ˇ
derived from cosmology and origin of special relativity and QM
G. CAVALLERI
Università Cattolica del Sacro Cuore - via Trieste 17, 25121 Brescia, Italy
(ricevuto il 14 Dicembre 1994; revisionato il 30 Ottobre 1995; approvato il 4 Marzo 1997)
Summary. — Electrons are assumed to have the speed c of light and an e.m. self-reaction perpendicular to the velocity so that they perform a circular motion (spin or real zitterbewegung) that generates the zero-point field (ZPF), hence special relativity (SR). The spin angular frequency vs4 cORs, where Rsis the radius of the
spin orbit, is obtained from the Hubble constant H and the average density N of electrons in the universe by equating the powers Pradiated4 Pabsorbed. Used in the
reverse way the new relationship leads to values of H and of Vb4 rmbOrmc (where
rmb is the average barionic mass density of the universe and rmcthe critical mass
density) in agreement with recent observations. Quantum mechanics (QM) is derived as well, although the Schrödinger equation is found to be an approximate regime equation not valid for scattering. Even for diffraction of electrons there are different predictions between QM and the proposed stochastic electrodynamics (SED) implemented by spin. A relevant experiment to discriminate between the two theories is proposed.
PACS 03.65 – Quantum mechanics.
PACS 14.60.Cd – Electrons (including positrons). PACS 03.50 – Classical field theory.
PACS 98.80 – Cosmology.
1. – Main assumption for the spin of an elementary charged particle
Barut and Zanghi [1] have presented a covariant, sympletic, classical dynamical system whose quantization gives the Dirac equation. The classical system undergoes a real zitterbewegung, i.e. a motion at the speed of light c of an almost point-like electron whose orbital angular momentum represents its spin. An electron at rest should actually move with c on a circular orbit having the Compton radius
Rc4 ˇ( 2 mc)21. (1)
This conflicts with common “classical” dynamical ideas as well as with electromag-netism (e.m.) and special relativity (SR). Actually, there should be a continuous e.m. radiation whose power diverges if the gyration speed is c. The inertial mass should also be infinite.
1194
In a paper with some collaborators [2] I have proposed a new model for a charged elementary particle that is assumed to emit filaments any point of which moves on a straight line at the speed of light c. The oriented filaments (there is an arrow on them) materialize the electric flux lines. Positive and negative charges emit filaments with outward and inward arrows, respectively. No SR should be present at the true particle level and the electric field E obtained does not satisfy Maxwell equations. Any particle is then assumed to have, on the average, an orbital, circular motion of revolution around an ideal center called “spin”, or “gyration” motion. The gyration speed v1 is assumed to be constant (and it can also be equal to the speed of light c) and a fluctuation of its direction is superposed to the average circular motion so that the plane of revolution slowly changes and the probability density of its axis direction is isotropic. If the center of revolution is at rest, the average generated field is Coulombian and, if it has a velocity and an acceleration, the average field is that of Lienard-Wiechert. The vector addition of the fields Eidue to many particles is inherent to this model and even the Lorentz force is derived theoretically. Consequently, all classical electromagnetism is obtained without the need for any experiment. For v14 c the gyration motion becomes self-sustained and the second assumption (i.e. the circular motion) becomes a consequence of the first one (the filament emission). For v14 c the gyration motion explains the properties of the particle spin.
Notice that the constancy of the gyration speed v1is relative to a local (Galaxy size) observer S for which the cosmic background microwave radiation at 2.73 K is isotropic. This constancy, and at the speed of light (v14 c), must not be confused with the invariance of the light speed for different observers (which, in SR, occurs only if the internal synchronization is used [3]).
It is to be emphasized that the electromagnetic (e.m.) self-reaction occurs only for
v14 c and points towards the instantaneous center of the osculator circle of the trajectory. As a consequence, a charged particle performs a circular motion (gyration) at the speed of light which is the realistic interpretation of the zitterbewegung [1]. A particle is said to be at rest when the center of the spin (or, rather, gyration) orbit is stationary. If the center has velocity v the particle is said to move with v while it actually follows a helix with speed c. The pitch of the helix tends to infinity when v Kc. SR arises because the point of reference is the velocity v of the centre of the spin orbit and not the true velocity of the particle (that has magnitude v14 c). The Lienard-Wiechert fields derived in ref. [2] have been obtained by applying the Lorentz transformations to the centre of the gyration motion (i.e. using v and not v14 c).
