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The de Broglie relations derived from the electron and proton coupling to the Planck vacuum state.

1 Force transformation

The electron and proton cores, (-[e.sub.*], [m.sub.e]) and ([e.sub.*], [m.sub.p]), exert the two-term coupling forces [1]

[+ or -] ([e.sup.2.sub.*]/[r.sup.2] - [mc.sup.2]/r) (1)

on the Planck vacuum (PV) negating-energy continuum, where the plus and minus signs refer to the electron and proton respectively and [mc.sup.2] represents the rest energy of either particle. The bare charge [e.sub.*] is assumed to be a massless point charge. The massive particle cores, however, possess a small spherical extension due to the zero-point formation of their derived masses [2].

The coupling force vanishes

[e.sup.2.sub.*]/[r.sup.2.sub.c] - [mc.sup.2]/[r.sub.c] = 0 (2)

at the Compton radius [r.sub.c](= [e.sup.2.sub.*]/[m.sub.e][c.sup.2]) of either particle, leading to the Compton relations

[r.sub.c] x [mc.sup.2] = [e.sup.2.sub.*] [right arrow] [r.sub.e][m.sub.e][c.sup.2] [r.sub.p][m.sub.p][c.sup.2] = [e.sup.2.sub.*] (3)

for the electron ([r.sub.e][m.sub.e]) and proton ([r.sub.p][m.sub.p]), and the (reduced) Planck constant [??] = [e.sup.2.sub.*]/c. It is noted that (1) is a force acting between a free-space particle and the vacuum state--it is not a free-space/free-space force as are the Coulomb and Newton forces. The Compton relations and [??] = [e.sup.2.sub.*]/c are used throughout the following calculations.

The vanishing force (2) can be expressed as a tensor 4-force difference. In the primed rest frame of the particle where these static forces apply, this vanishing force difference [DELTA][F'.sub.[mu]] is ([mu] = 1,2,3,4)

[DELTA][F'.sub.[mu]] = [0, i([e.sup.2.sub.*][r.sup.2.sub.c] - [mc.sup.2]/[r.sub.c])] = [0, 0, 0, i 0] (4)

where i = [square root of -1]. Thus the vanishing of the component [DELTA][F'.sub.4] = 0 in (4) can be thought of as the source of the Compton relations in (3).

The force difference in the laboratory frame (in which the rest frame travels at velocity v along the z-axis) [3]

[DELTA][F'.sub.[mu]] = [a.sub.[micro]v] [DELTA][F'.sub.v] = [0.sub.[mu]] (5)

follows from the tensor nature of (4) and the Lorentz transformation [x.sub.[mu]] = [a.sub.[micro]v] where [x.sub.[mu]] = (x, y, z, ict),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

and [mu], v = (1,2,3,4). Thus (5) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where

rd = [r.sub.c]/[beta][gamma] and [r.sub.L] = [r.sub.c]/[gamma] (8)

are the de Broglie radii for the space and time directions respectively; and where [beta] = v/c < 1 and [gamma] = 1/ [square root of (1 - [[beta].sup.2])] .

The force difference [DELTA][F'.sub.3] = 0 in (7) gives the de Broglie relation

[r.sub.d] x cp = [e.sup.2.sub.*] or [r.sub.d] = [??]/p (9)

in the space direction, where p = m[gamma]v is the relativistic particle momentum. The force difference [DELTA][F.sub.4] = 0 gives the de Broglie relation

[r.sub.L] x E = [e.sup.2.sub.*] or [r.sub.L] = [??]/m[gamma]c (10)

in the time direction, where E = m[gamma][c.sup.2] is the total relativistic particle energy.

The momentum and energy in equations (9) and (10) are derived from nothing more than the vanishing of the Lorentz transformation of (2), whose results can be taken a step further:

E = [e.sup.2.sub.*]/[r.sub.L] = [e.sup.2.sub.*][gamma] = [mc.sup.2][gamma]

= [mc.sup.2][(1 + [[beta].sup.2]/1 - [[beta].sup.2]).sup.1/2]

= [([m.sup.2][c.sup.4] + [c.sup.2][p.sup.2]).sup.1/2] (11)

showing that this well known equation has its source in the two-term particle/PV coupling force.

