ArticlePDF Available

Abstract and Figures

This paper introduces a Zitterbewegung (ZBW) model of the electron by applying the principle of Occam's razor to Maxwell's equations and by introducing a scalar component in the electromagnetic field. The aim is to explain, by using simple and intuitive concepts, the origin of the electric charge and the electromagnetic nature of mass and inertia. A ZBW model of the electron is also proposed as the best suited theoretical framework to study the structure of Ultra-Dense Deuterium (UDD), the origin of anomalous heat in metal-hydrogen systems and the possibility of existence of "super-chemical" aggregates at Compton scale.
Content may be subject to copyright.
J. Condensed Matter Nucl. Sci. 25 (2017) 76–99
Research Article
The Electron and Occam’s Razor
Francesco Celani
Istituto Nazionale di Fisica Nucleare (INFN-LNF), Via E. Fermi 40, 00044 Frascati, Roma, Italy
Antonino Oscar Di Tommaso
Università degli Studi di Palermo – Department of Energy, Information Engineering and Mathematical Models (DEIM), viale delle Scienze,
90128 Palermo, Italy
Giorgio Vassallo
Università degli Studi di Palermo – Department of Industrial and Digital Innovation (DIID), viale delle Scienze, 90128 Palermo, Italy
Abstract
This paper introduces a Zitterbewegung (ZBW) model of the electron by applying the principle of Occam’s razor to Maxwell’s
equations and by introducing a scalar component in the electromagnetic eld. The aim is to explain, by using simple and intuitive
concepts, the origin of the electric charge and the electromagnetic nature of mass and inertia. A ZBW model of the electron is also
proposed as the best suited theoretical framework to study the structure of Ultra-Dense Deuterium (UDD), the origin of anomalous
heat in metal–hydrogen systems and the possibility of existence of “super-chemical” aggregates at Compton scale.
c
2017 ISCMNS. All rights reserved. ISSN 2227-3123
Keywords: Compton scale aggregates, Electric charge, Elementary particles, Electron structure, LENR, Lorenz gauge, Occam’s
razor, Space–time algebra, Ultra-dense deuterium, Vector potential, Weyl equation, Zitterbewegung
Nomenclature (see p. 77)
1. Introduction
The application of Occam’s razor principle to Maxwell’s equations suggests a Zitterbewegunga(ZBW) interpretation
of quantum mechanics [1] and a simple electromagnetic model for charge, mass and inertia. A new, particularly simple
ZBW model of the electron is proposed as the best suited one to understand the structure of the Ultra-Dense Deuterium
(UDD) [2,3] and the origin of Anomalous Heat in metal–hydrogen systems.
Also at: International Society for Condensed Matter Nuclear Science (ISCMNS)-UK. E-mail: francesco.celani@lnf.infn.it.
E-mail: antoninooscar.ditommaso@unipa.it.
Also at: International Society for Condensed Matter Nuclear Science (ISCMNS)-UK. E-mail: giorgio.vassallo@unipa.it.
aGerman word for “tremble” or “shaking motion”.
c
2017 ISCMNS. All rights reserved. ISSN 2227-3123
F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 77
Nomenclature
Symbol Name SI units Natural units (NU)
AElectromagnetic four-potential V s m1eV
AElectromag. vector potential V s m1eV
GElectromagnetic eld V s m2eV2
FElectromagnetic eld bivector V s m2eV2
BFlux density eld V s m2=T eV2
EElectric eld V m1eV2
SScalar eld V s m2eV2
JeFour-current density eld A m2eV3
JCurrent density eld A m2eV3
vFour-velocity vector m s11
vVelocity vector m s11
ρCharge density A s m3=Cm3eV3
x, y, z Space coordinates m (1.97327 ×107m1 eV1) eV1
tTime variable s (6.5821220×1016 s1 eV1) eV1
cLight speed in vacuum 2.99792458 ×108m s11
Reduced Planck constant 1.054571726 ×1034 J s 1
µ0Permeability of vacuum 4π×107V s A1m14π
ϵ0Dielectric constant of vacuum 8.854187817 ×1012 C (V m)11
/4π
eElectron charge 1.602176565 ×1019 A s 0.085424546
meElectron mass at rest 9.109384 ×1031 kg 0.510998946 ×106eV
λcElectron Compton wavelength 2.4263102389 ×1012 m1.229588259×105eV1
PEnergy-momentum four-vector kg m s1eV
PMomentum vector kg m s1eV
U, W Energy J = kg m2s2eV
One of the most detailed and interesting ZBW electron models has been proposed by David Hestenes, emeritus
of Arizona State University. He rewrote the Dirac equation for the electron using the four dimensional real Clifford
algebra Cl1,3(R)of space–time with Minkowski signature “+− −”, eliminating unnecessary complexities and
redundancies arising from the traditional use of matrices. The Dirac gamma matrices γµand the associated algebra
can be seen as an isomorphism of the four-basis vector of space–time geometric algebra. This simple isomorphism
allows a full encoding of the geometric properties of the Dirac algebra, and a rewriting of Dirac equation that does
not require complex numbers or matrix algebra. In this context the wave function ψis characterized by the eight
real values of the even grade multivectors of space–time algebra Cl1,3(STA). Even grade multivectors of STA can
encode ordinary rotations as well as Lorentz transformations in the six planes of the space–time. Hestenes associates
the rotations encoded by the wave function with an intrinsic very rapid rotation of the electron, the ZBW, that is
considered at origin of the electron spin and magnetic moment. The word Zitterbewegung was originally used by
Schrödinger to indicate a fast movement attributed to an hypothetical interference between “positive” and “negative”
energy states. Kerson Huang later, more realistically, interpreted the ZBW as a circular motion [4].
In particular, B. Sidharth states that “The well-known Zitterbewegung may be looked upon as a circular motion
78 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99
about the direction of the electron spin with radius equal to the Compton wavelength (divided by 2π) of the electron.
The intrinsic spin of the electron may be looked upon as the orbital angular momentum of this motion. The current
produced by the Zitterbewegung is seen to give rise to the intrinsic magnetic moment of the electron. [5].
Hestenes considers the complex phase of the wave function solution of the traditional Dirac equation as the phase of
the ZBW rotation, showing “the inseparable connection between quantum mechanical phase and spin” consequently
rejecting the “conventional wisdom that phase is an essential feature of quantum mechanics, while spin is a mere detail
that can often be ignored” [6]. Using the space–time algebra in [7] Hestenes denes the “canonical form” of the real
wave function ψ:
ψ(x) = (ϱeiβ)1
/2R.
In the above equation xis a generic space–time point, ϱ=ϱ(x)is a scalar function interpreted as a probability density
proportional to charge density, iis the spatial bivector i=γ2γ1,β=β(x)is a function representing the value of a
rotation phase in the plane γ2γ1and Ris a rotor valued function that encodes a Lorentz transformation. In the STA
canonical form for Dirac’s wave function the imaginary unit iis replaced by a bivector that generates rotation in a
well dened space-like plane and not in a generic undened “complex plane”. This simple approach clearly reveals
the geometric meaning of the imaginary numbers in the wave functions of quantum mechanics [7]. In agreement with
the most common interpretations of quantum mechanics Hestenes associates the probability density function with a
point-like shaped charge. In Eq. (48) of [7], by applying the relativistic time dilation to the ZBW period, Hestenes
predicts a ZBW angular frequency that slows as the electron speed increase.
According to the model proposed in this paper, the electron characteristics may be explained by a massless charge
distributed on the surface of sphere that rotates at the speed of light along a circumference with a radius equal to the
reduced electron Compton wavelength (0.386159 pm), a value that is two times the one proposed by Hestenes in
Eq. (33) of a relatively recent work [7]. The electron mass–energy, expressed in natural units, is equal to the angular
speed of the ZBW rotation and to the inverse of the orbit radius (i.e. 511 keV), whereas the angular momentum is
equal to the reduced Planck constant . Consequently, unlike the Hestenes prediction, our model proposes a relativistic
contraction of the ZBW radius and a corresponding instantaneous ZBW angular speed that increases as the electron
speed increases.
The inter-nuclear distance in UDD of 2.3 pm, found by Holmlid [2], seems to be compatible with proton–electron
structures at the Compton scale [8,9] where the ZBW phases of neighbor electrons are correlated. These structures may
generate unusual nuclear reactions and transmutations, considering the different sizes, time-scale and energies of these
composites with respect to the dimension of the particles (such as neutrons) normally used in nuclear experiments.
By using the electromagnetic four-potential as a “Materia Prima” a natural connection between electromagnetic
concepts and Newtonian and relativistic mechanics seems to be possible. The vector potential should not be viewed
only as a pure mathematical tool to evaluate spatial electromagnetic eld distributions but as a real physical entity, as
suggested by the Aharonov–Bohm effect, a quantum mechanical phenomenon in which a charged particle is affected
by the vector potential in regions in which the electromagnetic elds are null [10].
