Expanding Spacetime Theory

Explaining quantum mechanics.

Quantum theory in a different light

Modern quantum theory is rather strange since it makes no attempt to explain why the submicroscopic world is discrete or provide a physical explanation the quantum mechanical wave functions. Starting with the assumption that the metrics of spacetime oscillate at very high frequencies one can show that the quantum mechanical wave functions might be spatial modulations of very high frequency oscillations in the metrics of spacetime. The formerly unexplainable double slit interference experiment, which has confounded generations of students, can now be explained in a simple and direct way. I will try to give you and idea of the explanatory power provided by the Expanding Spacetime theory as well as a new appreciation for the quantum world.

A quick review of the development of quantum theory.

Most everyone knows from any basic physics course that light as well as radio waves is electromagnetic radiation and that light waves will interfere and cause interference fringes. However, Newton thought that light comes as particles rather than as waves and ever since his time people sometimes have favored the wave picture, sometimes the particle interpretation. This is the situation even today. By the end of the last century the emphasis was firmly on light as a wave phenomenon, since this explained interference and diffraction. But, in the beginning of this century, after Planck had suggested his black body radiation law, which was based on light as quanta, the particle interpretation returned on the scene. It was Einstein who suggested that light came in particles later called photons, realizing that this could explain the photoelectric effect, which cannot be explained by the wave theory.

Louis de Broglie then made a bold speculation: If light really is a stream of particles why shouldn't other particles also have their own waves? He used Einstein's relation for the relationship between the energy of a photon and its frequency, E=m·c2=h·f, suggesting that the corresponding relation also might apply to particles like the electron, proton and neutron. However, he had no idea what these waves might be. Erwin Schrödinger, one of the founders of quantum theory, picked up de Broglie's idea and carried it further to arrive at his famous equation, which today more than anything is associated with quantum mechanics. De Broglie tried to further develop his proposal that all particles also behave as waves, but his effort dissipated in the whirlwind of the rapid development of a new quantum theoretical formalism in the 1920: ties.

This mathematical formalism, which was championed by Werner Heisenberg, Max Born, P. A. M. Dirac, Pascual Jordan and others initially was based on Heisenberg's algebraic method and became the dominant approach for the school of thought gathered around Niels Bohr in Copenhagen. Later it became known as the Copenhagen Interpretation of quantum theory.

Faced with the enigmatic problem that matter seemingly behaves both like particles and waves, which became known as the particle-wave duality, Bohr developed his philosophy of “complementarity” whereby no attempt is made to explain this strange dual property of matter. Instead Bohr insisted that we should accept as a fact that matter has two different faces that cannot be reconciled. This way of dealing with an unresolvable problem might perhaps be acceptable while awaiting some resolution to the puzzle, since it allows developments to continue without getting sidetracked by philosophical speculation. However, such a position clearly is inadequate in the long run. Adopting this pragmatic view of complementarity the outcome of an experiment depends on what we are looking for. If we are looking for a particle we will see a particle and if we are looking for a wave we will see a wave. Unfortunately for the future development of quantum theory, Bohr, who by then had become a dominant force, carried his philosophy of complementarity beyond its initial pragmatic position. He argued that the quantum world is such that two complementary viewpoints is needed and always will be needed in dealing with quantum phenomena. To him it was not only futile, but also wrong to try to find a deeper explanation. Although many prominent people of his school did not subscribe to Bohr's view, his influence was such that they often officially supported the Copenhagen Interpretation. This support might perhaps be seen as a closing of ranks in the struggle between two groups, Bohr's school in Copenhagen, and a loose alliance of prominent but dissenting scientist most notable Schrödinger and Einstein.

Schrödinger was convinced that his wave equation could describe both the particle and wave aspect of nature and suggested different ways to incorporate the particle into the wave equation. Einstein, with his unerring feeling for physics and honest intellect, objected to the very suggestion that no deeper explanation for quantum phenomena was possible, insisting that quantum theory in its current form must be incomplete. He strongly felt that something important was missing. Although Einstein could not tell exactly what was wrong he sensed that the mathematical formalism and Bohr's complementarity fell far short of a complete theory. He repeatedly challenged Bohr over many years. But, Bohr's dominating influence is evidenced by the common perception today that Bohr won these arguments with Einstein and that quantum theory in its current form is a complete theory that tells us all we ever need to know about the submicroscopic world. However, it now appears that Einstein was right and Bohr was wrong. Not only can the quantum world be understood, but it also has a simple and beautiful structure of its own.

