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# Wave interpretation of special relativity and quantum michanics.

June 2002, Moscow, Zelenograd, Spassky Stanislav.

The model in principle was building on EXTREMELY simple presumptions to be EXTREMELY transparent.

There are not any mathematical calculations, but as to mathematics, there is enough.

## Brief content.

Already almost for century STR exists without model. As result we have awful gamble on concepts "space" and "time". Author asserts, that according to the modern publications many aspects STR remain not clear.
IT IS OFFERED simple model of STR, "within the framework of common sense", in which the ether is supposed three-dimensional wave medium in usual sense. Small difference of medium from classical one allows the steady existence on it two types of wave objects, which we conventional will call: "a standing wave" (StW) and "traveling wave" (TrW). The first type corresponds to micro-objects with non-zero rest mass, second - with zero one.
The objects of the first type can become displaced and at movement are distorted. Complexes of micro-objects of the first type (i.e. macro-objects, people), their movement further are considered. Micro-objects at motion in group are deformed equally. But it is possible to connect with each group the deformed system of co-ordinates (in space and time), which by its own distortion compensates distortions of micro-objects. It is shown, that in this case "macro-objects" will perceive the world, as we perceive it now (with the account of STR). All these deformed systems are completely symmetric in relation to each other. Relativistic effects, invariance of light velocity and the paradoxical properties of an ether become clear.
Of course, the specialist will find some "subtle moments", for example, as strict proof of possibility of StW-object existence.
But it is amazing, first, how simply by means of this simple model it can be explained all moments of special relativity even in school.
And secondly, how simultaneously THIS simple model can explain (see last part) the sense of wave-function of quantum mechanics ( "pilot-wave" of de Broglie ).
This can not be only the coincidence.
The model can be used in system of secondary school and higher institutes.

## Introduction.

Author had found out the considered model somewhere in 1985. He was very much surprised, when had met skeptical attitude of the familiar scientists. Their response was: "Physics has advanced far forwards, the problems of STR already are not actual". "If this model does not yield in essence new conclusions - it is not necessary".
Recently author has come across a controversy on STR in the modern popular scientific log-book. On the one hand inacceptance of the theory built on postulates, furthermore not understandable, and without a physical model. On the other hand - mastication for century of formulation, that "physics have achieved a level, on which it is impossible to explain its phenomena within the framework of common sense".

Let us briefly enumerate the difficult moments of STR:

• Why light propagation velocity is limiting and particular?
• Light has wave character. Very difficult to imagine a wave without the medium. But if there is the medium and if you move in respect to it, the velocity of the travelling wave on it should for you vary. But it does not vary. What is this medium?
Usually in this place is traditionally spoken, that "ether" is not wave medium in the generally accepted sense, but a certain special physical reality etc.
• The time is for us a category, penetrating ALL. How this category can vary in all world from our motion? What, then, IT is? How to work with IT?
Here supporters respond, that "a modern physics reached the level..."

Proposed very simple model displays, at first, that physics not yet "has reached the level..." And, secondly, how much even simple model gives more for understanding in comparison with the theory, built on strange postulates, founded on a bare symmetry. In addition, it explains initial moments of a quantum mechanics, which were substantiated formally, as well as in STR ("Physics has reached the level..."). It magnifies probability that the model is adequate to reality.

Now let's proceed to model.

• Model assumes existence of some three-dimensional wave medium in customary for us conception (it is the analog of the ether). The medium is supposed nonabsorbing energy, almost classic. Its small difference from classic is those, that allows the steady existence on it two kinds of the wave solutions. The first kind is the solution, which we conventional will call as "standing wave" (StW). It corresponds to micro-objects of our world with a non-zero rest-mass. The second kind is the solution such as "traveling wave" (TrW). It corresponds to micro-objects with a zero rest-mass.
• It is supposed (and there is some logical basis for this supposition), that the motion of objects of the first kind (StW) changes neither amplitude of oscillations, nor transversal to motion sizes.

