Maxwell's Equations and Occam's Razor

Abstract

In this paper a straightforward application of Occam's razor principle to Maxwell's equation shows that only one entity, the electromagnetic four-potential, is at the origin of a plurality of concepts and entities in physics. The application of the so called "Lorenz gauge" in Maxwell's equations denies the status of real physical entity to a scalar field that has a gradient in space-time with clear physical meaning: the four-current density field. The mathematical formalism of space-time Clifford algebra is introduced and then used to encode Maxwell's equations starting only from the electromagnetic four-potential. This approach suggests a particular Zitterbewegung (ZBW) model for charged elementary particles. .

Full text

J. Condensed Matter Nucl. Sci. 25 (2017) 100–128
Research Article
Maxwell’s Equations 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
In this paper a straightforward application of Occam’s razor principle to Maxwell’s equation shows that only one entity, the electro-
magnetic four-potential, is at the origin of a plurality of concepts and entities in physics. The application of the so called “Lorenz
gauge” in Maxwell’s equations denies the status of real physical entity to a scalar field that has a gradient in space-time with clear
physical meaning: the four-current density field. The mathematical formalism of space-time Clifford algebra is introduced and
then used to encode Maxwell’s equations starting only from the electromagnetic four-potential. This approach suggests a particular
Zitterbewegung (ZBW) model for charged elementary particles.
c
⃝2017 ISCMNS. All rights reserved. ISSN 2227-3123
Keywords: Coulomb gauge, Clifford algebra, Electric charge, Electron structure, Elementary particles, Maxwell’s equations,
Lorenz gauge, Occam’s razor, Space–time algebra, Vector potential, Zitterbewegung
Nomenclature (see p. 101)
1. Introduction
Science is the creation and validation of models of abstract concepts and experimental data. For this reason it is
important to examine the rules used to evaluate the quality of a model. Occam’s razor principle emphasizes the
simplicity and conciseness of the model: among different models that fit experimental data, the simplest one must
∗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.
c
⃝2017 ISCMNS. All rights reserved. ISSN 2227-3123

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 101
Nomenclature
Symbol Name SI units Natural units (NU)
A�Electromagnetic four-potential V s m−1eV
r�Four-position vector m eV−1
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 A m−2eV3
v�Four-velocity vector m s−11
A′Electromagnetic eight-potential V s m−1eV
PPseudoscalar field V s m−2eV2
J�mMagnetic four-current density A s m−3eV3
ρElectric charge density A s m−3=C m−3eV3
ρmMagnetic charge density A m−2eV3
x, y, z Space coordinates m (1) eV−1
tTime variable s (2) eV−1
cLight speed in vacuum 2.99792458 ×108m s−11
µ0Permeability of vacuum 4π×10−7V s A−1m−14π
ϵ0Dielectric constant of vacuum 8.854187817 ×10−12 A s (V m)−11
/4π
P�Electromag. four-momentum kg m s−1eV
SGeneralized Poynting vector W m−2eV4
wSpecific energy J m−3eV4
(1) 1.9732705 ×10−7m≈1eV−1;
(2) 6.5821220 ×10−16 s≈1eV−1.
be preferred, i.e. the model that does not introduce concepts or entities that are not strictly necessary. The following
sentences in Latin briefly illustrate this principle:
Pluralitas non est ponenda sine necessitate.
Frustra fit per plura quod potest fieri per pauciora.
Entia non sunt multiplicanda praeter necessitatem.
[1], which can be translated respectively as “plurality should not be posited without necessity”, “it is futile to do with
more things that which can be done with fewer” and “entities must not be multiplied beyond necessity”.
According to this principle, the quality of a model can be measured by means of two fundamental parameters:
(1) Good agreement of a model’s predictions with experimental data and/or with other expected results.
(2) The simplicity of a model, a value that is inversely related to the amount of information, concepts, entities,
exceptions, postulates, parameters and variables used by the model itself.

102 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128
These rules are universal ones and can be applied in many contexts [2]. From this point of view, the intuitive and
simple framework of Clifford algebra is a natural choice.
In this paper, we introduce and use the space–time Clifford algebra, showing that only one fundamental physical
entity is sufficient to describe the origin of electromagnetic fields and charges, i.e. the electromagnetic four-potential.
The vector potential should not be viewed only as a mathematical tool 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 [3]. Actually, many papers deal with the application of
geometric algebra to Maxwell’s equation (see [4–9] and many others), but few of them deal with the concept of scalar
field. Among the most interesting works we can find a paper by Bettini [5], two papers written by van Vlaenderen
[10,11] and two papers of Hively [12,13].
In this paper we propose a reinterpretation of Maxwell’s equations which does not use any gauge: the unique
constraint is that the electromagnetic four-potential must be represented by a harmonic function, as proposed by Bettini
[5]. This fact gives rise to an electromagnetic field composed not only of the classical electric and magnetic flux density
fields, but also by a scalar field. The scalar field will be here investigated and it will be shown that its existence produces
many interesting implications and consequences on the essence of electrical charges and currents. A brief and simple
but essential introduction on the main fundamental properties of Clifford algebra is given preliminarily in this paper in
order to encourage a particular interpretation of Maxwell’s equations at the picometric scale.
This paper is composed of the following parts: Section 2 is a short introduction to Clifford algebra and its fun-
damental properties; Section 3 illustrates how Maxwell’s equations can be derived from a four dimensional vector
potential without using the Lorenz gauge; Section 4 deals with the main properties of the electromagnetic field, the
derivation of Maxwell’s equations from the Lagrangian density, the Lorentz force, the generalized Poynting vector, the
symmetrical Maxwell’s equations and, finally, in Section 5 some essential points are summarized.
2. The Language of Scientific Knowledge
Scientific knowledge is expressed mathematically, but the importance of the optimal choice of the appropriate mathe-
matical language is often underestimated [4–6,14]. The geometric algebra (Clifford algebra) formalism, according to
Occam’s razor principle, is by far the best choice for modern physics. Clifford algebra provides a simple and unify-
ing mathematical language for coding geometric entities and operations [8,9,15]. It integrates different mathematical
concepts highlighting geometrical meanings that are often hidden in the ordinary algebra.
A particular Clifford Clp,q algebra is defined in a space with n=p+qdimensions with an orthonormal base of
nunitary vectors. The first pvectors of this base have positive squares, whereas the remaining ones have negative
squares, as shown by the following equations:
γ2
i= 1 with 1≤i≤p, (1)
γ2
i=−1with p+ 1 ≤i≤p+q, (2)
γiγj=−γjγiwith i̸=j, (3)
where γiare the unitary orthogonal vectors.
The geometrical product γiγjrepresents a “segment” of a unitary “area” of undefined shape in the plane identified
by unitary vectors γiand γj. The product γ1γ2...γnrepresents a unitary, n-dimensional volume segment identified by