In the beginning of the 1950:ties a remarkable scientist and philosopher, David Bohm, embarked on a lonely quest in search for the underlying explanation to the quantum world, which he sensed must be there. He showed that all results of quantum mechanics might be explained if particles move under the influence of a certain guiding function derived from the Schrödinger waves. Just before publishing a paper on this in 1952 he found that the same idea earlier had been suggested by de Broglie, who now joined Bohm in a revival of his “pilot wave” theory from the Solway conference of 1927. Einstein, stayed out of this development. Perhaps he felt that introducing this rather magical guiding function wasn't much of an explanation. However, the importance of Bohm's and de Broglie's work is that it shows that there is at least one other possible approach to quantum theory other than the Copenhagen interpretation, thereby opening up the possibility of further research and deeper understanding.

Unfortunately, Bohm's new theory was not well received by the scientific community, who felt no reason to abandon the standard interpretation in favor of a new theory unless it could offer some practical advantages. Furthermore, the spirit of Bohm's theory went against the by then dogmatic Copenhagen Interpretation, since it suggested a possible physical explanation for the quantum world based on particle motion.

Today discontent with quantum mechanics in its current form is evident and is growing stronger day by day. This theory, strongly influenced as it is by the Copenhagen Interpretation, simply cannot be understood as a physical description of the world. People using quantum mechanics are like practitioners of ancient Chinese medicine. They know what works and what does not work, but they do not really know why it works or the nature of the underlying physical process. It is a way of doing physics that delivers the right results but does not explain why. Even the most competent specialists, for example Richard Feynman, have admitted this fundamental shortcoming. Recently a growing movement has been building aimed at introducing the de Broglie-Bohm theory and other approaches into the mainstream epistemology by having it taught as an alternate approach to quantum theory. This is a sound development that hopefully will gain impetus from the new insights presented below.

The problem with understanding quantum mechanics.

Much of what has been written about the problem with understanding quantum mechanics might perhaps be blamed on a general misunderstanding of what the quantum mechanical wave functions really are. Textbooks typically avoid any ontological explantion and merely state that the wave functions express probability densities and have no real existence beyond this. Yet, these wave functions are solutions to the Schrödinger equation. They behave like real, physical, waves showing interference and reflection. This similarity becomes even stronger when noting that the Schrödinger equation is very closely related to the Hamilton-Jacobi equation of classical physics. It is very natural to think of the wave functions as being physical entities governed by dynamic differential equations. It is this conceptual image that seems to be causing the perceived strangeness of the quantum world. However, if we approach quantum mechanics from a different perspective most of the interpretation problems disappear.

The perspective changes dramatically as soon as we consider a different interpretation of the wave functions. The reader with traditional schooling might feel somewhat uneasy about the proposition that the wave functions are modulations of high frequency oscillations of the metrics of spacetime. However, if this were the case it would explain quantum mechanics. Louis de Broglie came close to discovering the real nature of the wave functions with his dual wave model proposing that a particle is formed by a physical wave, which is accompanied by second “probability wave” the Schrödinger wave functions. In de Broglie's model the first wave associated with the particle has real physical existence while the second wave is the traditional quantum mechanical wave function. One might make the objection that a particle cannot be formed by a linear superposition of waves since it will not remain local but will disburse with time. However, in the context of the EST theory I am not considering linear superposition of wave functions but rather high frequency oscillations in the coordinate metrics of spacetime. Such oscillation can generate spatial energy density given by the energy-momentum tensor of general relativity. Here you might object that this energy is undefined since it changes with the reference frame, but in the EST universe there exists a cosmological reference frame and the energy density is well defined in this frame. Furthermore, the energy-momentum tensor is non-linear and by modulating the four spacetime metrics independently it is not unlikely that spatially confined particles might form in particular resonance states.

