Content uploaded by A. I. Arbab
Author content
All content in this area was uploaded by A. I. Arbab on Mar 03, 2018
Content may be subject to copyright.
Content uploaded by A. I. Arbab
Author content
All content in this area was uploaded by A. I. Arbab on Oct 11, 2017
Content may be subject to copyright.
Modified electrodynamics for London’s superconductivity
A. I. Arbab (a)
Department of Physics, College of Science, Qassim University, P.O. Box 6644, 51452 Buraidah, KSA
PACS 03.50.De – Classical electromagnetism
PACS 74.20.De – Phenomenological theories
PACS 03.75.-b – Matter waves
Abstract –A spinless massive scalar particle (boson) is found to emerge from minimally modifying Maxwell’s
equations. This occurs inside a medium whether charges are static or moving. It interacts with static charges in
the medium, and satisfies the Klein-Gordon equation. A pure electric wave with vanishing magnetic component is
predicted. The modified Maxwell’s equations yield London’s equations of superconductivity, and generalize some
of them. The electric field, magnetic field, charge and current densities are found to satisfy the Klein-Gordon
equation whenever London’s equations are satisfied. The medium inside the superconductor is shown to exhibit
fluid nature. The gauge invariance has been shown to be intact. A magnetic field arising from the motion of the
boson fluid is found, that generalizes the Biot-Savart law. A relativistic London’s moment is introduced.
Introduction. – Maxwell’s equations govern the electromagnetic phenomena ever known. These
equations are successful in all physical and mathematical respects. The electromagnetic wave is composed
of electric and magnetic field propagating in free space at right angles (transverse). Hertz had shown
experimentally that they really exhibit this nature. At the same time Tesla showed that these waves
are longitudinal (non-Hertzian) electric waves [1]. Unfortunately, this discovery was deemed not to
be justified by theory, and was accordingly rejected by the scientific media. In this paper, we will
extend Maxwell’s equations to encompass this longitudinal property of the electromagnetic wave. Any
modification of Maxwell’s equations will have immense consequences in fields relying on it, e.g., condensed
matter, magnetohydrodynamics applications, etc. However, when the electromagnetic wave travels in a
medium a longitudinal wave can arise.
A massive spinless boson partaking in the electromagnetic interactions in the medium was already
observed in the phenomenon of superconductivity where Cooper pairs of spin equals to zero are predicted
[2]. The Cooper pairs are two electrons of opposite spin mediated by a phonon. It has been shown
by Proca that massive spin-1 photons can be obtained by extending Maxwell’s equations [3]. However,
in our present formulation we have both ordinary massless photon besides a massive spinless particle
(a)arbab.ibrahim@gmail.com
p-1
A. I. Arbab2
(boson). We aim here at a massive spinless boson. The photon with spin equals to unity characterizes
the photon in free space, while a photon becomes a spinless boson in a medium. This spinless boson will
be the mediator of the electromagnetic interactions inside a medium, where the photon can no longer
be able to communicate. Thus, the particle nature of the photon is exhibited when the photon enters a
medium. Within this framework, the photon exhibits the material nature when it encounters a medium.
Interestingly, the modified Maxwell’s equation are found to be effective when applied inside a medium.
They yield London’s equations of superconductivity [4]. Note that these modified Maxwell’s equations
are gauge invariant. However, despite the fact that the Proca theory for spin boson is not gauge invariant,
but we have recently introduced new transformations that render it gauge invariant [5].
To account for the superconductivity, London’s theory employed the Newton’s second law in addition
to Maxwell’s equations to obtain his equations. By introducing a scalar function in Maxwell’s equations,
we derive the London’s equations without any additional assumption. The scalar function satisfies the
Klein-Gordon equation. This means that it represents a spin zero particle that has been dubbed the
Cooper pairs in the Bardeen-Cooper-Schrieffer (BCS) theory [2]. In the BCS theory, the Cooper pair has
not been formally shown to satisfy the Klein-Gordon equation for spin zero particles. The superconducting
state is found to be a macroscopic phase coherent state with order parameter [6].
The paper is structured as follows: We introduce in Section 1 the Maxwell’s modified equations by
adding a scalar field that interacts with the electromagnetic field. In Section 2 we propose a modified
charge conservation equation in which the current and charge density interact with the scalar field men-
tioned above. This field is defined in such a way to keep the total charge conserved, and is shown to
lead to a longitudinal wave accompanying the ordinary transverse wave. In Section 3 we introduce the
superconducting electrodynamics that is compatible with the London’s equation of superconductivity.
In this Section several superconducting properties are derived. In Section 4 we derive the London’s two
equations from the modified Maxwell’s equations, and extend the the Lorentz force. It is shown that the
new scalar field gives rise to force and power associated with a moving charge particle. We finally end
our paper with some concluding remarks.
