This is the second part of a three-part article focusing on Lewis Little's revolutionary Theory of Elementary Waves. It is prefaced by a short digest, an "Executive Summary" highlighting key elements of the article, leaving out technical details and substantive information. It may also be helpful to establish the overall context before reading the entire article. ----------------------------------------------------------------- "Executive Summary" - Part 2 ---------------------------- Schroedinger's wave equation is the mathematical cornerstone of quantum mechanics. Heisenberg's uncertainty principle represents the intellectual underpinnings of the standard theory. The standard theory: if anything is definite in quantum mechanics, it is mathematical; if anything is physical, it is indefinite; there are uncertainties in the nature of matter that cannot be overcome. The essence of the standard theory: It is the measurement of particles that create their reality. Heisenberg's uncertainty principle: there are fundamental limits to which the accuracy of pairs of quantum variables can be specified and measured. Examples: the uncertainty relation between energy and time means that if you were able to localize a particle at a given instant, it would not have any definite energy; the more accurately you are able to determine the location of a particle the less precise you can be about its momentum. The TEW identifies that the uncertainty is a consequence of the anti-concept of wave-particle, along with the mistaken notion of the forward motion of the wave. Part 1 explained the TEW which posits the existence of real fundamental particles and real fundamental elementary waves which travel in the reverse direction of what has previously been considered to be the case. The standard theory interpretation of the Schroedinger wave equation relies on probabilities which inherently possess uncertainties which, in logic and according to the TEW, do not exist for the real particle and the real wave. The uncertainties are due to a mathematical construct and an illogical physical interpretation. In the TEW, the elementary waves exist as real objects. In the standard theory, a particle exists as ghost-like particles, giving rise to the alleged uncertainties. In the TEW, the elementary waves exist as independent objects and there are no uncertainties for any quantity or quality of real particles and real waves. The TEW refutes the Heisenberg uncertainty principle by essentially exposing its false philosophical base and illogical acausal premises. The EPR experiment: Einstein and colleagues use a 'thought experiment' to question the lack of causality in quantum mechanics. EPR becomes a 'real' experiment and appears to violate causality. The standard theory interpretation of the experiment says that even if two particles are separated by half the distance of the entire universe there is a 'connectedness' between them. The theory requires 'spooky action-at-a-distance' to explain interactions that occur instaneously regardless of the distance of separation. Another theory: 'hidden variables', a kind of reality that lurks behind quantum reality. To explain the EPR experiments our choice seems to be between the acausal ghost-like reality of the standard theory and the causal view of a 'hidden' reality that cannot be seen. The TEW says: again the assumed forward direction of the quantum waves have misled the theorists; with that assumption causality was doomed. The TEW explains the experiment with reference to real particles, real waves, and the identification that the waves move in the opposite direction of what has been supposed. There is no 'spooky action-at-a-distance' required and strict causality is restored. ----------------------------------------------------------------- PART 2 ------ In the late 1970's, I attended Richard Feynman's graduate seminar at Caltech which was based on his book "Quantum Mechanics and Path Integrals". Feynman told me that there were only two ways to unseat Heisenberg's uncertainty principle: either you defeat it experimentally or you find some other way to explain the results of quantum mechanics. Feynman was right - Lewis Little accomplished it by the latter of the two methods via his Theory of Elementary Waves (TEW). Just as Schroedinger's wave equation is the mathematical cornerstone of quantum mechanics, Heisenberg's uncertainty principle represents the intellectual underpinnings of the standard theory. But, while Schroedinger had some interest in physical reality, Heisenberg was not confined to such restrictions. Heisenberg once said: "The very fact that the formalism of quantum mechanics cannot be interpreted as visual description of a phenomenon occurring in space and time shows that quantum mechanics is in no way concerned with the objective determination of space-time phenomena". More succinctly, the standard theory says, in effect, that if anything is definite in quantum mechanics, it is mathematical, and if anything is physical, it is indefinite. Quantum variables are measurable things pertaining to the field of quantum mechanics which can change, such as position, momentum, energy, and time. Heisenberg's uncertainty principle states that there are fundamental limits