Nov 27, 2016 - Index Terms—electron wave packet, energy-momentum, ... De Broglie waves exist in a distributed oscillatory system ... way, a statistical domination of gas molecules moving ..... λ and e λ are rewritten below in expanded form to emphasize their ...... in what part of this energy-momentum may be found as.
De Broglie Waves as Dynamic Wave Packets of Natural Oscillatory Systems Alexander Gritsunov Abstract—De Broglie matter waves are interpreted as real oscillations of generalized coordinates of some natural oscillatory systems with distributed parameters (NOSs), not as Born’s “probability waves”. In particular, electrons are considered as excited modes of natural electron-positron oscillatory system (NEPOS), not as “hard” particles. The quantum kinematics (spatio-temporal evolution of NOS wave packets) and the quantum dynamics (interaction by means of stochastic exchange with random energy-momentum quanta between wave packets of different NOSs) are considered in this paper. The energymomentum is assimilated with the geometry of NOS eigenmodes in Minkowski spacetime. So, their conservation, forbidding any objective “uncertainty”, must be a result of only trigonometric relations. The Wheeler-Feynman’s concept of “direct interparticle action” is developed for both the quantum radiation-absorption and the Coulomb interaction. The spatiotemporal localization of NEPOS wave packets and Heisenberg’s “uncertainty principle” are supposed to be a result of permanent stochastic exchange with random quanta of energy-momentum between NEPOS and other NOSs, mainly, electromagnetic one. The absence of “zero-point oscillations” of the natural oscillatory systems is asserted. The new physical sense of de Broglie wavefunctions is illustrated with the simplest quantum systems “electrons in potential wells”.
concepts of natural electromagnetic (EM) and electronpositron (EP) distributed oscillatory systems (NEMOS, NEPOS) respectively, as real physical bases for the de Broglie matter waves. Moreover, NEMOS and NEPOS are also alternatives to the “physical vacuum” of the quantum electrodynamics [4]. The statistical method of the second quantization of NEMOS and NEPOS (quantum dynamics) was described in [5, 6]. In [7], some additional problems of the quantum kinematics and the quantum dynamics of electron waves and wave packets in vacuum and matters are discussed. The present paper is a summary and generalization of the ideas and achievements in the hypothesis of quantized natural oscillatory systems (NOSs) with distributed parameters as an alternative to the “particle-wave dualism” and to the complex “probability wavefunctions” of the “Copenhagen school”.
Index Terms—electron wave packet, energy-momentum, matter wave, natural oscillatory system, second quantization, zero-point oscillation 1
Each wave process has real base (distributed oscillatory system) periodically changing its physical characteristic(s).
I. PREFACE This is so-called “online paper”. “Online” means that the development process is exhibited in internet in “real-time mode”, not ready paper is presented, as usually. The reason is that the subject of study is too large-scale for a single author; so this article will never be finished. Please, keep up with the upcoming updates regularly… II. INTRODUCTION In spite of striking achievements in the engineering applications of the quantum mechanics (QM) and the quantum electrodynamics (QED), there is no consensus in understanding the theoretical bases of quantum world behavior yet. Such thought is confirmed by the existence a number of interpretations of the quantum theory other than so-called “Copenhagen interpretation”. Serious problems of the “Copenhagen school” are expressed in the best way in the notorious “Schrödinger’s cat” paradox and David Mermin’s “Shut up and calculate” sentence. Reasonable alternatives to the “Copenhagen interpretation” of the quantum theory were proposed in [1 – 3]. Those are 1
Paper version 2016/11/27.
III. INTRODUCTORY PHYSICAL ISSUES The logic of our hypothesis can be expressed by the following quasi-syllogism: If an interference figure appears during an experiment, a wave process occurs in spacetime.
De Broglie waves exist in a distributed oscillatory system
Indeed, all known natural waves pass through some media, which have oscillatory properties (mathematically, it means that the Lagrange equations for ones have oscillating solutions, not only decaying). De Broglie waves in their Born’s interpretation are an inconceivable exception, as there is no known material object for physical realization of these “probability” oscillations. In our opinion, physics does not give a place for “preternatural” objects like “immaterial ghosts,” but Born’s “probability waves” are, apparently, just such objects… The above is obvious for many researchers; over hundred various interpretations of QM and QED do exist, other than the “orthodox” Copenhagen interpretation. As a radical, yet logical solution, let’s assume that there are no “hard” particles in atomic world at all, only vibrations and waves. Electron is neither small sphere nor any other clot of charged substance. All observable effects produced by “electrons” or “positrons” are results of NEPOS oscillations. Coordinates and velocities of the wave packets (“particles”) have no strict sense, the occupation numbers for NEPOS eigenmodes must be considered instead. Thus, there is no principal difference between quanta of NEMOS (“photons”) and “electrons”. Three facts are known from a lot of physical experiments:
1. Energy-momentum four-vector is uniquely associated with the wave four-vector of some harmonic process in spacetime; 2. Rigorous conservation of energy-momentum occurs in each act of substance interaction; 3. There are no pure harmonic processes in nature. As it can be seen, these facts are mutually exclusive. For solving this contradiction, let’s assume that each nonharmonic process in nature is, in fact, a statistical ensemble of a quantity of harmonic processes. Such ensemble cannot be realized as a simple superposition of excited eigenmodes of a single NOS, because of the mutual orthogonality of the eigenmodes. A permanent 2 nonlinear exchange with random energy-momentum quanta between, at least, two different NOSs must take a place. The energy-momentum rigorous conservation assumes that energies-momenta of interacting NOS modes are strictly defined. Hence, those modes are pure NOS eigenmodes. If so, the spatio-temporal coordinates of the energy-momentum quantum exchange are absolutely indefinable; the interaction between NOS eigenmodes occurs in the whole 4D Universe. The energy and momentum of excited NOS eigenmodes are also absolutely nonlocalized, but the eigenmode interference creates localized spatio-temporal areas where the eigenmode ensembles (wave packets) can interact with one another; those are spatio-temporal equivalents of the spectral representation. Also, there are no natural oscillations of NOSs; all wave packets are stochastic combinations of their forced oscillations. So, the principal physical objects in the Universe are excited or unexcited eigenmodes of various NOSs. The principal physical process is the stochastic exchange with random energy-momentum quanta between eigenmodes of different NOSs. Time evolutions of NOS spatially localized wave packets (e.g., their mutual “attraction” or “repulsion”) are only some stable trends in the quantum chaos. In the same way, a statistical domination of gas molecules moving backward the gradient of their concentration is considered macroscopically as gas flow from areas with higher pressure. The statistical (probabilistic) nature of QM and QED is caused by a permanent stochastic exchange with random energy-momentum quanta between different NOSs, not by Heisenberg’s “uncertainty principle” or “zero-point oscillations of vacuum”
The detection of an “electron” means, in fact, exchange with the quantum of energy-momentum between a NEPOS eigenmode initially having occupation number of 1 and another NEPOS eigenmode originally having occupation number of 0 (through NEMOS as a coupling system accepting or supplying with the difference in energy-momentum). The 2 The term “permanent” is used as a synonym of “continuous” or “unceasing” to underscore that respective process cannot be considered as “passing through time”. This random process, probably, takes a place “over” the spacetime, stochastically changing the state of all 4D Universe (like the “many-worlds interpretation” of QM [8]). From the point of view of a moving in time 3D observer, all replacing one another NOS eigenmodes exist “at the same time,” but with different probabilities.
actual coordinates of this process cannot be ascertained in principle, so ones have no sense. Orthogonality or nonorthogonality of NEPOS modes does matter only. In particular, the interplay between electron as a quantum object and a classical apparatus is, in fact, interaction between a “poor-localized” NEPOS mode (e.g., mode of free space) and its “well-localized” mode (e.g., mode of crystal lattice) using NEMOS as intermediate. The interaction is a random process based on the Einstein coefficients. The dispersion of NEPOS wave packets does not matter, because the packet, regardless of its spatial extension, always becomes excited or unexcited as a single whole. Note that “photon” emitted from an atom may also spread over a wide wave front in fundamentally non-dispersive NEMOS. Nevertheless, all energy-momentum of this “photon” immediately transfers to another atom, if the interaction with that atom occurs. I.e., the “quantum jump” may be, in fact, the specificity of the interaction between NEPOS and NEMOS in Minkowski spacetime. If a “wide” (almost planar) electron wave packet creates a small light spot on a fluorescent screen, it means that the poorlocalized wave process in NEPOS interacts with the welllocalized “electron shell” of an atom of the screen. If a wide wave packet runs into another wide wave packet, the interchange of the both wave packets via a NEMOS eigenmode of a small spatial wavenumber is more probable, so the momenta of the NEPOS wave packets vary slightly (“electrons pass too far from one another”). Only in rare cases, the spatial wavenumber of the intermediate NEMOS eigenmode is large and the momenta of the NEPOS wave packets change greatly (“electrons collide”). The idea that only fields do exist, not particles, periodically is discussed by various authors (see, e.g., [9, 10]). However, no one of them has proposed a vector wavefunction for “electrons” like generalized coordinates of incompressible elastic media (at least, as an opposition to the coordinates of compressible elastic media for “photons”); all authors consider some scalar field like Schrödinger’s function Ψ (t , x, y , z ) , which, nevertheless, has rather physical than probabilistic character. IV. NOTATIONS AND ABBREVIATIONS A. Mathematical Notations The Cartesian coordinate system is used in this paper; x, y, and z are the spatial coordinates; t is the temporal coordinate with dimension of length, which is defined as product of the time (in seconds) and the light velocity c in vacuum. The 4D pseudo Euclidean formalism is assumed on default, so, four-vectors, four-tensors and four-operators have no special indications. “Physical” vectors in the Minkowski spacetime are marked with upper arrows ( a ). “Mathematical” vectors-columns in the Hilbert space are in bold ( a ). The braces mean the combining of scalar values into a vector: a = {at , a x , a y , a z } . The vectors are not divided into covariant and contravariant; instead, the minus signs are placed before
the corresponding terms in operators and scalar products. E.g., the scalar product of a and b = {bt , bx , by , bz } is of a ⋅ b= at bt − a x bx − a y by − a z bz , and squared vector a is of ( a ) 2 = at2 − a x2 − a 2y − a z2 . Four-matrices and four-tensors are enclosed in square brackets: ctt c [ c ] ≡ c xt yt c zt
ctx cxx c yx czx
cty cxy c yy czy
Klein-Gordon ∇02 =∇2 + k02 operators are used, where k0 ≥ 0 is a cutoff wavenumber. Antisymmetric second-rank tensor
0 ∂a y ∂x
−
∂a x ∂y
∂a z ∂a x − ∂x ∂z
0 ∂a z ∂a y − ∂y ∂z
[ e]
is a sum of products of their respective components
(taking into account the opposite signs for the temporal and the spatial components):
−c yt e yt + c yx e yx + c yy e yy + c yz e yz − czt ezt + czx ezx + czy ezy + czz ezz ) .