Modified Maxwell’s equations. – If we assume the Lorenz gauge is relaxed to be given by [7]
⃗
∇ · ⃗
A+1
c2
∂φ
∂t =−S , (1)
where Sis a function of space and time, then Maxwell’s equations will read [8]1
⃗
∇ × ⃗
E=−∂⃗
B
∂t ,⃗
∇ × ⃗
B=µ0⃗
J+1
c2
∂⃗
E
∂t −⃗
∇S , (2)
1Here, S=−Λ.
p-2
Modified electrodynamics for London’s superconductivity
and
⃗
∇ · ⃗
E=ρ
ε0
+∂S
∂t ,⃗
∇ · ⃗
B= 0 .(3)
The Lorenz gauge is normally adopted in electromagnetism, firstly because it has a covariant form,
and secondly it makes Maxwell’s equations, in terms of the vector and scalar potentials, having simple
forms. It is interesting that Eqs.(2) - (3) are gauge invariant. Notice that the Ampere’s law and the
divergenceless nature of the magnetic field are not influenced by this scalar field. It is to be noticed that
a quantum equation employing eq .(1) has already be considered that reflects the quantum (particle)
nature of the electromagnetic wave [9]. In that work the wavefunctions of the photon are the vector and
scalar potentials, ⃗
Aand φ, respectively.
Extended gauge transformations. – Let us now consider a more general gauge transformation
that takes into account the current (charge) density as well as the vector (scalar) potential fields. To this
aim, we allow the continuity equation to take the form
⃗
∇ · ⃗
J+∂ρ
∂t =µ2
µ0
S , (4)
so that the divergence of Eq.(2) together with Eqs.(3) and (4) yield the massive spinless Klein - Gordon
equation, i.e.,
1
c2
∂2S
∂t2− ∇2S+µ2S= 0 ,(5)
where µ=mc
¯his a measure of the scalar field mass. Now add Eqs.(1) and (4) to get
⃗
∇ · ⃗
J+µ2
µ0
⃗
A+∂
∂t ρ+ε0µ2φ= 0 .(6)
It is interesting to see that in the absence of electric and magnetic fields, i.e.,⃗
E=⃗
B= 0, Eqs.(1) - (4)
reveal that Ssatisfies the Klein-Gordon equation, Eq.(5).
We can define the total current and charge densities of the new system as
⃗
JT=⃗
J+µ2
µ0
⃗
A , ρT=ρ+ε0µ2φ . (7)
Hence, the the new continuity equation becomes
⃗
∇ · ⃗
JT+∂ρT
∂t = 0 ,(8)
We can define here, ⃗
JS=µ2
µ0
⃗
Aand ρS=ε0µ2φas the current and charge densities, respectively, due
to the scalar field. It is evident from Eq.(8) that the combined current is conserved, but not each one
separately. This is so since the scalar field interacts with the electromagnet filed.
Let us now apply the gauge transformations
⃗
A′=⃗
A+∇λ , φ ′=φ−∂λ
∂t ,(9)
p-3
A. I. Arbab3
in the continuity equation and require it to be invariant. Thus, Eq.(6) becomes
⃗
∇ · ⃗
J′+µ2
µ0
⃗
∇λ+µ2
µ0
⃗
A+∂
∂t ρ′+ε0µ2φ−ε0µ2∂λ
∂t .(10)
This requires the current and charge densities to transform as
⃗
J′=⃗
J−µ2
µ0
⃗
∇λ , ρ ′=ρ+ε0µ2∂λ
∂t .(11)
Now let us assume
S=−µ2λ , (12)
so that
⃗
J′=⃗
J+1
µ0
⃗
∇S , ρ ′=ρ−ε0
∂S
∂t .(13)
The transformations in Eq.(13) has been obtained using different approach [10]. Applying Eq.(12) in
Eq.(9) yields
⃗
A′=⃗
A+1
µ2∇S , φ ′=φ−1
µ2
∂S
∂t ,(14)
We now call Eqs.(9) and (13) the extended gauge transformations [5]. The transformations in Eqs.(13)
and (14) can be expressed in a covariant form as
J′
µ=Jµ−1
µ0
∂µS , A′
µ=Aµ−1
µ2∂µS .
These transformations are shown to make Proca-Maxwell’s equation invariant under Lorenz gauge trans-
formations [3]. If we apply the extended gauge transformations, Eq.(13), in the electromagnetic energy
equation [11]
⃗
∇ · ⃗
Sem +∂uem
∂t =−⃗
J·⃗
E , uem =1
2ε0E2+B2
2µ0
,⃗
Sem =⃗
E×⃗
B
µ0
,(15)
we will obtain, using Eq.(3), the new energy equation involving the scalar boson, S, as
⃗
∇ · ⃗
E×⃗
B−S⃗
E
µ0+∂
∂t 1
2ε0E2+B2
2µ0
+S2
2µ0=−⃗
J·⃗
E−c2S ρ . (16)
It is also true that an equation of the form in Eq.(16) is also valid with −Sreplacing S. If one equation
represents a wave moving to the right, the second one will be that wave moving to the left, and vice versa.
This equation tells us that the scalar wave (S) is physical, since it has both energy and momentum
densities. This scalar boson travels along the direction of the electric field. It could be the mediator
between the two cooper pairs. Or equivalently, the Cooper pairs themselves. While −⃗
J·⃗
Eaccounts for
energy dissipation due to interaction of the electric field with the free moving charges, the term −ρc2S
account for the interaction of the static charges in the medium in which the scalar boson exits.
p-4
Modified electrodynamics for London’s superconductivity
This scalar field has a dimension of magnetic field density. It therefore better to be called the magnetic
scalar field density. It flows along the direction of the electric field as evident from Eq.(16). It is worth
mentioning that in [7] Sis associated with massless filed (wave). The energy equation in Eq.(16) is
identical to that obtained by [7] (Eq.(33)), however. We assume that S(or λ) is a kind of wave associated
with the particle nature of the electromagnetic wave as exhibited by de Broglie. This wave shows up
whenever the charge is not conserved in the medium in which the electromagnetic wave is propagating.