This model answered to author on all acute problems of STR:

• from it are easily derived Lorentz transformation laws;
• the formulas of STR mechanics, including a ratio of Planck (without value of Dirac constant, of cause) are derived;
• it yields the basis for GR;
• the sense of the wave function of the quantum mechanics is displayed, allowing easy to receive the formula of a wave function for isotropic space.

In STR the requirement of invariance of transversal to motion sizes follows from the requirement of a full symmetry of all inertial systems. Or, that is equivalent, from the statement that inverse transformation is the same, as direct one (with precision to sign). In our model this requirement is formulated by separate point, to not misuse the principle of symmetry.

What is the wave solution such as StW (standing wave)? You saw its two-dimensional analog in a glass with water if correctly to select with the help of a spoon a resonant frequency at center of a water surface. The oscillations in all points are synchronous. The central wave (from center) interferes with wave, reflected by walls of a glassful. In our model there is no defined wall. But taking into account presence of some nonlinearity of medium, it is possible to assume, that big vibration amplitudes in the middle of structure can do the StW-type energy-wise more favourable, then TrW.

Wave objects on medium, near to classical one, can be represented by set of plane harmonics, which weakly, but interact with each other. In both structures the mechanism, supporting these objects as a single whole, takes place.

As to a question of key opportunity of existence of returning wave, it is possible to refer to TrW-structure, which for us simulates a micro-object such as a quantum. This structure is more easy (more habitually) accepted by us. It is represented by a wave packet with the strongly expressed one component. Therefore transversal components are obliged to be present too. And, therefore, there should be mechanisms, in accordance with which the energy from the waves, traveling from center, is gradually transformed into energy of the wave, traveling to center. Otherwise this structure would be dissipated.

The same mechanism, but more sharp expressed, works and in structure StW. The absence of strongly expressed one component can be explained by large wave energy at center and influence of center on harmonics.

If the big amplitudes in wave structure are energy-wise more favorable, than the structure "standing wave" is the most favorable under condition of definite energy. Two oppositely directed plane waves with equal amplitudes are more favorable, than one plane wave with the same total energy. It is not question of this article, how specifically is realized the mechanism of gradual transformation of energy from the waves, traveling from center, to one traveling to center. The conservation laws in both cases are not broken. The author would not like to clutter this article with own conjectures concerning interior mechanism working in structures StW and TrW.

For the same reason the question of invariance of transversal to motion sizes and invariance of vibration amplitudes at a motion of objects we shall restrict by the following:
The realization of requirement of invariance transversal to motion the sizes results in Lorentz transformation law and is similar to same requirement in STR.
The requirement of invariance of vibration amplitudes is similar to the requirement of symmetry of inertial systems (systems of Lorentz) for physical phenomena. In our model it results in relativistic mechanics.
Pay attention, we nothing postulate. The closer are properties of medium to alleged, the closer are the properties of our objects to properties of objects of STR.

For the analysis we use the most simple wave StW-object, but we shall represent it in two ways. If it is required, it, at first, can be represented as a superposition of two waves - from center and in center (recoll a glass with water). Secondly, it can be represented as a superposition of plane traveling wave set of all directions with an identical wavelength and inphase in some center. Now we are ready to the logical analysis of difficult for understanding moments of STR.

