But a new solution avoids the pitfalls. it is suggested that the wave function of a particle is a separate entity and not part of the particle itself! This idea avoids the conceptual difficulties of the quantum theory which arise from considering the wave function as an alternative description of its associated particle hence its assumed dual nature.
But suppose they
are two separate entities. The particles now have no alternative description as a wave but are just objects made from energy. The wave function is now exactly that: a superimposition of number-like quantities having no real existence. They are analogues and not an energy form. They map out alternative paths in space depending on the geometrical distribution of other objects. As the particle moves, it is controlled by chance like the throwing of a weighted dice.
This idea is most easily understood by thinking of waves on a pond. Ripples from a falling pebble spread out uniformly in growing rings. If another pebble is thrown, however, then the two sets of ripples cross over each other and interfere. The superimposed waves double up in some places and cancel out in others. But let us consider the way such a happening could be represented by sets of numbers. The problem is best considered step by step and so first of all the effect of the first pebble alone will be imagined.
It would be possible to represent the height of any wave by a number. Let us call this "WN". Then these letters can represent any number and the larger it is, the higher the wave height it represents. Then to represent the place to which this point relates on the surface of the pond, two other numbers can be allocated, "X'. and "Y". These are the "spatial coordinates" measured in two directions at right angles to one another. It could be a square pond. for example, with its banks running East -West for the "X" direction and North-South for "Y". Looked at in plan from above we could decide to measure every point from the lower lefthand corner. This is our "origin", labelled "X = W' and "Y = W'. By giving two numbers, the first for "X" and the second for "Y", any point on the pond can be specified. The numbers would relate to some unit of distance. Each unit could for example be one metre (1m). First we need to specify the place where the pebble falls. This might be at "X" = 10 and "Y" = 10. This is the "source" we will call "S". Then a different set of coordinates, say 13 and 14, could mark off distances to some point we will represent by letter "P". Here wave height is to be calculated and represented by the number "WN" measured in centimetres (cms). In this way a point at any place can be specified and then the way "WN" varies there can be found by using arithmetic.
First we need to find the straight-line distance between "S" and "P". we can do this by measuring the distance directly. It will be found to be 5. More readily this can be calculated using a simple formula. If now it is known, for example, that at a distance of one metre from the source the wave height is 10 (cms), which means the value above the undisturbed surface, then the wave height at "P" can be worked out. But to do this further knowledge is required. it might also be known, for instance, that the wave height halves every time the distance from source is doubled; then when the wave reaches point "P" the height the water level will reach can be worked out. In this case the wave height would fall to 5 cms at 2m from "S", 2.5 cms at 4m and 1.25 cms at 8m. So at our point at 5m distance the wave height would be between the last two values; in fact it would be about 2.2
This would represent the maximum height of the wave at "P". But this would be reached at some time after the wave height peaked at the source, because of the time taken for the wave to travel. If the wave moved at 0.5 m/s it would take 5/0.5 or 10 seconds to reach "P,, from "S". Hence time has to be incorporated as another dimension and has to be represented by a further set of numbers. If the way the wave height varies with time is specified at a place close to "S", then by arithmetic the way the height varies with time at point "P" can be calculated. In our example, at a time 2 seconds from start at "S" a peak wave height of 10 cms would be
reached at 1 m distance and at 10 seconds the peak wave height of 2.2 cms would be reached at "P". If a snakelike "waveform" of up-and-down motion occurs at the lm position, then a similar waveform will appear at "S", though scaled down and occurring with a time-delay of 8 seconds.
So far only a single point "P" has been considered. In order to create a wave map the entire area of the
pond needs to be dotted with closely-spaced points like our friend "P',. They can be imagined set out in straight lines in both the "X" and "Y" directions and uniformly spaced. They would be set out like conifers in a planted woodland. Calculations of the kind described need to be made simultaneously at every specified location, so at each the numbers are seen to rise and fall in rhythmic fashion. But the maximum heights and the times at which these maxima occur vary progressively from point to point so that when viewed as a whole the number- sequences represent a flowing wave Pattern.
By an extension of the method the wave heights Produced by the interference resulting from two pebbles thrown in together can be calculated for any point such as "P". The resulting heights will be obtained by adding the heights of the two found separately. Unless the distances from the two sources to point "P" happened to be equal, the time of arrival of the waves would differ. It could differ so much that the peak of one, represented by a positive number of cms, could meet the trough of the other represented by a negative number. This is because the trough represents a level below the undisturbed surface. when added, the numbers nearly cancel out in this case, representing destructive interference.
In a very similar way numbers can be used to represent the abstract waves which we are now saying control the way particles move. The paths they are most likely to travel lie where the wave amplitudes, analogous to the heights we have been discussing, are greatest. The paths are chosen at random, however, just as if amplitudes represented the weightings of a loaded dice. It has been found, however, that these weightings are not simply proportional to the total wave amplitudes. To get the proper weightings they have to be multiplied by themselves. They need to be squared. So "wave functions" can be defined as the squares of the total amplitudes of summated waves. The chances of particles travelling along given paths are then directly proportional to these wave functions.
