Axial vector magnetic charge and magnetic moment.
Maxwell’s equations and Lorentz force law.

Eugenio Olmedilla Moreno a

Departamento de F́ısica Teórica, Facultad de Ciencias F́ısicas,
Universidad Complutense, 28040 Madrid, Spain

(Dated: January 25, 2023)

We relate the non null mass of the elementary fermions (leptons and quarks) with their spins 1
2
.

We replace the Amperian concept in the definition of the intrinsic magnetic dipole moment for every
elementary fermion by a related concept, the axial vector magnetic charge. In order to accommodate
it to the electromagnetic interaction we define an extension of the Maxwell’s equations and of the
Lorentz force law. In this way we incorporate the spin (a vector) in the Maxwell’s equations.

The neutrinos, electrically uncharged leptons, are a special case. We suggest:
the chargeless electrical but with spin neutrinos are under the action of the electrical fields.

Keywords: Elementary fermions (leptons, quarks). Mass. Maxwell equations. Lorentz’s force law. Neutrinos.

PRELIMINARY.

• The electron, a very small sphere with a surface charge, would rotate at 1
α ≈ 137 times the velocity of the

light over some axis of rotation of the sphere (about a hundred years ago).
• The massless photons and later the neutrinos, also massless in former times, would have an axis of rotation

parallel to their direction of motion. (See Figure 1 in the Appendix F: Graphics).
• The massless neutrinos would be subjugated only by the weak interaction. Even today, most physicists working

with them treat them in such way, although we admit their no null masses.
But, the Special Relativity prohibits such velocity and the Quantum Mechanics do not let massive elementary

fermions with a definite axis of rotation [1], [2]. The spin [3] underneath, but without the surface, without the axis
of rotation and with a 4π rotation for the returning to a previous same three dimensional position:

a 4d-geometry without geometry. What is the solution?
The content of the previous paragraphs are a common place for the physicists, clearly stated by Gerard ’t Hooft

in his “In search of the ultimate building blocks”, published in 1997 [1]. He concludes for the spin: “an audacious
idea” and “such objections are simply ignored” (for the velocity). On other side, at present time some researches are
looking for “new physics” in the possible electromagnetism of the neutrinos due to their no null masses.

Photons have spin 1 without the zero projection and with zero rest mass. All the elementary fermions have spin 1
2 ,

there is no zero projection and they are massive, including the neutrinos (the neutrino solar problem behind),
could we relate the non null masses of the elementary fermions with their spins 1

2? [4].
This author puzzled over this question while working on a geometric model for the elementary fermions [5]. What

follows is an attempt for answering it, with an unexpected result: an extension of the Maxwell’s equations and of the
Lorentz force law. In the pathway we have to enlarge the idea of an “internal space” that would contend the previous
“audacious idea” of the spin (a scalar with two values) and the new idea of characteristic axes, a discrete set of
axial vectors: these axes are fixed distinct axes for every elementary fermion.

I. THE MAGNETIC MOMENT.

1845. “Faraday rotation effect”. The light changes its plane of oscillation under the magnetic fields.
1896-97. “ Zeeman effect” and “anomalous Zeeman effect”. The appearance of splittings in the spectroscopy

lines of some materials in the presence of magnetic fields. These splittings appeared in two different forms with
various materials. [3]

At the same time (1897), Lorentz explained the first one assuming vibrating electrons and using arguments of
classical electrodynamics. The role of a magnetic moment related to a mechanical angular momentum (rotations)
is relevant, with several approaches for its introduction. In the Bohr Sommerfeld conception of the electrons in the
atoms (1913-16), the electrons “circulate” around a nucleus producing an “Amperian dipole magnetic moment”.
Electrons are in “orbitals” (quantization of the energy) and the orientation of these orbitals are discrete: quantization
of the orbital angular momentum (a first quantization of the space).

The understanding of the anomalous Zeeman effect, the second one, took more time. It implied the revolutionary
idea of Kronig and of Uhlenbeck and Goudsmit of a rotation over an “internal axis” (an “internal space”), with a half
unit of angular momentum (the spin was there). The two-valuedness of something intrinsic to the electron observed
in relation to the magnetic fields in the Stern Gerlach experiment. It was justified with the introduction of a fourth
(angular) quantum number. Finally, the development by Pauli of a formalism of spin matrices [6]. Quantization of
the space definitely established.

a email: eugenio.o.m@ucm.es



2

The magnetic dipole moment.

Assume the definition of the vector:

−→µ =
1

2
(−→r ×−→j ) =

1

2
q−→r ×−→v =

q

2 m
(−→r × m−→v ) . (1)

This vector, the magnetic dipole moment, appears in a formula for the magnetic field produced by a stationary
moving electrical charge q at a large distance. This is the Amperian model for the magnetic dipole moment.
The formula for the magnetic field (at a large distance) produced by this moment has the same structure as the

one for the electric field produced by an electric dipole. This is significant for the term “dipole”, visualizing it as the
existence of two separated magnetic charges (North and South poles). This is the Gilbertian model. But without the
existence of magnetic charges (monopoles) we can consider these magnetic dipole moments in relation to a first non
vanishing term in a certain expansion which drives to the formula for the magnetic field [2], [7], [8], [9].

With v ≪ c, the vector mechanical angular moment is:
−→
L = −→r ×−→p = (−→r × m−→v ) and we rewrite (1):

−→µ =
q

2m

−→
L . (2)

We use this for the electrons in the orbitals of the atoms, and we get the above mentioned first discrete set of values
for explaining the Zeeman effect, with: [10]

−→µ orb, l,m =
−e
2me

−→
L l,m . (3)

In order to explain the anomalous Zeeman effect it was added a term representing “more rotation” (geometry -
mechanical angular moment) in an “internal space”, more magnetic moment due to another rotation, a spinning
(“the spin”), with only two values:

−→µ orb + “ −→µ spin ” , (4)

which under the action of a magnetic field
−→
B , in relation to the energy, drives to a relation of the type:

e
2me

(
−→
L“orb” + 2

ge
2
“
−→
S ” ) . −→B . (5)

With “
−→
S =

−−−→
1
2ℏσ ” (

−→σ also a -spin- vector), µ
B

= e ℏ
2me

and a gyromagnetic ratio ge = g0e+∆ge = 2+∆ge. it is:

“ −→µ spin = −→µe =
ge
2

µ
B

(
−−→
−σ) ” . (6)

This expression of the (intrinsic) magnetic dipole moment has three striking features:

a) with (5), the energy is a scalar and
−→
B is a vector.

−→
L l,m and −→µ orb, l,m have to be vectors, they are in

a scalar product, and they are so (“quantization of the space”). Therefore, the spin beneath the internal magnetic

dipole moment producing only two discrete values has to be a vector: “
−→
S ”. But, in the same way as Gerard ’t Hooft

writes in [1], quantum mechanics would not let us assign a direction for a rotation axis in the three dimensional space.
With another gesture of recklessness (audacity?), could we assume the existence of discrete specific characteristic

vectors (spinning) in an “internal space” in a similar way as the spin (a scalar), driving to only two values?,

b) the velocity in the current
−→
j in (1) has disappeared in −→µ spin in (6),

c) it is proportional to the electric charge and inverse proportional to the mass (via µ
B
), the constant ℏ

involved, although the dimensional analysis of [µ
B
] = [e ] L2T−1 seems to indicate that the mass has no role.

Which one is the dimension of the electric charge? See Appendix A.
With the Lorentz force law, we can add in the SI of units, to include the electromagnetic phenomena, any one of

the following three:

or with [e ] = C (eventually [ I ] = A), or with [
−→
E ], or with [

−→
B ] = B (the tesla, t̂ ); ([

−→
E ] = LT−1[

−→
B ]).

All the elementary fermions (leptons and quarks) have spin ( 12 ), but one of them has 0 electric charge. Nowadays
there are researches concerning the possibility of a non zero magnetic dipole moment (electromagnetic properties) for
the chargeless neutrino due to its non null mass [11], [12] and [13]. This suggests us to choose the following extension
for the SI of units, as more convenient for our elementary fermions:

{ kg , m , seg , tesla = t̂ } ({m, l , t , b }) for the dimensions {M , L , T , B }.



3

With this election we have for the electric charge and for the magnetic dipole moment:{
[ e ] = M T−1B−1

[−→µ ] = ML2T−2B−1 = [mc2 ] B−1 = [e ] L2T−1 = [e ] [ c ] L
.

On the other hand, Planck introduced “the constant of nature h” in the study of the black body radiation [14].
The starting point for the Quantum Mechanics. This constant explaining a discretization of certain energies (levels):

ϵ = h ν. h with the dimensions of an angular momentum: [h] = ML2T−1 = [c] M L = [c2] M T ,

so that we formally write: L = [ ℏ
c
] M−1 and T = [ ℏ

c2
] M−1, with the constants of nature ℏ

c
and ℏ

c2
(ℏ = h

2π ).

Proceeding in an opposite way to Planck, we assume for the “discrete structures of nature”, our elementary

fermions, the relationships: l = ℏ
c m−1 and t = ℏ

c2
m−1, even if we do not know what is the geometrical or

physical meaning of these l and t. In a similar way to the “spin” we can take them as belonging to the “internal
space”. We include the neutrinos and therefore with their non null masses.

With these elements and the previous units, we write the magnetic dipole moments for the elementary fermions in
the following way:

1) we associate, we define, a characteristic length and a characteristic time, in terms of the characteristic mass
of a fermion, for every elementary fermion (f):

ℏ = mf lf c = mf tf c2 =⇒
{
lf =

ℏ
c

1

mf

, tf =
lf
c

=
ℏ
c2

1

mf

}
, (7)

with the values of ℏ = 1.05457173 10−34 Js and of c = 2.999792 108 m/s. In particular for an electron:

me = 9.109382 10−31 Kg , =⇒
{
le = 3.8592 10−13 m , te = 1.2865 10−21 s

}
, (8)

2) the charge of an electron is, without writing the sign (Kg s−1 t̂−1 = C - Coulomb):

e = me t
−1
e b−1

e = c me l
−1
e b−1

e = 1.60210 10−19 C , (9)

and the Bohr magneton is:

µ
B

=
e ℏ
2 me

=
e
2

cme le
me

=
le
2
ce =

1

2
mec

2 b−1
e = 9.2740997 10−24 Jt̂−1. (10)

From (9) or (10) we obtain the characteristic b−1
e for the electron (with the inverse of the magnetic field unit):

b−1
e =

e te
me

=
2µ

B

me c
2 = 2.2626 10−10 t̂−1, (11)

3) we define the electric charge of any elementary fermion (up to the sign), with another “constant of nature e”:

(a) qk = |qf,k| = k
3 e (C) with k ∈ {0, 1, 2, 3} and C − Coulomb ,

where it is (b ) e = mf t−1
f b−1

f ,
(12)

with k = 3 for the charged leptons (cl), k = 2 for the family of the u-quarks, k = 1 for the family of the d-quarks
and k = 0 for the neutrinos. The factor b−1

ν of the neutrinos can not be determined with formula (a)-(12).

We can write k
3 in terms of an angle: k

3 = 4
π φ

f,k
, obtaining: qφ

f,k
= |qf,k| = ( 4π φ

f,k
)mf t−1

f b−1
f , with

φ
f,k

= π
12 k ∈

{
0
∣∣
(k=0)

, π
12

∣∣
(k=1)

, π
6

∣∣
(k=2)

, π
4

∣∣
(k=3)

}
.