to which the accuracy of pairs of these quantum variables can be specified and measured. That is the uncertainty _principle_. There are many particular uncertainty _relations_. For instance, the uncertainty relation between energy and time means that if you were able to localize a particle at a given instant, it literally would not have any definite energy. There are similar relations between other pairs of quantum variables. The most well-known uncertainty relation is between position and momentum, where momentum is usually understood to be the product of mass times velocity. The more accurately you are able to determine the location of a particle the less precise you can be about its momentum. More formally stated, the uncertainty in location multiplied by the uncertainty in momentum can never be less than a certain quantity. That quantity, which is extremely small, is known as Planck's constant. As an illustration of how the standard theory views these uncertainties, consider the single-slit experiment shown in Figure 4. In this essentialized model our source consists of electrons; the single slit in the wall has a width which is denoted by 'Y'. The (by now) familiar interference pattern is shown as the intensity experienced at the detector. When an electron goes through the slit we then know its position, at least to the accuracy determined by the width 'Y'. Since the electron is considered to also be a wave, when it goes through the slit, the wave interference results in the pattern seen at the detector. Due to the probabilistic nature of the solution to the wave equation we discussed in Part 1, the intensity due to that particular electron is only known within a given uncertainty. If we try to make a more precise determination of the position of the electron by reducing the width of the slit, the pattern on the detector widens, increasing the uncertainty in the determination of the intensity due to this particle at this particular location. The more precisely we are able to measure the _position_ of the particle, the less precisely we can determine its intensity. This is a concretization of Heisenberg's uncertainty principle. F I G U R E 4 -------------- | IIIIII| | IIIIIII| | IIIIII| | IIII| | II| | III| | IIIII| | slit IIIIIII| - - IIIIIIIII| IIIIIIIIIII| source- - -> Y (width) IIIIIIIIIIII| (electrons) IIIIIIIIIII| - - IIIIIIIII| | IIIIIII| | IIIII| | III| | II| | IIII| | IIIIII| | IIIIIII| | IIIIII| WALL DETECTOR These uncertainties are neither a consequence of our inability to build more precise instruments for measurement nor a limitation of the resolution of the devices that are used. The basic idea, according to the standard theory, is that there are uncertainties in the nature of matter that, _in principle_, cannot be overcome. That is the intellectual cornerstone of modern quantum mechanics. Heisenberg's uncertainty principle, with its various uncertainty relations, embodies the essence of Kantianism: there are barriers to our knowledge, limitations on what is measurable, and it is impossible even to speculate on what cannot be measured. In the TEW, Little identifies the fact that this whole view of uncertainty is a consequence of, an artifact of, the anti-concept of wave-particle, along with the mistaken notion of the forward motion of the wave. When the particle is thought to be the wave and when the wave is thought to move in a forward direction from the source to the detector, the interpretation of the wave function solution to the Schroedinger wave equation becomes a probabilistic function that _inherently_ possesses uncertainties. But these uncertainties do not exist for the _real_ particle and the _real_ wave; the uncertainties are due to the mathematical construct and physical interpretation underlying the standard theory. Recall that in the TEW, the elementary waves exist as real objects and are available in a complete range of quantum states. The waves travel from the detector towards the source, through the slit, interfere and induce the emission of particles at the source. The particle then follows the path of the waves back to the detector. The intensity pattern has already been determined by the dynamics of the waves prior to the particles even reaching the detector. In the standard theory a particle exists in multiple states, the ghost-like particles we discussed previously, and gives rise to the uncertainties inherent in that theory. In the TEW, it is the elementary waves that exist in multiple states, but they exist as independent objects and do not gain their reality by the 'collapse of the wave function'. In the TEW there are no uncertainties for any quantity or quality associated with the real particles and the real waves. Little is careful to distinguish 'unpredictability' from 'uncertainty'. We may not know which particular wave leads to particle emission at the source and, therefore, we may not know, in advance, the particle momentum. But this fact is just a consequence of our ignorance, our lack of knowledge, of not knowing in advance the particular parameters which determine the particle and which characterize the source. There is nothing that is inherently uncertain in this process, nothing which by its nature is indeterminate. The entire