The present paper is structured in the following manner: Section 2 deals with a brief presentation of Maxwell’s
equations that does not use Lorenz gauge; Section 3 presents a new simple ZBW model of the electron with a list of the
main parameters that can be deduced by applying this model; Section 4 describes an original method to easily derive
the Lorentz force law from the electromagnetic eld; Section 5 consists of a short introduction to the concept of “quanta
current” and it also presents the relation between the ZBW modeling and Heisenberg’s uncertainty principle; Section 6
summarizes other main models of the electron based on the concept of spinning charge distributions and, nally,
Section 7 presents some preliminary hypotheses on UDD, Compton scale aggregates and the origin of anomalous heat
in condensed matter.
F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 79
In this paper all equations enclosed in square brackets with subscript “NU” have dimensions expressed in natural
units.
2. Maxwell’s Equations in Cl3,1
The space–time algebra is a four dimensions Clifford algebra with Minkowski signature Cl1,3(“west coast metric”) or
Cl3,1(“east coast metric”) [11,12].
In Cl3,1algebra, used in this work, calling {γx,γy,γz,γt}the four unitary vectors of an orthonormal base the
following rules apply:
γiγj=γjγiwith i̸=jand i, j {x, y, z, t},(1)
γ2
x=γ2
y=γ2
z=γ2
t= 1.(2)
Maxwell’s equations can be rewritten considering all the derivatives of the electromagnetic four-potential A:
A(x, y, z, t) = γxAx+γyAy+γzAz+γtAt.(3)
Each of the vector potential components Ax,Ay,Azand Atis a function of space and time coordinates and has
dimension in SI units equal to V s m1.Ais a harmonic function [13] that can be seen as the unique source of
all concepts-entities in Maxwell’s equations. Using the following denition of the operator in space–time algebra,
where
=γx
x+γy
y+γz
zand c=1
ϵ0u0
,
=γx
x+γy
y+γz
z+γt
1
c
t=+γt
1
c
t,(4)
the following expression can be written (see Table 1):
A=·A+A=S+F=G,(5)
where
S=·A1
c
At
t(6)
is the scalar eld,
F=1
cEγt+IBγt=1
c(E+IcB)γt(7)
the electromagnetic eld and
I=γxγyγzγt(8)
is the pseudoscalar unit.
80 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99
Table 1. Relation between electromagnetic entities
and the vector potential.
AγxAxγyAyγzAzγtAt
γx
xS1Bz1By1
1
cEx1
γy
yBz2S2Bx1
1
cEy1
γz
zBy2Bx2S3
1
cEz1
γt1
c
t
1
cEx2
1
cEy2
1
cEz2S4
The electromagnetic eld Gcan be expressed in the following compact form
G(x, y, z, t)=·A1
c
At
t+Atγt1
c
A
tγt+I×Aγt,(9)
and the expression
G=2A= 0,(10)
represents the four Maxwell’s equations.
By applying, now, the operator to the scalar eld S, we obtain the expression of the four-current as
1
µ0
S=1
µ0γx
S
x+γy
S
y+γz
S
z+γt
1
c
S
t=Je,(11)
where Je=γxJex +γyJey +γzJez γtcρ=Jγtcρ=ρ(vγtc)is the four-current vector and v=
γxvx+γyvy+γzvzγtc=vγtcis a four-velocity vector. The operator applied to the four-current gives the
charge–current conservation law
1
µ0
·(S) = ·Je=Jex
x+Jey
y+Jez
z+∂ρ
t= 0,(12)
which can be written alternatively as
·(S) = 2S=2S
x2+2S
y2+2S
z21
c2
2S
t2=2S1
c2
2S
t2= 0.(13)
The charge is related to the scalar eld according to
1
c
S
t=µ0Jet =µ0cq
xyz=µ0cρ,(14)
so that, by applying the time derivative to (13) and remembering (14), the wave equation of the charge density eld
ρ(x, y, z, t)can be deduced:
t(2S)=2S
t=2(µ0c2ρ)=µ0c22ρ= 0,(15)
whose last equality gives
F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 81
2ρ=2ρ1
c2
2
t2ρ= 0.(16)
A more detailed development of Maxwell’s equations in Cl3,1algebra, made by the authors, can be found in J.
Condensed Matter Nucl. Sci. 25 (2017) entitled “Maxwell’s Equations and Occam’s Razor”.
3. Electron Zitterbewegung Model
The concept of charge that emerges from this rewriting of Maxwell’s equations has a non-trivial implication: the
analysis of (13) and (16) shows that the time derivative of a eld Swhich propagates at the speed of light, must
necessarily represent charges that are also moving at the speed of light.
This observation advises a pure electromagnetic model of elementary particles based on the ZBW interpretation of
quantum mechanics [1,14]. According to this interpretation, the electron structure consists of a massless charge that
rotates at the speed of light along a circumference equal to electron Compton wavelength λc[15,16]. Calling rethe
ZBW radius, ωethe angular speed and Tits period we have:
re=λc
2π3.861593 ×1013 m,(17)
ωe=c
re= 2πc
λc7.763440 ×1020 rad s1,(18)
T=2π
ωe=2πre
c8.093300 ×1021 s.(19)
The value of the electron mass, expressed in SI units, can be derived from the following energy equations [1]
Wtot =mec2=ωe=c
re,(20)
from which
me=ωe
c2=
cre=h
cλc9.109383 ×1031 kg (21)
is obtained. Using natural units with =c= 1 the electron mass (in eV) is equal to the angular speed ωeand to the
inverse of re:
[me=ωe=1
re0.511 ×106eV]NU
.
Recently, a connection between frequency and mass, in agreement with De Broglie’s formula f=mc2
/h, has been
experimentally demonstrated [17].
82 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99
3.1. Simple electron model
A charge rotating at speed of light generates a current Iethat is equal to the ratio of the elementary charge eand its
rotation period T[18]:
Ie=e
T=ec
2πre=eωe
2π19.796331 A.(22)
The electron magnetic moment µB(Bohr magneton) is equal to the product between the current Ieand the enclosed
area Ae
µB=IeAe=eωe
2ππr2
e=ec
2re=ec2
2ωe=e
2me9.274010 ×1024 A m2.(23)
Occam’s razor is an effective epistemological instrument that imposes to avoid as much as possible the introduction
of exceptions. Following this rule a pure electromagnetic origin of the electron’s “intrinsic“ angular momentum should
be found.
Consequently, the canonical momentum Ptof the rotating massless charge may be seen as the cause of the intrinsic
angular momentum:
=Ptre,
where the canonical momentum Ptof e, in presence of a vector potential A, generated by the current Ie, is
Pt=eA.
Imposing the constraint that =we can compute Aas function of Ie
=eAre=eAc
ωe=e2cA
2πIe=,(24)
from which it is possible to derive the expression of the vector potential seen by the spinning charge
A=2π
e2cIe=
ere=ωe
ec =mec
e1.704509 ×103V s m1.(25)
From (25) it is possible to derive the Fine Structure Constant (FSC)
α=µ0
4π
ce2
=µ0
4π
eωe
A7.297352 ×103.(26)
Using natural units we get these simple relations:
A= 2πα1IeNU ,
eA =ωe=r1
e=me=PtNU ,
F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 83
where α1, the inverse of the FSC, is a pure number and eis the elementary charge expressed in natural units
α1=e2137.035989NU .
3.2. Spin and intrinsic angular momentum
The intrinsic angular momentum of the electron model (see (24)) is compatible with the spin value /2if we consider
the electron interaction with the external ux density eld BE, as in the Stern–Gerlach experiment. We can interpret
the spin value ±/2as the component of the intrinsic angular momentum =aligned with the external ux density
eld BE. In this case the angle between the BEvector and the angular momentum have only two possible values,
namely π/3and 2π/3while the electron is subjected to a Larmor precession with angular frequency ωp=dϑp/dt. The
Larmor precession is generated by the mechanical momentum
τ=|µB×BE|=BEµBsin π
3.(27)
But
d=dϑp=sin π
3dϑp,
where is the component of the intrinsic angular momentum orthogonal to BEand, therefore, it is possible to write
τ=d
dt=sin π
3dϑp
dt.(28)
By equating (27) and (28) we get
BEµB=ωp,
from which it is possible to determine the precession angular frequency
ωp=BEµB
=BEµB
.(29)
3.3. Value of the vector potential, cyclotron resonance and ux density eld
The pure electromagnetic momentum eA of the spinning charge of an electron at rest can be seen as it were the
momentum of a particle of mass meand speed cin classical Newtonian mechanics. Considering ωeas the cyclotron
angular frequency (which is coincident with the ZBW angular speed) given by the ux density eld Bgenerated by
the current Ie
ωe=eB
me=eBc2
ωe,
it is possible to deduce the magnetic ux density produced by the electron
84 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99
Be=ω2
e
ec24.414004 ×109V s m2.(30)
This very high ux density value (also known as Landau critical value) seems to be related to the physics of neutron
stars and pulsars [19–21] or to that of superconductivity [22,23].