4

4) Among the elementary fermions, the electrons are the best known experimentally. We take them as reference
for the others. The masses, the times, the inverse magnetic fields and the magnetic dipole moments are related in the
following ways:

mf = Af me ( this defines the value ofAf )

tf = A−1
f te ( ℏ = mf c2 tf = Afme c

2 A−1
f te ). Similarly lf .

b−1
f = A−2

f b−1
e ( qf,k

k
3∼ e = mf t−1

f b−1
f = Afme (A

−1
f te)

−1 (A2
f be)

−1 ) (k3 apart)

and

µf

k
3 ,gf∼ A−1

f µ
B

( e ℏ = 2me µB

k
3 ,gf∼ 2mf µf

k
3 ,gf∼ 2Afme A

−1
f µ

B
) (k3 , gf apart)

, (13)

g
f
a gyromagnetic ratio. We write it in the form: g

f
= g0

f
+∆g

f
; in particular for an electron: g0

e
= 2. Extrapolating

the third line for the neutrinos we could formally write: b−1
ν = A−2

ν b−1
e ; but it is k = 0 (qf,0 = qf,ν = 0),

5) the definition of the magnetic dipole moment for any elementary fermion, ruling out the neutrino family, is:

µf,k =
g
f

2

k

3

1

2
mfc

2 b−1
f =

g
f

2

qk ℏ
2mf

=
g
f

2

k

3

e ℏ
2Afme

= A−1
f

g
f

2

k

3
µ
B

Jt̂−1 , k ∈ {1, 2, 3} , (14)

which is similar (based in) to the one obtained with the Amperian concept for the (internal) dipole magnetic moment
of the electron (in formulas (4) - (6)). The electric charge involved (up to the sign).

We set the notation:

a subindex to the left of a magnitude (scalar or vector) to indicate

{
an axial (an ‘a’)

a polar ( a ‘p’)
character

(see Appendix B). And,

6) we introduce the definition of the axial vector magnetic charge a

−→
Jmf :

a

−→
Jmf = s (mf t−1

f b−1
f ) a

−→σf (C) , with the norm: ∥ a

−→σf ∥ = 1 and C − Coulomb , (15)

with a

−→σf a specific axial vector in the “internal space”. s = s(f) ∈ {+, −} related to the sign of the electric charge
of every elementary fermion. For k ∈ {1, 2, 3} we fix: s = sign(qef ), and pending of its determination for k = 0
(neutrinos). Look that, for example, for the electrons in a reference frame we have two possible opposite values of the

helicity, one corresponding to a spin and the other one to the opposite spin ( a
−→σe and − a

−→σe).

The magnitude e is an axial scalar magnitude (Appendix B): ae. Using (b)-(12) in (15):

a

−→
Jmf = s ae a

−→σf (C) = s ae
2

ℏ a

−→
Sf (C). (16)

The norm of these vectors is a fix magnitude for all the elementary fermions, the unit of magnetic charge:

Jm = ∥ a

−→
Jmf ∥ = ae =

2

ℏ
µ
B
me = 1.60210 10−19 C (Kg s−1 t̂−1). (17)

This is not the charge of a magnetic monopole which would not be an elementary fermion. In this way ae has a
double role: it is the basic unit for all the{

electro (with the axial scalar charge) −
− magnetic (with the axial vector charge)

phenomena of the elementary fermions,

and their interrelations (Maxwell’s equations).
Now, with this unit, our previous comments for b−1

ν =A−2
ν b−1

e , make sense.

We generalize the equation (14) to account for all the elementary fermions ( k ∈ {0, 1, 2, 3} ):

µf,k =
g
f

2
(
k

3
) m−1

f

ℏ
2
∥
−→
Jmf ∥=

g
f

2
(
k

3
) A−1

f µ
B
, with µ

B
=

ℏe
2me

= m−1
e

ℏ
2
∥
−→
Jm,f ∥ . (18)

With: µ
B

= K e it is K =
µ
B

e
(

Jt̂−1

Kg s−1 t̂−1

)
= ℏ

2 me
= 0.58 10−4 m2s−1.

For the neutrinos (k = φν= 0) we have: µν,0= 0.
Later on, we will define possible vector forms with the last formula in (39).



5

II. ELECTROMAGNETIC INTERACTION OF THE ELEMENTARY FERMIONS.

01: Maxwell’s and Lorentz’s equations. [2]
Maxwell’s equations: p∇. p−→E = 1

ϵ0
aρe , − a∂t p

−→
E + c2 p∇× a

−→
B = 1

ϵ0
p

−→
je

p∇. a−→B = p0m , a∂t a

−→
B + p∇× p

−→
E = a

−→
0 m

.

The source is a distribution of electrical charges and electrical currents.
Lorentz force law (a charge aqe under the action of electromagnetic fields):

p

−→
Fe = aqe ( p

−→
E + p

−→v × a

−→
B ) , aqe =

y
aρe dV ,


aqe p

−→v = p

−→
Je

aρe p
−→v = p

−→
je

[ aqe ] = MT−1B−1

.

(we have used a simplified notation, which it is not too precise).

02: Maxwell-Dirac’s and Lorentz-Dirac equations. [7] [15]

Polar scalar magnetic charge, monopole: pρm, pqm .

The source: a distribution of electrical charges, electrical currents and magnetic charges (monopoles) and magnetic
currents.  p∇. p−→E = 1

ϵ0
aρe , − a∂t p

−→
E + c2 p∇× a

−→
B = 1

ϵ0
p

−→
je

p∇. a−→B = µ0 pρm , a∂t a

−→
B + p∇× p

−→
E = −µ0 pρm p

−→v ρm

.

Formula for a Lorentz-Dirac force law:

p

−→
F =

aqe [ ( p

−→
E ) + (β p

−→uv )×(c a

−→
B) ]+

+ pqm

c [ (c a

−→
B)− (β p

−→uv )×( p

−→
E ) ]

,


p
−→v ρm

= c β p
−→uv = p

−→v , , c β = ∥ p
−→v ρm

∥

pqm =
t

pρm dV

[ pqm ] = c [ aqe ] = MLT−2B−1

.

A: In the pathway of the Maxwell’s equations.

The form of the fields created by a lepton or a quark (the source).

We modify the Maxwell’s or the Maxwell-Dirac’s equations to accommodate them with an axial vector magnetic

charge density a

−→
jmf (and [ a

−→
Jmf ]=L3[ a

−→
jmf ]), but without a scalar magnetic charge (a monopole). See Appendix B

and figure 3 in Appendix F.

For any one of the leptons or of the quarks (f fermion), the source of the fields, we propose:
p∇. p−→E f =

1

ϵ0
aρef , − a∂t p

−→
E f + c2 p∇× a

−→
Bf =

1

ϵ0
p

−→
jef =

1

ϵ0
aρef p

−→vf

p∇. a

−→
Bf = p0mf , a∂t a

−→
Bf + p∇× p

−→
E f = − s′

ϵ0
a

−→
jmf

, (19)

s′∈{1,−1} still undetermined, although Maxwell-Dirac equations suggest: s′=1.

In particular for the neutrinos:


p∇. p−→E ν = a0eν , − a∂t p

−→
E ν + c2 p∇× a

−→
Bν = p

−→
0eν

p∇. a

−→
Bν = p0mν , a∂t a

−→
Bν + p∇× p

−→
E ν = − s′

ϵ0
a

−→
jmν

. (20)

In the third equation in (19) and in (20): if different from zero, then there would be a magnetic monopole charge,
neither a lepton nor a quark. Therefore it has to be with zero polar scalar magnetic charge (the monopole).

With the massive neutrinos subject to electromagnetism and therefore ruled by possible electromagnetic Maxwell
and Lorentz type equations, we need to force the fourth equation of the system, possibly like in (20). In (20) but

without the term in the right hand side in the fourth equation ( a
−→
jm,ν) we would be driven to the same equations

that provide the electromagnetic waves in empty space. We admit this term to distinguish them, and afterwards we
implement the same right term of the fourth equation in (20) for any elementary fermion (fourth equation in (19)).



6

The divergence in the fourth equation in (19), with ∇.(∇×−→E ) = 0 and the third equation drive to:

∇.−→jmf = 0mf ⇐⇒
s

⃝S
−→
jmf.−→ds =

t
⃝V ∇.−→jmf d3r = 0mf , (21)

like with the steady currents in magnetostatics. Any arbitrary region V . The circles “⃝” indicating a volume
enclosed in a surface, or that closed surface, also a closed curve enclosing a surface.

But an elementary fermion is taken here as a ‘punctual’ charge. We should write our equations in integral form.
δ(3) the Dirac’s delta measure in the three dimensional space, with the convention for its unit dimensions [δ(3)] = L−3.

And there are not velocities for defining a

−→
jmf and a

−→
Jmf ; there are not (scalar) magnetic monopoles. Later on

we will impose velocities associated to a

−→
Jmf . Under this assumption (‘punctual’ charges) we should read:

a1) a

t
⃝V aρef (

−→r ′) d3r′ = a

t
⃝V aqef (

−→r ′) δ(3)(−→r ′−−→r ) d3r′ = aqef (
−→r ) ,

a2) a

t
⃝V p

−→
jef (
−→r ′) d3r′ = a

t
⃝V aqef (

−→r ′) p
−→vf (
−→r ′)δ(3)(−→r ′−−→r ) d3r′= aqef (

−→r ) p
−→vf (
−→r ) ,

and: b ) a

t
⃝V a

−→
jmf (
−→r ′) d3r′ = a

t
⃝V a

−→
Jmf (

−→r ′) δ(3)(−→r ′−−→r ) d3r′ = a

−→
Jmf (

−→r ) ,

(22)

in the second one we set: aqef p
−→vf =p

−→
Jef . But at this point there is not an involved velocity with a

−→
Jmf (

−→r ) .

The extra term a

−→
jmf or a

−→
Jmf (

−→r ) makes advisable further research (conservation laws). [16] [17] [18] [19]

We have added a source for the electromagnetic fields of the leptons and the quarks with:

an axial vector magnetic charge ( a

−→
Jmf ), and

without a polar scalar magnetic charge (a monopole: pρmf ≡ p0mf ).

We write equations (19), with the ‘punctual’ source of the fields (an elementary fermion), in integral form:
p

s
⃝S p

−→
E f.−→ds′ = 1

ϵ0
aqef , − a∂t a

s
A p

−→
E f.−→da′ + c2p

∫
⃝C a

−→
Bf.−→dl ′= 1

ϵ0
aqef a

s
A
δ(3)(−→r ′−−→r )

[
p
−→vf .−→da′

]
p

s
⃝S a

−→
Bf.−→ds′ = p0mf , a∂t a

s
A a

−→
Bf.−→da′ + p

∫
⃝C p

−→
E f.−→dl ′= −s′ϵ0

[(
a

s
A
δ(3)(−→r ′−−→r ) a

−→
Jmf .−→da′)] , (23)

the last term in the second equation (relate it to the intensity of a current) and especially the one in the fourth
equation (relate it to (21)) require more research. Formally, we have written:

aρef (
−→r ′) = aqef (

−→r ) δ(3)(−→r ′−−→r ) and p

−→
jmf (

−→r ′) = p

−→
Jmf (

−→r ) δ(3)(−→r ′−−→r ).

For the charged leptons: see the comment after equation (30) and later (39) and figure 1 in Appendix F.

Also, if we choose a surface with
−→
da′ parallel to p

−→vcl it is
[
a

−→
Jmcl.−→da′] = 0.

It is implicit our geometrical interpretation of the elementary fermions for this last result. [5]

For the regions with volumes ⃝V which satisfy:

a

t
⃝V a∂t p

−→
Ef d

3r′= a∂t a
t

⃝V p

−→
Ef d

3r′= p0 and a

t
⃝V a∂t a

−→
Bf d

3r′= a∂t a
t

⃝V a

−→
Bf d

3r′= a0 ,

the second and the fourth equations in (19) would drive to the equations:

a

t
⃝V p∇× (c a

−→
Bf ) d

3r′ = p
−→vf
c

aqef

ϵ0

, a

t
⃝V p∇× p

−→
E f d

3r′ = −s′ a

−→
Jmf

ϵ0

.

