Symmetry Slip Knot Picture-Photon as "Electron Eating It's Tail"

Symmetry Slip Knot Picture-Photon as "Electron Eating It's Tail"

Geometric Model for a Photon and an Electron

by Richard Richardson

The following article "Microvita Physics" is excerpted from a larger article "Microvita: a holistic paradigm for a new science of matter, life and health" that was published in the Proceedings of the 2^{nd Gauss Symposium.}

The proposed models of the photon and electron below (only the electron model is given in the associated computer graphics on this web site) are based on the concept of microvita given by P.R. Sarkar in 1986. Microvita were described as fundamental entities of life, smaller than an electron, and at that time millions of microvita were understood to compose an electron, while billions compose a carbon atom. There is presently no scientific proof of microvita. But the geometrical models of the photon and the electron inspired by this microvita concept may have some relevance to Dan Winter's scientific work, so they are included in this web site. The complete article from which the present article was excerpted, and other articles about microvita by the author and by others can be found at the web site: www.home.zonnet.nl/microvita ("Microvita Physics" was written in 1993. Currently (2002) the author understands that while atoms may be structured by the action of microvita within atoms, individual electrons are not composed of microvita. But electrons could be composed of other minute particles that are not microvita, so that the geometrical photon and electron models described here could still be useful for physics).

Summary of the "Microvita Physics" article:

To make the geometrical model for the photon, the following experimental facts about photons were used:

1) the known dependence of a photon's linear momentum on its wavelength

2) the known value of the angular momentum (spin) of a photon

3) the fact that with circularly polarized light, the electric and magnetic vectors of the light rotate 360 degrees when the light wave moves forward one wave-length

The mathematical result derived from combining the above 3 facts in the geometrical photon model is that the microvita composing a photon travel along a helical path whose wave length is equal to the circumference of the helix.

In the geometrical model for the electron, in addition to the 3 above facts, there is also included:

4) the known value of the angular momentum (spin) of an electron.

The geometrical electron model is based on the idea that an electron is "a photon chasing its tail" given by David Bohm. To create the geometrical electron model, the center line or axis of the helix of the geometrical photon is curved into a double loop (a circular loop that doubles over itself before closing ), so that the microvita, instead of moving in a open helical path like in the photon, move in a specific closed helical path, where the radius of the closed helical path (as the radial distance measured from the double-loop axis) is twice the radius of the double-loop axis around which the microvita move. The double-loop center line or axis of the closed helical path is used instead of a single loop so that the calculated spin of the geometrical electron will be equal to the experimental value of the electron's spin, which is half the experimental value of the photon's spin.

In the "Microvita Physics" article below, equations 51, 52, and 53 give the formulas for the x, y and z coordinates used for generating the closed spiral path for the electron in the associated computer graphics figures presented on this web site, while equation 54 gives the length value re of the constant for this microvita electron model:

x = re (1 + 2 cos(q/2)) cos(q) (51)

y = re (1 + 2 cos(q/2)) sin(q) (52)

z = 2 re sin(q/2) (53)

where q (theta) is an angle running from 0 to 720o (once around each loop of the double-looped circle generating the helix, for a single closed loop path) and

re = (1/4p) h/mc (54)

where h is Planck's constant, m is the mass of an electron, and c is the speed of light.

Microvita Physics, by Richard Richardson

The concept of microvita can lead to useful quantitative descriptions in the area of sub-atomic physics, and particularly by providing new models describing the size and structure of elementary particles such as the electron and photon. The microvita approach also leads to a new interpretation of quantum physics. Quantitative dynamic models of the electron and the photon have been developed based on the concept of microvita and combining known experimental facts about electrons and photons.[9] The models are presented and further developed below.

According to Sarkar, "each and every existence has its own peculiar wavelength and its peculiar rhythm." So this is applies to microvita as well. We may interpret rhythm here as vibrational frequency, measured in cycles per second. Wavelength might be measured in metres. So any particular microvitum will have its own frequency and wavelength. The photon or quantum of light energy is also described in terms of frequency and wavelength. Electrons and other sub-atomic particles with a "rest mass" also have a wave nature and can also be described in terms of a frequency (which depends on a particle's total energy) as well as a wavelength, called the de Broglie wavelength (which depends on a particle's velocity or its momentum.)

