Emergence of Quanta in a Wave Model  Working paper

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    Summary

This essay provides an outline of how we can go about setting up a locally causal continuum model for a wave system and then shows how it can give rise to the emergence of quantal phenomena. This is done with reliance upon no more than a very sparse set of original structural features in the model. It pursues reasoning in support of the idea that the quantal phenomena as evident in nature can be accounted for in terms of a locally causal continuum model, and therefore it departs immediately from the approach in conventional Quantum Mechanical theories.

    Introduction and Motivation

The essay does not set out to challenge the accuracy of prediction possible with conventional Quantum Mechanical theories. It seeks only to use a different model exhibiting what are arguably attractive alternative features as the basis of the explanation of those phenomena. It is concerned with some issues of theoretical awkwardness and anomalies at large in modern physics, but in particular an improved conceptual continuity with the models used in engineering is in prospect and as such is therefore a major motivation to this approach. If new extrapolations or predictions should happen to become possible through such a model that would be just an interesting bonus.

It is written in conjunction with a number of other related pieces that take different approaches to the same model and deal with different aspects of it. In its ramifications it is a rather big subject! Some of these working papers are publicly accessible on WWW at http://wavemodel.org. This piece is also intended primarily for internet publication. Links to related articles will be given throughout the text and also I hope to build up a few other book and paper references at the end.

The wave model we are discussing here is developed and, as far as possible, justified in an essay titled "Smooth and Quantal Properties of the Complex Wave" [ABoSQ]. For completeness (e.g. for the sake of printed copies) the basic definitions of the model are repeated here in appendix A.

To discuss the issues of quantal nature and its emergence in a wave-continuum process model I shall split the subject into two parts. First any system that occupies only distinct states out of some continuous range of possible state descriptions could in a loose sense be called quantal, but that is really only in the sense of discreteness. As a second point we are interested too in the apparent fact that our physical processes quantise in ways where the differences between possible naturally persisting states always involve changes of spatial charge disposition in amounts made up from a universal charge quantum. There is a type of magnetic quantisation linked to this effect too.

First I shall discuss the issues of discreteness and distinctness of the sorts of wave structures that are candidates as quantal waves, and then I shall deal with the nature of wave systems giving rise to universal constancy of the charge quantum. Finally I shall explore specifically the elements of reasoning about the wave model that give support to the notion of emergence of discrete and quantal phenomena.

I shall attempt to link to relevant mathematical terminology either by non-specialist clarificatory expansion in the line of the text or, where maths inscrutability makes it necessary, by bracketed comments.

As caveat the reader should at least note that the word "space" is used to refer variously to things like the co-ordinate systems of time and physical space and also to the basis for identifying sets and ranges of other things like wave functions. The idea of a space occupied by lots of functions (typically an infinite dimensional Hilbert space) is NOT the same thing as the basis space ON which such a function is defined (called its domain, and usually four dimensional physical time-space or just the Euclidean physical 3-space) NOR is it the space OVER which the function produces its results (called its range, and for purposes here usually in the real or complex numbers). I shall try to avoid ambiguity, but care may be needed by the reader depending upon their degree of familiarity with these usages, and on my grasp too!

    Discrete Forms in Wave Function Space

Given a model for a continuum wave process defined in terms of local differentials (often expressed as a partial differential equation) and that purports to explain quantisation, we need at least to prove that only certain forms of wave function can persist. Then by virtue of this potential for persistence we can say that these particular functions "exist" as possible products of the wave process (i.e. as eigen-function solutions of the differential equation model). Many of the excluded states may operate temporarily as parts of a transient sequence in transitions between persistent states, but that is all. We do not think of these as existing in the persistent sense.

Non-linearity.
Nonlinear processes are involved and because of this we have to take care to note that changes of amplitude have significance beyond that which is the case for the so-called "eigen-functions" of a linear system. In the systems with which we are concerned here wave amplitudes can affect coupling coefficients, and in that way non-linearity comes about because there is a multiplication of variables involved. So if the idea of eigen-function is to be used it has to incorporate a sense of specificity of amplitude in addition to that of shape as would be the simpler case for linear systems.