An interesting direct consequence of (19) in the equations:
□2 −→E f =

[
− ∂tt + c2∇2

] −→
E f = 1

ϵ0

(
∂t
−→
jef +

[
s′ c2 ∇×−→jmf

]
+ c2 ∇ρef

)
□2 −→Bf =

[
− ∂tt + c2∇2

] −→
Bf = 1

ϵ0

( [
s′ ∂t
−→
jmf

]
− ∇×−→jef

) , (24)

Also, with the standard Maxwell’s equations we have: ∇× (
−→
E f + ∂t

−→
Af ) =

−→
0 =⇒

−→
E f + ∂t

−→
Af = −∇φf ;

but here it is: ∇× (
−→
E f +∂t

−→
Af ) = − s′

ϵ0

−→
jmf . At the macroscopic level this can have important implications

if the directions of the
−→
jmf are not randomly distributed, for example with the ferromagnetism.

We have obtained the previous modifications in a straightforward way. This is not the case for the following ones
(next subsection), leaving various open possibilities, and therefore the necessity of an even deeper research. From
here on (and mainly after (37)) the above mentioned geometrical hypothesis for the elementary fermions is crucial.
Obviously we require experimental verification or either disapproval by looking at already known experimental results.



7

B: In the pathway of the Lorentz force law.

We are interested in the force exerted by the electromagnetic fields over an elementary fermion. We propose:

generalized formula for the force over a lepton or a quark due to the electromagnetic fields:

p

−→
Ff =

[
p

−→
F elec +

{
p

−→
F amp

]
+ p

−→
F spin

}
+ p

−→
Fm,E =

=
[

p

−→
F−→

E,qe
+

{
p

−→
F−→

B,qe

]
+ p

−→
F−→

B,
−→
Jm

}
+ p

−→
F−→

E,
−→
Jm

.

(25)

The square brackets indicating the classical Lorentz force law. The curly brackets: the previous form of the action
of the magnetic field over the electrical currents (here denoted Amperian) and the new one due to that action, but
now over the axial vector magnetic charge (the vector spin) via the “intrinsic” magnetic dipole moment. The fourth
term, the action of electrical fields over the axial vector magnetic charge.

Classical electromagnetism.

We write the Lorentz force law
−→
F = q (

−→
E +

−→v × −→B ), for a localized continuous distributions of charge

(classical electromagnetism) with electric charge density ρ and with a current density
−→
je = ρ−→v , in a spatial

volume V , due to the electromagnetic fields
−→
E and

−→
B in the following form: [20]

−→
F (⃝V (r), t) =

t
⃝V

{
ρ
[−→
E +

−→v ×−→B
]}

(−→r ′, t) d3r′ =
−→
F e +

−→
F b . (26)

We consider a region of the space where the magnetic field is a smooth function of the space, with a Taylor series:

−→
B(−→r ′) =

−→
B(−→r ′)|−→r ′=−→r + (−→r ′−−→r ).

(
∇−→r ′′

−→
B(−→r ′′)|−→r ′′=−→r

)
+ . . . , (27)

for every point −→r which eventually can be taken −→r =
−→
0 .

For the essentially ‘punctual’ charges of the previous subsection we write (26) in the standard Lorentz form:

p

−→
F ef = aqef p

−→
E , p

−→
F bf = aqef p

−→vf × a

−→
B

(
a

−→
B =

−→
B(−→r ′)|−→r ′=−→r =

−→
P

)
. (28)

But, with the second term in (27), we write
t

⃝V
−→
je×

[
(−→r ′−−→r ).(∇−→r ′′

−→
B(−→r ′′)|−→r ′′=−→r

)]
d3r′, which lets us describe

another type of force: the force on a magnetic moment (problem 6.22 in [2], [9] and equations (2)-(15) in [21]).
We now impose localized and steady currents, obtaining:

a) ∇.−→je = 0 and b)
t

⃝V
−→
je d

3r′= 0 =⇒
(t

⃝V
−→
je d

3r′
)
×
(−→
B(−→r ′)|−→r ′=−→r

)
= 0 , (29)

and driving to: a
−→µamp = 1

2

t
⃝V p
−→r ′× p

−→
je(
−→r ′) d3r′

aU = − a
−→µamp . a

−→
B(−→r ′′)

p

−→
F b

µ
= − p∇−→r ′′ aU|−→r ′′=−→r

, (30)

This result relies in two conditions: continuity of the mathematical functions, up to a certain border for
−→
je, and

steady currents. It works for macroscopic media, for example, currents in wires. But, with the elementary fermions
(point charges here), which belong to the domain of quantum mechanics, we can write in the same sense expressed
by Griffiths (first paragraph in page 224 in [2]):

in what way could a point charge because of its movement give rise to a steady current, . . . ?

See also, in relation to the magnetic field, the concept of “point magnetic dipole” in: subsection 11.2.3 and chapter 13
in [20] and also [9]. Even though, its works; an example in the quantization of the angular momentum of the orbitals

in the atoms (see formula (3)). Naively, imposing p

−→
jef (
−→r ′) = aqef p

−→vf δ(3)(−→r ′) = p

−→
Jef (
−→r ′) δ(3)(−→r ′),

we can infer from formula (30) for the “punctual charged leptons”:

p

−→
F bf

µ
= p∇−→r ′′

[
a
−→µef . a

−→
B(−→r ′′)

]|−→r ′′=−→r ; a
−→µef =

1

2
p
−→r × p

−→
Jef (
−→r ) ; p

−→
Jef = aqef p

−→vf . (31)



8

Proposal for the electromagnetic force, including the axial vector magnetic charge.

In a first step, tentatively, we propose the followings third and fourth forms for the electromagnetic force over the
elementary fermions in (25), trying to follow closely the forms of the first and second ones:

(1) p
−→
F elec : p

−→
F ef = aqef p

−→
E = (αf ae ) p

−→
E

(2) p

−→
F amp :

↗ p

−→
F bf = p

−→
Jef × a

−→
B = [ aqef p

−→vf ]× a

−→
B = [ p

−→uv,f (β αf ae ) ]× ( c a

−→
B )

↘ p

−→
F bf

µ
= p∇−→r ′′

[(1
2 p
−→r × p

−→
Jef (
−→r )

) . a−→B(−→r ′′)
]|−→r ′′=−→r (Amperian )

(3) p

−→
F spin :


Amperian (A) : p

−→
FAmp

spin = [ p
−→uv,f× (

g0
f

2 a

−→
Jmf )]× ( c a

−→
B )

Gilbertian (G) : p

−→
F Gilb

spin = [ p
−→uv,f

⊥ . ( g0
f

2 a

−→
Jmf ) ] ( c a

−→
B )



(4) p
−→
Fm,E :


(a) − (β) s′′s

{
a
−→uv,f . ( g0

f

2 a

−→
Jmf )

}
p

−→
E

(b) − (β) s′′s
{

a
−→uv,f

⊥
×(

g0
f

2 a

−→
Jmf )

}
× p

−→
E

. (32)

αf = k
3s = k

3 sign(qef ). ∥ p
−→vf ∥= β c and p

−→vf = β c p
−→uv,f . p

−→uv,f

⊥
in the plane containing a

−→
Jmf and p

−→uv,f .

s′′∈{1,−1} pending of an experimental fixing. a
−→uv,f the corresponding axial unit vector (same triad of coordinates).

p
−→vf the velocity of the fermion in the laboratory frame, where we measure the electric and magnetic fields. The

axial vector magnetic charge a

−→
Jmf and the unitary polar vector p

−→uv,f in general in different directions (see Figures

1 and 2). For the neutrinos we set a

−→
Jm,ν parallel to p

−→uv,ν , even being massive, one axial and polar the other.
The charges (scalar and vector) in red color.

B1: The first term (1)-(32) .

• p

−→
F elec = p

−→
F−→

E,qe
. In (28). Nothing new.

B2: The second term (2)-(32) .

• p

−→
F amp = p

−→
F−→

B,qe
:

↗ p

−→
F bf

↘ p

−→
F bf

µ
.

For p

−→
F bf : directly a force over a moving electric charge due to a magnetic field. In (28). Nothing new.

For p

−→
F bf

µ
: we add a restriction, a global null electrical charge and we do account for electrical currents (currents

in neutrally charged electrical wires). Besides, these results applies to one electron in an orbital in an atom, bounded

due to p

−→
F elec . We apply it exclusively to the charged leptons with the Amperian concept for current loops. It

does not account for the interpretation of the spin in the sense of producing an “intrinsic” dipole magnetic moment.
Let us look at it in detail for the magnetic moment (see [2], [7], [9], [20], [21]), and here (31):

p

−→
F amp = − p∇−→r ′′

(
− a
−→µamp . a

−→
B

)∣∣∣−→r ′′=0
= − p∇−→r ′′ aU

∣∣∣−→r ′′=0
,

(
a
−→µamp =

1

2

∫
p
−→r × p

−→
J (−→r ) d3r

)
. (33)

In particular, we write for an electron (k = 3) in an orbital in an atom, still without spin (formula (3)):

a
−→µamp =

1

2 p
−→r ×

(
qf,k p

−→vf
)∣∣∣f=e

k=3
= sign(qf,k)

k

3
µ
B

1

ℏ a

−→
L“orb ”

∣∣∣f=e

k=3
,

(
a

−→
L“orb ” = p

−→r×(me p
−→ve )

quantified

)
. (34)

Up to this point, in a similar way as in the previous part (classical electromagnetism).



9

The relationship of this second term p

−→
F bf

µ
with the third term p

−→
F spin = p

−→
F−→

B,
−→
Jm

((3)-(32)).

Pay attention to equations from (2) to (6). In these equations we add a term in the following form, the Amperian
concept of the intrinsic magnetic dipole moment, with only two possible values:

µ
B

( 1

ℏ a

−→
L“orb ” + “

1

ℏ
g0e
2 a

−→
Le,spin”

)
,

(
“
−→
Le,spin∼

−→
r̃×

(
me

−→
ṽ

)
” ?

)
. (35)

But, neither
−→
r̃ nor

−→
ṽ seems to have any physical or geometrical meaning; they could be related with what it

was denoted an “internal space” (unknown). In this study, the unit space vector a
−→u r also in the “internal space”.

The term ‘spin’ associated to some kind of rotation, and used with different connotations:

as a ‘scalar’ with two assigned values, ‘up’ or ‘down’, intrinsic, without specific three dimensional
orientation. Once an electron with one of the values, such value does not change, without minding in what way
(direction) the electron could be moving, while not interacting,

with a g0e [22], when subject to a magnetic field it makes its appearance through the ”internal” vector
magnetic dipole moment for its interaction with such external magnetic field. This vector magnetic dipole moment
with an orientation in the standard three dimensional space: the mixture of equation (33) with equation (35). See
later, with the Stern Gerlach experiment (only 2 values), the positions of Einstein, Ehrenfest, Schwinger and Feynman,

also with a ge, in Quantum Electrodynamics (QED) there appear spin matrices representing directions
in relation to the calculation of the form factors, with σµν (pν

′−pν) . It is illustrative to look at equations from (6.35)
to (6.37) in Peskin [23], particularly the two previous to (6.37). Here, equations (5) and (6).

In our equations, the characteristic vector (“internal”) corresponds to the spin, but it avoids the concept of a three
dimensional axis of rotation. Remind the very illustrative paragraphs in Gerard ’t Hooft’s book [1] and the preliminary
section.

In brief, the magnetism is related to two different physical entities: first, the Amperian type (currents in wires)
which is applicable to orbital electrons in atoms, here expressed in equations (33) - (34) (after (2)-(32)), and second,
the spin type. Customarily this second one represented also as Amperian in the way of equations in (35) with (6), with
the definition of the Bohr magneton µ

B
and only two possible values. A guidance in the Stern Gerlach experiment.

Here, we explore it in the form developed next.

B3: The third term (3)-(32) .

• p

−→
F spin = p

−→
F−→

B,
−→
Jm
. Instead of the usual interpretation, in this study we propose for the magnetic part of

the electromagnetic force over an elementary fermion, due to its axial vector magnetic charge, the two possibilities in
(3)-(32). They are simple formulations for this type of force, once assumed that the elementary fermions can not be
monopoles (see Appendix B and equations (19)) or their combination, and after including the axial vector magnetic
charges. This procedure entails the inclusion of the neutrinos as it avoids the usage of the electrical charge concept.
The geometric arguments play an essential role ( p

−→uv,f).