## The verbal description of model operation logic.

• What happens to StW-object at movement? It is distorted. In a state of rest all its parts oscillated synchronously. It is possible easily to show, that in a state of motion oscillations of different parts of object will cease to be synchronous; that the oscillation frequency of StW-object (wave pattern) should decrease; the StW-object should be reduced along a line of motion.
Pair of words on these distortions.
• Delay of a phase in forward parts.
In the rest the oscillations of SRT-object are synchronous in all its parts. Two any opposite plane harmonicses are cophased in immovable center and should have an identical wavelength. In moving StW two longitudinal to motion of object harmonics should be cophased in moving center of a picture. It means, that the overtaking wave should be shorter of counter wave. But then oscillation in different parts of a picture cannot already be synchronous. The oscillations in back points should lead the vibration phase in forward points of a picture.
It is possible to take two opposite longitudinal harmonics, close in frequency and to transform their sum into two multipliers. You will see, that the speed of "wave of establishing of a phase" is in inverse proportion to velocity of a motion of "the node" in relation to wave velocity (velocity of light).
• Introduction of "artificial time".
The delay time of a phase depends only on a speed of wave object and displacement along an axis of motion. It does not depend on the sizes of objects (in smaller objects an oscillation frequency is more). It allows to introduce for group equally moving objects their group (artificial) time, in which the delay of oscillations is compensated, and oscillation in each separately taken StW-object of group are looked synchronous. In this artificial time all picture looks such, as it was when the group was in rest relatively to medium.
• Declination of a perpendicular harmonics.
In artificial time the harmonics, which for moving in group objects "seems" to be directed perpendicularly to motion (phase of this harmonics along an axis of motion is identical in artificial time), for external observer it "seems" to be directed a little forwards (phase in "external" time on an axis of motion is not identical). The sinus of inclination is equal to the ratio of velocity of object to velocity of light.
• Decrease of an oscillation frequency.
At supposed invariance of cross sectional dimensions, the sizes of interference zones from transversal harmonics should be saved too. Because of an actual declination forwards of harmonics, which "seems" to be directed perpendicularly for moving objects, the cycle of their motion inside a zone is lengthened. It means, that the oscillations in moving objects will be "looked" slowed. The frequency of these harmonics will increase for same reason to save the width of interference zone.
• The reduction of structure length along the motion is result of all listed above distortions. To some extent it is possible to explain, that moving structure (its beginning and the end) external observer watches at the different moments of interior time.
• How to describe distorted moving StW-structure?
It is best to enter distorted, moving with the structure the coordinate system and distorted time to compensate the distortions of structure. As you guess, the matter concerns the Lorentz transformation. In this coordinates the wave structure will look same, as in rest variant.
For mathematicians or physicists all this can be formulated in another way. The moving coordinate system can be distorted so that the classical wave equation looked in it like in rest relatively to medium system. Then also the solutions in distorted coordinates will have the same view, that it was in rest coordinates.
• The distortions depend only on velocity. Therefore for group of objects such as StW (may be, different), the distortion will be the same. We already noted, that in distorted coordinates, tied with group, all group looks the same as if it all was in rest (relativ to medium). For the description of behavior (physics) of group of particles it is convenient to choose a distorted coordinates, in which the objects are considered in easiest way, as in rest relative to medium. Imagine, what will take place, if we with you, being macro-object consisting of micro-objects such as StW, shall try to construct the physics? Early or lately, consciously or unconsciously we shall come to the destorted systems with "artificial time", as most simple and convenient. Yes, namely we select destorted coordinates and "time". But this choice is caused by a wave nature of micro-objects, of which we consist and our world is constructed.
• If the restriction, specified by us, on invariance of transversal to motion sizes of wave structures is fulfilled, than is realized one interesting quality of considered systems. They will be completely symmetrical to each other.
As in any such system all looks, as in system tied with medium, it is possible to choose each such system as initial, and all our former logic to repeat in it. All such systems are completely symmetrical. Indeed, it is valid only from the point of view of wave objects. But there are not others in the world, constructed by us. We with you (complexes of wave micro-objects) are not capable to distinguish, what of our systems are in rest relative to medium. It for us as though is not existance. It is of another physical reality level. We are wave (oscillatory, energy) pictures on medium. From some point of view the medium, generating us, is "more material", but is perceived by us as the emptiness. From the point of view of its physics it is a wave medium, well known for us, (instead of certain not clear to common sense reality).
So, all viewed systems are completely symmetrical each other, but only from the point of view of wave objects. In relation to wave medium they are not symmetrical. The part from them is tied with medium. That is the paradox of ether, which was giving no rest to Einstein, demanding formally the complete symmetry.
• "Time" in any considered system is only convenient parameter coincided with "actual" time (of medium, ether) only in systems, which are tied with medium. But than what is bad in this parameter for wave macro-objects (for us)? It is very convenient, and behaves itself as "real" time. Understanding this it is possible to work with it without fear.
• Since all wave structures in considered distorted systems look, as in systems, tied with the medium, the traveling waves look in all these systems as in system, tied with medium. Therefore velocity of light (the wave velocity) in all these systems has identical value. No matter how you measure it in the world of "only wave objects", you always receive identical result and in any directions. Namely this was received by Michelson in his experience.
• So any object such as StW is as a matter of fact the interference, it is clear why its velocity can not exceed wave velocity. That is why velocity of light is limiting and particular.