2. – Connections between ˇ and cosmological quantities
Since no SR is assumed at the particle level the electric field E can be obtained by the emission theory [2] and the radiated power due to the gyration motion turns out [4] to be 3 327 O4times that given by the Larmor formula, i.e.
Prad4 2 3 27 O4 e2 c3
g
c2 Rsh
2 , (2)1195 Let us compare eq. (2) with the radiated power PBohr of an electron in Bohr’s first orbit PBohrC 2 3 e2 c3
g
a2c2 RBohrh
2 , (3)where a 41O137.036R is the fine-structure constant. Since Rs4 aRBohr, it is
PradOPBohr4 2 a26D 1012. The e.m. field radiated by all the particles of the universe is therefore due almost exclusively to the gyration (spin) motion.
If N is the average number density of the electrons, in a hypothetical static universe the spectral density of the specific power would be given by r(v) 4
NPradd(v 2vs) (if we do not consider the small spread in v around the gyration angular frequency vs4 cORs). In the real expanding universe, the frequency received at the observation point 0 and coming from a thin shell concentric with 0 and receding with speed bc, is reduced because of the Doppler-Fizeau effect. Consequently, there is a spread in the received frequency from vs(due to near electrons) down to v 40 (due to electrons lying on the universe horizon of 0 and whose gyration center recedes with
b 41).
The universe is homogeneous on the large-scale structure (if the average is performed over spheres having a radius larger than 2 3108light years) and it is known that a monochromatic source with luminosity L
(
in our case L 4NPrad with Prad given by eq. (2))
uniformly distributed in a uniform, steady-state universe produces a power spectral density given by [5]r(v) 4 N HPrad v3 v4s , (4)
where H is the Hubble constant. In the standard, evolutionary model of the universe, taking also into account the effect of the electron-positron pairs present in the early universe, eq. (4) turns out to be somewhat modified. As will be shown in a future paper, we have r(v) 4 N HPrad v3 v4s
k
1 1g
v vsh
2l
7 O4 . (5)For v b vseqs. (4) and (5) give the same result that may be compared to the standard expression of the zero-point field (ZPF), i.e.
r(v) 4ˇv3( 2 p2c3)21, (6)
thus allowing one to derive ˇ from the cosmological quantities.
Since vs4 cORscontains ˇ, it is simpler to obtain the gyration radius Rs directly by equating Pradto the absorbed power
Pabs4 n 2 3 e2 m *p 2r(v s) , (7)
where n 42 are the degrees of freedom for a plane motion (as is a circular motion) and m * the inertial mass when the external force is perpendicular to the gyration velocity v1.
1196
PradP Rs22
(
see eq. (2))
and PabsP Rs23(
due to eqs. (6) and (7))
so that, if Rsdecreases,Pabs increases more than Prad and a balance is reached. If Rs increases the diffusing absorption decreases less than Prad that tends to produce a spiraling toward the center of gyration. The dynamical equilibrium is therefore stable. Notice that it is not the initial assumed force pointing to the center of the gyration orbit that determines Rs. This assumed self-reaction force is only “qualitative” (no mathematical expression is demanded for it) although it is necessary to have a circular motion. Since the gyration speed is constant (and equal to c) the acceleration acof the gyration center turns out to be zero if the external force is parallel to the gyration plane (see appendix). In other words, ac is parallel to the gyration axis n×, i.e. if F is the force acting on the particle, it is
ac4 F Q n× n× Om *, (8)
where m * is the mass one measures when F is parallel to n×. With a uniform distribution of n× (it is sufficient in a hemisphere) it is (see the appendix) aacb 4FO3m * so that the experimental mass is
m 43m *.
(9)
This result could be obtained immediately by considering that ac has only one degree of freedom.
By m * 4mO3 and Prad4 Pabswe get by eqs. (7) and (5) with v 4vs
Rs4 mcH( 4 p227 O4e2N)21, (10)
where we remind that N is the electron number density.