It is now worth to mention that the scalar function in the gauge transformations, λ(or S), represents
a physical particle (boson) whose current and charge densities are the vector and scalar potentials, ⃗
Aand
φ, respectively. Thus, λis not just a mere scalar function as has been dealt with before. It can be seen
from Eq.(16) that even if there are no free charges in the medium, the scalar field interacts with the static
charges present. It also acquires a mass as demonstrated in the Higgs mechanism for gauge bosons [12].
We may assume that inside a medium the massive scalar boson mediates between the electromagnetic
interactions making them of short range nature. Moreover, this scalar field can acquire a charge at the
expense of removing it from the medium in which it travels, as allowed by Eqs.(1) and (4).
It is interesting to observe that a massless particle (boson) is associated with conservation of energy,
as evident from Eq.(3). Moreover, whenever the charge is not conserved a massive component satisfying
the Klein - Gordon equation is developed, as evident from Eqs.(1), (4) and (5). Such a situation occurs
in superconductivity where a spinless massive Cooper pair is created in the superconductor [2]. We can
associate the scalar boson energy density to the thermal energy imparted to the medium. In this case
one has the relation S2
2µ0=n kBT, where Tis the absolute temperature, nis the number density, and
kBis the Boltzman constant. This magnetic field scalar results from the motion (current) of the charged
bosons. Now if n∝T3, then S∝T2.
A medium with static charges in which the magnetic field is zero, its electric field is static and is
therefore described by the energy equation, see Eq.(16),
⃗
∇ · −S⃗
E
µ0+∂
∂t S2
2µ0=−ρ c2S . (17)
This equation clearly shows that the scalar field with pure electric field can be generated. It is of an
identical form to matter wave energy equation found in [9]. This once again proves that the scalar wave
is in essence a particle (de Broglie) wave. In this sense, way can say that Eq.(16) represents the energy
equation for the electromagnetic wave due to the motion of the charge and mass of the boson. We see
from Eq.(17) that when ρ= 0 (vacuum), the scalar wave will propagate without loss. Moreover, as
evident from Eq.(16), when S=−⃗
J·⃗
E
c2ρ, the scalar field propagates without scattering or energy loss. And
in particular, if ⃗
J=ρ ⃗v, then S=−⃗v·⃗
E
c2, where ⃗v is the particle velocity. This agrees with our recent form
of a longitudinal wave resulting from a different formulation of Maxwell’s equations using biquaternions
p-5
A. I. Arbab5
[10].
Superconducity electrodynamics: Charge and current densities waves. – If we now differ-
entiate Eq.(4) partially with respect to time and use Eq.(3), we obtain
1
c2
∂2ρ
∂t2+⃗
∇ · 1
c2
∂⃗
J
∂t −µ2
c2µ0
⃗
E+µ2ρ= 0 .(18)
The charge density will satisfy the Klein-Gordon equations if
∂⃗
J
∂t +⃗
∇ρc2=µ2
µ0
⃗
E , (19)
and therefore
1
c2
∂2ρ
∂t2− ∇2ρ+µ2ρ= 0 .(20)
Let us now take the gradient of Eq.(4) and use Eqs.(19) and (2) to obtain
1
c2
∂2⃗
J
∂t2− ∇2⃗
J+µ2⃗
J=⃗
∇ × ⃗
∇ × ⃗
J+µ2
µ0
⃗
B.(21)
Once again the current density will satisfy the Klein-Gordon equation
1
c2
∂2⃗
J
∂t2− ∇2⃗
J+µ2⃗
J= 0 ,(22)
provided that 4
⃗
∇ × ⃗
J=−µ2
µ0
⃗
B . (23)
It is interesting to remark that Eqs.(19) and (23) are but the London’s equation for superconductivity
[4]. Therefore, Eqs.(2) and (3) in addition to the modified Lorenz gauge condition in Eq.(1), are the
appropriate Maxwell’s equations for superconductivity. Thus, not only the scalar field satisfies the Klein-
Gordon equation but the charge and current densities as well. If we use the definitions of the electric and
magnetic fields, i.e., ⃗
B=⃗
∇ × ⃗
Aand ⃗
E=−⃗
∇φ−∂⃗
A
∂t , the Eq.(19) leads to
∂
∂t ⃗
J+µ2
µ0
⃗
A+⃗
∇c2ρ+µ2
µ0
φ= 0 .(19a)
This has a special solution ⃗
J=−µ2
µ0
⃗
Aand ρ=−µ2ε0φ. The first part is the one considered by London.