The frequency of a photon is directly proportional to its energy, while its wavelength is inversely proportional to its momentum. The wavelength of an electron is inversely proportional to its momentum, like the photon. The frequency of an electron is directly proportional to its total energy (its rest energy plus its kinetic energy or energy of motion.) So for both photons and particles with rest mass, the formulas for energy and momentum are

E=hf and P=h/L (1)

where h is a fundamental physical constant called Planck's constant. These formulas are also true relativistically, i.e. for particles with velocities approaching the speed of light. Photons move at the speed of light, independent of their energy. In a vacuum their velocity is c = 300,000 kilometers per second, approximately.

For particles with rest mass, the energy contained in the particle is given according to Einstein by

E = mc2 (2)

where m is the total mass of the particle, which depends on its velocity. The momentum of a particle with mass is given by

P = mv (3)

where m also depends on the velocity of the particle.

Since microvita are said to compose electrons and atoms, we can assume they compose photons as well. We can assume that microvita have a frequency and a wavelength, and that they carry energy and momentum related to their frequency and wavelength, respectively, similar to elementary particles. But the proportionality constant is not h but a constant much smaller than h, which may be called S (for Sarkar).

h = N S (4)

where N is the number of microvita in an electron, which may be in the millions or less. S is a fundamental constant that relates energy and momentum to the spacio-temporal pattern or form of movement of microvita. We may assume for now that the number of microvita in an electron and in a photon is the same, given by N.

So for microvita in a photon

E = S f and P = S/L (5)

for the energy and forward momentum of a microvitum, respectively, where f is the frequency and L is the wavelength of a microvitum, and S = h/N .

THE MICROVITA STRUCTURE OF A PHOTON

A photon has a momentum given by

p = h/L (6)

(where L is the wavelength) and angular momentum or spin is given by

s(photon) = h/(2p) (7)

The spin of a photon may be positive (in the direction of motion) or negative (in the opposite direction.) Light that is right circularly polarised consists of photons with one type of spin, while left circularly polarised light consists of the other. With circularly polarised light, the electric field direction (perpendicular to the direction of propagation) rotates 360 degrees as the light moves ahead one wavelength. The tip of the electric field vector follows a helical path. So we can assume that the microvita making up a photon also move in a helical path which rotates 360 degrees clockwise or counterclockwise for each advance of one wavelength.

So the rotational movement of the microvita around the axis of the helix accounts for the spin of the photon, while their forward motion parallel to the axis of the helix gives the photon its linear momentum.

There will be a relationship between the radius of the microvita helix for a photon and the wavelength of the photon. (For a photon, a radius is not defined in current physics, only a wavelength.) This radius will tell us something of the microvita structure of a photon.

Assume that the total momentum vector P for the photon is directed along the helical path of the microvita at each point on the helix. Let A be the angle that this total momentum vector makes with the direction perpendicular to the photon's linear motion. Then the component of P that contributes to the spin is P cosA.

The component of P (the horizontal P vector in Figure 1) that produces the linear momentum of the photon is P sinA.

So if r is the radius of the helix, then the angular momentum is

Angular momentum = rP cosA=h/(2p) (8)

while the linear forward momentum is given by

Linear momentum = P sinA = h /L (9)

where L is the wavelength (distance between turns of the helix.) Dividing (9) by (8), we get

(P sinA)/(rP cosA) = (h / L)/(h/2p) (10)

(sinA)/(cosA) = (2pr) / L (11)

tanA = (2pr) / L (12)

When the microvita move around one circumference (2pr), they also move forward one wavelength L. So according to this geometrical relationship in Figure 1,

tanA = L / (2pr) (13)

Comparing equations (12) and (13) we see that

(2p r) / L = L / (2p r) (14)

This will only be true if

tan A = 1 and A= 45o (15)

for photons of any wavelength, and

L = 2pr (16)

that is, the circumference of the microvita helix of a photon equals the wavelength of the photon.

Therefore, the radius of the microvita helix of a photon is

r = L/(2p) (17)

where L is the photon's wavelength.

So if the wavelength of a photon increases or decreases, (corresponding to a decrease or increase in the energy and momentum of the photon, respectively,) the radius of its microvita helix changes proportionately.

Mathematically, the coordinates of position of one microvitum moving along its helix of radius R are

x = r cos(2p z/L) (18)
= (L/2p) cos(2p z/L) = (1/k) cos(kz) (19)

y = r sin(2p z/L) (20)
= (L/2p) sin(2p z/L) = (1/k) sin(kz) (21)

where k is the wave number (k=2p/L), z is the distance along the helix, and x and y are directions perpendicular to z. The maximum slope of these sine and cosine curves is 1, that is, 45 degrees, independent of the wavelength of the photon.