Thus Rs (that contains ˇ) is derived from cosmological quantities. In practice, since
Rsis known with nine significant figures, eq. (10) is used to increase the accuracy of H and N. Now N is related to the barionic average mass density rmb by N 4
rmb( 1 . 114 mp)21, where mp is the proton mass and 1.114 the barion number per each electron (the relative abundance of4He2is [6] 0.228). In turn rmb4 Vbrmc, where rmc4 3 H2( 8 pG)21 (with G being the gravitational constant) is the critical mass density so that N 43H2Vb( 1.114 mp8 pG)21. Writing the Hubble constant H as H 4H50h50, the relative abundances of4He
2,3He2,2H1,7Li3give [6] Vbh5024 0.02(1 6 0.2). By the latter interval we get
N 43H502( 0.02 6 0.004)(1.114mp8 pG)21. (11)
Substituting eq. (11) in eq. (10) gives
h504 3 p 23 O4cR LRSH50( 0.02 6 0.004) 1.114mpG , (12)
where RL4 e2Omc24 2.81794 3 10213cm is Lorentz’ electron radius. Rs may be obtained by equating eq. (2) to eq. (7) with r(v) given by eq. (6) multiplied by 27 O4 as given by eq. (5) when v 4vs. We obtain
mcRs4 ˇ , (13)
1197 1.95562 3 1010
y40.6171356 3 1018, eq. (12) gives h
504 1.35(1 6 0.2), whence
H C67.5(1 6 0.2) km s21 Mpc21, or 54EHE81 , (14)
a value that is compatible with the most recent data that give [7] H 4736 10 kms21Mpc21. Then by eqs. (10) and (11) we obtain
N 4531028( 1 60.2) cm23; V
b4 1.1(1 6 0.2) Q 1022.
(15)
3. – The factor 1O2 for the spin of leptons, the doubling of their gyromagnetic ratio and the solution of the Einstein-Podolsky-Rosen paradox
The spin angular momentum given by eq. (13) is the value measurable in a time interval of A10217s (in the 1S state of a hydrogen atom). But the unit vector n× perpendicular to the spin orbit plane precesses around the magnetic field B seen in the system S 8 instantaneously at rest with the center of the spin orbit so that only the component of ˇ n× along B is measured. The probability of finding the arrow of n× is uniformly distributed on a hemispherical surface having B as a symmetry axis. This is for a spin “up”. For a spin “down”, the symmetry axis is 2B. Consequently, the average value of the measurable angular momentum is
aˇ n× Q B×b 4ˇ
0pO2
dw sin w cos w 4ˇO2 , (16)
which is the standard expression. In other words, eq. (16) explains why Rs4 2 Rc with
Rcgiven by eq. (1).
Pitowski [8] and Barut [8] have shown that a uniform distribution of n× in a hemisphere leads to the same predictions of QM regarding Aspect’s experiments [9]. In other words, a hidden-variable theory with already quantized n× (or spin) is at variance with QM. On the contrary, a uniform distribution of n× in a hemisphere (so that its mean value coincides with that of QM) can violate Bell’s inequalities.
The doubling of the “classical” gyromagnetic ratio is simply due to the two useful degrees of freedom for the precession (see appendix). The corresponding inertial mass for precession is mpr4 3 m * O2 4 mO2 so that
gc4 e 2 mprc 4 e mc . (17)
The radiative corrections are due to the diffusion produced by the ZPF so that the instantaneous gyration radius is R 4Rs1 DR with aDRb 4 0. With a fluctuating R the mass mfobtainable from (13) leads to agcb4aeOmfcb4eaRs1 DRb Oˇ 4 eRsOˇ. However, the experimental mass m
(
still obtainable from (13))
ism 4 amfb 4ˇc21a(Rs1 DR)21b 4ˇ(cRs)21
»
!
i 40 Qu
21 iv
(DRORs) i«
. (18)1198
4. – Derivation of special relativity
The obtained power spectral density (4) is a consequence of both the gyration radiation and the expansion of a homogeneous universe. This spectrum r(v) Pv3has been shown to be relativistically invariant by Boyer [10]. I report the much simpler proof given by Landsberg [11] in a still shorter form.