In continuing this line of argument I will not further explore the creation of particles from spacetime oscillations (which is an interesting subject, see further below), but simply assume that particles in essence are spacetime oscillations at the Compton frequency. I will leave open the details of exactly how the particles are formed and simply investigate the consequences of this assumption. According to this scenario a moving particle will generate spatial modulations of the metrics of spacetime, which are the wave functions of quantum mechanics. If a particle is confined to a tiny region in space with metrics oscillating at the Compton frequency and if this region starts moving, applying the Lorentz transformation to the line element of general relativity tells us that this motion generates a spatial modulation of the phase of the oscillating metrics. This modulation looks exactly like the de Broglie matter wave! This would explain why a moving particle always is accompanied by a wave. The de Broglie matter wave simply is spatial modulation of a “carrier” oscillation of the metrics at the Compton frequency that generates the rest mass. The conjecture that wave functions are modulations of the metrics gains support from the observation that the corresponding geodesic equation of general relativity at low velocities is identical to the de Broglie/Bohm guiding function! David Bohm and others have shown that quantum mechanics follows directly if particles move according to this guiding function (i.e. the geodesic) and the Schrödinger equation applies.

We now arrive at a most important new insight.

Particles generate spacetime resonance, which are the quantum mechanical wave functions. The wave functions do not have separate and independent existence.

This simple statement goes a long way toward explaining quantum mechanics. Wave functions express the response of the spacetime metrics in the presence of an oscillating the particle. The wave functions in an atom are resonance states of an electron that are activated when the electron is at a certain location moving with a certain velocity. Without the presence of a particle the wave functions do not exist physically; they are no more than dormant resonance conditions. A particle's Compton oscillation resonates with the geometry and the field potential creating resonance described by the wave functions, which are modulations of the spacetime metrics of the particle. One can show that this resonance depends on the position and velocity but not on the direction of particle's motion. The dependence on the velocity occurs via the particle's relativistic Compton frequency that changes with the velocity. Thus, the spacetime resonance conditions (i.e. the wave functions) only depend on the particle's position and frequency of oscillation. Since the wave functions do not exist independently but are generated by the moving particle, the various solutions to the Schrödinger equation correspond to different possibilities that cannot be simultaneously active. The value of a wave function at a certain location describes the amplitude and phase modulation of a particle's metrics oscillating at the Compton frequency. Thus, the quantum mechanical wave function cannot be separated from the particle; it describes how the Compton oscillation that sustains the particle depends on its location and velocity.

Now we clearly see the weakness of the orthodox interpretation of quantum mechanics by which a “pure state” typically is represented by a superposition of contributions from several orthogonal wave functions. However, this conception has no basis in physical reality, in fact it leads to the enigmatic measurement problem and the problem with “collapse” of the wave function. Since a particle only can be in one of several possible resonance states at a certain moment it is no mystery that it always is found in one of these states. Only one of the branches of the wave function corresponding to a certain eigenvalue of an observable is “active” at any time. In the orthodox interpretation the wave functions are taken to represent all our knowledge of quantum systems and the evolution of the wave functions as given by the Schrödinger equation is of central importance. But, it now seems that the particle's motion should be the focal point since wave functions merely express the passive response of spacetime to a moving particle. Or rather, the wave functions tell us how the oscillating region of spacetime that is the particle resonates with the force field and the geometry depending on its velocity and position.

However, this fundamental insight needs some elaboration because the quantum world is different from the classical world. In classical dynamics we only have to be concerned with the particle and the force field that influences its motion. But, in the quantum domain the presence of a particle modulates spacetime and since the geodesic responds to this modulation we are often dealing with feedback where a particle's motion might be influenced by the response of spacetime to its motion. This feature is unique to quantum mechanics and accounts for much of its mystery.

Part of the reason for the orthodox interpretation is the need to keep the various wave functions active in order to be able to account for interference. At the beginning of the development of quantum theory this might have been the strongest reason for considering the superposition of wave functions. But, we now realize that interference between particles only can take place when their oscillation frequencies match. We realize that the idea that the state of a quantum system for a single particle is to be described by a superposition of several wave functions, some of them corresponding to different energies (Compton frequencies), does not make much sense.

The EST interpretation of the matter wave and the particle-wave duality.

