Smooth and Quantal Properties of the Complex Wave  Working paper

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    Summary

Complex scalar waves in continuous space are capable of displaying a special kind of smoothness of dynamic properties, and this feature applies in particularly interesting ways to the Minkowski relativistic sorts of 4-spaces and the emergence of quantal phenomena.  One way of viewing the smoothness is as a tendency to concentrate energy in oscillatory modes at lower frequencies to a degree that is greater than is the case in purely (mathematically) real systems.  In these terms we may see why the famous "ultraviolet catastrophe" of early twentieth century physics models does not occur in nature.

The conventional approach to physics based upon primacy of the Lagrangian model of dynamics obscures this characteristic smoothness and consequently leaves an impression that the ultraviolet catastrophe is bound to occur.  Having started with such "noisy" models this impression is then very hard to dispel by reasoning, a problem that is rampant in late twentieth century quantum electrodynamics.  In contrast we pursue here an Eulerian model of continuous complex fields until, in the simplest sufficient form compatible with observed reality, it reveals the emergence of quantal phenomena in its behaviour.

The objective here is to explore and develop the especially smooth nature of interactions in the class of complex waves in a continuum so that models may be created that do not start with the disadvantage of implausibility at high frequencies.  Where such models are applicable, and that appears to be the case in the complex variable wave dynamics of sub-quantal physical processes, the resulting approach is often simpler both to understand and to manipulate than are models based on Lagrangian ideas.

    Introduction and Motivation

A wish to find better working models in electrodynamics motivates this work.  There are problems with the basic paradigm of twentieth century physics, in which a deterministic wave process provides the wave function from which is derived the probabilities of the state in which "each particle may be found".  An implicit abandonment of the principle of causality in such a model leads to difficulties in physics where it shows up as observable phenomena demanding the status of non-local causation.  Also in applications such as engineering the departure from underlying rigorous causal logic impedes comprehension and development by the individual non-specialist worker.

Here we study a model that is deterministic and continuous throughout, and in which the essentials of uncertainty and quantal phenomena arise as emergent in the inherent and unavoidable characteristics of the processes of observation by an observer who is also part of the wave continuum.  This is done at least in the hope of providing some useful approximate models that are more logically tractable than usual physics theory, and of exploring to expose and extend the limits of development of continuum wave models of this sort.

This approach does not set out to be different in its predictions about phenomena.  It is merely an alternative model in which the convenience of use is improved by greater intuitive value regarding matters of causality.  It will, however, require that the user of such a model remove the idea of "particle" from its primal position and replace it with something like "wave" or "field".  The photon, though having quantal energy, will be never more than an illusion created by the process of observation.  The electron will have no more than partial status as "substance", taking approximately stable form only when it is part of an atomic, molecular or crystalline structure, and behaving as no more than a special kind of waves when in free flight.  The nuclear particles such as baryons and hadrons will be regarded as having substantial form as solitons, but will not be dealt with in this essay beyond providing point nuclear charges upon which electronic fields may be centred.

The basis of analysing atomic electrodynamics is developed using a Klein-Gordon like equation to the point where the spontaneous emergence of charge and magnet quantisation is made evident.  The use of a Hilbert transform in deriving the charge and current origination from the ψ-field is essential, and is the seat of the non-Lagrangian "smoothness" referred to in the title.

So we set out by describing matter as De Broglie's waves in a non-Lagrangian locally causal continuous complex scalar wave model.  Within this model interactions, occurring via Maxwell electromagnetic waves, are described in terms of functions of the De Broglie wave state and its derivatives.  This is done on a basis that uses no real frame of reference in the complex plane.  In other words only relative values of the complex argument (phase) are involved in bringing about these interactions.  Such "relativity of phase" is important, and although easy enough to realise in a complex wave model such as when using the Hilbert transform, it presents great difficulties with concepts of either the Lagrangian or particle based sorts.  These problems are currently handled by means of gauge theories, whereas no such construct is necessary in the approach taken here.  From this model with its built-in Minkowski geometry all the necessary effects are shown to emerge that produce the non-linearity essential to give rise to spontaneous quantisation of electronic fields.

Triplet interactions of wave modes, usually between two resonant complex matter waves and one electromagnetic wave either resonant or propagating ... see Allen and Eberly [AE87], also form an important concept and this is developed in Appendix B.

For an introduction to some basic ideas about observation that offer a platform for this approach to modelling see the accompanying paper "Essential Structure in Physical Observation" [ABoES].

For those who have a conventional background in physics based upon use of the Lagrangian model there is further discussion in Appendix A of why its use is rejected here.

Apart from the nature of Cauchy-Riemann differentiability I have not yet found out and would like to know if and how the mathematical fraternity refer to this "smooth" nature that is peculiar to these sorts of complex waves.  They are characteristically smoothly rotative in the complex plane (N.B. that does not mean they necessarily rotate in space) with more or less uniform magnitude whereas real waves must oscillate in their real scalar dimension with repeated transitions through zero.  The difference from dynamic systems described by real variables is for this reason dramatic.

    Electronic Interactions

By way of our experiences of basic physical interactions at the atomic level we are most aware of the atomic spectral interactions with an electromagnetic field.  Each such interaction usually involves a Maxwellian radiation field and the simultaneous perturbation of the excitation of two atomic electron modes ... see [AE87].  This is the simple case, and the two electronic modes can alternatively be associated with larger structures such as molecules or crystals.  The process remains the same in that each interaction involves three elemental field structures, two of them De Broglie matter waves and the other an electromagnetic wave.  We can, if we choose, complete the model of interaction by terminating the electromagnetic wave at its "other end" in some other nearby resonant structure of matter.  Although this extends the range of dynamics possibilities the elements of the process remain similar.

Viewed in the above way the model of an atom may be seen as continuous in its space of possible electronic excitations, but discrete only in its state of charge quantisation ... say for instance its state of ionic charge.  By inducing shifts of the excitation state progressively away from equilibrium an otherwise stable atom can be made to increase its tendency to break from its existing ionic state and change to another.  This model accords with quite usual sorts of systems models with continuous state and discrete attractor points near to which the system fluctuates and between which it sometimes jumps.  The attractors are associated with the charge structure state, and not with any other aspect of the atom.  The excitation levels of the individual electron modes in such a model tend to their Fermi distribution levels (ref here) only as the result of electromagnetic interactions in a thermal milieu.  If that thermal interaction is strong then following a disturbance they will return quickly, and if not so strong, then slower.  They will always suffer some degree of thermal perturbation about their respective Fermi levels.

It is a widespread usage to consider these states of excitation in atoms to take only quantal values, but in fact there is no direct evidence of this since the only ways in which the quantisation can be observed is by means of the rather more explicit quantal changes of charge state within the processes that are essential to the structure of the observation apparatus ... see [ABoES].  The rules that dictate the relationships of these quantal effects are strict but have a form that allows us to remove them from the electron mode excitations without altering the extent of and limitation to prediction of behaviour of the system.  That is what I shall do here.  This possibility is rather widely believed not to be true, but that is the result of other assumptions about the possibility of objective observation, and I shall not make those assumptions here.  For many purposes this can actually simplify our model and is at least an interesting alternative to the so called "Standard Model" and the non-local "Copenhagen interpretation" that are current in quantum mechanics.