We assume for the elementary fermions a point-like structure in the sense that they do not have any further
“physical structure”, they are not composed of other “even more elemental particles”, although we conjecture for
them a ‘geometry’.

In order to write (3)-(32) we have established the following correspondences with previous expressions in (32):

↗ (A) :
{
[ p
−→uv,f× (

g0
f

2 a

−→
Jmf )]×

}
Amperian, like in (2)-(32) a vector ( p

−→uv,f (β αf ae) )

↘ (G) :
{
[ p
−→uv,f

⊥ . ( g0
f

2 a

−→
Jmf )]

}
Gilbertian, like in (1)-(32) a scalar (αf ae )

.

We have suppressed the speed factor ( β ) to express the no dependence in the speed, like with the electric charge in
(1)-(32) and also like in equation (6).

(A) The Amperian form. In a similar way to the one used to define (26), we define:

−→
FAmp

spin (⃝V (r), t) =
t

⃝V

{
[ p
−→u v,f× (

g0
f

2 a

−→
jmf )]× ( c a

−→
B )

}
(−→r ′, t) d3r′ . (36)

Again, we proceed as in [9] and in (2)-(15) in [21], with the implementation of the Taylor series in (27), and we

express:
−→
FAmp

spin (V (r), t) =
−→
F 11 +

−→
F 12 + . . . (the first and the second orders):

i)
−→
F 11 =

t
⃝V

{
[ p
−→u v,f× (

g0
f

2 a

−→
jmf )]×

(
c
−→
B(−→r ′′)|−→r ′′=−→r

)}
(−→r ′, t) d3r′

ii)
−→
F 12 =

t
⃝V

{
[ p
−→u v,f× (

g0
f

2 a

−→
jmf )]×

(
c (−→r ′−−→r ).(∇−→r ′′

−→
B(−→r ′′)|−→r ′′=−→r

))}
(−→r ′, t) d3r′

. (37)



10

ii) We are able to write
−→
F 12 like the usual one (

−→
F = ∇(−→µ .−→B)), the Amperian one like (15) in [21] and here

(30), driving to (31) for the punctual charge. We make the substitution p

−→
Jef (
−→r ) −→ [ c p

−→uv,f× (
g0
f

2 a

−→
Jmf ) ] :

−→
F 12 = p∇−→r ′′

[{
1
2 p
−→rf ×

(
c
g0
f

2 s ae [ p
−→uv,f× a

−→σf ]
)}. a−→B(−→r ′′)

]∣∣∣−→r ′′=−→rf
=

= p∇−→r ′′

[ {
s
g0
f

2 µ
B

p
−→rf
le
×[ p
−→uv,f× a

−→σf ]
}. a−→B(−→r ′′)

]∣∣∣−→r ′′=−→rf
=

= p∇−→r ′′

[ {
s
g0
f

2 A−1
f µ

B p
−→u r,f ×[ p

−→uv,f× a

−→σf ]
}. a−→B(−→r ′′)

]∣∣∣−→r ′′=−→rf
=

= p∇−→r ′′

[
s
g0
f

2 A−1
f µ

B
sin(2φf,k) a

−→uµφ
. a−→B(−→r ′′)

]∣∣∣−→r ′′=−→rf
= p∇−→r ′′

[
a
−→µφ,f. a−→B(−→r ′′)

]∣∣∣−→r ′′=−→rf
,

(38)

like in (31). We have used (10) ce=µ
B

2
le
=µ

B
A−1

f
2
lf
. All the ′′−→u ’s′′ indicating unit vectors.

We have imposed and defined the following geometrical conjectures:

p
−→rf = lf p

−→u r,f = A−1
f le p

−→u r,f , p
−→u r,f = p

−→uv,f (both imposed)

p
−→u r,f ×[ p

−→uv,f× a

−→σf ] = sin(2φf,k) a
−→uµφ

p
−→uv,f

⊥
= p
−→u r,f

⊥
∥ a
−→uµφ

a
−→µφ,f = s

g0
f

2 A−1
f µ

B
sin(2φf,k) a

−→uµφ

. (39)

Where p
−→rf ( p

−→u r,f ) are in the “internal space”; their meaning different to the one imposed for the angular momentum.
In favor of this geometrical interpretation the essentially orthogonal polarizations of electrons and neutrinos, The

spin vectors (the characteristic axes) for the neutrinos ′almost′ in the direction of their velocities due to its ′almost′

zero mass, and accordingly to the polarizations, the spin vectors (the characteristic axes) of the electrons in a plane
′almost′ perpendicular to their velocities.

For the neutrinos it is: φ
k=0

= 0 , and therefore a

−→σν is parallel to p
−→uv,ν. This implies: a

−→µφ=0 = a

−→
0 . In this

way, the neutrinos are included, obtaining for them: a
−→µν = a

−→
0 .

For the other elementary fermions: φ
k={1,2,3}

∈
{
π
12 , π6 , π4

}
. For all these values of k it is a

−→uµφ perpendicular to

p
−→uv,f, with the first one axial and the second one polar. In the definition of the magnetic dipole moment in equation

(6), in a
−→µe we replace in the right hand side term the (

−−→
−σ) by plus or minus a

−→uµπ
4
. See the graphic in Figure 1.

With (36)-(39) we express: p

−→
FAmp

spin =
−→
F 11 − p∇ aUf + . . . with aUf = − a

−→µφ,f . a

−→
B .

i) Let us now inquire about
−→
F 11. We have to do

−→
F 11 =

−→
0 . The parallel result to (29) is:

a) ∇.−→jmf = 0 in (21),

but we need to prove here: b)
(t

⃝V [ p
−→u v,f×

−→
jmf ] d

3r′
)
×
(−→
B(−→r ′′)|−→r ′′=−→rf

)
=
−→
0

. (40)

We have not proven b) (compare with b) in (29)). In a preliminary step for the proof, with −→r ′ a position vector

and
−→
F a vector function, we could use: ∇−→r ′.{−→u r′× [r′

−→
F (−→r ′)]} = 0. For the neutrinos it is: [ p

−→u v,ν×
−→
jmν ] =

−→
0 ;

and for the charged leptons it is:
−→
jm,cl perpendicular to p

−→u v,cl = p
−→u r,cl .

In a different way, acting like in (28), which inherits the essentially ‘punctual’ character of the particles:

−→
F 11 = s

g0
f

2 c ae [ p
−→uv,f× a

−→σf ]× a

−→
B(−→r ′′)|−→r ′′=−→rf

=

= s
g0
f

2 c ae sin(2φf,k) p

−→
ũv,σ,f × a

−→
B(−→r ′′)|−→r ′′=−→rf

=

= s 2
lf
∥a
−→µφ,f ∥ p

−→
ũv,σ,f × a

−→
B(−→r ′′)|−→r ′′=−→rf

.

(41)

For electrons with perpendicular trajectories to the magnetic field and with
−→σe parallel to

−→
B , we have the

behavior: ∥
−→
F 11 ∥ = 2

le
∥
−→
B ∥

∥∇
(−→uµe.−→B)

∥
∥
−→
F 12 ∥ ∼ 1013 ∥

−→
B ∥

∥∇
(−→uµe.−→B)

∥
∥
−→
F 12 ∥ . The action of the magnetic field

could produce a very large positive or negative acceleration of the particle in the tangent of the trajectory.

The mathematical treatment involved in differentiating (40) from (41) have to be clarified. Should we include a

Dirac δ to cancel the
−→
F 11 term?



11

(G) For the second expression in (3)-(22):

p

−→
F Gilb

spin =
−→
F 22 +

−→
F 21 + . . .


−→
F 21 = c

g0
f

2 s ae [ p
−→uv,f

⊥ . a

−→σf ]
(

a

−→
B(−→r ′′)|−→r ′′=−̃→rf

)
−→
F 22 = c

g0
f

2 s ae [ p
−→uv,f

⊥ . a

−→σf ]
(
( p
−̃→rf. p∇−→r ′′ ) a

−→
B(−→r ′′)|−→r ′′=−̃→rf

) . (42)

Our handling of the Gilbertian term follows similar steps, now the similitude with (1)-(32). For
−→
F 22 :

−→
F 22 = s

g0
f

2 µ
B
A−1

f
2
lf

cos(π2− 2φf,k) ( p
−→uv,f

⊥. a−→uµφ
) ( p
−̃→rf. p∇−→r ′′ ) a

−→
B(−→r ′′)|−→r ′′=−̃→rf

=

=
(
s
g0
f

2 A−1
f µ

B
sin(2φf,k) a

−̃→u r,f . p∇−→r ′′

)
a

−→
B(−→r ′′)|−→r ′′=−̃→rf

=
(

a
−→µφ,f . p∇−→r ′′

)
a

−→
B(−→r ′′)|−→r ′′=−̃→rf

,

(43)

with p
−→uv,f

⊥. a−→uµφ
= p1 and a

−̃→u r,f = p1
2
lf p
−̃→rf = a

−→uµφ
(imposed).

We easily observe the similarities with classical electromagnetism. In order to obtain the same magnetic field, at
least at large enough distances of the magnetic dipole (moments), the r in m = 1

2

∫
(r × J)d3r and the r̃ in

m = (+g− (−g))r̃ (two opposite magnetic monopoles) are perpendiculars to each other. See equations (12) and (19)
in [21] and the graphic representations of the fields of a “pure” and of a “physical” dipole in Figure 5.55 in [2] (page
255).

And the term
−→
F 21 :

−→
F 21 = s

g0
f

2 c ae sin(2φf,k) p1 a

−→
B(−→r ′′)|−→r ′′=−̃→rf

= p

{
s 2
lf
∥a
−→µφ,f ∥

}
a
−→
B(−→r ′′)|−→r ′′=−̃→rf

, (44)

Using a similar argument to the one after (41), this force would have a very large action in the direction of the
magnetic field, which we know it is not the case. This can be an argument against the Gilbertinan interpretation of
the magnetic moment. Also, experiments with neutrons (Section VII in [21] and [24]) suggest the Amperian character

for the force p

−→
F spin. Therefore we study p

−→
F spin = p

−→
FAmp

spin , even without electric charge.

We will not further discuss these questions. One way to avoid them consists in establishing, instead of the first
expression in (3)-(32), a fundamental law for this force, also valid for neutrinos, in the form:

p

−→
F spin = p

−→
FAmp

spin = − p∇ aUf with aUf = − a
−→µφ,f . a

−→
B (equation (38)),

and to reject the (G) interpretation.

Griffiths [2] (page 378) writes about a term in the Lorentz force law with such form ( ∇ (m.B) ) :

“in any event it is not classical electrodynamics”.

Obviously this is the case: the spin is not classical physics.

We can compare the formula (14), prior to the radiative corrections (∆g
f
= 0), with the last formula in (39), the

absolute values and the norms (see Appendix C):

| µf,k |∆g
f

=0
|

∥a
−→µφ,f ∥

=
k
3

sin(2φf,k)
=


1/3

sin(π/6)
= 2

3 = 0.66 d-quarks

2/3

sin(π/3)
= 4

3
√
3
= 0.77 u-quarks

1

sin(π/2)
= 1 charged leptons

a
−→µν =

−→
0 neutrinos

. (45)

Let us interpret these equations for the elementary fermions:

a) for the quarks. Unluckily, the results of the equation in (38) and particularly (45) tell us their irrelevancy for the
quarks: at the experimental level and at least at present day, the quarks under the strong force are always confined
and we also do not know their masses with enough precision (see Appendix C). This argument applies to the fourth
term (4)-(32) too. Therefore we are not going to consider them any further,

b) for the neutrinos. With a

−→σν parallel to p
−→uv,ν , which implies φk=0 = 0 and therefore sin(2φk=0) = sin(0) = 0,

it is: p

−→
F spin|ν = p

−→
0 the magnetic fields do not have action over the neutrinos,

c) for the electrically charged leptons (cl): 2φcl,3 = π
2 =⇒ sin(2φcl,3)=sin(π2 )=1 , with ∥a

−→µ π
4 ,f ∥=

g0
cl

2 A−1
cl µ

B
,

and therefore a
−→µ π

4 ,cl = s ∥a
−→µ π

4 ,cl ∥ a
−→uµπ

4

= s µcl,3 a
−→uµπ

4

, with µcl,3 in (14) (equal results).