If the Expanding Spacetime theory is right the universe expands by changing the metrics of both space and time with the pace of time changing in discrete steps. This mode of expansion cannot be perfectly smooth. It sets up high frequency oscillations in the metrics of spacetime throughout the universe relative to an observer participating in the cosmological expansion. If we model such high frequency oscillations by general relativity we find that these oscillations might exist at infinitely many frequencies without generating gravitating energy. However, we also find that there might be modes of oscillation within small regions that could create spatially confined energy - a particle's rest mass. If such a small region were to move through space it follows directly from the line element of general relativity that it will be accompanied by a spatial phase modulation of the spacetime metrics. Furthermore, if the oscillation frequency is such that it multiplied by Planck's constant equals the rest mass energy (i.e. E=mc2=h·f) then this spatial modulation of the metrics of spacetime looks exactly like the de Broglie matter wave. Thus, oscillating metrics might generate both the particle's rest mass energy and its accompanying matter wave! Those familiar with the problem of creating a “particle” by a linear superposition of matter waves should note that the particle's energy might be created from the non-linear terms of the energy-momentum tensor. This would explain the nature of the matter wave as being spatial modulation of the amplitude and phase of very high frequency oscillations in the metrics of spacetime that create the particle's rest mass energy. When the particle moves through space, the phase, and sometimes also the amplitude, of the high frequency oscillation resonates with its position creating a matter wave with a wavelength that decreases with increasing velocity. Thus, the rest mass energy is created out of nothing but oscillating spacetime metrics and as a consequence of this oscillation a matter wave is created when the particle moves. In this view the particle and wave aspects are inseparable since they are created by the same phenomenon - oscillating spacetime metrics.

With this explantion the enigmatic particle-wave duality has found a natural explanation. Moreover, the generally accepted but still quite mystical interpretation of wave functions as being both physical, since they can interfere, and non-physical, since they express probabilities, can now be understood. The wave functions do not exist by themselves; they are modulations “experienced” by a particle that influence is motion via the geodesic of general relativity. If the particle is not present, they do not exist. They represent “potentialities” rather than physical waves in three-dimensional space. This also explains the use of configuration space with 3N coordinates to model N particles. The wave field created by these N particles at a certain location describes the modulation of the metrics of a particle should it happen to be at this location.

The quantum mechanical wave equation.

The matter wave is obtained directly from the line element of general relativity. To find the rest mass energy of a particle we have to look at the so-called Ricci tensor. This is a mathematical object from general relativity consisting of two main parts, a linear contribution and a nonlinear contribution. When evaluating the linear part we find that it takes the form a wave equation for the oscillating spacetime metrics. Setting the linear part of the Ricci scalar equal to zero we get a wave equation that applies to all coordinate systems.

The second, non-linear, part of the Ricci tensor could generate the rest mass energy. Thus, the Ricci tensor not only determines the wave equation but also supports the proposition that spacetime oscillation is the mechanism that creates the rest mass energy and the quantum world.

Particle interference and the double slit experiment.

One of the first things a beginning student of quantum theory encounters is the double slit particle interference phenomenon. Understanding this phenomenon is the key to understanding quantum mechanics. Particles pass through two narrow slits in a screen and then impinge on a second screen making small dots. After many particles have hit the second screen one sees an interference pattern develop in the form of light and dark stripes similar to light interference fringes. It looks like we are dealing with waves passing through the two slits together with the particles, but the interference pattern appears even if the particles arrive at the screen one by one. It seems that a particle somehow interferes with itself, but how can this be possible? It was this strange phenomenon and the inability to explain it that motivated Bohr to develop his idea of complementarity. After having struggled with this riddle and discussed it with leading scientists over a long time he came to believe that it was impossible to explain the particle-wave duality and that we just have to accept that Nature is strange. Somehow matter is both particle and wave. He thought that we have to regard these two descriptions as being complementary. Ever since the 1920:ties people working with quantum theory have simply accepted this unsatisfactory situation and students first encountering quantum theory have to struggle to accept, or if this is impossible, to live with this state of affairs. Bohr and his followers came to believe that the inability to explain this aspect of quantum mechanics is an unavoidable fact and that the quantum world has it own strange rules that are very different from our macroscopic world.

In standard quantum mechanics the double slit problem is handled by assuming that the particle passes through both slits simultaneously so that it can interfere with itself. Although this doesn't make much physical sense it seems to give the right answer. The standard interpretation by which a wave function expresses probability helps, since it can be argued that we are dealing with probabilities rather than with actual events. Still, this doesn't really explain why interference appears when just a single particle at a time passes through the slits.