To see that the Fermi distribution comes about, and with it the related Planck distribution of mean energy levels as associated with their respective electromagnetic frequencies, we need only to recognise that non-linear interactions between three lossless complex wave systems at different frequencies produce signed power transfers into the respective systems that are proportional to the respective frequencies and depend upon the phase relationships (see Appendix B).  That causes the De Broglie standing mode excitation levels to conform to Fermi-Dirac statistics and produce the Fermi distribution whilst the Maxwellian processes that mediate the energy interchanges follow Bose-Einstein statistics producing spectra that correspond to the Planck distribution of radiation intensity versus frequency.

Herein lies a further heretical point of this model.  This is that the principle of power proportional to frequency in interactions applies in terms of "signed" frequencies ... we shall not be free to associate negative frequencies with anti-matter in this model, but fortunately that is not a fatal problem.  It does mean that we cannot use the usual ideas surrounding the Schrödinger and Dirac equation models as basic in this respect, but that difficulty will be solved by means of what is in fact a simplification and generalisation in our choice of equation form.  The conventional equations remain as approximations which are sometimes useful.

    The Causally Completed Model

In order to have a closed loop of effects constituting a causal model for electrodynamics we need to depict the origination of charge and magnetic field.  This will complete a cycle of causal influences whereby the Klein-Gordon equation mediates the form of electron wave in a given coupling field and the model of charge origination developed here below operating through Maxwellian propagation mediates the form of coupling field for the given electron wave and atomic nuclei.  The causal cycle is thereby closed.

Figure 1 -- Causal Linkages in the Wave Model

Each nodal dot represents a variable that has a value at every point in the four dimensional space-time.  The ψ-wave is complex scalar, the charge density is real scalar, the current density is real 3-vector and the EM (electromagnetic) wave is generally thought of as a related pair of real 3-vectors.

The four-space in which all these fields exist as wave functions is referred to as the Minkowski space to denote the way that one dimension, that associated with time, is treated differently from the other three "physical" space dimensions with their Euclidean geometric properties.  This is the basis of special relativity in the model.

The inner loops marked E and M deal with the dynamics of charge and current regulation.  Indeed, it is only the E loop that takes effect in determining the distinctions between the discrete attractor states of different ionic charge distributions.  Magnetic quantisation involving loop M is more subtle.  The remaining causal paths at the lower part of the diagram are via Maxwellian fields.  These latter effects are mainly observable as interactions between separate charge or current bearing entities.  The processes supporting inter-atomic molecular bonding lie in the hinterland where either or both of these sorts of causal systems may be involved, the upper loops dominating in what are called valence bonds and the lower loops in ionic bonds.

Forces of attraction and repulsion between otherwise intact substantial entities can arise with both sorts of interactions, but the steeply rising potential fields of repulsion (i.e. steeper than inverse square law with distance) typical of the van der Waals forces for close proximity must involve the upper loops in the diagram.

What we know as "force" is manifested in this model by the way that the overlapping of one intact substantial entity by the fields of another inherently introduces an acceleration of its wave structure that can only be nullified by the introduction of some other such overlapping field from yet another substantial entity.  It is the systematic quantification of the means for creating and maintaining these field overlaps that we refer to as "force".  These acceleration inducing, or momentum transferring, or, as we would account for them in this model, field deforming influences are mutual in certain effects between any two substantial entities, and fortunately these effects can be quantified consistently and abstractly, but nevertheless simply, by vector addition in physical space (Newton's laws).  However, it is only in respect of the interactions of supposedly otherwise intact separate entities that the concept can be given a reality.  In this model the moving waves are the reality and although lumpy, they are all of a piece and move under local causal field rules.  The forces are an abstraction only relevant to an observer that is itself a self identifying part of the system of waves ... you may hypothesise or postulate something or other involving forces, but with this sort of model you do not need to, and indeed cannot, observe (in the explicit sense of "measure") from outside of the system any forces, because there are none described by the model.  It is basically a kinematic model.

Thus we have here a type of abstraction that, in spite of having its own sense of absoluteness, is the opposite of the notion in the direct sense of an "objective" model.  In this model it becomes clear that we could never in practice be privy to the entire picture simply because we always have to be part of it.  The uncertainties in processes of both the quantal and thermodynamic sorts, both of which are fundamental to the processes of observation ... see [ABoES], arise from this nature.

By reference to this sort of diagram we may distinguish two meanings to the word "coherence".  The first, which I like to refer to as "quantal coherence", is the locking together into a substantial and persisting whole of a system of wave modes by virtue of their interwoven and shared charge distribution structure.  Such is the nature of physical solidity, and physicists refer to such a state as "condensed matter".  This is not the normal usage of the word "coherence" in physics, but may well suit the general conception of its lay meaning.  After all, at root to cohere means to "stick together".  The second, we could call it "wave coherence" and it is the more common meaning in physics, is the sense in which waves of either the De Broglie or the Maxwell sorts may form concentrations in bands of frequency sufficiently narrow that they do not change in relative phase by more than about a radian throughout the space and time of some sympathetic process of interaction.  They can then produce exaggerated effects, called interference, through "square of sum of amplitudes" as compared to the smaller "sum of squares of amplitudes" which is all that can be effective in processes where the phases are randomised or chaotic.  Seen on this basis the modal waves underlying the quantal coherence of a solid have to be wave-coherent within each mode, but are not so between the modes.

The details of this model must now be painted in.

    Charge and Magnet Origination

To make our closed causal model possible we must define how electric charge and current are manifested in terms of the De Broglie matter wave field.  The coefficients of the Klein-Gordon equation are influenced by the distributed form of the charge, and are therefore functions of space-time co-ordinates.  Thus we might refer to this form of the equation as a Modulated Klein-Gordon equation, shortened here to MKG.  In different forms of approximation the modulated coefficients may be introduced either in the matrix of coefficients of the second differential terms or in the non-differential term in the equation, as discussed below.  Such "modulated" forms of the equation are sometimes referred to elsewhere as "Non-linear Klein-Gordon equations", but they are in fact linear so long as the modulated coefficients can be taken as independent of the wave variable.  As they are used here below that is a valuable approximation.

Physics and engineering concepts of electric potential often appear different, one being absolute and the other relative.  In its effects within an atom as described by Schrödinger and Dirac equations, electric potential acts as though it is an absolute quantity whereas in relation to phenomena at greater distances it is manifested as being relative ... a difference between different points in space (consider the enormous potential differences of megavolts involved in a thunder storm that have no absolute effects upon local chemistry).

"Potential" is a concept applicable only in system states with zero relative velocities.  Such invariance of integral over different paths cannot be applied generally in the Minkowski 4-space.  Therefore it is not suitable as part of a relativistic dynamics model.

Here we deal with these coupling effects only via the Maxwellian field strengths and their rates of change.  These involve only local potential gradients and not absolute potentials.  To do this the couplings are all introduced into the constitutive Klein-Gordon equation via its differential terms, and in particular its mixed space/space and space/time terms.  The Schrödinger and Dirac equations are approximations to this model in which these gradients are pre-integrated to give potentials before installing them in the constant term in the differential operator of the equation.  Under those approximations the only possible form of motion consists of steady oscillations (i.e. having zero real part to their exponential characteristic) and at zero group velocity.

We may first describe the way that the electric and magnetic effects (charge and current) arise from the ψ-wave field.  Afterwards we shall look at the nature of the relative effects acting at a distance, and that will be in terms of the Maxwellian system of wave propagation.

