Serious problems of the “Copenhagen school” itself are expressed in the best way in the notorious. “Schrödinger's cat” paradox and David Mermin's sentence:.
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”. Index Terms—electron wave packet, energy-momentum, matter wave, natural oscillatory system, second quantization, zero-point oscillation
I. INTRODUCTION In spite of striking achievements in the engineering applications of the quantum mechanics and the quantum electrodynamics, there is no consensus in understanding of the theoretical bases of quantum world behavior yet. Such thought is confirmed by the existence a number of interpretations of the quantum theory different from so-called “Copenhagen interpretation”. Serious problems of the “Copenhagen school” itself are expressed in the best way in the notorious “Schrödinger’s cat” paradox and David Mermin’s sentence: “Shut up and calculate”. As reasonable physical alternatives to the “Copenhagen interpretation” of quantum theory, concepts of natural electromagnetic (EM) and electron-positron (EP) distributed oscillatory systems (NEMOS, NEPOS) as real physical bases for De Broglie matter waves were proposed in [1, 2] and [3] respectively. Those 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” of the “Copenhagen interpretation”. II. 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. Each wave process has real base (distributed oscillatory system) periodically changing its physical characteristic. De Broglie waves exist in a distributed oscillatory system.” Let’s assume that there are no “hard” particles in atomic world, 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”. The idea that only fields do exist, not particles, periodically is discussed by various authors (see, e.g., [8, 9]). However, no one of them has proposed vector wavefunction for “electrons”; all authors consider some scalar field like Schrödinger’s function Ψ (t , x, y , z ) , which, nevertheless, has rather physical than probabilistic character. The detection of an “electron” means, in fact, exchange with the quantum of energy-momentum between the NEPOS mode initially having occupation number of 1 and another NEPOS mode originally having occupation number of 0 (through NEMOS as a coupling system accepting or supplying with the difference in energy-momentum). The actual coordinates of this process cannot be ascertained in principle, so ones have no sense. Orthogonality or non-orthogonality 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 wave packets does not matter, because the NEPOS mode 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
wavefront in principally non-dispersive NEMOS. However, if the interaction with another atom occurs, all energymomentum of this “photon” transfers to the atom immediately. I.e., the “quantum jump” may be, in fact, the specificity of the interaction between NEPOS and DEMOS in Minkowski spacetime. If a “wide” (almost planar) electron wave packet creates a small light spot on a fluorescent screen, this means that the poor-localized NEPOS mode has interacted with the welllocalized electron shell of an atom. If a wide wave packet runs into another wide wave packet, the interchange of both wave packets with a NEMOS mode of a small spatial wavenumber is more probable, so the momenta of the wave packets vary slightly (“electrons pass too far from one another”). Only in rare cases, the spatial wavenumber for the intermediate NEMOS mode is large and the momenta of the wave packets change greatly (“electrons collide”). III. 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 natural oscillatory systems (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; h = c is the Plank constant; 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. 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: a three-vector (fermion) system with three generalized coordinates; a four-vector (boson) one with four the coordinates; and a four-tensor (also boson) NOS with ten generalized coordinates. 2. All “fundamental particles” are, in fact, “quasi particles”, i.e., wave packets of respective NOSs. The spatial (spatio-
temporal) 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). 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 GRT 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.