Consider two inertial observers S and S 8 with relative velocity v. If v is the angular frequency for S of a light ray forming the angle u with v, the frequency v 8 received by
S 8 is v 84
g
1 2 v ccos uh
k
1 2g
v ch
2l
21 O2 v 4Fv . (19)Then, the energy flux per unit solid angle per unit frequency range
(
which is proportional to r(v))
transforms asr 8(v8) 4F3r(v) . (20)
By substituting v84Fv into eq. (20) we see that r84r if rPv3
(hence r8PF3v3). Now, the sizes of the atoms and the periods of revolution of the electrons around the nuclei are determined [12] by r. If an atom is at rest with the inertial system S and another atom is at rest with S 8, each observer measures the same sizes and the same period of revolution for its atom at rest if r 4r8. This is true if we transform the r for S to the r 8 for S 8 by means of Lorentz transformations. The sizes and the revolution periods of the two atoms are therefore connected via Lorentz transformations.
If we consider two electrons instead of two atoms, what is at rest with each observer is the center of the gyration orbit and it is interesting and very simple to show how the relativistic transformation of time is generated in this case. The gyration period for the electron at rest for S 8 is T 842pR 8s Oc , where R 8s is the gyration radius. Let, for simplicity, the plane a of the gyration orbit be perpendicular to v. For the “privileged” observer S the speed of the electron is still c and its component c» on a is c»4
k
c22 v2so that the period for S is
T 4 2 pRs c» 4 2 pR 8s c( 1 2v2 Oc2)1 O2 4 gT 8 , (21)
where T 8 is the period for the observer S 8 at rest with the gyration center and
g 4 (12v2
Oc2)21 O2 is the usual relativistic factor here derived from Galilean kinematics by the Pythagoras’ theorem.
5. – Derivation and modification of QM
The division of the power spectral density expressed by eq. (4) by the mode number density v2(p2c3)21gives the e.m. energy per mode
U 4p2c3NP
rad(Hv4s)21v , (22)
1199 which, by eqs. (2), (10) and (13), is equal to
U 4 3 2 m 2c2H( 8 p2e2N)21v 4 1 2 ˇv , (23)
called the “half-photon per normal mode”.
The effect of this zero-point field (ZPF) on an electron is usually very small since the ZPF consists of trains of sinusoidal oscillations so that the velocity acquired by an electron in a half-wave is lost in the subsequent, opposite half-wave. Even the radiation pressure has usually a small effect since, according to eq. (8), the velocity variation Dv due to E is parallel to n× so that the Lorentz force (responsible for the radiation pressure) is zero: ec21Dv n× 3BQn× n× 40 if n× does not change appreciably during a period of the e.m. wave. However, if the electron is confined, for instance in an atom, n× tends to point to the nucleus. If n× rotates, Dv is no longer parallel to n× and the Lorentz force no longer vanishes. The diffusion velocity vD due to e(E 1c21v 3B)Qn× n× is maximum when the angular velocity vn×of n× is equal to that of the acting e.m. wave. Let
R be the distance from the center of confinement (for instance a nucleus), v the particle
velocity so that vn4 vOR. Now eq. (23) gives the following average kinetic energy per
each absorbed mode:
mv2O2 4 ˇvO2 R , i.e. mvR 4ˇ ,
(24)
which is Bohr’s condition for the fundamental state.
We may obtain eq. (24) also by Prad4 Pabs in the case of circular orbits. To get a connection between the most probable R appearing in (24) and the probability density
r(r) to find the electron in r, we observe that in a stochastic motion with an attracting
center, r is distributed as the atmosphere around the Earth r(r) Pexp [22rOR] so that
R21
4 N˘rNO2 r. By this expression the velocity v appearing in (24) (which, in steady conditions, is the diffusion speed vD) may be written as
vD4 ˇ( 2 m)21N˘rNOr . (25)
Obviously, this expression turns out to be rigorously valid when the electron moves on a circular orbit having the most probable distance R from the nucleus in a 1 S state only. Yet, eq. (25) leads to the Schrödinger equation since, owing to Konig’s theorem, the local density of kinetic energy turns out to be the sum of the kinetic energy of the centre-of-mass (having the local average velocity v) and the kinetic energy relative to the local centre-of-mass, i.e.
1
2mr[ (˘W) 2
1 vD2] . (26)
v 4˘W because the friction force on an oscillator of proper frequency v0, charge e, mass m, given by the Einstein-Hopf formula F 42 (4O5) p2e2(mc2)21v[r(v
0) 2 1 O3NdrOdvNv0] vanishes if the power spectral density r(v) Pv
3.