However, a more interesting solution is that
⃗
J=−µ2
µ0
⃗
A+⃗
∇χ , ρ =−µ2ε0φ−1
c2
∂χ
∂t ,(19b)
where χis some space-time scalar function. It can be manipulated to guarantee that Eq.(19b) is gauge
invariant. This is unlike the original London’s gauge that is not gauge invariant. If we compare Eq.(19b)
4The general solution is that ⃗
∇ × ⃗
J+µ2
µ0
⃗
B=⃗
∇ζ, for some scalar function ζsatisfying Laplace equation.
p-6
Modified electrodynamics for London’s superconductivity
with that of a quantum mechanically treated wavefunction involving a phase angle θ[6] (ψ=nsexp(iθ)),
where ns=m
q2
µ2
µ0, then one finds that, θ=q
¯h
µ0
µ2χ.
It is interesting to see that Eqs.(1), (2) and (3) yield
1
c2
∂2φ
∂t2− ∇2φ=ρ
ε0
,1
c2
∂2⃗
A
∂t2− ∇2⃗
A=µ0⃗
J . (19c)
Thus, the potentials (φand ⃗
A) satisfy the same equations as those of the ordinary Maxwell’s equations
with Lorenz gauge condition satisfied. Therefore, the potentials in the modified Maxwell’s equations are
the same as those in ordinary Maxwell’s equations.
Taking the curl of Eq.(22) and using Eq.(23), and taking the curl of Eq.(2) using Eqs.(3) and (19),
reveal that electric and magnetic fields satisfy the Klein-Gordon equations, i.e.,
1
c2
∂2⃗
E
∂t2− ∇2⃗
E+µ2⃗
E= 0 ,1
c2
∂2⃗
B
∂t2− ∇2⃗
B+µ2⃗
B= 0 .(a)
Apparently, stationary fields decay exponentially with distance inside the superconductor.
Substituting Eq.(19b) in Eq.(4) and using Eq.(1) show that χsatisfies the wave equation traveling
at speed of light. Equation (19b) is consistent with Eqs. (19) and (23). Hence, Eq.(19b) can be seen as
the extended London’s current and charge densities, or a transformation of supercurrent and supercharge
densities.
The Lorentz force density, ⃗
f= (ρ⃗
E+⃗
J×⃗
B), due to the ⃗
Eand ⃗
Bfields, as described in Eqs.(19) and
(23), can be expressed as
⃗
f=ρµ0
µ2∂⃗
J
∂t −⃗v ×(⃗
∇ × ⃗
J) + c2⃗
∇ρ.(23a)
Equation (23a) can be seen as representing the force density (Euler’s force) acting on a fluid (bosons)
moving with velocity, ⃗v. If we now let ⃗
J=ρ ⃗v and apply it in Eq.(23a), we obtain
⃗
f=ρµ0
µ2∂(ρ⃗v)
∂t +⃗v ·⃗
∇(ρ⃗v)−v2⃗
∇ρ−ρ
2⃗
∇v2+⃗
∇(ρ c2).(23b)
Let us consider the case when ρ= const.and substitute it in Eq.(23b) to obtain
ρ2µ0
µ2∂⃗v
∂t +⃗v ·⃗
∇⃗v=ρ(⃗
E+⃗v ×⃗
B) + ⃗
∇µ0
2µ2ρ2v2.(23c)
The first term in Eq.(23c) represents the Lorentz force density while the second term represents a pressure
force due to bosons. However, the second term may also be related to the negative of the chemical potential
of the bosons. Hence, one can associate a mass density of the quantum fluid given by,
ρm=µ0
µ2ρ2,(23d)
p-7
A. I. Arbab6
and a pressure of the fluid given by
Pm=µ0
2µ2(ρ v)2,(23e)
or Pm=µ0
2µ2J2. These give the equation of state of a non-relativistic gas, Pm=1
2ρmv2.Equation (23d)
gives a relation between the mass of the bosons as, m=µ0nbq2¯h2
c21/3where nbis the number density
of the bosons. Equation (23d) and (23e) are very interesting since they connect the electrical properties
of the bosons to their inertial properties. The number density of the bosons becomes nb=m
µ0q2mc
¯h2.
This can be expressed as nb=1
λqλ2, where λ=¯h
mc and λq=µ0q2
mis some length scale of the theory
(bosson classical radius). It may pertain to a length per area relevant to the superconductor. Let us now
consider the ration, β=λ
λq. This is given by
β=¯h
µ0cq2.(a)
For q= 2e n, where nis an integer, and α=ke2
¯h c , the above equation yields
β= (16πα n2)−1.(b)
It is evident that β > 1 for n= 1, or q= 2e, and β < 1 for n≥2 . Writing ⃗
J=ρ ⃗v in Eq.(19b) and
taking the curl of the resulting equation bearing in mind that ρ= const., give the relation
⃗ω =−ρ
ρm
⃗
B . (23f)
where ⃗ω =⃗
∇ × ⃗v is the vorticity of the boson fluid. This equation is very interesting since it relates the
inertial property of the fluid with its electromagnetic one. Hence, the application of magnetic field gives
rise to vortices inside the superconductor. It follows the direction of the magnetic field for ρ < 0 and
opposite to it for ρ > 0. Because of such vortices, part of a magnetic field applied to the superconductor
is allowed to pass through [13]. Now integrating Eq.(23g) yields
⃗ω ·d⃗
S=−ρ
ρm⃗
B·d⃗
S . (23g)
Therefore, the above equation can be written as
Γ = −ρ
ρm
ϕB,
and hence, if the magnetic flux, ϕB, is quantized the circulation, Γ, is quantized too. Notice that the
circulation can be clock-wise or counter clock-wise depending o the charge sign. Since for Cooper pairs
ρ < 0, then Γ >0.