D’Alembert ∇ 2 = ∂ 2 / ∂t 2 − ∂ 2 / ∂x 2 − ∂ 2 / ∂y 2 − ∂ 2 / ∂z 2 , and
∂at ∂a y + ∂y ∂t ∂a x ∂a y − ∂y ∂x
etc. The scalar product of a four-tensor [ c ] and a four-tensor
= ( + ctt ett − ctx etx − cty ety − ctz etz − cxt ext + cxx exx + cxy exy + cxz exz −
acceleration ( d 2 ξ / dt 2 ) is measured in m −1 . The “action” (Hamilton’s first principal function) [11] is defined as the integral of the Lagrange function over the temporal coordinate (t) and is measured in J ⋅ m . This is also relating to the quantum of action h = c , where is the reduced Planck constant. The energy and the momentum are joined into an energy-momentum four-vector W with dimension of J, so, no special symbol for the momentum like P is used. Other physical values are as commonly accepted. Four-gradient ∇ = {∂ / ∂t , −∂ / ∂x, −∂ / ∂y , −∂ / ∂z} , four divergence ∇ ⋅ a = ∂at / ∂t + ∂a x / ∂x + ∂a y / ∂y + ∂a z / ∂z ,
∂at ∂a x + ∂x ∂t
d x = cxt bt − cxx bx − cxy by − cxz bz ,
[ c ] ⋅ [ e] =
ctz cxz . c yz czz
Generic symbols may be used instead the spatio-temporal coordinates. τ is a generic symbol for t, x, y, or z. ξ is a generic symbol for x, y, or z. If one of the generic symbols appears in the summation sign ( Σ ), it means summation over all respective coordinates. The velocity ( d ξ / dt ) is a dimensionless variable; the
0 ∂a x ∂at − − ∂t ∂x ∇×a ≡ ∂a y ∂a − t − ∂ ∂y t ∂a ∂a − z − t ∂z ∂t
d t = ctt bt − ctx bx − cty by − ctz bz ,
∂at ∂a z + ∂z ∂t ∂a x ∂a z − ∂z ∂x ∂a y ∂a z − ∂z ∂y 0
is interpreted as four-curl of a . It is known that the curl tensor in n-dimensional space (n = 2, 3, 4, …) can be treated as an ordered set of vector field rotations in all mutually orthogonal 2D crosscuts of this space. There are six such crosscuts in Minkowski spacetime. The product of four-tensor [ c ] and four-vector b is a four vector d = [ c ] b with components defined as the scalar products (in the above sense) of [ c ] respective rows and b :
The normalized metric tensor of the pseudo Euclidean spacetime is defined in [11] as symmetric four-tensor g tt g [ g ] ≡ g xt yt g zt
g tx g xx g yx g zx
g ty g xy g yy g zy
g tz g xz . g yz g zz
The unit four-tensor of the same metrics is 1 0 0 0 0 −1 0 0 . I ≡ [ ] 0 0 −1 0 0 0 0 −1
For the “flat” pseudo Euclidean spacetime, [ g ] = [ I ] . For the “curved” spacetime, [ g ] ≠ [ I ] . The principal outcome from our hypothesis is: there are no real particles in nature, only “quasi-particles”, i.e., wave packets of natural oscillatory systems (NOSs). However, we will often use for simplicity the traditional denominations of ones, but enclosed in quotation marks, e.g., “particle,” “electron,” “photon,” etc. B. Abbreviations EM means electromagnetic; EP means electron-positron; GR is the general relativity; NEGOS is a natural electro-gravitational oscillatory system; NEMOS is the natural electromagnetic oscillatory system; NEPOS is the natural electron-positron oscillatory system; NGOS is the natural gravitational oscillatory system; NOS is a natural oscillatory system; QED is the “traditional” quantum electrodynamics; QHO is the quantum harmonic oscillator; QM is quantum mechanics. V. USED HYPOTHESES Our theory is based on two main postulates: 1. “Hard” particles do not exist in nature; all physical objects and phenomena are results of oscillations of NOSs with distributed parameters along their generalized coordinates. The “particle-wave dualism” does not have a place.
2. Quantization of total energy-momentum four-vector W of NOS eigenmodes Wm = hK m km is a general principle of nature. Here m is a NOS eigenmode number; km is the wave
four-vector (wavenumber) of m-th eigenmode; K m = 0, 1, is m-th eigenmode occupation number. This postulate assimilates energy and momentum as physical values with the pseudo Euclidean geometry in Minkowski spacetime. In particular, the energy-momentum conservation law becomes only a result of trigonometric relations, so, cannot be void in principle (even during Heisenberg’s “uncertainty interval”). In spite of seeming obviousness of the second postulate, consistent application of one may result in unexpected outcomes. E.g., time-independent EM potential around rest “electron” does not contain energy; all electron self-energy must be a result of NEPOS oscillation. Another outcome is the absence of the zero-point oscillations of vacuum (because the zero-point energy hkm / 2 of eigenmodes does not satisfy the quantization principle). Only the zero-point oscillations of “composite” oscillators (like crystal lattices), based on the interaction between NEPOS and NEMOS, exist. Besides the above postulates, other surprising assumptions are used in our theory. Some of ones are, nevertheless, logical consequences of the postulates. These are: 1. There exist three kinds of NOSs: three-vector (fermion) systems with three generalized coordinates; four-vector (boson) ones with four the coordinates; and four-tensor (also boson) NOSs with ten generalized coordinates. The existence of scalar (“Higgs”) systems with one generalized coordinate is unlikely from the author’s point of view. 2. All “fundamental particles” are, in fact, “quasi-particles”, i.e., wave packets of respective NOSs. The spatial (spatiotemporal) localization of wave packets and Heisenberg’s “uncertainty principle” both are results of permanent stochastic exchange with random quanta of momentum (energy-momentum) between different NOSs, e.g., NEPOS and NEMOS. So, all wave packets are fundamentally dynamic, not static. The wave packet localization is only some approximation; in the strict sense, each “particle” occupies the entire Universe. 3. De Broglie wavefunctions are continuous real-valued generalized coordinates of NOSs, not Born’s complex-valued “probability waves”. The physical sense of the wavefunctions is deviations of NOSs from their “undisturbed” states along the generalized coordinates in all points of Minkowski spacetime. All wavefunctions are gauge-dependent, but this is not of matter for the quantum theory. 4. The Lagrange equations for NOSs are covariant secondorder partial derivative differential equations, having oscillating solutions (e.g., the wave equation for NEMOS or Klein-Gordon equation for NEPOS). 5. The free oscillations of NOSs as well as pure harmonic processes do not exist in nature. All de Broglie waves are the forced non-harmonic oscillations (wave packets). The forced oscillations of NOSs are an equivalent of the “virtual particles” of “traditional” QED [4] while the “real particles”,
precisely obeying the relativistic energy–momentum relation, do not exist at all. 6. All quantum effects are, in fact, non-classical specificities of NOS eigenmodes interaction. The transfer of energymomentum quanta from one eigenmode to another is an “overspacetime” act having no coordinates, not continuously passing process in spacetime. All energy and momentum of nature are distributed uniformly over the entire 4D Universe as strictly defined energy-momentum of NOS eigenmodes. The wave packets indicate areas where that energy-momentum can “appear”. 7. The “conversion of fundamental particles” is a transfer of energy-momentum quanta between wave packets of different NOSs following by the excitation and the extinction of the packets as wholes. Quark NOSs must be considered instead of “meson” and “hadron” ones. 8. EM interaction is, in fact, a universal property of all fermion wave packets, not of some “privileged” “electrically charged” ones. “Positive” and “negative” charges do not exist. The direction of Coulomb interaction (attraction or repulsion) is defined by the relative orientation of wave packet streamlines (average four-vectors) in spacetime. The “elementary charge” and the “magnetic flux quantum” existence is a seeming effect caused by the quantization of NEMOS linear deviation and twisting respectively. 9. EM interactions always occur both in “positive” and “negative” directions of all four coordinates in Minkowski spacetime. The direction of energy-momentum transfer is defined by the “photon’s” wave four-vector orientation. In particular, there are “photons” with kt > 0 and “antiphotons” with kt < 0 . 10. The gravitation interaction is a result of excitation of some natural gravitational oscillatory system (NGOS), not the consequence of the “spacetime curvature”. An “empty” spacetime cannot be considered as “flat” or “curved” in principle. Moreover, after the elimination of the “fundamental particles” as some “material points”, the Euclidean geometry lose its strict physical base, even for “filled” with NOSs spacetime. 11. Like EM interaction, gravitation one is a permanent stochastic exchange with random energy-momentum quanta between the wave packets via NGOS. So, the gravitation affects “sizing tools” (“rulers” and “clocks”), not the spacetime itself. “Non-inertial” frame systems of general relativity (GR) are only curvilinear coordinates in the flat spacetime. Usage of such coordinate systems is allowable but not advisable in the most of cases. 12. NEMOS and NGOS may be only different degrees of freedom of the same boson NOS. The gravitation interaction may be a result of existence of ten extra quadratic (“flexural”) degrees of freedom of NEMOS in addition to four linear (“displacing” and “torsional”) EM ones. A next (“cubic”) interaction may also exist, etc.