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 103
Table 1. Blades of space–time algebra (Cl3,1).
Blade Bit mask Grade hex.
10000 0 (scalar) 0
γx0001 1 (vector) 1
γy0010 1 (vector) 2
γxγy=γxy 0011 2 (bivector) 3
γz0100 1 (vector) 4
γxγz=γxz 0101 2 (bivector) 5
γyγz=γyz 0110 2 (bivector) 6
γxγyγz=γxyz =I△0111 3 (pseudovector) 7
γt1000 1 (vector) 8
γxγt=γxt 1001 2 (bivector) 9
γyγt=γyt 1010 2 (bivector) A
γxγyγt=γxyt 1011 3 (pseudovector) B
γzγt=γzt 1100 2 (bivector) C
γxγzγt=γxzt 1101 3 (pseudovector) D
γyγzγt=γyzt 1110 3 (pseudovector) E
γxγyγzγt=γxyzt =I1111 4 (pseudoscalar) F
the unitary vectors γ1,γ2, . . . , γn. In the n-dimensional space no more than 2nelementary distinct “components”
exist. Each of these entities corresponds to a particular subset of the orthonormal base vectors. The “grade” of these
entities (called blades) is equal to the number of base vectors which are present within the subset. The blade of grade
zero (empty set) is the dimensionless scalar unit. The number of blades of kth-grade in a n-dimensional space is equal
to the binomial coefficient
Nk=(n
k)=n!
k! (n−k)!.(4)
Using Clifford algebra there are two possible choices for the metric of space–time coordinates, namely Cl1,3and
Cl3,1. For Cl1,3(signature “+− − −”) we have
γ2
t=−γ2
x=−γ2
y=−γ2
z= 1,
whereas for Cl3,1(signature “+ + + −”)
γ2
x=γ2
y=γ2
z=−γ2
t= 1.
In Cl3,1algebra, which is used in this work, the spatial coordinates can be viewed as the familiar Cartesian coor-
dinates of the Euclidean space. The 24components of the C l3,1space–time Clifford algebra are listed in Table 1.
Each blade is associated to a real number, the blade value. The expression aγiγj, in which ais a real scalar,
represents an area aof an undefined shape in the plane identified by vectors γiand γj.
Summing can be carried out only between coefficients of identical blades. If used for different blades the sum must
be reinterpreted as a composition, in analogy with the concept of sum between real and imaginary parts of a complex
number.
A multi-vector is a generic composition of one or more blades. Within an n-dimensional space a multi-vector is
composed by no more than 2nblades. The geometrical product between two blades gives a blade which is obtained
from the application of the bitwise exclusive OR operation between the bit mask of blades to be multiplied. The value

104 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128
of the resultant blade is equal to the product of the values of the operands multiplied by a sign which is a function of
the two blades. For C l3,1algebra the sign can be easily determined by means of Table 2, identifying the blades to be
multiplied through their hexadecimal “label” taken from Table 1. In general, this product is not commutative. The sign
table is obtained by the direct application of Eqs. (1)–(3).
There are other types of products in Clifford algebra, two of them are the wedge (symbol ∧) and the scalar product
(symbol ·). The result is computed following the same rules of geometric product, but is zero in some cases:
(1) the wedge product is always zero if the intersection between the set of base vectors of the first operand blade
and the set of base vectors of the second operand blade is not empty;
(2) the scalar product is always zero if the blade of the first operand is different from that of the second operand.
The geometric product of two vectors in Clifford algebra can be decomposed in a scalar product and a wedge product
according to the relation
uv =u·v+u∧v.(5)
It is important to note that the space–time algebra of the four γivectors is isomorphic to the algebra of Majorana
matrices. The Majorana matrices are the Dirac gamma matrices times the imaginary unit.
Example 2.1. Some examples of the application of products are here reported referring to Cl3,1:
γxγyγxγy=−γxγxγyγy=−1,
γxγy·γzγt= 0,
Table 2. Signs of the geometrical product in the space–time algebra Cl3,1.
I=γxyzt =γxγyγzγt,I△=γxyz =γxγyγz.
Grade 0 1 1 2 1 2 2 3 1 2 2 3 2 3 3 4
hex. 0 1 2 3 4 5 6 7 8 9 A B C D E F
Label 1γxγyγxy γzγxz γyz I△γtγxt γyt γxyt γz t γxzt γyzt I
1+ + + + + + + + + + + + + + + +
γx+ + + + + + + + + + + + + + + +
γy+ – + – + – + – + – + – + – + –
γxy + – + – + – + – + – + – + – + –
γz+ – – + + – – + + – – + + – – +
γxz + – – + + – – + + – – + + – – +
γyz + + – – + + – – + + – – + + – –
I△+ + – – + + – – + + – – + + – –
γt+ – – + – + + – – + + – + – – +
γxt + – – + – + + – – + + – + – – +
γyt + + – – – – + + – – + + + + – –
γxyt + + – – – – + + – – + + + + – –
γzt + + + + – – – – – – – – + + + +
γxzt + + + + – – – – – – – – + + + +
γyzt + – + – – + – + – + – + + – + –
I+ – + – – + – + – + – + + – + –

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 105
aγxγy∧bγz=abγxγyγz,
γxγy∧γy= 0,
γxγy∧γzγt=γxγyγzγt,
γx·γy= 0,
(γx+γt)2= (γx+γt)(γx+γt) = γ2
x+γxγt+γtγx+γ2
t= 1 + γxγt−γxγt−1 = 0
(example of light-like vector, the square is 0),
(γx+γy)2= (γx+γy)(γx+γy) = γ2
x+γxγy+γyγx+γ2
y=1+γxγy−γxγy+ 1 = 2
(example of space-like vector, the square is >0),
(aγt)2=−a2
(example of time-like vector, the square is <0),
(aγx+bγy)2= (aγx+bγy)(aγx+bγy) = a2γ2
x+abγxγy+baγyγx+b2γ2
y=a2+abγxγy−abγxγy+b2=a2+b2
(always a space-like vector),
(aγx+bγt)2= (aγx+bγt)(aγx+bγt) = a2γ2
x+abγxγt+baγtγx+b2γ2
t=a2+abγxγt−abγxγt−b2=a2−b2
(light-like if a=b, time-like if a < b, space-like if a > b).
In these examples aand bare generic real scalars.
2.1. Reflection and rotation of vectors
In order to perform a mirror reflection of a vector with respect to a plane, the following formula holds in Clifford
algebra:
a′=−mam,(6)
where mis the unitary vector orthogonal to surface α, as shown in Fig. 1. As a matter of fact, if
a=a⊥+a∥,