A photon whose energy equals the rest energy of an electron, has a wavelength equal to h/mc. So the radius of the microvita helix for this photon would be, from the above results,

r = L/2p = h/(2pmc) (22)

The length h/mc is well known in physics and is called the Compton wavelength. It equals .0243 x 10-8 cm for an electron. A photon with this wavelength has an energy corresponding to the rest mass of an electron.

We may model a photon as a spiral of N microvita. All N microvita move along the same spiral path one after the other, like beads sliding along a string. The microvita may be considered to be equally spaced along the photon spiral. If the photon is many wavelengths long, the N microvita would be spread out evenly over the total length, while if the photon is only a few wavelengths long, the N microvita would be concentrated along this shorter total length. In each case the formula E = hf for total energy as a function of frequency would apply. A long and a short photon having the same wavelength and frequency have the same total energy. Let us see how this microvita model for a photon compares with that of an electron composed of microvita.

THE MICROVITA STRUCTURE OF AN ELECTRON

An electron is known experimentally to have its own spin or angular momentum. In quantum theory the electron is described as having "intrinsic" spin and no size, although it does have a measurable wavelength which varies inversely with its velocity. The spin is thought to be too small to arise from the orbital motion of particles. This leads to a lack of a visual model for an electron as well as for other particles in quantum theory. But the microvita model below shows that the electron's measured value of spin can arise from a closed spiraling movement of microvita. The microvita model of an electron accounts for the electron's spin, even though the moving microvita themselves are point entities. These microvita have no internal structure or spin, but carry momentum and energy due to their movement in space.

The Radius of a Microvita Electron

A particle with a rest mass can be first modeled as a closed circular path of vibrating energy, with microvita travelling around the circle in one rotational direction. Assume that all the microvita in the particle vibrate in parallel in a single circular pattern. The wave motion of the microvita moves around the circle at the speed of light.

Now assume that the circle's circumference, the distance around the vibrating circle of microvita, is the minimum length for a stable structure. That means the circumference of the circle is one wavelength. Assume that all the microvita in the particle vibrate together at a common frequency and common wavelength, so that the particle's wave pattern is a simple circular one. The radius of the particle can then be calculated by knowing the circle's circumference.

Assume that the rest mass of the particle is m. Then it contains energy E = mc2, by Einstein's equation. This energy is the total energy of all the microvita vibrating together in the particle. From that total energy, the frequency of the vibrating microvita can be calculated by

E total = N(Sf) (23)

for N microvita vibrating at frequency f. Therefore

E total = hf (24)

since

NS = h (25)

mc2 = hf (26)

for the vibrating particle. Since the microvita wave travels in a circle at the speed of light, the relationship of the wavelength L to frequency f is

Lf = c (27)

f=c/L (28)

m c2 = h c/L (29)

m c = h/L or L = h/mc (30)

but

L = 2p r (31)

where r is the radius of the circle. So

2pr = h/mc (32)

r = h/(2pmc) (33)

So the radius of this circle depends on Planck's constant h, its rest mass m, and the speed of light c.

The angular momentum or spin of the particle is calculated by

Spin = r P (34)

where r is the radius and P is the momentum at the distance r,

P = E/c = m c2/c (35)

P = mc (36)

Spin = r P = h/(2pmc) mc (37)
= h/(2p) (38)

In units of h/(2p), this particle would be said to have spin equal to 1 unit.

But an electron is known to have spin of 1/2 unit. So the above description cannot be that of an electron. How can a particle be structured from microvita so that its spin equals 1/2?

The simplest way is that the microvita move around in a double loop before joining together. The total length of the double loop is still one wavelength L, and the wave motion still moves around its length at velocity c. But the particle radius re is now smaller because the structure is two loops instead of one. So the circumference of this smaller loop is L/2. The radius of the particle is then given by

2p re= L/2 (39)

re= (1/2p) (h/mc)/2 (40)

where L = h/mc as before, or

re = (1/4p) (h/mc) (41)

That is, the radius re of the circle for a spin 1/2 particle (the electron) is half that of a particle of the same mass with spin 1. Confirming the spin of the electron, we see that

Spin(electron) = re P = (1/4p) (h/mc) (mc) (42)

where

P=mc (43)

as before, or

Spin(electron) = 1/2 h/(2p) (44)

So to obtain a spin of 1/2 for an electron, the microvita structure requires a double circular loop of radius

re = h/4pmc (45)

What is the calculated value of this radius re of an electron made from microvita in the double loop structure described? Since the actual structure of the microvita electron, as shown in the detailed microvita electron model below, is neither circular nor spherical, it does not have a single radius. But we can take r = h/4pmc (the radius of the double-looped circle generating the electron's closed helix) as a measure of the radius.

