Dynamics of Complex Waves           Working Paper

 

A.G.Booth		Original on WWW 6 February 2005		Copyright © A.G.Booth 2005 All rights reserved
Document ident:		Last updated 1 September 2005		Dynamics of Complex Waves.  A.G.Booth
Keys:		quantize quantized quantization wave equation continuum theory

 

   Summary

The zero d'Alembertian wave equation has a particularly simple structure.  With the added stipulation of rotational symmetry of interactions to comply with the Euclidean nature of our physical 3-space and the presence of an endemic noise field the model gives rise to behaviour corresponding to many natural physical phenomena.  The picture is developed by applying stochastic correlation analysis to a complex scalar noise field in the Minkowski 4-space.  The results are developed by following these steps:

• Discuss the zero d'Alembertian complex scalar wave model with its waves at speed of light.
• Link its singularities with spin and postulate a set of slower eigenmodes supported by noise.
• Analyse the combined field in terms of its correlations and spectra by Wiener methods.
• Separate the coupling terms to correspond to the conventionally recognised physical forces.
• Use these couplings to build dispersive wave models (Klein-Gordon or Schrödinger equations).

These dispersive equations then form the basis to vindicate the postulated existence of slow and standing wave eigenmodes.  An example of how that remaining step is done for electrodynamics can be found in Smooth and Quantal Properties of the Complex Wave.

 

   Introduction and Motivation

Whilst thinking about engineering wave models as candidates suitable for use in electrodynamics the following structure of dynamics arising from a very simple initial model became apparent.  The ideas moved something like this:

Occam's Razor
The existence of the same constant (the speed of light) appearing in two structurally distinct dynamic systems (Maxwellian and De Broglie waves) in our models of natural dynamics is enough to arouse the parsimonious sense of Occam's Razor in us.  It leads us to suspect that there must be some supervening common model.  Further, just looking at the form of the Klein-Gordon or Schrödinger equation and wondering about its constant term in the operator (the one that causes dispersion of wave velocity) suggests gravitational mechanisms through induced phase tilts by making its value a function of spatial position.  Here then is, perhaps, a path to the supervening model.  Can an underlying field with simple structure embrace both of these wave models (Maxwell and De Broglie) whilst manifesting the properties we observe in physics?

The search went like this:
Einsteins's gravitational light bending trick is right.
My own wave model estimate of Sommerfeld's fine structure constant is on the high side.
So substrate spectrum and/or velocity range might be "sub-light-cone" in order to get coupling, but as developed hereunder that looks unlikely.  Rather the involvement of singularities (in the nature of spin) may be key.  Still some form of finite intensity field may be involved.  From prediction errors for the fine structure constant or some other marginal effects we may then be able to deduce this coupling margin, and interpret it into nuclear mechanics to produce the value of the neutron mass.
For greatest simplicity we need reasoning, if possible, to show that there are not two independent functions, one spectral and the other velocity distribution, in determining the nature of this substrate field.

Time survives as a reference parameter
In studying correlations in a wave field we do not necessarily have to think of cross correlation functions between first derivatives as referred to translations of their respective dimensional co-ordinates.  Rather we can form correlations always over time as the correlation parameter between each pair of local partial derivatives in the four space.  Then the resulting 16 element correlation matrix can be converted to a hybrid frequency domain form by taking the Fourier transform, over the domain of its time parameter, of each of its element correlation functions.

In its unique position as contrary to the three Euclidean dimensions time serves to take on the negative characteristic in the second differential, and hence develops the steady oscillatory nature that gives it long term correlation/persistence, unlike the Euclidean relationships amongst the spatial dimensions.

Substrate noise would seem to enter the dynamics model only in the effect of its fluctuation variances and correlations, and that is at a level of modelling much lower than the uncertainties associated with quanta (at least, it is at the lower frequencies).  This is more than merely a matter of zero point fluctuation entering the dynamics at the first hand by being there in the Maxwell field.

 

   Note on Vectors and Tensors

 

   Simplest Possible Equation

In the study of wave systems in physical space 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:

... where the matrix K defines the Minkowski metric in its diagonal and the coupling terms from the electromagnetic field in its off-diagonal elements.

Study of the model generated by this second order differential equation and its somewhat less general first order cousin the Schrödinger equation leads to properties in the solutions that correspond valuably to the phenomena we observe around us that we classify as "electrodynamics".  That is discussed and demonstrated in an accompanying essay Smooth and Quantal Properties of the Complex Wave.

Here the exploration is taken to a lower level of causal processes based upon an even simpler wave model.

Given that we appear to get fruitful results from the kinds of models cast in the form of a complex variable that is a function of position in physical space, we might ask what is the simplest constraint on mathematical good behaviour of such a function and see how useful that might be.  We quickly arrive at the desirability for some sort of smoothness in the function, in particular that it should not have precipitous sharp steps with infinite gradient anywhere.  This idea is motivated as part of the wish to allow causality to work always locally.  If the rules for such local relationship are to remain as simple as possible then this sort of smoothness can make mathematical and rational life a lot easier to lead.

A very simple form of constraint rule is to equate the sum of the second partial derivatives to zero.  You might be tempted by the even simpler first order differential constraint, but its only solutions are flat ... too boring.

So we take up the second order option, but there is a tricky bit about this in that we are permitted either sign for the terms in this summation of second derivatives.  The arrangement that seems to fit our requirements best gives a reversed sign to that term for the time dimension.  If we did not do this time would be part of our static geometry (Pythagoras' theorem would work symmetrically in all four dimensions), and we know anyway that it does not do that.  Having this sign reversal present produces what we talk of as a wave equation (it has solutions that are wave-like).  The geometry that results from this sign reversal of the time related second order term is attributed to Minkowski, and we therefore talk of this as the four dimensional Minkowski space.  It is the basis of special relativity.