106 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128
Figure 1. Reflection of a vector with respect to plane α.
where a⊥and a∥are the orthogonal and the parallel components of vector arespectively, then
a′=−ma⊥+a∥m=−ma⊥m−ma∥m
=a⊥m2−a∥m2=a⊥−a∥.
This operation is justified by the fact that the product between parallel vectors commutes, i.e.
ma∥=a∥m,
whereas the product between orthogonal vectors anti-commutes, i.e.
ma⊥=−a⊥m.
If vector a′in now reflected again with respect unitary vector nrotated with respect to mby an angle θ/2we obtain
a′′ =na′n=nmamn.(7)
Vector a′′ is rotated, with respect to vector a, by an angle θon the common plane of the two-vector mand nas shown
in Fig. 2.
The rotation of vector acan be described also by the following formula
a′′ =Ra ˜
R=e−bθ
2aebθ
2,(8)
where R=nm,˜
R=mn and the bivector b=m∧nis a segment of the surface on which vector ais rotated.
The product of two vectors is called rotor. We remember that
R=nm =m·n−m∧n
and
˜
R=mn =m·n+m∧n

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 107
Figure 2. Rotation of vector adue to two subsequent reflections
It is important to note that these rules are independent from the signature, and for this reason they are also valid for
four-vectors of the space–time algebra. In particular, rotors with pure spatial bivector parts (such as γxγyϑ/2) generate
ordinary rotations, whereas rotors containing bivectors with the term γt(such as γzγtϑ/2) generate hyperbolic rota-
tions. Rotors operations are a very powerful geometric tool and some hardware implementations have been attempted
[16].
3. The Electromagnetic Field and the Wave Function
The behavior of electromagnetic waves was described in 1865 by James Clerk Maxwell in his work “Dynamical Theory
of the Electromagnetic Field”. Maxwell’s equations are a system of partial differential equations, where different
concepts are employed: electric field, flux density (or magnetic) field, charge density and current density [4–6].
In order to study the undulatory behavior of particles, the concept of wave function was introduced. Following
the interpretation of Born, the square of this function represents the probability density to find a particle in a point
of the space, just like the undulatory theory of light, whose intensity is given by the square of the electromagnetic
wave amplitude. Now, following the principle of Occam’s razor, which suggests carefulness in the introduction of new
concepts, we consider two interesting possibilities:
(1) find a common origin of the conceptual entities used in Maxwell’s equations;
(2) consider the wave function as a particular reformulation of concepts/entities already present in Maxwell’s
equations.
3.1. The electromagnetic potential
Maxwell’s equations can be reinterpreted by means of a unique entity, namely, the vector potential with four compo-
nents, as defined by the following equation:
A�r�=γxAxr�+γyAyr�+γzAzr�+γtAtr�,(9)
where each of the vector potential components Ax,Ay,Azand Atare functions of the space–time coordinates and
r�(x, y, z, t) = γxx+γyy+γzz−γtct =r△−γtct is the position vector in space–time. From now on in the four-

108 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128
potential and in other field quantities the variable r�will be omitted for simplicity. The four-potential has dimension
in SI units equal to V s m−1. Two basic assumptions are made:
(1) the vector potential field A�is represented by a harmonic function;
(2) the space is homogeneous, linear and isotropic.
Therefore, we assume a function that links a vector of four components to each point of the space–time as the unique
source of Maxwell’s equations entities.
We use the following definition of the operator ∂in space–time algebra
∂=γx
∂
∂x+γy
∂
∂y+γz
∂
∂z+γt
1
c
∂
∂t=∇+γt
1
c
∂
∂t,(10)
where
∇=γx
∂
∂x+γy
∂
∂y+γz
∂
∂zand c=1
√ϵ0µ0
.
If A�is the vector potential defined by (9) the following expression can be written:
∂A�=∂·A�+∂∧A�=S+F=G,(11)
where
G(x, y, z, t)=S+γxγtFxt +γyγtFyt +γzγtFzt +γyγzFyz +γxγzFxz +γxγyFxy .(12)
Expanding (11), by considering the products as shown in Table 3 and by collecting all terms with the same blade, the
following set of equations is found:
∂·A�=S=∂Ax
∂x+∂Ay
∂y+∂Az
∂z−1
c
∂At
∂t,(13)
γxγtFxt =γxγt
1
cEx=γxγt(∂At
∂x−1
c
∂Ax
∂t),(14)
γyγtFyt =γyγt
1
cEy=γyγt(∂At
∂y−1
c
∂Ay
∂t),(15)
γzγtFzt =γzγt
1
cEz=γzγt(∂At
∂z−1
c
∂Az
∂t),(16)
γyγzFyz =γyγzBx=γyγz(∂Az
∂y−∂Ay
∂z,),(17)
γxγzFxz =−γxγzBy=γxγz(∂Az
∂x−∂Ax
∂z),(18)

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 109
Table 3. Products ∂A�.
∂A�γxAxγyAyγzAzγtAt
γx∂
∂x
∂Ax
∂xγxγy
∂Ay
∂xγxγz
∂Az
∂xγxγt
∂At
∂x
γy∂
∂y−γxγy
∂Ax
∂y
∂Ay
∂yγyγz
∂Az
∂yγyγt
∂At
∂y
γz∂
∂z−γxγz
∂Ax
∂z−γyγz
∂Ay
∂z
∂Az
∂zγzγt
∂At
∂z
γt1
c
∂
∂t−γxγt1
c
∂Ax
∂t−γyγt1
c
∂Ay
∂t−γzγt1
c
∂Az
∂t−1
c
∂At
∂t
γxγyFxy =γxγyBz=γxγy(∂Ay
∂x−∂Ax
∂y),(19)
where S=S1+S2+S3+S4is a scalar field, whose meaning will be clarified later. It is to be noted that equating
(13) to zero, i.e. S= 0, gives an expression that takes the form of the “Lorenz gauge” condition if At=−φ/c, where
φis the scalar potential of the electric field [4,8,10,17].
Equation (13) can be rewritten as
S=∇·A△−1
c
∂At
∂t,(20)
where A△=γxAx+γyAy+γzAzis the usual three-component vector potential.
Using the so-called “Lorenz gauge” the scalar field Sis considered zero everywhere, denying its status of a real
physical entity [5]. Same consideration can be done for the “Coulomb gauge” that assign zero value to each addendum
Si. We simply do not apply any “gauge”, apart from defining A�as a harmonic function. According to our point
of view, both Lorenz and Coulomb “gauges” should be considered just as boundary conditions and the scalar field S,
although not directly observable, has a gradient in space–time with a clear physical meaning. Similar considerations are
normally presented in electromagnetism to introduce the concept of vector potential, that is a not directly measurable
field. The components of the geometric product ∂A�are shown in Table 3. An electromagnetic field Gwith seven
components emerges, composed by one scalar and six bivectors.
Table 4 represents the relation between the fundamental electromagnetic entities and the space–time components
of the vector potential A�.
Table 4. Relation between electromagnetic entities
and the vector potential A�.
∂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 set of equations from (14) to (19) can be rewritten also in the following way:

110 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128
Ex=c∂At
∂x−∂Ax
∂t(21)
Ey=c∂At
∂y−∂Ay
∂t(22)
Ez=c∂At
∂z−∂Az
∂t(23)
Bx=∂Az
∂y−∂Ay
∂z(24)
By=−∂Az
∂x+∂Ax
∂z(25)
Bz=∂Ay
∂x−∂Ax
∂y,(26)
where
E=γxEx+γyEy+γzEz=c∇At−∂A△
∂t,(27)
B=γxBx+γyBy+γzBz=∇×A△.(28)
The sum of all diagonal elements in Table 3 represents the scalar product
S=∂·A�,(29)
whereas the sum of all extra-diagonal elements gives the six components of electromagnetic bivector F
F=∂∧A�.(30)
Referring to the function G, it is possible to note that the “electromagnetic field” is characterized by seven values:
three for the electric field, three for the flux density field and one for the scalar field S.
With reference to Table 4 the electromagnetic field Gcan also be expressed as
G(x, y, z, t)=S+F=S+γxγt
Ex
c+γyγt
Ey
c+γzγt
Ez
c+γyγzBx−γxγzBy+γxγyBz
=S+1
cEγt+IBγt=S+1
c(E+IcB)γt,(31)
where

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 111
I=γxγyγzγt(32)
is the unitary pseudoscalar and
F=1
cEγt+IBγt=1
c(E+IcB)γt.(33)
On the other hand, with reference to Table 3, 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,(34)
which again results in Eqs. (20), (27) and (28) by taking (31) into account.
3.2. Maxwell’s equations
Now, by applying the operator ∂to the multivector G(11) and equating it to zero, a new expression is found, i.e.
∂G=∂2A�= 0,(35)
whose components are shown in Table 5. The equation ∂G= 0 can be seen as an extension in four dimensions of the
Cauchy-Riemann conditions for analytic functions of a complex (two dimensional) variable [15,18]. In [18] Hestenes
writes: “Members of this audience will recognize �ψ0= 0 as a generalization of the Cauchy–Riemann equations to
space–time, so we can expect it to have a rich variety of solutions. The problem is to pick out those solutions with
physical significance.”. In fact, if A�is harmonic then
∂2A�=∇2A�−1
c2
∂2A�
∂t2= 0,(36)
which represents the wave equation of the four-potential and where
∂2=∂2
∂x2+∂2
∂y2+∂2
∂z2−1
c2
∂2
∂t2=∇2−1
c2
∂2
∂t2.
It should be noted that in our case, considering the scalar field S̸= 0 and A�harmonic, (36) is always homoge-
neous.
By collecting all common factors contained in Table 5 the following equations are derived:
γx(∂S
∂x−∂Bz
∂y+∂By
∂z+1
c2
∂Ex
∂t)= 0,(37)
γy(∂Bz
∂x+∂S
∂y−∂Bx
∂z+1
c2
∂Ey
∂t)= 0,(38)
γz(−∂By
∂x+∂Bx
∂y+∂S
∂z+1
c2
∂Ez
∂t)= 0,(39)

112 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128
Table 5. Products ∂G=∂(∂A�).γij =γiγj,γijk =γiγjγk.
∂2A�Sγxt 1
cExγyt 1
cEyγzt 1
cEzγyz Bx−γxz Byγxy Bz
γx∂
∂xγx∂S
∂xγt1
c
∂Ex
∂xγxyt 1
c
∂Ey
∂xγxzt 1
c
∂Ez
∂xγxyz
∂Bx
∂x−γz
∂By
∂xγy
∂Bz
∂x
γy∂
∂yγy∂S
∂y−γxyt 1
c
∂Ex
∂yγt1
c
∂Ey
∂yγyzt 1
c
∂Ez
∂yγz
∂Bx
∂yγxyz
∂By
∂x−γx
∂Bz
∂y
γz∂
∂zγz∂S
∂z−γxzt 1
c
∂Ex
∂z−γyzt 1
c
∂Ey
∂zγt1
c
∂Ez
∂z−γy
∂Bx
∂zγx
∂By
∂zγxyz
∂Bz
∂z
γt1
c
∂
∂tγt1
c
∂S
∂tγx1
c2
∂Ex
∂tγy1
c2
∂Ey
∂tγz1
c2
∂Ez
∂tγyzt 1
c
∂Bx
∂t−γxzt 1
c
∂By
∂tγxyt 1
c
∂Bz
∂t
γt
1
c(∂Ex
∂x+∂Ey
∂y+∂Ez
∂z+∂S
∂t)= 0,(40)
γyγzγt
1
c(∂Ez
∂y−∂Ey
∂z+∂Bx
∂t)= 0,(41)
γxγzγt
1
c(∂Ez
∂x−∂Ex
∂z−∂By
∂t)= 0,(42)
γxγyγt
1
c(∂Ey
∂x−∂Ex
∂y+∂Bz
∂t)= 0,(43)
γxγyγz(∂Bx
∂x+∂By
∂y+∂Bz
∂z)= 0.(44)
Rearranging all equations from (37) to (44) the following are derived:
∂Bz
∂y−∂By
∂z=∂S
∂x+1
c2
∂Ex
∂t,(45)
∂Bx
∂z−∂Bz
∂x=∂S
∂y+1
c2
∂Ey
∂t,(46)
∂By
∂x−∂Bx
∂y=∂S
∂z+1
c2
∂Ez
∂t,(47)
∂Ex
∂x+∂Ey
∂y+∂Ez
∂z=−∂S
∂t,(48)
∂Ez
∂y−∂Ey
∂z=−∂Bx
∂t,(49)

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 113
∂Ex
∂z−∂Ez
∂x=−∂By
∂t,(50)
∂Ey
∂x−∂Ex
∂y=−∂Bz
∂t,(51)
∂Bx
∂x+∂By
∂y+∂Bz
∂z= 0,(52)
which are coincident with Maxwell’s equations if
∂S
∂x=µ0Jex =µ0
∂q
∂y∂z∂t=µ0
∂q∂x
∂x∂y∂z∂t=µ0ρvx,(53)
∂S
∂y=µ0Jey =µ0
∂q
∂x∂z∂t=µ0
∂q∂y
∂x∂y∂z∂t=µ0ρvy,(54)
∂S
∂z=µ0Jez =µ0
∂q
∂x∂y∂t=µ0
∂q∂z
∂x∂y∂z∂t=µ0ρvz,(55)
1
c
∂S
∂t=µ0Jet =−µ0c∂q
∂x∂y∂z=−µ0cρ,(56)
where ∂qis the differential of a generic charge [4,17]. Equation (56) can be also written as
∂S
∂t=cµ0Jet =−µ0c2∂q
∂x∂y∂z=−µ0c2ρ=−ρ
ϵ0
.(57)
By taking into account (53)–(56), the following relation holds for the current density field,
1
µ0
∂S=1
µ0(γx
∂S
∂x+γy
∂S
∂y+γz
∂S
∂z+γt
1
c
∂S
∂t)=J�e,(58)
where
J�e=γxJex +γyJey +γzJez +γtJet =γxJex +γyJey +γzJez −γtcρ
=J△−γtcρ=ρ(v△−γtc)(59)
is the four-current vector,
v�=γxvx+γyvy+γzvz−γtc=v△−γtc(60)
is a four-velocity vector and v△is the speed in the ordinary space.
In this formulation the partial derivatives of the scalar field Swith respect to time and space coordinates can be
interpreted as charge density and current density, respectively. As a matter of fact (45)–(47) represent the spatial
components of the Ampere’s law, i.e.

