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12.02 So what purpose do these pictures serve, and what can be gained by adding one more to the list? They exist only to satisfy our curiosity about the reasons for things, to provide us with explanations and understanding. The blind formulae of quantum mechanics give correct answers in the laboratory, but the various theories attempt to provide interpretation. Galileo knew how to calculate the distance a falling body would move in a given time, but did not know why it fell. Newton gave us a partial understanding of why a body falls when he showed that the rules it follows also govern the motions of the moon and planets. Then Einstein carried this process further by showing that the changes in such motions, which Newton had ascribed to some intangible and unexplained gravitational force, were due solely to properties of space and time, and required the postulation of no mysterious forces acting at a distance.

12.03 Perhaps some similar all-embracing principle will one day give a generally agreed explanation for the behaviour of particles in the micro-world, but it seems unlikely that any such principle will remove entirely the perplexity which we feel when contemplating this behaviour. The quantum world is so different from anything we experience directly that it will always seem strange. The controversy which persisted through most of the twentieth century between the various interpretations of quantum phenomena was centred on the relative plausibility of the contending explanations, on their intellectual economy and tidiness, rather than on any differing predictions they made. It is into this arena of conflicting ideas that we pitch the Nodal Interpretation, believing that its simplicity, and the relatively few directions in which it challenges our credulity, make it worthy of consideration and comparison with the established theories.

12.04 While it seems unlikely that we can ever test experimentally these various interpretations, it is possible to argue against several of them on other grounds, particularly where they come into conflict with the fundamental properties of time which we developed in Chapter 2. The time-asymmetry of the collapse of a particle's wave function contrasts sharply with the dynamical symmetry of the particle's behaviour when seen from a mechanical point of view, and becomes an outright inconsistency when we recall that the wave and the particle are attempts to explain the same phenomena.

12.05 This inconsistency is again evident when we view the history of a quantum system as a static picture on the four dimensional canvas of space-time. The conventional wave-form describing the state of the system during the time interval between two events is based on the supposition that, because the first has already occurred, it is a certainty, while because the second lies in the future, and because we know the quantum world not to be deterministic, it is still undecided, and may yet be one thing or another. But, as we have stressed throughout the book, the dividing line between past and future, the moment we call "now", does not exist in the real world, and is a purely subjective phenomenon within individual brains, so there can be no intrinsic difference between future events and past ones. The future seems to us less certain than the past only because we can remember the one but not the other. We are less certain of the future than the past; the future itself is no less certain. The wave-form of conventional quantum mechanics is a probability amplitude, providing a measure of our ignorance of a future event until it occurs, and collapsing when that ignorance is replaced by certain knowledge as the result of observation. But for Nature herself there is nothing about which to be ignorant. Past and future events are all laid out in four-dimensional space-time with nothing to distinguish between them, and the probability of future events is an unnecessary and meaningless idea. Those features of the wave-form which appear to collapse at the moment we make an observation are wholly in the mind, with no counterpart in the real world.

12.06 The present theory, by basing itself on the picture of time which we presented in Chapter 2, forces us to ignore what our intuition tells us about the nature of past and future, the concept of "the present moment", and the apparent flowing of time. Renunciation of these beliefs may provide the chief stumbling block to a ready acceptance of Nodal Theory, but this new view of the nature of time is now becoming widely accepted, or at least widely discussed, by philosophers and physicists, including many who have no special interest in, or knowledge of, quantum mechanics. We shall quote two passages, written recently by a philosopher and a physicist respectively, which discuss the main thesis on which the Nodal theory is founded, that there is no such thing as "now" in the world outside ourselves, that the nature of future events can therefore not differ intrinsically from the past, and that the "flow" of time must be an illusion.