It is also possible to calculate the ux density at the center of the electron orbit by the following expression derived
from the Biot–Savart law
B0=µ0
2
Ie
re32.210548 ×106Vsm2.(31)
Considering that
dA=AdϑdA
dt=Aωe,(32)
where dϑ=ωedtis the differential of the ZBW phase, and considering that the magnetic force FBmust be equal to
the time derivative of the canonical momentum, it is possible to write
FB=Beec =edA
dt=eAdϑ
dt=eAωe0.212014 N.(33)
Finally, by manipulating the previous equation, it is possible to recompute by another method the module of the vector
potential
A=Bec
ωe=ωe
ec =
ere.
3.4. Value of magnetic and electrostatic energy, magnetic ux quantization and radius of the elementary charge
Once we obtaine the expression for the vector potential it is possible to determine the magnetic ux produced by the
rotating elementary charge by applying the circulation of the vector potential A:
ϕe=Iλc
Adλ=2π
0
ereredϑ= 2π
e=h
e4.135667 ×1015 V s,(34)
i.e., the magnetic ux crossing the surface described by the charge trajectory is quantized (ux quantum). This expres-
sion has been found with a different approach with respect to [24], i.e., with the application of the vector potential.
This ux quantum, though different from the value given in CODATA 2014, is compatible with h/2efor the same
reasons explained in Section 3.2, with reference to . Now it is possible to calculate the magnetic energy stored in the
eld produced by the spinning charge
Wm=1
2ϕeIe=1
22π
e
ec
2πre=c
2re4.093553 ×1014 J (35)
which is equal to half the electron rest energy Wtot as can be seen from (20). The other half part can be attributed to
electrostatic energy, i.e.,
F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 85
Wtot Wm=We=V
wedV, (36)
where weis the electrostatic energy per unit of volume and Vis the volume in which the whole energy Weis stored,
and whose expression is given by
we=1
2ϵ0E2=ϵ0
21
4πϵ0
e
r22
=1
32π2ϵ0
e2
r4.(37)
By expanding and integrating (36), with dV= 4πr2dr(the generic elementary volume of a spherical thin shell
centered in the middle of the electron trajectory) we obtain
We=e2
32π2ϵ0
r0
1
r44πr2dr=e2
8πϵ0
r0
1
r2dr=e2
8πϵ0
1
r
r0
=e2
8πϵ0r0
.(38)
Now, by taking into account that We=Wmwe get the radius
r0=e2
8πϵ0We=e2
8πϵ0Wm=e2
8πϵ0
2re
c=e2re
4πϵ0c2.817940 ×1015 m,(39)
whose value is coincident with the classical electron radius [25]. The upper equation states that the rotating charge
must have a nite dimension, in particular it may be visualized as a sphere with charge equal to euniformly distributed
over its surface. The charge cannot be concentrated in a point in order to exhibit a nite electrostatic energy. It is
interesting and at the same time very important to note that the ratio re/r0is exactly equal to the inverse of the FSC,
i.e.,
re
r0
=4πϵ0c
e2rere=4πϵ0c
e2=α1137.035999.(40)
The expression of the ratio ϕe/T has in SI the dimension of a voltage:
Ve=ϕe
T=h
e
c
2πre=c
ere5.109989 ×105V,(41)
where Tis dened by (19). Now, dividing the above voltage by the current generated by the rotating charge expressed
by means of (22), we nd the von Klitzing constant or quantum of resistivity, related to the quantum Hall effect [24]
RK=Ve
Ie=h
e
c
2πre
2πre
ec =h
e2=2π
e225812.807 .(42)
An alternative expression of the von Klitzing constant can be derived from the electrostatic potential φeand the current
Ie
RK=φe
Ie=1
4πϵ0
e
r0
2πre
ec =1
2cϵ0
re
r0
=µ0c
2α25812.807 .(43)
86 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99
Finally, it is possible to deduce the values of two interesting electrical parameters, namely the inductance Le, the
capacitance Ceof the electron and the frequency fe. In fact
Le=ϕe
Ie= 4π2re
e2c2.089108 ×1016 s,(44)
Ce=e
φe= 4πϵ0r03.135381 ×1025 F (45)
and
fe=1
LeCe1.235590 ×1020 Hz.(46)
3.5. Electron kinematics
The nite dimension of the elementary charge imposes the constraint that all points of the surface of the spinning
charged sphere must have the same instantaneous speed of light c(see Eq. (16)) and the same angular speed. In a
frame rotating with the ZBW frequency the spinning charged sphere rotates around its center with opposite speed with
respect to the ZBW angular frequency:
ω0=ωe.(47)
The new (not point-like) electron model and the speed diagrams are shown in Fig. 1a. Here the charge rotates with
angular speed ω0around the axis passing through the center of the sphere and, therefore, all points of the sphere have
the same absolute speed c.
During the revolution around the origin C the charge describes a torus whose cross section is equal to πr2
0and
having a volume equal to 2π2rer2
0. In Fig. 1b, the elementary charge is represented as a charged sphere.
(a) (b)
Figure 1. (a) ZBW model and speed diagrams of the electron charge (e). All points of the sphere have an absolute speed equal to c. (b) 3D
representation. The charged sphere is rotating with the relative angular speed ω0=ωeon the trajectory having radius rearound the vertical axis
passing through the center of the sphere.
F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 87
3.6. Electron and electromagnetic Lagrangian density
From (25), (22) and (39), with the hypothesis that the electron is characterized by a uniform current density, we get the
rst term of the interaction part of the Lagrangian density
Lint1=JA=JA =Ie
πr2
0
mec
e1.352604 ×1027 J m3.(48)
By integration over the volume described by the electron toroidal trajectory, it is possible to recompute its rest
energy:
Wtot =V
J A dV=Ie
πr2
0
mec
e2π2rer2
08.187106 ×1014 J= 510.998946 keV,(49)
which gives the same result calculated by means of (36). The same results are obtained, apart from a sign –, by the
following relations
Lint2=ρφe=e
2π2rer2
0
e
4πϵ0re=e2
8π3ϵ0(rer0)2≈ −1.352604 ×1027 J m3,(50)
Wtot =V
|ρφe|dV=e22π2rer2
0
8π3ϵ0(rer0)2=e2
4πϵ0re8.187106 ×1014 J= 510.998946 keV.(51)
All parameters that can be deduced by the application of the present ZBW model are resumed in Table 2, where
the rst three rows are referred to the model’s input parameters.
3.7. ZBW and a simple derivation of the relativistic mass
With the ZBW model it is possible to show a simple, original and intuitive explanation of the relativistic mass concept.
For an electron moving at constant speed vzalong the z-axis orthogonal to the rotation plane, calling vthe component
of the velocity of the rotating charge in the γxγyplane we can nd the value of the ZBW radius rof the moving electron.
In fact, assuming a constant value of ωe, we have
v2
z+v2
=c2,(52)
that can be written as
v2
z+ω2
er2=ω2
er2
e=c2.
Therefore
r2
r2
e
= 1 v2
z
c2
or
88 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99
r=re1v2
z
c2.(53)
Finally, by considering that the mass is inversely proportional to r, it is possible to write the relativistic expression
of the mass as
m=me
1v2
z
c2
,(54)
where meis the electron mass at rest. Fig. 2 represents the ZBW trajectory of the spinning charge of an electron sub-
jected to an acceleration directed along the positive z-axis. Due to the acceleration the radius reduces itself according
to (53).
3.8. Dirac equation and spinor representation of motion
By using space–time algebra and following the idea of Hestenes–Dirac equation
i
ψmψ= 0 (55)
becomes the Hestenes–Dirac equation [26]
Table 2. Parameters of the Zitterbewegung model.
Item Symbol Value (SI) Unit (SI)
Charge e1.602176565 ×1019 C=As
ZBW orbit radius re=λc/2π3.861593 ×1013 m
Intrinsic angular momentum ==h
/2π1.054571726 ×1034 J s
Spin1
/20.527285863 ×1034 J s
Angular speed ωe7.763440 ×1020 rad s1
Mass me9.109384 ×1031 kg
Current Ie19.796331 A
Magnetic moment (Bohr magneton) µB9.274010 ×1024 A m2
Vector potential A1.704509 ×103Vsm1
Magnetic ux density Be4.414004 ×109Vsm2
Magnetic ux ϕe=h
/e4.135667 ×1015 V s
Magnetic energy Wm4.093553 ×1014 J
Electrostatic energy We4.093553 ×1014 J
Electron energy at rest Wtot =mec28.187106 ×1014 J
Charge radius r02.817940 ×1015 m
Inverse of the FSC α1=re/r0137.035999 1
Von Klitzing constant RK=h
/e2=µ0c
/2α25812.807
Inductance Le=4π2re/e2c2.089108 ×1016 s
Capacitance Ce= 4πϵ0r03.135381 ×1025 F
1-st part of Lint JA 1.352604 ×1027 J m3
Electron energy at rest ∫∫VJA dV510.998946 keV
Electron energy at rest ∫∫VρφedV510.998946 keV
1Component of the angular momentum due to Larmor precession along the external magnetic eld BE(see
(27)).
