Δ Δ = =

,4xhx pπ

Δ Δ ≥

yΔ ypΔ

.4yhy pπ

Δ Δ =

Cλ

!y!py = (! / 2mc)(mc / 2 ) = .707(! / 2) = .707 h

4" (10)

The relations in (9) and (10) for the electron model contain a value .707 times the minimum value found in the Heisenberg uncertainty relation. Therefore these position/momentum relations for the electron model would not be experimentally detectable in principle, according to the Heisenberg uncertainty relation. Furthermore, since these relations are not detectable in principle, the TEQ electron model would not violate conservation of linear momentum, even though its internal momentum vector rotates at the zitterbewegung angular frequency . Yet the electron model has the electron’s spin value

s = Rzitt p = (! / 2mc)(mc) = ! / 2 , which is detectable. This rotating internal Compton momentum pC = mc could give rise to the TEQ electron’s rest mass m, and therefore the electron’s inertia.

7. The Electron Model and Inertia

The TEQ electron model may provide a new approach to understanding the nature of inertia. The electron is modeled as a double-looping circulating photon-like object having an internal Compton wavelength

and therefore a Compton momentum . This internal Compton momentum is

rotating at the zitterbewegung angular frequency

! zitt = 2mc2 / ! = 1.6 "1021 / sec . The well-known

relativistic equation relating the total energy E of an electron to its linear momentum p and its mass m is

. In the

TEQ electron model, an electron with a ‘rest mass’ m is never internally at rest. The TEQ electron model has a rotating internal linear momentum which is mathematically on an equal footing with the electron’s three external linear momentum components , and . (It is the rapidly rotating internal linear momentum pC = mc in the TEQ electron model that gives the electron its spin ! / 2. ) The total relativistic energy E of the TEQ electron model is then given by

.

It may be that the rotation of an electron’s internal Compton momentum pC = mc at the zitterbewegung frequency ! zitt = 2mc

2 / ! = 1.6 "1021 / sec is what gives an electron its inertia, that is, its resistance to

being accelerated by an applied external force. The inertia of an electron (as measured by its mass m) is then related directly to its internally rotating Compton momentum pC = mc and only indirectly to the electron’s “rest energy” (when

). Momentum is often described as

‘inertia in motion’. With the TEQ electron model this can now be turned around: inertia is ‘momentum at rest’, where the ‘at rest’ momentum is pC = mc .

8. Testing the Photon and the Electron Models

Since equations (6) and (9) show that the variation in the transverse position and momentum components of the TEQ photon model is at the exact limit of the Heisenberg uncertainty relation, it appears that the photon model can at least in theory be tested by measuring a photon’s transverse position and momentum. Knowledge of the phase of the photon’s TEQ in equations (1) and (2) would permit a theoretical specification of the TEQ’s instantaneous position and momentum. Perhaps such phase relations could be tested experimentally using two-photon coincidence counts as proposed by Sivasubramanian13. Another approach to testing the proposed ! / 2 radius of the photon model is by analyzing the cutoff frequencies for microwaves transmitted in waveguides of different sizes, as did Ashworth11. The prediction is that a waveguide will not transmit microwaves as well, as indicated by the waveguide cutoff frequency, if the diameter of the waveguide is less than the diameter of a microwave’s photon. Although the cutoff frequencies of waveguides are related to wavelength for rectangular waveguides, the possible relation of cutoff frequencies to ! / 2" is not so straightforward for other geometries and could be researched further, perhaps using waveguides with non-standard geometries.

There could be tests of the TEQ electron model’s closed double-looped helical structure. A TEQ electron and a TEQ positron would differ in the direction of their internal helicities. If the electron were structured like a circulating left-handed photon, then a positron would be structured like a circulating right-handed photon (see Figure 1). Electrons and positrons could therefore perhaps differentially absorb, scatter or otherwise interact with differentially polarized gamma photons, for example with differentially polarized gamma photons having energies corresponding to the rest mass of electrons.