## Aproach to relativistic mechanics. Energy.

It is not difficult to calculate wave energy for a fragment of space volume. It is proportional to volume of a space fragment, squared amplitude of oscillations and squared frequency of oscillations at the certain moment of time.

Every plane harmonic, composing StW-object in rest, at movement is transformed in another plane harmonic.

It is simple enough to calculate the energy of a concrete convenient fragment of plane harmonic in state of rest and then to calculate the energy of relevant fragment of another garmonic, in which the first is conversed at movement of StW. Certainly, taking in account, that longitudinal to a motion the sizes are reduced, but cross sizes and amplitudes are invariant. You remember, that the transversal harmonic is conversed in declined forword harmonic with sinus, equal to the ratio of velocity of object to velocity of light. Frequency of this harmonic ought to be increased to save the width of interference zones. If to take into account all these requirement, then the energy of fragment will change in prorortion to change of frequancy. In general, it is valid for any plane harmonic (and TrW). This is Planck relation. As to our transversal harmonic, its energy will be increased by the factor of Lorentz. It is possible to show, that the energy of any pair of equal opposite harmonics (transformated are not opposite) is incremented by the factor of Lorentz. Therefore it is valid for any StW-object, which can be represented as set of pairs of plane harmonics.

StW-type objects can be represented as the certain curled mode of TrW-object. At the inceasing of frequency of an generating StW-object wave, the amplitude of StW will be increased. And the size of StW-object in this case should be reduced. At a motion of StW, its amplitude is not changed.

To connect energy of wave object with concept of mass of usual physics is not difficult, as it is traditionally done, expressing excess of wave energy moving wave object through a kinetic energy.

## Impulse

In the world of "only of wave objects" there are some difficulties with definition of an impulse of wave object. Let's definite it as a stream of wave energy. In usual physics the impulse is definite as a stream of mass. As mass can be connected with energy, our definition is valid. For a fragment of a traveling wave the vector of impulse of this fragment is directed along the motion of a wave and equal to energy of this fragment multiplied by velocity of light. The object StW can be presented by set of plane harmonics. Then the energy StW is the total energy of plane harmonics, and its impulse is the vector total impulse of plane harmonics. In the rest StW-object the total vector is compensated and equal to zero. In moving StW the total vector no more compensated.
The impulse of any pair of equal opposite harmonics equal to zero. So for our transversal harmonics. But impuls of this transformed pair is proportional to velocity of StW.
It is possible to calculate the value of the vector total of harmonics, but it is possible to use the definition, given by us.
The result: the impulse of StW-object is equal to energy of this object, taking into account its motion, multiplied by velocity of a motion of this object.

## Approach to GR.

If to suppose, that the medium, generating wave objects, has slightly different optical density from place to place and time, it is possible to assume, that this optical density has an influence on a motion of wave objects. It is possible also to suppose, that the optical density of medium can depend on presence of wave objects and from their density function. The sizes of wave objects too can vary depending on an optical density of medium. For example, if the group of StW-objects moving, enters in a zone with other density of medium or density of medium for some reason varies with time and all members of group, suppose, decrease in the sizes, they will "perceive" it as general expansion of space.