F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 89
Figure 2. Zitterbewegung trajectory during an acceleration of the electron in the z-direction.
∂ψ mψγtγxγy= 0,(56)
where is the same operator used in Maxwell’s equations G= 0 (see (4)), but using the space–time Minkowski
signature of Cl1,3+− − −”. For a massless particle, where m= 0, (56) becomes the Weyl equation
∂ψ = 0,(57)
which is formally identical to (10). In all cases the solution is a spinor eld. A spinor is a mathematical object that
in space–time algebra is simply a multivector with only even grade components. The geometric product of an even
number of vectors is always a spinor. A spinor that is the geometric product of two unitary vectors is a unitary rotor.
The movement of a point charge that rotates in the plane γxγyand at the same time moves up along the γzaxis can be
seen as the composition of an ordinary rotation in the plane γxγyfollowed by a scaled hyperbolic rotation in the plane
γzγt. The composition of these two rotations can be encoded with a single spinor of Cl3,1. Therefore, if re0is the
coordinate of the center of the charged sphere at t=t0,rethe Compton radius we have
re0=γxre+γtct0,
re0=γxre+γtt0NU .
By introducing the rotor Rxy, that generates ordinary rotation in the γxγyplane, and remembering that ωeis the ZBW
angular frequency, with
Rxy (t) = cos ωet
2+γxγysin ωet
2= exp γxγy
ωet
2,
we obtain the instantaneous position of the center of the charged sphere:
90 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99
re(t)=Rxy (re0+γtct)g
Rxy.
We introduce now the rotor Rzt, that generates a hyperbolic rotation with rapidity φin the γzγtplane, in order to
encode the motion at speed vzalong the γzdirection
Rzt = cosh (φ)1/2cosh φ
2γzγtsinh φ
2= cosh (φ)1/2exp γzγt
φ
2,
where
φ= tanh1vz
c.
The instantaneous position r
e(t)of the center of the rotating and translating sphere can be obtained by applying Rzt:
r
e(t)=Rztreg
Rzt.
With the denition of spinor R
R=RztRxy = cosh (φ)1
2exp γzγt
φ
2exp γxγy
ωet
2,
we can rewrite the instantaneous position in a compact form as
r
e(t)=R(re0+γtct)e
R.
The instantaneous coordinate of a generic point on the surface of the charged sphere can be obtained adding a
specicxed vector ro
rsurf(t)=r
e(t) + ro.
The module of vector rois equal to ro=αr
e, a value equal to the classical radius of the electron for non relativistic
speeds.
It is important to note that, according to Hestenes, the ZBW angular frequency is two times the De Broglie value
mec2/used in our model: “The diameter of the helix is the electron Compton wavelength 2λ0=2c
/ω0=
/mc[27].
The value 1.93079 ×1013 m of the “zitter-radius” of Hestenes’ electron model is conrmed in Eq. (33) of a more
recent work [7].
4. Electromagnetism, Mechanics and Lorentz force
The “pure electromagnetic” vector eAmay be interpreted as the momentum–energy Pof a particle with electric
charge e, momentum Pand energy U=Ptc:
P=eA,(58)
F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 91
P=γxPx+γyPy+γzPz+γt
U
c=P+γt
U
c.(59)
For a particle that moves with speed valong a direction zorthogonal to the ZBW rotation plane, the momentum Pcan
be decomposed in two vectors, one parallel and one orthogonal to v. The orthogonal component is a rotating vector,
that indicates the component of the momentum due to the angular frequency ωein the spatial plane xy orthogonal to z
P=P+P,(60)
where
|P|=ωe
c=meωere=mec,
[|P|=1
re=ωe=me]NU
.
The Pcomponent can be seen as the usual three components momentum of a particle with mass at rest me. For
simplicity of notation, from now, we will call Pthis component, so that
P2
=e2A2
=P2
U
c2
2
=P2+m2
ec2U
c2
2
.
The relativistic mass mcan be derived directly by the application of the Pythagorean theorem
m2c2=m2
ec2+P2=P2
+P2=m2
ec2+m2v2.
Consequently this electromagnetic four-momentum P, for electrons moving with uniform velocity, is a light-like
vector:
P2
=m2c2U
c2
2
= 0.
An electron that moves with velocity vchas an approximate momentum Pgiven by
P=eAP
v
c=mev,
and a variation of speed
a=dv
dtimplies a force f=dP
dt=edA
dt=mdv
dt.
Now recalling that the bivector part of (5) is
A=F
after multiplying both sides by the charge e, it becomes
92 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99
eA=eA=eF.(61)
By considering (58) and (59) this equation can be rewritten as
P+γt
U
c=eF,(62)
or, by means of (60), as
[(P+P)+γt
U
c]=eF.(63)
The term Pcan be carried out because the average value of P, in a scale time much larger than the ZBW
period, is zero:
P+γt
U
c=eF,(64)
P+γt
U
c=e
cEγt+eIBγt.
Equating only the components that contain bivectors with γtterms we obtain
P
tEU
γt+Uγt=eEγt
or
P
tEU
=eE− ∇U. (65)
In (65) (P/t)EU is the force acting on the charge edue both to the electric eld E(Coulomb force) and to the
gradient of the “potential energy” U. Instead, by equating only the components that contain pure spatial bivectors we
get
∇ ∧ P=eIBγt=eIγtB=eIB,(66)
where the term Iγt=γxγyγz=Iis the unitary volume of the three dimensional space. Left-multiplying both
sides of (66) by Igives
I∇ ∧ P=eB,(67)
which is equivalent to the two following equations in the ordinary algebra
×P=eB.(68)
F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 93
Table 3. Products v×(×PeB).
v×(×PeB)γx(Pz
yPy
zeBx)γy(Px
zPz
xeBy)γz(Py
xPx
yeBz)
γxvx0γz(Pz
txy evxBy)γy(Py
txz evxBz)
γyvyγz(Pz
tyx evyBx)0γx(Px
tyz evyBz)
γzvzγy(Py
tzx evzBx)γx(Px
tzy evzBy)0
As an example, the component of the above equation along the xaxis is
γxPz
yPy
z=γxeBx.
Now, by applying the cross product of the velocity vof charge eto both terms in (68) we obtain
v×(×PeB) = 0.(69)
The components of (69) are represented in Table 3 considering that
vi
Pj
i=i
t
Pj
i=Pj
t,
vj
Pj
i=j
t
Pj
i=j
i
Pj
t= 0 fori̸=j, where i, j {x, y, z }.
For these reasons (69) leads to the usual form of the force contribution due to the magnetic ux density eld B
P
tB
=ev×B.(70)
Finally, we get the whole force contribution by summing up the forces
dP
dt=P
tEU
+P
tB
given respectively by (65) and (70)
dP
dt =e(E+v×B)− ∇U. (71)
5. Energy, Momentum and Quanta Current
The nature of energy and momentum can be understood if we consider the quantum of action as a “physical object”
that moves in space–time. The Planck relation Wφ=ω=2π
/Tφtells us that photons with energy Wφcan transmit
quanta of actions with a quanta current Qc(number of quanta of actions per time unit)
94 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99
Qc=n
Tφ
=nWφ
2π.(72)
The same quanta current can be obtained in a time unit by a large number of low energy photons or by a small number
of high energy photons.
Calling WHthe energy of high energy photons and WLthe energy of low energy ones, and nHand nLtheir
number, we observe that the same Qccan be obtained with the same total energy following two different ways if
nHWH=nLWL:
n
Tφ
=nHWH
2π,(73)
n
Tφ
=nLWL
2π.(74)
In the rst case we have few “high speed” – high energy photons that guarantee the required current. In the latter
case the same result is obtained by many “low speed” – low energy photons. An information technology analogy can
be given if we consider a bus with few wires driven by a high frequency clock compared to a large bus with many
wires driven by a low frequency clock. The information per unit time is the same in both case. Now, following the
above considerations the equation Pφ=k= 2π/λφ, where k= 2π/λφis the wave number, should be viewed as
the direction of quanta current in space. For photons the four-momentum Pis a light-like vector
P2
= 0.(75)
In this case the momentum is a vector that gives the direction of quantum of action in space and the module of
momentum Pφis the energy Wφ:
P2
φW2
φ= 0NU .
