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Understanding the Dirac equation and the electron-vacuum system.

1 Introduction

The Dirac electron defined here is a massive "point" charge (-[e.sub.*], m) that obeys the Dirac equation and is coupled to the negative-energy Planck vacuum (PV) continuum via the two-term coupling force [1]

[e.sup.2]/[r.sup.2] - [mc.sup.2]/r (1)

the massive charge exerts on the PV. The electron Compton radius [r.sub.c](= [e.sup.2.sub.*]/[mc.sup.2]) is that radius from the center of the massive charge (in its rest frame) to the radius [r.sub.c] where the coupling force vanishes. The bare charge (-[e.sub.*]) itself is massless, while the electron mass m results from the bare charge being driven by the zero-point electromagnetic field [2] [3]; corresponding to which is a vanishingly small sphere containing the driven charge whose center defines the center of both the driven charge and its derived mass. It is from the center of this small sphere that the position operator r for the massive charge and the electron-vacuum complex is defined and from which the radius r in (1) emerges.

The PV model of the complete electron consists of two interdependent dynamics, the dynamics of the massive charge in the previous paragraph and the dynamics of the PV continuum to which the massive charge is coupled. An example of the latter dynamic is the (properly interpreted) zitterbewegung [4] [1] that represents a harmonic-oscillator-type excitation taking place at the r = [r.sub.c] sphere surrounding the massive point charge, an oscillation resulting from the vacuum response to the vanishing of (1) at [r.sub.c]. The point-like nature of the massive charge, in conjunction with the continuum nature of the PV, are what give the electron its so-called waveparticle-duality. Mathematically, the electron's wave nature is apparent from the fact that the spinor solutions to the Dirac equation are spinor fields, and it is upon these fields that the covariant gradient operator

[[partial derivative].sub.[mu]] = [partial derivative]/[partial derivative][x.sup.[mu]] ([partial derivative]/c[partial derivative]t, [DELTA])

operates. Thus the spinors are associated with PV distortion --with no distortion the gradients vanish, resulting in null spinors and the dissolution of (3).

The free-particle Dirac equation can be expressed in the form (from (A10) in Appendix A)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

in terms of the single constant [r.sub.c], a constant that normalizes the operator in (2). The free-space particle solution [phi], and the negative-energy vacuum solution X, for this electron-vacuum system are 2 x 1 spinors and [??] is the Pauli 2 x 2 vector matrix. The spinor solutions from the two simultaneous equations in (3) are strongly coupled by the inverted X - [phi] spinor configuration of the second term, showing the vacuum state to be an integral part of the electron phenomenon. (It will be seen that this coupling is even present in the nonrelativistic Schrodinger equation.) The negative spinor (-X) on the right is a manifestation of the negative-energy nature of the vacuum. Equation (3) expresses the Dirac equation in terms of the normalized PV gradients on the left of the equal sign.

What follows illustrates the previous ideas by reiterating the standard development of the free-particle Schrodinger equation and the minimal coupling substitution leading to the Pauli equation.

2 Schrodinger equation

The Dirac-to-Schrodinger reduction [5, p. 79] begins with eliminating the high-frequency components from (3) by assuming

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [[phi].sub.0] and [X.sub.0] are slowly varying functions of time compared to the exponentials. Inserting (4) into (3) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where the 0 on the right is a 2x1 null spinor. This zero spinor indicates that the mass energy of the free particle is being ignored, while the effective negative-mass energy of the "vacuum particle" has been doubled. In effect, mass energy for the particle-vacuum system has been conserved by shifting the mass energy of the free particle to the vacuum particle.

The lower of the two simultaneous equations in (5) can be reduced from three to two terms by the assumption

[absolute value of [ir.sub.c] [partial derivative][X.sub.0]/c[partial derivative]t] [much greater than] [absolute value of -2[X.sub.0]] (6)

if the kinetic energy (from the first equation in (A2)) of the vacuum particle is significantly less than its effective mass energy. Inserting (6) into (5) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

as the nonrelativistic version of (3). The mass energy of the free particle, and the kinetic energy of the vacuum particle (associated with the lower-left null spinor), are discarded in the Schrodinger approximation.