Periodic Motion in a "Steady State".
For most purposes here the conditions will be reduced to the "steady state" case, which means that there is no relative motion between the parts of the system, including the observer, nor any other progressive change that would lead to transitions between the discrete states. It should be noted that this term "steady state" does not mean that there will be no sense of motion at all, but only that motion is restricted to that which is describable in terms of sums (superpositions) of distinct temporally periodic functions each with its own fixed temporal frequency. Even then it still transpires that there are continuous spaces of possible wave states corresponding to different amplitudes of the wave function components (referred to as quantum superpositions in conventional physics terms) as produced by the wave model.

Where a wave function can be considered as this kind of superposition of variable magnitudes of the eigenmodes of a quasi-linear system then the term "mode" in place of "eigenmode" will often be used herein. So a wave function typically contains terms in a number of different modes. It must be remembered that these modes interact, and any persistent state subsumes these effects.

Complex Rotative Motion.
These periodic motions are something like what are normally called oscillations in dynamic systems except that here the basic construct involves complex rather than real values. Because a complex value has two parts, the real and the imaginary, the basic form of motion concerned is where these two values move in quadrature, that is, they describe the two components of a circular motion in the complex plane (also called the Argand diagram). Thus for these sorts of periodic motion the magnitude of the value at a given point in physical space is constant but the phase continually advances to produce the circular motion of the complex value. The rotation that this involves is in the 2-space of the complex plane and concerns a single point in the physical 3-space. The frequency and magnitude determine an electric charge density ... see Echarge. There is not necessarily any rotation of the wave structure in the physical space associated with this, though when there is it determines a corresponding circulating electric current density.

Temporal Smoothness of Energy and Charge Distributions.
Because of the constancy of magnitude associated with this sort of motion there is a remarkable temporal smoothness in the energy distribution in the physical space. This cannot be achieved in a system of wave functions over a real variable, as is for instance the case in Lagrangian models, because the magnitude necessarily oscillates at twice the frequency of the wave motion. You would need at least two related real variables in order to smooth it out, but then you are back to the equivalent of a complex variable.

Thus because of this temporal smoothness the dynamics of certain types of functions over a complex variable have a markedly different nature from that with the real variable. Indeed, with such a system of complex waves the charge and current origination can be modelled (and this makes it the simplest way!) so that it displays no fluctuations of charge or energy resulting from the complex rotative motion.

To grasp fully how freedom from radiation can occur at all frequencies involves the wave mechanism of what is conventionally called "spin", and that will be dealt with elsewhere ... see Wave Topology of a Spin Mode. The form of the symmetry in the spin mechanism also nullifies the direct interaction between superposed modal waves within a single atom, molecule or crystal (Note in passing that each distinct wave mode has also as a factor a spin mechanism with real parts, i.e. involving interlocked balanced components having oppositely signed frequencies).

In this way it is only the energy flows between the modes caused by electromagnetic fields, including those imposed from outside of any particular region of modal superposition, that bring about changes and fluctuations of the otherwise steady state. That even remains true with superposition of modes in respect of the way they produce far electromagnetic field, and hence of radiated energy effects. So this model accounts for why the energy of an electron does not leak away as particle orbital models suggest it should.

As regards near field effects, there is a difference between the way that magnetic field motivated from within the spherical form of a field can affect it compared to fields induced from outside. There is more to be said about this yet.

De Broglie and Maxwell Waves.
The flows of coupling energy in physical space occur in the form of what we call Maxwell's waves and they involve real variables only but in a vector form. It is the scalar complex waves that provide charge and current origination and these are the matter waves first introduced as an idea by De Broglie.

Perturbed Linear System.
To rescue us from total chaos in non-linearity, at least in the domain of practical electrodynamics, we can use the fact that the couplings induce changes that are slow (involving frequencies of orders seldom higher than 10¹⁴ Hz and usually much lower) by comparison with the frequencies of steady periodic complex variable motions that constitute the material system (in the order of the Compton frequency at 1.2356×10²⁰ Hz). This gives us a way of separating from the model a linear part with relatively slowly modulated parameters. In the completed model these parameters are then driven by the system state producing a closed loop of causation (see Figure 1 in Appendix A). The model, though approximated to deal with the difficulties of non-linearity, is thereby kept totally locally causal. It involves no instantaneous action at a distance in physical space.