5.1. ZBW and Heisenberg’s uncertainty principle
The concept of “measure” is strictly related to the quantum of action as the concept of information is related to the
binary digit (bit). In natural units the quantum of action is a dimensionless (i.e., scalar) value, as it is always the ratio
(measure) of two values of the same nature. For this reason the concepts of “energy”, “momentum”, “space” and
“time” cannot be separated and space and time can be measured, using natural units, in eV1.
The product of the momentum Pof the rotating charge and the radius rof the orbit in the proposed ZBW model is
always equal to :
P r =.(76)
This expressions points out that the intrinsic momentum of a particle conned in a spherical space of radius rcannot be
less than /r. Calling tr=T /2π=ω1the inverse of the ZBW angular frequency, and remembering that mc2=ω,
we can observe that
F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 95
mc2tr=ωtr==P r. (77)
This formula just states that the energy of a particle at rest (such as an electron) conned in a spherical space of radius
rcannot be less than /tr=c/r. Both (76) and (77) can be viewed as a particular reformulation of Heisenberg’s
uncertainty principle. From this point of view, energy can be seen as strictly related to the concept of quantum of
action density in space–time. We remember that in quantum mechanics the concept of “particle at rest” cannot be con-
sidered as realistic, because the assumed point-like model of elementary particles implies that, due to the Heisenberg’s
principle, the momentum is not determined as the precision in position tends to zero.
6. Some other Spinning Charge Models
In 1915 Alfred Lauck Parson published “A Magneton Theory of the Structure of the Atom” in the Smithsonian Mis-
cellaneous Collection, Pub 2371 [28], where he proposed a spinning ring model of the electron. Various forms of the
spinning charge model of electrons have been rediscovered by many authors. However, the incompatibility with the
most widely accepted interpretations of quantum mechanics prevented them from receiving proper attention.
According to Randell L. Mills the free electron is “is a spinning two-dimensional disk of charge. The mass and
current density increase towards the center, but the angular velocity is constant. It produces an angular momentum
vector perpendicular to the plane of the disk” [29]. As in our proposed model the intrinsic angular momentum of free
electron is but there is an important difference in charge distribution shape and speed. A constant angular velocity
for a at charge distribution implies that the charge speed is not always equal to the speed of light as strictly demanded
by our model [30]. Mills’ theory [31,32] “assumes physical laws apply on all scales including the atomic scale” in
agreement with Occam’s razor principle, is based on simple fundamental physical laws and is highly predictive.
Using geometric algebra and starting from Dirac theory, David Hestenes has proposed a ZBW model according to
which “the electron is a massless point particle executing circular motion in the rest system” and “with an intrinsic
orbital angular momentum or spin of xed magnitude s=/2” [1]. The phase of the probability amplitude wave
function is related to the ZBW rotation phase, a concept usually hidden in the traditional mathematical formalism
used in quantum mechanics based on complex numbers and matrices. However, we should remark that the concept of
point-like charge in quantum mechanics should be considered unrealistic. It violates Occam’s razor principle and may
be used only as a rst approximation. We remember also that in our model the value of intrinsic angular momentum
for a free electron is and that the “spin” is interpreted as the component of the angular momentum along an external
magnetic eld as in Stern–Gerlach experiment (see Section 3.2). Another interesting electron model has been proposed
by David L. Bergman [15,16]: according to this model the electron is a very thin, torus shaped, rotating charge
distribution with intrinsic angular momentum of the electron equal to its spin value s=/2.The torus radius has a
length R=/mc and half thickness r= 8Reπ/α, where αis the ne structure constant.
7. Electromagnetic Composite At Compton Scale
If the electron is a current loop whose radius is equal to the reduced electron Compton wavelength, it is reasonable
to assume the possibility of existence of “super chemical” structures of pico-metric (1pm = 1012 m) dimensions.
These dimensions are intermediate between nuclear (1fm = 1015 m) and atomic scale (1Å= 1010 m).
A simple ZBW model of the proton consists in a current loop generated by an elementary positive charge that
rotates at the speed of light along a circumference with a length equal to the proton Compton wavelength (1.32141×
1015 m) [33]. According to this model the proton is much smaller than the electron (re/rp=mp/me1836.153).
A hypothetically very simple structure formed by an electron with a proton at his center would have potential energy
96 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99
Figure 3. UDH protons distance.
of e2/re≈ −3.728 keV corresponding to a photon wavelength of λφ3.325 ×1010 m. This structure may be
created starting from atomic hydrogen or Rydberg State Hydrogen only in specic environments, as materials with high
free electron density and with lattice constants and energy levels allowing a resonant absorption of 3.7keV photons.
A high electron density can be obtained in “swimming electron layers” formed when a metal is heated in contact with
materials, such as SrO, with low work functions [34,35].
The hypothesis of existence of Compton-scale composites (CSC) has been experimentally conrmed by Holmlid
[2,3,36]. The inter-nuclear distance in Ultra-Dense Deuterium (UDD) of 2.3pm, found by Holmlid [2], seems
compatible with deuteron–electron (or proton–electron in Ultra-Dense Hydrogen(UDH)) structures where the ZBW
phases of adjacent electrons are correlated. Such distance may be obtained imposing, as a rst step, the condition that
the space–time distance dbetween adjacent electrons rotating charges is a light-like vector:
d2
=d2
c2δt2= 0,(78)
d2
=d2
δt2= 0NU ,
where dis the ordinary euclidean distance in space. This condition is satised if dis equal to electron Compton
wavelength (d=λc), δt=Tis the ZBW period and the phase difference between adjacent electrons is equal to π.
In this case from a direct application of the Pythagorean theorem we can nd the internuclear deuteron distance dias
shown in Fig. 3.
di=λc
ππ212.3×1012 m,(79)
Figure 4. UDH model.
F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 97
Figure 4 shows a hypothetical chain of these deuteron–electron pairs. We must remark that the hypothesis of
existence of exotic forms of hydrogen is not new and has been proposed in different ways by many authors (Mills [31],
Dufour [9], Mayer and Reitz [8,37], Krasnoholovets, Zabulonov and Zolkin [38] and many others). A very interesting
result has been obtained by Jan Naudts starting from the Klein–Gordon equation for the hydrogen atom. Naudts found
a low-lying eigenstate in witch “hydrogen” has a deep energy level E0≈ −mec2α≈ −3.728 keV and a radius
re=/mec3.9×1013 m(0.0039 Å)[39].
Indirect support for these hypotheses comes also from the numerous claims of observation of anomalous heat
generation in metal–hydrogen systems. We must remark that these hypothetical “Compton Scale Composites” should
be electrical neutral or negatively charged objects that cannot be stopped by the Coulomb barrier. For this reason they
may generate unusual nuclear reactions and transmutations, considering the different sizes, time-scales and energies
of this composites with respect to the particles (such as neutrons) normally used in nuclear experiments.
Mayer and Reitz, starting from a ZBW model of the electron, propose a three body system model at the Compton
scale, composed by a proton and two electrons [8]. F. Piantelli, in patent application WO 2012147045 “Method and
apparatus for generating energy by nuclear reactions of hydrogen adsorbed by orbital capture on a nanocrystalline
structure of a metal”, proposes an orbital capture of “H- ions” by nickel atoms in nano-clusters as a trigger for Low
Energy Nuclear Reactions [40]. The orbital capture of the negatively charged structures at pico-metric scale described
by Mayer and Reitz may be viewed as an alternative explanation to the capture of the much larger H- ions.
8. Conclusions
In this paper the authors want to underline that simplicity is an important and concrete value in scientic research.
Connections between very different concepts in physics can be evidenced if we use the language of geometric algebra,
recognizing also the fundamental role of the electromagnetic four-vector potential in physics.
The application of Occam’s Razor principle to Maxwell’s equations suggests, as a natural choice, a Zitterbewe-
gung interpretation of quantum mechanics, similar but not identical to the one proposed by D. Hestenes. According
to this framework, the electron structure consists of a massless charge distribution that rotates at the speed of light
along a circumference with a length equal to electron Compton wavelength. Following this interpretation the electron
mass–energy, expressed in natural units, is equal to the angular speed of the ZBW rotation and to the inverse of the
orbit radius. Inertia has a pure electromagnetic origin related to the vector potential generated by the ZBW current.
Moreover, in this framework the Heisenberg “uncertainty principle” derives from the relation between a particle ZBW
radius and its angular momentum. The proposed model supports the ideas of some authors [3,8] that the ZBW may
explain the existence of “super-chemical structures,“ such as ultra-dense deuterium, at pico-metric scale. A prelimi-
nary hypothesis on the structure of Holmlid’s UDD, in which the ZBW phase of adjacent electrons are synchronized,
has been presented demonstrating with good agreement Holmlid’s experimental results. Pico-chemistry reactions and
composites with intermediate energy values between nuclear and chemical ones can emerge as a key concept in un-
derstanding the origin of anomalous heat and the unusual nuclear reactions seen in many metal–hydrogen systems, as
already suggested by some researchers in the eld of condensed matter nuclear science.