Let us now consider Eq.(23) and use ⃗
J=ρ⃗v and Eqs.(19), where we have assumed that ρis a function
of space, to obtain
⃗
B=⃗v ×⃗
E
c2−ρµ0
µ2⃗ω −µ0ρ
µ2c2⃗v ×∂⃗v
∂t .(23h)
p-8
Modified electrodynamics for London’s superconductivity
Using Eq.(23d), Eq.(23h) can be written as
⃗
B=⃗v ×⃗
E
c2−ρm
ρ⃗ω −ρm
ρ⃗v
c2×∂⃗v
∂t .(23i)
Recall that ρm/ρ =m/q. The above equation can be used to evaluate the magnetic field inside a neutron
or white dwarf stars that are thought to have superconducting core [14, 15]. It can also be applied to
investigate the magnetic field due to plasma. Defining an effective field, ⃗
Beff.=⃗
B−⃗v×⃗
E
c2, Eq.(23i) can
be written as
⃗
Beff.=−ρm
ρ⃗ωeff., ⃗ωeff.=⃗ω +⃗v
c2×∂⃗v
∂t ,
where ⃗
Beff.is the magnetic field as seen in a moving (rotating) frame. Equation (23i) can also be written
as
⃗
B=⃗
J×⃗
Eeff.
ρc2−ρm
ρ⃗ω , ⃗
Eeff.=⃗
E−⃗
Ei,⃗
Ei=ρm
ρ
∂⃗v
∂t ,
here ⃗
Eiis the induced electric field due to boson motion. It depends on the mass of the boson and
therefore can be thought to be related to inertial effect. The above equation expresses the evolution of
the magnetic field inside a superconductor that is evidently changing with space and time [16]. This
magnetic field can vanish if ⃗ω =⃗
J×⃗
Eeff.
ρmc2. Noting that for a rotating fluid with angular velocity, Ω, one
has ⃗ω = 2 ⃗
Ω, and hence Eq.(23i) can be written as
⃗
B=⃗v ×⃗
E
c2−2ρm
ρ⃗
Ω−ρm
ρ⃗v
c2×∂⃗v
∂t .(23j)
The first term expresses the magnetic field as seen in the rotating frame. The second term is usually
referred to London’s moment [17]. Because of the factor c2in the numerator of the first and third
terms in the above equation have small contribution, our formula approximates to London’s result. The
third term can be seen as a magnetic field that arises due acceleration of the charge. Therefore, Eq.(23j)
generalizes the London’s moment to include high speed rotation. Hence, Eq.(23j) can be thought of a
relativistic London’s moment. The last term in Eq.(23j) can be expressed as
⃗
Bi=−m
q⃗v
c2×⃗a, ⃗a =∂⃗v
∂t .
For a circular motion, one can write, a=v2/r, and therefore the above equation yields an inertial
magnetic field
Bi=m
q
v3
rc2, Bi=mr2
qc2Ω3
which is a magnetic field that is developed at a distance rfrom a rotating superconductor with velocity
v. It is directed inward the conductor for clock-wise rotation. Under quasi-state (stationary) conditions
∂⃗v
∂t = 0 . In this case, one has
⃗
B=⃗v ×⃗
E
c2−2ρm
ρ⃗
Ω.(23k)
p-9
A. I. Arbab7
In this case, ⃗
Bcan be positive or negative depending on whether the first term is greater or less that the
second. Recall here that the electric and magnetic fields are not identically zero but decay exponentially,
as can be seen from Eq.(a). If the electric field is radial one can write, ⃗
E=kq ⃗r
r3, where kis the Coulomb
constant, so that Eq.(23k) can be written as
⃗
B=−2m
q⃗
Ω + ⃗
L
I, I =2r
rqmr2, rq=kq2
mc2,m
q=ρm
ρ(23l)
where ⃗
Lis the angular momentum of the boson, and rqis its classical radius. Here Ican be seen as a
moment of inertia of the boson. The magnetic moment associated with ⃗
Lis defined as ⃗µ =q
2m⃗
L.
Equation (23i) can be inverted to express the vorticity developed inside a superconductor due to ⃗
E
and ⃗
B. This can be written as
⃗ω =−q
m⃗
B−⃗v ×⃗
E
c2−⃗v
c2×∂⃗v
∂t ,(23m)
where ⃗
Eand ⃗
Bare given by Eq.(a). In the steady state, ∂⃗v
∂t = 0 so that
⃗ω =−q
m⃗
B−⃗v ×⃗
E
c2.
The equation of motion of the vorticity is obtained by taking the curl of Eq.(23c), viz.,
ρm
d⃗ω
dt =⃗
∇ × ⃗
f , (23n)
If the charge of the boson is equal to twice the charge of the electron, then
nb=ε0c4
4e2¯h2m3.(23o)
Equation (23c) is a very interesting equation, since it exhibits the fluid nature of superconductivity.