VI. KINDS OF NATURAL OSCILLATORY SYSTEMS
“Stiffness factors” Rγ , Re couple the local deviations of There are three kinds of NOSs differing in the number N NEMOS and NEPOS in some point with three-densities of of their generalized coordinates: three-vector (fermion) system energy-momentum in this point (like 1 / µ 0 factor does for the with N = 3 ; four-vector EM system (NEMOS) with N = 4 ; EM potential). Three-densities of the Lagrange function and four-tensor gravitational system (NGOS) with N = 10 . λ(t , x, y, z ) (relativistic invariants) for non-interacting NOSs Because of additional relations (like the Lorenz gauge), the may be written as: numbers of fully independent generalized coordinates N i are Rγ of 2, 3, and 5 respectively, therefore, corresponding maximal −(∇ℵtγ ) 2 + (∇ℵγx ) 2 + (∇ℵγy ) 2 + (∇ℵγz ) 2 ; (VII.1) = λγ spins of NOS quanta s are of 1/2, 1, and 2 respectively 2 according to the known relation N= 2 s + 1 [4]. i Re 2 e 2 e 2 e 2 e 2 e 2 k0 e (ℵ ) − (∇ℵt ) + (∇ℵx ) + (∇ℵy ) + (∇ℵz ) . (VII.2) λe The natural distributed oscillatory systems differ also in = 2 their cutoff wavenumbers k0 determining kinematics and dynamics of wave packets. These relativistic scalars are full- Total actions of NOSs (also relativistic invariants) are value replacements for the Newton’s “particle rest masses” respectively: concept. All fermion systems have k0 > 0 , so, their wave (VII.3) Λ γ = ∫ λ γ dtdxdydz ; Λ e = ∫ λ e dtdxdydz , V V packets are “particles” (or, more strictly, quasi-particles) with non-zero “rest masses” (leptons, quarks and neutrinos). For where the integrals are taken over all 4-volume of the four-vector and four-tensor systems k0 = 0 , accordingly, Universe. quanta of EM and gravitation interactions have no “rest mass”. Expressions for λ γ and λ e are rewritten below in expanded form to emphasize their perfection and four-symmetry: VII. ELECTROMAGNETIC AND ELECTRON-POSITRON 2 2 2 2 OSCILLATORY SYSTEMS Rγ ∂ℵtγ ∂ℵtγ ∂ℵtγ ∂ℵtγ γ λ = − + + + + A. General Consideration 2 ∂t ∂x ∂y ∂z Let’s generalize both the EM potential four-vector and de 2 2 2 2 ∂ℵγx ∂ℵγx ∂ℵγx ∂ℵγx Broglie EP wavefunction as some real-valued aleph-functions + − − − + ℵγ (t , x, y , z ) and ℵe (t , x, y , z ) respectively 3. EM aleph ∂t ∂x ∂y ∂z 2 2 2 2 ∂ℵγy ∂ℵγy ∂ℵγy ∂ℵγy function ℵγ is a four-vector restricted with the Lorenz gauge + − − − + ∇ ⋅ℵγ ≡ 0 (note that ℵγ differs from the EM potential four ∂t ∂x ∂y ∂z 2 2 2 2 vector A(t , x, y , z ) only in the measure unit). EP aleph ∂ℵγz ∂ℵγz ∂ℵγz ∂ℵγz e ; − − − + function ℵ is also a four-vector restricted with both the ∂t ∂x ∂y ∂z e Lorenz gauge ∇ ⋅ℵ ≡ 0 and a spatial three-solenoidality Re 2 (“media incompressibility”) condition ℵte ≡ 0 in some k0 e (ℵte 2 −ℵex2 −ℵey2 −ℵez 2 ) − = λe γ e 2 “privileged” rest frame system. In contrast to ℵ , ℵ has no
“potential” eigenfunctions (with spin zero) in one’s Fourier expansion, therefore, according to the angular momentum quantization rules, spins of NEPOS modes are of ±1 / 2 , not of −1, 0, +1 as for NEMOS. The physical senses of ℵγ and ℵe are local deviations of NEMOS and NEPOS respectively from their “undisturbed” states along respective coordinate axes. According to such interpretation, both ℵγ and ℵe must be gauge-dependent (i.e., to tend to zero far off from a matter), but this is insignificantly in the quantum theory, because any invariable in the space time addition to ℵγ or ℵe has zero energy-momentum quantum, so, cannot be involved in the interaction. 3
“Aleph” symbol is chosen for our interpretation of de Broglie wavefunctions partly because of overusing Latin and Greek symbols and partly due to some similarity of NOSs to His essence in Judaism and Christianity (“I will become what I choose to become,” “He who causes to exist,” “He who is, who exists”).
2
2
2
2
2
2
2
2
2
2
∂ℵe ∂ℵe ∂ℵe ∂ℵe − t + t + t + t + ∂t ∂x ∂y ∂z
∂ℵex ∂ℵex ∂ℵex ∂ℵex + − − − + ∂t ∂x ∂y ∂z 2
2
+ 2 2 2 2 ∂ℵez ∂ℵez ∂ℵez ∂ℵez + − − − . ∂t ∂x ∂y ∂z
∂ℵey + ∂t
∂ℵey − ∂x
∂ℵey − ∂y
∂ℵey − ∂z
For the rest in some coordinate system NEPOS wave packet, ℵte ≡ 0 ; so, the expression for λ e is even simpler:
= λe
Re −k02e (ℵex2 +ℵey2 +ℵez 2 ) + 2 2
2
2
2
∂ℵex ∂ℵex ∂ℵex ∂ℵex + − − − + ∂t ∂x ∂y ∂z ∂ℵey + ∂t
2
∂ℵey − ∂x
2
∂ℵey − ∂y
2
∂ℵey − ∂z
2
+
2 2 2 2 ∂ℵez ∂ℵez ∂ℵez ∂ℵez + − − − . ∂t ∂x ∂y ∂z
Let’s suppose that the EM interaction, like the gravitational one, is not a peculiarity of specific “privileged” (“electrically charged”) “particles”, but universal property of all fermion NOS wave packets (leptons and quarks), except for neutrinos. The spatio-temporal direction of NEMOS deviation coincides with k direction of a deviating fermion wave packet. The seeming existence of “elementary charge” is only a result of quantization of NEMOS deviation. Neutrinos cannot deviate NEMOS, possibly, because of their vanishingly small k0 . The reason for the quantization of NEMOS integral “deviation” and integral “twisting”, causing imaginary existence of the “elementary charge” and the “magnetic flux quantum” respectively, has to be explained in the future. The cause why fermion NOSs cannot have K m > 1 is the inconsistence of the quantum levels for energy-momentum, on the one hand, and angular momentum (spin), on the other one. Let’s assume that the amplitude of aleph-function for m-th eigenmode ℵem is of ℵem 0 for K m = 1 . The corresponding spin of this mode is of –1/2 or +1/2. If to try for excitation of the same eigenmode with K m = 2 , its amplitude must be increased to 2 ℵem 0 and its spin must become of –1 or +1. However, such values of s are forbidden by the angular momentum quantization rules. The next allowed spin values are of –3/2 or +3/2. Respective amplitude of the eigenmode must be of 3 ℵem 0 . This amplitude corresponds to K m = 3 . But one cannot excite m-th eigenmode from K m = 1 to K m = 3 overstepping K m = 2 , because the energy-momentum quantization rules prohibit such quantum jump. Therefore, accessible values of K m for fermions are only of 0 or 1.
B. Eigenfunctions of NEMOS Full set of NOS eigenfunctions is a Fourier basis for harmonic decomposition of solutions of the NOS Lagrange equation. Doubly orthogonal [12] in the Universe four-volume V eigenfunctions ℵmγ (t , x, y, z ) of D’Alembert operator −∇ 2 are defined as non-trivial solutions of an equation 2 2 2 are ∇ 2ℵmγ + km2ℵmγ =0 , where km2 = kmt2 − kmx − kmy − kmz respective eigennumbers (squared wave eigenvectors km = {kmt , kmx , kmy , kmz } ) of NEMOS. For a huge yet limited in the spatio-temporal directions Universe, km and ℵmγ are
discrete sets of enumerable four-vectors and functions. For free oscillations, km2 = 0 ; for forced ones, km2 ≠ 0 . Note that the free oscillations turn the Lagrange function three-density of NOS λ averaged over their spatio-temporal period into zero, so, their total action in the Universe four-volume is also zero. Let’s write the full set of NEMOS eigenfunctions for some four-parallelepiped in pseudo Euclidean space with dimensions of T, X, Y, and Z along the respective axes and periodical boundary conditions on all its borders. In this case, kmt = 2πmt / T , kmx = 2πmx / X , kmy = 2πm y / Y , and kmz = 2πmz / Z , where mτ = , − 2, − 1, 0, + 1, + 2, are integer numbers. This full set can be divided into four subsets differing in number of spatio-temporal dimensions necessary to close the “field lines” of respective eigenfunctions (or, in other words, number of nonzero components of these functions). Those subsets are well known from the classical electrodynamics [13]. 1. Eigenfunctions of Zero Magnetic (ZM) or Potential (P) γ type ℵmZM (t , x, y, z ) , which are solenoidal in four dimensions, γ γ γ γ i.e., ∂ℵmZMt / ∂t + ∂ℵmZMx / ∂x + ∂ℵmZMy / ∂y + ∂ℵmZMz / ∂z ≡ 0 :
γ ℵmZMt = exp(ikm r ) ; kmt kmx γ ℵmZMx = exp(ikm r ) ; 2 2 2 kmx + kmy + kmz
kmt kmy γ ℵmZMy = exp(ikm r ) ; 2 2 2 kmx + kmy + kmz kmt kmz γ ℵmZMz = exp(ikm r ) , 2 2 2 kmx + kmy + kmz where r = {t , x, y, z} is the coordinate four-vector in pseudo
Euclidean space. If kmx , kmy , and kmz all are zero, kmt also γ γ γ ≡ 0 , ℵmZMy ≡ 0 at that. must be zero; ℵmZMx ≡ 0 , and ℵmZMz
2. Eigenfunctions of Transverse Magnetic (TM) or Electric γ (E) type ℵmTM (t , x, y, z ) , which are solenoidal in three (x,y,z) γ γ γ dimensions, i.e., ∂ℵmTMx / ∂x + ∂ℵmTMy / ∂y + ∂ℵmTMz / ∂z ≡ 0 : γ ℵmTMt = 0;
γ exp(ikm r ) ; ℵmTMx =
kmx kmy γ ik ℵmTMy = − 2 exp( mr ) ; 2 kmy + kmz
k k γ ℵmTMz = − 2 mx mz2 exp(ikm r ) . kmy + kmz
If kmy and kmz both are zero, kmx also must be zero; γ γ ≡ 0 at that. ℵmTMy ≡ 0 and ℵmTMz
3. Eigenfunctions of Transverse Electric (TE) or Magnetic γ (H) type ℵmTE (t , x, y, z ) , which are solenoidal in two (y,z) γ γ dimensions, i.e., ∂ℵmTEy / ∂y + ∂ℵmTEz / ∂z ≡ 0 :
γ ℵmTEt = 0; γ ℵmTEx = 0;
γ exp(ikm r ) ; ℵmTEy = kmy kmz γ ℵmTEz = − 2 exp(ikm r ) . kmz
γ γ γ dimensions, i.e., ∂ℵmTMx / ∂x + ∂ℵmTMy / ∂y + ∂ℵmTMz / ∂z ≡ 0 :
γ If kmz is zero, kmy also must be zero; ℵmTEz ≡ 0 at that.