“It is a delusion to think of electrons and elds as two physically different, independent entities. Since neither can
exist without the other, there is only one reality to be described, which happens to have two different aspects; and the
theory ought to recognize this from the outset instead of doing things twice!” – A. Einstein, cited in [41].
“In atomic theory, we have elds and we have particles. The elds and the particles are not two different things.
They are two ways of describing the same thing, two different points of view” – P.A.M. Dirac, cited in [42].
98 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99
Acknowledgements
Thanks to Salvatore Mercurio, former professor of physics at the North University of China (NUC), Taiyuan, Shanxi,
for interesting discussions on the nature of electric charge and on hypothesis of existence of “super-chemical” reactions.
Many thanks also to the reviewers for their interesting advice and benecial suggestions and to Jed Rothwell for the
English revision of the manuscript.
References
[1] D. Hestenes, Quantum mechanics from self-interaction, Found. Phys. 15(1) (1985) 63–87.
[2] S. Badiei and P.U. Andersson and L. Holmlid. High-energy Coulomb explosions in ultra-dense deuterium: time-of-ight-
mass spectrometry with variable energy and ight length, Int. J. Mass Spectrometry 282(1–2) (2009) 70–76.
[3] L. Holmlid and S. Olafsson, Spontaneous ejection of high-energy particles from ultra-dense deuterium D(0), Int. J. Hydrogen
Energy 40(33)(2015) 10559–10567.
[4] K. Huang, On the zitterbewegung of the Dirac electron, Amer. J. Phys.20(8) (1952) 479–484.
[5] B.G. Sidharth, Revisiting Zitterbewegung, Int. J. Theoret. Phys. 48 (2009) 497–506.
[6] D. Hestenes, Mysteries and insights of Dirac theory, Annales de la Fondation Louis de Broglie 28 (2003) 3.
[7] D. Hestenes, Hunting for Snarks in Quantum Mechanics, In P.M. Goggans and C.-Y. Chan (Eds.), American Institute of
Phys. Conf. Series, Vol. 1193, pp. 115–131, December 2009.
[8] F.J. Mayer and J.R. Reitz, Electromagnetic composites at the Compton scale, Int. J. Theoret. Phys.51(1) (2012) 322–330.
[9] J. Dufour, An introduction to the pico-chemistryworking hypothesis, J. Condensed Matter Nucl. Sci. 10 (2013) 40–45.
www.iscmns.org/CMNS/JCMNS-Vol10.pdf.
[10] Y. Aharonov and D. Bohm, Signicance of electromagnetic potentials in the quantum theory, Phy. Rev. 115 (1959) 485–491.
[11] D. Hestenes, Reforming the Mathematical Language of Physics, Oersted Medal Lecture 2002, 2002.
[12] D. Hestenes, Spacetime physics with geometric algebra, Amer. J. Phys. 71(3) (2003) 691–714.
[13] G. Bettini, Clifford algebra, 3- and 4-dimensional analytic functions with applications, Manuscripts of the Last Century.
viXra.org, Quantum Phys.:1–63, 2011. http://vixra.org/abs/1107.0060.
[14] D. Hestenes, Zitterbewegung modeling, Found. Phys. 23(3) (1993) 365–387.
[15] D.L. Bergman and J.P. Wesley, Spinning charged ring model of electron yielding anomalous magnetic moment, Galilean
Electrodynamics 1(5) (1990) 63–67.
[16] D.L. Bergman and C.W. Lucas, Credibility of common sense science, Found. Sci., Vol. 6, No 2 (2003) 1–17.
[17] Shau-Yu Lan, Pei-Chen Kuan, Brian Estey, Damon English, Justin M. Brown, Michael A. Hohensee, and Holger Müller. A
clock directly linking time to a particle’s mass. Science, January 2013.
http://science.sciencemag.org/content/early/2013/01/09/science.1230767.full.pdf.
[18] F. Santandrea and P. Cirilli, Unicazione elettromagnetica, concezione elettronica dello spazio, dellenergia e della materia,
2006. Atlante di numeri e lettere. http://www.atlantedinumerielettere.it/energia2006/labor.htm.
[19] R.A. Treumann, W. Baumjohann and A. Balogh, The strongest magnetic elds in the universe: how strong can they become?
Frontiers Phys. 2(2014) 59.
[20] Q.-H. Peng and H. Tong, The Physics of strong magnetic elds in neutron stars, Mon. Not. Roy. Astron. Soc. 378 (2007) 159.
[21] D. Dew-Hughes, The critical current of superconductors: a historical review, Low Temperature Phys.27(9) (2001) 713–722.
[22] J. Dolbeault, M.J. Esteban and M. Loss, Relativistic hydrogenic atoms in strong magnetic elds, Annales Henri Poincaré 8
(2007) 749–779.
[23] J. Dolbeault, M. J. Esteban and M. Loss, Characterization of the critical magnetic eld in the Dirac–Coulomb equation.
ArXiv e-prints, December 2007.
[24] O. Consa, Helical model of the electron, General Sci. J. (2014) 1–14.
http://gsjournal.net/Science-Journals/Research%20Papers/View/5523
[25] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics: Mainly Electromagnetism and Matter, Vol.
2, Basic Books, 2011.
[26] D. Hestenes, Real spinor elds, J. Math. Phys. 8(4) (1967) 798–808.
F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 99
[27] D. Hestenes, The Zitterbewegung interpretation of quantum mechanics, Found. Phys. 20(10) (1990) 1213–1232.
[28] A.L. Parson and Smithsonian Institution, A magneton theory of the structure of the atom (with two plates), Number v. 65 in
Publication (Smithsonian Institution), Smithsonian Institution, 1915.
[29] BlackLight Power, Periodic table of elements of the rst twenty-electron-atoms solved with the grand unied theory of
classical Physics. http://www.millsian.com/images/theory/Periodic-Table-Poster-light.pdf.
[30] A. Rathke, A critical analysis of the hydrino model, New J. Phys. 7(2005) 127.
[31] R.L. Mills, J.J. Farrell and W.R. Good, Unication of Spacetime, the Forces, Matter, and Energy, Science Press, Ephrata, PA
17522, 1992.
[32] R.L. Mills, The Grand Unied Theory of Classical Physics, Blacklight Power, 2008.
[33] D.L. Bergman, The real proton, Found. Sci. Vol. 3, No 4, (2000).
http://commonsensescience.org/pdf/articles/the_real_proton.pdf.
[34] F. Celani, E. Purchi, F. Santandrea, S. Fiorilla, A. Nuvoli, M. Nakamura, P. Cirilli, A. Spallone, B. Ortenzi, S. Pella, P.
Boccanera and L. Notargiacomo, Observation of macroscopic current and thermal anomalies, at high temperature, by hetero-
structures on thin and long Constantan wires under H2gas, Int. Conf. on Condensed Matter Nucl. Sci., ICCF-19, Padua, Italy,
13–17 April 2015.
[35] H. Hora, G.H. Miley, J.C. Kelly and F. Osman, Shrinking of hydrogen atoms in host metals by dielectric effects and inglis-
teller depression of ionization potentials, Proc. 9th Int. Conf. on Cold Fusion (ICCF9), pp. 1–6, 2002.
[36] L. Holmlid, Excitation levels in ultra-dense hydrogen p(-1) and d(-1) clusters: structure of spin-based rydberg matter, Int. J.
Mass Spectrometry 352(2013) 1–8.
[37] F.J. Mayer and J.R. Reitz, Thermal energy generation in the earth, Nonlinear Processes GeoPhys.21(2) (2014) 367–378.
http://www.nonlin-processes-geophys.net/21/367/2014/.
[38] V. Krasnoholovets, Y. Zabulonov and I. Zolkin, On the nuclear coupling of proton and electron, Universal J. Phys. Appl.
10(3) (2016) 90–103.
[39] J. Naudts, On the hydrino state of the relativistic hydrogen atom. ArXiv Phys. e-prints, July 2005.
[40] F. Piantelli, Method and apparatus for generating energy by nuclear reactions of hydrogen adsorbed by orbital capture on a
nanocrystalline structure of a metal, Nov. 2012. WO Patent App. PCT/IB2012/052,100.
[41] C. Mead, Collective Electrodynamics: Quantum Foundations of Electromagnetism, MIT Press, Cambridge, 2000.
[42] A. Babin and A. Figotin, Neoclassical theory of electromagnetic interactions: a single theory for macroscopic and micro-
scopic scales, Theoretical and Mathematical Physics, Springer, London, 2016.