Separating the two equations in (7) produces

[ir.sub.c] [partial derivative][[phi].sub.0]/c[partial derivative]t + [??] x [ir.sub.c][DELTA][x.sub.0] = 0 (8)

and

[??] x [ir.sub.c] [DELTA][[phi].sub.0] = -2[X.sub.0] (9)

and inserting (9) into (8) leads to

[ir.sub.c] [partial derivative][[phi].sub.0]/c[partial derivative]t - [([??] x [ir.sub.c][DELTA]).sup.2]/2 [[phi].sub.0] = 0. (10)

Finally, inserting the Pauli-matrix identity (A12)

[([??] x [ir.sub.c][DELTA]).sup.2] = I[([ir.sub.c][DELTA]).sup.2] (11)

into (10) yields the free-particle Schrodinger equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where the two spin components in [[phi].sub.0] are ignored in this approximation. The scalar harmonic function

[[phi].sub.0] [right arrow] exp [-i(Et - p x r)/[??]] (13)

satisfies both equations as it should, and leads to the nonrelativistic energy-momentum relation E = [p.sup.2]/2m, where p = mv. The equation on the left in (12) expresses the Schrodinger equation in terms of PV gradients.

The vacuum property implied by (11), and the fact that [[phi].sub.0] is a spinor field, show that the vacuum state is a significant (but hidden) part of the nonrelativistic Schrodinger equation. The Dirac-to-Pauli reduction leads to the same conclusion.

3 Minimal coupling

By itself the coupling force (1) is insufficient to split the twofold degeneracy of the spinors in the free-particle Dirac (3) and Schrodinger (12) equations. It takes an external field to effect the split and create the well-known 1/2-spin electron states. The following illustrates this conclusion for the case of the minimal coupling substitution.

The minimal coupling substitution [5, p.78] is

[p.sup.[mu]] [right arrow] [p.sup.[mu]] - e[A.sup.[mu]]/c (14)

where e is the magnitude of the observed electron charge, [p.sup.[mu]] = (E/c, p) is the 4-momentum, and [A.sup.[mu]] = ([A.sub.0], A) is the electromagnetic 4-potential. Inserting (14) with (A1) and (A2) into the Dirac equation (A3) leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

which can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

in the 2x1 spinor formulation, where [a.sub.0] [equivalent to] e[A.sub.0]/[mc.sup.2] and a [equivalent to] eA/[mc.sup.2]. Then proceeding as in Section 2 produces

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

The Compton radius in (16) and (17) has been accounted for as a gradient normalizer. The remaining constants (e and m) appear only in association with the 4-potential [A.sup.[mu]]--if the external potential vanishes, the electron charge and mass are removed ([a.sub.0] = 0 and a = 0) from the equations, and (16) and (17) reduce to (3) and (7) respectively. Furthermore, the energy eA0 appears to increase the energy level of the negative-energy PV continuum. This latter conclusion can be appreciated by combining the two terms on the right side of (17):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where [a.sub.0] has been replaced by its definition. With a constant potential energy e[A.sub.0] = 2[mc.sup.2], the lower parenthesis vanishes and the free-space electron energy and the vacuum-energy spectrum just begin to overlap [1]. This latter result is the phenomenon that leads to the relativistic Klein paradox [5, p. 127].

If it is further assumed that

[absolute value of [a.sub.0] [x.sub.0]] [much less than] [absolute value of -2[X.sub.0]] (19)

then (17) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

which is the nonrelativistic version of (16). Then eliminating [x.sub.0] from the two simultaneous equations in (20) leads to the equation

[ir.sub.c] [partial derivative][[phi].sub.0]/c[partial derivative]t = [??] x ([ir.sub.c] [DELTA] + a) [??] x ([ir.sub.c][DELTA] + a)/2 [[phi].sub.0] + [a.sub.0][[phi].sub.0] (21)

for the spinor [[phi].sub.0]. Equation (21) then leads to the Pauli equation [5, p.81].