211.6 10 / seczittω = ×

/C h mcλ =/C Cp h mcλ= =

2 2 2 2 2 2 2 2/ ( ) ( )x y zE c p mc p p p mc= + = + + +

Cp mc=

xp yp

zp

2 2 2 2 2 2 2 2/ C x y z CE c p p p p p p= + = + + +

2CE mc p c= =

0x y zp p p= = =

λ

It is proposed that the TEQ electron may generate its inertia by the rapid rotation of its internal Compton momentum at the electron’s zitterbewegung frequency. This rotating momentum is associated with the circulation of the TEQ’s point-like charge at the same high frequency. Subatomic effects could show themselves at the macroscopic level, for example as in magnetic materials. The rotation of a macroscopic physical object at a particular frequency could shift the internal rotational rate of the Compton momentum of electrons by a corresponding frequency, depending on the alignment of the electrons with the rotational direction of the macroscopic object. This leads to a testable prediction that the inertia of the rotating object could change with its angular velocity. The object could become more or less massive, with a correspondingly larger or smaller weight. The inertia explanation in the TEQ electron model can also lead to new proposals for experiments in inertia alteration, which could be then subjected to experimental tests. Positive results would of course lead to a better understanding of inertia, while negative results would also be informative.

9. Conclusions

The photon and the electron are modeled as helically circulating point-like transluminal energy quanta having both particle-like (E and

!P ) and wave-like (

and ) characteristics. The number of quantitative similarities between the Dirac equation’s electron and the TEQ electron, and between the Heisenberg uncertainty principle and the TEQ photon model of the photon are remarkable, given the relatively simple mathematical form of the TEQ models of the electron and the photon. A new approach to understanding inertia as “momentum at rest” is presented.

Appendix

Finding the TEQ Electron’s Helical Radius that Gives the Known Magnetic Moment of the Electron

For the TEQ electron model’s closed, double-looped helical trajectory, the model’s magnetic moment Mz is

calculated from the formula for the total magnetic moment

!M of a 3-dimensional closed current loop. The

loop carries a current I caused by the motion of the TEQ’s point charge –e around its closed trajectory in a period . This gives a current I =Q /T = !e / (2" /# o ) = !e# o / 2" . The position and

velocity components of the TEQ are given in equations (3) and (4). Using these equations, in vector notation the position

!r (t) and velocity !v(t) of the circulating

charged TEQ in the electron model are given by

!r (t) = x(t)i + y(t) j + z(t)k

=Rx (t)i + Ry(t) j + Rz (t)k

and

!v(t) = vx (t)i + vy(t) j + vz (t)k

=Vx (t)i +Vy(t) j +Vz (t)k The magnetic moment

!M and its component Mz for a

rapidly circulating charge in a current loop are given by

!M = I

2 !r (t) ! d!r

t=0

T

" (t) where T = h / mc2

!M = I

2 !r (# ) !

d!r (t)dt#=0

2$

"dtd#

d#

!M = I

2 !r (# ) !

!v#=0

2$

" (# ) 1%0

d# since # =%0t

!M = 1

2!

-e%0

2$ !r (# ) !

!v(# )#=0

2$

"1%0

d# since I=-e%0

2$!

M = 12!

-e%0

2$ !r (# ) !

!v#=0

2$

"1%0

d#

!M = -e

4$ !r (# ) !

!v(# )#=0

2$

" d#

Mz =-e4$

(Rx (# )Vy (# )#=0

2$

" & Ry (# )Vx (# ))d#

(12)

To obtain the radius R of the electron model’s

closed helix that would yield the electron’s Dirac magnetic moment , the value of Mz in

equation (12) was set to be Mz = !e! / 2m , which is the

negative of the value of the Bohr magneton MB = 9.274 !10

"24 JT"1 (which is the magnitude of the magnetic moment of the Dirac electron). Equations (3) and (4) for the TEQ’s position and velocity coordinates were then used with 2R0 replaced by the variable R

(since the TEQ helical radius value 2R0 required to get the electron’s magnetic moment was initially unknown). The equation for Mz in equation (12) was then solved for the value of R that is the radius of the TEQ’s closed helical trajectory that yields

Mz = !e! / 2m for the TEQ electron model. The

Cp mc=

ωλ

T = h / mc2 = 2! /" o

Mz = !e! / 2m

solution is R = 2R0 where R0 =12 ! / mc . It is this

helical radius R = 2R0 that was then finally included

in equation (3) to obtain the trajectory of the TEQ model of the electron.

Using equation (3), the values for Mx and M y

components of !

M for the TEQ electron model can then be found from

Mx =-e4!

(Ry (" )Vz (" )# Rz (" )Vy (" ))d""=0

2!

$ = 0

M y =-e4!

(Rz (" )Vx (" )# Rx (" )Vz (" ))d""=0

2!