Then by a variational principle and eqs. (25) and (26) two hydrodynamic, real equations equivalent to the single, complex, Schrödinger equation are easily obtained [13]. This derivation gives a full physical meaning to eq. (25) even for a single electron and also gives the relevant coefficient, which was impossible in the preceding attempts [14].
1200
stochastic process since the average motion of a particle is equal to the classical motion at least for linear forces (Ehrenfest’s theorem), the quantum effects reducing to stochastic fluctuations whose entity little changes the average, measurable effects.
Actually, a better approximation should be obtained during an electron’s half-generic-revolution by giving the electron a random impulse with a value obtainable from eq. (23), i.e. mvr4 ˇvnO2 c, with vn4 vOr, taking for r the average distance from
the nucleus corresponding to the considered half-revolution. In the 1 S state of a hydrogen atom, where vOcCa, we obtain vrC aˇ( 2 mr)21.
Comparing vr with the regime value given by eq. (24) and taking r CR gives vrC
vaO2 or, more in general, vrC v(vO2 c). Consequently, eq. (25) becomes, for scattering,
vD4 (aO2 ) ˇ( 2 m)21N˘rNOr as if ˇ were reduced by a factor a (or, more in general, by
vO2c). The same occurs for the Schrödinger equation
(
that is a consequence of eq. (25))
which, for scattering, should become, if U is the potential (aO2)2ˇ2( 2 m)21˘2
c 1Uc4i(aO2) ˇ¯cO¯t .
(27)
This strong modification of the Schrödinger equation when applied to scattering implies no modification for the Rutherford cross-section. Actually, even classical physics (corresponding to ˇ K0), gives the same expression as given by usual QM. However, in the case of a screened potential some differences should appear.
This could explain the differences which can be as high as 60% between the experimental and theoretical (obtained by the Schrödinger equation) values of the ro-vibrational cross-sections for collisions between hydrogen molecules and free electrons [15]. We have considered n× distributed between 0 and p and no SR. With SR and n× between 0 and pO2 (with respect to B) one should obtain the Dirac equation because of the correspondence shown in ref. [1]. The approximation regarding the diffusive aspect (not the zitterbewegung) is the same as in Schrödinger’s equation and is good for the states in atoms. Errors in excited states, due to the use of eq. (25) rigorously valid for the 1 S state, should be very small, at maximum of the order of 20% of the Lamb shifts as observed [16].
The ZPF absorbed by an electron is only a thin slice of the spectrum around vn, so
that practically all r(v) still has to be considered. It causes the spontaneous decay of the excited states, which are actually decays stimulated by the ZPF. Moreover, the ZPF is the cause of the radiative corrections [17].
6. – Proposal of a new experiment to discriminate between QM and SED with spin
Let us consider electrons passing through one slit. The ZPF is modified by the conducting wall up to the plasma frequency of the metal. (A modification of this kind but relevant to two parallel conducting plates explains the Casimir effect [18]).
The Maxwell equations with E 40 on the walls and Ec0 in correspondence of the slit give a spatial Fourier transform for the ZPF amplitude proportional to [19] (kyb)21sin (kyb). The corresponding spatial distribution of the energy modes allowed
by the slit is proportional to (kyb)22sin22(kyb) with intensity maxima for ky4 0 and kyb 4 (n11O2), n41, 2, 3 .
(28)
These are just the intensity maxima for a plane wave of either e.m. radiation or of a large beam of electrons according to QM. But why does an electron passing through the slit feel only these standing waves of ZPF and not the more intense ZPF far from
1201 the slit walls? The reason is that an electron approaching the wall has n× pointing to the nearest edge (on which there is an induced charge) of the slit which it is going to traverse. If v is the speed of the electron and r the minimum distance from the nearest edge the effective frequency of the ZPF is v 4vn4 vOr. The electron can receive a
maximum transversal impulse mav2
»b1 O24 ˇvnO2 c and the consequent deviation is
sin u 4 av»
2b1 O2Ov 4 ˇv
n( 2 mvc)21.
(29)
Substituting eq. (28) into vn4 kc of eq. (29) gives the intensity maxima in
correspondence of
sin uM4 0 and sin uM4 ˇp 2 bmv
g
n 11 2
h
, (30)which is the usual expression. However vn4 vOr can be equal to only one of the kbc
values given by eq. (30) so that, for a given r, only three maxima should appear: u 40 and two lateral ones corresponding to the integral number n that appears in eq. (30) satisfying vOr4kyc. These details are beyond QM.