Continuous Sets of Waves in Discrete Sub-spaces.
So we might show that distinct persistent wave functions exist under some particular set of conditions. We may classify their existences as continuous sets of wave functions, each set comprising a discrete sub-space of functions, and excluding others in the intervening space that are in some sense describable but which fail as naturally persistent wave modes. In the space of all describable wave functions we might expect that this condition for existence is almost everywhere not met (the conditionally valid subspace has measure zero), because without that surrounding space the clusters of viable wave functions could hardly be discretely separate from each other. In spite of the latter, still the regions of such discrete existence are neither necessarily nor typically mere points in the space of wave functions, but rather form isolated (i.e. disjoint) continuous sets.

As a very simple example of a continuum of wave existence in our four dimensional time-space consider the set of functions produced by all possible constant velocity translationally moving versions of a given wave structure. These are related by what are called Lorentz transformations. Since the sorts of models considered here are "Lorentz invariant", then if one exists as a possible product of the defined model process we know all exist ... a kind of local continuum. Perhaps more importantly the steady state superpositions of modes mentioned above provide cases of continuous spaces of wave functions and do not involve any motion of the wave structure. These individual sub-spaces each span a reduced dimensional order and therefore can remain distinct within the space of all possible wave functions.

Fluctuations.
Further to our concern for describing the distinctions between these different classes of wave function states, we shall be interested in the way that energy transfers bring about relatively slow changes of state amongst the wave functions within each continuous set. We can think of a set of states that belong as a continuum to a given particular discrete state. In other words, each island of discrete state contains a continuous set of wave states (i.e. having neighbours differing by only indefinitely small amounts) for which energy movements can bring about gradual changes between states within the island. These movements constitute gyrations about particular states in terms of energy and its thermal fluctuations.

State Attractors.
In so far as the system can fluctuate around particular states as though it is attracted towards them like bees around flowers or planets around stars, some states are called "attractors" in the state space. Persistence then consists in the system making small gyrations about such an attractor until some relatively rare circumstance induces a change large enough to make the system undergo a discrete change and settle to gyrations about a different attractor point. In physical systems such fluctuations and gyrations of state are usually associated with energy storage in the form of heat, and due to the random nature of the fluctuations occasional discrete transitions can be induced by peaks of disturbance.

In the space of possible states of the system there are unstable as well as stable equilibrium points. Because of the symmetry in the nature of the system there are equally many repeller points as there are attractors. Naturally any system that operates near to such repeller points is not likely to stay in that region of the state space for very long, and systems are likely only to be rarely observed close to such unstable equilibria. However, the processes of evolution determining the rate of convergence to an attractor or divergence from a repeller essentially involve the proximity of the De Broglie matter waves in order for quantal groupings of charge to undergo change (called "inelastic collisons" in kinetic models). The Maxwellian processes of energy coupling can occur over greater distances (and involve the so called "elastic collisions" in kinetic models) so that removal of energy (cooling) from a system to bring it towards its equilibrium point is possible to a limited extent between systems of opposite charge type (i.e. matter and anti-matter). See Appendix A regarding the distinction between De Broglie and Maxwellian wave couplings.

The separation of interaction between the De Broglie and Maxwellian types gives rise to a particular type of equilibrium process in which the rate of its convergence/divergence evolution depends differently upon the two types of interaction. Because of this we may distinguish between trans and auto types of equilibrium processes to help in the explanation of what is involved, and Appendix B takes this issue further.

Parsimony of the Model -- Occam's Razor.
As a corollary to this requirement for a restricted set of possible persistent wave function states the wish for parsimony of the model drives us to insist that their persistence can be proved in individual cases by correlation of plural observations (which, of course, necessitates persistence of the observer too). Without inference following the correlations that are made possible with two or more observations the states would be fictitious, and a model with fictitious states is conceptually wasteful. Hence our concern.