Thus the simple arrangement of equating this form of second derivative of the complex scalar variable to zero presents itself as about the simplest basis for a causal model we could wish for.  This essay explores the forms of solutions of that equation, though we shall also consider the effects of allowing certain well behaved singularities to enter the game too.  By "well behaved" here I mean that these singularities are still limited to finite squared magnitude, which along with a finite temporal frequency means finite energy density.  In fact, since the restriction applies jointly across the four dimensions including time we may regard this restriction as a finite action density, action being measured in units of energy multiplied by time.  These singularities break up the monotony of an otherwise totally smooth solution and, as we shall see, by seeding the structure of spin they make possible the existence of matter.

This sum of the second derivatives over the four dimensions with sign of the time element reversed is given the name "the d'Alembertian", and is traditionally denoted by   , a symbol referred to variously as "square", "box" or "squabla".

So the most basic form of the equation constituting dynamic complex waves in three physical dimensions consists of nothing but the d'Alembertian operation on the scalar complex wave function equated to zero which we may write in the appealingly compact form of  ψ = 0 .  Writing this in the vector notation we are using here gives:

◊T·M·◊ ψ(t, x)   =   0

EZdA

It is true that for this equation there is a set of solutions in plane waves that spans any physical Euclidian 3-space in our Minkowski 4-space, and all of these solutions imply four dimensional solutions (by extension under the zero d'Alembertian) that have the same characteristic speed (speed of light).  These waves are all mutually orthogonal, they do not interact with each other at all.  The general form of such a single plane wave with amplitude α, angular frequency β, 3-vector direction ζ and phase γ may be expressed as:

ψ(t, x)   =   α.cis(β[−1, ζ]T.[t, x] + γ)           where |ζ|   =   1

Enat

I shall refer to this three dimensional set of four dimensional wave solutions below as the "native" set.  They have the nature of a complex Fourier basis for a continuous space of three dimensions.  Because of the conical 4-space geometry of the vector directions of these native waves they are said to exist "in the light cone".

What we shall think of as the "power density" of these waves is proportional to the squared magnitude of the first derivative, not to the squared magnitude itself.  On this basis, if we wish to form the total energy of a set of waves with different frequencies by summing their squared magnitudes then each such square must be multiplied by the square of its corresponding frequency.

There is an alternative transformation of this native basis set into the set of all radial waves with magnitude asymptotically inversely proportional to distance from a given centre point and with radial speed equal to the speed of light.  The elements of this basis each have a finite maximum magnitude at the centre, and for a given magnitude asymptote at great distance this central maximum has a magnitude that is proportional to the frequency.  Alternatively, for a given distant field time derivative magnitude the amplitude at the centre is independent of frequency, but still the central maximum always has a width that is inversely proportional to frequency.  The phase velocity of these radial waves is also in the light cone as were the plane waves mentioned above, but there are some difficulties in interpreting phase velocity proximate to the centre of the radial pattern.

These waves so far described correspond closely to those used by both Milo Wolff and Chris Hawkings in their respective discussions of modelling matter by means of waves.  Those authors present descriptions of matter based upon wave structures.  Here the picture of the waves will be taken further by introducing the rôle of singularities as means to obtain interactions before seeking the forms of solutions.  That way the wave structures are not merely described but emerge as naturally produced solutions of the wave model equations.  They are shown as having natural forms and frequencies and natural levels of amplitude.  Also the set of wave function structures produced this way is richer in detail, as it will transpire.

 

   Singularities and Interactions

Both the plane wave basis and this radial basis set of waves span the same three dimensional space, so either may be used for forming finite three dimensional integrals that are stochastic (i.e. summed over random or incoherent sets).  So both the plane and radial forms of this (native) Fourier basis can form valid integrals as wave functions (both, of course, because they are equivalent bases) so long as the sum of the squared derivative (i.e. total energy density) is finite.  This amounts to their having finite spectral density and bandwidth.  But we still have no mechanism of interaction between the elements of this set of waves in either of its forms.

To achieve interaction it is necessary to introduce a special kind of wave function that from the outset relies for its existence upon interaction.  Such a wave can be constructed from components that would each alone violate the defining wave equation (and therefore could not exist individually), but when taken together cancel their illegal effects.  In these particular cases of combination a persistent wave may exist, but only when in addition it produces the scattering of a background field of noise-like plane waves.  Scattering corresponds to converting waves of the plane kind to those of the radially propagating kind.  Such combination waves are not of themselves solutions of the original wave equation, but so long as the scattering they cause has certain stochastic equilibrium properties then these special additional forms of waves can exist in persistent forms.  By this means interactive processes can arise in this complex scalar wave system on Minkowski space.  Any of the simpler systems of scalar waves that are real and/or on Euclidean space cannot produce these properties.

(One might be tempted to conjecture the existence of further more complicated interactive extensions to sets of wave solutions on Minkowski-like spaces of sufficient dimension over quaternion or octonian scalar variables ... I wonder!)

It is as though a kind of boundary effect on stochastic eigen functions can exist as distributed throughout the physical space (rather than being independently defined by some bounding surface).  Also where these terms persist so they can mediate alternative overall solutions in that they scatter the waves of the native system.  It turns out that these additional terms are necessarily bound up with the possibility of "spin" topology (the native set are all spinless), and where they exist at all they display the tendency to settle into fluctuations around discrete attractor states.

Let us refer to this stochastically based extension of the set of solutions of the zero d'Alembertian rule as the "derived" set.

There is no mutual interaction between waves in the native set (at least almost none, c.f. the gravitational light bending effects demonstrated by Einstein ... we must also ask "Does light gravitationally attract light?", so there remains a question regarding the exact truth of this statement, at least physically even if not mathematically), but this is not true of waves in the derived set which interact strongly (consider van der Waals forces).  Also the native waves are not generally orthogonal to the derived waves, and therefore interactions are possible between element pairs with one each from the native and derived sets.  Indeed, without this interaction the derived set could not exist at all ... alone they cannot maintain equilibrium as solutions of the zero d'Alembertian wave equation.  Presumably it is for this reason that I find they are omitted in the usual mathematical descriptions of the solutions of this basic wave equation (still searching for refs!).  It also transpires that there is no such derived set of wave solutions if the wave variable is real; it is only possible for the necessary structures to exist with a complex variable.