To comply with the Planck constant proportionality between (modulus of) frequency and observed energy we need a simple model whereby charge will be generated locally as dependent upon intensity (i.e. squared modulus) of ψ excitation in proportion to both this intensity and to its (modulus of) frequency.  The following model does this and also suggests a form whereby the field can give rise to a magnetising vector in a related manner.  The resulting electromagnetic field can then return as a causal term in the mixed time-space coupling terms in the Klein-Gordon differential operator matrix.  It introduces a fixed linkage (that is chirally handed) between sequence of complex value of ψ and handedness of rotation of any directionally oriented form of motion.

Taking H_t( . ) to denote a time domain Hilbert transform (π/2 phase advance) we may see the densities of electric charge and magnetising vector (i.e. electric current) as generated by:

	Electric charge density Electric current density q(t, x)   =   ψ*(t, x).Ht(∂ψ(t, x)/∂t)                     − Ht(ψ*(t, x)).∂ψ(t, x)/∂t i(t, x)   =   ψ*(t, x).Ht(ψ(t, x))                     − Ht(ψ*(t, x)).ψ(t, x)	EHt

Use of a Hilbert transform in the time domain (a non-causal operator that is equivalent to advancing the phase by π/2 at each frequency in the Fourier transform domain) has been introduced in the model in the process of charge and magnet origination as dependent upon local ψ-wave.  Its inherent combination with the temporal derivative and final self product returns the overall process to causal form.

So although a non-causal operator (i.e. one having output dependent upon future input) is used, it occurs only in combinations in which the overall effect is causal ... it is no more than a local mathematical construct.  An alternative "all causal" model could be built using an explicit pair of real variables throughout in place of the parts of the complex variable, and prescribing the causal relationships using, where necessary, either of the two reals.  Use of the Hilbert transform with the complex variable can be regarded as just a shorthand way of doing the same thing.

The phase shifts in the Hilbert and differentiation operations are both π/2 and occur in pairs so that the operations are real overall.  This is sufficient to establish the causal nature (in the sense of requiring only temporally one sided integration) of the integral operators for charge and magnet origination.  (Additional proof would be valuable here.  Otherwise there would remain a temporal type of non-locality at this point in the model.) See below where the equivalent temporal frequency domain expression of this model indicates convergence to locality for evaluation with high temporal bandwidths.

In fact we can go farther over this issue of keeping the causality local.  The bilinear Hilbert/integration operation is instantaneous in so far as it can be for any particular temporal wave frequency.  Whereas either the integration or the Hilbert transform alone involves temporal non-locality (i.e. memory or prediction), the two together do not.  This, in spite of being unworldly, makes it seem curiously more tractable even though we are not used to it in macroscopic physics.  It is made possible by the polyphase nature of an "actual" complex variable.  It brings about the uncoupling of conjugate wave motions (i.e. those having oppositely signed frequencies).  Coupling between electronic waves is then related to algebraic difference of signed frequencies, not the difference of absolute frequencies as must be the case for a system of non-complex (monophase) variables.  The complex conjugate symmetry is thereby split and the existence of the so called "spin" is made manifest in individual wave modes (though spin involves also paired components of frequencies, and that will be discussed later).

The expressions for charge and current densities negate with relative time reversal.  The oddness of the Hilbert transform kernel function (which for relative time reversal does not reverse) sees to this.  This is consistent with anti-matter being reality-convergent in reversed time since negation of its real part is required in order to maintain the convergence.

We might feel concern over the way that these deliberations about charge involve only the time dimension and lack the usual relativistic symmetry across the four dimensions of space-time.  However the basic Planck relationship between frequency and energy for quanta is also asymmetrical in this same way, so we appear to be discussing an issue that is properly specific to the time dimension.  See Appendix E for further discussion of this.

Separation of the handling of terms at different temporal frequencies is rather more intuitive and convenient in the one dimensional frequency (Fourier or Laplace transform) domain than it is with time and space remaining as represented in a four dimensional partial differential equation.  This can be readily understood by visualising the atom in its four dimensional space-time using the natural units.  For all practical time periods between, and even within, state transitions the atom is much longer in the time dimension than in any space dimension, and is also strongly periodic along the time axis.  Even for a state duration of 10^-15sec the atom is still some thousands of times longer in time than its equivalent width in space.  Writing this in spatial Laplace transform form would require a convolution in the spatial frequency (wave vector) domain.  However the inter-term products of differing resonant frequencies are relatively small and for many purposes (e.g. electrodynamics) have little significance.  In particular these high frequency cross terms converge rapidly with increasing frequency, and this justifies the use of the Hilbert transform in the time domain representation because convergence of this integral in the frequency domain shows that the evaluation of the time domain function converges to local dependence as the bandwidth of the evaluation is increased.

So freed from any severe problems with these high frequency terms in our electrodynamics mission and presuming steady state ("eigen" conditions) with s = jω we may now express the charge and current origination formulae in their temporal frequency domain equivalent form:

	Electric charge density Electric current density q(x)   =   − ∫  |ω|.Ψ*(jω, x).Ψ(jω, x) dω i(x)   =   −j. ∫  sgn(ω).Ψ*(jω, x).Ψ(jω, x) dω	Echarge

See Appendix C for a discussion of how physical phenomena relate to this formulation of charge and current origination.
See also Appendix F for comments regarding the involvement of modulus and sign of frequency in these expressions.

The variables q(x) and i(x) are here depicted as having no dependence on time ... they are constant fields.  However, there is no reason why this model should not be used to accommodate relatively slow variations of these fields, and here "slow" means much slower than the oscillations of the order of the Compton frequency (1.2356×10²⁰ Hz) in the eigenmodal components making up Ψ(jω, x).  Thus to do this we may write the same equations in terms of short term averaged values Ψ(jω, x, t), q(x, t) and i(x, t).

From the above expressions the electromagnetic field strengths may be derived in terms of a Huygens wave integration of all the sources over space.  Here is an expression based on speed of light Maxwellian propagation delays:

	Electric vector near field Magnetic vector potential near field u(t, x)   =     q(t − |x − ξ|, ξ)  d3ξ |x − ξ|.(x − ξ) w(t, x)   =     i(t − |x − ξ|, ξ)  d3ξ |x − ξ|.(x − ξ)	E5

Note the divisions with vectorial divisor and a dividend that is scalar in the electric case and vector in the magnetic.  For a description of the form of this operation and further notes on the meaning of this please refer to Appendix D.

These equations define the Maxwellian system of coupling wave propagation in free space for near fields.  We can use the free space form because our model includes the sources that would create or disturb any such waves.  Screening and increase of refractive index occurs through the interaction of matter in the propagation path.  It is no more than the presence of an interactive response term in the matter that acts to scatter, reflect and delay the incident Maxwellian wave.

For atomic structures the propagation delay of field perturbations is very small in relation to the periods of state evolution (i.e. in relation to the reciprocals of real parts of the system poles), so the time delay terms can usually be omitted for these cases and a simpler prompt model can be used instead.

	Prompt electric vector near field Prompt magnetic vector potential near field u(t, x)   =     q(ξ)  d3ξ |x − ξ|.(x − ξ) w(t, x)   =     i(ξ)  d3ξ |x − ξ|.(x − ξ)	E6

Regarding the vectorial divisions, as was the case for E5, for a description of the form of this operation and further notes on the meaning of this please refer to Appendix D.