So "persistence" and also some sort of associated "identification" of states are basic requirements, or else some alternative equivalent account similarly in terms of interactions is needed. Indeed, the objects, be they waves or whatever, that occupy these states cannot be known directly at all but only by the agency of processes that support correlations and inferences following from quantal changes induced in observer sub-systems with which they are connected.

Inference of the Existence of Matter and its States.
We may assert the presence of an object by inference from the energy that it radiates, or, more subtly, merely that which gets reflected from the region of physical space that we claim by inference that it occupies. The latter case of reflection is particularly interesting because it need not involve any explicitly demonstrable discrete changes in the object of our observation ... state continuum fluctuations are enough to make the model. But nevertheless such detection still always involves persistent changes of state which that reflected energy induces in our observational sub-system. Without those changes there is no observation possible. The inferences in which observation culminates come via correlations of the changes induced in the observer sub-system, and for that to be possible some of those changes must be transitions between persistent states ... continuum fluctuations alone, such as marginal changes to electronic excitation levels within atoms, are not enough.

The observation process depends for its existence upon amplification of the signal energy. There exist no processes yielding the unilateral (i.e. acting in one direction only) amplification essential to provide information useful to the purpose of the observer that do not rest upon changes between persistent discrete states at the heart of the observer's amplifier system. For further discussion of this refer to "Essential Structure in Physical Observation" [ABoES]

Objectivity.
There is a sense of independent continuity associated with the meaning of the word "exist", and I suggest that we would prefer that in our model the parameter with respect to which this independence is asserted should be something simply related to the basis space of our model, such as one of its dimensions. Some care is needed. The most common basis for the independence aspect of "existence" is the time dimension, but even in that case care is needed in physical models because any given temporal independence may require different interpretation as seen from different physical frames of reference (various sorts of relativity).

In order that an approach may be made to achieving some degree of objectivity in observation we must accommodate the unavoidable paradoxical fact that the basis of each observation differs in some way from any other. It is only the possibility, and always a limited one, that correlation can be proved through repeated sets of observations that gives us any basis at all for the notions of persistence of properties and thereby predictability that underlie what we call "objectivity".

    Quanta Emerge with Universal Values

Inspection of either the causal diagram Figure 1 or the partial differential equation form of the model Emode along with Echarge reveals a distinction between two sorts of closed causal loops. One sort of loop involves a second order integration and the other only first order. It is intuitive to see that where a second order integration loop is disturbed by the action of a singly integrated term then in the steady state the second order loop will determine the result. At sufficiently low frequencies the second order integration can always "beat" the first order. Thus the influences contributing effects in the second order loop can be expected to determine attributes that remain qualitatively and sometimes also quantitatively constant under sufficiently slow perturbations occurring in the first order loop.

All such talk of loops, integrations and low frequencies is normally most readily handled intuitively in the single dimension of time or its frequency transform, but there is still something valid about it in the 3-space physical model we are considering here. The mode equation Emode mediates solutions that are simultaneous across 3-space as defined locally differentially, but if a causal sense is invoked then it is possible to see that the settling to a persistent solution to satisfy such an equation, involving spatial propagation of residual unbalanced influences near to the speed of light, suggests that something of the simplified low frequency validity for competing integration loops survives. It remains for us to establish the same reasoning in a more formal way.

If we look at the equation Emode we can see the nature of the terms that contribute to such second order integration loops. The original form of the equation written for wave functions on physical time-space is a Klein-Gordon equation with the electromagnetic interactions introduced via a matrix of coefficients for its second order differential terms over the four dimensions. However, we are working with its time dimension transformed into frequencies so that in Emode we are left with a Helmhotz form of equation that is rather simpler to handle because it lends itself to a linearised form of model for relatively slow perturbations within a steady quantal state.

The Charge Quantisation Case.
Consider contributions coming from the zero order differential term in the differential operator. The universal constant of the Klein-Gordon is still there in the Helmholtz format, but the squared frequency for any given mode now also appears as a constant applicable throughout physical space for that mode. In addition to these two constants we also have the influence of charge (which also may be viewed as electric field divergence) that appears as a function of spatial position. It is only this latter term that can perform the role of the influence that will "win" in determining any charge quantal situation.

