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The Planck Vacuum Physics Behind the Huygens Principle and the Propagator Theory for the Schrodinger Electron.

1 Introduction

The theoretical foundation [1-3] of the PV theory rests upon the unification of the Einstein, Newton, and Coulomb superforces:

[c.sup.4]/G (= [m.sub.*][c.sup.2]/[r.sub.*]) = [m.sup.2.sub.*]G/[r.sup.2.sub.*] = [e.sup.2.sub.*]/[r.sup.2.sub.*] (1)

where the ratio [c.sup.4]/G is the curvature superforce that appears in the Einstein field equations. G is Newton's gravitational constant, c is the speed of light, m, and [r.sub.*] are the Planck mass and length respectively [4, p. 1234], and [e.sub.*] is the massless bare charge. The fine structure constant is given by the ratio [alpha] = [e.sup.2]/[e.sup.2.sub.*], where (-e) is the observed electronic charge.

The two particle/PV coupling forces

[F.sub.c] (r) = [e.sup.2.sub.*]/[r.sup.2] - m[c.sup.2]/r and [F.sub.*](r) = [e.sup.2.sub.*]/[r.sup.2] - [m.sub.*][c.sup.2]/r (2)

the electron core (-[e.sub.*], m) and the Planck-particle core (-[e.sub.*], [m.sub.*]) exert on the PV state, along with their coupling constants

[F.sub.c] ([r.sub.c]) = 0 and [F.sub.*] ([r.sub.*]) = 0 (3)

and the resulting Compton radii

[r.sub.c] = [e.sup.2.sub.*]/m[c.sup.2] and [r.sub.*] = [e.sup.2.sub.*]/[m.sub.*][c.sup.2] (4)

lead to the important string of Compton relations

[r.sub.c] m[c.sup.2] = [r.sub.*][m.sub.*][c.sup.2] = [e.sup.2.sub.*] (=c[??]) (5)

for the electron and Planck-particle cores, where [??] is the reduced Planck constant. The electron and Planck-particle masses are m and m, respectively. To reiterate, the equations in (l) represent the forces the free electron and Planck-particle cores exert on the PV space, a space that is itself pervaded by a degenerate collection of Planck-particle cores [5].

The Planck constant is a secondary constant whose structure can take different forms, e.g.

[??] [erg sec] = [r.sub.c]mc = [r.sub.*][m.sub.*]c = ([e.sup.2.sub.*]/[r.sub.*]) [t.sub.*] = [m.sub.*][c.sup.2]t, (6)

that are employed throughout the following text, where [t.sub.*] (= [r.sub.*]/c) is the Planck time [4, p. 1234].

Furthermore, the energy and momentum operators expressed as

[mathematical expression not reproducible] (7)

and

[mathematical expression not reproducible] (8)

will be used freely in what follows.

Section 2 re-examines the Schrodinger equation in light of the PV theory, the calculations concluding that the pervaded vacuum state is the source of the scattering in the propagator theory. Section 3 presents a nonrelativistic look at the Huygens principle and the propagator theory for the electron core.

2 SchrOdinger equation

The inhomogeneous Schrodinger equation, where H = [H.sub.0] + V is the Hamiltonian operator, can be expressed as

(i[??] [partial derivative]/[partial derivative]t - H) [psi](x, t) = 0. (9)

The free-space Hamiltonian is [H.sub.0] and V is some position and time-dependent potential that is assumed to slowly vanish in the remote past (t [right arrow] -[infinity]) and in the remote future (t [right arrow] +[infinity]). In free space V = 0 and (9) becomes

(i[??][partial derivative]/[partial derivative]t - [H.sub.0]) [phi](x, t) = 0. (10)

For t' > t, the formal solution to (9) or (10) takes the form [6]

[psi](x, t') = T exp [-i [[integral].sup.t'.sub.t] dt" H(t")/[??]] [psi](x, t) (11)

where T is the time-ordering operator whose details are unimportant here (see Appendix A). What is important is the decomposition of [??] (= [m.sub.*] [c.sup.2] [t.sub.*]) in the exponent of (11), leading to

[[integral].sup.t'.sub.t] dt" H(t")/[??] = [[integral].sup.t'.sub.t] dt"/[t.sub.*] H(t")/[m.sub.*][c.sup.2]. (12)

From the perspective of the PV theory, the normalization of dt" by the Planck time [t.sub.*] and H by the Planck-particle mass energy [m.sub.*][c.sup.2] strongly suggest that the scattering in the quantum-electro-dynamic propagator theory is caused by the Planck-particle cores that pervade the vacuum state. This conclusion will be reinforced by the calculations to follow.