Using (A11) to calculate the square of the numerator in the first term on the right of the equal sign in (21) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

remembering that [[phi].sub.0] post-multiplies the square before calculation. The first term in (22) contains the electron's orbital angular momentum; and the second its spin, as manifested in the scaler product of [??] and the curl of the vector potential A. Using (A1), the corresponding spin operator can be expressed as 2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

where one of the charges (-[e.sub.*]) in (23) belongs to the massive point charge (-[e.sub.*], m) and the other to the separate Planck particles (-[e.sub.*], [m.sub.*]) within the PV. The product [e.sup.2.sub.*] suggests that the spin may be related to the interaction of the massive point charge with the PV charges when the vacuum is under the influence of a magnetic field B (= [DELTA] x A).

4 Conclusions and comments

The physics of the PV state [1,6] has provided a simple intuitive explanation for the Dirac, Schrodinger, and Pauli equations in terms of the massive point charge (-e,, m) and its interaction (1) with the PV. It is the ignorance of this coupling force that has obscured the meaning of the Dirac equation since its inception and, as seen in the next paragraph, the meaning of the zitterbewegung frequency.

The electron Compton relation [r.sub.c]m = [e.sub.*] in (A1) holds for both combinations ([- or +] [e.sub.*], [+ or -] m); so the vacuum hole ([e.sub.*], -m) exerts a coupling force on the vacuum state that is the negative of (1). The combination of the two forces explain why the zitterbewegung frequency (2c/[r.sub.c] [1] [4]) is twice the angular frequency ([mc.sup.2]/[??] = c/[r.sub.c]) associated with the electron mass (from Appendix B).

The purpose of this paper is to illustrate the massive-charge-PV nature of the electron phenomenon; and to reestablish the vacuum state as an essential and necessary part of a complete electron theory, that part that has been superseded by the idea of the quantum field. While the quantum field formalism, like the Green function formalism, is an important tool [5, p. 143] [7], the present author believes that the corresponding quantum field does not constitute an essential physical phenomenon apart from the dynamics of vacuum state (from Appendix C).

Appendix A: Dirac equation

The PV is characterized in part by the two Compton relations [1]

[r.sub.c][mc.sup.2] = [r.sub.*][m.sub.*][c.sup.2] = [e.sup.2.sub.*] (= c[??]) (A1)

connecting the massive point charge (-[e.sub.*], m) of the electron to the individual Planck particles (-[e.sub.*], [m.sub.*]) within the degenerate PV, where [r.sub.c] and m, and [r.sub.*] and [m.sub.*] are the Compton radius and mass of the electron and Planck particles respectively. The bare charge (-[e.sub.*]) is massless and is related to the observed electronic charge (-e) via the fine structure constant [alpha] = [e.sup.2]/[e.sup.2.sub.*]. From (A1), the energy and momentum operators can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A2)

the parenthetical factors implying that the operators, operating on the Dirac spinors, provide a measure of the gradients within the PV continuum. In the present free-electron case, these gradients are caused solely by the coupling force (1) and its negative (Appendix B).

The upper and lower limits to the PV negative-energy spectrum are -[mc.sup.2] and -[m.sub.*][c.sup.2] respectively, where m is the Planck mass. The continuum nature of the vacuum is an approximation that applies down to length intervals as small as ten Planck lengths (10 [r.sub.*]) or so; that is, as small as ~ [10.sup.-32] cm.

Using (A1) and (A2), the Dirac equation [5, p.74]

[ie.sup.2] [partial derivative][psi]/c[partial derivative]t + [alpha] x [ie.sup.2.sub.*] [DELTA] [psi] = [mc.sup.2][beta][psi] (A3)

can be expressed as

[ir.sub.c] [partial derivative][psi]/c[partial derivative]t + [alpha] x [ir.sub.c] [psi] = [beta][psi]

where the 4x4 vector-matrix operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A5)

where [??] = ([[sigma].sub.1], [[sigma].sub.2], [[sigma].sub.3]) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A6)

are the three 2x2 Pauli matrices. The 4x4 matrix operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A7)

where I represents the 2x2 unit matrix and the zeros here and in (A5) are 2x2 null matrices. The covariant gradient operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A8)

is seen in (A2) and (A4) to have its differential coordinates normalized ([partial derivative][x.sup.[mu]]/[r.sub.c]) by the electron Compton radius.