The first indication that there after all might be possible to explain single particle interference came from David Bohm who used his guiding function to find the trajectory of a particle. This was a significant step since it showed that interference fringes would appear if the particle went through one of the two slits being guided by a “pilot wave” emanating from the two slits. However, it went only part of the way, since the matter wave seemingly still had to go through both slits, which would imply that the matter wave and particle must follow different trajectories.

In our interpretation the matter wave is a relativistic phenomenon generated by the particle's motion. It is a phase modulation of a very high frequency oscillation in the metrics. However, this wave modulation might extend beyond the particle, spreading out both in front of and behind it. Let's assume that the particle just has passed though one of the two slits. Its matter wave extends backward to the screen with the two slits. Let's further assume that this matter wave does not vary at the screen. This is a boundary condition that fixes the phase of the wave along the screen. One way to see that this must be true is to consider an observer moving with the particle who will see the screen with the two slits moving. The metrics of spacetime relative to this observer oscillate at the screen and points along the screen will be in phase. However, the two open slits do not constrain the phase and because of this an interference pattern in the metrics of spacetime develops in the space behind the screen. As a result the amplitude and phase of the matter wave varies slightly when the particle moves from point to point. At locations where the particle's distances to the two slits differs by an integer number of wavelengths the amplitude of the matter wave is larger than at locations where the interference is destructive. As a consequence the particle resonates with a matter wave (i.e. a modulation of its Compton oscillation) that varies with its position. Note that this matter wave is the resonating response at the moving particle and that the pattern develops because of the double slit geometry. Thus, we are dealing with self-interference. But, how can this affect the particle's motion?

The generalized guiding function.

If the particle changes direction under the influence of the wave field we should be able to see this in the geodesic. However, a position that changes with time implies a time dependent wave function. Assuming oscillating spacetime metrics, one can derive a generalization of the de Broglie/Bohm's guiding equation from the line element of general relativity corresponding to a situation where both the low frequency wave function and the high frequency “carrier” excitation are functions of both space and time. Since this new guiding relation is derived from general relativity it is covariant, which means that it can be used to find the motion of the particle in any coordinate system. In particular, we may select the coordinate system that coincides both with the particle's motion at a particular instant. In technical terms this is a Lorentz transformation into the particles instantaneous inertial frame. Performing this transformation we find that the generalized geodesic gives a velocity that depends on the derivative of the wave function. In classical physics this velocity would be zero since we have matched the instantaneous velocity by making the transformation to the inertial frame. Instead, we might possibly see acceleration between the inertial frame and the particle frame. But, here in the quantum world we see a velocity instead of acceleration. How should this be interpreted?

If the cosmological expansion occurs in steps so that the pace of time changes discretely as suggested by the Expanding Spacetime theory, the velocity obtained from the geodesic might indicate where the particle will be at the next step. In this discrete expansion mode, events progress like frames in a movie. There is no motion or velocity at any particular instant, velocity and acceleration are concepts constructed “after the fact” from the particle's trajectory. This is a very old idea. Over two thousand years ago Greek philosophers argued that continuous motion is impossible. They used Zeno's paradox and reasoned that if a particle were to move between two locations continuously it would have to pass over infinitely many points between these two locations in a finite time, which in their view was impossible. They therefore concluded that the nature of all motion must be discrete. With the Expanding Spacetime theory we arrive at the same conclusion by a different line of reasoning based on spacetime equivalence. If motion is discrete, instantaneous velocity does not exist. Motion and velocity is associated with a sequence of locations. This might explain why velocity and position are incompatible concepts in quantum theory, where it is impossible to simultaneously determine both velocity and position. We see that velocity is associated with a minimum of at least two positions. At a certain position velocity does not exist more than as a potential. Thus, given only a single position the velocity is undetermined. Conversely given a velocity, the position is undetermined. This is the essence of Heisenberg's uncertainty principle.

Returning to the double slit experiment, we find that the generalized geodesic makes the particle avoid regions where the wave function's magnitude is small, guiding it to regions where it is large. The motion of the particle creates matter waves that “senses” the geometry and guides the particle accordingly. This action is based on feedback response. If the particle initially by chance moves into a region where interference is destructive and the magnitude of the wave function small, it automatically changes direction moving into a region with constructive interference. Viewed from the moving particle it appears that the magnitude of the surrounding (self induced) wave function increases with time because its changing position increases the matter wave resonance. Since the particle prefers regions where the magnitude is large it will together with other particles form an interference pattern at the screen.