... Therefore, an alternative realistic approach that fully addresses these very basic problems is indispensable. A possibility is given by a Zitterbewegung interpretation of quantum mechanics, according to which charged elementary particles can be modeled by a current ring generated by a massless charge distribution rotating at light speed along a circumference whose length is equal to particle Compton wavelength [3,4]. As a consequence, every elementary charge is always associated with a magnetic flux quantum and every charge is coupled to all other charges on its light cone by time-symmetric interactions [2]. ...
... These constants have a simple and clear interpretation if one accepts a particular electron model consisting of a current ring generated by a massless charge rotating at the speed of light along a circumference whose radius is equal to the electron reduced Compton wavelength, defined as r e = λc 2π ≈ 0.38616 × 10 −12 m [3][4][5][6]. According to the model described in [4] the charge is not a point-like entity, but it is distributed on a spherical surface whose radius is equal to the electron classical radius r c ≈ 2.8179 × 10 −15 m . ...
... These constants have a simple and clear interpretation if one accepts a particular electron model consisting of a current ring generated by a massless charge rotating at the speed of light along a circumference whose radius is equal to the electron reduced Compton wavelength, defined as r e = λc 2π ≈ 0.38616 × 10 −12 m [3][4][5][6]. According to the model described in [4] the charge is not a point-like entity, but it is distributed on a spherical surface whose radius is equal to the electron classical radius r c ≈ 2.8179 × 10 −15 m . In Eq. (4) ω e is the angular frequency of the rotating charge, r e is its orbit radius and T e its period. ...
Article
Full-text available
In this paper, a simple Zitterbewegung electron model, proposed in a previous work, is presented from a different perspective based on the principle of mass−frequency equivalence. A geometric−electromagnetic interpretation of mass, relativistic mass, De Broglie wavelength, Proca, Klein−Gordon, Dirac and Aharonov−Bohm equations in agreement with the model is proposed. A non-relativistic, Zitterbewegung interpretation of the 3.7 keV deep hydrogen level found by J. Naudts is presented. According to this perspective, ultra-dense hydrogen can be conceived as a coherent chain of bosonic electrons with protons or deuterons located in the center of their Zitterbewegung orbits. This approach suggests a possible role of ultra-dense hydrogen in some aneutronic and many-body low energy nuclear reactions.
... Therefore, an alternative realistic approach that fully addresses these very basic problems is indispensable. A possibility is given by a Zitterbewegung interpretation of quantum mechanics, according to which charged elementary particles can be modeled by a current ring generated by a massless charge distribution rotating at light speed along a circumference whose length is equal to particle Compton wavelength [3,4]. As a consequence, every elementary charge is always associated with a magnetic ux quantum and every charge is coupled to all other charges on its light cone by time-symmetric interactions [2]. ...
... These constants have a simple and clear interpretation if one accepts a particular electron model consisting of a current ring generated by a massless charge rotating at the speed of light along a circumference whose radius is equal to the electron reduced Compton wavelength, dened as r e = λc /2π ≈ 0.38616 · 10 −12 m [3,5,6,4]. According to the model described in [4] the charge is not a point-like entity, but it is distributed on a spherical surface whose radius is equal to the electron classical radius r c ≈ 2.8179 · 10 −15 m . ...
... These constants have a simple and clear interpretation if one accepts a particular electron model consisting of a current ring generated by a massless charge rotating at the speed of light along a circumference whose radius is equal to the electron reduced Compton wavelength, dened as r e = λc /2π ≈ 0.38616 · 10 −12 m [3,5,6,4]. According to the model described in [4] the charge is not a point-like entity, but it is distributed on a spherical surface whose radius is equal to the electron classical radius r c ≈ 2.8179 · 10 −15 m . In equation (4) ω e is the angular frequency of the rotating charge, r e is its orbit radius and T e its period. ...
Preprint
Full-text available
In this paper, a simple Zitterbewegung electron model, proposed in a previous work, is presented from a different perspective based on the principle of mass-frequency equivalence. A geometric-electromagnetic interpretation of mass, relativistic mass, De Broglie wavelength, Proca, Klein-Gordon, Dirac and Aharonov-Bohm equations in agreement with the model is proposed. A non-relativistic, Zitterbewegung interpretation of the 3.7 keV deep hydrogen level found by J. Naudts is presented. According to this perspective, ultra-dense hydrogen can be conceived as a coherent chain of bosonic electrons with protons or deuterons located in the center of their Zitterbewegung orbits. This approach suggests a possible role of ultra-dense hydrogen in some aneutronic and many-body low energy nuclear reactions.
... where d is the distance between plates and c is the light speed in vacuum. Maruani considers a Zitterbewegung [27,26,24,9] electron model where the reduced Compton wavelength is the electron diameter . In this case the plate area in (3) becomes A = π ( λe /4π) 2 and the attractive Casimir Force F C (d) between electrons can be computed and compared with the Coulomb repulsion force F e (d) : ...
... According to another Zitterbewegung electron model [9,16], the electron can be modeled by a current loop, with radius r e , generated by a charge distribution that rotates at the speed of light. This current loop is proposed as the origin of the electron's mass, inertia, angular momentum, spin and magnetic momentum. ...
... According to [9,11,32], the electron is associated with a magnetic ux Φ M = h /e equal to the ratio of the Planck constant h and the elementary charge e. Consequently, the possible role of a magnetic attraction in charge connement cannot be dismissed a priori. ...
Preprint
Full-text available
Abstract Some theoretical frameworks that explore the possible formation of dense exotic electron clusters in the E-Cat SK are presented. Some considerations on the probable role of Casimir, Aharonov-Bohm, and collective effects in the formation of such structures are proposed. A relativistic interaction Lagrangian, based on a pure electromagnetic electron model, that suggests the possible existence of very low entropy charge aggregates and that highlights the primary role of the electromagnetic potentials in these clusters is presented. The formation of these cluster may be associated to a localized Vacuum polarization generated by a rapid radial charge displacement. The formation of these dense electron clusters are introduced as a probable precursor for the formation of proton-electron aggregates at pico-metric scale, stressing the importance of evaluating the plausibility of special electron-nucleon interactions, as already suggested in [#GullstromRossi]. An observed isotopic dependence of a particular spectral line in the visible range of E-Cat plasma spectrum seems to confirm the presence of a specific proton-electron interaction at electron Compton wavelength scale.
... 14 Not shown. 15 500-700 °C. 16 The test coil comprises a 350 µm, 160 cm long wire. ...
...  Perhaps due to an enhancement of electron screening effects on nuclear fusion reactions and/or due to the formation of dense charge clusters and pico-metric structures [15] [16]. ...
Presentation
Full-text available
[Enriched version of the original presentation] This short presentation introduces an experimental design for the enhancement of the anomalous thermal phenomena (AHE) observed since 2011 in Constantan3 wires exposed to a deuterium or hydrogen atmosphere, and heated by direct current. In fact, the occurrence of AHE requires specific conditions such as deuterium/protium absorption in the wire, sufficiently high temperature, as well as presence of strong non-equilibrium conditions such as those induced by thermal gradients, variations of pressure, and electric/magnetic fields. Previous experiments provided a strong evidence for the role of a flux of active species through the wire or at the wire surface. Though various techniques to induce a flux were tested before, and have been instrumental for a phenomenological understanding of AHE occurrence, they could not provide a solution for a sustained and exploitable energy production. For instance, wires loaded with deuterium at 700 °C and 1 Bar, may show the occurrence of AHE when the pressure is slowly reduced to 1 mBar. During this process, anincrease of wires electric resistance is observed and corresponds to the out-gasing of deuterium or hydrogen; this creates a flux associated with AHE. Nonetheless when the out-gasing ceases, the phenomenon tends to vanish. Similarly, simple knots along the wires, can be used to create hot-spots and corresponding thermal gradients; this approach proved quite effective and AHE could exceed 40% with respect to the electric input to heat the wires. Notwithstanding, the method was affected by a cumbersome preparation of the wires and their frequent breaks. In 2016 a second non heated wire was positioned near the active hot Constantan and, at a sufficiently reduced pressure did reveal a thermionic emission of electrons from Constantan. The empiric correlation between this electron emission and AHE occurrence was initially puzzling, but soon lead to experiments where the electron emission was enhanced by mean of an external power supply to sustain an active voltage bias among Constantan and the second wire (anode). This approach also proved able to trigger AHE though not indefinitely. The next step was using an alternating current between the two wires, an approach which led to observe the occurrence of Paschen or dielectric barrier discharges (DBD) when voltage and pressure were in the appropriate range. Interestingly in presence of these discharges, we recorded high and long lasting AHE as observed by other authors before. An intense scrutiny of the data collected from the experiments mentioned above led us to design an updated setup which could take into account the learning of almost ten years of Constantan wires studies. This setup includes the implementation of pulsed power supply based on a previous concept used by some of the authors of this presentation in electrolytic experiments with palladium. This power supply configuration and overall circuitry, is capable of providing powerful negative pulses along the wire, and positive pulses among the wire and an iron tube anode on which the wire (insulated with a special glass fiber sheath) is coiled. The positive pulses among the Constantan and the tube are used in particular to trigger Paschen or DBD discharges, while the tube can also be used as support for a multilayer coating of nickel-copper and low work-function oxides described in previous works.