From these two classes of solutions we can begin to see the emergence of the distinction between the Maxwellian phenomena (as waves at speed of light) and the De Broglie phenomena (where geometric mean of phase and group velocities is speed of light).  However, neither the Maxwellian nor the De Broglie waves can be directly manifested as the wave processes I have been describing.  The De Broglie waves are indeed the derived set, but for these to exist there must be a background field of native waves present.  This background field of native waves must occupy a broad band of frequencies and I shall call it the "substrate field".  As for the Maxwellian phenomena, these vector waves are realised as correlations amongst the native plane scalar wave elements of the substrate field.  In free space (with native waves only ... no matter present), as a result of the total absence of interactions amongst native waves, these correlations decay in intensity according to the inverse square law of distance.  Again, such correlations could not exist unless there were a substrate field to support them.

We need to understand how the native and derived waves might interact, in spite of the wave equation being linear in the sense of the terms in the equation all being of only first degree in the wave variable and its derivatives.  The only prospect then is to uncover some sort of interaction based upon a singularity at the centre of a spherical wave structure.

For a spherically symmetrical wave field in time plus three spatial dimensions the integral of an inbound linearly superposing wave from a finite spherical symmetric source in space produces only finite values around the spherical centre.  The intensity at the central point is finite.  Most importantly, the superposed wave components in this system do not interact.  So in the absence of any singularity of a simple radial type we are left to look for some other structure as the basis for interactions between plane and spherical wave systems.  Thus we turn to the notion of spin structure in the wave fields.  We may be tempted to explore the possibility of singularities with infinite densities at the centre but finite integrated energy.  However, we shall not even need to go that far ... the spin fields can be constructed around an axial singularity of a line rather than a point type, and that has finite energy density everywhere, and indeed, a zero energy density at the central axis.  For further description of the form of the spin singularity please refer to the accompanying essay Wave Topology of a Spin Mode.

 

   Superposed Stochastic Waves

With spin singularities present the most general solutions to the zero d'Alembertian equation involve interactions, and consequently to model them in full generality is transcendentally complicated.  Therefore we must seek a basis of workable approximation that will remain applicable over the range of solutions with which we hope to work.  So we start on such a limited basis by seeking separability of terms and factors in the solutions.

Separability Postulates:
Consider a substrate endemic noise field of native waves ψ_S(t, x), that, because it is noise, we conceive of stochastically.  Consider also another wave field that is not expressible in terms of the native set except by interaction with the substrate field (i.e. it is a "derived wave").  It is called ψ_D(t, x), is modelled as deterministic, and is very small in intensity compared to the substrate except in sparsely located small regions where it can be of similar or greater intensity than the substrate.

The total field ψ_T(t, x) = ψ_S(t, x) + ψ_D(t, x) may be described in correlative terms, given sufficient averaging time, as a local to (t, x) matrix autocorrelation (a 4x4 tensor) φ_TT(τ, x, t) relating time shifted pairs of elements (including the self pairs) of the 4-vector gradient ◊ψ_T by the following time averaged direct (i.e. Cartesian) product:

From symmetries in the definition of the element correlation functions in Ecorr we have φ_TThk(τ) = φ_TTkh*(−τ).  So for τ = 0 the matrix φ_TT is Hermitian.  For other values of τ we have φ_TT(τ) = φ_TT*^T(−τ).

There is a corresponding spectral matrix Φ_TT(ξ, x, t) obtained from φ_TT by Fourier transformation of the individual elements.

The symbol ξ is used here to represent angular frequencies of components of the substrate field, whilst the symbol ω is reserved for the typically much smaller angular frequencies of the derived waves.  This is helpful in maintaining clarity when both are appearing in the same formula.  The wave functions of the derived field usually occur as eigen functions and their eigen frequencies are therefore discrete and often appear with a suffix as ω_n to indicate one particular such frequency for the nth mode.

The symmetries in this spectral formulation are again strong so that for ξ = 0 the total spectral matrix Φ_TT is Hermitian and for other values of ξ we have Φ_TT(jξ) = Φ_TT*^T(−jξ).

We may alternatively think of the entire problem in terms of the hybrid frequency domain representation.  In that case the basic scalar wave equation, the "zero d'Alembertian" EZdA above, is replaced by a corresponding scalar equation in terms of the Laplace transformed variable where the transformation is made with respect to the time dimension only.  These frequency domain variables are denoted here by upper case symbols to distinguish them, and that is similar to the way that the spectral (frequency domain) relationships are distinguished from their corresponding correlative (time domain) relationships.  For the general case of the total field the constitutive equation is then:

We maintain the option to keep time as a further argument placed at the end, as, for example ◊_jξΨ(jξ, x, t).  This is to allow for non-stationary statistics ... the statistical averages are then formed as centred upon the value of the time argument.  For the cases where the statistics can be considered as stationary then the original time argument can be omitted.  This means that in the hybrid formulation the frequency replaces time as the first vector element and appears in the function arguments as the first in the list.

Taking advantage of the similarity in form between the correlation integral and a convolution with the time parameter reversed in sign in one of the two factors we may alternatively re-express the correlation spectral result directly in terms of the Fourier transform of the original scalar stochastic wave function (with suitable boundaries to Fourier integration as per the Gabor transform, and normalised to power as commented in the notes on "Averaging and Normalising" above).  In these terms the statistics of the wave are summarised in their auto-spectral matrix Φ_TT(jξ, x) as:

The angle brackets around this expression are to convey the need to use some form of averaging or smoothing in the frequency domain in order to avoid pathological degrees of narrow band structure in the spectra produced.

These stochastic matrices are a dimensionally asymmetrical formulation of the statistics with local time as the basis of the frequency transform, a kind of inertially based parameter for the statistics, but with space handled directly rather than in any equivalent sort of wave number transformation.  The correlation integrations (of which the possible forms of averaging are not detailed here) are over offsets in time only, not in space.  Such an asymmetrical approach is useful, and indeed necessary, because of the special status of the time dimension in the Minkowski geometry.

Because of the quasi-linear nature we are dividing the problem into components that are narrow bands of locally defined temporal frequency, as is typically done in engineering dynamical analysis.  It does not invalidate the relativistic requirements (Lorentz invariance) of the resulting analysis so long as any such co-ordinate transformations are performed properly, and anyway, many analyses need no such transformations.