Thus our mission becomes that of showing that the only possible steady state solutions of such an equation determine quantal values of the integral of this charge density over the whole of space. In fact it leads to reasoning that goes a bit further by showing how the distribution of charge is divided into segmented regions of space each of which contains that basic quantum of charge.

We must pay attention to the superposition of modes and the resulting addition of their respective contributions of charge density. The quantisation process, depending as it does on the second order integral of charge density, is determined by the sum of these contributory charge densities. Thus the equation for each individual mode remains linear but is subject to a spatial modulating parameter function that is shared in common with all modes (each with its respective frequency and spatial wave function) occupying the region of space in question.

The Magnetic Quantisation Case.
Further inspection of the Helmholtz equation Emode reveals a second order integral of a more subtle form. It is the magnetic term that appears in the first order differential part of the differential operator, but contains within itself another differentiation operating upon the wave function variable. It has a div operation acting upon the cross product of the vector potential and the grad of the wave function. Since taken together this is also a second order differentiation, the result of its integration will be capable of determining a state that for sufficiently low frequencies will be independent of disturbances coming via terms within a first order integration loop. Because the loop involves a cross product, it is the integrated intensity of the curl of the vector potential that is thereby determined, and this is the means by which magnetic field is restricted to a form consisting of quantal bundles of magnetic flux.

Static Electric Couplings.
From further inspection of our basic equation Emode we may see that it is also sensitive to the influences of static electric fields. In the differential operator there is an explicit term in electric field strength operating on the first differential (grad) of the wave function. Through this we have a basis for the Stark effect phenomena. Although the discrete quantal state remains unchanged the field produces offsets in the modal frequencies and their concomitant energy levels.

There are effects upon the modal form via the electric field term in the first derivative part of the operator, and these arise both from the charge within the locally active modes and through fields that are externally imposed. As regards any influence via the electric field term in the non differential part of the operator, it has none because this term is formed only of the divergence of the electric field, and that is zero except to the extent that charge is present locally. So we may see that whatever direct effects the electric field might have via this path, its integration in the first order will not affect the invariant nature of the outcome of the second order integration loop mediating charge.

Static Magnetic Couplings.
Regarding the equation Emode I have already described above how the magnetic vector potential term in the differential operator produces at low frequencies an invariance of the magnetic quantisation as regards disturbing fields. However, that does not mean that static magnetic fields have no other effect at all. Indeed, the influence of the magnetic field enters the causal loop at the same stage of the two stage integration as does the second differential operator. That was not the case for the electric field that entered at the opposite stage of the two. Thus the magnetic field, although of itself bringing no violation of the magnetic quantal condition (it is only the divergence of its cross product with grad of the wave field that does that), still directly influences the value of the second derivative needed to balance the equation, and in doing that it affects the frequency of each mode.

Because of this direct influence in terms of the second derivative of the wave, the shift in frequency produced by a static magnetic field is linearly proportional to the strength of the applied magnetic field. This is what is called the Zeeman effect. This is generally different from the Stark effect produced by an imposed electric field operating on the other stage of the two stage integration. The result there is broadly a quadratic effect in the strength of the applied electric field.

Couplings of Energy.
The charge and vector potential influences and also the electric and magnetic fields described above were effective in a static situation and were not dependent upon wave function state or on energy other than in a static sense. There were no effects involving progressive transfers of energy (i.e. power transfers). This is not the case if an oscillatory electric and/or magnetic field is applied to the space containing the wave function.

Rabi Processes.
If the frequency of this oscillatory Maxwellian field is matched in some degree of closeness to the difference in frequency between two wave function modes that operate as superposed in the spatial region concerned then given suitable geometric conditions between the related waves a transfer of energy amongst the three waves (two De Broglie modes and one Maxwell wave, either resonant modal or a far field radiation) can come about. This process can be modelled on the basis of a three dimensional state space called a Bloch space. Two of the three co-ordinates are energy levels (say of the two De Broglie modes concerned) and the third characterises the phase relationship between the three waves. The energy transfers that occur tend to be oscillatory, and are often referred to as Rabi oscillations. They correspond to state gyrations in the Bloch space [AE87].