4. Eigenfunctions of Transverse Electric and Magnetic γ (TEM) type ℵmTEM (t , x, y, z ) , which are solenoidal in one (z) dimension, i.e., ∂ℵ
γ mTEMz
/ ∂z ≡ 0 :
γ 0; ℵmTEMx =
ℵ
= 0;
If kmy and kmz both are zero, kmx also must be zero; γ γ ≡ 0 at that. ℵmTMy ≡ 0 and ℵmTMz
kmz always is zero.
It is obvious that the condition kmz ≡ 0 can be matched with the condition of ℵγ → 0 at far distances from a NEMOS spatially localized wave packet only in the trivial case of identically zero amplitude of all TEM eigenmodes. So, TEM eigenmodes are never presented in Fourier decompositions of real EM potentials. Arbitrary oscillation of NEMOS can be expanded in a Fourier series ∞
∑u
m = −∞
mZM
γ ℵmZM +
∞ γ + ∑ umTEℵmTE + m = −∞
γ ℵmTMx = exp(ikm r ) ;
k k γ ℵmTMz = − 2 mx mz2 exp(ikm r ) . kmy + kmz
γ ℵmTEMz = exp(ikm r ) .
= ℵγ
γ ℵmTMt = 0;
kmx kmy γ exp(ikm r ) ; ℵmTMy = − 2 2 kmy + kmz
γ ℵmTEMt = 0;
γ mTEMy
Let’s write the full set of NEPOS eigenfunctions for the same four-parallelepiped in pseudo Euclidean space with dimensions of T, X, Y, and Z along the respective axes and periodical boundary conditions on all its borders. Obviously, components of km stay to be unchanged. However, the full set of NEPOS eigenmodes can be divided only into three subsets, not four, as for NEMOS. “Zero Magnetic” (“Potential”) eigenfunctions cannot be excited in “incompressible” NOS, only solenoidal in 3D or less space. It is not unreasonable to keep names of remaining NOS eigenmodes unchanged. 1. Eigenfunctions of Transverse Magnetic (TM) or Electric γ (t , x, y, z ) , which are solenoidal in three (x,y,z) (E) type ℵmTM
mTM
γ ℵmTM +
mTEM
γ ℵmTEM ,
∞
∑u
m = −∞ ∞
∑u
m = −∞
(VII.4)
C. Eigenfunctions of NEPOS Doubly orthogonal [12] in the Universe four-volume V eigenfunctions ℵem (t , x, y, z ) of D’Alembert operator −∇ 02 are defined as non-trivial solutions of an equation 2 2 2 are ∇ 02ℵem + km2ℵem =0 , where km2 = kmt2 − kmx − kmy − kmz wave
γ γ dimensions, i.e., ∂ℵmTEy / ∂y + ∂ℵmTEz / ∂z ≡ 0 : γ ℵmTEt = 0;
γ ℵmTEx = 0;
γ ℵmTEy = exp(ikm r ) ;
kmy kmz γ ℵmTEz = − 2 exp(ikm r ) . kmz γ If kmz is zero, kmy also must be zero; ℵmTEz ≡ 0 at that.
where umZM , umTM , umTE , and umTEM are dimensionless coefficients (amplitudes of m-th eigenmodes from respective subsets).
respective eigennumbers (squared km = {kmt , kmx , kmy , kmz } ) of NEPOS.
2. Eigenfunctions of Transverse Electric (TE) or Magnetic γ (H) type ℵmTE (t , x, y, z ) , which are solenoidal in two (y,z)
eigenvectors
3. Eigenfunctions of Transverse Electric and Magnetic γ (TEM) type ℵmTEM (t , x, y, z ) , which are solenoidal in one (z) γ / ∂z ≡ 0 : dimension, i.e., ∂ℵmTEMz
γ ℵmTEMt = 0; γ ℵmTEMx = 0; γ ℵmTEMy = 0;
γ ℵmTEMz = exp(ikm r ) . kmz always is zero.
The condition kmz ≡ 0 can be matched with the condition of e ℵ → 0 at far distances from a NEPOS spatially localized wave packet only if amplitudes of all TEM eigenmodes are identically equal to zero. So, TEM eigenmodes are never presented in Fourier decompositions of NEPOS wave packets. As it can be seen, eigenfunctions and eigenvalues of D’Alembert and Klein-Gordon operators are the same. The only difference is that the Lagrange function three-density of NEMOS turns into zero at km2 = 0 , while for NEPOS it occurs at km2 = k02e . VIII. THE CLASSICAL INTERPRETATION OF NOSS A. The Wave Kinematics of NEPOS The quantum kinematics describes the spatio-temporal evolution of NOS wave packets irrespective of the mechanisms causing their specific behavior (those mechanisms are objects of the quantum dynamics). The Lagrange equation for free oscillations of NEPOS can be formally derived from the expression for non-interacting NEPOS Lagrange function three-density λ e as [14]: ∂λ e ∂λ e d 0. ∑τ d τ ∂(∂ℵ e / ∂τ) − ∂ℵ e = That is the Klein-Gordon equation for four-vector ℵe : ∇2ℵe + k02eℵe =0 .