All this follows from the generalized guiding function, which is sufficient to explain it. But what is the basic principle behind this guiding action? General relativity can be derived from the minimum action principle, which says that Nature always selects the path that minimizes the action (action is an energy-type function). This is the golden rule of all physics dealing with motion and is “built into” general relativity. A freely moving particle always moves in the way that minimizes the action. However, the modulation of the metrics “curves” spacetime relative to an outside macroscopic observer and from this external viewpoint the particle's trajectory might appear curved. This also is the explanation to the elliptic planetary orbits that are created by the curving of spacetime by gravitation. Similarly, the motions of the electrons in an atom could be explained by spacetime modulation in the form of the quantum mechanical wave functions. Motion on a geodesic explains why no energy is lost by electromagnetic radiation.


The just described action where a particle interferes with its own matter wave in a feedback fashion is a unique and central feature of the quantum world. Particles continuously sense their environment. Electrons in an atom move in regions where self-interference maximizes the amplitude of the wave function and are constrained to remain within these regions by the guiding action of the geodesic. Matter wave interference takes place instantaneously; it does not propagate at the speed of light. It can span vast distances, which explains the non-local feature of quantum theory.

The collapse of the wave function.

The philosophically problematic collapse of the wave function, where the mere act of observation selects one of its possible branches while the other branches “collapse” and seemingly alters the predicted future course of events, is caused by a misconception regarding the nature of the wave function. Like Einstein suspected, quantum theory is incomplete since it does not recognize the possibility that wave functions actually might be spatial modulations of a particle's oscillating metrics. If so, the wave functions do not exists as physical entities; they merely describe how a particle would resonate if it were in a certain energy state at a certain spatial position. Generally, different branches of the wave function correspond to different energies (with corresponding “carrier” frequencies) that cannot be simultaneously active. Each will have its own unique geodesic. Thus, the branches of wave functions represent different possibilities. At any particular time only one basis function can be activated by the presence of a particle, since a single particle cannot simultaneously have different energies. The other branches are dormant. As a consequence the measurement problem with the collapse of the wave function simply disappears. The particle will be in one of the basis resonance states regardless of whether or not an observation is made.

Let me repeat the crucial point that modulation of a particle's metrics given by the wave functions do not exist unless a particle is present. The different wave functions that are solutions to a wave equation are no more than possibilities; most of them are “empty” in Bohm's language. They do not exist in reality other than as possibilities, or “propensities”, with certain probabilities. This sheds an interesting light on the nature of the quantum mechanical wave functions. Like a hole in a flute determines a certain note, a particular solution to the wave equation might correspond to a certain energy level. A tone is heard only when playing the flute and a quantum state is activated only in the presence of a particle with the right energy (oscillating at the right frequency) at the right location. Thus the wave functions with their corresponding energy states represent potential possibilities that might be activated by the presence of a particle. With this insight it is clear that the wave function for a pure state does not describe the motion of a particle and consequently most people would not consider it to be a complete description of the quantum world.

What is the meaning of the Schrödinger equation?

Much of the confusion surrounding the collapse of the wave function is caused by the presumption that the Schrödinger equation expresses a dynamic process similar to the dynamic equations of motion familiar from classical physics. However, the derivation of the Schrödinger equation from oscillating spacetime metrics shows that the quantum mechanical wave functions might be spacetime resonance conditions in response to the presence of one or more particles. The wave functions do not model the independent evolution of probability functions, they tell how the amplitude and phase of the oscillating metrics of spacetime would respond if a particle were at a certain location with a certain energy at a certain time. In the time dependent Schrödinger equation the energy and the location of the particle varies with time and in response to the moving particle the spacetime metrics change accordingly, as expressed by the wave function solutions to the Schrödinger equation. Various wave function solutions express different possibilities depending on a particle's energy, location, the geometry of the surrounding environment etc. Thus, the Schrödinger equation does not model the evolution of “probability waves” but describes various possible resonance conditions that could be stimulated by a particle. Although it is true that it is more likely to find a particle in regions where the wave function is maximum or minimum, this phenomenon is not a property of the wave function itself but the result of self-interference. If a particle by chance moves into a region with increased resonance it will tend to remain within this region. If it starts drifting away, the resonance amplitude decreases, which will prevent it from leaving. This feedback mechanism follows directly from the geodesic of general relativity applied to the oscillating line element. Heuristically this action might be explained by speculating that a particle prefers regions where spacetime resonates with its Compton oscillation since it takes less energy to sustain the particle in these regions.