12

The Stern Gerlach experiment.

We now apply these results to the classical Stern Gerlach type experiments. Instead of the original uncharged silver
atoms, we initially consider electrons, with their trajectories perpendicular to the magnetic field and taking apart the

cyclotron effects due to the force p

−→
F amp = p

−→
F bf . The Lorentz force law drove for the magnetic dipoles to a force

of the form p

−→
F amp = p

−→
F bf

µ
= −∇(−→µ.−→B). This is also the result obtained with (38).

But, already in 1922, Einstein, Ehrenfest and later Schwinger and Feynman pointed out the difficulties for explaining
the only two specific values [25] , [26]. The question relies on whether the dominance of:

(a) the rotation of the magnetic dipole moment direction, originally in an arbitrary direction, around the lines of
the magnetic field due to a torque (dynamical) with the Larmor frequency, or

(b) over a different type of rotation that tends to align the intrinsic magnetic moment with the magnetic field, not
rotating any longer once aligned, likewise a compass in the magnetic field of the Earth.

The role of the measurement process is important and it is treated in [27], [28] with a spin relaxation and in a
different way in [29], [30] (with the wave function).

There is a noticeable difference in our treatment: we have imposed the inclusion of the characteristic axis
and therefore of the intrinsic magnetic moment in a plane perpendicular to the velocity of the charged
lepton. We write some first approach qualitative comments.
• In the experiment, in the limiting zones, the magnetic field varies in a small region from zero to the values inside

the magnet. At first the magnetic field is zero and there is not a “Larmor rotation”. The electrons (or the silver
atoms) are not relativistic.
• As the particle moves to enter in the magnet, the magnetic field increases and it starts the alignment (only two

final possibilities, up and down) competing with the “Larmor rotation”. But this rotation would force a change in the

direction of the velocity due to our definition of a
−→uµe

, always perpendicular to p
−→uv,e . This change could imply an

′arbitrary′ large energy to produce such a change in the direction of the velocities as they depend in the ′arbitrary′

directions of the internal magnetic moment:

could we conclude the predominance of the alignment over the “Larmor rotation”,
prior to the entrance in the magnet (a zone with a weak and increasing magnetic field)?

• In what way do we have the “Larmor rotation”? We can introduce it via QED with the action of the radiative
corrections. We assume a modification of the vector intrinsic magnetic moment by adding a very small vector in
any direction. Experimental results with the anomalous magnetic moments tells us of a small modification related
via the gyromagnetic ratio ge = g0e+∆ge = 2+∆ge (see equation (6) and in [23] with σµνqν after their equation
(6.32) ). The result of this modification can be in any close direction to the original one. Once the particles enter
the magnetic field and with the predominance of the alignment described above, we have the “Larmor rotation” of
a slightly modified axis (of a

−→µφ̃,cl ) around the lines of the magnetic field, with a very small angle between them.
This one is the fact that permits this “Larmor rotation”, which is independent of the value of the angle.

Schematically: with 2̃φ = π
2 +∆φ we have a “Larmor rotating” vector a

−→µφ̃,cl. Again, with equation (6) we

wrote ge = g0e+∆ge; we now write: a
−→µφ̃,cl = a

−→µ π
4 ,cl+∆ a

−→µ∆φ,cl with a
−→µ π

4 ,cl parallel to a

−→
B . The condition of

the orthogonality of the vector intrinsic magnetic moment and the velocity forces a helical trajectory for the charged
lepton. The helix with an irregular but periodic pitch and a very small radius. Perhaps this reminds of the old concept
‘Zitterbewegung’ [31].

The above explanation would also justify the results of Rabi (1929) [25] (pages 24, 25) and [26] (pages 58, 59) using
a homogeneous magnetic field but with the bean in a non perpendicular direction to the magnetic field.

Original experiments with neutral atoms, complex systems, still require to add some more explanations.

B4: The fourth term (4)-(32) .

• p

−→
Fm,E = p

−→
F−→

E,
−→
Jm
.

(a) Now the guidance is with p
−→
F elec = (αf ae) p

−→
E (the first one in (1)-(32)), and we write it in the form:

p

−→
Fm,E = −(β) s′′

g0
f

2 ae
[
a
−→uv,f . a−→σf

]
p

−→
E = −(β) s′′

g0
f

2

2

c le
µ
B

[
a
−→uv,f . a−→σf

]
p

−→
E . (46)

It is not clear the β dependence; in any case, experiments would decide. A possible experiment for neutrinos in
Appendix E.

For the charged leptons it is: p

−→
Fm,E|cl = p

−→
0 , due to the perpendicularity of a

−→σcl with a
−→uv,cl .



13

And for the neutrinos, due to the parallelism of a

−→σν with a
−→uv,ν , we obtain:

generalized formula for the electromagnetic force over a neutrino:

p

−→
Fν = − (β) s′′

g0ν
2 ae p

−→
E . (47)

(b) This second one towards the form of the Lorentz-Dirac force law, although without monopoles. We need some

other rule to fix a
−→uv,f

⊥
. For the neutrinos with: p

−→
Fm,E = −(β) s′′ g

0
ν
2 ae

[
a
−→uv,ν

⊥
× a

−→σν

]
× p

−→
E ( in (4)-(32)).

Three more simple possibilities with:

(4) p
−→
Fm,E :



(c) − (β) s′′s
{

a
−→uv,f

⊥ . ( g0
f

2 a

−→
Jmf )

}
p

−→
E

(d) − (β) s′′s
{

a
−→uv,f × (

g0
f

2 a

−→
Jmf )

}
× p

−→
E

(e) − (β) s′′s
{

(
g0
f

2 a

−→
Jmf )

}
× p

−→
E

(c) and (d) in order to follow the forms expressed in (3)-(32) and (e) again with the Lorentz-Dirac form.

We summarize: charged leptons neutrinos

p

−→
Fm,E : −s′′(β)

g0
f

2 ae



{
a
−→uv,f . a−→σf

}
p

−→
E −→ p

−→
0 , , p

−→
E (a)[

a
−→uv,f

⊥
×a

−→σf

]
× p

−→
E −→ p

−→
0 , ,

[
a
−→uv,ν

⊥
×a

−→σν

]
×p
−→
E (b){

a
−→uv,f

⊥ . a−→σf

}
p

−→
E −→ p

−→
E , , p

−→
0 (c)[

a
−→uv,f×a

−→σf

]
× p

−→
E −→

[
a
−→uv,cl×a

−→σcl

]
× p

−→
E , , p

−→
0 (d)

a

−→σf × p

−→
E −→ a

−→σcl×p

−→
E , , a

−→σν×p

−→
E (e)

.

There are reasons against the last three: electrons would be notoriously affected in arbitrary electric fields (apart
from (1)-(22)). On other side, (c) and (d) would represent a null force on the neutrinos.

Two different types of research deserve mention in relation to our proposed generalization of the Lorentz force law:

first, some researchers studying the force on magnetic dipoles, using internal forces and “hidden momentums”,

add a term and they write for the force the formula:
−→
F = ∇(−→µ.−→B) − d

dt
(c−2−→µ×

−→
E ) . This has been throughly

examined by Griffiths [16], and also in [17], [18], [19],
second, the model proposed by Rashba (1957) for the action of electric fields over spin-orbit couplings, [32], [33];

also including a relation with spintronic [34] [35].

The following commentaries are pertinent for the neutrinos, [36], [37], [38]

i) an important consequence of the massive neutrinos is their possible electromagnetic interaction, “a new
physics”. At present time, experimental results are negative: there is not a magnetic dipole moment up to 10−11µ

B
or

10−12µ
B
. The frontier value for not having this “new physics ” in the order of 10−19µ

B
, still far from reachable in any

experiment [36]. Let us pay attention to µ
B
. It was obtained in relation to the electric charge and mass of an electron

and used for the muon and quarks (in the way of (13)). Continuing that process, the value for the µν should be 0
as we would write k = 0 in (14) (electrically chargeless neutrino) even with A−1

ν very large, a consequence of a mass
of, at most, a few electron-volts for a neutrino. This means that µ

B
is taken exclusively as a unit of measurement.

In brief: A−1
ν k µ

B
(k = 0) or

qνℏ
2mν

(qν = 0) either are 0 or have no meaning for neutrinos,

ii) in this research, the neutrinos have an axial vector magnetic charge a

−→
Jm,ν with norm Jm,ν = e ̸= 0 but with

null magnetic moment a
−→µν= a

−→
0 (see Appendix F, figure 1). This is due to the characteristic axis (spin) always in

the direction of the velocity and both under a vector product (equations in (39)). Afterwards (perhaps) it could be

added a magnetic moment, due to virtual processes, with: µν=(
gν
2 ) µ

B
< (10−11) 10−23 = 10−34 Jt̂−1.

iii) the neutrinos are under the action of electrical fields p

−→
F ν (in (4)-(32)), although their electrical charge is 0,

but they are not affected by magnetic fields ( p
−→
F−→

B,
−→
Jm,ν

=p

−→
0 ), although their magnetic charge (Jm,ν = e) is not 0.

This is so with the ruling equation (47), up to the possible effect of virtual processes (previous comment).



14

III. DISCUSSION. CONCLUSIONS.

We can adopt one of these two positions, to keep some of the defined magnitudes in an “internal space” (unknown)
in a similar way as in the Amperian model, or to implement a model which would explain these magnitudes. This
last is the position of this author as mentioned above envisaging a geometrical model for the elementary fermions [5].

The pathway, in this research, is as follows.

In order to answer the question in the Preliminary we proceeded to the construction associated to equations (7)-(18).
We have started with the conjecture in (7). This first conjecture suggests the non null masses of every elementary
fermion, including the neutrinos. And we also obtain unexpected results: there are hints for some features of the
electromagnetic phenomena that were not accounted previously. Is this a possible “new physics”, beyond the Standard
Model? The initial key point is in the concept of an axial vector magnetic charge, our second conjecture (in (15)).

The conjecture in (7) jointly with our definition of electric charge (9) and (b)-(12) involve the definition of four
characteristic magnitudes mf , lf , tf , bf , corresponding to {M , L , T , B }, for every elementary fermion. A
question lays in: what are the actual values and the physical or geometrical significance of these four magnitudes?
We have seen the results for the electrons (charged leptons), hardly for the quarks, and nothing for the neutrinos
(we do not know their masses); there are not enough experimental data. More or less we understand the significance
of mf ; that is not the case for lf , tf and bf .

We have been guided by the following idea: there is a vector magnetic charge (spin vector) which we conjecture
in equation (15), and an electric charge (scalar), both axial sources and independent of each other, although deeply
interconnected in the physical phenomena: the electromagnetic fields. The neutrinos are singular cases: they do

not have electric charge and therefore they can not produce an electric current ( a
−→
Je,ν = a

−→
0 ). The definition of an

intrinsic magnetic moment for them as an Amperian concept is questionable.

The elementary fermions have four different values of the electric charge, without taking account of the sign, which
we relate with four different values of a constant (and of an angle variable) to a unit of electric charge ae.
But, they have only one value of the spin with two projections (as scalars). Its measurement involves the intrinsic

dipole magnetic moment, customarily considered as Amperian, which implies a non null electric charge, therefore it
should not be valid for the neutrinos. In order to solve this problem, it is in our aim to avoid the electric charge as
the source for the intrinsic dipole magnetic moments. This motivates the definitions (15)-(18) in a first step, with a
vector character for these sources.