....[We shall discuss] the old philosophical controversy about the status of the distinctions between past, present and future. One side takes the view that these distinctions represent objective features of reality, the other that they rest on a subjective feature of the perspective from which we view the world. On the latter view, the notion of the present, or now, is perspectival in much the way that here is. Just as there is no absolute, perspective-independent here in the world, so there is no absolute, perspective-independent now. Obviously a view of this kind does not have to say that our ordinary use of "here" and "now" is mistaken ... Once we understand the perspectival character of these terms, we see that their ordinary use does not involve a mistaken view of the nature of reality; it doesn't presuppose that there is an objective here and an objective now. (Time's Arrow and Archimedes' Point, Huw Price, OUP, 1996)

We remember the past, we plan the future, but we act now. The present moment is our moment of access to the universe, -- we can always change the world at this instant. But what is "now"? There is no such thing in physics; it is not even clear that 'now' can ever be described, let alone explained, in terms of physics. For example, suppose the following is tried. 'Now' is a single instant of time. The response 'which instant' yields the answer 'every instant'. Each instant of time becomes 'now' when 'it happens'. But this is going in circles. ... To counter that all time is 'now' eventually, but not all at the same time, is mere tautology. (Space and Time in the Modern Universe, P.C.W. Davies, CUP, 1977)

12.07 Let us examine again some of the problems which the quantum world presents to us, and assess the plausibility of the explanations which the Nodal theory offers. This theory, like all the others, has its own wave function, but it assigns a different function to represent each segment of a particle's world-line, that part of the world-line lying between each consecutive pair of nodes. The wave is independent of any observer, and is determined wholly by the characteristics of the particle and the positions in space-time of these nodes. Furthermore, it is time-symmetric; nodal theory claims to overcome the temporal contradictions which plague other theories.

12.08 The classical experiments which appear to show quantum bodies possessing simultaneously the incompatible properties of particles and waves are readily explained by the nodal theory. We claim that particles exist only at nodes, the points where they seem to come into contact; they do not move from one to the next. But as we are prevented from gaining any direct knowledge of a particle except where it experiences collisions, its non-existence between them can have no effect on what we observe. We can consider the wave to move from node to node, but those features of the traditional wave of quantum mechanics which are time-asymmetric, such as the radiation of the wave representing a photon emission, are not real, and are not represented in the wave form described by the new theory.

12.09 Before the advent of quantum theory, the only real, substantial constituents of the universe were supposed to be the particles from which matter is built, but the Nodal theory and the Copenhagen theory both question the reality of these particles. However, whereas Nodal theory claims that such particles exist only at nodes, and cease to exist between one node and the next in a particle's life history, several other interpretations question the reality of particles between one observation and the next. The consequences of this latter view take their most extreme form when observations or measurements of a system are separated by considerable intervals of time, and during these intervals all the possible ways the system could develop are said to exist simultaneously as a vast superposition of waveforms. Only when the system is next observed do the waves collapse, and one out of the many possible outcomes crystalises out. Until this occurs, the particles are believed not to have unique positions and momenta, but to be in a superposition of incompatible states, and can not be said to exist in the usual sense of the term.

12.10 The best known illustration of this principle is provided by the famous Schrodinger Cat paradox. This unfortunate creature is incarcerated in an opaque box, which also contains a flask of poisonous gas, and a trigger mechanism capable of breaking the flask to release the gas, thereby killing the poor animal. The trigger is actuated by a small radioactive source which has a 50% probability of releasing an alpha-particle within the next hour, and if this does happen the cat will die. The box is not opened until the end of the hour, and so the cat has a probability of one-half of being found dead when the lid is lifted. Now the paradox concerns the state of the cat during that hour. Because no observation can be made of the animal until the box is opened, the waveform describing the box and all its contents must be in a superposition of two states, in one of which the cat is alive, and in the other it is not. The cat itself must be in a superposition of a live state and a dead state.

12.11 We can understand why conventional quantum theory forces us to this bizarre conclusion, for Schrodinger's Cat illustrates on a large scale a principle which we see in a less dramatic form in the familiar two-slit experiment of Thomas Young. The two possible states of the cat correspond with the two routes a particle can follow through Young's apparatus. We would like to say that the cat must be either alive or dead, and that the particle goes through one slit or through the other, but this leads to a contradiction, for the probability of the particle reaching any particular point on the screen would then be the sum of the probabilities that it got there via the one slit or via the other. This would not give the interference effect we observe; we know we must not add probabilities here, but rather the complex wave functions corresponding to the two routes, which differ not only in magnitude but also in phase angle. If the phase angles differ by about 180^o, the superposition can result in a zero magnitude at certain points on the screen, and only thus can we explain the dark bands in the interference pattern. In the same way, we are told, it is possible that, when we open the Schrodinger box, interference effects could be observed between the waveforms resulting from the alpha-particle's creation and its non-creation. So we must not try to imagine the cat existing, in either a live or a dead state, until the box is opened, just as we must not visualise the particle passing through either the one slit or the other.