114 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128
∇×B=µ0J△+1
c2
∂E
∂t,(61)
where J△=γxJex +γyJey +γzJez is the three-component vector of current density, (48) is the Gauss’s law for the
electric field
∇·E=ρ
ϵ0
,(62)
(49)–(51) represent the spatial components of Faraday–Neumann–Maxwell–Lenz law
∇×E=−∂B
∂t(63)
and (52) the Gauss’s law for the flux density field
∇·B= 0.(64)
Finally, by applying the ∂·operator to (58) and setting the result to zero, the equation representing the law of
electric charge conservation is obtained
1
µ0
∂·(∂S) = ∂·J�e=∂Jex
∂x+∂Jey
∂y+∂Jez
∂z+∂ρ
∂t= 0.(65)
It is important to note that the wave equation of the scalar field Scan be deduced from the charge–current conservation
law:
∂·(∂S) = ∂2S=∂2S
∂x2+∂2S
∂y2+∂2S
∂z2−1
c2
∂2S
∂t2=∇2S−1
c2
∂2S
∂t2= 0.(66)
Now, by applying the time derivative to (66) and remembering (57), the wave equation of the charge field ρr�can
also be deduced, i.e.
∂
∂t∂2S=∂2(∂S
∂t)=∂2−µ0c2ρ=−µ0c2∂2ρ= 0,(67)
which gives
∂2ρ=∇2ρ−1
c2
∂2
∂t2ρ= 0.(68)
Clearly, both (66) and (68) represent, respectively, fields (Sand ρ) that must necessary propagate at the speed of light
[17,19]. Equation (58) means also that the 4-vector current density field can be derived directly from the scalar field
S. The hypothesis of existence of scalar waves has been recently explored at the Oak Ridge laboratories: “The new
theory predicts a new charge-fluctuation-driven scalar wave, having energy but not momentum for zero magnetic and
electric fields. The scalar wave can co-exist with a longitudinal-electric wave, having energy and momentum. The new
theory in 4-vector form is relativistically covariant. New experimental tests are needed to confirm this theory.” [13].

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 115
Figure 3. Helical motion of an elementary charge qmoving at the speed of light, with v2
z+v2
⊥=c2.
The proposed reinterpretation of Maxwell’s equations in this paper is in agreement with the principle of Occam’s
razor: the concepts of charge and current density are not inserted “ad hoc” but are deduced from a single more
fundamental entity, the four dimensional vector potential field A�r�=A�(x, y, z, t).
Equation (68) imposes a precise condition on charge dynamics, describing only distributions of charge density
moving in vacuum at the speed of light c. At first glance, this result seems to be incompatible with experimental
observations, with the usual concepts of charge and current and with the traditional way of working with Maxwell’s
equations. In fact, with reference to this perspective, a big advantage in using Maxwell’s equations is the ability to
simply specify both current density and charge density distributions and then see what fields result. Nevertheless, in
the model proposed in this paper, the added constraint on the charge and current density seems to imply that one is
no longer free to specify charge and current density distributions at will, because this information is indeed included
within the electromagnetic four potential A�.
As will be shown later, we can interpret (68) as a constraint for the definition of models of elementary charges (or
particles). This constraint, however, can be removed when considering macroscopic electromagnetic systems or even
the dynamics of a single elementary charge at a spatial scale greater than the particle Compton wavelength λcand at a
time scale greater than the Compton period λc/c. In this case static elementary charges can be seen as charge density
distributions moving at the speed of light on a closed trajectory but with a zero average speed (this generalization
would be consistent with static charge densities, electrets, dielectrics), whereas currents can be considered as an ordered
motion of charge density distributions moving with an absolute velocity equal to the speed of light but with an arbitrary
average speed lower than c.
As an example, referring to Fig. 3, the electromagnetic effects generated by an elementary charge q, moving
at instantaneous speed cin a helical motion of radius ≤λc/2πwith average velocity vzalong the helix axis zand
tangential velocity v⊥, can be approximated, on a spatial scale ≫λcand a temporal scale ≫λc/c, to those produced
by the same elementary charge qmoving at uniform velocity vz, creating the current density
Jz=Jzγz≈q
δxδyδz
dz
dtγz=q
δV
dz
dtγz=ρvzγz=ρvz,(69)
where δV=δxδyδz≈λ3
c. In this view and at a macroscopic level the here proposed new interpretation of Maxwell’s
equations remains compatible with the traditional way of working with them, i.e. by assigning the sources and deter-
mining, as a consequence, both the electric and the flux density (magnetic) field.
The new formulation of Maxwell’s equations expressed by (35) is quite similar to the Dirac–Hestenes equation for

116 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128
m= 0 (Weyl equation). In all cases the solution is a spinor field. A spinor is a mathematical object that in space–time
algebra is simply a multivector of even grade components. The motion of a massless charge that moves at speed of
light can be described using a composition of a rotation in the γxγyplane followed by a scaled hyperbolic rotation in
the γzγtplane and can be encoded in Cl3,1with a single spinor.
At this point the Authors are encouraged by an interesting sentence of P.A.M. Dirac. In fact, in his Nobel lecture
[20], held in 1933, Dirac proposed an electron model in which a charge moves at the speed of light: “It is found that an
electron which seems to us to be moving slowly, must actually have a very high frequency oscillatory motion of small
amplitude superposed on the regular motion which appears to us. As a result of this oscillatory motion, the velocity of
the electron at any time equals the velocity of light.”
4. Properties of the Electromagnetic Field
In this section, the main properties of the electromagnetic field will be presented and discussed by means of Cl3,1
Clifford algebra.
4.1. Lorentz force
A very compact and elegant form for the expression of the Lorentz force can be achieved in Cl3,1, in terms of a generic
charge qmoving at a generic speed v�=v△−γtc, extracting from the expression qGv�the blades of degree 1, and
considering the four-momentum P�, i.e.
(dP�
dt)q
=qGv�1=q(E−I△v△∧B−γt
1
cv△·E+Sv△−γtcS)
=q(E+v△×B+Sv△)−γtq(1
cv△·E+cS),(70)
where I△=γxγyγz=−Iγtis the unitary volume of the three dimensional space. In the last member of (70) the first
term represents the Lorentz force acting on the charge qplus a force acting on the same charge but depending on the
scalar field and on the speed and directed along the motion, whereas the last term in γtrepresents the work carried out
by the electric and scalar fields in moving the charge along a unitary distance.
In terms of force density (in N m−3) the above expression becomes
dP�V
dt=ρGv�1=ρE−I△J△∧B−γt
1
cJ△·E−γtρcS
=ρE+J△×B+SJ△−γt(1
cJ△·E+ρcS),(71)
where P�Vis the four-momentum spatial density, J△is the generic 3-D current density and ρthe spatial charge
density. The term SJ△is the contribution, in terms of force per volume, due to the scalar field; this force density has
the same direction of the 3-D current density. The last term in (71), with the unitary vector γt, represents the work
density produced by the electric and scalar fields when moving the spatial charge density ρalong a unitary distance.
4.2. Derivation of Maxwell’s equations from Lagrangian density
Maxwell’s equations can be derived considering the following Lagrangian density, in form of a composition of a scalar
and a pseudoscalar part:

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 117
L=1
2µ0
∂A�g
∂A�=1
2µ0
Ge
G=1
2µ0∥G∥2=1
2µ0
(S+F) (S−F) = 1
2µ0S2−F2
=1
2µ0(−E2
c2+B2+S2−2
cIE·B),(72)
where, bearing (33) in mind,
F=1
cEγt+IBγt=1
c(E+IcB)γt(73)
is the bivector part of the electromagnetic field and erepresents the conjugation operator. Expanding (72), and taking
equations from (20) to (26) into account, we obtain the Lagrangian density as a function of the derivatives of the
electromagnetic four-potential components, i.e.
L=1
2µ0{−(∂At
∂x−1
c
∂Ax
∂t)2
−(∂At
∂y−1
c
∂Ay
∂t)2
−(∂At
∂z−1
c
∂Az
∂t)2
+(∂Az
∂y−∂Ay
∂z)2
+(∂Ax
∂z−∂Az
∂x)2
+(∂Ay
∂x−∂Ax
∂y)2
+(∂Ax
∂x+∂Ay
∂y+∂Az
∂z−1
c
∂At
∂t)2
−2I[(∂At
∂x−1
c
∂Ax
∂t)(∂Az
∂y−∂Ay
∂z)+(∂At
∂y−1
c
∂Ay
∂t)(∂Ax
∂z−∂Az
∂x)+
+(∂At
∂z−1
c
∂Az
∂t)(∂Ay
∂x−∂Ax
∂y)]}.(74)
In Cl3,1algebra the Euler–Lagrange equations can be expressed, considering as variables the electromagnetic
four-potential components Ax(x, y, z, t),Ay(x, y , z, t),Az(x, y, z, t)and At(x, y , z, t), in the following way:

j=x,y,z,t 

i=x,y,z,t
γi
∂
∂i
∂L
γiγj∂∂Aj
∂i
−∂L
γj∂Aj
= 0,(75)
which reduces itself to

j=x,y,z,t 

i=x,y,z,t
γi
∂
∂i
∂L
γiγj∂∂Aj
∂i

= 0,(76)
considering that in this case

j=x,y,z,t (∂L
γj∂Aj)= 0,(77)

118 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128
because in (74) only the derivative terms of the four-potential (∂Aj/∂i) appear. By expanding (76), for example with
j=t, we achieve, after some trivial calculation steps,
−γt
∂L
∂At
=−γt
∂
∂x
∂L
∂∂At
∂x−γt
∂
∂y
∂L
∂∂At
∂y−γt
∂
∂z
∂L
∂∂At
∂z−γt
∂
∂t
∂L
∂∂At
∂t
=γt
1
µ0(1
c
∂Ex
∂x+I∂Bx
∂x+1
c
∂Ey
∂y+I∂By
∂y+1
c
∂Ez
∂z+I∂Bz
∂z+1
c
∂S
∂t)= 0,(78)
and this equation returns Gauss’s laws for the electric field (see Eq. (40)) and for the flux density field (see Eq. (44)),
respectively:
γt
1
c(∂Ex
∂x+∂Ey
∂y+∂Ez
∂z+∂S
∂t)= 0,
γxγyγz(∂Bx
∂x+∂By
∂y+∂Bz
∂z)= 0.
Now, if we expand (76) with j=x, we obtain
γx
∂L
∂Ax
=γx
∂
∂x
∂L
∂∂Ax
∂x+γx
∂
∂y
∂L
∂∂Ax
∂y+γx
∂
∂z
∂L
∂∂Ax
∂z+γx
∂
∂t
∂L
∂∂Ax
∂t
=γx
1
µ0(∂S
∂x−∂Bz
∂y+I
c
∂Ez
∂y+∂By
∂z−I
c
∂Ey
∂z+1
c2
∂Ex
∂t+I
c
∂Bx
∂t)= 0.(79)
Equation (79) gives (37) and (41):
γx(∂By
∂z−∂Bz
∂y+∂S
∂x+1
c2
∂Ex
∂t)= 0,
γyγzγt
1
c(∂Ez
∂y−∂Ey
∂z+∂Bx
∂t)= 0.
If we carry on the above procedures with j=yand j=zthe other remaining components of Maxwell’s equation can
be determined, i.e (38), (42), (39) and (43):
γy
∂L
∂Ay
=γy
∂
∂x
∂L
∂∂Ay
∂x+γy
∂
∂y
∂L
∂∂Ay
∂y+γy
∂
∂z
∂L
∂∂Ay
∂z+γy
∂
∂t
∂L
∂∂Ay
∂t
=γy
1
µ0(∂Bz
∂x−I
c
∂Ez
∂x+∂S
∂y−∂Bx
∂z+I
c
∂Ex
∂z+1
c2
∂Ey
∂t+I
c
∂By
∂t)= 0,(80)

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 119
γz
∂L
∂Az
=γz
∂
∂x
∂L
∂∂Az
∂x+γz
∂
∂y
∂L
∂∂Az
∂y+γz
∂
∂z
∂L
∂∂Az
∂z+γz
∂
∂t
∂L
∂∂Az
∂t
=γz
1
µ0(−∂By
∂x+I
c
∂Ey
∂x+∂Bx
∂y−I
c
∂Ex
∂y+∂S
∂z+1
c2
∂Ez
∂t+I
c
∂Bz
∂t)= 0,(81)
that give, as expected, respectively
γy(∂Bz
∂x+∂S
∂y−∂Bx
∂z+1
c2
∂Ey
∂t)= 0,
γxγzγt
1
c(∂Ez
∂x−∂Ex
∂z−∂By
∂t)= 0,
γz(∂Bx
∂y+∂S
∂z−∂By
∂x+1
c2
∂Ez
∂t)= 0,
γxγyγt
1
c(∂Ey
∂x−∂Ex
∂y+∂Bz
∂t)= 0.
By analyzing the above-reported equations, it is possible to reach some conclusions. First of all the Lagrangian
density, as defined in (72), can be divided in the sum of two parts
L=Lfield +Lint.(82)
The first part
Lfield =1
2µ0(−E2
c2+B2)=−1
2µ0
(F·F)(83)
represents the “field part” of the Lagrangian density, as known in literature, and the second
Lint =1
2µ0(S2−2
cIE·B)(84)
represents the “interaction term” of the Lagrangian density, that takes the interaction of the electromagnetic field with
the sources into account, remembering, in addition, that the derivatives of the scalar field S, with respect to the four
dimensional space coordinates x,y,zand t, are bounded respectively to the sources Jex,Jey ,Jez and Jet =−cρ(see
Eq. (58)). Indeed, by deriving only the interaction terms of the Lagrangian density with respect to the four-potential,
i.e. by performing the operation ∂Lint/∂Aj, it is possible to derive the term J�e·A�. In fact, for the component along
γtwe find