Previous comment, at once with the form of the equations and the dimensional analysis of the formulas in (19),
suggest the vector character for the basic magnitude representing the internal magnetic property (the spin) of the
elementary fermions. This vector in the definition (15), in terms of the characteristics mass, length - time and magnetic
field; and in (16)-(17) with the same unit: the electric-magnetic charge ae. But there are various specific (discrete)

values of this vector, with the characteristic axes
−→σf (φk). Behind the definition: k

3 = 4
π φ

f,k
, up to a cylindrical

symmetry. They are graphically represented in the Figures 1 and 2 in Appendix F. Our analysis forbids the inclusion
of polar scalar magnetic charges (monopoles) in the definition of the elementary fermions (see Appendix B).

Let us return to the intrinsic magnetic moment µf,k (scalar). In a first approach we define it in terms of the
parameter imposed for the electric charge, k, in formula (18); this is coincident with the results of the Amperian
treatment. Later on, by using a geometrical argument we propose its vector definition in equations (39), this time in

terms of a new parameter, an angle φ
f,k

. The relation between them, thanks to the vector character: k
3 ←→ sinφ

f,k
.

We write it in the form a
−→µφ,f . Up to this point our arguments are based only in the electromagnetism with a

generalization of the Maxwell’s equations and of the Lorentz force law.

Afterwards, the necessity for a rectification of this value due to the inclusion of the radiative corrections, with
implications for the value of the gyromagnetic ratio. The radiative corrections permit slights vector corrections of
the internal dipole magnetic moment in any close direction to the exclusively electromagnetic one with an average
⟨ a
−→µφ,f (gf ) ⟩ assumed in the original direction a

−→µφ,f (g
0
f
); except, perhaps, for the neutrinos (due to the zero value).

The vector aspect implies:

∥a
−→µφ,f (g

0
f
)∥ − ∥ ∆ a

−→µφ,f (∆g
f
)∥ ≤ ∥a

−→µφ,f (gf )∥ ≤ ∥a
−→µφ,f (g

0
f
)∥+ ∥ ∆ a

−→µφ,f (∆g
f
)∥ .

See the chapters 6 and 7 in [23] with the QED treatment for the electrons. In particular we are interested in equations
from (6.33) to (6.35) and (7.46). At the present stage it is not clear the relationship of σµν qν and of our geometrical
hypothesis. Could this be relevant in the specific calculation of the g-factor and its concomitant α? (apply it to the
g-2 muon problem).



15

With the excuse of studying the relationship of the masses of the elementary fermions and their spins, we have been
driven to a generalization of the classical electromagnetism. The spin is a quantum mechanical concept. For us it
is a bridge from the classical to the quantum. We need it in order to explain the magnetism (the Amperian apart)
but we can not understand it classically: the obscure concept of “rotations in an internal space”. In this study we
have based the (internal) magnetism upon the concept of a characteristic axis, avoiding any reference to rotations,
although its defining direction is expressed in terms of the Pauli matrices, which are deeply involved with rotations.
We do not need the standard three dimensional rotations any more for this magnetism (the spin).

We have tried to show up the study as simple and far reaching as possible, with two important drawbacks:

the anomalous magnetic moment. We need to study it with the techniques of the QED which is out of
scope at this level, motivating many of the questions without answer. Here we have posted a possible answer to the
reticence expressed by Einstein and Ehrenfest, later on considered by Schwinger and Feynman; and

a deeper justification of the geometrical foundations and also of their consequences. In this sense, it
reminds the “obscure internal space” concept [39]. An exposition of the geometrical structure of the elementary
fermions is in another study [5]. In addition, the metrical structure of the time space would be very relevant.

In brief, the main points of this study are:

the conjecture in ( 7 ) with the characteristic mass, length, time and subsequently the magnetic part,

the conjecture in (15) for a discrete axial magnetic vector ( a
−→
Jmf ); discrete in two aspects: a magnitude and a

direction in space, an axial characteristic axis (with the a

−→σ vectors),

we define the charges:

{
aqef , axial scalar − (electric)

a

−→
Jmf , axial vector − (magnetic)

, with a unique unit ae

without a pqmf , polar scalar magnetic charge (a monopole),

the generalizations: straight ahead for the Maxwell’s equations and complex for the Lorentz force law,

the definition of a vector dipole magnetic moment, in terms of an axial vector magnetic charge and of a unit
vector velocity (in (39)), and perpendicular to this vector,

a special consequence, the unique behavior of the neutrinos:

they have magnetic charge but they do not respond to magnetic fields,

up to a possible anomalous magnetic moment;

although they do not have electric charge they are under the action of electric fields,

with a suggestion for an experimental verification. See Appendices E and F (figure 4),

our results, from equation (38) on, deeply depend on geometrical conjectures for the leptons and quarks (equations
in (39)). [5]

The exposition raises many questions along the text. Even more we have to search that we are not driven to
erroneous answers to what it is already known. Finally, with Griffisth, after his equation (5.2) in the page 212 in
[2], we write:

a theory needs validation with experiments.

The author of these pages is conscious of the unusual of the proposed model. Even if it is wrong or there are
mistakes, the motivation to present it is based in the hope that there are enough ideas for a better development by
other researchers. We need a model for the

electromagnetism of the elementary fermions.

IV. ACKNOWLEDGMENTS

This author is most grateful to Gabriel Alvarez Galindo and Luis Martinez Alonso. Gabriel Alvarez communicated
the ref. [21] and maintained many useful discussions.



16

APPENDIX A: PHYSICAL CONSTANTS AND DIMENSIONS. [2] [40]
c = 2.999 792 458 10 8 ms−1 , [ c ] = L T−1

ℏ = 1.054 571 817 10−34J s , [ ℏ ] = ML2T−1

e = 1.602 176 634 10−19 Kg s−1 t̂−1 (C) , [e ] = MT−1B−1

.

1 MeV = 1.602 176 634 10−13 J , 1 J = 6.241 509 074 1012 MeV .

me = 9.109 383 701 10−31Kg , me (c
2) = 0.510 998 950MeV .

[ a

−→
B ] = B , [ p

−→
E ] = [ p

−→v × a

−→
B ] = LT−1B .

ϵ0 = 8.85 10−12 C2N−1m−2 , [ ϵ0 ] = [C2N−1m−2 ] = ML−3B−2 .
c2 = (ϵ0 µ0)

−1.
µ0 = 4π 10−7 NA−2 (C−2N t 2) , [µ0 ] = [ c−2 ϵ−1

0 ] = M−1LT 2B2 .


[
−→
L ] = [−→r ×−→p ] = [m −→r ×−→v ] = ML2T−1 = [ ℏ ] , Mech. ang. moment

[−→µ ] = [ e2
−→r ×−→v ] = ML2T−2B−1 = M−1 [e ] [ ℏ ] , Magnetic moment

.



2 = g0e g−factor (gyromagnetic ratio), ge = 2.002 319 304 361 53

µ
B

= e ℏ
2me

= 9.274 099 7 10−24 Jt̂−1 (m2A), t̂ = Tesla , [ t̂ ] = B

µe = −928.476 43 10−26 Jt̂−1

µµ = − 4.490 44 10−26 Jt̂−1 ≈ 1
206.768 µe mµ = 206.768 me

µp = 1.410 606 10−26 Jt̂−1 ≈ − 1
658 µe

.

[ p

−→
j e ] =

{
[µ−1

0 p∇× a
−→
B ]

[ aρe p
−→vρ ]

}
= ML−2T−2B−1 , in

{
Maxwell’s equations

current density def.

}
.

[ p

−→
J e ] = [ aqe p

−→v ] = MLT−2B−1 = L3 [ p

−→
j e ] = [e ] [ c ] , Lorentz equation .

[ a

−→
jm ] = [ ϵ0 a∂t a

−→
B ]=ML−3T−1B−1= [ 1c p

−→
j e ], in generalized Maxwell’s equations (19) .

[ a

−→
Jm ] = MT−1B−1 = L3 [ a

−→
jm ] = [e ] , in generalized Lorentz equation (22) .

[ a
−→µ ]=ML2T−2B−1=[mc2]B−1=[ a

−→
Jm]T [ c2 ] ( a

−→µ =
−−→
I
∫
da , U=− a

−→µ . a−→B) .

[
−→
de ] = [−→q r ] = MLT−1B−1 = [ p

−→
J e ] T , Electric dipole moment .

[ a
−→µ ] = [

−→
de ] [ c ] , Magnetic dipole moment .



17

APPENDIX B: A DIGRESSION. ELECTRO (SCALAR) - MAGNETIC (VECTOR) CHARGES
FOR THE ELEMENTARY FERMIONS HAVE AN AXIAL CHARACTER. [39]

Some naive considerations about geometry (in R
3) and physics.

We start situating two observers O1 and O2 in symmetrical positions with respect to a central point, an origin for
a couple of coordinate systems S1 and S2, also the point for our test magnitudes. Both coordinate systems related
by a parity transformation, opposition of the space coordinates. Associate O1 with S1 and O2 with S2.

1) We kick (a force) a mass-point at the origin. This mass-point moves in any direction. We represent the
force and the movement with arrows starting in the origin, we denote them as polar vectors. Both observers
see a vector, but the second observer sees it in an opposite way to the first one. By using the term see we mean:

the vector “ −→ ”, which is represented by
−→v

∣∣
S1

= (x1, x2, x3)
∣∣
S1

and also by
−→
v′
∣∣
S2

= (x′
1, x

′
2, x

′
3)
∣∣
S2

with

{ x′
1=−x1, x′

2=−x2, x′
3=−x3 }. The electric field p

−→
E belongs to this kind of vectors.

2) Consider the temperature at any point, and in particular also in the origin. We associate to these points
numerical values, we denote them as axial scalars. Both observers see a value at a point, and in both coordinate
systems we have a unique value v

∣∣
S1

= v
∣∣
S2
. The ′punctual′ electric charge

a
q belongs to this kind of scalars.

The work aW =
p
−→
F. p−→d is an axial scalar. The electric current vector p

−→
J e = a

qe p
−→v is a polar vector.

3) We define the mechanical angular moment
a
−→
L = p

−→r × p
−→p . A product of two polar vectors, therefore it

has only one representation in the two coordinate systems, there is not a change of sign (a mathematical operation).
We denote these type of vectors, axial vectors.

Two polar vectors, one rotated of the other one, have as reference for the rotation an axis of rotation and an angle,
with a rule for doing it (usually the righthand rule). We can fix this axis with a vector. Both observers see a unique line
(the axis of rotation), and in both coordinate systems we have a unique vector represented by −→a

∣∣
S1

= (a1, a2, a3)
∣∣
S1

and also by
−→
a′
∣∣
S2

= (a′1, a
′
2, a

′
3)
∣∣
S2

with { a′1 = a1 , a
′
2 = a2 , a

′
3 = a3 } ( a

−→a = pα p
−→n = (−pα) (−p

−→n ) (= aα a
−→n )).

pα a polar scalar (see -4)- below). In this sense it is of interest the reading of Klein and Sommerfeld [41].

The spin a
−→σ and the magnetic field a

−→
B belong to this kind of vectors.

In electromagnetism the concept of charge connotes a non zero divergence of the corresponding field. This is
depicted for the electric charge and the field in the left part of the figure 3 (Appendix F). For the electrical charge

density there is a continuity equation p∇. p−→je + a∂t aρe = a0e .
Customarily we establish a zero magnetic charge with the zero divergence of the magnetic field. This can be

exemplified with an imam. For any closed surface with an imam inside, the balance of the incoming and outgoing
lines of the magnetic field is zero (a total null flux). Now, accept the axial vector magnetic charge as a source for a
magnetic field and relate it to a defining axial characteristic vector (the spin). This is depicted in the right part of
the figure 3 (Appendix F). With these assumptions we tentatively establish:

the axial vector a

−→
Jmf (therefore the related spin) is the vector magnetic charge and a source for a magnetic field.

We obtain magnetic fields in two different ways (two origins): in an Amperian way with a moving electrical charge,
and in a “spin” way (intrinsic) with this vector magnetic charge (see formulas in (19)).

