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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 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.

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)

A�Electromagnetic four-potential V s m−1eV

A△Electromag. vector potential V s m−1eV

GElectromagnetic ﬁeld V s m−2eV2

FElectromagnetic ﬁeld bivector V s m−2eV2

BFlux density ﬁeld V s m−2=T eV2

EElectric ﬁeld V m−1eV2

SScalar ﬁeld V s m−2eV2

J�eFour-current density ﬁeld A m−2eV3

J△Current density ﬁeld A m−2eV3

v�Four-velocity vector m s−11

v△Velocity vector m s−11

ρCharge density A s m−3=Cm−3eV3

x, y, z Space coordinates m (1.97327 ×10−7m≈1 eV−1) eV−1

tTime variable s (6.5821220×10−16 s≈1 eV−1) eV−1

cLight speed in vacuum 2.99792458 ×108m s−11

�Reduced Planck constant 1.054571726 ×10−34 J s 1

µ0Permeability of vacuum 4π×10−7V s A−1m−14π

ϵ0Dielectric constant of vacuum 8.854187817 ×10−12 C (V m)−11

/4π

eElectron charge 1.602176565 ×10−19 A s 0.085424546

meElectron mass at rest 9.109384 ×10−31 kg 0.510998946 ×106eV

λcElectron Compton wavelength 2.4263102389 ×10−12 m1.229588259×10−5eV−1

P�Energy-momentum four-vector kg m s−1eV

P△Momentum vector kg m s−1eV

U, W Energy J = kg m2s−2eV

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 deﬁnes 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 deﬁned space-like plane and not in a generic undeﬁned “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 ﬁeld 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 ﬁelds 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 ﬁeld; 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, ﬁnally,

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 m−1.A�is a harmonic function [13] that can be seen as the unique source of

all concepts-entities in Maxwell’s equations. Using the following deﬁnition 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=∇·A△−1

c

∂At

∂t(6)

is the scalar ﬁeld,

F=1

cEγt+IBγt=1

c(E+IcB)γt(7)

the electromagnetic ﬁeld 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.

∂A�γxAxγyAyγzAzγtAt

γx∂

∂xS1Bz1−By1

1

cEx1

γy∂

∂yBz2S2Bx1

1

cEy1

γz∂

∂z−By2Bx2S3

1

cEz1

γt1

c

∂

∂t

1

cEx2

1

cEy2

1

cEz2S4

The electromagnetic ﬁeld Gcan be expressed in the following compact form

G(x, y, z, t)=∇·A△−1

c

∂At

∂t+∇Atγt−1

c

∂A△

∂tγt+I∇×A△γt,(9)

and the expression

∂G=∂2A�= 0,(10)

represents the four Maxwell’s equations.

By applying, now, the ∂operator to the scalar ﬁeld 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=J�e,(11)

where J�e=γxJex +γyJey +γzJez −γtcρ=J△−γtcρ=ρ(v−γtc)is the four-current vector and v�=

γxvx+γyvy+γzvz−γtc=v−γtcis a four-velocity vector. The ∂operator applied to the four-current gives the

charge–current conservation law

1

µ0

∂·(∂S) = ∂·J�e=∂Jex

∂x+∂Jey

∂y+∂Jez

∂z+∂ρ

∂t= 0,(12)

which can be written alternatively as

∂·(∂S) = ∂2S=∂2S

∂x2+∂2S

∂y2+∂2S

∂z2−1

c2

∂2S

∂t2=∇2S−1

c2

∂2S

∂t2= 0.(13)

The charge is related to the scalar ﬁeld according to

1

c

∂S

∂t=µ0Jet =−µ0c∂q

∂x∂y∂z=−µ0cρ,(14)

so that, by applying the time derivative to (13) and remembering (14), the wave equation of the charge density ﬁeld

ρ(x, y, z, t)can be deduced:

∂

∂t(∂2S)=∂2∂S

∂t=∂2(−µ0c2ρ)=−µ0c2∂2ρ= 0,(15)

whose last equality gives

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 81

∂2ρ=∇2ρ−1

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 ﬁeld 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

2π≈3.861593 ×10−13 m,(17)

ωe=c

re= 2πc

λc≈7.763440 ×1020 rad s−1,(18)

T=2π

ωe=2πre

c≈8.093300 ×10−21 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

cλc≈9.109383 ×10−31 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

re≈0.511 ×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

2π≈19.796331 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�

2me≈9.274010 ×10−24 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:

Ω=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 Ω=�we can compute Aas function of Ie

Ω=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

e≈1.704509 ×10−3V s m−1.(25)

From (25) it is possible to derive the Fine Structure Constant (FSC)

α=µ0

4π

ce2

�=µ0

4π

eωe

A≈7.297352 ×10−3.(26)

Using natural units we get these simple relations:

A= 2πα−1IeNU ,

eA =ωe=r−1

e=me=PtNU ,

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99 83

where α−1, the inverse of the FSC, is a pure number and eis the elementary charge expressed in natural units

α−1=e−2≈137.035989NU .