quantum process as described by the TEW is determined by the nature of the waves and the particles and by how they interact. There is nothing _in principle_ that stops us from identifying the nature of the parameters involved in the exact determination of particle emission - it is just our current ignorance, not some inherent uncertainty, of what strictly determines this level of quantum processes. Heisenberg published his uncertainty principle in 1927 and, along with Niels Bohr, he became one of the founders of the standard theory, which is sometimes referred to as the Copenhagen interpretation of quantum mechanics. At that time some theorists, most notably Albert Einstein, expressed concern over the lack of causality in the theory. This concern of Einstein engendered a debate with Niels Bohr in 1927 which, amazingly, lasted until Einstein's death, almost three decades. Unfortunately, very little was ever resolved. In 1935 Einstein, along with Boris Podolsky and Nathan Rosen, published a paper attacking the standard theory's view of physical reality. As they stated in their paper, they concluded that "the description of reality as given by a wave function is not complete." Although their paper cannot be considered to be a fundamental or devastating attack, the 'thought experiment' they offered underscored a major problem with the standard theory. In honor of the authors Einstein-Podolsky-Rosen, this experiment was dubbed EPR and it has persisted these many decades through today. There are elements in this 'thought experiment' which are somewhat subtle, but EPR can generally be thought of in the following way. Assume we have two particles (1 and 2) which have interacted with each other and are now moving in opposite directions at the same speed. If you can measure (without disturbance) the position of particle 1, then you will automatically know the position of particle 2, since they are both moving at the same speed. According to EPR, if we can predict with certainty "the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." Therefore, our knowledge of the position of particle 1 itself establishes the reality of the position of particle 2 without in any way making a measurement or observation of particle 2. But this is contrary to the standard theory which holds that the reality of particle 2 exists only when an act of measurement or observation of the particle is performed. In addition, since we already have an accurate measurement of the position of particle 1 (which also then tells us the position of particle 2) we could then accurately measure the momentum of particle 2 directly. These independent measurements can be made accurate to whatever precision we like. But this would be contrary to the uncertainty principle which places limits on the degree of our accuracy in measuring these combined quantities. Bohr used his response to Einstein to further entrench his wave-particle view of the quantum world; his idea of 'complementarity', the joining of wave and particle, was a scientific pluralism applied to matter. Bohr firmly implanted his view that an act of measurement, an observation, was what made a particle 'real'. He identified an 'ambiguity' in the EPR idea; namely that it presumed the ability to measure _without any disturbance_ to the overall system. He argued that since no measurement was taken of particle 2 (which act would have established it's own reality) then the measurement of particle 1 must be responsible for the 'reality' of particle 2. The presumption, therefore, is that a wave function must exist that applies to particles 1 and 2, and the 'collapse of the wave function' upon measuring particle 1 gives 'reality' to particle 2. This principle of Bohr's applies whether the particles are separated by an inch or by a light-year. This is the source of the famous 'spooky action-at-a-distance' (Einstein's words) idea associated with the standard theory of quantum mechanics. It represents the complete destruction of the idea of local causality. While the EPR 'thought experiment' is interesting from a philosophic point of view, one might ask in what way does this relate directly to science and to the TEW? The reason we have spent time outlining the 'thought' experiment is that in the decades since the original paper on EPR, a whole family of _actual_ experiments has been devised. As with the double slit experiments, the EPR experiments highlight the absurdity of the standard theory and afford the TEW another opportunity to present a rational explanation for observed events. In order to understand the basis for these experiments, we must first introduce and explain what is known as Bell's theorem, or more precisely, Bell's inequality. (With apologies to the physicist Heinz Pagels, who has presented a similar treatment, albeit with a completely different objective in mind.) Consider the essentialized model shown in Figure 5. Here we have a source, similar to the ones we have used previously, except that this particular source (a gun) is unique - it is a 'No. 2 pencil with eraser' gun. When the gun shoots, two pencils are ejected simultaneously, each going in opposing directions. The pencils move sideways; that is, rather than the eraser or the pointed end facing