No associate wave to the electron has been necessary since what feels and keeps the connection between the two slit edges is the ZPF. The slit diffraction pattern (with only two lateral maxima) should appear even if an electron beam much narrower than the slit size is used. This prediction is at variance with QM since, according to QM, no diffraction should occur when the transverse size of the electron beam is much smaller than the slit size [19]. On the contrary, if the narrow beam is made of photons (instead of electrons) no interference is produced since the photons do not interact with the standing ZPF waves. My prediction is that there is a fundamental difference between photons and electrons with regard to diffraction, and a relevant experiment could discriminate between the present interpretation and QM.
The reason why the experts of stochastic electrodynamics (SED) have not reached this conclusion and have not even explained the diffraction with electrons, is that they did not introduce the spin (or gyration) and in particular eq. (8). They knew very well that the ZPF is more intense further away from the slit. Actually, it is not the decrease of the ZPF near the slit that produces the transversal deviation but the p rotation of n× which makes an electron receive a transversal impulse from the ZPF. The reduction of the ZPF only implies its “quantization” due simply to the solution of the Maxwell equations with the boundary conditions (the walls of the slit).
7. – Conclusions
Classical physics with the only additional assumption that the charged elementary particles move at the speed of light along circular trajectories leads, because of the expansion of the universe, to a power spectral density r(v) Pv3 for the electromagnetic (e.m.) field radiated by the above motion which is the real version of the zitterbewegung. The derived r(v) has the same v-dependence of the zero-point field (ZPF) of QED and equating their constants of proportionality a connection between ˇ, the Hubble constant H and the average concentration N of the electrons in the universe is obtained. This connection may also be derived by equating the power radiated by an electron because of its spin (or gyration) motion to the average power absorbed by an electron from the random e.m. field (or ZPF) radiated by the gyration
1202
motions of all the electrons inside the horizon of the events. The connection is given by eq. (10) where Rs4 ˇO(mc) is the gyration (or spin) radius. Taking values for the Hubble constant H and the average concentration N of the electrons in the universe inside their uncertainties, eq. (10) leads to a value for Rs (or for ˇ) around the correct one. Using eq. (10) in the reverse way, i.e. inserting the value of Rs with its nine significant figures in eq. (10) an improvement in the accuracy of H and N is derived.
A consequence of the assumed constancy of the gyration speed is that the acceleration undergone by a charged elementary particle is perpendicular to the plane of its gyration orbit
(
as expressed by eq. (8))
. This consequence explains the doubling of the gyromagnetic ratio (sect. 3). The “classical” distribution of the spin axis in a hemisphere having the symmetry axis parallel to the external measuring magnetic field B for spin up and antiparallel for spin down, immediately gives ˇO2 for the average observable spin angular momentum(
eq. (16))
although mcRs4 ˇ. This distribution of the spin axes also solves the Einstein-Podolsky-Rosen paradox since the same predictions of QM are obtained [8]. The radiative corrections to the gyromagnetic ratiogcgiven by eq. (17) are calculable by eq. (18).
The random e.m. field radiated by all the electrons of the universe has a power spectral density r(v) Pv3 which is the only one that gives no friction for a particle moving through it. Moreover, a r(v) 4Av3reproduces itself with the same coefficient if we pass from one inertial observer to another by Lorentz transformations applied to the centres of the gyration orbits. Consequently, special relativity (SR) is derived in sect. 4 since the periods and the sizes of the atoms depend [12] on r(v). The Lorentz transformations arise because of r(v) Pv3 and of the reference to the ideal centre around which an elementary particle revolves at the speed of light. In other words, we must not apply the Lorentz transformations to the actual motion of an elementary particle, but to the centre around which it revolves.
In sect. 5 QM is derived, at the same time showing its approximations and limits of validity. In particular, QM is a consequence of r(v), the energy (23) per normal mode, and of eq. (8). QM works rather well for regime conditions when the average kinetic energy of an electron tends to equate the energy per normal mode corresponding to its most probable orbit. However, for a nonregime situation, as occurs for scattering, the Schrödinger equation should deeply be modified. The suggested modification could explain the 60% discrepancy between the theoretical and experimental ro-vibrational cross-sections for free electrons colliding against hydrogen molecules [15].