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 ﬂux density ﬁeld BE, as in the Stern–Gerlach experiment. We can interpret

the spin value ±�/2as the component of the intrinsic angular momentum Ω=�aligned with the external ﬂux density

ﬁeld 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

dΩ=Ω⊥dϑp=Ωsin π

3dϑp,

where Ω⊥is the component of the intrinsic angular momentum orthogonal to BEand, therefore, it is possible to write

τ=dΩ

dt=Ωsin π

3dϑp

dt.(28)

By equating (27) and (28) we get

BEµB=Ωωp,

from which it is possible to determine the precession angular frequency

ωp=BEµB

Ω=BEµB

�.(29)

3.3. Value of the vector potential, cyclotron resonance and ﬂux density ﬁeld

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 ﬂux density ﬁeld Bgenerated by

the current Ie

ωe=eB

me=eBc2

�ωe,

it is possible to deduce the magnetic ﬂux density produced by the electron

84 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 76–99

Be=�ω2

e

ec2≈4.414004 ×109V s m−2.(30)

This very high ﬂux 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 ﬂux density at the center of the electron orbit by the following expression derived

from the Biot–Savart law

B0=µ0

2

Ie

re≈32.210548 ×106Vsm−2.(31)

Considering that

dA=Adϑ⇒dA

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=eAωe≈0.212014 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 ﬂux quantization and radius of the elementary charge

Once we obtaine the expression for the vector potential it is possible to determine the magnetic ﬂux produced by the

rotating elementary charge by applying the circulation of the vector potential A:

ϕe=Iλc

Adλ=2π

0

�

ereredϑ= 2π�

e=h

e≈4.135667 ×10−15 V s,(34)

i.e., the magnetic ﬂux crossing the surface described by the charge trajectory is quantized (ﬂux 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 ﬂux 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

ﬁeld produced by the spinning charge

Wm=1

2ϕeIe=1

22π�

e

ec

2πre=�c

2re≈4.093553 ×10−14 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

4πϵ0�c≈2.817940 ×10−15 m,(39)

whose value is coincident with the classical electron radius [25]. The upper equation states that the rotating charge

must have a ﬁnite 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 ﬁnite 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

=4πϵ0�c

e2rere=4πϵ0�c

e2=α−1≈137.035999.(40)

The expression of the ratio ϕe/T has in SI the dimension of a voltage:

Ve=ϕe

T=h

e

c

2πre=�c

ere≈5.109989 ×105V,(41)

where Tis deﬁned by (19). Now, dividing the above voltage by the current generated by the rotating charge expressed

by means of (22), we ﬁnd 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�

e2≈25812.807 Ω.(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

2α≈25812.807 Ω.(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= 4π2�re

e2c≈2.089108 ×10−16 Ωs,(44)

Ce=e

φe= 4πϵ0r0≈3.135381 ×10−25 F (45)

and

fe=1

√LeCe≈1.235590 ×1020 Hz.(46)

3.5. Electron kinematics

The ﬁnite 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

ﬁrst term of the interaction part of the Lagrangian density

Lint1=J△A△=JA =Ie

πr2

0

mec

e≈1.352604 ×1027 J m−3.(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

0≈8.187106 ×10−14 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 m−3,(50)

Wtot =V

|−ρφe|dV=e22π2rer2

0

8π3ϵ0(rer0)2=e2

4πϵ0re≈8.187106 ×10−14 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 ﬁrst 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 v⊥the component

of the velocity of the rotating charge in the γxγyplane we can ﬁnd 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=re1−v2

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

1−v2

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✁

∂ψ−mψ= 0 (55)

becomes the Hestenes–Dirac equation [26]

Table 2. Parameters of the Zitterbewegung model.