However, as it is shown below, all NOS wave packets are fundamentally dynamic composite objects, the existence of which is possible only due to the permanent interaction between different NOSs. There are no fermion wave packets without “clouds” of NEMOS and NGOS wave packets and vice versa. In particular, free oscillations of NEMOS (“free photons”) do not exist, all “photons” are “envelopes” of fermion wave packets in the spacetime, as it was suggested in [15, 16] for atoms. So, any material “particle” must be considered as localized irregularity of several NOSs, not one. NOS wave packets may be single- and multilocalized. The former exists, e.g., immediately after emission of the “electron” from small cathode. The latter may be result of splitting a wave packet after its passing through two or more slots in a diaphragm. The multilocalized NEPOS wave packets have other spectra in spatial domain than single-localized ones; nevertheless, this does not affect fundamentally their interaction with NEMOS. As it is shown in the next chapter, the interaction between harmonic components of wave packets (eigenmodes) of various NOSs occurs “at once” in whole fourvolume of the 4D Universe. So, the spatial disconnection of the parts of multilocalized wave packet does not prevent one from the “quantum collapse”, if the four-intervals between all those parts are equal to zero. Three different kinds of NOS wave packets velocity can be distinguished from the position of the quantum kinematics. Those are: wave packet group four-velocity v g ; components
of “particle” classic three-velocity vξ ; and components of wave packet average phase three-velocity v f ξ . If an “electron” or “positron” is motionless in some frame system, this is a pure stationary wave. All spatial components of one’s group four-velocity v g = k / k0 e are zero, where k is average wave four-vector of the wave packet. For the “moving particle”, a regular trend is superposed on k , so the wave packet becomes a mixed stationary-travelling wave. In any case, v gt ≤ −1 for “electrons”, v gt ≥ +1 for “positrons”. The “electron” and “positron” classic three-velocity is the scaled by ck0 e / kt set of the spatial components of the wave packet group four-velocity, which is calculated as vξ ≡ c
ckξ ∂kt k = ± c 0 e vg ξ = . 2 ∂kξ kt k0 e + k x2 + k y2 + k z2
The plus sign is for “positrons”, the minus is for “electrons”. The wave packet average phase three-velocity is defined as
vf ξ
c k02e + k x2 + k y2 + k z2 kt = c = ± . kξ kξ
The choice of signs is the same as for vξ . A relation has a place vξ v f ξ = c 2 , where vξ < c ; v f ξ > c for the fermion NOSs. Note that the three-velocity components vξ are observable values (they are interpreted as classic “particle” velocity), but give no possibility to distinguish “electron” and “positron’, as kt and kξ change their signs together if a “particle” is replaced with its “antiparticle”. Only four-velocity v g direction analysis can be used for that. Such analysis can be performed by means of study of NEMOS deviation direction. Other kinematic relations can be derived from the above:
vξ
kξ = ± k0 e
vg = ±
2
2 x
c − v − v y2 − vz2
1 2
2 x
c − v − v y2 − vz2
and
{c, v , v , v } , x
y
z
where the plus sign is also for “positrons”, the minus is for “electrons”. B. Energy-Momentum Relations for NEMOS and NEPOS The three-velocity components vξ are no more interpreted as “energy-momentum transfer velocity”. As this is explained below, all energy and momentum of nature are distributed uniformly over the entire 4D Universe as energy-momentum of NOS eigenmodes. The NOS wave packet is only finite area of spacetime where these energy and momentum can “appear”, i.e., to participate in interaction with another NOSs. The destructive interference of NOS eigenmodes forbids doing
this in other areas, as amplitude of the aleph-function is zero in those. The “particle” three-velocity cannot exceed c, but this does not mean if the “wave function collapse” occurs with a multilocalized wave packet. e+ = hkt e + > 0 The positive sign of “positron” energy W t together with the negative one for energy of “electron” e− W = hkt e − < 0 do not mean that the energies mutually cancel t while the “particles” annihilate. The “total” frequency of generated at the annihilation “photons” is determined by the “beating” between kte + and kte − , so, the summary energy Wt γ± of the “photons” is equal to the difference between original energies of the annihilated fermions: Wt γ± = ±(Wt e + − Wt e − ) , therefore Wt γ± ≥ 2hk0 e . Certainly, the same principle has a place for the resulting momentum. The above illustrates as “geometric” definition of energy-momentum prevails over traditional “physical” comprehension of ones. If a transfer to “classic” (multi-photon or multi-graviton) description of boson NOSs (NEMOS or NGOS) is performed, a four-vector of matter flow density j (t , x, y , z ) and a fourtensor of energy-stress density
[ w] (t , x, y , z )
[11] must be
introduced. These values are placed in the right-hand sides of the Lagrange equations for NEMOS (D’Alembert’s equation) and NGOS (Einstein’s equation) respectively. Let’s define a contribution of each wave packet of a fermion NOS (e.g., NEPOS) to j and [ w] as j = cµv g and v gt v gt v v cµ [ w] = 2 k k T = cµ vgx vgt k0 e gy gt v gz v gt
v gt v gx v gx v gx v gy vgx v gz v gx
vgt vgy v gx v gy vgy vgy v gz v gy
vgt vgz v gx v gz vgy vgz vgz vgz
respectively, where µ(t , x, y , z ) is a relativistic scalar describing spatio-temporal distribution of the “wave packet density” (“rest mass density”) in the “own” frame system of this wave packet; the upper index “T” means the transpose of the vector-column k . After comparing the above expressions for j and [ w] , a hypothesis comes to mind: NEMOS and NGOS may be only different degrees of freedom of the same boson NOS. If so, the gravitation interactions may be treated as a quantum kinematics and dynamics of ten extra quadratic (“flexural”) degrees of freedom of NEMOS in addition to four linear (“displacing” and “torsional”) EM ones. A next (“cubic”) interaction caused by a third rank tensor may also exist. Whereas tensors of odd ranks can produce both “attraction” and “repulsion” of fermions depending on the signs of their v gt , tensors of even ranks don’t distinguish those. Unfortunately, the “cubic” and next interactions, if ones exist, may be too weak for any measurements. As an illustration of the above, the matter flow density fourvectors (“world lines”) for native and generated “electrons”
Fig. 1. The matter flow density four-vectors (“world lines”) for fermions.
and “positrons” together with ones for “protons” are schematically shown in Fig. 1. IX. THE QUANTUM DYNAMICS OF NEPOS AND NEMOS A. Preface There are two methods for a distributed oscillatory system excitation, known as “parametric” and “force” ones [17]. The former is based on varying the system eigenfunctions by an external influence. The latter does not change the eigenfunctions, but varies the occupation numbers of normal modes of the system. The difference is noticeable, e.g., for atomic systems. “Parametric” approach assumes that the stationary electron shells are new eigenmodes of NEPOS, which were “deformated” by the static EM potential of the nucleus. The “force” point of view explains the spatial localization of the electron shell as a result of permanent exchanging with random quanta of momentum between the nucleus and the “electron” via NEMOS, so, the stationary electron shells are only wave packets of forced eigenmodes of NEPOS with ∇02ℵe ≠ 0 , not free eigenmodes with ∇02ℵe =0 . We accept the “force” approach as more consistent with the quantum principles. B. General Consideration As it follows from our second postulate, the energymomentum conservation is a fundamental law of the pseudo Euclidian Universe; any objective “uncertainty” for ones is impossible. E.g., for an insulated physical system “charged particle in its own EM potential”, the total energy-momentum is objectively strictly defined. The uncertainty has a place only in what part of this energy-momentum may be found as located in NEPOS and what part of one is reside in NEMOS at the specific measurement. Also, mechanism of the quantum EM interaction is supposed to be the same for both timedependent (e.g., radiating-absorbing atoms) and static (e.g., mutually repulsive “electrons”) systems. Total momentum of a solitary system “spatially localized electron in its own EM potential” is strictly defined and unchanging. Only a stochastic reassignment of the momentum between parts of this isolated physical system occurs
The interaction process cannot be described in the temporal domain (consequently, also in the spatial one) in principle [4].
So, the Lagrange equations for NEMOS and NEPOS (the wave equation and the Klein-Gordon equation respectively) can be written only for their non-interacting (free) vibrations, with ∇02ℵ = 0 . However, free oscillations of NEMOS and NEPOS do not occur at all. Single “pure” eigenmodes of these systems cannot be excited or “discovered” as having infinite spatio-temporal spread. On the other hand, harmonic components of localized wave packets are not independent (because energies-momenta of the components taken separately cannot satisfy the quantization principle: if energymomentum of each separate m-th harmonic component is of hkm , their total energy-momentum would be infinite, according to the Parseval’s identity). So, a localized NEPOS wave packet (“electron” of “positron”) can be linked together only by permanent interaction of one’s spectral components with each other by means of NEMOS (their direct interaction is impossible because of the orthogonality of these components). Thus, the spatial localization of NEPOS wave packets and the Heisenberg’s “uncertainty principle” both are results only of NEPOS and NEMOS nonlinear interaction. E.g., a sole NEPOS wave packet in its “own” EM potential stochastically exchanges with NEMOS by random quanta of momentum (not energy) producing continuous ℵe spectrum of finite width in the spatial domain. The energy-momentum spectral density describes the probability that respective (m-th) eigenmode has a non-zero occupation number K m = 1 . The “associated” with an “electron” NEMOS wave packet has another envelope of its energy-momentum spectrum than the NEPOS one, because NEPOS is the fermion NOS, while NEMOS is the boson NOS; so, the latter can “accumulate” several energy-momentum quanta in the same eigenmode, while the former cannot. However, the wavenumbers of NEMOS harmonics always may be treated as the differences between wavenumbers of NEPOS wave packet harmonics. Therefore, one more essential distinction exists between the both spectra. The “central” wavenumber in the spectrum of moving NEPOS wave packet is proportional to the average energy-momentum of the “electron” hk , therefore, this is not zero. But the “central” wavenumber in the spectrum of the “coupled” NEMOS wave packet is zero. This means that EM potential of a single “electron” does not hold average energymomentum at all. After the inverse Fourier transform, the amplitude spectra of e ℵ and ℵγ provide spatially localized dependencies for the both functions. However, the amplitude spectra of NEPOS and NEMOS are essentially differing. The reason is the different dependences of renormalization functions for ℵe and ℵγ on k . E.g., for kt = 0 , the amplitude spectrum of NEMOS can be qualitative estimated from the same spectrum of NEPOS by means of multiplication by
k0e2 + k x2 + k y2 + k z2
k x2 + k y2 + k z2 .