spin zero) in its 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. “Stiffness factors” Rγ , Re couple the local deviations of NEMOS and NEPOS in some point with three-densities of energy-momentum in this point (like 1 / µ 0 factor does for the EM potential). Three-densities of the Lagrange function λ(t , x, y, z ) for non-interacting NOSs may be written as:
IV. KINDS OF NATURAL OSCILLATORY SYSTEMS There are three kinds of NOSs differing in the number N of their generalized coordinates: three-vector (fermion) system with N = 3 ; four-vector EM system (NEMOS) with N = 4 ; and four-tensor gravitational system (NGOS) with N = 10 . Because of additional relations (like the Lorenz gauge), the numbers of fully independent generalized coordinates N i are of 2, 3, and 5 respectively, therefore, corresponding maximal spins of NOS quanta s are of 1/2, 1, and 2 respectively according to the known relation N= 2 s + 1 [4]. i The natural distributed oscillatory systems differ also in their cutoff wavenumbers k0 determining kinematics and dynamics of wave packets. These relativistic scalars are fullvalue replacements for the Newton’s “particle rest masses” concept. All fermion systems have k0 > 0 , so, their wave packets are “particles” (or, more strictly, quasi-particles) with non-zero “rest masses” (leptons, quarks and neutrinos). For four-vector and four-tensor systems k0 = 0 , accordingly, quanta of EM and gravitation interactions have no “rest mass”. V. ELECTROMAGNETIC AND ELECTRON-POSITRON OSCILLATORY SYSTEMS Let’s generalize both the EM potential four-vector and De Broglie EP wavefunction as some real-valued aleph-functions ℵγ (t , x, y , z ) and ℵe (t , x, y , z ) respectively. EM aleph-function ℵγ is a four-vector restricted with Lorenz gauge ∇ ⋅ℵγ ≡ 0 , where ∇ ⋅ℵ = ∂ℵt / ∂t + ∂ℵx / ∂x + ∂ℵy / ∂y + ∂ℵz / ∂z is the four-divergence ( ℵγ differs from the EM potential four-vector A(t , x, y , z ) only in the measure unit). EP aleph-function ℵe is also four-vector restricted with both Lorenz gauge ∇ ⋅ℵe ≡ 0 and a spatial three-solenoidality (“media incompressibility”) condition: ℵte ≡ 0 in some “privileged” rest frame system. In contrast to ℵγ , ℵe has no “potential” eigenfunctions (with
= λγ
Rγ
−(∇ℵtγ ) 2 + (∇ℵγx ) 2 + (∇ℵγy ) 2 + (∇ℵγz ) 2 ; 2
Re 2 e 2 e 2 e 2 e 2 e 2 k0 e (ℵ ) − (∇ℵt ) + (∇ℵx ) + (∇ℵy ) + (∇ℵz ) , 2 where ∇ℵτ = {∂ℵτ / ∂t , ∂ℵτ / ∂x, ∂ℵτ / ∂y , ∂ℵτ / ∂z} is the
= λe
four-gradient; the braces mean the combining of scalar values into a vector; τ is a generic symbol for t, x, y, or z. Let’s suppose that 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. VI. THE QUANTUM 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 [10]:
d
∂λ e ∂λ e 0. − = e / ∂τ) ∂ℵe
∑ d τ ∂(∂ℵ τ
That is the Klein-Gordon equation for four-vector ℵe : ∇ 2ℵe + k02eℵe =0 , where ∇ 2 = ∂ 2 / ∂t 2 − ∂ 2 / ∂x 2 − ∂ 2 / ∂y 2 − ∂ 2 / ∂z 2 is the D’Alembertian. 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 [11, 12] 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 k 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 , where ξ is a generic symbol for x, y, or z. 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 k ∂kt c 0 e vg ξ = . = ± ∂kξ kt k02e + 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
c k02e + k x2 + k y2 + k z2 kt = ± . 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 one. 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”. 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 NOS. 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+ The positive sign of “positron” energy W = hkt e + > 0 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 )
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 [ w] = cµ vgx vgt gy gt v gz v gt
v gt v gx v gx v gx v gy vgx v gz v gx
v gt 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 assumption of NGOS existence means that we leave the concept of “curved” spacetime and pass to so-called field theory of gravitation [13]. In other words, we suppose that gravitation, like EM interaction, performs a “force” influence on the matter, not “parametric” (see below). The spacetime cannot be considered as “flat” or “curved” until it is not filled by “sizing tools” (“rulers” and “clocks”). Like NEMOS does, energy-momentum quanta of NGOS exert stochastic influence on those “tools” (not on the spacetime itself) causing change k and shape of spectral envelope of fermion 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. Other outcomes are out of the objectives of this paper. 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.
Fig. 1. The matter flow density four-vectors (“world lines”) for fermions.