The normalized Hamiltonian operator [H.sub.0] can be expressed as

[mathematical expression not reproducible] (13)

where the equalities in (5) are used. Then the normalized Schrodinger equation becomes

i[r.sub.*] [partial derivative][phi]/c[partial derivative]t - (i[r.sub.c][nabla]) x (-i[r.sub.*][nabla])/2 [phi] = 0 (14)

or

(i[t.sub.*] [partial derivative]/[partial derivative]t + [r.sub.c][r.sub.*][[nabla].sup.2]/2) [phi] = 0 (15)

where [t.sub.*] (= [r.sub.*]/c) is the Planck time and the equations are dimensionless. The dimensionless aspect of the equations here and in what follows will help in recognizing the relationship between the Huygens principle and the propagator formalism. The normalized inhomogeneous equation (9) becomes

(i[t.sub.*] [partial derivative]/[partial derivative]t + [r.sub.c][r.sub.*][[nabla].sup.2]/2) [psi] = V/[m.sub.*][c.sup.2] [psi] (16)

where again the equation is dimensionless.

3 Electron-core propagator

Roughly speaking, the Huygens principle states that every point on a wavefront is itself the source of a spherical wavelet. In the present context, the Huygens principle takes the form [7, eqn. 6.29]

[mathematical expression not reproducible] (17)

and

[mathematical expression not reproducible] (18)

where the Green function propagators [G.sub.0] and G have the units "erg-sec per unit volume". In the present paper, equations (17) and (18) are associated with what are defined as internal- and external-scattering processes respectively. The internal scattering refers to the free electron [phi](x, t) scattering off the pervaded PV space. The external scattering refers to the electron [psi] (x, t) scattering off the pervaded PV space with an external potential V(x, t) perturbing that space. It will be seen in what follows that the units "erg-sec per unit volume" almost define the "pervaded vacuum space".

Now begins the calculation of the wave function [psi] resulting from the continuous interaction of the free-electron wave function [phi] with the perturbed vacuum state. The calculation will not be carried to completion, but only far enough (equation (25)) to suggest that the wave scattering takes place between [phi] and the pervaded vacuum space. Furthermore, many of the details in the following calculations based on reference [7] are unimportant to the present needs' so the calculations are heavily referenced in case the reader is interested in those details.

For t = [DELTA][t.sub.1] [7, eqn. 6.30]

[mathematical expression not reproducible] (19)

and

(i[t.sub.*][partial derivative][partial derivative][t.sub.1] + [r.sub.c][r.sub.*][[nabla].sup.2]/2) [psi]([x.sub.1], [t.sub.1]) = 0 (20)

for t [not equal to] [DELTA][t.sub.1]. Equation (19) refers to an external scattering as defined above.

The new wave function due to the external perturbation V in (19) can be expressed as [7, eqn. 6.31]

[psi]([x.sub.1], [t.sub.1]) = [phi]([x.sub.1] [t.sub.1]) + [DELTA][psi]([x.sub.1], [t.sub.1]) (21)

so the Schrodinger equation yields (using (15) for [phi])

[mathematical expression not reproducible]. (22)

It can be shown that the second terms on the left and right sides of (22) can be dropped [7, eqn.6.35], leading to

i[t.sub.*][partial derivative]/[partial derivative][t.sub.1] [DELTA][psi] ([x.sub.1], [t.sub.1]) = V([x.sub.1], [t.sub.1])/[m.sub.*][c.sup.2] [phi]([x.sub.1], [t.sub.1]) (23)

which to first order in A[t.sub.1] yields

[DELTA][psi]([x.sub.1], [t.sub.1] + [DELTA][t.sub.1]) = -i V([x.sub.1], [t.sub.1])/[m.sub.*][c.sup.2] [phi]([x.sub.1], [t.sub.1]) [DELTA][t.sub.1]/[t.sub.*] (24)

where the differential [DELTA][psi] ([x.sub.1], [t.sub.1]) coming from the approximation is ignored compared to the [phi]([x.sub.1], [t.sub.1]) on the right side of (24).