When needing to deal with time varying systems then so long as the rate of variation is slow compared to the bandwidth of Φ_TT then the stochastic matrices can also be made functions of time as Φ_TT(jξ, x, t) on a relatively low frequency basis.  This is a useful approximation at least for most purposes in electrodynamics, including the interesting dynamics underlying quantal state transitions and the related issues of Rabi processes.

A mathematical framework for this combined time/frequency basis of analysis with the necessary rigour can be found in the work by Gabor and to some extent in its more recent developments in wavelet theory, but beyond this mention and the foregoing notes that will not be developed further here.  It is however interesting to note in passing that Norbert Wiener, who was the originator of the style of stochastic time series analysis used here, and Dennis Gabor were both cyberneticians ... indeed, it was Wiener that coined the term "cybernetics".

Constraints on Derived Waves:
We seek to account for the way that deterministic waves may display the physical coherence and persistence of matter whilst satisfying the zero d'Alembertian equation, and to that end we invoke the supportive influence of an endemic noise substrate field.  Such matter must be capable of moving arbitrarily slowly with respect to our reference co-ordinate system.  However, we know that any suitable solution must remain Lorentz invariant and thereby accommodate changes of velocity of co-ordinates, and we expect that to be possible by adhering to second order equation structure (Helmholtz or Klein-Gordon) in any form of approximation we set up.  Thus we may confine our attention to proving the existence, that is the physical viability, of static forms of wave modal systems as attractors about which stochastically supported persistent wave processes can centre their fluctuations.

In order that such an attractor may be static requires that the intensity distribution of the wave mode(s) we propose shall stand still in our co-ordinate system.  This suggests that the wave mode(s) should have zero group velocity.  But we shall also need to include the possibility that there can be a closed and steady circulation of what we think of as charge (i.e. spatial integral of intensity times modal frequencies), and still regard the system as standing still.  This steady current circulation is a quasi-dynamic feature forming the origin of magnetic effects.  So really for any single mode it is something like the vectorial mean of the intensity weighted group velocity that we need to remain at zero.  This corresponds to the idea of a "centre of gravity".

Note here how the complex variable permits at least topologically a steady wave flow with constant intensity at all points ... indeed, without even any group velocity.  That is something that cannot be achieved in a Lagrangian model with real variables unless you superpose two part-models and constrain them to a similar complex structure.  But by comparison with that the model used here is built more simply, more directly.

For any given wave eigenfunction of magnitude μ and phase angle α we may write down a Stationary Intensity Criterion at any given point, but still allowing circulatory flow of charge, as the rule:

This rule is not additional to the basic wave equation except in so far that it is essential for persistent existence of any derived field eigenmode.  In fact it is not strictly necessary that it shall be accurately fulfilled.  Rather it is a rule governing the form of an attractor wave function, whilst the actual wave can fluctuate around that form with small perturbations induced by substrate field interaction over a wide range of frequencies. However, further development and checking of this idea is due in order to deal with the combination of two frequencies involved in a spin based wave mode.

The rule says that for a mode in the steady state its charge is allowed to flow by wave propagation, but only along paths of constant intensity of the said mode (i.e. perpendicular to its local magnitude gradient vector).  Whilst this is possible in a complex wave model it cannot happen in the real variable models built on the Lagrangian or Newtonian basis such as with particulate electrons.  In these conventional physics models all acceleration of charge causes radiation, so that circulating charge, say like electrons orbiting a nucleus, must radiate energy and thereby collapse to low energy.  The wave model makes a different prediction on this as on several other points.  These paths of steady state flow can be curved in space yet so long as they comply with this rule there will be no radiation of energy caused by the circulating charge.  In order that energy shall be radiated by an individual wave mode acting alone it is necessary for either the paths of such flow or the locations of the wave intensity to undergo accelerative change.  The smooth sort of circulatory motion of which complex waves are capable is not a basis for such radiation.  More will be said about this below when the nature of propagative field induction has been further developed.  Our task remains to set up the reasoning that field components satisfying this criterion of persistence can exist.

This rule requires that charge flow shall always be spatially orthogonal to any gradient of magnitude.  For some reason (find this reason!) this rule can only be satisfied by solutions with paired frequencies of which the following factor wave function is an example:

It is the viability of modes with spin structure (following the mathematical existence of spin structure in wave functions) that allows this last condition to be met, but it will only ever be possible subject to a residual fluctuation about this kind of modal form ... it is only the description of an attractor.  Also, in meeting this condition the wave functions are therefore inherently conformal with regard to spatial shape deformations caused by adjacency of such wave systems.

We may note that low pressure acoustic waves in a non-flowing inviscid fluid obey the same form of zero d'Alembertian scalar equation for their dynamics (exclude solids as such a model because of the additional effects of shear and rotative forces).  In that acoustic model plane waves propagate in a manner similar to the substrate waves discussed here.  The puzzle then arises as to why they do not display the full richness of behaviour of the waves considered here including relativistic effects.  The difference is that acoustic waves occur with a real variable (consider pressure), and that cannot satisfy the above Stationary Intensity Criterion over flows of wave energy.  To go beyond that requires the variable to be complex.  This also means that the acoustic system is amenable to analysis under the Lagrangian model, while the system discussed here is not.  From that difference arises the possible existence of spin and along with it the enormous extension to the range of solutions that display the properties of substantial matter arising from complex wave structures.

 

   Building the Interactions

The correlations and also the spectra of the field are bilinear functions of the gradients ... they are actually direct products of gradient vectors.  Thus by applying a divergence operation we can bring into existence the form of the original wave equation in each of the two linear factors of the bilinear product, except that each of the linear divergence operations must be compensated for the operation of the other because the wave invariance is only mono-linear.  This compensation can be applied in the form of a matrix embedded into the additional bilinear divergence operator.  Since the bilinear operations are themselves matrix rather than vector operations we may use a matrix to contain the two vector compensation effects whilst holding these effects independent of each other.  We may then use algebra to manipulate the expressions as a whole and to study the cross terms which represent interactions between the various parts of the field.