forward, the length of the pencils face towards the direction they move. The orientation of the pencils, as they are ejected, is completely random, meaning that the eraser end can be pointed anywhere within a circular direction in the plane of its motion. Each successive _pair_ of pencils, however, will have exactly the same orientation. Each of the pencils, call them pencil A and pencil B, move in a direction towards wall A and wall B respectively. Each wall has a wide cutout in them allowing pencils of a certain orientation to fit through. We will call these wall-cutout arrangements 'polarizers', since they allow pencils with a certain orientation to pass through but effectively block out others. The angle of these 'polarizers' can be changed during the course of our experiment, but initially both have the same alignment. Behind each wall are counters which keep records of the pencils that get through the 'polarizers' and those that do not. F I G U R E 5 -------------- _ _ | | | | _ | | | | _ | | | | | | | | | | | | | | | | | | | | A * B | | | | | | --- | |---------<----- *** ----->---------| | --- | | | | | | * | | | | | | | | | | | | | | | | Source | | | | |_| | | (pencils) | | |_| Counter A | | | | Counter B |_| |_| Wall A Wall B With the 'polarizers' aligned in the same direction, we would expect the counters to record pencil events that looked something like this. A 010010010001100000100100000100... B 010010010001100000100100000100... The counters record a '1' for each pencil going through the 'polarizer' and a '0' for an event where the pencil did not make it through. As would be expected, the events recorded for A precisely match the events recorded for B, since the pair of pencils are correlated (that is, they both have the same orientation) and each of the 'polarizers' have the same alignment. Now let us change the alignment of the 'polarizers'. We will rotate 'polarizer' A clockwise through an angle (say 20 degrees) relative to B. Since each of the 'polarizers' do not have the same alignment anymore, some of the pairs of pencils will not be recorded the same for each counter. Some pencils getting through A will not get through B, and vice versa. The recorded events might look like this. A 000010011000001010001010100010... B 010010010000001010001010000010... As indicated, the counting record shows there are events that no longer match up between A and B due to the different alignment of the 'polarizers'. In fact, the mismatch is 1 in every 10 events, a 10% mismatch rate. Had we rotated 'polarizer' B counterclockwise by the same angle, instead of 'polarizer' A clockwise, we would expect the results to be essentially the same. By fixing one 'polarizer' and rotating the other we are creating results where the mismatches between the two act in an independent manner. This makes sense. Why should the outcome of the counter at A have any effect on the counter at B? One of the counters is always used as the standard to judge the mismatches with the other counter. If we were to double the angle of 'polarizer' A (from 20 degrees to 40 degrees) we would reasonably expect the mismatch rate to exactly double. If, however, we rotate A an angle of 20 degrees clockwise and then rotate B an angle of 20 degrees counterclockwise, we introduce a new possibility. By rotating A we lose the standard for B and by rotating B we lose the standard for A. That means when we compare the two counts for A and B, we will completely overlook any double mismatches that may have occurred. That is, if a particular event would show up as recording a pencil to go through the 'polarizers' for both A and B when the 'polarizers' are perfectly aligned, it might be that they both record a failure to see the pencils at both A and B when the 'polarizers' are both rotated in opposite directions. Therefore what might have been a 1 and a 1 at both A and B is now a 0 and a 0; i.e. a double mismatch has occurred. Therefore, the best case for mismatches at a double angle would be twice that of the single angle. The worst case, however, would not include the double mismatches which would be undetectable. Therefore, stated more formally, the mismatch for a double angle is always less than or equal to twice the mismatch for a single angle. Mathematically, this is: M(2*angle) <= 2*M(angle) This is known as Bell's inequality. These are the results we would expect from a system that was governed by local causality: that is, influences in the state of a system are a consequence of a change in the system itself or are due to energy changes transmitted into the system. In other words, Bell's inequality should hold without regard to, or consideration of, 'spooky action-at-a-distance'. With this as background we can now look at an actual EPR experiment. We require pairs of photons that are to be polarized; that is, similar to the orientation of the pencils, there is a precise direction in space associated with each photon. Figure 6 shows a typical setup which appears similar to the essentialized model we used with the pencil gun. The purpose of our source is to generate photon pairs, moving in opposing directions, which have their polarizations correlated, i.e., photons which have the same orientation in space. The orientation of each