## Wave interpretation and quantum mechanics.

The wave conception cannot but lead to the quantum mechanics.

This is de Brogle’s conception of quantum mechanics, which is 80 years old.

• The frequency of some wave ("de Broglie wave") is calculated from energy (mass) of a particle (only from idea of ANALOGY with photon).
• It is supposed, that the velocity of this wave is in an inverse proportion to velocity of a particle in relation to velocity of light (c).
• The particle is represented by package from these waves (set of waves, close in frequency and directed in a course of moving particle).
• There is strange dispersion, generating the "group velocity", which coincides with velocity of a particle.
• If to connect by rigid relation the length of this wave with an impulse of a particle, the requirement of the self-consistency of this wave at a motion of an electron around of nucleus gives a quantum condition of Bohr.

Too many of imposed requirements. Besides there are many problems with localization of this wave function and its obligatory expression through the complex value.

## In the offered wave conception it is supposed:

• The frequency of wave process is calculated from energy (mass) of a particle (as well as at de Broglie's model, but it does not start from analogy. We represent particle, as alternate to quantum the wave mode, "standing wave", convoluted "traveling wave").
• The velocity of all waves, composing a particle, coincides with velocity of light, as they are common nature with waves of quantums. And they compose not "package", but are directed everywhere ("bouquet"), producing "standing wave".

That's all. Practically anything new above the requirements of the first part of this work.

## Conception.

In our conception the particle in rest is the "standing wave". The oscillations in all its parts are synchronous and its function can be represented by two multipliers. First (spatial) one shows allocation of function in space. Second (time) one is a harmonic function (sin, cos) of time. At motion of a particle its oscillations cease to be synchronous, the back oscillations are phase leading. Therefore second multiplier becomes the "wave of establishing of a phase", directional as the particle. The speed of this wave propagation is in inverse proportion to velocity of a motion of the particle in relation to velocity of light. The same velocity as "de Brogie wave" has. But the essence of this velocity in our conception is absolutely another. The first maltiplier (spatial) at a motion begins to depend on time, the wave pattern is moving forward.

If particle pass through various obstacles or is situated in "potential well" (electron in a neighborhood of nucleus), the different parts of our particle begin to interact. The self-consistency of its oscillations phase becomes important. Therefore the second multiplier becomes very important, that is "the wave of an establishing of a phase". It is so-called "wave function" of the quantum mechanics.

How to receive it with the help of mathematics? We already have proved identity of "artificial private time of a particle" with the time in the moving Lorentz's co-ordinates. Let's take advantage of it. In "artificial" system of a particle its time multiplier is a harmonic function of time with frequency expressed through its rest energy. Let's return again in system in rest, substituting instead of t' its expression through t and coordinate of system in rest. It is possible to express the coefficient at coordinate containing velocity of a particle through the impulse (in its relativistic form) of a particle. Our harmonic multiplier has become a well familiar wave function of a particle, moving in free space. But, in real value, instead of complex one.

When our particle bends the obstacle from different sides samultaneously (interference) and the particle keeps itself, as a wave complex, as a single whole, "the wave of phase installation" begins to play a major role. Those directions, in which phases of different parts coincide, have advantages, when the particle, not having the right to decompose, "should" select for itself (wave complex) one from all possible directions.

You have noted, that we considered only time multiplier. If we do not forget about the first (spacial) multiplier, we cannot have problems with localization of all wave function, localization of a particle.

There are also no problems with a complex value of a wave function. It has the same sense, as in the rest of physics.

Our deduction of a wave function is not restricted by small velocities in any way and remains valid for relativistic velocities.

That for a deduction of "wave function" it was not required anything in addition to preceding part, gives hope, that the offered concept correctly reflects the physics of phenomena under consideration.