For two consecrative time periods [DELTA][t.sub.1] [DELTA][t.sub.2], with an infinite past [where [psi](x') [right arrow] [phi](x')], it can be argued that [7, eqn. 6.43]

[mathematical expression not reproducible] (25)

where the obvious notations (x) [equivalent to] (x, t) and [phi](2) [equivalent to] [phi]([x.sub.2]) are used. The four terms in (25) represent respectively the propagation from (x, t) to (x', t'): a) as a free particle with no external scatterings; b) with one scattering at ([x.sub.1], [t.sub.1]); c) with one scattering at ([x.sub.2], [t.sub.2]); and d) with a double scattering at ([x.sub.1], [t.sub.1]) and ([x.sub.2], [t.sub.2]) in succession. The representations of these scatterings in Figures 1-4 are called Feynman diagrams [7, eqn. 6.43], where the horizontal axis represents space and the vertical axis represents time.

4 Conclusions and comments

A close examination of the previous calculations strongly suggests that the PV theory, which envisions a vacuum space pervaded by a degenerate collection of Planck-particle cores, provides a fundamental explanation for the Huygens principle and the scattering associated with the quantum-electro-dynamic propagator formalism.

The retarded Green function Go+ associated with the Green function [G.sub.0](x', t'; x, t) in equation (17) and in Figure 1 is given by the equations [7, eqn. 6.6o]

[mathematical expression not reproducible] (26)

for t' > t and [G.sup.+.sub.0] (x'; x) = 0 for t' < t, where x' = (x', t') and x = (x, t); or

[mathematical expression not reproducible] (27)

where the parenthesis on the left and the bracket on the right of (27) are dimensionless.

Appendix A: Time-ordering operator T

The time-ordering operator [6] is defined by

T exp [-i [[integral].sup.t'.sub.t] dt" H(t")/h [equivalent to] (A1)

[mathematical expression not reproducible] (A2)

[mathematical expression not reproducible] (A3)

where the final equality comes from the decomposition of the Planck constant, [??] = [m.sub.*] [c.sup.2] [t.sub.*], in (A2).

Submitted on April 14, 2018

References

[1.] Davies P. Superforce: the Search for a Grand Unified Theory of Nature. Simon and Schuster, Inc., New York, 1984.

[2.] Daywitt W.C. A Model for Davies' Universal Superforce. Galilean Electrodynamics, 2006, v. 5, 83.

[3.] Daywitt W.C. The Trouble with the Equations of Modern Fundamental Physics. American Journal of Modern Physics. Special Issue: "Physics without Higgs and without Supersymmetry", 2016, v. 5(1-1), 22. See also www.planckvacuumDOTcom.

[4.] Carroll B.W., Ostlie A.D. An Introduction to Modern Astrophysics. Addison-Wesley, San Francisco, Boston, New York, Cape Town, Hong Kong, London, Madrid, Mexico City, Montreal, Munich, Paris, Singapore, Sydney, Tokyo, Toronto, 2007.

[5.] Daywitt W.C. The Planck Vacuum. Progress in Physics, 2009, v. 1, 20.

[6.] Fradkin E. Chapter 10: Observables and Propagators. Internet website: "Observables and Propagators, Fradkin".

[7.] Gingrich D.M. Practical Quantum Electrodynamics. CRC, The Taylor & Francis Group, Boca Raton, London, New York, 2006.

William C. Daywitt

National Institute for Standards and Technology (retired), Boulder, Colorado, USA

E-mail: wcdaywitt@me.com

Caption: Fig. 1: The Feynman diagram for the propagation of the electron core (-[e.sub.*], m) from (x, t) to (x', t') with no external scattering.

Caption: Fig. 2: The Feynman diagram for the propagation of the electron core (-[e.sub.*], m) from (x, t) to (x', t') with one external scattering at ([x.sub.1], [t.sub.1]).

Caption: Fig. 3: The Feynman diagram for the propagation of the electron core (-[e.sub.*], m) from (x, t) to (x', t') with one external scattering at ([x.sub.2], [t.sub.2]).

Caption: Fig. 4: The Feynman diagram for the propagation of the electron core (-[e.sub.*], m) from (x, t) to (x', t') with a double external scattering at ([x.sub.1], [t.sub.1]) and ([x.sub.2], [t.sub.2]).
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Author:Daywitt, William C.
Publication:Progress in Physics
Date:Jul 1, 2018
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