A new experiment capable of discriminating between QM and the new stochastic electrodynamics (SED) implemented by the gyration motion is proposed in sect. 6. The diffraction of free electrons passing through one slit is explained by the standing e.m. waves of the ZPF and eq. (8). Actually, an electron undergoes a transversal impulse proportional to the angular velocity with which the spin axis rotates. This behaviour, a consequence of eq. (8), was unknown to the experts of SED and that is why they did not succeed in explaining the electron diffraction.
The new prediction is that a partial diffraction should occur even if the size of the electron beam is much smaller of the size of the slit.
* * *
I thank Dr. F. BORGONOVI and Dr. E. TONNIfor helpful discussions and the editing of this work.
1203 AP P E N D I X A
The self-reaction occurs only when v1K c so that the particle can reabsorb a small part of the emitted filaments [2]. The number of the reabsorbed filaments per unit time is proportional to the spin angular velocity cOR and it is therefore PR21 as the centripetal acceleration. If only this self-reaction were present, the dynamical equilib-rium would be satisfied for any R value, i.e. the equilibequilib-rium would be indifferent. There is, however, the radiation damping that can be derived from the expression of the radiated power Prad
(
given by eq. (2))
dividing Prad by c and is PR22. Unlike in the purely classical treatment, the radiation damping acts along the gyration radius (and not along the tangent to the trajectory on which it would be ineffective since v14 c). There is also the diffusion term (that acts along the gyration radius, as well) due to the action of the zero-point field (ZPF) radiated by all gyrating particles of the universe and given by eq. (5). This diffusion term corresponds to what is called the power Pabs absorbed from the e.m. stochastic field. In precise terms, it is given by PabsOc with Pabs given by eq. (7). If there is an external uniform and constant electric field E lying in the plane of the gyration orbit, projecting both the inertial term and the forces on R, the equation of motion turns out to be, in polar coordinates:m(RO2 Rv2) 42A R 2 B R2 1 D R3
k
1 1g
Rs Rh
2l
7 O4 2 eE cos w , (A.1)where A, B, and D are constants, Rsis the gyration radius in equilibrium conditions (at regime with E 40) and R is the perturbed gyration radius. The first term at RHS of eq. (A.1) is due to the reabsorbed filaments and is the one that produces the gyration. Inertia arises because of it and it cancels the inertial centripetal term in conditions of dynamical equilibrium (RO4 0) since v2
R 4c2
OR. To linearize the resultant equation, small initial perturbations around Rsare considered, i.e.
R 4Rs( 1 1e) (A.2)
with e b 1. Then eq. (A.1) becomes
mRse n n 4 D Rs3( 1 1e)3
y
1 1 1 ( 1 1e)2z
7 O4 2 B Rs2( 1 1e)2 2 eE cos w . (A.3)Expanding to first order in e gives
mRse n n 22 D Rs3
g
1 2 19 4 eh
1 B Rs2 (1 22e) 42 e mE cos w . (A.4)For E 40 the equilibrium (en n
4 e 4 0) gives 2 D 4 BRsso that eq. (A.4) reduces to
mRse n n 111 4 B Rs2 e 42 e mE cos w . (A.5)
Considering e as a small quantity means that the driving term at RHS of eq. (A.5) is of e order. Consequently, since w 4vt1o(e), the first term only of the expansion of w in power of e can be retained, and the solution of the simple resultant differential equation is e 4 eE m cos vt mRsv22 ( 11 O4 ) BRs22 , (A.6)
1204
which manifests an oscillating behaviour without any systematic (secular) increase. The conclusion can be drawn that a force acting in the plane of the gyration orbit produces accelerations useful for the precession but no net acceleration (averaged over a period) for the centre of the gyration orbit. Whence eq. (8) follows.
The maximum acceleration (averaged over a period) aMof the centre of the gyration orbit occurs when the external force F is parallel to the unit vector n× perpendicular to the plane of the gyration orbit. The corresponding mass m * is defined as aM4 FOm *. From eq. (8) we derive the acceleration aab averaged over all the direction when n× is isotropically distributed:
aa Q F×b 4 aaVb 4 a(F× Qn×)2b FOm *4aM
0 p dwsin w 2 cos 2 w 4 aM 3 4 F 3 m * 4 F m . (A.7)Consequently, the average, experimental mass for translation m, i.e. the inertia to the acceleration (averaged over a period) of the centre of the gyration orbit, is three times the minimum mass m * corresponding to an acceleration parallel to n×.