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” and “positrons” together with ones for “protons” are schematically shown in Fig. 1. VII. THE QUANTUM DYNAMICS OF NEPOS AND NEMOS There are two methods for a distributed oscillatory system excitation, known as “parametric” and “force” ones. 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 was “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 forced modes (wave packets) of NEPOS, not eigenmodes. We accept the “force” approach as more consistent with the quantum principles. As it follows from our second postulate, the energymomentum conservation is a fundamental law of the pseudoEuclidian 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 as 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. 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. However, free oscillations of NEMOS and NEPOS do not occur at all. “Pure” eigenmodes of these systems cannot be excited 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 energy-momentum 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 its spectral components with each other by means of NEMOS (their direct interaction is impossible because of the orthogonality of this components). Thus, the spatial localization of NEPOS wave packets and the Heisenberg’s “uncertainty principle” both are results only of NEPOS and NEMOS 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 “electron” NEMOS wave packet has similar (mirrored) envelope of its energy-momentum spectrum ensuring total energy-momentum conservation. Wavenumbers of its harmonics are the differences between wavenumbers of NEPOS wave packet harmonics. So, an 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 , so, this is not zero. But the “central” wavenumber in the spectrum of the “coupled” NEMOS wave packet is zero. So, the energy momentum spectrum of ℵγ is the mirrored energy-momentum spectrum of ℵe transferred to the zero “central” wavenumber. This means that EM potential of a single “electron” does not hold average energy-momentum at all. After the inverse Fourier transform, amplitude spectra provide spatially localized functions ℵe and ℵγ . However, in spite of similar shapes of the energy-momentum spectra of NEPOS and NEMOS, their amplitude spectra are essentially differing. The reason is the different dependences of renormalization functions for ℵe and ℵγ on k . E.g., for kt = 0 amplitude spectrum of NEMOS can be obtained from amplitude spectrum of NEPOS by means of multiplication by
k0e2 + k x2 + k y2 + k z2
k x2 + k y2 + k z2 .
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 space-
time. 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 (see Chapter VI). 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 [11, 12] 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. 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. 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, ones are operators “collinear” to the respective products of k components of the “deviating” wave packet. VIII. 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 second quantization of
EM potential performs in [4]. Using the wave equation for ℵγ and the Fourier method ∞ ℵγ (t , x, y , z ) = ∑ um (t )ℵmγ ( x, y , z ) , m = −∞
a Lagrange equation d 2 um / dt 2 + kmt2 um = 0 is derived for m-th eigenmode instantaneous value um (t is assumed as ct; c is the light velocity). This equation is similar to the mechanical quantum harmonic oscillator (QHO) Lagrange equation, so, the Hamilton function of m-th eigenmode is written as 2 du R du Η m um , m = γ m + kmt2 um dt 2 dt
2
γ 2 ∫ ℵm dxdydz , V
where the integral is taken over all 3-volume of the Universe. The Hamilton operator for m-th eigenmode
(
)
ˆ Uˆ , Pˆ = Η m m m
Rγ Pˆm 2
( )
2
2 2 + kmt2 Uˆ m ∫ ℵmγ dxdydz
( )
V
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 Pˆ =−ih∂ / ∂u . But such m
m
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. The argument “contra” our hypothesis is the existence of
Fig. 2. “Classical” models of “electrons” (NEPOS wave packets) in rectangular (a) and parabolic (b) potential wells.
Fig. 3. The lowest NEPOS wavefunctions (solid) and their squares (dashed) for rectangular (a) and parabolic (b) potential wells.
the Casimir effect. However, an attempt of the interpretation of one without using the zero-point oscillations of vacuum is made in [14]. 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 . IX. SIMPLE QUANTUM OSCILLATORS Let’s compare infinitely deep rectangular (Fig. 2, a) and parabolic (Fig. 2, 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. 3, 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. X. SHORT 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. XI. 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 natural. Clarification of the fundamental laws of the quantum world behavior clears the way to further progress in nanoelectronics and nanotechnology. 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] [2] [3] [4]
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