pair of photons is random, but the correlation requires the particular pair to possess the exact same orientation. In general, these kinds of EPR experiments use sources such as calcium or positronium atoms to generate the polarized photon pairs. Actual polarizers replace our wall-cutout apparatus to permit passage of photons possessing the correct orientation, and photomultipliers (the counters) are used to detect and record the photon events. There are many more details to these types of experiments, but the above reflects their essence, which is all we need in order to understand what is going on. F I G U R E 6 -------------- _ _ | | | | _ | | | | _ | | | | | | | | | | | | | | | | | | | | A * B | | | | | | --- | |---------<----- *** ----->---------| | --- | | | | | | * | | | | | | | | | | | | | | | | Source | | | | |_| | | (photons) | | |_| Counter A | | | | Counter B |_| |_| Polarizer Polarizer A B As in our 'pencil gun' experiment, we first keep both polarizers aligned with each other so that the angle between them is zero. The source generates pairs of photons which go off in opposing directions towards each polarizer. Each pair of photons has the same orientation, but the particular orientation of each pair is random. If the orientation of the photon aligns to the orientation of the polarizer, then the photon passes through and is counted as a '1' event. If the photon does not pass through, it is a '0' event. The results of such an experiment are shown below and are very much like the results from the 'pencil gun'. As we would expect, since each pair of photons has the same orientation and the A and B polarizers are aligned the same, the counts for A and B match perfectly. A 001000110001001100000100001000... B 001000110001001100000100001000... When we rotate polarizer A a small angle, say 25 degrees, we find results that are very similar to the 'pencil' experiment; that is, we find a small number of mismatches between the A and B count. The differences between the A and B count occur 1 in 10, a 10% mismatch rate. A 001100110001001100000100001000... B 001000110000001100000100001010... When we double the angle of rotation for polarizer A to 50 degrees, the event counts are as indicated below. A 101000110001011000010100001010... B 001010110000001101000101001000... The mismatches that have been counted are now 3 out of 10, a mismatch rate of 30%. According to Bell's inequality, the mismatch rate for twice the angle should be less then or equal to twice the mismatch rate for the single angle. In other words, since the single angle had a 10% mismatch rate we would expect the double angle mismatch rate to be less then or equal to 20%. The experiment shows, however, that the rate observed is not 20%, but 30%. Bell's inequality has been violated! Recall that this means, philosophically, that the idea of local causality has been violated. This experiment has been performed repeatedly and the results are essentially the same in all cases. This appears to be a substantial confirmation of the 'weirdness' of quantum reality. The standard theory is perfectly poised to explain this strange phenomenon; it is, after all, just another example of the ideas developed by Heisenberg and Bohr. We can speak of a 'connectedness' between these pairs of photons that have interacted, whether their separation be measured in inches or they are distanced by the size of the entire universe. We can speak of how changing the alignment of polarizer A influenced the polarization of the photons counted at B even if the polarizers were light-years away. We can speak of the 'collapse of the wave function' and the role it plays in the final result. We can speak of these, and many other things, but the real essence of the explanation of the standard theory lies in the _fundamental_ ideas of Bohr and Heisenberg - speculation about the underlying reality of this experiment is nonsense. It is a fantasy to think of the photons existing in a definite state. It is the measurement we make of the particles that create their reality and such measurements change the conditions of the experiment. We really do not know that the polarizations of the pairs of particles were originally correlated and the act of measurement changes the very conditions upon which Bell's inequality was derived. Bell, like the physicist David Bohm, did not subscribe directly to all these views; he thought that local causality could be saved by postulating 'hidden variables', a kind of reality that lurks behind quantum reality. Our choice seems to be between the acausal ghost-like reality of the standard theory and the causal view of a 'hidden' reality that cannot be seen. In the six decades since Einstein and his colleagues published the EPR, it was not until Lewis Little and the TEW that a rational explanation could be given to this 'weird' EPR phenomenon. Recall that in the TEW, elementary waves exist for all quantum states which include all values of polarization. There are elementary waves that flow from the 'counter' as well as waves that flow from the polarizer itself. The polarizer is able to impose a coherence, a similarity, on some of the 'counter' waves with the same orientation as the polarizer. An opposite orientation wave flows directly