Substituting the values for Planck's constant h, the mass m of the electron and speed of light:

h = 6.63 x 10 -34 joule sec (46)

m(electron) = 9.11 x 10 -31 kg (47)

c = 3.00 x 10 8 meters/sec (48)

we find

r(microvita electron)= (1/(4p)) (h/mc) (49)

= 1.9x10 -13 meters (50)

This calculated value for the radius of a microvita electron contrasts with the radius of an atom of about 10-10 meters and the radius of a small atomic nucleus of around 10-15 meters. That is, the electron's radius is calculated to be roughly 1/500 times the radius of an atom but around 200 times the size of a small nucleus.

In this preliminary model therefore, an electron consists of a quantum of energy (proportional to the mass of the electron) shared by perhaps millions of microvita circling at the speed of light on a double-loop path of total length equal to one Compton wavelength.

Detailed Microvita Structure of the Electron

The actual motion of microvita for the double loop particle above will not be a circular motion but a closed spiral motion with the double circular loop as the center line of the closed spiral. What is the amplitude of the microvita motion around the double loop? We can consider that the microvita structure of an electron is created when the helical path of microvita making a photon is curved so that the centre line of the microvita spiral follows a double-looped circular path and the spiral closes on itself after travelling a centre-line distance of one wavelength (the distance between two successive turns of the corresponding microvita photon spiral.) The circumference of the circle of each single loop is therefore one-half the wavelength of the corresponding microvita photon spiral.

The microvita electron structure is therefore generated by rotating the helix radius, whose length is (1/2p) h/mc, around a point moving along a circle of radius (1/4p) h/mc, rather than rotating it along a straight line, as with a microvita photon helix. The radius of the helix generating the microvita electron is therefore twice the radius of the double-looped circle. The point travelling along the circle moves twice around the circle, or 720o, while the radius of the helix rotates through 360o about the moving point.

The above operation creates a closed-loop three-dimensional path for microvita (see computer graphics display) whose shape is uniquely specified (except for mirror image reversal) by the geometrical parameters of the photon (spin 1) and the electron (spin 1/2), and whose size is determined by the electron's mass (as well as by Planck's constant h and the speed of light c.)

We can assume for our microvita model that a number N of microvita move spread out along the three-dimensional pathway all in the same rotational direction. This microvita electron has several interesting features. First, if it is turned upside down (rotated 180 degrees about the x-axis) the same form is obtained, but with the microvita moving in the opposite direction. This corresponds to an electron whose spin is down (the original orientation is the spin up orientation.) Second, a mirror image of the form gives a "left-handed" version of the form, which cannot be made to coincide with the original form. This corresponds to a positron (the anti-particle of the electron) if the original form is that of an electron.

The double-looped microvita model of an electron is consistent with an aspect of quantum theory of electrons related to spin. In the double-looped model, the helical path of microvita composing an electron closes on itself after one turn of the helix spread over two turns around a circle, i.e. after 2x360o or 720o. In quantum theory, the mathematical wave function describing the spin of the electron also has a rotational periodicity of 720o.

The coordinates of the above three-dimensional closed loop microvita structure for an electron can be derived straightforwardly from the above geometrical relationships and are given below. The xy, xz, and yz projections of the three dimensional form are shown in Figure 2.

x = re (1 + 2 cos(q/2)) cos(q) (51)

y = re (1 + 2 cos(q/2)) sin(q) (52)

z = 2 re sin(q/2) (53)

where q (theta) is an angle running from 0 to 720o (once around each loop of the double-looped circle generating the helix) and

re = (1/4p) h/mc (54)

the previously calculated microvita electron radius from equation (45). This microvita electron structure can also be expressed in radial notation:

r = re (1 + 2 cos(q/2)) (55)

z = 2 re sin(q/2) (56)

where goes from 0 to 720o (i.e. 4), before the pattern repeats itself. In these expressions, the coordinates are expressed in units of re, the radius of the circle which the helix is rotating around, or (1/4p) h/mc. So in units of re the generating circle has radius of 1 and the helix has radius of 2. The generating circle is in the xy plane with its center at x = 0 and y = 0. For example, at q = 0, the above expressions give x = 3, y = 0, and z = 0. Here x is the sum of the radius of the generating circle (1 unit) and the radius of the helix (2 units). As increases, the y and z components increase as the helix rotates about the circle. The resulting form represents the path of flow of microvita in an electron.