The 4x1 spinor wavefunction [psi] can be expressed as [5, p. 79]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A9)

where [phi] and x are the usual 2x1 spinors, and where the two components in each represent two possible spin states. The spinor [phi] is the free-space particle solution and x is the negative-energy hole solution. Inserting (A9) into (A4), and carrying out the indicated matrix operations, yields the Dirac equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A10)

in terms of the 2x1 spinors.

The following is an important property of the Pauli matrices, and the PV state (because of [??]): the vector Pauli matrix [??] obeys the identity [5, p.12]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A11)

where a and b both commute with [??], but are otherwise arbitrary three-vectors. Using (A11) (with a = b = [r.sub.c] [DELTA]) leads to

[([??] x [r.sub.c][DELTA]).sup.2] = I [([r.sub.c][DELTA]).sup.2] (A12)

which connects the normalized [DELTA] operator in the relativistic Dirac equation to the same operator in the nonrelativistic Schrodinger equation.

Inserting the operators from (A2) into (A10) and rearranging the result leads to the two simultaneous equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A13)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A14)

Then, pre-multiplying (A13) by ([??] + [mc.sup.2]) and using (A14) and (A11) leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A15)

and, after reversing the process, to an identical equation for. Thus both [phi] and X separately obey the Klein-Gordon equation [5, p.31].

Appendix B: Zitterbewegung frequency

The following rough heuristic argument identifies the two coupling forces that explain why the zitterbewegung frequency [1,4] is twice the angular frequency ([mc.sup.2]/[??] = c/[r.sub.c]) associated with the electron mass energy.

The force the massive point charge (-[e.sub.*], m) exerts on the PV is given by equation (1) which, using r = [r.sub.c] + [DELTA]r and [r.sub.c] = [e.sup.2.sub.*]/[mc.sup.2], leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B1)

for small [DELTA]r/[r.sub.c]. This yields the harmonic oscillator motion from Newton's second law

[d.sup.2][DELTA]r/[dt.sup.2] = ([e.sup.2.sub.*]/[mr.sup.3.sub.c]) [DELTA]r = -[(c/[r.sub.c]).sup.2][DELTA]r (B2)

with the "spring constant" ([e.sup.2.sub.*]/[r.sup.3.sub.c]) and oscillator frequency c/[r.sub.c]. The corresponding motion that is due to the vacuum hole ([e.sub.*], -m) (whose charge and mass fields exert a force that is the negative of (1)) is

- [d.sup.2][DELTA]r/[dt.sup.2] = + [(c/[r.sub.c]).sup.2] [DELTA]r (B3)

showing that the massive free charge and the vacuum hole cause identical accelerations within the PV continuum. The total vacuum acceleration is the sum of (B2) and (B3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B4)

with the corresponding harmonic oscillator frequency

[square root of 2[e.sup.2.sub.*]/[mr.sup.3.sub.c]] = [square root of 2] c/[r.sub.c] (B5)

which is [square root of 2] times the angular frequency associated with the electron mass energy. Given the roughness of the calculations, this result implies that the combined massive-charge forces, acting simultaneously on the PV continuum, are the source of the zitterbewegung with its 2c/[r.sub.c] frequency.

Appendix C: Quantum field

The PV is envisioned as a degenerate negative-energy sea of fermionic Planck particles. Because of this degeneracy, the vacuum experiences only small displacements from equilibrium when stressed. Thus the displacements due to the coupling force (1) are small, and so the potential energy corresponding to the stress can be approximated as a quadratic in those displacements. This important result enables the vacuum to support normal mode coordinates and their assumed quantum fields, as explained in the simple demonstration to follow.