Primarily we seek to extract the Helmholtz equation with matrix coefficients that is the hybrid transformed version of the required matrix Klein-Gordon equation.  That also gives rise to the Schrödinger equation as an approximation.

Our primary wave equation in its hybrid form as EZhybrid above states: ◊_jξ^T·M·◊_jξ Ψ(jξ, x)   =   0

A Minkowski vector divergence operation column by column or row by row respectively can be applied to the auto-spectrum from equation EFspect above and the result is a zero vector where ψ_T is analytic:

But we can apply both of these operations together and thereby achieve a scalar zero result so long as we allow that the combination of the left and right lozenge operations needs to be corrected on a matrix element by element basis for interaction effects of commutation and multiplication to preserve their respective individual operations.  Using this device we may establish as no more than a consequence of the axioms of the model that overall continuity of correlation, expressed here in terms of the auto-spectrum, is maintained according to this particular sort of constraint on divergence.

Commutation of differentiators introduces the off diagonal chequer board pattern of sign reversals from the anticommutation effect of mixed partial derivatives (see AMPD above).  For the hybrid type of operators these arise in the three space dimensions but not in the frequency dimension because there they operate through simple multiplication that is commutative.  However this remaining dimension still involves a sign reversal (Clarify this).  Combining these requirements we can build a matrix L defined thus:

This matrix can then be applied element by element to the direct (Cartesian) product of a pair of operators to generate a bilinear divergence operator (it contracts a 4×4 square matrix to a scalar).  This achieves the application of two divergence operators simultaneously to a spectrum that is itself a bilinear product of two gradients.  Thus we need to extend the vector notation to carry two vectors simultaneously as a direct product along with a modifying matrix.  This cannot be reduced to successive operations of simple vectors and matrices.

We could resort to tensor notation at this point, but in view of the desirability of using hybrid vectors that would be awkward too.  So I shall extend the vector notation a little so as to handle the excursion into the second degree brought by correlations and spectra and the way that then paired vectors operate simultaneously in direct product as matrices.

Operators of this bilinear type must be applied by means of a matrix inner product, shown by symbol here, to the auto-spectrum matrix to yield a scalar invariance equation.  To do this also needs symbols for the operation of direct vector product modified by a matrix, and is shown here in the form of the direct product symbol split to enclose the matrix symbol as in L , producing a scalar invariance equation thus:

For general convenience we may allow that any number of matrices may be incorporated into the product as a succession of factors separated by the symbol so that the extended inner product result is the sum of all of the corresponding element by element products over all of the factor matrices.  Where elements are sensitive to commutation (as is the case for differentiator elements) then any commutative manipulation of the factor matrices is only permitted subject to the commutation restrictions of the elements within the respective matrices.  Thus the above equation may alternatively be written as:

For the sum of our substrate and derived fields ψ_S and ψ_D taken together the hybrid wave equation then says:

So we may state an invariance involving separated auto and cross spectral terms making up the total auto-spectrum for frequency ξ thus:

We know from symmetries in the definition of the correlation and spectrum functions that Φ_SS(jξ) and Φ_DD(jξ) have the same kind of symmetries as Φ_TT(jξ) in equation EFspect above.  Additionally for the individual matrix elements of the cross-spectra we have the symmetry Φ_DSkj(jξ) = Φ_SDjk*(−jξ).   Therefore for the cross-spectrum matrices we know Φ_DS(jξ) = Φ_SD*^T(−jξ) so that defining the sum of the cross-spectral terms from Ψ_S to Ψ_D as Φ_K we have:

Thus the same kind of time/frequency reversal, transposal, complex conjugation symmetry exists throughout the correlations/spectra.

So we can write a Whole Field Spectral Invariance Equation as:

 

Before taking separate paths of approximation we may analyse the nature of the coupling correlations and spectra.

Substrate Scattering Effects:
Consider the steady attractor wave function for a single mode n being a derived field with discrete frequency of the form:

So the (column vector) gradient for this single mode is:

Perhaps it is helpful to see how the correlation generating φ_SD for the single mode n may be transformed:

The correlation in the above expression is equivalent to a convolution with the sign of time reversed in one of the two functions.  This reversal can be implemented by conjugating the cis factor in ψ_D.  By writing the relationship in the frequency domain the convolution transforms to a multiplication, and the overall multiplying cis factor takes effect as a translation in frequency.

Transforming these time domain expressions into the frequency domain in this way and considering just the steady state for Ψ_n and statistically also for Ψ_S (because we can easily put back in the temporal perturbations later when necessary) gives:

So the substrate coupling matrix can be built for the case of a single modal wave as:

We may also substitute for this single mode derived field auto-spectrum a spike function δ(ξ − ω_n) defined as being narrow in ξ and with unit integral (a Dirac impulse function or a unit Gaussian pulse approaching that limit):

Substituting back to expand ESspec we have:

For the purposes of a wave model of electrodynamics we must derive from this equation two reduced equations.  One must be derived by incoherent summation of substrate power and coherent summation of coupling effects to yield a Helmholtz equation in Ψ_n and the other must be derived by coherent summation to describe the charge and current origination effects of Ψ_n driving the Maxwell wave propagation process.

We can substitute   ◊_jξ ≡ ◊_j(ξ−ωn) + [jω_n, 0]   where   0 ≡ 0, 0, 0.
Making substitutions and expanding builds the Substrate Spectral Divergence Equation thus:

The divergence balance equation is bilinear in the amplitudes and as such it has things to say about power flow balances.  These effects are statistically incoherent.  They can lead to gradients of the ambient intensity of the substrate, and thereby to the general relativistic aspects of the model.  For the special relativistic cases such gradients are, by definition, negligible.

The equation also produces couplings in the form of coherent scattering of substrate amplitudes due to the presence of the derived field.  It is these latter coherent correlative effects in the substrate that bring about the Maxwell wave system.  They propagate merely as part of the substrate field, decaying in intensity according to the inverse square law with distance.  They are induced by the terms in Ediv.3 that represent charge and current origination.  They affect matter via the terms in Ediv.2.