For the precession, all the accelerations perpendicular to v1 are effective so that there are two useful degrees of freedom.
The corresponding acceleration of precession is therefore aaprb 4 2 3 F m * 4 F mpr , (A.8)
so that, by eqs. (A.7) and (A.8) we obtain
mpr4 3 2m * 4 m 2 , (A.9) used in eq. (17). R E F E R E N C E S
[1] BARUTA. D. and ZANGHIN., Phys. Rev. Lett., 52 (1984) 2009.
[2] CAVALLERI G., GRASSOTTI C. and TONNI E., Proceedings of Physical Interpretation of Relativity Theory Conference (PIRT), London, 6-9 September 1996, edited by M. C. DUFFY (University of Sunderland) 1996, p. 71; CAVALLERIG., submitted to Phys. Found.
[3] MANSOURIR. and SEXL R. V., Gen. Relativ. Gravit., 8 (1977) 497, 515, 809; PODLAHAM., Indian J. Theor. Phys., 26 (1978) 189; SJO¨DIN T. S., Nuovo Cimento B, 51 (1979) 229; CAVALLERIG. and SPINELLIG., Phys. Found., 13 (1983) 122; CAVALLERIG. and BERNASCONI C., Nuovo Cimento B, 104 (1989) 545; WILLC. M., Phys. Rev. D, 45 (1992) 403; CAVALLERIG. and TONNIE., Ch. J. Syst. Engin. Electron., 6 (1995) 147.
[4] CAVALLERIG., FRANCESCHINIS. and TONNIE., submitted to Phys. Rev. D.
[5] WEINBERG S., Gravitation and Cosmology (J. Wiley, New York) 1972, pp. 451-454. This result has been noticed by SURDINM., Phys. Lett. A, 58 (1976) 370 who assumed for vsthe value vs4 3 cO2 RLwhere RLis the Lorentz electron radius instead of obtaining it by Prad4
Pabs. The value of vsassumed and not derived by Surdin is 3 a21O2 C 205.5 times larger than
cORs. Moreover, he assumed an expression for r(v) twice the correct one given by eq. (6) and
a value for the luminosity L given by L 4rmc2where rmis the matter density of the universe
instead of taking L 4N0Prad. Surdin did not specify what kind of matter density (barionic,
1205 corresponding to a continuous disintegration of all matter. Obviously, he did not perform any numerical calculation to compare his predictions with the values of N , rmand H very little
known in 1976. In the Introduction of CAVALLERIG., Phys. Rev. D, 23 (1981) 363, I wrote: “A more plausible justification of the Surdin approach will be given in a future paper”. This better justification is the one given in the present paper.
[6] SONGALIAA., COWIEL. L., HOGANC. J. and RUGERSM., Nature, 368 (1994) 599; COPIC. J., SCHRAMMD. N. and TURNERM. S., Science, 267 (1995) 192.
[7] FREEDMANW. L., Proceedings of “Critical Dialogs in Cosmology”, Princeton, 1996, edited by N. TUROK(Cambridge University Press) 1997.
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[14] CAVALLERIG. and MAURIG., Phys. Rev. B, 41 (1990) 6751; CAVALLERIG. and ZECCAA., Phys. Rev. B, 43 (1991) 3223.
[15] BUCKMANS. J., BRUNGERM. J., NEWMAND. S., SNITCHLERG., ALSTONS., NORCROSSD. W., MORRISONM. A., SAHAB. C., DANBYG. and TRAILW. K., Phys. Rev. Lett., 65 (1990) 3253; CROMPTONR. H. and MORRISONM. A., Aust. J. Phys., 46 (1993) 203; MORRISONM. A. and TRAILW. K., Phys. Rev. A., 48 (1993) 2874.
[16] BARWICKJ. T. F., J. Math. Phys., 24 (1983) 2776 (Table I); SANSONETTIC. J., GILLASPYJ. D. and CROMERC. L., Phys. Rev. Lett., 65 (1990) 2539.
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