To set up the nuclear charges upon which to base simple electrodynamics models equations E5 or E6 can be evaluated with the requisite fixed concentrations of charge located at the desired nuclear centre(s) to be added to any fields produced by the electrons via the charge and current formulae EHt or Echarge.

The combination of the Klein-Gordon equation governing electron wave and the charge origination formulae with Maxwell equations for propagation of electromagnetic field stress is the complete definition of the electrodynamics model as used here.  Our purpose is now to explore features and properties of its dynamic behaviour in the vicinity of atomic nuclei (but not inside those nuclei ... that remains for further extension work on the model).

    Charge Quantisation

We now set out to describe both structurally and quantitatively the way in which the continuum charge in the electron fields of atoms, molecules and crystals becomes organised so as to behave in quantal units.

Quantal behaviour occurs in phenomena of both the original charge and the original magnetising current kinds and they are both produced in the same type of wave field.  The charge quantity is connected with the number of half wave ripples in a real wave (with actual zero crossings) along a path starting at a positive nucleus and ending by tapering away at great distance, and the magnetic quantisation is connected with the number of ripples (or rather, argument rotations of a complex variable having no zero crossings) along a path that forms a closed loop.

For an initial grasp we can concentrate upon the charge phenomena alone.  This will allow us to make non-relativistic approximations whilst we come to a picture of the "orbitals" surrounding static structures of atomic nuclei.  We shall then be in a position to augment this picture with the quasi dynamics of wave-orbital magnetic effects whilst still avoiding total generality, but with a view to how this might then be generalised by various forms of perturbations, both static and dynamic, for yet further extensions of validity of the working approximations.

About the wave equations:
Here are examples of the Modulated Klein-Gordon (MKG) equation with the units scales, including charge, explicit in their conventional form and omitting the magnetic terms as unnecessary for a simple static model of an atomic system.  This omission would affect resonant frequencies (c.f. Lamb shift), but not the value of the charge quantum.

We could (but I do not) use the Cartesian MKG equation to define an electron wave system that is structured only by the Coulomb potential of a nuclear point charge at the origin.  That would be reminiscent of the potential notion of a Lagrangian model giving a conservative field where energy level for a charged particle would be a single valued function of spatial position.  Such an equation, having the spherically enclosed charge q as merely a function of spherical radius |x| and independent of ψ, would look like this:

E7

That sort of equation can yield eigen functions that are broadly similar in frequencies and spatial geometries to what follows, but it omits some essential form, and in particular it does not quantise spontaneously.  It is a second order version of the usual first order temporal differential Schrödinger equation model for a single atomic electron mode.

For the electrodynamics modelling purposes of this essay we stay with a point centric approximation for the nuclear charge, but we add the effects of distributed charge of the electronic waves themselves throughout the space that they occupy.  We can introduce these effects via the coefficients of the second order spatial derivative terms as a tensor matrix that ultimately reduces in its effects to the magnetic vector potential w(x), and via the coefficients of the zero and first order terms as the electric field u(x).  That is a model capable of handling magnetic and dynamic effects.

For simple steady state cases only (continuous oscillatory and no externally applied fields) we may introduce the effects solely via the non-differential term in the equation.  When this is done the requisite function div u(x) ≡ ·u(x) has the mathematical properties of a potential in that it is a single valued scalar field.  In these simple cases it has a form roughly similar to the Coulomb field for a nuclear charge but it is not the electric potential ... a confusion that must be avoided.  This is not a Coulomb potential (stressing again that this is not a Lagrangian model).  The divergence of the electric field div u(x) is in fact just −4π times the charge density function.  The static local electric potential p(x) would need to satisfy grad p(x) = −u(x), but we do not use that.

So we can then write the Cartesian Modulated Klein-Gordon (MKG) equation with distributed charge density equal to the divergence of the electric field as:

[D2 − c22 + ωC(ωC − c.div u(t, x))] ψ(t, x)   =   0

E8

Remember however that we are working here below with the charge in basic (hypothetical) units of q_C = √(4π/z₀) and in basic (hypothetical) underlying units = ω_C = c = z₀ = 1.  These units must be hypothetical for these original definitions because they remain to be adjusted for scale as we arrive at the effects the model has on the processes of measurement of these very constants.

I shall for general purposes later use the wave equation written in terms of the field strengths.  For the electric field these are the gradient of potential, and for magnetic effects they are the so called "vector potential".  However for evaluations limited to the steady state (steady oscillatory) condition a compact form of the wave equation written in terms of charge density div u(x) with Modulated Klein-Gordon (MKG) form in Cartesian co-ordinates is then:

[D2 − 2 + 1 − div u(x)] ψ(t, x)   =   0

E9

Note that if we could remove the dependent variable charge term in E9 then that equation could take the Helmholtz differential equation form.  So we attempt this by setting up the wave variable as the product of two independent wave functions.  We then have the opportunity to collect the differential terms of the product in such a way that the effects of the distributed charge appears through u(x) only in one of the equations, and that one is then of a Non-Uniform or Modulated Helmholtz (MH) form that will be described here below.  The result is not as simple as the conventional Helmholtz form that we would have liked, but it still has certain valuable properties about it, and they are enough for our present purpose.  This leaves the remaining factor wave equation for each individual electron field (orthogonal modes) as a separate equation in which the solution of the former equation appears as a structure defining constant factor field.

Seen this way the process is bilinear. We shall break it into two linear processes, each depending for its form on the other, with the two sub-processes thus forming a circular causal loop.  One of these processes describes the distinguishing part of the complex variable motion of any one individual electronic mode, we may call it g_n(t, x) for the nth mode, and the other is the aggregate of the charge origination effects of the totality of electron modes and nuclear charges present that we shall call H(x).  The latter factor is common to all of the electron wave modes in the given atom, molecule or crystal.  Thus, through this bilinear product ψ_n(t, x) = g_n(t, x).H(x) and the quadratic charge origination process as described in Echarge we have set up a basis of third degree nonlinearity that is essential if spontaneous emergence of quantisation in a solitonic form is to be possible, yet the model is partitioned into an appealingly simple pair of quasi-linear factor systems.

For electrodynamics modelling this is an attractive proposition because the periods of the electronic oscillations are very much shorter than those of the charge fluctuations, so the aggregate charge effects act like a nearly constant structure controlling the form of the individual electron mode waves.  Also the charge distribution is formed as the sum of the individual electronic modal charge distributions each of which displays the smoothness described above, i.e. they do not oscillate at double frequency in their self quadratic effects as a Lagrangian model would tend to suggest.  Further this aggregation of mode charges tends to remove the effect of charge fluctuations by averaging, especially for heavy atoms or high order modes in molecules, crystals etc.  (Compare this simplicity with the difficulty of the modelling approach through the "many body system" concept under a Lagrangian particle model.)

Types of Equation
To clarify the type of equation we are about to use first consider these basic forms of differential equations describing a three dimensional scalar field.  We omit time from this discussion since it will be taken care of in the other equation of the pair, and indeed is reduced in effect to a complex constant for the steady state oscillatory solutions:

2 y(x)   =   0

... Laplace equation -- altogether uniform and symmetrical - produces harmonic functions.

2 y(x) + f(x)   =   0

... Poisson equation -- adds the effect of non-uniformly distributed divergence (e.g. charge).

[2 + k] y(x)   =   0

... Helmholtz equation -- incorporates a scalar operator on the field but is spatially uniform, typical for the free motion of a resonator with uniform medium and no other structure.