Item Symbol Value (SI) Unit (SI)

Charge e1.602176565 ×10−19 C=As

ZBW orbit radius re=λc/2π3.861593 ×10−13 m

Intrinsic angular momentum Ω=�=h

/2π1.054571726 ×10−34 J s

Spin1

/20.527285863 ×10−34 J s

Angular speed ωe7.763440 ×1020 rad s−1

Mass me9.109384 ×10−31 kg

Current Ie19.796331 A

Magnetic moment (Bohr magneton) µB9.274010 ×10−24 A m2

Vector potential A1.704509 ×10−3Vsm−1

Magnetic ﬂux density Be4.414004 ×109Vsm−2

Magnetic ﬂux ϕe=h

/e4.135667 ×10−15 V s

Magnetic energy Wm4.093553 ×10−14 J

Electrostatic energy We4.093553 ×10−14 J

Electron energy at rest Wtot =mec28.187106 ×10−14 J

Charge radius r02.817940 ×10−15 m

Inverse of the FSC α−1=re/r0137.035999 1

Von Klitzing constant RK=h

/e2=µ0c

/2α25812.807 Ω

Inductance Le=4π2re/e2c2.089108 ×10−16 Ωs

Capacitance Ce= 4πϵ0r03.135381 ×10−25 F

1-st part of Lint JA 1.352604 ×1027 J m−3

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 ﬁeld 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 ﬁeld. 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 re�0is the

coordinate of the center of the charged sphere at t=t0,rethe Compton radius we have

re�0=γxre+γtct0,

re�0=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 (re�0+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

2−γzγtsinh φ

2= cosh (φ)−1/2exp −γzγt

φ

2,

where

φ= tanh−1vz

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)=Rztre�g

Rzt.

With the deﬁnition 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(re�0+γtct)e

R.

The instantaneous coordinate of a generic point on the surface of the charged sphere can be obtained adding a

speciﬁcﬁxed vector ro△

rsurf�(t)=r′

e�(t) + ro△.

The module of vector ro△is 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 ×10−13 m of the “zitter-radius” of Hestenes’ electron model is conﬁrmed in Eq. (33) of a more

recent work [7].

4. Electromagnetism, Mechanics and Lorentz force

The “pure electromagnetic” vector eA�may be interpreted as the momentum–energy P�of a particle with electric

charge e, momentum P△and 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 P△can

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 P∥component 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

ec2−U

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

�=m2c2−U

c2

2

= 0.

An electron that moves with velocity v≪chas an approximate momentum Pgiven by

P=eA∥≃P⊥

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

e∂∧A�=∂∧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 ∂∧P⊥can 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 ﬁeld 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=eI△B,(66)

where the term −Iγt=γxγyγz=I△is the unitary volume of the three dimensional space. Left-multiplying both

sides of (66) by I△gives

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×(∇×P−eB).

v×(∇×P−eB)γx(∂Pz

∂y−∂Py

∂z−eBx)γy(∂Px

∂z−∂Pz

∂x−eBy)γz(∂Py

∂x−∂Px

∂y−eBz)

γxvx0γz(−∂Pz

∂txy −evxBy)γy(∂Py

∂txz −evxBz)

γyvy−γz(∂Pz

∂tyx −evyBx)0γx(−∂Px

∂tyz −evyBz)

γzvz−γy(−∂Py

∂tzx −evzBx)−γx(∂Px

∂tzy −evzBy)0

As an example, the component of the above equation along the xaxis is

γx∂Pz

∂y−∂Py

∂z=γxeBx.

Now, by applying the cross product of the velocity vof charge eto both terms in (68) we obtain

v×(∇×P−eB) = 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 ﬂux density ﬁeld 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 ﬁrst 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 P�is 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

φ−W2

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 eV−1.

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 conﬁned in a spherical space of radius rcannot be

less than �/r. Calling tr=T /2π=ω−1the 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) conﬁned 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 ﬂat 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 ﬁxed 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 ﬁrst 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 ﬁeld 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= 8Re−π/α, where αis the ﬁne 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 = 10−12 m) dimensions.

These dimensions are intermediate between nuclear (1fm = 10−15 m) and atomic scale (1Å= 10−10 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×

10−15 m) [33]. According to this model the proton is much smaller than the electron (re/rp=mp/me≈1836.153).

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 λφ≈3.325 ×10−10 m. This structure may be

created starting from atomic hydrogen or Rydberg State Hydrogen only in speciﬁc 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 conﬁrmed 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 ﬁrst step, the condition that

the space–time distance d�between adjacent electrons rotating charges is a light-like vector:

d2

�=d2

△−c2δt2= 0,(78)

d2

�=d2

△−δt2= 0NU ,

where d△is the ordinary euclidean distance in space. This condition is satisﬁed if d△is 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 ﬁnd the internuclear deuteron distance dias

shown in Fig. 3.

di=λc

ππ2−1≈2.3×10−12 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=�/mec≈3.9×10−13 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 scientiﬁc 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 ﬁeld of condensed matter nuclear science.

“It is a delusion to think of electrons and ﬁelds 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 ﬁelds and we have particles. The ﬁelds 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 beneﬁcial suggestions and to Jed Rothwell for the

English revision of the manuscript.

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