The term Ediv.4 labelled as "Compton constant" is the non-differential term that appears in Klein-Gordon or Schrödinger wave equations and is responsible for defining the Compton frequency, and thereby the electron rest mass.

The term Ediv.1 labelled as "Strong coupling" is the source of the so called "strong force" that appears as part of the conventional description of nuclear processes (chromodynamics).  Note that it is of quadratic degree in the intensity of the derived field whereas the electrodynamic effects (Ediv.2, .3 and .4) are all of linear type.  To a first approximation is has no effect upon electrodynamics models because their derived fields are much weaker than the substrate.

This equation is a divergence balance that takes a given derived mode wave function ψ_n with its discrete frequency ω_n and applies it to determine statistical differential divergence contributions to the substrate field linking all pairs of substrate frequencies that satisfy ξ₁ + ξ₂ = −2ω_n.  If we were to maintain the discreteness of this frequency interval in our analysis, especially when handling many modes and therefore many different such intervals together, the analysis would become intolerably complicated (perhaps analogously with the chaotic nature of a "many body problem" as discussed in Newtonian particulate physics).  However, we are interested in the cases where the bandwidth of the substrate spectrum Φ_SS is very much greater than any |ω_n| we shall encounter (even in atomic nuclei), so a continuous differential approximation in the frequency domain offers us a great simplification.

We can take advantage of some approximations as follows:

• Substrate has wide bandwidth - The substrate spectrum is finite but its bandwidth is many decimal orders of magnitude greater than any of the values of derived mode frequencies we shall encounter and the spectrum is smooth so that over intervals equal to derived mode frequencies we are able either to use a first order slope approximation for intensity versus frequency or for some purposes even to neglect the slope altogether.
  
• Substrate has high intensity - For electrodynamic processes the ratio of derived field intensity to substrate field intensity is very small.
  
• Substrate is near equilibrium - The substrate spectrum remains very close to being an even function of frequency.  It is small coherent deviations from this statistical equilibrium symmetry that convey the Maxwellian radiation, and they are induced by charges and currents.  When the effects are integrated across the very wide band of the substrate frequency ξ the statistical fluctuations remaining as effects upon the derived field are very small.  (We might query the relationship between the minimum random noise coming from the substrate and the electromagnetic zero point fluctuations caused by the ubiquitous restriction to quantal interactions.  Are they just two views of the same thing?)

Rather than attempt to use all of this equation Ediv as a single piece we may benefit by separating approximations that describe sub or factor processes more or less independently.  The conventional processes to consider in this way are electromagnetics, electrodynamics, chromodynamics (i.e. nuclear) and gravity (i.e. general relativity).  These respective sub-models can be extracted from the Ediv equation by interpreting its terms as follows.

Substrate Energy Conservation

This is the energy divergence of the substrate wave which must remain at zero in free space propagation (i.e. where there are no modal/derived/matter waves).

Modal Energy Conservation

This term balances with the substrate intensity in nuclear processes.  Because it is quadratic in the modal amplitude and the fact that electronic fields are much less intense than the substrate, this term is very small in electrodynamics modelling.

Electromagnetic Couplings

Coherent correlations in the substrate field act to generate reactive (imaginary) biases to the coefficients of the equation determining the modal resonance mechanism.  The biases can be summarised in the coupling matrix generating the differential operator of the modal Klein-Gordon wave equation.

Electromagnetic Induction

The four vector gradient of the modal wave acts to induce coherent patterns of divergence in the substrate with no more influence on the modal wave than a noise fluctuation.  These inductions in the substrate field can be summarised as the charge and current originations driving the Maxwell field of electromagnetic radiation, and the form of that induction is expressed in equation Echarge.

The induction when expressed in terms of the time domain (as opposed to the more usual frequency domain in these essays) involves the appearance of a Hilbert transform in the formula.  That occurs as a result of the phase relationships inherently present in this wave process ... see equation EHt.  An explanation of this otherwise tricky notion needs to be expanded here.

De Broglie Wave Dispersion

This is the term that allows the intensity of the substrate wave to provide support for existence of a derived wave (a modal wave) with its group velocity below the speed of light.  By integrating across the bandwidth of the substrate this term allows the dispersive loading term in the Klein-Gordon equation for electrodynamics to be obtained from the mean energy density of the substrate.

 

   Electromagnetics

For free space we may omit all derived field terms from Ediv leaving just the Ediv.0 term.  This leads to the Maxwell system of wave propagation for free space.  For the substrate field alone, in the absence of the derived field corresponding to matter, the self spectrum matrix Φ₀₀ of the pure native wave field must satisfy:

The dynamic correlations of the substrate field can be interpreted as the basis of the electromagnetic field.

The interaction effects carried by modulations of the substrate field are in a general sense extremely complex.  A first approach to bringing some simplification to this situation as an approximation is to study the 4×4 substrate correlation matrix φ_SS of which the sixteen elements are temporal correlation functions between pairs of the four substrate field derivatives, as discussed here above.  That reduces the task to studying the interaction dynamics in terms of the statistical correlation parameters rather than the underlying wave variable itself. However we need to go further than that.

Reciprocal Moments of Correlation Functions:
So long as the bandwidth of the substrate field is much greater than the modal frequencies of any derived field (matter waves) then a good approximation of these interactions can be constructed in terms of the first reciprocal moments of these same correlation functions that link pairs of the four partial derivatives.  The reciprocal relationship arises because the spectral shifting effect induced into the substrate field by scattering from a derived wave mode with a given amplitude is proportional to the frequency of that mode. It is only this induced spectral shift that carries the causal effect of this interaction; the specific value of modal frequency inducing it does not otherwise matter. This is a result of the reliable degree to which the substrate noise is incoherent with any modal motion we may consider.

Here we may see that the first degree in frequency that multiplies this interaction produces the nature of charge, whilst the concept of energy involves a multiplier in the second dgree of the modal frequency. The ratio of the two carries the Planck proportionality between energy and frequency for a given charge.