[2 + f(x)] y(x)   =   0

... This is a generalisation combining the nature of both Poisson and Helmholtz equations -- it is the form of a resonator with non-uniform density of its wave medium.  Until I can find a mathematician's name for this equation let us call it the Modulated Helmholtz (MH) equation.

It is the modulated Helmholtz (MH) form of equation that we shall use to carry the effects of the irregular charge distribution as a factor in the model.  We can take a look at some properties of the waves that are its solutions.

We shall be particularly interested in those cases of the MH type of equation where the modulating coefficient function f(x) is real.  In such cases the solution y(x) is also real and may be expressed as the sum of a pair of complex conjugate components y₊(x) + y₋(x).  Can we establish a relationship for one such complex term as developed along any contour S of maximum gradient of y(x) that is:

This looks like a one dimensional Poisson differential equation in the log of the wave function with a real function of space as the source term.

If there exists another function u(x) that is a vector satisfying div u(x) ≡ ·u(x) = f(x) then from the original MH equation we can also write:

y(x)   =   u(x) − u0(x)

By treating u(x) as the electric stress field then f(x) is the charge density.  Thus y(x) takes the form of that part of the electric potential that is the negative integral of the vector field u_D(x) plus an arbitrary constant.

So what is this special part of the electric field u_D(x) ?

The solution of this real scalar MH equation in three dimensions is dependent only upon div u(x).  Interference between the two complex conjugate terms forms ripples in y(x) with distinct zero surfaces, i.e. where the complex arguments of the two conjugate components are ±π/2.  In terms of either complex component part of the real solution, there are surfaces of constant complex argument.  These surfaces are always simply closed, and disjoint from any other such surfaces, except for those surfaces with the argument ±π/2, the zero surfaces, that may intersect at lines.  Intersecting the constant argument surfaces perpendicularly we may construct our S-contours.

In order to demonstrate that quantal resolution must occur within the model propounded here we shall need to show that due merely to the properties of the MH equation the rate of rotation of the argument of y₊(x) is everywhere the product f(x).y₊(x) divided by the growth rate in the local direction of the S-contour of the log of the right cross-sectional area of any small S-contour bundle.  We need proof of validity for this conjecture.

Using a construct of this MH sort the following procedure extracts the effects of charge as a separate factor with ripple form, leaving behind a residual equation for each electronic mode that, although subject to ripple distortions due to the separated charge factor, still has a linear wave equation form.  This relies upon the effect (a constitutional necessity when the equation is factored in this way) that the zero surfaces of ripple caused by the common charge distribution shall be identical for all modes.  There can be additional zero surfaces in those modes having principal quantum number n > 1, but these do not affect the reasoning here.

Consider a factor differential process that produces ripple with zero surfaces that partition the charge spatially.  This will only produce the correct results if the actual modal distributions are also calculated and used, and the execution of such a calculation is a recursive procedure.  However, for the purpose of deducing the magnitude of quantal charge segmentations by the ripple the reasoning can be based upon the phase of the ripple itself.  So long as this can be shown to segment the charge in a manner that is invariant over the space of possible electronic modal forms in all their additive combinations then the basis for constant levels of quantisation may be thereby established without resort to the calculation of actual modal distributions or relative excitations.  Given such invariance the ripple function can be separated as a factor common to all electron modes in the given atom, molecule or crystal.  Only the total charge origination will then be of importance, and is in any case all that is relevant to the quantisation proof.

Such a ripple factor process will require that the qualitative properties at its boundaries regarding charge state attractor stability and convergence lead to stable quantal charge operation.  It is this which determines the permitted stable states of ionisation.  The outermost zero surface of the ripple for ionised structures will take the form of amplitude convergence as the first order exponential of distance (quadratic exponential for convergence of wave intensity), or to a higher (exponential) order of exponential convergence for electrically neutral structures.

We could establish a picture of the quantal formation process by means of the scalar modulated equation in E9 converted into spherical radial co-ordinates, and that has a certain attractive simplicity.  However, the interpretation of deviations of the structures from pure spherical symmetry and the generalisation to dynamic situations are not then readily evident.  So I shall use the more general form of the wave equation here in which the coupling terms appear only as field strength coefficients in the elements of a differential operator coefficient matrix.  This approach has the further advantage that it can be used in a similar form for derivation of the magnetic quantisation process where a simple potential function cannot in any case be used.  Unfortunately this way the one dimensional radial form of the especially simple (near) spherical symmetry of atomic cases is not so readily visualised and needs a little more thought.

Vector Differential Operators:
Many issues herein could be expressed very compactly in tensor notation, but our mission here is topological comprehensibility and not conciseness or mathematical generality for its own sake.  Instead I shall mostly adhere to the layman's sense of distinctness between time and space and treat the co-ordinates as a compound 2-vector.  The first element of this vector is a real variable ... time as we all know it, and the second is a 3-vector of space co-ordinates.  Thus together they make a 4-vector of real scalar elements.

Such a 4-vector is heading in the direction of similarity to a quaternion.  Indeed, it might be possible to use that notation as a close parallel with the notation used here, but the technique used with quaternions of combining the Minkowski geometry into the symbolism again reduces clarity of visualisation.  I believe it to be better left explicit for the purposes here.

Thus the separation into time as a scalar real variable and space as a 3-vector real variable is both symbolically practical and is more suited to commonplace visualisation, which is very desirable.  The incorporation of the Minkowski metric signature (− + + +) is then left to the signs of elements in the matrices used to create the equations.  A slight awkwardness arises in that the two sorts of elements ... namely scalar and vector ... call for different symbolic representation, and that leads to tricks like attempting to use bold font for the elements that are vectors.  I shall try to do that.

Because we need to use both four dimensional expressions as basic definitions and three dimensional expressions where static approximations occur we need to make sense of the distinction between the respective differential operators.  First we combine the D ≡ ∂/∂t and differentiations into a single four dimensional operator denoted by the lozenge symbol ◊.  I avoid here the square symbol called "box" or "squabla" because, at least sometimes, it is used to represent specifically the Minkowski four dimensional second order differential operator ... the d'Alembertian.  By embedding constants in this ◊ operator (and, indeed, also in D and when necessary) the time and space units can be normalised.  So we use τ_C = 1/ω_C to represent the reciprocal of the Compton angular frequency and c as the speed of light.  However, as explained above, the formulation used here does not go as far as putting the Minkowski metric into the vector differential operator (it would require a multiplication of √(−1) in the first element).  So the normalised 4-vector differential operator in terms of equivalent conventional symbols is:

◊   ≡   [D, ]   { ≡   τC [∂/∂t, c.∂/∂x, c.∂/∂y, c.∂/∂z]in practical units}

Eloz

... where no transpose symbols T are used because vectors written in line of text are taken to have column form.

Using this symbolism the second order Minkowski differential operator (the d'Alembertian) can be written as ≡ ◊^T·M·◊, where the matrix M contains the Minkowski metric thus:

M   =     −1,   0  0 ,   I

Emink

For cases where we wish to analyse a system that can be described as linear with constant or perhaps very slowly varying parameters then to achieve improved separability of its dynamic terms we shall wish to express it in terms of the frequency domain transform of its wave variable.  Thus a complex wave variable ψ(t, x) is replaced for these static or very slowly varying systems by its Laplace transform Ψ(s, x) where s ≡ σ + jω is the complex variable of the generalised frequency domain corresponding to the time dimension.  More general vectorial transformation involving the spatial dimensions might also be entertained, but that is not the case here.