12.12 Now the Nodal theory does admit that interference effects can result from the coming together of two parts of a wave function which, since they separated, have passed through intermediate nodes on their journey. We discussed two examples in Chapters 6 and 7, one relating to Young's experiment when the particles collided with low-energy photons, and the other when photons are reflected from a pair of mirrors. From the Nodal point of view, the only link between one node and the next is information, and in cases where all the necessary information can be passed from one segment of a particle's world-line to the next via a node, then interference effects can still follow. This is what happens in the case of the particle collisions and the mirror reflections, and in principle could arise when the lid of the Schrodinger box is opened. In practice, though, this would never occur, for any coherent information must certainly be lost when the flask is shattered; this is Bohr's "irreversible amplification", the moment at which he believed the waveform to collapse. Only later writers have tried to delay this moment of collapse until an observation is made, but in the Nodal interpretation neither of these moments has any special significance. Schrodinger's cat seems irrelevant to believers in the Nodal doctrine.

12.13 The Heisenberg uncertainty principle is easily explained by the new theory, but its implications are rather different from those arising in the traditional interpretation. The usual description of the position-momentum uncertainty of a particle tells us that the product of the uncertainties of these two quantities is of order h, Planck's constant, and assumes that both quantities have their own degree of uncertainty. But the fact that either can be measured as accurately as we wish by designing a suitable experiment, provided the other is not measured at the same time, led Bohr to suppose that it was meaningless to speak of such exact values except in relation to the apparatus used to measure them, and he sidestepped the issue of whether these values both had an objective existence if they were not measured. From our viewpoint a particle's position has no meaning between one node and the next, and so the question of accuracy of measurement does not arise. On the other hand we can say that the momentum has a precise value between one node and the next, for the wave corresponding to the particle, the NWF, does have an exact frequency. In principle we can measure the positions of nodes as accurately as our methods allow, but there is no direct way of measuring momentum. Any attempt to measure simultaneously the position of a node and the momentum with which the (imagined) particle approaches that node must involve apparatus to change the position of the node, for example by passing the particle through an aperture. It is not surprising that in these circumstances we must trade knowledge of the one against the other, and in simple cases the Heisenberg relation is easily verified.

12.14 What can we say about the indeterminacy of quantum phenomena? We know the future behaviour of a system is not determined uniquely by its past history, and so in general we cannot accurately predict the future by calculation, but to Nature herself this is of no importance. Our capacity to store information about the past, as memories in the brain or as records on paper or in computers, is one of the miracles of life on earth which may have no parallel anywhere else in the universe, but because our brains are thermodynamic mechanisms, driven by the inexorable entropy gradient without which we could not live, we will never be able to gain direct knowledge of the future in the same way. In the absence of life, Nature keeps no detailed records of either the past or the future, and so it is rather strange that we are puzzled more by our inability to know the future than by our remarkable capacity of knowing the past.

12.15 It follows that Einstein's belief in quantum systems' possessing some system of hidden variables of which we are ignorant, but which nevertheless determine their future, this belief also was unnecessary. There may indeed be processes at work in the micro-world of which we still know nothing, but if such new secrets are revealed in the future it seems unlikely they will change the way we view the unpredictability of quantum phenomena.