Since the sum of the three power flows must be zero we may see that for half of the Rabi phase cycle energy will be absorbed from the electromagnetic wave and also transferred from the lower to the higher frequency mode, whilst for the other half there will be emission and transfer from the higher to the lower.

Absorption and Stimulated Emission.
When separate means are used to maintain the ratio of excitation of the higher to the lower frequency mode at a level above that which they would achieve in thermal equilibrium (i.e. the Fermi distribution) then the net effect of impinging radiation in the relevant frequency band will be to release more energy than is absorbed. This phenomenon of sympathetic reinforcement of an electromagnetic wave is called stimulated emission.

Transition Selection Rules.
It is the geometric conditions mediating these transfers of energy that give rise to the selection rules (Hund's rules) for atomic and molecular state changes. It is necessary that the stress vector of the electromagnetic field shall align with the gradient of the conjugate product of the two complex scalar wave functions that are being coupled in order that the Rabi process shall operate, and the strength of that coupling determines the rate of progress of the process, either absorption, emission or gyration. Because the electronic modes in atomic structures take on spherical harmonic mode forms, the couplings that are possible depend upon the nature of the products of the respective wave functions in any given pair. For the simplest cases of energy transfer to occur the conjugate product of the wave functions must produce a dipole that can couple to the Maxwell field, and the effect of the electric and magnetic couplings must not cancel. The selection rules can be derived from this.

The Atomic Excitation Continuum.
In this model we may see that the energy couplings affect atomic electron excitation levels continuously, they are not quantised in energy. In an atom undergoing these sorts of interactions each of the superposed modes transfers energy at a rate proportional to its modulus of frequency, but always with one electron mode gaining and the other losing, and with the energy balance made up by the electromagnetic wave. So the coupling effects induced by low frequency (which includes luminous at around 10¹⁵Hz) and static electromagnetic fields operate between superposed modes whose frequencies are close to equal values but with opposite spins. (This links to the "opposite spins" transition selection rule of the ordinary physics model.)

Eigen-modes in Radiative Equilibrium.
By comparison with the couplings induced by imposed electromagnetic fields, the direct interaction of superposed modes with roughly similar frequency (including having like signs) causes charge and current intensity fluctuations that we might expect to produce far field electromagnetic radiation in any cases where the mode product has a dipole form, but the necessary condition of orthogonality of the superposed modes determines that no such far field effects can arise. There are also near field electromagnetic effects to be considered, and these along with the charge and current distribution of the atom or molecule pulsate at the difference of frequencies. Since this is brought about by a motion entirely complying with the modal wave equation, and it is a linear equation, we can rest under the assurance that the combination of the near electromagnetic field and the pulsating charge or current distributions so produced will act entirely equivalently to viewing the individual modes as present and additive without interaction. Only the mean densities of charge and current appear to matter.

There is more to be said in this respect regarding how the structure of the spin wave factor nullifies such direct interactions ... see Wave Topology of a Spin Mode.

Essential Role of the Observation Process.
Because of the energy to modulus-of-frequency proportionality of Planck's constant manifested in equation Echarge inter-mode couplings caused by imposed electromagnetic fields leave the charge on the atom or molecule unchanged. In this model the reason that the coupling processes appear to be quantised is merely that every possible way of observing these effects introduces an essential involvement of quantal interaction in the process of observation. In those cases where an atomic or molecular state of excitation is driven sufficiently far from its equilibrium then the stability of the system is compromised and change of the charge or magnetic quantal state of atom, molecule or crystal then follows. That is the only sort of quantal change that occurs in this model of electrodynamics, namely to rearrange discrete charge distributions in pursuit of achieving stability by gravitating towards local energy minima. The detailed processes of energy change that lead to this are continuous, not quantised.

    Uniqueness of the Quantal Charge and Flux Values

Given the above we need to prove the invariance of the charge and magnetic quantal values and to determine what these values are.