Applied to Young's two-slit experiment a wave pattern like that shown in FIG.1 will appear. A particle passing through either slit would then have, in the example shown, five possible paths from that point to the screen. These would lie where the waves add up as marked by the rows of short thick lines. one of these paths would be chosen at random in a manner similar to throwing a five-sided dice. This of course is an over-simplification because the position on each of the columns of short thick lines needs to be chosen as well.
The important feature to be noticed is that we now have a model which explains how an interference pattern can be built up, even though only one particle passes through the slits at a time. This fits the experimental observation. The two-slit experiment is, however, a simple case which can be represented in two spatial dimensions alone. In the general case motion occurs in three spatial dimensions.
The model can be readily extended to suit the general case. So far the model can be represented by a two-dimensional wave map of points on a thin sheet. Now sheet upon sheet of such wave maps need piling on top of one another to fill a volume of space. Motion can now be represented by numbers varying at each point so that wave motion in any direction both along each sheet or from sheet to sheet is possible. There can be no real sheets obstructing motion, but somehow the points need some form of physical location.
For the universe to behave the way it does, the entire volume of all space extending for at least ten billion light-years in all directions needs to be filled with points at which wave-function-numbers can be manipulated. Since similar abstract wave-functions need to control the electrons of an atom for a consistent explanation of reality, the spacing of these points will need to be small, even on an atomic scale. The amount of calculation going on all the time to control the motion of all sub-atomic particles in the universe has to be unimaginably huge. The mind boggles at such an extraordinary picture, yet no other solution to the problems raised is any less difficult to accept. Indeed all others so far introduced seem to me far less acceptable because they demand impossibilities like an infinite number of systems of matter coexisting in the same places. The imagination jibes at the prospect of new universes instantly created every time a minute particle has a choice of route for its travel. Then again how could quasars possibly be created in the past by observation in the present?
If the resulting wave functions are controlling the Position of particles, then most will be observed at the high crests and low troughs. None will be seen where the waves cancel.
Clearly if particles are controlled by non-material wave functions, then the latter must be acting like pure numbers. Hence some kind of gigantic computer is being specified which fills the entire universe. This idea cannot be dismissed as absurd. Paul Davies(107) states that others have already come to the conclusion that computers are required for the control of electrons. But each tiny electron has been envisaged as housing its own individual unit. IS it not more reasonable to suggest that space itself could behave as an information network having computer-like properties?
Let us call it the "Grid". The required information link between particles would be difficult to conceive if an attempt were made to provide one in any other way. For example, how could electrons, each wielding computer power, obtain information without such a Grid? With a Grid accepted, electrons themselves do not need such properties.
Furthermore Wheeler's idea postulates that the brains of life-forms co-operate to structure matter. For this to have validity a Grid needs to exist to connect all brains to each other and to some kind of matter-forming substrate. However one looks at the problem, it appears that some form of interconnecting Grid permeating all space just has to exist.
With this new concept, however, particles are always particles made from energy. They can be either real, so that they exist permanently, or can be virtual, so that lifetime is short. In either case, whilst they exist, they can be regarded as imaginable objects with true surfaces. There is no longer any need for the inverted commas around this word "surfaces". They were only needed when duality had the particles existing as alternative ghost-like waves whilst in transit. Both the virtual particles of space and the real ones of matter now have a common explanation for their control.
In a real situation a barrier like the two-slit mask shown in FIG.1 is made from atoms having their plans mapped out by wave-functions patterned on the Grid. The sub-atomic particles, representing the reality of matter, only exist in the places allowed them by this plan. Then if freely moving particles like photons or electrons are introduced, they will follow moving plans formed by interfering wave functions. These allow paths which take account of the geometrical distribution of other atoms by taking their wave functions into account as well. Hence moving particles then either reflect from the mask or go through the slits. The wave-plan maps out all alternative routes by its interference patterns but any particle chooses only one path at random. In this way experimental observation is satisfied.
This model, however, does not yet explain the way permanent electrons could fit in with the Schrodinger wave description of the atom. To match this, electrons need somehow to be given a means for jumping about at random all over the ball-shaped space or "orbital" allowed by the waves. Deprived of this extra mobility they would still be committed to circular or elliptical orbits like planets going round the Sun. So we are not yet "out of the wood", so to speak.
Before addressing this problem we will take a further look at gravitation. It is a related problem, as we have shown already. So by trying to move toward a solution in this area of physics we can hope to throw light on problems relating to wave-particle duality. That everything is related to everything else is an axiom worth keeping in mind.
We need clues to help us build up a physical model able to explain how countless billions of points can be located in space for use as number depositories. Space
must not be blocked solid, otherwise nothing would be able to move. Furthermore, the structure involved needs to be composed of zero net energy, because unless everything has always existed without a beginning, then the whole of creation must have arisen from pure nothingness.