The above approximation for substrate induction is made particularly powerful in the electrodynamics case because then it involves only very small perturbations about a statistically isotropic and equilibrated form of the substrate correlations. This approximation corresponds to the electromagnetic field as it is conventionally considered.  In terms of the 4×4 substrate correlation matrix φ_SS on time delay τ the approximation can be built as 4×4 matrix θ(t, x) of imaginary scalar elements:

The same result can alternatively be built from the equivalent frequency domain spectral matrix Φ_SS on frequency ξ as:

The variable t is left present to accommodate the relatively slow variations of these statistical parameters in a slowly perturbed steady state approximation form of model.  It is reintroduced as the first element of the argument in θ(t, x) so that it is ready to be used in general analyses of the Maxwell field at these lower frequencies. However this sense of "slow" is not very restrictive since the averages leading to these results need to be taken over periods that are long only as compared to the correlation time (reciprocal bandwidth) of the substrate field itself, and that is of the order of the Planck time around 10⁻⁴³ seconds.

These expressions amount to pre-integrating the coupling effects across the substrate spectrum before applying them to any or every derived wave component, each with its particular value of frequency.  To do this they rely upon a certain amount and type of smoothness of the substrate spectrum ... a moderately good approach to its statistical equilibrium form.  Such is the aggregated nature of the substrate spectral influences summarised in the Maxwell field.  The resulting variable θ is a 4×4 matrix of imaginary scalars corresponding to the elements of the electromagnetic field vectors and composed as follows:

... where [u₁, u₂, u₃] is the electric field vector and [v₁, v₂, v₃] = [θ₂₃, θ₃₁, θ₁₂] / j is the curl of what is conventionally called the magnetic vector potential.  Seen in terms of this way of modelling the term "vector potential" is an unsuitable name left over from the Lagrangian way of looking at things so does not really fit in here, but we still use it.  Each element of the matrix θ(t, x) represents a component of aggregated nutative motion of the correlation function across mixed pairs of dimensions.

The diagonal elements of this matrix are inherently zero due to the way it is derived from the correlation matrix. The diagonal correlation elements are autocorrelations and are therfore real even functions, and they produce a zero result under the operation of reciprocal moment evaluation.

To handle the presence of matter the charge and current source terms from Ediv.3 can also be added, but in ordinary electromagnetics these are usually expressed explicitly rather than as obtained from the distributed form of the modal wave functions.  So expressed in this more comprehensive form it will look more complicated if the detail is expressed. Unless spin based effects are at stake, such as in the Aharonov-Bohm effect, there is little reason to go to that extra trouble.

 

   Electrodynamics

For the purposes of electrodynamics the term in Ediv.1 is negligible at least for a first approximation.  The term depicted in Ediv.3 is balanced by adjacent pairs with respect to couplings between the substrate and derived fields whereas those in Ediv.2 and Ediv.4 are not.  This .3 term has effect upon the substrate in coherent and therefore strong ways involving oppositely signed frequencies.  By contrast its effect on the derived field, because of the balance, is of zero mean and incoherent across the substrate frequencies.  Thus it has no more effect upon the derived field than to create a fine high frequency fluctuation about its steady value with no net mean effect.  Integrations are required here to show this point clearly and rigorously.

So, neglecting a wide band noise term, the immediate effects upon the derived field can omit the .3 term leaving the overall process to bring the influences back via the .2 term after propagation in the substrate.  There are however divergence terms, both electric div and magnetic div curl, buried in the structure of the .2 term, and these will take effect upon the derived field without any propagation delay at all.  These provide the causal basis for spontaneous quantisation ... see Smooth and Quantal Properties of the Complex Wave.

Using these ideas we can separate the charge/current part of the model from the electron wave dynamics part in building the overall causal electrodynamics model.  I shall show here how this generates the coefficient matrix for the Klein-Gordon wave equation.

Couplings to Matter:
The electromagnetic field elements of the matrix θ can be applied to express couplings between derivatives of the derived wave that depend upon the substrate spectral form.  They enter as the off diagonal elements of the matrix defining the 2-form of the differential operator for the derived wave constitutive equation.  However, if we wish to work with the hybrid form of vector expressions then the electric field terms in the leading row and column must have the frequency of the respective mode to which they are being applied divided out.  This is because ...

 

   Scaling the Fields

The system defined in terms of the native substrate field plus the derived field solitonic components is based upon coordinates with no particular definition of scale because it is on a projective space, which means that proportionality is all that is inherent.  The only source of scale in such a model arises from the bandwidth (a time/space gauge) and intensity (an energy density gauge) of the ambient substrate field.  Thus in our model here it is natural to normalise the scales in any region of space in some way based upon these two attributes of the substrate.  This can in fact be done, but for the purposes of establishing length and energy scales in electrodynamics the two constants are best involved in a somewhat indirect way that will now be described.

It must be remembered that isotropy and uniformity of the substrate are not inherent.  This is unlikely to be of great importance except in large structures (cosmology) where a general relativity is needed or in intensely excited matter (nuclear plasma) which is another more general area of stochastic analysis altogether!

Notice that thinking of general relativity in this way suggests variable scale gauge factors over a special relativistic uniform (flat) coordinate system as opposed to the more conventional scheme of curved space with fixed scale gauge factors.  These reciprocally conceived schemes of general relativistic modelling can be made equivalent at least for approximations under moderate degrees of gauge variation without singularities.  Nevertheless the two schemes cannot ever be completely equivalent because the substrate field establishes two scale parameters whilst the conventional general relativity normalises only one.  In its raw form this extra parameter sets the very small constant that in the Standard Model of conventional physics is referred to as the Planck time/length.

Rather than using the Planck time as a basic scale in electrodynamics it is more attractive to use the Compton frequency.  From ???? this can also be seen to be dependent upon that very small time but in combination with the other scale constant for the energy density.

 

   Nuclear Dynamics

The quadratic dependence in Ediv.1 upon the derived field allows this term to compete with and thereby locally to attenuate the intensity of the substrate field in those regions where the derived field reaches very high intensities.  Such is the basis for nuclear, and particularly quark, structures.  Suitable approximation to convey the nature and stability of such structures will be left for development elsewhere.