Scheme of Proof for Charge Quantum.
To do this for charge alone we first show that there exists a separable real ripple function h(x) spatially modulating the intensity of any superposed set of electron modes in the steady state that delimits the charge in quantal units within each cycle of ripple (which is a half cycle of the underlying pair of additive complex conjugate terms). Since this function carries all of the dependence upon the charge distribution, then once separated as a factor it leaves the electron modes as determined by a residual equation that, although having its coefficients modulated by h(x), nevertheless for a given steady definition of the charge distribution throughout space, is linear.

We then prove that the charge delimited by a radial h-ripple is invariant under all possible charge distributions that can be associated with steady state field forms that we can encounter by showing that:

Then by solving for the charge contained for a simple case within a single ripple we know the quantum for all cases in terms only of our basic space, time, energy and impedance units.

The result obtained in this way is then subject to adjustment to allow for differences in both geometry and scale between our basic definitions and the conventions in determining the standardised expression of these units.

    Discussion

    Appendices

Notes here.

    Appendix A: Definition of the Wave Model

The wave model we are discussing here is developed and, as far as possible, justified in an essay titled "Smooth and Quantal Properties of the Complex Wave" [ABoSQ]. Please refer to that if a more complete picture is desired.

A Causal Flow Diagram.
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.

The diagram in Figure 1 shows the general disposition of causal linkages for the model. This type of diagram summarises the fields and their interactions that make up the entire model.

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.

Partial Differential Equations.
In this electrodynamic model each nuclear charge is taken to be specified a priori, and usually to exist at a point. Atomic electronic charge and current, which act as sources bringing changes to the electromagnetic field, then originate from the electron wave field as follows:

	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

Each electron mode develops under the influence of both externally and locally generated electric field u(x) and magnetic vector potential field v(x) thus:

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

To explore charge quantisation 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:

Refer to Wave Topology of a Spin Mode for further detail on the form of the spin mode-factor. Here we leave it merely as built into G_n(s, x).

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

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

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

    Appendix B: Attractors - Types of Equilibrium Processes

First we may distinguish between stable (attractor) and unstable (repeller) forms of equilibrium. Then we may distinguish between cases where the state drift influence operates independently (auto-attractor) and those where it is linked to the same motion as gives rise to the state dispersion process (trans-attractor).

The interpretation of discrete electron emission in the photo-electric effect and of atomic state changes e.g. in the avalanche photo diode is conventionally that both the luminous radiation and the states of atomic excitation are quantised. Contrary to this it would appear that this deduction is not necessarily true. There are apparently compact and tractable models available that will account for the universally quantal results of observation in which both the operative quantities of luminous radiation and the states of energetic excitation of atoms, molecules and crystals occupy continuous ranges of value.

Attractors of Different Kinds
There are questions about the concept of attractors as means to the realisation of quantal behaviour. We may distinguish between two sorts of attractor mechanism that can be classified respectively as:

♦ AUTO-attractors
There is an ever present drift of state towards the attractors independent of the intensity of causal interaction. This covers also what is called hysteresis.

♦ TRANS-attractors
Negligible rate of spontaneous drift occurs, but only evolution of state directed towards attractors at a rate depending upon both state and intensity of causal input activity.

It is the latter which is needed to account for the Einstein photo electric effect in which quantised effects occur with mean frequencies in proportion to a causal radiation influence down to very weak levels of radiation.

We can characterise these two mechanisms by way of distinct models derived from the Fokker-Planck type. The Fokker-Planck equation balances the effect of dispersion, usually seen as "diffusion", in slowly destroying the structure of a function of space and time as by melting its form, against a so called "drift" term which acts to drive the function into some structured form. It is natural in the Fokker-Planck model for the strength of the drift term to follow the intensity of the driving activity, whereas the diffusion term does not necessarily do so. If the diffusion term has a form dependent solely upon state then we have the basis for an AUTO-attractor model. If the diffusion term contains very little spontaneous continuous part and is constituted mainly from chaotic forces resulting from the drift forcing activity then the model has the form of a TRANS-attractor system.