12.16 The Nodal Theory throws no new light upon the EPR experiments and the phenomenon of entanglement. Modern interpretrations of these experiments all seem to offer no escape from the belief that the probabilities associated with entangled particles can be transmitted non-locally. By this we mean that the occurrence of one event can affect the probability of another even when the two are not within sub-luminal range. To observe these phenomena we must set up the same experiment many times, compare observations of entangled pairs of events, and assess the frequencies of related results. Then we find correlations which appear unlikely unless they are influenced by these non-local effects. It is only in this rather abstract way that the EPR results appear strange. But the concept of non-local influences is an essential part of the Nodal Theory in a much wider sense, for the Nodal Wave Form is "aware" of the layout of the surroundings of a system without restriction of space or time, and so perhaps the non-local conclusions of entanglement experiments are easier to accept, now that they are seen to be just one manifestation of a much wider principle. If we think such non-local influences are counter-intuitive, it is only because we never witness them in everyday life, and indeed can never be directly aware of them. The present theory reduces the behaviour of particles and radiation to relationships between nodes, and nothing else, but all the classical rules which we had come to believe such particles to obey are reflected in these nodal relationships. We still never observe particles, radiation or information travelling at speeds greater than that of light; it is just the transmission of probabilities which can occur at such speeds.

12.17 Much work remains to be done before the Nodal Theory is sufficiently robust to challenge all its alternatives. These chapters attempt only to sow the seeds, which now need cultivation. Perhaps there is here a fruitful field of research for young physicists seeking unexplored territory within which to work and make a worthwhile contribution to our understanding of the physical world. Were the author fifty years younger he would gladly lead the expedition, but he is now at the wrong end of his life to engage in such an adventure. What he can do is point out the directions in which it seems exploration should be profitable, to give Nodal Theory a wider base of knowledge and understanding from which to argue its case.

12.18 We took the first few tentative steps along this journey of exploration in Chapter 11. We tried there to separate those features of conventional quantum theory which are purely subjective from those which correspond to features of the real world, and which the Nodal Theory attempts to represent. We decided that, among the objective characteristics of the Nodal Wave Function, its frequency must be real, for it bears a simple relationship to the rest mass or energy of the particle it represents. And the orientation in space-time of its wave-fronts determine the reference frame in which the particle is at rest and, together with the energy, decides the momentum which the particle will display in any other frame. But we decided that several aspects of the CWF must be purely subjective, either because they depend on our own velocity relative to the particle's world-line, and so upon the Einsteinian reference frame we happen to occupy, or because they reflect the imperfect knowledge we can have of a particle's present state or future history. In the first category we place the wavelength of the waveform we observe, for this depends wholly on the relative velocity of the particle's nodes and ourselves. We used the NWF to calculate the wavelength associated with a particle, as observed in the reference frame of any particular observer, and obtained the well-known de Broglie formula w = h/mv. Into the second category we must place any feature of the wave description which is time-asymmetric, and also any which reflects our own ignorance, such as the spread in the values of w and v when a particle is represented by a wave-packet, and various manifestations of the Heisenberg uncertainty relation. Again using only our definition of the NWF, we derived the Heisenberg relation between our knowledge of the position and momentum of a particle.

12.19 We discovered that a sort of probability significance can be attributed to the Nodal wave. Interference effects are certainly real and independent of any observers; when Young's experiment is set up there is no doubt parts of the screen lie in dark bands and parts in light ones, indicating clearly that the probability of any particular photon reaching certain points of the screen varies from place to place. When we make the hypothetical supposition that the time and place of the particle's starting node, (r₁,t₁), is fixed, while its ending node, (r₂,t₂), can take a range of values, we can consider the density distribution of points (r₁,t₁) if the experiment is repeated many times. With these conditions we find that the appropriate frequency distribution is given by |Y|² where Y describes here the NWF, just as it does for the CWF.

12.20 Finally in Chapter 11 we considered briefly the NWF for massless particles such as the photon, and we derived the correct relativistic formula for the Doppler shift in wavelength when a source of light and an observer are approaching or receding from each other.

12.21 We still do not know the nature of the nodes from which our world is built, nor of the NWF which connects them. We can picture the nodes as points where information is exchanged, like miniature telephone exchanges, and we can imagine the waves to resemble the electromagnetic variety with which we are more familiar, but it is unlikely that these images come close to the things themselves. Perhaps we shall never improve on such pictures, for we are dealing here with entities so unlike those of the familiar world, and our sensory perceptions of it, that our representations may always remain no more than mere analogies. But there are aspects of the present picture which are obviously incomplete, and we must hope someday to be able to add the missing features. We have, for example, some understanding of how the NWF transfers information about energy and momentum, but we cannot say the same about electric charge and particle spin, and those other characteristics of nuclear particles which are conserved at the interactions taking place at nodes. How does the wave represent these features, and transfer them from node to node?