It is remarkable that massive bosons provide the full behavior that Cooper pairs are deemed to do in
the theory of BCS. In the BCS theory phonons are thought to play the crucial rule in maintaining the
Cooper pairs. However, in our present theory, it is the massive boson that is governed by Klein-Gordon
equation, that is responsible for electromagnetic interaction inside the superconductors, that endows the
superconducting state.
Notice that Eq.(23) can be obtained by taking the curl of Eq.(19) and using the Faraday’s equation.
By taking the curl of Eq.(2) and using Eqs.(2), (19) and (23), we obtain
1
c2
∂2⃗
E
∂t2− ∇2⃗
E+µ2⃗
E= 0 ,(24)
and
1
c2
∂2⃗
B
∂t2− ∇2⃗
B+µ2⃗
B= 0 .(25)
p-10
Modified electrodynamics for London’s superconductivity
Therefore, ⃗
E,⃗
B,⃗
J,ρand Ssatisfy Klein-Gordon equation. Taking the dot product of Eq.(19) with ⃗
J
and using Eq.(4) we arrive at the energy conservation equation of the system as
∂
∂t µ0
2µ2J2+ρ2
2ε0µ2+⃗
∇ · ρ
ε0µ2⃗
J=⃗
E·⃗
J+c2Sρ . (26)
The right hand side in Eq.(26) is positive which represents a system with energy gain, while Eq.(16) is
a system with energy loss (dissipation). The term µ0
2µ2J2is the kinetic energy of the persistent current
inside the medium (superconductor), and the term ρ2
2ε0µ2is the electric potential energy connected with
the charge density in the superconductor [4].
Un-dissipated Magnetoscalar wave.Let us see what happens when no electric or magnetic fields are
present, i.e., ⃗
E=⃗
B= 0. In this case, the current and charge densities can arise from the spatial and
temporal variations of the boson field, S,viz., Eqs.(2) and (3) yield
⃗
J=1
µ0
⃗
∇S , ρ =−ε0
∂S
∂t .(I)
Therefore, it is interesting that one can generate electric and charge densities out of the boson field
variations. However, the current and charge densities don’t create electric and magnetic fields. The
energy equation in Eq.(26) will be
∂
∂t |⃗
∇S|2
2+1
2c2∂S
∂t 2
+µ2
2S2+⃗
∇ · −∂S
∂t ⃗
∇S= 0 .(26a)
Here the energy flux is, ⃗
S=−1
µ0µ2
∂S
∂t ⃗
∇S, and the energy density is, u=|⃗
∇S|2
2µ0µ2+ε0
2µ2∂S
∂t 2+S2
2µ0.
Equation (26a) is the energy conservation equation of scalar field, S. Interestingly, in this case energy
flows without dissipation, as evident from Eq.(26a). Therefore, in the absence of the electromagnetic
field the scalar field provides the background contribution to the electromagnetic energy density in the
space. It this interesting to remark that our avove formulation is equivalent to an electromagnetic field
coupled to a scalar field.
It is interesting that in the absence of electric and magnetic fields, the modified Maxwell’s equations
can describe a charged scalar field. As evident from Eq.(I), the charge and current densities of this field
arise from the spatial and temporal variation of the scalar field. This field exits inside the superconductor.
Recall that Shas a dimension of Tesla satisfying the Klein-Gordon equation. Unlike others, I prefer to
call it a magnetoscalar wave. The current in Eq.(I) is that of a diffusion current. It is worth to remark
that Eq.(26a) may help explain the emission of Tesla’s wave. Such a theoretical formulation give the
Tesla experimental discoveries a legitimacy to work, and the technology associated with it a prosperity.
Equation (26a) generalizes the van Vlaenderen, and Monstein & Wesley equations to include massive
boson [7, 18]. They consider the wave to emerge from the temporal and spatial variations of the scalar
potential but not from the potential itself, however.
p-11
A. I. Arbab8
Waves with vanishing magnetic field.Let us consider now a wave with ⃗
B= 0. Substituting this in
Eq.(2) and doing some mathematical manipulations, we arrive at the energy equation
∂
∂t µ0
2µ2J2+1
2ε0µ2ρ2+1
2ε0E2+1
2µ0
S2+⃗
∇ · −S⃗
E
µ0
+ρ⃗
J
ε0µ2= 0 .(26b)
This represents a wave with vanishing dissipation term, i.e., a wave that travels without attenuation.
The energy density of this wave is contributed by ⃗
E,⃗
J,ρand S. Interestingly, in the absence of free
current, i.e., ⃗
J= 0, Eq.(26b) reduces to
∂
∂t 1
2ε0µ2ρ2+1
2ε0E2+1
2µ0
S2+⃗
∇ · −S⃗
E
µ0= 0 .(26c)
This latter wave flows in the opposite direction to that of the electric field. Here ⃗
E,S, and ρsatisfy the
Klein-Gordon equation.
It is interesting to see that Eqs.(19) and (23) are the London’s equations describing the electromagnetic
fields in superconductors if we set, µ2=µ0nse2
m, where mis the electron mass and nsis the super-electron
number density [4]. Recall that London derived his first’s equation using Newton’s second law of motion.