Since we may observe both the photo electric effect at the quantum level and hysteretic phenomena at a macroscopic level it is evidently possible to assemble the AUTO form from the TRANS form of attractors. To achieve this would seem to require in the model of the macroscopic process no more than the presence of some form of inherent agitation, such as zero point fluctuation. It appears that it is not possible to assemble TRANS-attractor from AUTO-attractor mechanisms except through a relatively elaborate structure comprising an ensemble of thermodynamic sub-processes. Is the concept of zero point fluctuation able to overcome this limitation too? I think not.

More subtly, the trans-attractor mechanism can also act on a localised basis in a suitable medium. This is because its diffusion process only occurs in those regions of the medium which undergo the driving process (such as radiative disturbance) ... the system remains in a frozen state everywhere else. Thus for instance, the electronic states of an ensemble of atoms may occupy frequencies such that most modes are well separated from the others except for a few modes around the Fermi upper limit of energy/frequency. Around this upper limit thermal blurring (22mV standard deviation of blur width at room temperature) of the Fermi distribution boundary between occupied and unoccupied states occurs. When EM radiation in a narrow frequency band is applied at intensity levels much higher than corresponds to the black body equilibrium spectral power density then it provides means for diffusion to occur between that selected few states which find themselves related by the radiation as a frequency interval (see Allen and Eberly "Two Level Atom"). Thus a narrow band of red light (about 2V quantum level) can link between a very few electronic mode pairs which straddle the Fermi limit. In each of these pairs the higher frequency one would find equilibrium as nearly empty and the lower excited according to its position in the Fermi distribution almost to Planck's constant times its resonant frequency.

Three Wave Interactions
When a Lagrangian model of interactions in real variables is used then so long as non-linear coupling exists energy equilibrates with a white spectrum, producing statistically equal energy in every degree of freedom involved regardless of its frequency. When resonators are coupled by means of weak bilinear product couplings then the statistical equilibrium brings each degree of freedom to an energy level proportional to its frequency. This can happen with complex variables but not with reals. The reason for this difference is that the nature of interaction between real dynamic variables involves symmetrical scattering of energy across the spectrum, and that inevitably whitens the spectrum. There is an up-conversion that is the seat of a so called "ultra-violet catastrophe". Approach to the bilinear coupling process can only ever be approximate with Lagrangian real variables (unless you force the complex structure back in between two such models ... a pointless complexification!). When the dynamic variables are complex, as are the modal electronic variables, then bilinear couplings lead to an equilibrium in which the energy levels of the degrees of freedom (i.e. electron modes) are proportional to their respective frequencies. This is because the rates of energy transfer to the individual modes (including the complex conjugate paired EM wave) remain proportional to moduli of their frequencies (Check it out with some trigonometric wave relationships). So three frequency interactions of this sort confine energy to the modal frequencies concerned, and because of the proportionality to frequency the net charge contained in the two electron modes involved stays constant too. We have energy conservation between the two modes and the EM wave without any unrelated up conversion, and we have conservation of total charge in the two electron modes.

Under these mechanisms there arises a process of state gyration in the domain of energies of the electron modes and a relative phase between the three frequencies concerned. This is usually characterised in a three dimensional space with two energy dimensions and a relative phase dimension. This is called a "Bloch space".

So far so good. I have pushed around the wave interaction model a bit to show the coupling power proportionality and need to do some work on presenting it properly. Perhaps the most worrying part left to do there is to prove that the coherent gyrations (i.e. pure three frequency) in the Bloch space have centres of rotation which always have both electron mode energies at the product of Planck constant and respective frequency whilst the incoherent state trajectories (integrated uniformly over all values of that relative phase) converge to the Fermi distribution (which is the same proportionality truncated at the higher energies). I am pretty sure it will happen, but have not yet either written or imagined why (according to the model rules) it must.

To see how electron modes are free to modulate their energies in this way whilst leaving the discrete quantal atomic structure topologically unchanged refer to the essay "Smooth and Quantal Properties of the Complex Wave" which can be found at the URL http://wavemodel.org .

Finally we note the case in which two processes compete in both dispersion and drift effects and relate this to the behaviour of anti-matter under the competing influences of De Broglie (charge) and Maxwellian (thermal) interchanges. Such a competition must be involved in any system of prolonged storage of anti-matter species.

    References and Background Reading

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