12.22 The NWF, as we have described it, serves two functions. Firstly it carries information, such as the mass of the particle it represents and the orientation in space-time of its world-line between pairs of nodes. Secondly it helps determine the position of these nodes; for example points which are inaccessible because of intervening matter or interference effects are indicated by the low intensity of the NWF there. But in both respects our descriptions are incomplete, for in many situations the intrinsic spin of a particle should feature in both these functions. The spin of an electron or the state of polarisation of a photon is sometimes preserved through a series of nodes, showing that the relevant information is passed on from node to node, just as is information about energy and momentum. And in certain experiments, such as those involving polarisation, the spin of the particles can result in some destinations being less likely than others. The necessary information must be carried by the NWF, just as in interference experiments it is phase information carried by the wave which gives rise to light and dark bands on the screen or photographic plate. These considerations show that the NWF we have described in previous chapters needs to be supplemented in order to incorporate this spin information. This supplementary information must share two essential features with the NWF that we have already developed, namely its independence of any observer and its time-reversibility. Further work is required to derive the most plausible form for the new wave, and to show how it can explain spin and polarisation phenomena within the same limitations as our previous work has explained momentum and interference effects. This is perhaps the most pressing problem for NWF theory in the days ahead.

12.23 Indeed we do not understand properly what this property of spin is in itself. We may have in the back of the mind a picture of a spinning top, but we know the spin of an electron cannot really be like this. For a body to spin, different parts of it must be moving in different directions. So if an electron has no component parts, it cannot spin in this sense. Nodal theory provides some relief from this dilemma, for the electron does not really exist between its nodes, and the spin is seen to be no more than a part of the message transferred from one node to the next by the wave function. But there are experiments in which the combined spins of a large number of particles can produce a macroscopic change of angular momentum. How can a nonsubstantial wave impart angular momentum to a material body? Equally puzzling, why is the angular momentum always transferred in integral multiples of h/2?

12.24 May we end on a more abstract note? The purpose of the Nodal Theory, as of all the other theories of quantum mechanics, is to provide explanations of the phenomena we observe at the atomic and nuclear scale, to provide understanding. What do we mean by explanation or understanding? These concepts clearly differ from knowledge or skill, which give us the tools we use with notable success and without controversy in our work in sub-atomic science. Acquiring understanding seems to be more difficult than acquiring knowledge. One can have a detailed knowledge of the topography of a country without understanding the geological processes which have formed it. One can be a skilful car driver without being able to explain the chemical and mechanical operations which enable the car to move.

12.25 It seems that the explanation of a fact involves a higher degree of abstraction, or generality, than the fact itself. One must ask if this process of generalisation can go on for ever, whether we are pursuing an infinite regress, or whether we will someday reach the end of the quest, a single overriding principle which will explain everything, and give us an understanding of all the processes which drive the universe. Many a parent of a precocious child must have pondered on this when confronted by an endless sequence of "why?" questions. "Daddy, why is it getting dark?" "Because the sun is setting." "But Daddy, why does the sun set?" "Because it goes below the horizon." "Why?" "Because the earth is turning on its axis." "But what keeps it turning?" At this stage most parents will admit defeat. Sadly some will reply, "Oh, do stop asking silly questions!"; only those blessed with the same childlike curiosity as their youngsters, and faith in their growing intellectual grasp of facts and principles, may answer, "I don't know, but perhaps someday you will, and will be able to explain it to me."

12.26 Some philosophers have expressed the view that mere scientists will never reach the ultimate truth that answers every "Why" question, and that only philosophy will reach this final point.

If one says to the physicist, "Now please tell me what exactly is energy? And what are the foundations of this mathematics you're using all the time?" it is no discredit to him that he cannot answer. These questions are not his province. At this point he hands over to the philosopher. Science makes an unsurpassed contribution to our understanding of what it is that we seek an ultimate explanation of, but it cannot itself be that ultimate explanation, because it explains phenomena in terms which it then leaves unexplained. (Confessions of a Philosopher, Brian Magee, Weidenfeld, 1997)

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