In the case of slowly changing linear systems then the system dynamic coupling coefficients can be treated as being slowly varying functions of time, and the analysis can follow as a slowly perturbed result derived from the steady state behaviour.  The restriction to "steady state" includes the cases of multiple modes each in steady oscillatory motion.  The question of how slow is "slow change" then rests upon whether the (eigenvalue) variable s_n associated with the motion of any such nth mode can be regarded as having the perturbation of its real part sufficiently small in absolute ratio to its imaginary part for all modal frequencies of interest in the analysis.

Hybrid Differential Operators:
To allow expressions corresponding to differentials treated symmetrically over space and time (as is typically done in relativistic considerations), but using the frequency domain for the time dimension, we shall need a hybrid form of the differential operator.  Thus a frequency domain hybrid lozenge operator ◊_s can be defined as:

◊s   ≡   [s, ]T   ≡   τC [s, c.∂/∂x, c.∂/∂y, c.∂/∂z]T

ElozF

      ... where s ≡ σ + jω, and s = jω is a valid substitution for a steady oscillatory state.

In this hybrid operator the first element operates multiplicatively upon the elements of left or right entities whilst the remaining differential elements operate each as though convolved with elements of its operand that is the neighbouring entity immediately to its right.  In this latter respect we consider a differential here to be the integral of the operand weighted by a smooth doublet function centred around the given coordinate value, with unit moment, that is then taken sufficiently close to its narrow limit to be valid.  So the first element of the hybrid operator commutes with elements to left or right which are functions (i.e. not differential operators), but the differential operator element, although in succession of differentiations still commutative over the same or anti-commutative (sign reversal) over different independent variables, cannot commute with its operand function that is to its right.

We may write a generalised form of the complex scalar Klein-Gordon partial differential equation with added coupling terms in both the space-space and the mixed space-time derivatives as:

I plan to work with the time dimension in the frequency transform domain whilst keeping the space dimensions untransformed.  Transforming all of the dimensions or none would also both be possible, but the hybrid arrangement is in many ways more convenient for visualisation.  We effectively think of the problem in terms of one component temporal frequency at a time along with its respective explicit physical spatial wave function geometry/topology.  We use s ≡ σ + jω (for our concerns here usually σ = 0) as the complex variable in the frequency domain, and the suffix n is used when we are considering in isolation the motion of the nth eigenmode.  This basis of analysis is valid so long as the resulting system parameters vary slowly enough to be regarded as changing negligibly over the period of any integrations that need to be performed in the time dimension.  For quantum electrodynamics analysis of interest here this is nearly always the case.

So we can write the hybrid frequency domain form of the equation for the nth eigenmode, and because time is no longer explicit it takes the matrix modulated Helmholtz equation form:

Then the hybrid frequency domain description of the full wave function is:

We may then dissect and define the parts of this matrix A as follows:

A   =       −1  ,   uT/s  u*/s , I + W

EmatA

W   =      0  , w12, w13 w21,  0  , w23 w31, w32,  0       where   wjk   =   wkj*

EmatW

Note that the overall matrix A is Hermitian, i.e. a_jk = a_kj*.  Furthermore (or at least for the cases in which we are interested here), u is all real and W is all imaginary.  In this model the electric stress field as usually handled in the macroscopic Maxwell wave system appears as the real vector u, and the magnetic stress field is brought about by the imaginary matrix W.  This matrix W is a tensor that operates on Ψ to generate the matter interaction effects of the magnetic stress field of the Maxwell equations.

Further development of the model regarding the origins and more general behaviour of this vector u and matrix W is possible, but for the purposes of the derivation of a quantisation model this description is sufficient.  I shall defer enunciation of the simplifications it involves for a separate presentation ... see Dynamics of Complex Waves.  However it is worth noting that the division by s = jω in the space-time coupling terms in A (see EmatA above) balances the multiplication by |ω| in the charge origination formula in Echarge above.  As indicated by the modulus, the charge effect is insensitive to the sign of the originating source wave frequency (the mandate of modal stability dictates that radiation must be that way), and the Maxwell wave as used here carries the coupling effects devoid of that information.  Nevertheless it enters into interactions in a way which is sensitive to the sign of the frequency of any such mode affected (there is no restriction due to any stability consideration there).  If the latter were not the case then the effects would have a dependence upon the absolute rather than the signed frequency of the affected mode, and that does not happen in reality (Zeeman effect).  Clarification of the model underlying these mechanisms will be left for development elsewhere.  Suffice it here that these relationships are defined.

Working in one particular temporal frequency with steady values of u and W replaces the modulated Klein Gordon by a modulated (and spatially damped) Helmholtz equation.  So expanding the terms in EMHM above we may write this Helmholtz equation for mode n under the effects of the total charge from all modes as:

[T·{(I + W(x))·} + (u(x) + u*(x))T· + ωn2 + (T·u*(x)) − 1] Ψn(s, x)   =   0

E10

Note particularly here that the differentiated product ^T·(u*(x).Ψ_n(s, x)) arising from equation EMHM produces two terms in the differential operator in E10, one a spatial derivative operator u*(x)^T· and the other a scalar constant term ^T·u*(x).  It is only this last term that becomes involved in the process that determines the magnitude of the quantal charge.

Now we collect the elemental differential terms, noting that u(x) which is the electric field strength vector is real, so that its divergence ^T·u(x) is −4π times the charge density.  Then using the inherent anti-commutativity of mixed second partial differentiation ∂²/(∂x∂y) ≡ −∂²/(∂y∂x) and the substitution of jv ≡ j[v₁, v₂, v₃]^T = [w₂₃, w₃₁, w₁₂]^T so that the real vector v(x) is the precursor of (it is the curl of) the magnetic vector potential field strength, we can build the identity ^T·W(x)· = j2^T·(v(x)^T×), and the equation for an electron mode becomes:

Electron Mode Equation [2 + 2u(x)T· + j2T·(v(x)T×) + ωn2 + (T·u(x)) − 1] Ψn(s, x)   =   0	Emode

Under some conditions we may take advantage of the vector identity  ·(v(x)×) ≡ ×v(x)· + v(x)·(×)  to simplify the above equation.  In any closed analytic region of the wave function the second of these two terms is identically zero.  There we are free to use solely the definition of the magnetic vector potential via curl of v(x) so that  a(x) = ×v(x) .  Then the divergence magnetic term in the differential operator may be replaced by an equivalent inner product as  ·(v(x)×Ψ) = a(x).Ψ .  This often produces a more readily grasped picture of the magnetic effects in a wave function for those cases where it is valid.  However this simplification is not valid when integration around a spin axis singularity is involved.  An example of this is the Aharonov-Bohm experiment.  In these more general cases it is necessary to perform the integrations via the agency of the function v(x) and taking account of the paths of integration.

The model includes no explicit expression of field induced from outside of the closed causal structure of the system being described.  These are, of course, the Maxwellian real vectorial fields that are the basis of electromagnetic phenomena.  It is indeed possible to add electric and magnetic field terms that are arbitrarily imposed, subject to their Maxwellian form, that result from charge and current structures outside of the system being directly modelled.  These electric field and magnetic vector potential field strength terms add respectively into u(x) and v(x).