We have suppressed the divergence requirement for the related field in the condition for a charge; different to the
first Maxwell’s equation in (19) with the electric field and the electric charge. We maintain the null divergence of the
magnetic field. And we can verify a ′continuity′ equation with null divergence for the axial vector magnetic charge

density p∇. a−→jmf = p0m.
Even for the Amperian magnetic dipole moment we can consider an axial vector, the area vector related to the

enclosed surface of a loop circuit (−→a = 1
2

∮−→r×−→dl). The problem with this is that it looses its meaning for the punctual
charge. (See in Griffitsh [2] equation (1.107) and pages 255 and 269). Actually, this is a question involved under the
named “rotations in an internal space”. Might this suggests a relation between an axial vector and a null divergence?

4) The Dirac monopole: a polar scalar. Mathematically there is not a question. But, what does it mean here
to see? With an analogy similar to that one of the temperature, O1 would obtain a value and O2 the opposite
value, at once in a point. What would other observers obtain? Nonetheless, the monopoles are not ordinary matter
(elementary fermions). An elaborate construction of a polar scalar (as a pseudoscalar) in [42].

Strange enough, several decades ago it was observed in some physical reactions (weak interactions) a lack of
symmetry in relation to space, which was explained with a polar - axial vector character of the interaction (V-A). Is
this only a characteristic of the interaction or is it telling us something intrinsic of the local time space of a particle?

Could we conclude from this type of observations the axial character of the charges, either scalar or vector?

We suggest: the electromagnetic charges have exclusively axial character, whatsoever scalar or vector like,
scalar for the electric part and vector for the magnetic part,

giving rise to polar electric fields and to axial magnetic fields.



18

APPENDIX C: A NOTE, THE ROUGHLY ESTIMATIONS WITH QUARKS.

Experimental values:

mp = 938.272 MeV
c2 , µp = 1.410606 10−26 Jt̂−1 ≈ 1.521 10−3 µ

B
Jt̂−1

mn = 939.565 MeV
c2 , µn = −0.966236 10−26 Jt̂−1 ≈ −1.042 10−3 µ

B
Jt̂−1

 .

Theoretical values of the magnetic moments for the u-quark and d-quark (after a model):

µp = 4
3 µ̃u − 1

3 µ̃d

µn = 4
3 µ̃d − 1

3 µ̃u

}
,

µ̃u = 1
5 (4µp + µn) = 0.935238 10−26 ≈ 1

991.192 µ
B
Jt̂−1

µ̃d = 1
5 (µp + 4µn) = −0.490868 10−26 ≈ − 1

1888.49 µ
B
Jt̂−1

 .

This accounts for various symmetries in a baryonic wave function with three particles, the “static” quarks, the only
structure considered. This was relevant in the origins, for the constituent quark model. This construction goes from
the protons and neutrons to the quarks, ignoring most of the complexities of the structure of the proton or of the
neutron.

Amperian definition of the intrinsic magnetic moments and related “effective” masses:

µ =
q ℏ
2m

= µ
B

q
e

me
m

=⇒ m =
µ
B
µ

q
e

me (ignoring a gyromagnetic ratio ).

m̃u =
µ
B

µ̃u

qu
e

me = 991.19 2
3 me ≈ 666 me ≈ 338 MeV

c2

m̃d =
µ
B

| µ̃d |
qd
e

me = 1888.49 1
3 me ≈ 639 me ≈ 322 MeV

c2

 .

A very simplified model:
mp ≈ 2m̄u + m̄d

mn ≈ m̄u + 2m̄d

}
m̄u ≈ 1

3(2mp −mn) ≈ 312 MeV
c2

m̄d ≈ 1
3(2mn −mp) ≈ 313 MeV

c2

 .

We can see: with m̄u ≈ m̄d ≈ m̃u ≈ m̃d and qu = −2 qd it is µ̃u ≈ −2 µ̃d .

Relation of the nucleon magnetic model as a “bare” proton (mp) and the previous value:

µN =
q ℏ
2mp

= µ
B

me
mp

= 5.05 10−27 Jt̂−1 we have:

{
µp = γp µN

γp = 2.7918
,

written in a similar form to the one with the gyromagnetic ratio. Is it a lack of knowledge of the structure of the
proton?

In our model, with sin(2φ) =
(
sin(2φ)
k/3

)
k
3 we have: sin(2φu)= sin π

3 ≈(1.3) qu , sin(2φd)= sin π
6 =(1.5) |qd|,

so that m̂u≈1.3 m̃u≈439 MeV
c2 , m̂d≈1.5 m̃d≈482 MeV

c2 ,

are too large values for the resulting masses of a proton or a neutron in a “static” quark model.

To introduce the values of the masses of the quarks provided by the Particle Data Group [40] is significant in the
opposite way, i.e. from quarks to protons and neutrons. Unlikely we do not have enough knowledge of the structure of
the protons and neutrons. And this is so even for the spin concept: “What we know and what we don’t know about
the proton spin after 30 years” [43]. Besides, their masses are not well precised, over all the ones of the u-quark and
d-quark [40] [44], which in the Standard model are the main components of protons and neutrons in their simplest
formulations. Therefore our results are of a theoretical kind that can help in an explanation.

With the values of the masses in the table in Appendix D, provided by the Particle Data Group [40]:

mu ≈ 2.29 MeV
c2 , md ≈ 4.59 MeV

c2 : qu m−1
u ≈4 |qd|m−1

d =⇒ µu,1 ≈ 4 µd,1 ,

and of sin(2φd) =
1
2 , sin(2φu) =

√
3
2 =

√
3 sin(2φd) , we obtain the magnetic moments

(absolute values) in the last column µf,2 (φ) in the table related to the values in the previous column µf,1 (k):

µd,2 = ( 1/21/3 ) µd,1 = 1.5 µd,1 = 1.5 A−1
d

1
3 µ

B
= 1

18 µ
B

≈ 36 µp

µu,2 = (
√
3/2
2/3 ) µu,1 = 1.3 µu,1 = 1.3 A−1

u
2
3 µ

B
= 1

3
√
3
µ
B
≈ 1

5.2 µ
B
≈ 2
√
3 µd,2 ≈ 125 µp

 Jt̂−1 .



19

The lack of knowledge of these two orders of magnitude are a consequence of our ignorance of a detailed model for
the proton. And in relation to the masses (values of Af ), in what way is it: 102 (md + 2mu) ≈ mp?

Therefore, what does it happen with the protons and neutrons? Even for just the spin? (see [43]).

We know for an electron: ∆µe = µe − µ
B
≈ 10−26 Jt̂−1, this value obtained after considering virtual processes.

We also know for protons and neutrons: µp,n ≈ 10−26 Jt̂−1 ≈ ∆µe ≈ 10−3µe.

The baryons have 0, ±1 electrical charges, and only a few have ±2. The baryons with 3
2 -spin are extremely

unstable, and the ones with 1
2 -spin (a minimum) are also unstable (some of them not so extremely), except the

neutron (‘almost’ stable) and the proton (stable). These last two are the baryons with the smallest masses.

An analogy. Let us consider a kind of “bag” model for the protons and the neutrons:

first: the electric charge (a scalar). The proton with q = +1 and the neutron with q = 0. These charges are
the sum of the charges of the component quarks. With the electric fields we have a superposition principle. We are
accustomed, in a first step, to simply add the values of the electric charges (at least for far enough distances). Multiple
pole developments appear in later stages. The value of the electric charge does not seem to have any relationship
with the stability of the baryons,

second: the spin (a scalar). In a similar way as with the electric charge we can obtain the spin scalar charge ( 12 )
in a proton or in a neutron summing projections of the spins (scalars) of the individual confined quarks. This was
assumed in µp = 4

3µu − 1
3µd , µn = 4

3µd − 1
3µu. Is the spin (the scalar, the ‘up’ and ‘down’) related with the

stability of the baryon? Perhaps just a little bit, protons and neutrons have a 1
2 -spin (a minimum with three quarks),

but also many other baryons. All the baryons with a 3
2 -spin seem to be even less stable.

In the framework presented here it remains:

third: the spin (a vector), the axial vector magnetic charge. In a similar way as for the electric charge and the
spin (a scalar) we sum the vectors, defining:

a

−→
Jp = a

−→
Ju + a

−→
Ju′ + a

−→
Jd and a

−→
Jn = a

−→
Ju + a

−→
Jd′ + a

−→
Jd.

Let us look for a minimum for the magnetic moment, perhaps for a proton or a neutron. With the conditions
defined in (39) applied to the three quarks, it is easy to obtain vectors which would have the exclusive global direction

p
−→u r = p

−→uv, which would imply a zero magnetic moment for a proton or a neutron µp,n (g
0
d,u) = 0, like with the

neutrinos. We are summing the axial vector magnetic charges (
−→
J ), not the magnetic moments (−→µ ), and having

assumed that this sum has the same direction as a resultant p
−→u r of the proton or of the neutron.

Is it possible to consider, in a similar way as with the virtual processes in the electrons and the muons, that:

|∆µp,n| = |µp,n − 0| ≈ 10−26 Jt̂−1 ≈ 10−3µ
B
?, (see bellow for neutrinos).

This time the {QED, Week, QCD} virtual processes are more significant. Could this be the reason for

102(md+2mu)≈mp? Perhaps the concept of “flipping” for the quarks is relevant, not only of a “flipp”.

Could the combination of these rules be good enough, as a first step, for justifying the order of magnitude of the
magnetic moments of the protons and neutrons? Are there other minimums? and, is this related to the stability of
the particle?

We have seen already with the charged leptons (k = 3 and φ = π
4 ) simple relationships, as for them it is

3
3 = sin π

2 = 1 (k3 = sin 2φ).

And, finally, the neutrinos. Present day experiments suggest that if the neutrinos have a magnetic moment (in
this research anomalous), this magnetic moment satisfies

∥ −→µ ν(gf ) ∥ = ∥
−→µ ν(g

0
f
) + ∆−→µ ν(∆g

f
) ∥ = ∥ ∆−→µ ν(∆g

f
) ∥ < 10−11µ

B
∼ 10−34 Jt̂−1.

Would we have virtual processes with neutrinos as to obtain it? As a vector, what direction would it have? Could
there be a precession of the Larmor’s type? Would we need the magnetic field in the direction of the movement for
such precession? Or, perhaps we need electric fields to obtain the precession?

Analogies?, or wanderings? Let us borrow the title of a Norton lecture (IV) at Harvard (1973) by Leonard Bernstein:

“The unanswered question. The Delights and Dangers of Ambiguity”.

Too many questions, and we need answers. This is, we need a model and experimental data. Once more, we submit
this research with the intention of signaling a different pathway for the approaching to a model for the electromagnetism
of the elementary fermions.



20

APPENDIX D: TABLE.

This table shows the values of the magnitudes we have been handling for the elementary fermions [40] [44]. The
signs of the charges are shown only in qk . We have chosen the masses of the quarks inside their ranks, in such a way
as to get afterwards simple results. The numbers written for these fermions (first column) have to be multiplied by
the numbers in the third line (e), after the fifth column onwards. The formulas of tf , lf , b

−1 were mentioned in

(13). Last two columns up to
gf/2 and related in (45).

- k
qk

C

mfc
2

MeV
Af

lf

m

tf

seg

b−1
f

tesla−1= t̂−1

µf,k

Jt̂−1

µf,φ

Jt̂−1

e ( 1
3
) (1.6 10−19) - (0.51−1) 3.86 10−13 1.29 10−21 2.26 10−10 9.27 10−24

neutrinos 0 0 ? ? ? ? 0

electron 3 −1 0.51 1 1 1 1

u 2 2
3

2.29 9
2

2
9

22

92
0.148 0.192

d 1 −1
3

4.59 9 1
9

1
92

0.037 0.055

muon 3 −1 105.66 207 1
207

1
2072

4.8 10−3

s 1 −1
3

102 200 5 10−3 2.5 10−5 1.7 10−3 2.5 10−3

c 2 2
3

1275 2500 4 10−4 1.6 10−7 2.7 10−4 3.5 10−4

tau 3 −1 1777 3484 1
3484

1
34842

2.9 10−4

b 1 −1
3

4250 8333 1.2 10−4 2.4 10−8 4 10−5 6 10−5

t 2 2
3

170000 333333 3 10−6 9 10−12 2 10−6 2.6 10−6

TABLE I. Various magnitudes for the elementary fermions.