While in the London’s theory mis the electron mass, in our present theory the mass of the scalar boson
is mS=µ¯h
c. We argue here that mis not the electron mass but the scalar boson mass that flows inside
the superconductor. The question arises whether the two masses are equal or not. The equality would
imply that
m=µ0nse2¯h2
c21/3
.(27)
It is very interesting that the mass of the scalar boson will be determined by the material properties. It
is the same as that in Eq.(23f).
It seems that equations (19) and (23) are generic equations valid for any medium in which the
current, electric field, magnetic field, current density and charge density satisfy Klein-Gordon equation
with a scalar boson mediating these interactions. They occur in a medium in which the electromagnetic
field is carried by massive boson (Conductor, superconductor, plasma, etc.). In such conditions the
electromagnetic interactions become of short range.
Generalized Lorentz force. – The scalar boson besides interacting with the electromagnetic field,
it induces a force on the particle. This makes Lorentz force to have the form [7, 8]
⃗
F=q(⃗
E+⃗v ×⃗
B)−q⃗v S . (28)
The bosonic scalar force ⃗
FS=−q⃗v S appears as a viscous (drag) force on the particle. It requires an
existence of a fluid in which the particle is immersed (vacuum fluid) even when no fields exist. This force
suggests that the viscosity coefficient is η∝S. It accelerates a negative charge (electron) while decelerate
p-12
Modified electrodynamics for London’s superconductivity
a positive charge (ions) cooling it down. Bounded electrons can migrate to conduction state causing the
temperature of the conduction medium to drop slightly, and increasing the current significantly. This
may help explain the cold current phenomenon occurring in some electronic systems. However, if S < 0 ,
then electrons are decelerated while positive ions are accelerated. This lower the velocity of conduction
electrons while increasing the velocity of the ions and nuclei. Thus, conduction electron temperature drops
significantly turning them into bound states. As the number of these electrons is reduced, their bonds
with their nuclei are weakened leaving their atoms vulnerable to rupture, without noticeable temperature
rise. Consequently, the material tends to melt or even explode [19].
It is interesting to note that in the absence of electromagnetic fields, i.e.,⃗
E=⃗
B= 0, there still exists
a force on the moving particle, −qS⃗v. This is a drag force that requires a fluid in which the particle flows.
It thus amounts to existence of an ether-like medium. Bear in mind that the electromagnetic wave has an
impendence (intrinsic) even in vacuum. This may provide a legitimacy for this impedance. The concept
and existence of ether was rejected by physicists but later introduces with a different name (vacuum).
Therefore, with this view, objects would inherently resist even it is moving with constant velocity. Thus,
this may explain the origin of inertia. This amounts to say that inertia is associated with moving objects
only. The force on an object can be expressed as
⃗
F=m⃗a +dm
dt ⃗v , (29)
where ⃗a is the particle acceleration. The second term in Eq.(29) is zero for an object (particle) with
a fixed mass. However, according to quantum mechanics, a microparticle exhibits a perpetual state of
creation and annihilation. Owing to this fact, the the second term in Eq.(29) would not in general vanish.
Thus, if we associate this term to the second force in Eq.(28), then
dm
dt =−q S . (30)
Depending on the the sign of the charge of the moving particle, Eq.(30) expresses the rate of cre-
ation/annihlilation of the particle mass. Therefore, the scalar Sis a measure of mass creation or annihi-
lation of the particle interacting with the electromagnetic fields. This rate may have a connection with
quantum mechanical nature of microscopic particles. Note that if a particle emitting an electromagnetic
waves, it may lose part of its mass. Hence, according to Eq.(30), a moving object may acquire or lose
energy from/to the medium in which it is moving. We assume here that the rate, dm/dt =m/τ, to be
constant, where τis the time interval during which the creation-annihilation occurs. This time is defined
by the Heisenber’s uncertainty relation. Therefor, S=m/(qτ). Owing to classical electrodynamics, an
accelerating charge radiates at a rate proportional to the square of its acceleration. This entitles us to
p-13
A. I. Arbab9
write this radiated power (Larmor formula) as
P=q2a2
6πε0c2=1
6πε0c
q4S2v2
m2c2,(31)
that amounts to a relativistic correction to the electric power (it contains the factor, v2/c2). Thus because
of this energy loss, the range of the electromagnetic interaction is shortened. While the Larmor power for
an accelerating charged particle is independent of the particle mass, the power in Eq.(31) depends on the
particle mass. The power in Eq.(31) is proportional to q4. Recall that the luminosity of a star is related to
its mass by a similar relation, L∝M4[20]. This amounts to say that gravitational and electromagnetic
systems are analogous. It is the Higgs field that exerts a force on any particle interacting with it [12].
This allows us to anticipate that Scould be the Higgs boson. Using the relation S=m/(qτ), Eq.(31)
can be written as
Pb=µ0
6πc
q2v2
τ2.(32)
Defining, I=q/τ , as the current developed during the creation-annihilation time, τ, one can write
S=mI/q2. It is interesting to note that the magnetic scalar, S, is determined by the mass and the
current of the boson. Moreover, Eq.(32) can be written as
Pb=µ0
6πI2v
cv . (33)
This can be associated with a force Fb, where Pb=Fbv(v/c) represents the relativistic power correction
such that Fb=µ0
6πI2. This can be compared with the longitudinal force arising from a massive photon
motion in the medium, F=1
2µ0I2[21]. Equation (33) also suggests that
Pb=I2Rb, Rb=µ0c
6πv
c2.(34)
The resistance Rbcan be thought as due to the resistance experienced by the bosons when moving in a
medium. This can be seen as a relativistic correction of the massless photon impedance when the photon
acquires a mass, noting that the vacuum impedance is given by Z0=µ0c. Hence, Rb<< Z0is the boson
impedance in a medium. Since bosons acquire charge and mass inside the medium, its motion experiences
a resistance Rb.