The normal mode connection [8, pp. 109-119] to the quantum field can be easily understood by examining a string, stretched between two fixed points in a stationary reference frame, that exhibits small transverse displacements from equilibrium. In this case, the corresponding potential energy can be expressed in terms of quadratic displacements. If the displacements are represented by the function $(t, x) at time t and position x along the string, then the quadratic assumption implies that the displacements must obey the wave equation

1/[c.sup.2] [[partial derivative].sup.2][phi]/[partial derivative][t.sup.2] = [[partial derivative].sup.2][phi]/[partial derivative][x.sup.2] (C1)

where c is a propagation velocity. The string geometry leads to the Fourier series representation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (C2)

for the standing wave on the string, where L is the string length. Inserting (C2) into (C1) produces

[[??].sub.n](t) = [[omega].sup.2.sub.n][a.sub.n](t) where [[omega].sub.n] = n[pi]c/L (C3)

and where the amplitude [a.sub.n](t) is that of a harmonic oscillator.

The constant characterizing the Dirac equation is the Compton radius [r.sub.c]. So it is reasonable to set the string length L ~ [r.sub.c] to determine the fundamental frequency [[omega].sub.1] = [pi]c/L in (C3). Furthermore, the harmonics of [[omega].sub.1] can have wavelengths of the order of the Planck length [r.sub.*] (antiparticle excitation is, of course, ignored in this rough argument); so the length L can be subdivided

N = L/minimum length division of string ~ [r.sub.c]/[r.sub.*]

= 3.86 x [10.sup.-11]/1.62 x [10.sup.-33] ~ [10.sup.22] (C4)

times, and [phi] in (C2) can be expressed as an integral if convenient since [r.sub.*] [much less than] [r.sub.c].

The total energy of the vibrating string can thus be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (C5)

which, inserting (C2) into (C5), results in [8, p.117]

E = L/2 [N.summation over (n=1)] [absolute value of [rho][[??].sup.2.sub.n]/2 + [rho][[omega].sup.2.sub.n][a.sup.2.sub.n]/2] (C6)

where the first and second terms in (C5) and (C6) are the kinetic and potential string energies respectively ([rho] is the string density).

The crucial significance of (C6) is that it is a sum of independent normal-mode energies, where the [a.sub.n](t) are the normal mode coordinates. From this normal mode setting, the quantum field energy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (C7)

is defined, where [n.sub.n] is the number of normal modes associated with the wavenumber [k.sub.n] = [[omega].sub.n]/c. In effect, the integers [n.sub.n] ([much greater than] 0) determine the quantized energy level of each normal mode oscillator [a.sub.n](t). The 1/2 component in (C7) is the zero-point energy of the string-vacuum system.

At this point the quantum-field formalism discards the preceding foundation upon which the fields are derived, and assumes that the fields themselves are the primary reality [8, p. 119]. Part of the reason for this assumption is that, in the past, no obvious foundation was available. However, the demonstration here provides such a foundation on the simple, but far-reaching assumption that the vacuum is a degenerate state which can sustain a large stress without a correspondingly large strain.

Submitted on September 11, 2013/Accepted on September 21, 2013

References

[1.] Daywitt W.C. The Electron-Vacuum Coupling Force in the Dirac Electron Theory and its Relation to the Zitterbewegung. Progress in Physics, 2013, v. 3, 25.

[2.] Puthoff H.E. Gravity as a Zero-Point-Fluctuation Force. Physical Review A, 1989, v. 39, no. 5, 2333-2342.

[3.] Daywitt W.C. The Source of the Quantum Vacuum. Progress in Physics, 2 9, v. 1, 27.

[4.] Barut A.O. and Bracken A.J. Zitterbewegung and the Internal Geometry of the Electron. Physical Review D, 1981, v. 23, no. 10, 2454-2463.

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

[6.] Daywitt W.C. The Planck Vacuum. Progress in Physics, 2 9, v. 1, 2. See also www.planckvacuum.com.

[7.] Milonni P.W. The Quantum Vacuum--an Introduction to Quantum Electrodynamics. Academic Press, New York, 1994.

[8.] Aitchison I.J.R., Hey A.J.G. Gauge Theories in Particle Physics Vol. 1. Taylor & Francis, New York, London, 2 3.

William C. Daywitt

National Institute for Standards and Technology (retired), Boulder, Colorado. E-mail: wcdaywitt@me.com
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