As it can be seen, the amplitudes of the spatial harmonics of NEMOS infinitely grow at k → 0 , so, EM potential is
“enriched” with the long-wave harmonics and decays more slowly at large distances from the “electron” than ℵe function of one. Because the transferred between NEPOS and NEMOS energy-momentum quanta are objectively strictly defined, we must assume that the interaction between the harmonic components of NEPOS and NEMOS wave packets occurs just in whole 4D Universe. Consequently, this stochastic process cannot be investigated, because it is placed “over” the spacetime. Only whole wave packets, produced by the interference of their harmonics, can be observed in experiment. If two or more immovable “electrons” are placed in their “common” EM potential, NEMOS performs a stochastic mutual exchange with momentum quanta between ones, i.e., the Coulomb repulsion, in addition to the “localizing” effect for each wave packet taken separately. For moving “electrons”, this exchange includes also energy quanta. Similar interaction process between atoms in the time domain transfers both energy and momentum and is known as “radiation-absorption”. We must put into our hypothesis the Wheeler-Feynman’s concept of “advanced” EM interactions along with “retarded” ones [15, 16] to explain the temporal localization of transferring energy EM wave packets. Whereas the “retarded” NEMOS eigenmodes with kt > 0 , containing quanta of positive energy, must be treated as the “ordinary photons”, “advanced” eigenmodes with kt < 0 are some “antiphotons” of negative energy. E.g., if an excited atom #1 has transferred a quantum of its extra energy-momentum to atom #2 via “photons”, this action may be still “rolled back” by “antiphotons” (“the Schrödinger’s cat can be revivified”). Only if atom #2 has retransmitted the obtained quantum to some atom #3, this quantum no longer can be returned to atom #1 (“a measurement has been made, the cat is dead, sorry…”). Why atom #3 cannot return the obtained quantum to atom #2, so, one will return it “then” to atom #1 (remember, these processes cannot be described as “flowing in time”)? Such situation is theoretically possible; however, the probability of the “rolling back” all chain of the events decreases dramatically as the number of events enlarges. Atom #3 can transfer the obtained quantum to an atom #4, or atom #5, etc, not necessarily to return it to atom #2. Figuratively, the unhappy cat is a victim of the second law of thermodynamics… The magnetic moment of “electron” is, probably, also similar result of the “uncertainty” in its own angular momentum. Stochastic changes of direction of an “electron rotation axis” are compensated by “mirror” variations in angular momentum of NEMOS, so, total angular momentum of the insulated system “electron in its own EM potential” remains objectively unchanged. A new physical sense must be supposed for the four-vector of matter flow density j . It can be treated only as a factor (rate) of energy-momentum interchange between localized wave packets of NEPOS and NEMOS, not as real physical object owning an “intrinsical” energy-momentum. After the
second quantization, j turns into a part of an operator that “shifts” the nonzero occupation number between different eigenmodes of NEPOS and NEMOS. The spatio-temporal components of j four-operator are collinear to the respective components of the central wavenumber four-vector k of NEPOS wave packet. Wavenumbers of j harmonics are the differences between wavenumbers of the stochastic harmonic components of NEPOS wave packet, on the one hand, and k for this wave packet, on the other hand. A hypothetic “pure” NEPOS eigenmode would not create j at all. The same physical sense, obviously, must be assumed for the four-tensor of energy-stress density [ w] components. In
particular, they are operators “collinear” to the respective products of k components of the “deviating” wave packet. Nevertheless, the “parametric” approach to the explanation of NOS excitation can be useful as some “homogenized” or “macroscopic” approximation of the described above “force” approach, like “refracting medium” concept in the geometric optics. E.g., for slowly varying in spacetime function ℵγ (t , x, y , z ) (when quantum fluctuations can be well smoothed), NEMOS may be regarded as some medium that “shifts” the “local” wavenumbers of NEPOS wave packets [18, 19], i.e., as “squeezing”, “stretching” or “twisting” object keeping the “flatness of spacetime” yet: k= km 0 + ℵγ , m where km 0 is the wavenumber of m-th NEPOS eigenmode in the absence of EM potential (when ℵγ ≡ 0 ); a system of units is used where the “elementary charge” is equal to one. Similarly, NGOS can be treated as some medium quadratically “curving” NEPOS and NEMOS eigenmodes (see below). C. Some Statistical Relations Unfortunately, there is no possibility of the strict direct deriving of statistical laws for the described above process of permanent exchange with random energy-momentum quanta between different NOSs, as the specific physical mechanism of this phenomena is unknown yet. However, we can make some reasonable assumptions concerning those laws and compare the outcomes from ones with known facts. Let’s consider the simplest closed physical system “moving alone electron in its EM potential”. The “central” wavenumber of respective NEPOS wave packet is of k . This localized wave packet stochastically transfers to NEMOS energy momentum quanta of h∆k with some probability P( ∆k ) , where 0 ≤ P( ∆k ) ≤ 1 ; −∞ < ∆k τ < +∞ . Here, ∆k = k − k , where k is a wavenumber of some harmonic component
presented in the NEPOS wave packet spectrum. For the considered system, energy is not transferred between NEPOS and NEMOS, only momentum; so, ∆kt ≡ 0 and P( ∆k ) ≡ 0 , if ∆k t ≠ 0 .
Because ∆k is a continual vector variable for the infinitely large Universe, P( ∆k ) → 0 . Therefore, let’s introduce a probability three-density p ( ∆k x , ∆k y , ∆k z ) = lim
δVk → 0
δP (0, ∆k x , ∆k y , ∆k z ) δVk
,
where δP(0, ∆k x , ∆k y , ∆k z ) is a total probability of transfer the momentum quantum located in a respective small threevolume δVk =δk x δk y δk z around ∆k x , ∆k y , ∆k z of the “wavenumber four-space” three-section at ∆kt = 0 . What may be the dependence p( ∆k x , ∆k y , ∆k z ) ? Some general assumption concerning
p ( ∆k x , ∆k y , ∆k z )
behavior can be made at first: 1. This function must be symmetric along each spatial wavenumber ∆k x , ∆k y and ∆k z with respect to ∆kξ = 0; 2. The probability density tends to zero at ∆k → ∞ , where ∆k =
∆k x2 + ∆k y2 + ∆k z2 ;
3. Overall probability of transfer any momentum quantum between NEPOS and NEMOS is equal to one:
∫∫∫ p( ∆k , ∆k x
y
, ∆k z )d ( ∆k x )d ( ∆k y )d ( ∆k z ) = 1,
∞
where the integral is taken over all three-volume of the “wavenumber four-space” three-section at ∆kt = 0 . The reason is that the amplitude of the “central” component in the momentum spectrum of NEPOS wave packet is the same as for the adjacent components, and this spectrum is continual. Correspondingly, the probability of no transfer any momentum between NEPOS and NEMOS tends to zero; 4. The top of function p( ∆k x , ∆k y , ∆k z ) (at low ∆k ) must be “flat”, because the relative amplitudes of the lowest spatial harmonics of NEMOS wave packet, specifying the behavior of EM potential at far distances from the NEPOS wave packet, have to be independent from the specific shape of the “electron”. Indeed, in extreme case, the latter can be the point particle, having the uniform spectrum over all spatial wavenumbers. So, the lowest spatial harmonics of ℵe always must be uniform. Also, the equality of amplitudes of these harmonics is essential for the enough fast relaxation of ℵe at large distances from the NEPOS wave packet according to general peculiarities of fermion NOSs and spectral analysis laws.
Within the above limits, shape of p( ∆k x , ∆k y , ∆k z ) may be quite intricate. Let’s consider the simplest case when p( ∆k x , ∆k y , ∆k z ) is the normal (Gaussian) distribution over all spatial wavenumbers with the same standard deviation of kσ on each coordinate and zero the mean (i.e., the spherically symmetric distribution): 1 = p 2 2 πk σ
3/2
exp −( ∆k x2 + ∆k y2 + ∆k z2 ) / 2kσ2 .
(IX.1)
As an example, the dependence of p on ∆k z at ∆k x = 0, 0 is shown in Fig 2, a. Formula (IX.1) assumes that the ∆k y = transfers of x-, y-, and z-components of momentum are the independent events. 1D (“radial”) dependence of the momentum quantum transfer probability density on ∆k also makes physical sense for the spherically symmetric distribution of p( ∆k x , ∆k y , ∆k z ) . It can be derived by integrating (IX.1) over all three-spherical surface of ∆k = const . The obtained dependence p1 ( ∆k ) is known as Maxwell’s distribution: 1 p1 = 4π 2 2 πk σ
3/2
exp −∆k 2 / 2kσ2 ∆k 2
(IX.2)
(see Fig. 2, b). The maximum of this dependence has a place at ∆kmax = 2kσ . As it was assumed above, from the point of view of a 3D observer, all excited with some probability eigenmodes of NEPOS and NEMOS wave packets exist “simultaneously”, but each with respective probability to be detected at the measurement. Because NEPOS is the fermion NOS, each act of transfer of the momentum quantum of h∆k from NEPOS to NEMOS is accompanied by a shift of “filled” (i.e., having a non-zero occupation number K m = 1 ) eigenmode in the spectrum of NEPOS wave packet for a value of −∆k . Only one harmonic component can exist at each instance of a fermion NOS wave packet spectrum. This component “wanders” in a random way over all area of possible values of ∆k according to p( ∆k x , ∆k y , ∆k z ) function. So, the probability density of existence of m-th eigenmode with wavenumber of= km ( kt , k x − ∆k x , k y − ∆k y , k z − ∆k z ) in that
spectrum is also described by (IX.1). In particular, the most probable deviation of km from k in the spatial domain for the spherically symmetric distribution of p (∆k x , ∆k y , ∆k z ) is of 2kσ . Description of NOS wave packets in terms of energiesmomenta of their sine and cosine spectral components (NOS eigenmodes) is more “fundamental” then description in terms of eigenmode amplitudes and phases. But amplitude spectra are necessary for transfer of the wave packets from the wavenumber domain to the spatio-temporal domain. Because of almost infinite size of the Universe, amplitude of each harmonic of a wave packet is vanishingly small. However, superposition of almost infinite number of the harmonics produces the wavefunctions of finite amplitude within the wave packet three-volumes for wave packets of limited spatial dimensions. These wavefunctions appear in the expressions for the four-vector of matter flow density j and the four-
tensor of energy-stress density [ w] . While the random “wandering” of the only spectral component of the fermion NOS wave packet, its amplitude does not change in the first-order approximation, because the amplitude defines the wave packet energy (via the four-tensor of energy-stress density), which is unchanged. But this amplitude varies in the second-order approximation due to the Lorentz contraction of three-volume of “yawing” in the spatial directions the wave packet. Amplitude spectrum of ℵe … NEMOS is the boson NOS; therefore, there is no a singlevalued correspondence between transferred to NEMOS momentum quantum and excited NEMOS eigenmode, as it has a place for fermion NOSs. In particular, the transferred quantum h∆k can be fragmented and distributed over several NEMOS eigenmodes. The specific probability distribution law can be derived from the classic approximation of NEMOS. The spatial distribution of EM potential from a rest point source is described by the function 1 ℵtγ (t , x, y , z ) = . 2 x + y2 + z2
(IX.3)
The amplitude spectrum of (IX.3) is 1 1 . ℵtγ (0, k x , k y , k z ) = 4π2 k x2 + k y2 + k z2
(IX.3)
γ or ℵ= 1 / 4π 2 k 2 , where k 2 = k x2 + k y2 + k z2 . t
As it follows from the Lagrange function for NEMOS, the ξ -th component of m-th eigenmode total momentum for this NOS is proportional to U m2 kξ2 , where U m is the amplitude of
Fig. 2. Gaussian distribution of momentum transfer probability density on ∆k z (a); Maxwell distribution of the same density on ∆k (b).
m-th eigenmode. can be derived by integrating (IX.3) over all three-spherical surface of k = const :
1 ℵtγ ( k ) = . π
(IX.4)
1D (“radial”) dependence of on k =
k x2 + k y2 + k z2 can be
derived by integrating (IX.3) over all three-spherical surface of k = const : 1 . ℵtγ ( k ) = π
(IX.4)
(see Fig. 2, b). Quantum fluctuations cause small stochastic deviations of the Universe from the principle of least action rigorous observance???