In short, the time dependent Schrödinger equation does not describe evolution of independent wave functions; rather it describes how the metrics of spacetime would react if a particle were a some location, oscillating at some specified relativistic possibly time dependent Compton frequency (i.e. a with time varying energy). The trajectory followed by the particle might then result from with time changing geometry and velocity. In deriving the Schrödinger equation from oscillating spacetime metrics one finds that the direction of motion has no influence on this derivation; only a particle's location and relativistic Compton frequency, or equivalently its relativistic mass, matters. Therefore, the Schrödinger equation does not model a particle's motion; the wave functions simply tell us how the surrounding geometry (for example the two slits) or a force field would modulate spacetime metrics oscillating at the Compton frequency as a function of the particle's location. Also, the presence of other particles might influence the wave function and interfere with its motion. Again, a particle “senses” its surroundings via its matter wave and reacts accordingly. Its detailed motion is very difficult to predict since it changes in response not only to the particle's own position and velocity but also to any changes in its surroundings that might induce a sudden shift in resonance state. This is an important new feature characteristic of quantum mechanics. Also, a particle's own matter wave might interfere with the geometry of its surroundings and influence its motion by feedback as in the double slit interference experiment. This insight sheds a new light on the quantum world.

A possible connection with string theory.

It might be interesting to take the idea of spacetime modulation a step further and take a look at possible further implications. It is a well-known fact that the Schrödinger equation breaks down at very small distances and high frequencies. Now we know why. Since the Schrödinger wave functions modulate a particle's high frequency Compton carrier excitation, the lower limit of what the wave functions possibly can modulate is set by the period and wavelength of the Compton oscillation. The Schrödinger wave functions break down at this level and cannot model the interior of particles. Although the Schrödinger equations no longer works, scientists have pushed beyond its lower limit and are now investigating the structure of fundamental particles using mathematical objects like twistors and strings. Nobody knows if this new mathematics corresponds to any physical “things”, but strangely, in many respects it seems to mirror the world of elementary particles. But, instead of the four spacetime dimensions used by general relativity and standard quantum mechanics, the leading string theories use ten, or more recently, eleven dimensions. A string is a set of ten or eleven oscillating parameters, or dimensions, that together seem to describe features of particles. However, nobody knows what these extra dimensions really correspond to physically.

We have seen that oscillating metrics could explain the quantum world. One important assumption was made when considering this possibility - that these oscillations excite all four metrics of spacetime identically. I assumed that all four metrics are excited by the same periodic wave-form at the Compton frequency and found that this high frequency “carrier” wave could be modulated by wave functions with longer wavelengths, which are the solutions to the Schrödinger equation. These wave functions modulate the phase as well as the amplitude of the excited metrics. Typically the amplitude of the modulation is very tiny. For example, if the Compton excitation generates the rest mass energy of a particle, the modulation amplitude is extremely small for fundamental particles like the electron, the proton and the neutron

However, if the metrics of spacetime actually oscillate it is not unreasonable to consider the possibility that they might oscillate with independent amplitudes and phase angles. Also, the oscillation could consist of a superposition of many frequencies. The base mode of oscillation might be the Compton frequency and there might be additional terms at higher frequencies. Since the net contribution to the energy-momentum tensor that would generate the rest mass energy is proportional to the square of these frequencies, the modulation amplitude for the “overtones” might be much smaller than the fundamental amplitude; yet, these overtones could curve spacetime within a particle and thus provide its spatial confinement. Furthermore, the in-phase and the quadrature components of the oscillations might be modulated independently. This would provide two additional degrees of freedom per spatial dimension. The three spatial dimensions would then give six additional dimensions, which together with the four spacetime dimensions could be the original ten dimensions of string theory. Including time we get a total of twelve dimensions. However, if we consider just a single particle, the temporal metric could be selected as the reference with zero quadrature component. This would give a total of eleven dimensions used in the most recent version of string theory - the M-theory.