$ = .25M B

(13)

The values of Mx and M y depend on the way the x and

y axes for of the electron model’s helical trajectory are defined in equation (3). A rotation of the x and y axes for these equations by +90 degrees around the z-axis would give Mx = !.25M B and

M y = 0 , while leaving

unchanged Mz = !M B . The value of the experimental helical radius Rexp of

the transluminal energy quantum electron model corresponding to the precise experimental value of the electron’s z-component of magnetic moment can also be calculated from equations (12). The electron’s precise experimentally measured value Mz of its z-component of magnetic moment is Mz = !(g / 2)MB where g / 2 =1.00115965219 and MB is the Bohr magneton. Set Mz in the last line of equations (12) above to !(g / 2)MB . To calculate Rexp set the helical

radius in the electron model equations (3) to Rexp = FexpR0 , where Fexp is the fraction that the

required new Rexp is, compared to R0 . F = 2 in the

electron model equations (3), so this means replacing 2 by Fexp in equations (3) in order to solve for the

value of Fexp corresponding to the electron’s experimental value of Mz . Set ! 0t = " in the x and y components of the electron model equations (3). Solve the bottom line in equations (12) for Fexp . The solution for Fexp using the equation for Mz in equation (12) is

found to be Fexp = 2 (g / 2)! .5 , or Fexp =

1.41585260842, as compared to F = 2 for the electron model having its magnetic moment’s z-component set exactly equal to !MB (i.e. with g/2 = 1 exactly). This new value of Fexp gives the percentage increase of the

helical radius of the electron model having the exact

experimental value of the electron’s z-component of magnetic moment, compared to the helical radius corresponding to F = 2 , corresponding to the exact Bohr magneton value, to be (Fexp ! 2) / 2 "100 =

0.115898058% .

References

1. Dirac, P.A.M., “The Quantum Theory of the Electron, Part 1,” Proc. Roy. Soc., A117, 610-624, (1928a).

2. Dirac, P.A.M., “The Quantum Theory of the Electron, Part 2,” Proc. Roy. Soc., A118, 351-361, (1928b).

3. Schrödinger, E., “Non powerful motion in the relativistic quantum mechanics,” Sitzungsb. Preuss. Akad. Wiss. Phys.-Math. Kl. 24, 418-428, (1930).

4. Barut, A.O., Bracken, A.J., “Zitterbewegung and the Internal Geometry of the Electron,” Phys. Rev.D 23, 2454-2463, (1981).

5. Barut, A.O., Thacker, W., “Covariant Generalization of the Zitterbewegung of the Electron and its SO(4,2) and SO(3,2) Internal Algebras,” Phys. Rev. D 31, 1386-1392, (1985).

6. Hestenes, D., “Local Observables in the Dirac Theory,” J. Math. Phys. 14, 893-905, (1973).

7. Hestenes, D., “Quantum Mechanics from Self-interaction,” Found. Phys., 15, 63-87, (1983).

8. Hestenes, D., “The Zitterbewegung Interpretation of Quantum Mechanics,” Found. Phys. 20, 1213-1232, (1990).

9. Hestenes, D., “Zitterbewegung Modeling,” Found. Phys. 23, 365-387, (1993).

10. Bohm, D., Hiley, B. J., The Undivided Universe: An Ontological Interpretation of Quantum Theory, Routledge, London, 1993, pp. 218-222.

11. Ashworth, R.A., “Confirmation of Helical Travel of Light through Microwave Waveguide Analyses,” Phys. Essays 11, 343-352, (1998).

12. Kobe, D.H., “Zitterbewegung of a Photon,” Phys. Let. A 253, 7-11, (1999).

13. Sivasubramanian, S., et al., “Non-commutative Geometry and Measurements of Polarized Two Photon Coincidence Counts,” Annals Phys. 311, 191-203, (2004).

14. Gauthier, R., “Superluminal quantum models of the electron and the photon,” (2006), http://www.superluminalquantum.org/dallaspaper   ,  (accessed 12/30/2012.)

15. Gauthier, R., “FTL quantum models of the photon and the electron”, CP880, Space Technology and Applications International Forum--STAIF-2007, edited by M.S. El-Genk, (2007), http://superluminalquantum.org/STAIF-2007article.

16. Bender, D., et al., “Tests of QED at 29 GeV Center-of-mass Energy,” Phys. Rev. D 30, 515-527, (1984).

17. De Broglie, L., L’Electron Magnetique, Hermann et Cie, Paris, France, 1934, pp. 294-295.