Since such remotely induced fields can have no divergence because they are not associated with local charge or current they do not affect the term in ·u(x) appearing in the constant part of the differential operator, so have no effect upon the process determining charge quantisation.  In contrast to this the addition to the magnetic term occurs before the associated divergence operation in  ·(v(x)×Ψ)  so the process of magnetic quantisation is not in the same way immune to these externally imposed fields, though it does end up keeping the magnetic field quantised.  The magnetic coupling effects come through the relevant parts in the first spatial derivative term in the operator, and there they influence the eigen frequencies of the modes.  By this means they cause current circulations (even for a paired mode state) which act to regulate the magnetic field passing through the modal system.  Also in the case of tapered magnetic fields they produce phase twist in the modes which leads via induced phase velocity to accelerative effects.

The most simple and important cases of externally induced fields to be considered are those that are essentially of uniform stress in a single spatial direction throughout the model, and either constant in time or subject to an oscillatory modulation that may be described in terms of their frequency domain spectra.  To check the validity of the equation Emode as a model of testable physical phenomena under these sorts of fields we may investigate its compliance with observed spectral absorption and emission effects and Stark, Zeeman, Paschen-Back and Stern-Gerlach effects as outlined in Appendix C.

We substitute a product of factors for the wave variable as Ψ_n(s, x) = F_n(x).(H(x, t).G_n(s, x)) where:

Please refer to Wave Topology of a Spin Mode for further detail on the form of the spin mode-factor.

We can now partition the differential operator terms into groups corresponding to these factors to produce a form of separation of variables.  Let us for the time being leave each F_n buried within its respective G_n.  The validity of this as an approximation remains for further study here. Then we can consider just a pair of factors H and G_n associated respectively with differential operator partitions A and B so that [A + B](H.G_n) = 0.  Then so long as we can rely upon the forms of H and G_n being incoherent, orthogonal or in some substantial degree uncorrelated then this implies that both A.H = 0 and B.G_n = 0.  We must assume this in what follows.

We might describe this model as having "stochastically separable variables".  It is justified because for a wide range of electrodynamics problems the temporal frequency spectrum of the modal form corresponding to G_n is around the Compton frequency (1.2356×10²⁰Hz) whilst the dynamic perturbational spectrum of H, being associated with the charge distribution that is static or only slowly varying, is at a minute fraction of that frequency (seldom significant above 10¹⁴Hz).  We have the spin factor F_n buried within G_n, but in any case it is a standing wave and orthogonal to H.  So this condition is almost always acceptable and is valid for the form of modelling used here.  (Work is planned to improve the handling of F_n here.)

In response to any challenge to the validity of this form of approximation involving different orders of dynamical rates of change or of widely differing frequency bands, we may counter with the defence that the notion of "quantised variable" depends upon a concomitant notion of "steady state".  In the states of transition between what we think of as steady states the meaning of quantal value is in any case severely weakened.  At best we then have an underlying continuity and conservation of charge (and something corresponding for the magnetic effects).  We are bound to approach considerations of quantal phenomena in terms inevitably somehow related to what we call "steady states", and allowably also their weak perturbations.

Further, we know that the G_n are a number of different functions, and can have their own zero surfaces in different and unrelated places.  Therefore the only way that the function H could encapsulate the effects of the charge distribution (that is common to all modes) is for it not to depend on G_n at all.  So it must be based upon the whole of the charge bearing term (i.e. the divergence of the electric field) and also needs to be a second order partition of the differential operator terms since nothing less could introduce the spatial oscillations in H necessary to produce its own zero surfaces.  The only way that it can meet all of these requirements is for it also to carry only terms in the zero order differential of G_n so that G_n in all its forms can then be cancelled out from this equation in H.  See also the inset section following equation E12 below where including fractions of the spatially constant part of the non-differential term of the mode equation within H is ruled out.  Regarding how the alternative "ionised" solutions can occur refer to the Discussion section below.

Leaving as a built in part of each individual electron mode its own spin factor but separating out the ripple factor H(x):

[2 + 2u(x)T· + j2T·(v(x)T×) + ωn2 + (T·u(x)) − 1] (H(x).Gn(s, x))   =   0

E10a

... where H(x) is real (i.e. expressible as the sum of complex conjugate terms) and G_n(s, x) is complex.  We may expand the derivatives of the product in abbreviations as:

HG″+2H′G′+H″G + 2u·(HG′+H′G) + j2·(v×(HG′+H′G)) + (ω2 + ·u − 1)HG   =   0

E11

So we next select a set from the terms in the zero differential order of G_n (so that we can cancel G_n right out) sufficient to define steady state charge-induced spatial ripple in common across all modes, that includes the influence of the electric field upon the real function H and that converges to zero amplitude at the origin and at infinite distance.

H″G + (·u)HG   =   0

E12

Alternative forms for H
Notice regarding E12 that if the terms in the first spatial derivative of H containing u and v were included here it would alter the way that this equation determines the fields so produced, but any such states for the model will drift under thermal interactions until these first spatial derivative terms vanish.  As an alternative choice of this sort we could include the terms in the first derivative of H as shown in E12a.

H″G + 2u·(H′G) + j2·(v×(H′G)) + (·u)HG   =   0

E12a

The function H may take different forms under the presence of differing amounts of these terms so long as the G_n make up the remainder.  We may see that all such solutions involving non-zero amounts of the first derivative terms in H would, all other things being equal, tend to have longer spatial periods in H.  This may be readily grasped by noting the way that adding a damping term into a second order ordinary differential equation always decreases the imaginary part (i.e. frequency) of the solution eigenvalue.  However, longer periods cannot solve for simultaneous zero values of H at the central and peripheral boundaries so instead these "impure" solutions must actually take the alternative of having higher energies to produce the necessary charge densities that can maintain the zero requirements at the boundaries.

Thus it becomes clear that the pure second derivative Helmholtz forms of the H equation are locally lowest energy solutions for the electronic system in the continuum space of possible H functions.  Any amplitude deviations of H from a simple modulated Helmholtz ripple form caused by non-zero terms in the first spatial derivative can only be transient solutions, because net energy radiation occurring in thermal interactions (how fast?) will succeed in bringing the system state toward its minimum energy.  In this process the H function will transition smoothly via various continuum states along with corresponding smooth changes to the G_n functions, taking H towards the form that has zero coefficients in the first spatial derivative term of its wave equation, and maintaining all the time a quantal overall charge.  This is true in both neutral and ionised cases.  (There remains some question regarding the strictly quantal value of the overall charge, particularly in the ionised cases, so long as the H function is not relaxed to its lowest energy form ... check this.)

So using E12 we write, with modulated Helmholtz form, a "Static Ripple Process Equation":

Static Ripple Process Equation [2 + (T·u(x))] H(x)   =   0	Eripple

The above equation leads to H(x), which is a real ripple multiplying the wave function (it is spatially radial in the near spherical cases of atoms) that produces zero surfaces of the charge density and therefore also in div u(x) ≡ ^T·u(x).  See Appendix E for further notes about this point.

From this we may seek to establish an equation applying to a contour S following the direction of vector |Ψ(s, x)|.  We may then use its necessarily integer multiples of phase rotation π to show that the quantal charge is constant and the charge density is subject to ripples.

Note that equation Eripple is not based solely upon electric potential p(x), because that would have to use u(x) = −grad p(x) alone.  In fact, neither p(x) nor its gradient have direct effect in this equation ... this is not a problem in potentials ... it is not a Lagrangian model.  Instead it is div u(x) that appears there, and that is directly related to the local charge density, as must be the case in order that quantal charge shall be delimited.