from the polarizer. These are real, independent waves which arrive at the photon source and independently induce emission of a photon. Each individual photon will follow its reverse wave back to the source of the wave, either the counter or the polarizer. The same process as described, of course, occurs from each of the two opposite sides (A and B) of the experiment. The question that now arises is: how does the whole process change when one polarizer is rotated at an angle relative to the other? Assume a photon is first induced by an elementary wave flowing from the detector; the photon will follow this reverse wave back to the detector. But recall that the photon source is such that it emits particles in pairs with the same orientation. Since the polarizers are angled apart, there may not be an elementary wave coming from the opposite direction that has the proper orientation required for the induced emission. The waves from the opposite direction may be offset in their polarization by the angle that the polarizer has been rotated. Recall from our discussion of the double slit experiment that it was the intensity of the elementary wave, as seen at the particle source, which accounted for the probability of stimulated emission of a particle. That is exactly the case here, where the offset of the polarizer reduces the probability of the particle emission at least as far as the required polarized orientation is concerned. The wave that corresponds to the opposite orientation associated with the polarizer will induce emission of a photon, but the photon will traverse its reverse wave back to the polarizer and is never seen by the 'counter'. The dependence of the mismatches on the angular rotation of the polarizer is therefore explained without reliance on 'spooky action-at-a-distance' or otherwise acausal behavior; the elementary waves _at the particle source_ determine the entire dynamic outcome of the experiment. The dependence on the degree of angular rotation of the polarizer turns out to be a consequence of the intensity of the elementary waves experienced there, whose characteristics have already been determined due to the reverse direction of the wave. Of necessity, there were a number of simplifications that were made in describing the above process. One area in particular, in which the TEW clarifies a particle physics process, is in the emission of the two correlated photons. The TEW represents the cascade, the rapid succession of particles, to be all of a single quantum process. This is important because it underscores that it is the elementary wave interactions which determine all of the dynamics prior to the emission of a single particle. This is lost in the standard theory because of the anti-concept of wave-particle, where the real elements of particle and wave physics are disembodied. It should be mentioned that there is a whole class of EPR experiments which change the structure of the experiment while the photons are in flight. For instance, the rotated polarizer may be brought back into alignment after the photon particles have been emitted. There is, in fact, one experiment, referred to as a double delayed choice (both polarizers independently rotated after particle emission), where the results, under certain circumstances, are predicted by Little to validate the TEW while contradicting the expectations of the current theory. It is beyond the scope of this article to detail the process involved. Suffice it to say that there is nothing fundamentally different required for the TEW to make sense of such experiments. I just want to note here that the TEW does not require a particle to always follow a single reverse wave back to its source in all instances. Such a rigid requirement could not and would not account for any dynamic system changes that might occur, which is the case for most conditions outside of the laboratory environment. The TEW details the physics of the 'jump' conditions which permit a particle to follow more than just one wave. This 'jump' also helps explain the scattering of particles where their direction changes at a point of interaction. The actual 'jump' of a photon requires the annihilation of the original along with the creation of a new particle, which in turn is dependent on the organization of the wave which produced the scattering in the first place. In summary, we have seen in Part 2 how the TEW refutes the Heisenberg uncertainty principle by essentially exposing the false intellectual base upon which it rests. Bell's inequality and a whole class of EPR experiments were explained without consideration of the philosophically illogical acausal premises. As in Part 1 with the explanation of the double slit experiment, a rational analysis is made possible primarily by positing the existence of real fundamental particles and real fundamental elementary waves which travel in the reverse direction of what has previously been considered to be the case. In Part 3 of this article we will focus on relativity and how it relates to the TEW. We will also briefly mention some of the philosophic and scientific concerns about the TEW theory. sjs@compbio.caltech.edu Copyright (C) 1998 Stephen Speicher

California Institute of Technology, Pasadena, CA 91125.