APPENDIX E: A PROPOSAL FOR AN EXPERIMENT WITH NEUTRINOS. FIGURE 4.

In the Fermilab website [45] we read about the following possibility: to “ ′see′ the sun with neutrinos ”.
Could we ′see′ the trajectories of the neutrinos? Consider neutrinos with the appropriate energy as to extract

electrons from their atoms in a detector and with the property: the angles of the trajectories of the neutrino and of
the extracted electron coincide. The question is: can a detector measure that outgoing electron angle and discriminate
the neutrinos?

The experimental device would consist of three parts. The first part, the source of the neutrinos (in principle -anti-
neutrinos from the sun). A second part, an electrical field for the possible deflection of the neutrinos going through.
The electrical field being null outside of this device. The third part, the detector.

With the source of neutrinos far enough we have neutrinos arriving to the deflection mechanism with parallel di-
rections and with the direction of the electrical field prepared perpendicularly to them. The equation (47) presented
in this research suggests for the neutrinos a parallel behavior to the one of the electrons in the Stern Gerlach experi-
ment. Therefore, substituting the magnetic field and the Amperian internal dipole magnetic moment (electric charge
current) by the electric field and the axial vector magnetic charge (the spin as a vector and with a unit of charge).

The neutrinos are ultrarelativistic, we do not know their masses but we can know their range of energies and we
can select a wide range of electric fields, in correspondence.

There are three different settings. Without electrical field (
−→
E =

−→
0 ), with electrical field (

−→
E ̸= −→0 ) and with

electrical field in the opposite direction (−
−→
E ̸= −→0 ). We could verify equation (47) if the three different types of

results shown in the Appendix F figure 4 (three different distributions of neutrinos) were differentiated.
Antineutrinos and neutrinos have opposite spins. With antineutrinos we should have similar results, but inverting

the senses of the electrical fields with respect to the neutrino case.



21

APPENDIX F: GRAPHICS.



22

APPENDIX G: PROGRAM OF THE STUDIES CONTAINING THIS RESEARCH.

On the fermionization of the XYZ spin Heisenberg chain (algebra).
(2022) https://eprints.ucm.es/id/eprint/72882/ Study -2 )

The JordanWigner transformations and the fermionization of the XYZ spin Heisenberg chain.
Algebra, geometry and physics? (2022) https://eprints.ucm.es/id/eprint/74550/ Study -1 )

A tentative model of creation and annihilation operators for neutrinos.
(2021) https://eprints.ucm.es/id/eprint/65151/ Study 0 )

Expression of the 3- and 4-dimensional vectors in total polar exponential form.
(2021) https://eprints.ucm.es/id/eprint/65825/ Study I,1)

Vectors. Dimensions 4 and 8. (2023) https://eprints.ucm.es/id/eprint/76327/ Study I,2)

Geometry of the time and the space. Study I )

Geometry of the symmetries in dimension 4=(1+[1)+“2”], and general Time-Space-Spin vectors.
(2023) https://eprints.ucm.es/id/eprint/76328/ Study II)

Geometry and Physics of the Elementary Fermions. (On pride of Jordan Wigner Pauli Weyl Dirac). 1.
(2021) https://eprints.ucm.es/id/eprint/69295/ Study III)

Geometry and Physics of the Elementary Fermions. 2. Study III)

Axial vector magnetic charge and magnetic moment. Maxwell’s equations and Lorentz force law.
(This study). (2021) https://eprints.ucm.es/id/eprint/69294/ Study IV)

Addenda. Study V )

[1] G. ’t Hooft. In search of the ultimate building blocks. Cambridge University Press (1997). (Pages 27 and 29).
[2] D. J. Griffiths. Introduction to Electrodynamics, fourth ed. . Pearson (2013). (Page 379, problem 8.17. Section 5.4.3 ).
[3] Nature Milestones: Physics is set spinning. March 2008. Nature Publishing Group. www.nature.com/milestones/spin.
[4] M. E. Peskin. Spin, Mass, and Symmetry. XXIst SLAC Summer Institute. Spin Structure in High Energy Processes.

Stanford California, July 26–August 6, 1993. arXiv:hep-ph/9405255v1 9 May 1994.
[5] E. Olmedilla. Geometry and Physics of the Elementary Fermions. 1 (On pride of Jordan Wigner Pauli Weyl Dirac).

https://eprints.ucm.es/id/eprint/69295/ (2021).
[6] W. Pauli Zur Quantenmechanik des magnetischen Elektrons. Z Phys. 43, 601-623. (1927). (On the quantum mechanics

of magnetic electrons). (Page 6).
[7] J. Schwinger et al.. Classical electrodynamics. CRC Press (2018). (Pages 325-327. Page 36. Pages 65-66).
[8] L. D. Landau. The Classical Theory of Fields, fourth ed.. Pergamon Press (1971). (Section 44).
[9] J. D. Jackson. Classical Electrodynamics, second ed.. Wiley (1975). (Sections 5.6 and 5.7).

[10] A. Galindo and P. Pascual. Quantum Mechanics I. Springer-Verlag (1990). (Section 1.9, after equation 48).
[11] S. Bilenky. Introduction to the Physics of Massive and Mixed Neutrinos. Second ed., Springer Int. Pub. (2018). (Pages

248-251).
[12] C. Broggini, C. Giunti, and A. Studenikin. Electromagnetic Properties of Neutrinos. Advances in High Energy Physics,

Vol. 2012, Art. ID 459526. Doi:10.1155/2012/459526.
[13] C. Giunti et al.. Electromagnetic neutrinos in laboratory experiments and astrophysics. Ann. Phys. (Berlin) 528, No.

1–2, 198–215 (2016) / DOI 10.1002/andp.201500211.
[14] M. Planck. Verh. Deut. Phys. Ges., 2, 237-45. (1900). Translated to English: On the Theory of the Energy Distribution

Law of the Normal Spectrum, in the book by D. ter Haar: The Old Quantum Theory (page 84). Pergamon Press (1967).
[15] P. A. M. Dirac. Quantised Singularities in the Electromagnetic Field. Proc. Roy. Soc. A 133, 60 (1931).
[16] D. J. Griffiths. A catalogue of hidden momenta. Phil. Trans. R. Soc. A.376. 0043. (2018).
[17] J. Franklin. What is the force on a magnetic dipole? arXiv:1508.01808v4 [physics.class-ph] 20 Sep 2017.
[18] T. H. Boyer. Classical interaction of a magnet and a point charge: The Shockley-James paradox. Phys. Rev. 91, 013201.

(2015).
[19] T. H. Boyer. Illustrations of Maxwell’s term and the four conservation laws of electromagnetism. American Journal of

Physics 87, 729 (2019). doi: 10.1119/1.5115339.
[20] A. Zangwill. Modern Electrodynamics. Cambridge University Press (2013). (Chapter 2, section 2.3 and section 11.2,

particularly 11.2.3).
[21] T. H. Boyer. The force on a magnetic dipole. American Journal of Physics 56, 688 (1988).
[22] B. Friedrich, G. Meijerand, H. Schmidt-Böking and G. Gruber. One hundred years of Alfred Landé’s g-factor. Nat Sci.

2021;1:e20210068. wileyonlinelibrary.com/journal/ntls . https://doi.org/10.1002/ntls.20210068.
[23] M. E. Peskin and D. V. Schroeder. An Introduction to Quantum Field Theory. Perseus, Reading MA. (1995). (Sections

6.2 and 6.3). (1995).



23

[24] J. D. Jackson. The nature of the magnetic dipole moments. CERN (1977). Summer Student Lecture Program. CERN.
77-17: 1–25. (1 September 1977).

[25] H. Schmidt-Böking et al. The Stern-Gerlach experiment revisited. The Eur. Phys. J. H, 41, 4, 327-364 (2016).
arXiv:1609.09311v1 [physics.hist-ph]. (Particularly pages 5-9 and Sections 3 and 4).

[26] B. Friedrich and H. Schmidt-Böking. Molecular Beams in Physics and Chemistry. From Otto Stern’s Pioneering Exploits
to Present-Day Feats. Springer (2021). https://doi.org/10.1007/978-3-030-63963-1. Chapter 5: Otto Stern’s Molecular
Beam Method and Its Impact on Quantum Physics.

[27] D. Bohm. Quantum Theory. Prentice-Hall, Inc (1951). Dover Pub. (1989). (Section 22.6).
[28] H. Wennerström and P. Westlund. The Stern-Gerlach experiment and the effects of spin relaxation. Phys. Chem. Chem.

Phys., 2012, 14, 1677–1684.
[29] M. Devereux. Reduction of the atomic wave function in the Stern-Gerlach experiment. Can. J. Phys. 93(11), 1382–1390.

(2015).
[30] H. Wennerström and P. Westlund. A Quantum Description of the Stern–Gerlach Experiment. Entropy 2017, 19, 186;

doi:10.3390/e19050186.
[31] D. Hestenes. Zitterbewegung structure in electrons and photons. arXiv:1910.11085v2 [physics.gen-ph] 24 Jan 2020.
[32] S. E. Barnes et al.. Rashba Spin-Orbit Anisotropy and the Electric Field Control of Magnetism. Scientific Reports.

February 2014, 4, 4105. arXiv:1312.1021v1 [cond-mat.mtrl-sci] 4 Dec 2013.
[33] A. Manchon et al.. New perspectives for Rashba spin–orbit coupling. Review Article. Published online: 20 August 2015.

DOI: 10.1038/NMAT4360.
[34] R. Ramesh and S. Manipatruni. Electric field control of magnetism. Proc. R. Soc. A 477: 20200942. (2021)

https://doi.org/10.1098/rspa.2020.0942.
[35] J. Krempaský et al.. Operando Imaging of All-Electric Spin Texture Manipulation in Ferroelectric and Multiferroic Rashba

Semiconductors. Phys. Rev. X 8, 021067 (2018).
[36] J. E. Kim. History of Neutrino magnetic moment. arXiv:1911.06883v1 [hep-ph] 15 Nov 2019.
[37] A. Erdas, Z. Metzle.r Magnetic field effects on neutrino oscillations. International Journal of Modern Physics A Vol. 34,

No. 23, 1950121 (2019).
[38] C. Giunti, A. Studenikin. Neutrino electromagnetic interactions: a window to new physics. Rev. Mod. Phys. 87, 531

(2015).
[39] E. Olmedilla. Vectors. Dimensions 4 and 8. https://eprints.ucm.es/id/eprint/76327/ (2023).
[40] Review of Particle Physics. Particle Data Group. DOI: 10.1093/ptep/ptaa104.
[41] F. Klein and A. Sommerfeld. The Theory of the Top. Vol I. Introduction to the Kinematics and Kinetics of the Top. R.J.

Nagem, G. Sandri, transl., Birkhäuser, Boston (2008). Original: Theorie des Kreisels. (From 1897 to 1910). (Page 36).
[42] W. E. Baylis, J. Huschilt, and Jiansu Wei. Why i?. Am. J. Phys. bf 60 (9) (1992). (At the end of section IV).
[43] Ji, X., Yuan, F. and Zhao, Y. What we know and what we don’t know about the proton spin after 30 years. Nat Rev

Phys (2020). https://doi.org/10.1038/s42254-020-00248-4.
[44] D. J. Griffiths. Introduction to Elementary Particles, second ed.. Wiley-Vch (2008).
[45] https://neutrinos.fnal.gov/types/energies and https://neutrinos.fnal.gov/types/energies/#moreinfo.