X. WHAT ARE THE ZERO-POINT OSCILLATIONS? The above concept generally adjusts with [4] except for a one point. Our hypothesis does not assume existence of the zero-point oscillations for neither NEPOS nor NEMOS taken separately. Respectively, the energy-momentum levels of their m-th eigenmodes start from zero, not from hkm / 2 . To explain this mismatch, let’s remember how the “canonical” second quantization of EM potential performs in [4]. Using the wave equation for ℵγ and the Fourier method ℵγ (t , x, y, z ) =
∞
∑u
m = −∞
3m
(t )ℵ3γ m ( x, y, z ) ,
a Lagrange equation d 2 u3m / dt 2 + kmt2 u3m = 0 is derived for mth eigenmode instantaneous value u3m . This equation is similar to the mechanical quantum harmonic oscillator (QHO) Lagrange equation, so, the Hamilton function of m-th eigenmode is written as du Η m u3m , 3m dt
Rγ du3m =2 dt
2
2 2 + kmt2 u3m ∫∫∫ ℵ3γ m dxdydz , V3
where the integral is taken over all spatial three-volume of the Universe. The Hamilton operator for m-th eigenmode
(
)
ˆ Uˆ , Pˆ = Η m m m
Rγ Pˆm 2
( )
2
γ
( ) ∫∫∫ ℵ
+ kmt2 Uˆ m
2
3m
2
dxdydz
V3
is similar to the Hamiltonian of mechanical QHO. For this ˆ is “declared” as coincident reason, the energy spectrum of Η m with the spectrum of mechanical QHO: W = hkmt ( K m + 1 / 2 ) . m ( Km )
An error has a place in such reasoning. In order to the ˆ coincides with the spectrum of eigenvalue spectrum of Η m mechanical QHO, the generalized momentum operator for mth eigenmode must be written as Pˆm =−ih∂ / ∂u3m . But such expression is not the generalized momentum operator.
Moreover, there is no any physical sense in one because there is no function undergoing this operator. A formal cause why the Hamilton operators mismatch, while the Hamilton functions are similar, is as follows. Generalized coordinates of mechanical QHO are the spatial coordinates; respective wavefunction is a function of x, y, z. On the contrary, wavefunctions of NEMOS ℵγ and NEPOS ℵe are also their generalized coordinates. The zero-point oscillations exist only in “mechanical” oscillatory systems, where the generalized and the spatial coordinates coincide. Such systems (e.g., crystal lattices) always are based on the interaction between NEPOS and NEMOS. The zero-point energy hkmt / 2 is “inserted” into mechanical QHOs during the creation of ones, when a parabolic potential well with confined inside “spreading” NEPOS wave packet is formed. But the “zero-point energy of vacuum” was not provided at the Universe creation… In simple phrases, NEPOS wave packet never can be squeezed to an infinitely small object “quietly lying” at the bottom of the potential well. But if at least a part of “positively charged pool” occupies an area with positive EM potential (as it has a place in the parabolic potential well), total energy of such system is greater than zero. That is the zero-point energy. In the rectangular potential well, all “charged pool” “floods” in area with zero EM potential. So, the zero-point energy is absent in this system. The zero-point oscillations of mechanical QHOs are stochastic “fluctuations” of spatially confined in the parabolic potential wells NEPOS wave packets. “Free” eigenmodes of NOSs, occupying the whole 4D Universe, do not perform the zero-point oscillations
The argument “contra” our hypothesis is the existence of the Casimir effect. However, an attempt of the interpretation of one without using the zero-point oscillations of vacuum is made in [20]. If that is right, the observations of the Casimir effect do not imply the reality of the zero-point energy. If the cancellation of the vacuum zero-point energy is done, quantization rules for energy-momentum of natural oscillatory system eigenmodes may be expressed in a obvious form: shift of the system in any spatio-temporal direction over the respective wavelength must cause the value of action equal to h, changing the aleph-function phase by 2π . XI. SIMPLE QUANTUM OSCILLATORS Let’s compare infinitely deep rectangular (Fig. 3, a) and parabolic (Fig. 3, b) 1D potential wells for an “electron” (NEPOS wave packet) created by a system of external “well charges”. As one can see, no principal difference between the both wells, except for the system total energy varies sharply or gradually with change of “electron” localization in z direction respectively. Because the energy in any case tends to infinity at z → ±∞ , the wave packet shapes should not differ essentially for both the wells. Note that, because both the systems are time-independent (stationary), a reversion from “energy” consideration back to
the Newton’s “force” description is suitable, where “force” is interpreted as quantized flow of momentum dP / dt . So, well shape may be defined as dependence of dPz / dt between the “electron” and the “well charges” on their relative position. dPz / dt depends on the rate of mutually non-orthogonal positive and negative harmonics in spectra of the “electron” and the “well charges” wave packets in z direction, which decreases as the wave packets move away one from another. The 3D solenoidality condition for fermion wavefunctions in the rest coordinate system ∂ℵex / ∂x + ∂ℵey / ∂y + ∂ℵez / ∂z ≡ 0 means that the boundary conditions for ℵe in the rectangular well must be “outwardly” similar to the ones for “magnetic field” in rectangular EM resonator. Normalized wavefunctions for ℵex , ℵey and their squares for four the lowest mz = 0,1, 2, 3 are plotted in Fig. 4, where (a) describes the rectangular well; (b) is for the parabolic one. A qualitative similarity of both kinds of the wavefunctions is obvious; the difference is only in “sharpness” of their decreasing with the distance. However, the aleph-function components ℵex , ℵey for the rectangular well differ fundamentally from the Schrodinger’s function Ψ (t , x, y , z ) behavior, which is zero at the walls. There is only nondegenerate wavefunction with mz = 0 in real (3D) rectangular well that is similar in appearance to TE (H) mode “magnetic field” of rectangular EM resonator with mx , m y ≥ 1 and mz = 0 . Note that this similarity is apparent; functions in Fig. 3 are not NEPOS eigenfunctions, because all those are not harmonic in the z direction. Those are the NEPOS wave packets. Only permanent exchange with momentum quanta between the “electrons” and the “well charges” via NEMOS can hold the “electrons” in the wells. Now, the origin of the “zero-point oscillations” in QHO is understandable. The NEPOS wavefunction in the parabolic well is “smeared” over a finite area even for mz = 0 . Energy of “diffused particle” is greater than zero due to the parabolic energy dependence on z. But for taken separately NEPOS and NEMOS this mechanism does not work, so, there are no “zero-point oscillations” of vacuum. XII. ILLUSTRATIONS A. NOSs in the Minkowski Spacetime Let’s consider finite 4D Universe as some fourparallelepiped in pseudo Euclidean space with dimensions of T, X, Y, and Z along the respective axes. This parallelepiped is filled with various NOSs having, e.g., the periodical boundary conditions on all the borders. Some eigenmodes of these
Fig. 3. “Classical” models of “electrons” (NEPOS wave packets) in rectangular (a) and parabolic (b) potential wells.
Fig. 4. The lowest NEPOS wavefunctions (solid) and their squares (dashed) for rectangular (a) and parabolic (b) potential wells.
NOSs are excited, i.e., have non-zero occupation numbers. These occupation numbers permanently “fluctuate” by increase and decrease, but the total wavenumber conservation law for all NOS eigenmodes is satisfying yet. Our 3D world can be treated as a three-section t = const of that 4D parallelepiped, which uniformly shifts along t axis with the unit velocity dt / dt = 1 (see bold line in Fig. 5, a). What occurs in this section while it traverses the NOSs is what we observe from our 3D point of view. The above is one more example as the “geometric” definition of energy-momentum prevails over its “physical” comprehension. Spatially localized “electron” stochastically changes its “observable” physical momentum due to the random exchange with its own EM potential. On the contrary, the “attendant” EM potential acquires only “unobservable” momentum, as a result. The reason is: the electron wave packet spectral components all have kt = − k0 e < 0 , while the respective EM wave packet harmonics have kt ≡ 0 around the rest “electron”. So, equal-phase surfaces for the NEPOS eigenmodes are moving in the 3D world, but similar surfaces for the NEMOS eigenmodes are static here.
B. Some Simple Examples Let’s consider a single rest spatially localized “electron” in its own EM potential in the free space. As it follows from previous chapter, this “electron” is surrounded by a “cloud” (wave packet) of “photons” having stochastic positive and negative momenta with kξ = 0 , but having no energy. If two “electrons” rest at some distance one from another, a part of the “photon cloud” spatial harmonics becomes “common” for the both “electrons”. So, the radiation and the absorption of momentum quanta by the NEPOS wave packets occur “in arbitrary order” (remember that, in fact, these processes do not pass sequentially in the spacetime). However, the statistical laws result in gradual transfer of an average momentum from one “electron” to another, i.e., in the Coulomb repulsion. If one turns the time axis of his four-frame system in the direction of the line passing through both the “electrons”, a spatial ℵγ component will appear in the “photon cloud” in addition to the temporal one for the rest “electrons”. Also, the spatial harmonics in the NEMOS wave packet spectrum will partly turn into the time ones. So, a stochastic exchange with energy quanta occurs between two moving one after another “electrons”, in addition to the momentum interchange. This mechanism is similar to the radiation-absorption between two atoms. The difference consists in the internal structure of both the objects. Whereas atoms are complicated systems, capable accumulating energy due to their “internal” degrees of freedom, “electrons” are no. Another corroboration of our hypothesis is the existence of electron waves in conductors and superconductors. Solid-state theory considers “electrons” in metal crystals as normal modes of “electron gas” rather than localized particles squeezing one’s way through atomic lattice. Why “electrons” in conductive media and “electrons” in vacuum exhibit different behavior? The reason is that all internal volume of the conductive or superconductive crystal is equipotential. High mobility of “electron gas” allows one effective smoothing any inhomogeneities of EM potential. Therefore, there are no harmonics of ℵγ differing from zero within a metal volume. If so, the described above mechanism of electron wave packet formation does not work. Only separate NEPOS eigenmodes can exist in the conductive media.