... • Perhaps due to an enhancement of electron screening effects on nuclear fusion reactions and/or due to the formation of dense charge clusters and pico-metric structures [9] [10]. ...
Presentation
Full-text available
This is a presentation held at the International LERN Workshop in memory of Dr. Mahadeva Srinivasan, Kanpur-India, on January 22-24, 2021. It is divided in two parts, the first describes a new experimental setup for enhancing LENR phenomena in Constantan wires and it is an extension of what presented already at the Workshop “Assisi Nel Vento 5” (ANV5), Assisi-Italy, December 18-19, 2020 [1]. Further, new, details on unconventional pulse excitation, with both new procedures and hardware are provided. In the second part instead, we summarize the findings on the association between LENR occurrence, thermionic phenomena, and the presence of low work-function materials (LWF). We also draw some parallelism with thermionic cathodes and industrial catalysts design. In fact, we believe that LENR studies may borrow extremely valuable information from the developments occurred in the past century in both fields. In that respect, we show how cermet thermionic cathodes, and composite catalysts, could be used as a base for the design of practical LENR devices.
Article
Full-text available
A new semiclassical model of the electron with helical solenoid geometry is presented. This new model is an extension of both the Parson Ring Model and the Hestenes Zitterbewegung Model. This model interprets the Zitterbewegung as a real motion that generates the electron’s rotation (spin) and its magnetic moment. In this new model, the g-factor appears as a consequence of the electron’s geometry while the quantum of magnetic flux and the quantum Hall resistance are obtained as model parameters. The Helical Solenoid Electron Model necessarily implies that the electron has a toroidal moment, a feature that is not predicted by Quantum Mechanics. The predicted toroidal moment can be tested experimentally to validate or discard this proposed model.
Article
Full-text available
L'Unificazione Elettromagnetica argomento di questo studio fornisce un modello atomico elettromagnetico utile alla comprensione delle reazioni Low Energy Nuclear Reaction (LENR) e della Propulsione Non Newtoniana Elettromagnetica (PNNE).Una concezione elettromagnetica di elettroni, protoni, nuclei come oscillatori/onde stazionarie giustifica l’apparente staticità della carica e della materia; il blocco coulombiano può essere visto semplicemente come una mancanza di sincronismo (non coerenza tra due oscillatori separati formanti i nuclei); ma non esclude anzi prevede una sintonizzazione / sincronizzazione a bassa energia dei nuclei rendendo quindi concepibile il superamento del blocco coulombiano stesso permettendo una fusione nucleare L.E.N.R. “morbida” o in risonanza esempio : da Ni58 x H1 → Cu59. Questa tesi è nata dopo molti anni di studi e approfondimento sulla costituzione della materia e costituisce una nuova sintesi di unificazione elettromagnetica alla base della quale c’è semplicemente lo spazio e le sue proprietà.Lo spazio viene visto come substrato base della totalità dell’Universo. Lo spazio è sostanzialmente un superconduttore dell’energia elettromagnetica ad esso inviata e può assumere forma traslante (onda)o stazionante (carica, materia).L’elettrone (carica) e il protone (massa) possono essere concepiti come risonanze in equilibrio dinamico con lo spazio, onde elettromagnetiche stazionarie, in perfetto accordo con la costante di Planck, con le equazioni di Maxwell e con la forza di Lorentz. Partendo dallo spazio e dalle sue proprietà caratteristiche elettromagnetiche epsilon0 e mu0, includendo il meccanismo di risonanza, questo lavoro vuole essere una nuova sintesi semplificante la QED.
Conference Paper
Full-text available
Since 2011, we introduced into the LENR Research field, the use of a Constantan alloy to absorb and adsorb proper amounts of H2 or D2 (concentrated and/or mixed with noble gases of low thermal conductivity) and to generate thermal anomalies even at low temperatures (>200°C wire temperature). Based on this idea, we developed a reactor with a core of sub-micrometric layered Constantan wires that produced measurable excess power and showed results with some reproducibility. During subsequent years, we modified this base configuration with the aim of improving both reproducibility and the Anomalous Heat Effect (AHE). We used fiberglass sheaths for ensuring electrical insulation and found out, by chance, that this material actually improves the performance of the reactor. In the most recent configuration, we studied the effects of adding Fe nanolayers to the Constantan wires and of including several small knots along their extension, actions that resulted in a larger excess power that grew with increasing wire temperature. Finally, we detected a new anomalous electric effect, comprising of the generation of a spontaneous voltage between the ends of a floating wire in the reactor, enhanced and stabilized by Fe presence. Lastly, we added some speculations/similarities regarding Rydberg matter, as developed by Leif Holmlid (Univ. Goteborg-SE) and Collaborators, based on some of our results.
Article
Full-text available
Magnetic fields in the universe are in general weak, of the order of µGauss only. However, in compact objects they assume extraordinarily large values. These are produced by gravitational collapse of massive magnetised objects. Clearly, fields in the massive progenitor are energetically limited by the available energy which can be fed into the generation of currents and magnetic fields. However, when collapsing down to small scales magnetic fields become superstrong exceeding any limits which can be reached in the laboratory. A brief review and discussion is given on the absolute limitation to the magnetic field strengths which can be obtained during such collapses.
Article
Full-text available
We show that a recently introduced class of electromagnetic composite particles can explain some discrepancies in observations involving heat and helium released from the earth. Energy release during the formation of the composites and subsequent nuclear reactions involving the composites are described that can quantitatively account for the discrepancies and are expected to have implications in other areas of geophysics, for example, a new picture of heat production and volcanism in the earth is presented.
Article
We study both experimentally and theoretically the creation of a new physical entity, a particle in which the proton and electron form a stable pair with a tiny size typical for a nucleon. A new theoretical approach to study atomic, sub atomic and nuclear systems is suggested. In the framework of this new approach, which takes into account a submicroscopic concept of physics, we discuss similar experimental results of other researchers dealing with low energy nuclear reactions in a solid, plasma, sonofusion and the electrostatic field generated by piezocrystals. It is shown that the formation of sub atomic particles, which we name subatoms, involves an inerton cloud of an atom from the environment. The inerton cloud, as a carrier of mass, is absorbed by the electron and proton, which strongly couples these two particles in a new stable entity - the subhydrogen. Besides, we have generated a subhelium and argue the existence of subdeuterium. In addition to these subatoms there exist also nuclear pairs formed by a subatom with proton, deuteron and neutron.
Article
The nuclear signatures that can be expected when contacting hydrogen with nickel were derived from thermal results recently obtained (Rossi energy amplifier), using the type of reaction paths proposed as the explanation of the energy produced. The consequences of either proton or neutron capture have been studied. It was shown that these consequences are not in line with the experimental observations. A novel tentative explanation is thus described. Should this explanation be true? It is proposed to call pico-chemistry the novel field thus opened.
Article
High-energy particles are detected from spontaneous processes in an ultra-dense deuterium D(0) layer. Intense distributions of such penetrating particles are observed using energy spectroscopy and glass converters. Laser-induced emission of neutral particles with time-of-flight energies of 1–30 MeV u−1 was previously reported in the same system. Both spontaneous line-spectra and a spontaneous broad energy distribution similar to a beta-decay distribution are observed. The broad distribution is concluded to be due to nuclear particles, giving straight-line Kurie-like plots. It is observed even at a distance of 3 m in air and has a total rate of 107–1010 s−1. If spontaneous nuclear fusion or other nuclear processes take place in D(0), it may give rise to the high-energy particle signal. Low energy nuclear reactions (LENR) and so called cold fusion may also give rise to such particles.
Article
Laser-induced Coulomb explosions in ultra-dense hydrogen clusters prove that the interatomic distance in such clusters is a few picometers, since the well-defined kinetic energy release (Coulomb repulsion energy) is many hundred eV. In the best characterized case which is ultra-dense deuterium D(-1) or d(-1), the kinetic energy given to the fragments in a cluster is normally 630 eV. This implies a D-D distance of 2.3 +/- 0.1 pm (2.15 +/- 0.02 pm measured in D-4 clusters). A description based on theoretical work by J.E. Hirsch using only the electron spins predicts that several excitation levels of spin quantum numbers exist in these quantum materials. The agreement with the experimental D-D distance is excellent at 2.23 pm for s=2. Theory is now verified by experimentally detecting states with spin values s=1 and s=3. The D-D distance is only 0.56 pm in the lowest excitation level s=1. The previously suggested "inverted" structure of D(-1) is obsolete since the new theory gives a better explanation of the accumulated experimental results.