120 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128
−γt
∂Lint
∂At
=−γt
∂
∂x
∂Lint
∂∂At
∂x−γt
∂
∂y
∂Lint
∂∂At
∂y−γt
∂
∂z
∂Lint
∂∂At
∂z−γt
∂
∂t
∂Lint
∂∂At
∂t
=γt
µ0(I∂Bx
∂x+I∂By
∂y+I∂Bz
∂z+1
c
∂S
∂t)=γt
µ0(I∇·B+1
c
∂S
∂t)
=γt
µ0c
∂S
∂t=γtJet =−γtcρ.(85)
Integration of (85) yields
Lint|t=∂Lint
∂At
dAt=1
µ0c
∂S
∂tdAt=1
µ0c
∂S
∂tAt=−1
µ0
µ0cρAt=−cρAt=JetAt.(86)
For the component along γxwe find
γx
∂Lint
∂Ax
=γx
∂
∂x
∂Lint
∂∂Ax
∂x+γx
∂
∂y
∂Lint
∂∂Ax
∂y+γx
∂
∂z
∂Lint
∂∂Ax
∂z+γx
∂
∂t
∂Lint
∂∂Ax
∂t
=γx
µ0(∂S
∂x+I
c
∂Ez
∂y−I
c
∂Ey
∂z+I
c
∂Bx
∂t)=γx
µ0
∂S
∂x=γxJex.(87)
Integration of (87) yields
Lint|x=(∂Lint
∂Ax)dAx=1
µ0
∂S
∂xdAx=1
µ0
∂S
∂xAx=JexAx.(88)
The same procedure is clearly valid also for the components in γyand γz. Finally, by integration of (77), we get
the Lagrangian density interaction term as
Lint =
j=x,y,z,t (∂Lint
∂Aj)dAj=JexAx+Jey Ay+Jez Az−cρAt=J�e·A�,(89)
which is the usual “source” term that is added in traditional Lagrangian theory for classical electricity and magnetism
in order to obtain the complete set of Maxwell’s equations [6-8,17]. The scalar product J�e·A�has a dimension
of energy per volume (J m−3); in particular, the contribution of the spatial components of vectors J�eand A�(the
scalar product J△·A△) can be considered as the specific “kinetic” energy of the electromagnetic field, whereas the
term JetAt=−cρAtthe “potential” energy. By virtue of (84), (89) becomes
Lint =1
2µ0(S2−2
cIE·B)=J�e·A�.(90)
The pseudoscalar term 2/cIE·Bis clearly null as the electric and the magnetic flux density fields are always or-
thogonal with respect to each other: indeed, this term contains information about (63). A direct consequence of (90)
is, therefore, the following relation between the scalar field, the electromagnetic four-potential and the four-current
density:
S2= 2µ0J�e·A�.(91)

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 121
By inspection of (85) and (87), and generalizing, it is possible to define the four-current vector J�efrom the
interaction Lagrangian term:

j=x,y,z,t (∂Lint
γj∂Aj)=γx
∂Lint
∂Ax
+γy
∂Lint
∂Ay
+γz
∂Lint
∂Az−γt
∂Lint
∂At
=γxJex +γyJey +γzJzx +γtJet =J�e.(92)
and, again, by virtue of the Noether’s theorem, the law of current and charge conservation
∂·

j=x,y,z,t (∂Lint
γj∂Aj)
=∂·J�e= 0,(93)
which returns, consequently, the wave equations (66) and (68), respectively.
As can be seen the definition of the electromagnetic field Gis complete and it includes itself the information of
both action and interaction, without the need of any additional term: this is in full accordance with the principle of
Occam’s razor.
Thanks to the Cl3,1Clifford algebra the Euler–Lagrange equations can be conveniently defined in in a very compact
form:
∂∂L
∂∂∧A�−∂L
∂A�
=∂(∂L
∂F)−∂L
∂A�
= 0,(94)
where, now, the scalar Lagrangian density is
L=Lfield +Lint =−1
2µ0
F·F+J�e·A�.(95)
Substituting (95) in (94) one can achieve directly Maxwell’s equations in Cl3,1in the form shown in the previous
sections (see Eq. (35)), i.e.
∂
∂−1
2µ0
F·F+J�e·A�
∂F
−
∂−1
2µ0
F·F+J�e·A�
∂A�
=−1
µ0
∂F−J�e= 0,(96)
which yields
∂F+µ0J�e=∂F+∂S=∂(F+S) = ∂G= 0.(97)

122 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128
4.3. Energy of the electromagnetic field
The energy density multivector of the field Gis given by
w=1
2µ0[S+(E
c+IB)γt]S+(E
c+IB)†
γt
=1
2µ0[S+(E
c+IB)γt][S+(E
c−IB)γt]
=S2
2µ0
+ϵ0E2
2+B2
2µ0−1
cµ0
IE∧B+1
cµ0
SEγt
=S2
2µ0
+ϵ0E2
2+B2
2µ0
+1
cµ0
(E×B+SE)γt
=ws+we+wm+1
c·Sγt,(98)
where
ws=S2
2µ0
=J�e·A�, we=ϵ0E2
2and wm=B2
2µ0
are the specific energies of the scalar, the electric and the magnetic flux density fields, respectively, †is the reversion
operator, whereas
S=1
µ0
(E×B+SE)(99)
is the generalized Poynting vector [10,11]. Beside the usual term E×Bhere a new energy term appears, namely SE,
which is associated to a longitudinal scalar wave [11] and that is not further investigated in the present work.
4.4. Electrostatic field and vector potential
In the case of non-time-varying potential ∂At/∂t= 0 the scalar field Sbecomes the divergence of the classical 3-D
vector potential A△:
S=∇·A△.
The time derivative of both sides gives
∂S
∂t=∂(∇·A△)
∂t=∇·∂A△
∂t.
Considering that ∂A△/∂t=−E(see Eqs. (21)–(23)) and that ∂S/∂t=−ρ/ϵ0we rediscover Gauss’s law,
∇·E=ρ
ϵ0
,
showing that even the electrostatic field may be seen as generated from time derivatives of a vector potential field!

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 123
4.5. Electric charge, antimatter and time direction
Feynman proposed to interpret the positron (a particle which is identical to the electron but with a positive charge)
as an electron traveling back in time. Such an interpretation seems to be perfectly compatible with the definition of
electric charge density given in (56):
∂S
∂t=−ρ
ϵ0
.(100)
By multiplying both sides of (100) by –1 we obtain
−∂S
∂t=ρ
ϵ0
or, equivalently
∂S
∂(−t)=ρ
ϵ0
.(101)
The positron traveling back in time is represented in the annihilation reaction diagram proposed by Feynman and
shown in Fig. 4 [21].
“I did not take the idea that all the electrons were the same one from him as seriously as I took the observation that
positrons could simply be represented as electrons going from the future to the past in a back section of their world
lines” [22].
4.6. Magnetic charges and currents
Starting from an hypothetical eight component “vector potential” that includes the four pseudovectors (trivectors) T
of space–time algebra, symmetrical Maxwell’s equations emerge. This new set of equations now include the magnetic
charge and magnetic current densities that are the time and spatial derivatives of a pseudoscalar field P. By considering
(9) and the four pseudovectors defined as
Figure 4. Feynman’s diagram of proton–electron annihilation.