For a given u(x) the above differential equation Eripple has the modulated Helmholtz form described above.  However, we shall be looking for those functions u(x) that are compatible with the product H(x).G_n(s, x) that generates them through the charge origination process.

Collecting the remainder of the original abbreviated product derivative terms we have:

HG″ + 2H′G′ + 2u·(HG′+H′G) + j2·(v×(HG′+H′G)) + (ω2 − 1)HG   =   0

E14

Dividing through by H this produces an "Electron Residual Equation":

Electron Residual Equation [2 + 2(HLT + u(x)T·( + HL)) + j2T·(v(x)T×( + HL)) + ωn2 − 1] Gn(s, x)   =   0	Eresid

... where H_L is an abbreviation for either of the expressions {(H(x)) / H(x)} ≡ {(sign(H(x))).ln|H(x)|}.  (The outer brackets { } on these expressions are to stress that they must be evaluated before being applied in the equation expression.)

From equation Eresid it can be seen that the dynamics of each electronic mode are linear with spatially varying but temporally constant parameters so long as the function H(x) remains effectively constant.  It therefore remains to grasp the nature of this function H(x).  In particular we may note that in those regions close to the zero surfaces of H the magnitude of G_n cannot vary very steeply, and precisely at a zero surface it is level.

As a consequence of this, and also on grounds of necessary smoothness in the solution of Eresid at the origin, we may deduce (perhaps with some surprise) that every G_n wave function assumes a constant non-zero value at the origin.  Presumably there is some corresponding constraint on the form of these functions at the periphery (what is it?).

Because equation Eripple has modulated Helmholtz form in three dimensions, the integral of charge density over each region that is bounded by zero surfaces is constant.  We need to establish this by proving that the solution is always an irrotational field (Gauss' divergence Theorem or the Poincaré lemma could be invoked here).  It can be approached by considering how:

• a) The zero Laplacian in three dimensions (and only in three) acts to modulate the intensity with a gradient that is normal to and inversely with curvature of the local isopotential surface (that is a mathematical idea of potential applied to the wave function, not the electrical potential as integrated from a Maxwellian electric field ... it is not an energy potential).  Thus only enclosed sources change the surface integral of intensity over any equipotential surface.
• b) The rate of phase rotation times the squared cosine of that phase for each complex conjugate solution is proportional to the local intensity of the oscillatory solution.

Note that positive charge contributes negative local second derivative to the potential gradient.

A similar form of reasoning then has to be used to establish the value of the quantum of magnetic field flux.  I hope, in the fullness of time, to add it here.  It is interesting to note from Emode or Eresid that the way v(x) interacts includes effects from both the locally driven and externally imposed parts of the magnetic field in cross product with the wave function gradient to get a divergence.  So magnetic quantisation will operate upon the total flux through the region of the wave function, not just that which is generated locally.  Thus whilst charge quantisation is "private" to the region of the group of superposed wave functions, magnetic field quantisation is by comparison a "social" business.

    The Fine Structure Constant

The quantal unit of charge in conventional units is expressed as q_e = √((4πα.)/z₀), i.e. α = z₀.q_e²/(4π.).  Using the basic (hypothetical) units, the Cartesian cuboid definition of charge unit used in the above model has value q_C = √(4π/z₀).  That is the charge density generated according to Echarge by a unit amplitude wave with unit temporal angular frequency.  The basic equivalent spherical shell definition (unit thickness at unit radius) would have value q_S = 4π.q_C .

There are three places where the "4π" multiplier enters this system of relationships:

1. The consequence of spherical integration in equation E5 of field generation by the originating charges and currents.
2. The definition of the basic unit of charge where it appears under the square root as multiplying Planck's constant .
3. The ratio between the two alternative basic charge unit definitions q_C and q_S.

In the first of these, because the field produced propagates in all directions of the three dimensional sphere, the divergence this produces is multiplied by this factor.  The second arises in either of the equivalent integrations EHt or Echarge and also corresponds to the way it appears conventionally in the relationships between the electronic charge and the fine structure constant.  The last of these three is not of any absolute significance because it is only a matter of convention, and if it is used then still its effects must cancel in the different parts of the model.

We must now apply the following principle before we can calculate the fine structure constant:

The topology associated with charge quantum regulation introduces a further change of scale, so further rescaling to the conventional effective units will be necessary because it is only the quantum charge unit itself that is absolutely defined, not the length or time scales that we choose for the original model.  We have deduced that the charge on each Klein-Gordon electron will be multiplied by a factor determined by the ripple phenomenon.  It is this that will need to be brought into the issue of scaling of units.

To set up the proportions of the system we use the static ripple process equation Eripple.  We can take advantage of the conformal properties of the solutions of the Helmholtz equation.  These are still true even in its modulated form.  Because of this we may ("without loss of generality" as they say) evaluate this factor by resort to the simple case of charge per unit thickness in a thin spherical shell wall at unit radius.  The logic is applied for just one of the two complex conjugate components making up the real ripple wave.

The following logic is not self-evidently accurate, though it is at least approximately correct in the value of the fine structure constant at which it arrives.  An improved account is being worked on, and what is sought is a proof that the charge quanta that are subdivided by the zero charge density surfaces are indeed all of equal magnitude.

Suppose a single complex wave component of the real solution of the ripple process equation Eripple has a uniform unit charge in the basic Cartesian hypothetical units per unit thickness of the shell wall (i.e. a total charge of one hypothetical unit q_C per unit thickness).  This charge density within the shell wall is then 1/(4π) units and this is the density (multiplying by −4π) that gives rise to unit divergence of the electric field within this shell wall.  As a matter of fact the amplitude of this component normalised to unit temporal angular frequency is then 1/√(4π).  At this strength of divergence the ripple phase rotates within the wall at one radian per unit radius.  Thus the radial spatial pitch of the zero surfaces is then π and remains so if we split the amplitude of the single component into two equal half amplitude complex conjugate components.  Doing this the mean charge per unit radius over the ripple cycle is halved because of its sinusoidal undulation.

Thus the enclosed charge between adjacent zero surfaces, being one quantum, is increased in the ratio π/2 in the basic Cartesian hypothetical units q_C.  So the system of equations has been based upon dimensions of space and time that lead to a natural geometrical form of quantum that is too small in charge value by this ratio π/2.  To make the same equations work properly it is then necessary to adjust the dimensional scale to correct for this, and that means that the linear scale units of both time and space need to be shortened (divided) in the ratio ³√(π/2).  This effect can be introduced to the model conversely by instead multiplying this same factor into the fine structure constant α.

This means that in the spirit of the Recursive Observation Lemma all possible electron quanta we might use for observations have already been increased in size by ripple effect in the ratio ³√(π/2) from their "geometrically natural value" (see the form of equation E7 above) in spherical geometry where α₀ = 1/(4π)².  Thus we arrive at a value for the Fine Structure Constant of α = ³√(π/2)/(16π²) = 0.0073613.  This is 0.876% greater than the observed value and corresponds to a charge quantum magnitude that is 0.437% greater than observed.

This has been evaluated with no allowance for the possible effect of the spin mode topology, so a further correction for that might yet be necessary (work ongoing).  However the discrepancy of a slightly too high value for the fine structure constant (and correspondingly of the charge quantum value) also may form part of the basis for a further development of the model, including in the matters of atomic nuclei referred to as quantum chromodynamics.

    Discussion

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