2 Conclusions and comments

The vast accumulation of electron diffraction data leaves no doubt that the electron and proton possess a wave nature. If the corresponding waves are roughly expressed in terms of planewaves, then it is reasonable to assign 2[pi][r.sub.d] and 2[pi][r.sub.L] as the wavelengths in the space and time directions respectively. As a first approximation then, the electron and proton de Broglie waves are planewaves propagating within the PV continuum.

The Synge primitive (or planewave) quantization of spacetime [4, p.106] is an independent calculation that parallels the ideas of the previous paragraph. That quantization divides the space and time axes of the Minkowski spacetime diagram into equal segments, where the space and time segments are [r.sub.d] and [r.sub.L] respectively (Synge actually multiplies these two segments by 2[pi] which defines a phase space).

The particle/PV coupling of the previous section provides the physical explanation for that quantization in terms of the coupling force (1).

Although the implied mathematics of the two previous paragraphs involves planewaves (which are global), the PV wave phenomenon must be a local property associated with the particle/PV interaction in the neighborhood of the particle cores (-[e.sub.*], [m.sub.e]) and ([e.sub.*], [m.sub.p]), with characteristic (radian) frequencies defined by

[[omega].sub.c] = [e.sup.2.sub.*]/[r.sub.c]/[??] = c/[r.sub.c] (12)

with

[[omega].sub.L] [e.sup.2.sub.*]/[r.sub.L]/[??] = [gamma][[omega].sub.c] and [[omega].sub.d] = [e.sup.2.sub.*]/[r.sub.d]/[??] = [beta][gamma][[omega].sub.c] (13)

for each particle. Then (11) yields

[[omega].sup.2.sub.L] = [[omega].sup.2.sub.c] + [[omega].sup.2.sub.d]. (14)

The preceding results offer a simple explanation for the double-slit thought experiment [5, p.85]. Consider a collimated beam of monoenergetic electrons that is directed at an opaque wall containing two narrow, parallel, and closely spaced slits A and B, with a detection screen at some distance beyond the slits. Being a particle (although with a wavelike nature), the electron cannot go through both slits at the same time. Now consider the two experiments: (1) with slit A open and slit B closed; and (2) with both slits A and B open. Assume that the slits are narrower than one de Broglie wavelength (2[pi][r.sub.d]) and that their separation distance is several wavelengths.

If the electrons are particle-like with no wave-like qualities, the screen would show a bell-shaped excitation curve in case (1) and two superimposed bell-shaped curves in case (2). But for case (2), however, the overwhelming diffraction evidence demands a well defined oscillatory excitation curve on the screen--because the particle exhibits a definite wave-particle nature. Since the electron must go through A or B, but not both, this result is difficult to understand [5, p.85] with present-day physics. But if the free-space particle is accompanied by a PV de Broglie wave, the diffraction of that wave through A and B, and its interaction with the particle core, easily explains the oscillatory curve on the detection screen.

Submitted on March 21, 2015 / Accepted on March 27, 2015

References

[1.] Daywitt W.C. The Strong and Weak Forces and their Relationship to the Dirac Particles and the Vacuum State. Progress in Physics, 2014, v. 11, 18. See also www.planckvacuum.com.

[2.] Daywitt W.C. Why the Proton is Smaller and Heavier than the Electron. Progress in Physics, 2014, v. 10, 175.

[3.] Jackson J.D. Classical Electrodynamics. John Wiley & Sons, Inc., 1st ed., 2nd printing, NY, 1962.

[4.] Synge J.L. Geometrical Mechanics and de Broglie Waves. Cambridge University Press, 1954.

[5.] Leighton R.B. Principles of Modern Physics. McGraw-Hill Book Co., New York, Toronto, London, 1959.

William C. Daywitt

National Institute for Standards and Technology (retired), Boulder, Colorado. E-mail: wcdaywitt@me.com
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Author:Daywitt, William C.
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Date:Apr 1, 2015
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