 

   Gravity

There are parts of the wave process that lead to bandwidth broadening and the consequent increase of entropy of the substrate field.  This occurs in some very weak degree associated with the modulation of the substrate by matter through its actions in the Ediv.3 term.  However, there is also an effect of "tearing apart the fabric of the wave" that must occur in a small measure close to every spin axis.  Apart from the super plasma situations that might occur in a star it is probably the latter that contributes most of the spectral up conversion producing the ordinary gravitational and apparent red-shift effects we encounter.  This is a big subject and is left for development elsewhere.

 

   Conclusions

A wave mechanics in which an energetic substrate field drives local interactions and that carries the symptoms of those interactions into the distance at the speed of light with decreasing strength (and is even symmetrical in it effects with reversal of time) appears very suitable as a basis for the under layer of physics.  To develop this into the areas of temporal irreversibility and the structure of observation refer to Essential Structure in Physical Observation and Smooth and Quantal Properties of the Complex Wave.

Until further checks are completed it remains unclear whether this model will be truly compatible with conventional quantum mechanics and the "Standard Model" of nuclear particles.  However, it is already clear that it has value as a means of comprehension, and brings into cohesion a variety of concepts that have traditionally appeared as arbitrary and disjoint.  The prospect for accounting for the superficial aspect of the appearance of particles looks to be a routine matter, dealing with such as Compton effect, cloud and bubble chamber tracks, Stern-Gerlach phenomena, photoelectric effect etc.  Similarly the otherwise anomalous effects of apparently non-local deviations of correlation in particle detection can be submitted to a new view in which there appears every possibility that the results will stay within the range of phenomena attributable to causal wave phenomena.  All of this work is, on my part so far, in a state of patchwork draft.

 

   Appendices

   A: Elements of an Emergent Non-linearity

Non-linearity emerges from the linear Minkowski wave equation model.  One might wonder just what we mean by "linear".  It can either be the restriction of the model equations to zero and first degree terms or it can mean the conformance of system behaviour to the superposition principle.  To my continuing surprise these appear to be subtly but importantly different.

As a good starting model to see this it might be useful to think of a linear 3D + time Maxwellian model bounded by reflective walls.  With fixed walls it is a linear system, but if you allow the walls to move in response to the radiation pressure or some such function of the wave amplitude, then the system becomes non-linear.  Now in allowing the walls to be movable in that way you can see how we have let the non-linearity into the model.  If we now use a Minkowski geometry and just use a complex scalar wave equation then we can show that reflective entities (field singularity centres of spin) can be realised within the wave system as nothing more than wave structures of the same sort.  These act like boundaries, except that they are at points within an otherwise unbounded physical space of the wave functions.  So because the resulting structures (that are what we refer to as "matter") can change shape under influence of the field, non-linearity has then been introduced, but without anything more than the same fabric as the wave function itself!

So the system behavioural non-linearity can arise in a system modelled by linear equations.  It took nothing more than Minkowski geometry (relativity) and a particular kind of singularity to make that happen.  We need a grasp of that singularity and its associated fields.

Two concepts of linearity: formal (zero and first degree) and observational (observed superposition).
The role of singularity (necessary for spin and thereby isolated eigenmodes and persistence of substance).
Observation is essentially based on 3D existence (persistent substance).
The relativistic distinction makes the difference - we live in a projection.

 

   B: Commutativity and Conformality

To enable certain deductive reasoning we are concerned to clarify where we have well behaved nature that we may call smoothness and analyticity of the wave functions in our model.  We have to deal with the occurrence of deviation from analytic nature in the form of line singularities.  The notions of Cauchy-Riemann differentiability in the plane and its possible or necessary existence in certain planes within the Minkowski 3+1 dimensional geometric space will be significant.

Scalars, real or imaginary etc., are commutative in multiplication, namely AB = BA.  The complex case vis-a-vis its real and imaginary parts is made commutative by the defined structure of complex multiplication.  It accords with the nature of commutativity of the operators on 2-vectors.  We are concerned to clarify and isolate the principle involved in this achievement of commutativity.

We may seek further entities that under multiplication produce a product in the same domain (so inner and direct products of vectors are excluded) and that are commutative.  The question can be applied in the domain of square matrices of scalar elements under the usual column and row rules for multiplication.  Thus we may recognise the possibility that a given pair of such elements may be either commutative or not.

Breaking apart these matrices we may see that the condition for commutativity is that the inner product between rows j of one element and columns k of the other shall obey the rule A_jB_k = A^T_jB^T_k for any pair of the indices j and k, where ^T denotes transposition of the matrix.  Except for trivial special cases this turns out only to be possible if the two matrices are both skew-symmetric, i.e. A_jk = A*_kj and B_jk = B*_kj, where the star symbol denotes complex conjugation.  The diagonal elements of such a matrix must be real and the off diagonal elements must comply with the rule of complex conjugated symmetry.  Each such matrix can be decomposed into the sum of a real symmetric matrix and an imaginary anti-symmetric matrix.  We refer to this sort of matrix as Hermitian.

We may apply a matrix coefficient to a pair of first order vector differential operators to produce a quadratic form.  We build our model from this type of compounded differential operator in which the matrix is permitted only to be Hermitian.

Non-commutative nature only appears when we do things that make the imaginary parts of the transformation matrices give rise to observable effects and that can only be in connection with processes involving the second differential order, and therefore in a general sense is not so readily evident.

Show that functions satisfying the MKG equation have at every analytic point a direction of amplitude gradient, and that the conjugate planes to these form closed surfaces, the two together forming a conformal system.

 

   References and Background Reading

This symbol denotes linkage to an independent web site:

[ABoES]	A.G.Booth "Essential Structure in Physical Observation" http://wavemodel.org/obs/observ.html
[ABoSQ]	A.G.Booth "Smooth and Quantal Properties of the Complex Wave" http://wavemodel.org/qed/smooth.html
[ABoWT]	A.G.Booth "Wave Topology of a Spin Mode" http://wavemodel.org/spin/spin.html
[ACo05]	A.J.Coleman "Whitehead's Principle of Relativity - Unpublished Lectures by J. L. Synge, FRS" Queen's University, Kingston Ontario May 2005   http://www.arXiv.org/abs/physics/0505027

 

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