With this model, the radial components of velocity of a single microvitum as a function of q is given by:

vr (q) = - re sin(q/2) dq/dt = - c sin(q/2) (57)

vq (q) = re (1 + 2 cos (q/2)) dq/dt = c (1 + 2 cos (q/2)) (58)

vz (q) = re cos (q/2) dq/dt = c cos(q/2) (59)

since dq/dt = 2 , where is the microvita angular frequency (q rotates through 720o while the microvita make one complete 360o cycle around their closed helical path) and 2re = c. Note that in this microvita electron model, as in the microvita photon model, the total speeds of the microvita along their closed or open helical paths exceed the speed of light.

Calculation of Electron Spin From the Microvita Model

With the above information, we can now calculate the exact spin of the microvita electron model. Assume that the electron is composed of N microvita, which are spaced at equal intervals of q in the interval 0 to 720o (e.g. if N = 10, then the 10 microvita will be spaced at every 72o around the figure, like beads on a string) starting at an arbitrary initial angle qo, and will travel along the path of the microvita electron figure according to the velocities given above, maintaining their equal angular separations. (Remember that the microvita figure closes after traversing a range of q of 720o). Assume that at q = 0, where the microvita velocity is maximum, the q component of momentum of the microvitum in the electron is also maximum and equal to the forward momentum of a corresponding microvitum in a photon containing N microvita, having the same total energy as the electron's energy E=mc2, e.g.:

Forward momentum of corresponding whole photon = m c (60)

Forward momentum of microvitum of corresponding photon = m c/N (61)

Maximum component of momentum of microvitum in electron Pqmax = m c/N (62)

Assume that the momenta of the N microvita composing the electron are proportional to the velocities of the microvita, and are directed along the path of movement of the microvita in the electron model. This implies that the momenta of the microvita combine by normal vector addition.

Pq( qi ), the q component of the momentum of the ith microvitum, is then given by

Pq( qi ) = (vq(qi)/vq(q =0o)) Pq max (63)

where vq(qi) is the q component of the velocity of the ith microvitum at qi. The contribution si to the electron's total spin, by the ith microvitum in the electron (calculated in relation to the vertical or z axis at r = 0) is given by

si = r(qi ) Pq( qi ) (64)

= r(qi ) (vq(qi)/vq(q = 0o)) Pq max (65)

where i goes from 1 to N, (for N microvita). Substituting into (64) the following values:

r(qi ) = re (1 + 2 cos (qi /2)) (66)

vq (qi ) = re (1 + 2 cos ( qi /2)) dq/dt (67)

vq(q = 0o) = re (1 + 2cos( 0o)) dq/dt = 3 re dq/dt (68)

Pq max = m c/N (69)

we obtain

si = ( re m c/N ) (1 + 2 cos (qi /2))2 / 3 (70)

The above equation must be summed for all N angles qi . This gives for the microvita electron model the total spin Se, where

Se = si = ( re m c/N ) (1 + 2 cos (qi /2))2 / 3 (71)

= ( re m c ) (1 + 2 cos (qi /2))2 / 3N (72)

Now it is a remarkable mathematical fact that

(1 + 2 cos (qi /2))2 / 3N = 1 (73)

when summed over N equally spaced angles qi in the range of q from 0 to 720o, (i.e. with angular separations of 720o/N) as long as N is 3 or more. So

Se = re m c (74)

Substituting from (41)

re = h/4pmc (75)

we find

Se = (h/4pmc) m c (76)

= h/4p = 1/2 h/2p (77)

which is exactly the experimentally measured spin of the electron. A similar calculation for the y components of the angular momentum of the microvita electron model gives a magnitude of 1/3 of the value of the z component above, while by symmetry the value of the x component of angular momentum in the microvita electron model is zero. (See Fig. 2) The magnitudes of the x and y components would be interchanged by a rotation of the microvita electron model by 90 degrees about the z axis.

If we check for conservation of total linear momentum for the N microvita in the microvita electron, using the microvita velocities and momenta that can be calculated from the model (Eq. 57-59), we find that the net linear momentum is zero in all directions for a "stationary" electron if N is 4 or more microvita in the electron, as long as the microvita retain their equal angular separation as they move along their common path in the microvita model. So the microvita electron model must have a minimum of 4 microvita, even though 3 are sufficient to conserve total spin. There is no mathematical limitation on the maximum number of microvita allowed in an electron.

Richard Richardson (richard@sfo.pl) holds a B.Sc. in physics from M.I.T., an M.Sc. in physics from the University of Illinois and a Ph.D. in psychology from Stanford. He currently resides in the U.S.A.