Concluding remarks. – A massive spinless boson can be generated within the framework of
Maxwell’s equations as a manifestation of the non-conservation of charge, and a relaxation of Lorenz
gauge condition. The charge carried by the scalar field is at the expense of being lost by the medium in
such a way the overall charge is conserved. However, if only Lorenz gauge condition is broken a massless
boson is generated. This agrees with spontaneous global breaking of a continuous symmetry where a
massless Goldstone boson is created [22,23]. Hence, when an electromagnetic wave enters a medium one
p-14
Modified electrodynamics for London’s superconductivity
of its components will acquire a mass if it can pick up a charge from the medium. This is in conformity
with the Higgs method that shows how a massless particle acquires a mass. It is worth to mention that
the scalar particle interacts with the static charges (and the electric field) in the medium. The proposed
modifications yield London’s theory of superconductivity, where the superconductivity is transported by
the scalar boson. The set of equations, Eqs.(2) and (3), are the Maxwell’s equations applicable in a
medium. In vacuum they reduce to the ordinary Maxwell’s equations since S= 0. In this case the charge
conservation and Lorenz gauge condition are satisfied. The scalar field is related to the gauge fixing scalar
by the relation S=−µ2λ. This scalar is a measure of the rate of creation or annihilation of the particle
mass.
Acknowledgements. – I would like to express my deep gratitude to Prof. A. J. Leggett for kind
hospitality at the Institute of Condensed Matter Theory (ICMT), University of Illinois, Champaign-
Urbana, where this work is carried out.
REFERENCES
[1] Tesla, N.,The transmission of electrical energy without wires as a means of furthering world, Electrical world
and engineer,–(1905) 21.
[2] Bardeen, J., Cooper, L. N., and Schrieffer, J. R.,Microscopic Theory of Superconductivity, Phys. Rev.,
106 (1957) 162.
[3] Proca, A.,Sur la Theorie ondulatoire des electrons positifs et negatifs,J. Phys. Radium,7(1936) 347.
[4] London, F. and London, H.,The Electromagnetic Equations of the Supraconductor, Proc. Roy. Soc. (Lon-
don) A,149 (1935) 71.
[5] Arbab, A. I.,The extended gaue tranformations, Prog. In Electromag. Res. M,39 (2014) 107.
[6] Ginzburg, V.L. and Landau, L.D.,Zh. Eksp. Teor. Fiz.,20 (1064) 1950.
[7] van Vlaenderen, K. J.,A generalization of classical electrodynamics for the prediction of scalar field effects,
http://arxiv.org/abs/physics/0305098.
[8] Arbab, A. I.,Complex Maxwells equations, Chin. Phys. B,22 (2013) 030301.
[9] Arbab, A. I.,The analogy between matter and electromagnetic waves, EPL,94 (2011) 50005.
[10] Arbab, A. I.,On the Generalized Maxwell’s Equations and Their Prediction of Electroscalar Wave, Prog.
in Phys.,2(2009) 8.
[11] Jackson, J. D.,Classical Electrodynamics, (Wiley, New York), 1999.
[12] Higgs, P. W.,Broken Symmetries and the Masses of Gauge Bosons, Phys. Rev. Lett.,13 (1964) 508.
[13] Abrikosov, A. A.,Journal of Physics and Chemistry of Solids,2(199) 1957.
[14] Page, D. et al.,Rapid Cooling of the Neutron Star in Cassiopeia A Triggered by Neutron Superfluidity in
Dense Matter Phys.Rev.Lett.,106 (081101) 2011.
[15] Arbab, A. I.,The Fractional Hydrogen Atom: A Paradigm for Astrophysical Phenomena J. Modern Physics,
3(1737) 2012.
[16] Graber, V., et al.,,Magnetic Field Evolution in Superconducting Neutron Stars,
http://arxiv.org/abs/1505.00124.
[17] London, F.,Macroscopic Theory of Superconductivity, (New York: Dover) 1961.
[18] Monstein, C. and Wesley, J. P.,Observation of scalar longitudinal electrodynamic waves Europhys. Lett.,
59 (514) 2002.
[19] see: http://www.signallake.com/innovation/GeneralizedClassicalElectrodynamics.pdf
[20] Eddington, A. S., On the relation between the masses and luminosities of the stars, MNRAS,84 (1924)
308.
[21] Arbab, A. I.,Electric and magnetic fields due to massive photons and their consequences, Prog. In Electro-
mag. Res. M,34 (2014) 153.
[22] Goldstone, J.,Field theories with (Superconductor) Solutions, Nuovo Cimento,19 (1961) 154.
p-15
A. I. Arbab10
[23] Goldstone, J., Salam, A. and Weinberg, S.,Broken symmetries, Phys. Rev.,127 (1962) 965.
p-16