One of the contributors to string theory has mused that string theory has been, and still is, developed “backwards” in the sense that instead of developing it as a model of some specific physical process it is a mathematical model in search of a physical application. For example, the ten dimensions of string theory were discovered after first investigating the properties of lower dimensional spaces beginning with the five dimensional Kaluza-Klein theory from 1926. Many physicists have been reluctant to take string theory seriously since ten dimensions seem quite strange. Perhaps they feel that a ten dimensional theory hardly can be more than a play with mathematics. One wonders if sting theory wouldn't have gained credibility quicker if people had discovered it while investigating the implications of independently oscillating spacetime metrics. It is indeed interesting that ten dimensions are needed for a theory that promises to explain the particle zoo. This might be one of the main reasons to why spacetime is four-dimensional.

The new superstring theory, which promises to model both fermions and bosons, is full of symmetries. There are gauge symmetries, super-symmetries, covariance, dualities, conformal symmetries and many more. Perhaps all these different symmetries might be traced to what could be the most fundamental symmetry of all, scale invariance, which preserves the topology of spacetime together with all physical phenomena.

All this is of course pure speculation, but it is consistent with the idea that quantum mechanics result from oscillating spacetime metrics. Thus, oscillating metrics might not only explain traditional quantum mechanics but also suggests what the ten (or eleven) dimensions of string theory might correspond to. Obviously, most of the string dimensions are “curled up” if they model very high frequency modulations of the metrics of spacetime.

Related work.

Recently several papers have appeared with ideas closely related to the EST interpretation.

Edward Nelson shows that the Schrödinger equation may be derived from the de Broglie/Bohm guiding function assuming that Markovian diffusion acting via a “background field” influences a particle’s trajectory.

Laurent Nottale has developed a theory based on the principle of scale invariance modeled by fractal spacetime. In this view spacetime is broken into fragments of different scale, which causes a particle to move unpredictably at small distances similar to the work by Nelson. Nottale assumes that a particle trajectory is continuous but not differentiable so that at every point on the trajectory the direction and magnitude of the velocity changes randomly. Based on this assumption he defines a complex velocity with two components and derives the Schrödinger equation by classical minimum action considerations. In this way quantum mechanics follows from the fractal structure of spacetime taking into account scale equivalence.

Enrico Santamato offers a different point of view in his geometric quantum mechanics. He combines the de Broglie/Bohm approach with Weyl’s geometry showing that the quantum behavior could be a consequence of the underlying spacetime geometry suggesting that the mysterious quantum potential of Bohm’s theory might arise from modulated spacetime curvature. Weyl geometry is closely related to the conformal transformations and thus to both Nottale’s theory and to the EST scale expansion.

Yet another example is the work by Sidharth who invokes fuzzy spacetime showing how Weyl’s geometry can result in non-commutative geometry with direct connections to quantum theory. He shows that there might be a direct link between phase of the wave function and the determinant of the metric tensor gij.

The common trend in these contributions is clear suggesting that quantum mechanics result from vibrating spacetime metrics, perhaps generated by the cosmological expansion as proposed by the EST theory.

A birds-eye view.

Quantum theory based on the Copenhagen interpretation is engulfed in mystery. It is a magic black box of unknown interior design that answers questions we put to it. By chance we have found this wonderful box, but what is inside it and how come it works so well?

Flying high over the ocean you might see a ship moving at a steady pace in a fixed direction. At a lower altitude you see the ocean waves and up close you notice how the ship moves up and down and sideways by the wave action. You now see that the location and speed is not constant but varies slightly. A Ping-Pong ball is thrown overboard. Following its motion you find that both its velocity and position change unpredictably when it rides on top of the waves.

Like a Ping-Pong ball on the ocean, particles moving through the fluctuating spacetime metrics don't move in straight lines with constant velocities but move in increments with changing directions influenced by local fluctuations. This is the world the black box of quantum mechanics describes. To find out what happens to the particle you could in principle either track the particle directly or follow it indirectly by noticing the spacetime fluctuations. Quantum theory uses the latter approach. It is based on the spacetime resonance waves, which are the solutions of wave equations. Thus, in quantum mechanics velocity is defined using the wave shape (the derivative of wave function) on which the particle rides rather than the actual particle velocity and the particle's position is on the average determined by the crests and valleys of the wave function. In principle this differs from the real motion of the particle, which always exists although it is quite unpredictable. Knowing what we really are dealing with, the modulation of the spacetime metrics, demystifies quantum theory and opens up a new world of possibilities.

Back to Top