Due to the spatial three-solenoidality condition for ℵe in the rest coordinate system: ∂ℵex / ∂x + ∂ℵey / ∂y + ∂ℵez / ∂z ≡ 0 ,
the spatial structures of NEPOS eigenmodes in the conductive crystal must be “outwardly” similar to the same for “RF magnetic field” in a microwave EM cavity resonator of the analogous shape. In particular, the same degeneration of TE and TM eigenmodes causes the distinguishing of “electrons” on their spins. XIII. THE QUANTUM DYNAMICS OF NGOS The described above theory of natural distributed oscillatory systems is imperfect until a place for the gravitation interactions is chosen. Einstein’s GR is, in fact, based on the “parametric” approach to consideration of NGOS excitation. “Curving” the spacetime by massive substance, supposedly, changes NOS eigenfunctions causing deformation of wavepacket “trajectories”. So, gravitation is assumed to be a peculiar kind of interactions different from all others. Such interpretation agrees with the quantum theory too poorly, as a “trajectory” is, in fact, only classic notion; no mechanism transferring the “gravitation” energy-momentum quanta; and change of NOS eigenvalues voids the energymomentum conservation in the Universe. Another, “force” approach to the gravitation interactions is developed in so-called field theory of gravitation [21, 22]. It supposes that gravitation, like EM interaction, performs a “force” influence on the matter, not “parametric”. The spacetime cannot be considered as “flat” or “curved” until it is not filled by “sizing tools” (“rulers” and “clocks”). The assumption of NGOS existence means that we have left the concept of “curved” spacetime and passed to the “force” interpretation. Like NEMOS does, energy-momentum quanta of NGOS exert stochastic influences on the “sizing tools” (not on the spacetime itself) causing change k and shape of spectral envelope of NOS wave packets. This varies spatial and temporal wavelengths, i.e., squeezes or stretches “rulers” and slows down or accelerates “clocks”. One of the consequences of such assumption is that all NOSs “fill” the flat pseudo Euclidean Universe. Let’s define a four-tensor aleph-function ℵG (t , x, y , z ) , which is a set of NGOS generalized coordinates. The physical sense of ones may be chosen ambiguously. If it is preferable to use the same criterion as for NEMOS, i.e., ℵG ≡ 0 for the undisturbed NGOS, this may be defined as the difference between the normalized metric four-tensor [ g ] (t , x, y , z ) of the pseudo Euclidean spacetime in the Einstein’s interpretation [11] and the unit four-tensor [ I ] of the same metrics: ℵG =
Fig. 5. 2D (t-z) section of finite 4D Universe considered as some fourparallelepiped in pseudo-Euclidean space filled with NOS eigenmodes and wave packets.
[g] − [I ] .
Four-tensor ℵG is symmetrical, so, there are only ten generalized coordinates of NGOS. Each the coordinate when varied in time or space creates energy or momentum
respectively. Energy-momentum of m-th NGOS eigenfunction is quantized according to the general quantization rule: Wm = hK m km . If all components of ℵG are constant (“the spacetime of uniform curvature”), NGOS does not interact with other NOSs. This is an analogue of a fixed addition to EM potential components. Like NEMOS, free oscillations of NGOS cannot be excited; all their vibrations are localized wave packets of forced eigenmodes, not free ones. “Gravitons”, like “photons”, are non-harmonic stochastic “clouds” surrounding one or more “exciting” wave packets, e.g., “electrons”. The quantum dynamics of NGOS is similar to the dynamics of NEMOS except for the sets of their generalized coordinates (four-tensor instead of four-vector). The “tensor direction” of NGOS deviation (“quadratic flexure”) coincides with the same “direction” of [ w] tensor of a deviating wave packet. More strictly, ℵG contains the same non-zero components as [ w] . Spatio-temporal harmonics of “deviated” ℵG components hold the energy-momentum quanta according to the general quantization rule. In the quantum dynamics, [ w] is a part of a tensor operator that stochastically “shifts” the nonzero occupation number between different eigenmodes of NEPOS or NEMOS, on the one hand, and NGOS, on the other hand. For example, if a single spatially localized rest wave packet of NEPOS (“immovable electron”) is placed in its own gravitation potential, only wtt ≠ 0 , therefore, only ℵGtt ≠ 0 ; all other components of ℵG are zero. This non-zero component of ℵG permanently “absorbs” and “gives back” random quanta of momentum (not energy) from/to the “exciting” wave packet of NEPOS; it seems like a “cloud of gravitons” around the “electron”. If the same “electron” moves in x direction, only wtt , wxx , and wtx = wxt are not equal to zero. Therefore, ℵGtt , ℵGxx , and ℵGtx ≡ ℵGxt are not equal to zero; all other components of ℵG are zero, etc. If two or more immovable localized wave packets of NEPOS are placed in their “common” gravitation potential, NGOS performs a stochastic mutual exchange with momentum quanta between ones, i.e., the gravitational attraction. For moving wave packets, this exchange includes also energy quanta. The “parametric” approach to consideration of NGOS excitation also can be used as some “macroscopic” approximation of the “force” one. E.g., the “local” squared wavenumber of NEPOS or NEMOS m-th eigenmode is calculated as km2= km2 0 + km kmT ⋅ ℵG , km2 0 km kmT ⋅ [ I ] is squared wavenumber of m-th where= NEPOS or NEMOS eigenmode in the “flat spacetime” (when
ℵG ≡ 0 ); km kmT is a symmetrical four-tensor of the wavenumber component products (like [ w] ).
If our hypothesis is true, NEMOS and NGOS may be consolidated in some natural electro-gravitational oscillatory system (NEGOS). Localized wave packets of, e.g., NEPOS can interact with all 14 continual degrees of freedom of NEGOS. Whereas the probability of interchange with an energy-momentum quantum between NEPOS and the “linear” (EM) degrees of freedom of NEGOS is about 1 − 10−42 , the likelihood of the interaction of NEPOS with the “quadratic” (gravitational) ones is as small as 10−42 . The total probability of relocation of some energy-momentum quantum from NEPOS to NEGOS is infinitely tending to 1 because the amplitude of the “central” component in the energymomentum spectrum of NEPOS wave packet is the same as for the adjacent components, and this spectrum is continual. The above does not mean necessarily “flat” (pseudo Euclidean) geometry of NOSs. E.g., NOS eigenfunctions may obey the periodical boundary conditions, not asymptotic ones. This assumes a confined Universe. But the spacetime curvature must be, probably, uniform to ensure the energymomentum conservation. If NEMOS and NGOS are only different degrees of freedom of NEGOS, its “quadratic” distorting (“curving”) may have the “parametrical” influence on its “linear” (“displacing” and “torsional”) eigenfunctions and eigenvalues. It seems like a curving of an initially flat rubber membrane distorts the “longitudinal” (“in-plane”) eigenmodes of one. Other NOSs (e.g., NEPOS) are not undergoing the “parametrical” influence from “curved” NEGOS, only the “force” one. If so, the Einstein’s concept of the “curved spacetime”, indeed, is almost incarnated, but only for a sole case. That is the gravitational influence on light. The EM wave packet (“light ray”) itself does not “curve” NEGOS, like as “inplane” deformations of the rubber membrane cannot cause one’s “flexure”. But the light ray deflects, if NEGOS is already quadratically distorted by a fermion (e.g., NEPOS) wave packet. The reverse impact of the light ray on the massive body is performed, probably, with the momentum quanta transfer via NEGOS while initially straight ray is deflected. Note that the interpretation of NGOS as “extra” degrees of freedom of NEMOS may result in the denial of the assumed by Einstein notorious NGOS nonlinearity. This issue needs an additional study. XIV. CONCLUSION Electrons, photons, and other “elementary particles” must be considered as spatially or spatio-temporally localized wave packets of natural distributed oscillatory systems, not as “hard” things. Such wave packets are composite dynamic objects; their existence is possible only due to the permanent stochastic interaction between different oscillatory systems
widening spectra of their modes and causing all oscillations to be forced, not free. The zero-point oscillations of NEPOS and NEMOS “taken separately” do not exist; this effect is a specific feature only of “mechanical” oscillatory systems, based on the interaction between NEPOS and NEMOS. Treating “electrons” and “positrons” as excited modes of a real distributed oscillatory system assumes another physical sense of de Broglie wavefunction than scalar function Ψ in the “Copenhagen” interpretation. A respective 4-vector aleph-function ℵe is illustrated for the simplest quantum systems “electrons in potential wells”. The most significant direction of further development of our new theory is derivation of both the “elementary charge” (the “fine-structure constant”) and the “magnetic flux quantum” values from the second (quantization) postulate and other hypotheses described in this paper. REFERENCES [1]
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Alexander V. Gritsunov was born in Merefa, Kharkiv Region, Ukraine, on January, 24 1959. He received the M.S. degree in electronics engineering from Kharkiv Institute of Radio Electronics in 1979, the Ph.D. degree in radio physics from the same institute in 1985, and the D.Sc. degree in physical electronics from Kharkiv National University of Radio Electronics (former Kharkiv Institute of Radio Electronics) in 2006. From 1982 to 1986, he was a Research Assistant with the “Electronics” Laboratory of Prof. Alexander G. Shein. From 1986 to 2008, he was a Junior Member of Teaching and an Associate Professor with the Electronics Engineering Department, Kharkiv National University of Radio Electronics. Since 2008, he has been a Professor with the same Department. He is the author of more than 100 articles and theses. His research interests include computer simulation of physical phenomena in vacuum microwave devices, matrix electrodynamics, and quantum electrodynamics. Prof. Gritsunov has no any memberships and no any official awards.