124 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128
T=γyγzγtTx+γxγzγtTy+γxγyγtTz+γxγyγzTt,(102)
a new vector potential can be defined as
A′=γxAx+γyAy+γzAz+γtAt+γyγzγtTx+γxγzγtTy+γxγyγtTz+γxγyγzTt(103)
from which we obtain
∂(A′) = ∂A�+T=S+F+P. (104)
Using SI units and following the same procedure as shown in Section 3 we can write:
S=∂Ax
∂x+∂Ay
∂y+∂Az
∂z−1
c
∂At
∂t,
γxγt
1
cEx=γxγt(∂At
∂x−∂Tz
∂y−∂Ty
∂z−1
c
∂Ax
∂t),
γyγt
1
cEy=γyγt(∂Tz
∂x+∂At
∂y−∂Tx
∂z−1
c
∂Ay
∂t),
γzγt
1
cEz=γzγt(∂Ty
∂x+∂Tx
∂y+∂At
∂z−1
c
∂Az
∂t),
γxγyγzγtP=γxγyγzγt(∂Tx
∂x−∂Ty
∂y+∂Tz
∂z−1
c
∂Tt
∂t),
γyγzBx=γyγz(∂Tt
∂x+∂Az
∂y−∂Ay
∂z−1
c
∂Tx
∂t),
γxγzBy=γxγz(−∂Az
∂x+∂Tt
∂y+∂Ax
∂z+1
c
∂Ty
∂t),
γxγyBz=γxγy(∂Ay
∂x−∂Ax
∂y+∂Tt
∂z−1
c
∂Tz
∂t).
By applying again the ∂operator to (104) and equating to zero:
∂2A′=∂(S+F+P) = 0.(105)
Here

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 125
∂F=−∂S−∂P=µ0J�e+1
ϵ0
J�m,(106)
where J�eis the four-current as defined in (58),
J�m=γxJmx+γyJmy+γzJmz+γtJmt=γxJmx+γyJmy+γzJmz−γt
1
cρm
is the magnetic four-current vector and ρmthe magnetic charge. By carrying out all calculation in (105) the
following set of equations is obtained:
γx(∂S
∂x−∂Bz
∂y+∂By
∂z+1
c2
∂Ex
∂t)= 0,
γy(∂Bz
∂x+∂S
∂y−∂Bx
∂z+1
c2
∂Ey
∂t)= 0,
γz(−∂By
∂x+∂Bx
∂y+∂S
∂z+1
c2
∂Ez
∂t)= 0,
γt
1
c(∂Ex
∂x+∂Ey
∂y+∂Ez
∂z+∂S
∂t)= 0,
γyγzγt
1
c(∂P
∂x+∂Ez
∂y−∂Ey
∂z+∂Bx
∂t)= 0,
γxγzγt
1
c(∂Ez
∂x−∂P
∂y−∂Ex
∂z−∂By
∂t)= 0,
γxγyγt
1
c(∂Ey
∂x−∂Ex
∂y+∂P
∂z+∂Bz
∂t)= 0,
γxγyγz(∂Bx
∂x+∂By
∂y+∂Bz
∂z+∂P
∂t)= 0.
This set of equations represents the symmetrical Maxwell’s equations considering the hypothesis (never confirmed up
until now by experiments) of existing magnetic currents and charges.

126 F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128
5. Conclusions
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. The application of Occam’s Razor principle to
Maxwell’s equations highlights some essential concepts:
(1) First of all, Clifford algebra is by far the most appropriate, simple and intuitive mathematical language for
encoding in general the laws of physics and in particular the laws of electromagnetism.
(2) A scalar field derives from the definition of “harmonic” electromagnetic four-potential and this scalar field
becomes the source of charges and currents.
(3) The charge density derived from the scalar field follows the wave equation with a propagation speed equal to
the speed of light.
(4) The Feynman model of the positron, seen as an electron traveling back in time, seems to be perfectly compat-
ible with the definition of electric charge density as the time derivative of a scalar field.
In particular, the important element emerging from the present paper is that (68) imposes a precise condition on
charge dynamics, describing distributions of charge density moving in vacuum at the speed of light. Indeed, van
Vlaenderen found the same condition on charge dynamics but with the difference that he considers both the conditions
E= 0 and B= 0 at the same time (no electromagnetic field) concluding that “a scalar field Scan be induced by a
dynamic charge/current distribution” [11].
In the model proposed here, the added constraint on the charge and current density seems to imply that one is
no longer free to specify charge and current density distributions at will, because this information is indeed included
within the definition of the four-potential A�. However, this constraint can be removed when considering macroscopic
electromagnetic systems or even the dynamics of a single elementary charge at a spatial scale greater than the particle
Compton wavelength λcand at a time scale greater than the Compton period λc/c. In this case static elementary
charges can be visualized as charge density distributions moving at the speed of light on a closed trajectory but with a
zero average speed (this generalization would be consistent with static charge densities, electrets, dielectrics), whereas
currents can be considered as an ordered motion of charge density distributions moving with an absolute velocity
equal to the speed of light but with an arbitrary absolute average speed lower than c. This observation favors a
pure electromagnetic model of elementary particles based on a particular Zitterbewegung interpretation of quantum
mechanics [23,24]. Therefore, the free electron, and perhaps all other elementary charged particles, can be viewed
as a charge distribution that rotates at the speed of light along a circumference whose length is equal to its Compton
wavelength [25].
Finally, a Lagrangian density equal to the square module of the seven component electromagnetic field reveals an
energy density formula for both fields and currents. Moreover, it has been demonstrated that Maxwell’s equations can
be explicitly derived in a simple way directly from the Lagrangian density of the electromagnetic field with the help
of Clifford algebra. An interesting consequence is also that the specific energy of the scalar field is deeply connected
to the interaction term of the Lagrangian density and, therefore, both to the electromagnetic four-potential and the
four-current density.
It is our opinion that the Zitterbewegung interpretation of quantum mechanics may give an important contribution
for understanding the structure of ultradense hydrogen and the origin of anomalous heat in some metal–hydrogen
systems. A Zitterbewegung electron model and a preliminary hypothesis for the structure of ultradense deuterium will
be treated more deeply in a paper written by the authors, entitled “The Electron and Occam’s Razor”, J. Condensed
Matter Nucl. Sci. 25 (2017).

F. Celani et al. / Journal of Condensed Matter Nuclear Science 25 (2017) 100–128 127
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. 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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