On massive photons inside a superconductor as follows from London and Ginzburg–Landau theory
A phenomenological derivation in the frame of London's and Ginzburg-Landau theories is given that photons behave inside a superconductor as if they have mass.
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| citation_txt | On massive photons inside a superconductor as follows from London and Ginzburg–Landau theory / R. de Bruyn Ouboter, A.N. Omelyanchouk // Физика низких температур. — 2017. — Т. 43, № 7. — С. 1109-1112. — Бібліогр.: 10 назв. — англ. |
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| container_title | Физика низких температур |
| description | A phenomenological derivation in the frame of London's and Ginzburg-Landau theories is given that photons behave inside a superconductor as if they have mass.
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| first_indexed | 2025-11-27T10:27:55Z |
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7, pp. 1109–1112
On massive photons inside a superconductor as follows
from London and Ginzburg–Landau theory
R. de Bruyn Ouboter
Kamerlingh Onnes Laboratory, Leiden Institute of Physics, Leiden University
P.O. 9506, 2300 RA Leiden, The Netherlands
E-mail: r.de.bruijn.ouboter@umail.leidenuniv.nl
A.N. Omelyanchouk
B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine
47 Nauky Ave., Kharkiv 61103, Ukraine
Received April 3, 2017, published online May 25, 2017
A phenomenological derivation in the frame of London`s and Ginzburg–Landau theories is given that pho-
tons behave inside a superconductor as if they have mass.
PACS: 03.50.De Classical electromagnetism, Maxwell equations;
31.30.J– Relativistic and quantum electrodynamic (QED) effects in atoms, molecules, and ions;
74.20.De Phenomenological theories (two-fluid, Ginzburg–Landau, etc.).
Keywords: massive photon, inside a superconductor, London and Ginzburg–Landau theory.
The research presented in this paper is triggered by the
reading the timelines in Wilczek’s recent book [1] on “a beau-
tiful question — finding nature’s deep design” on p. 335:
“1963 Philip Anderson [2] suggests the importance for parti-
cle physics of work on equations for massive photons that
arose in work by the brothers Fritz and Heinz London [3] in
1935 and Lev Landau and Vitaly Ginzburg [4] in 1950”.
A phenomenological derivation is given that photons
behave inside a superconductor as if they have mass by
comparison of the original first equation of the London`s
and the equations for the electromagnetic field with the
time-dependent relativistic Schrödinger equation. The pho-
tons move through a medium, the Ginzburg–Landau free
energy density, inside the superconductor in which they
acquire mass. The Compton wave length of the massive
photons is equal to 2π times the London penetration depth
and the mass of the photon is equal to / Lm c= λ .
The essential feature of superconductivity according
F. London [3] is a condensation of a macroscopic number
particles (bound electron pairs, with mass 2 em and charge
2e, first described by Cooper [5]) in the same single
quasiparticle quantum state and obtained a fundamental
relation for the generalized dynamical momentum sp of
the superconducting pairs,
(2 ) (2 ) ,s e sm e= + = ∇φp v A (1)
in which sv is the superfluid velocity, A is the vector po-
tential, and φ is the phase of the macroscopic wave func-
tion. Cooper pairs behave like bosons. The superfluid cur-
rent density (2 )
2
s
s s
n
e=I v in which sn is the superfluid
density. Taking the curl of Eq. (1) the well known first
relation of the London’s from 1935 is obtained for in simp-
ly connected isolated superconductor:
2
2
em
e
= −A v (2)
or
0 2
1
s
L
µ = −
λ
I A (3)
in which Lλ is the London penetration depth
2
2
0
e
L
s
m
n e
λ =
µ
, (4)
or
2
0 2
e
L
s
m
n e
µ λ = Λ = . (5)
Equation (3) is valid as long as s GL GLζ = ∇φ ζp
in which GLξ is the Ginzburg–Landau (GL) coherence
© R. de Bruyn Ouboter and A.N. Omelyanchouk, 2017
R. de Bruyn Ouboter and A.N. Omelyanchouk
length. When the wave length 1/ ∇φ is smaller or com-
patible to the coherence length the superconductivity dis-
appears. According to Ginzburg–Landau [4] is their mac-
roscopic theory reliable based on Eq. (3).
We restrict ourselves mainly to the case T = 0, and ne-
glect normal currents. We like to remark that at T = 0 the
London penetration depth
2
2
2
0
(0) e
L
pe
m c
n e
λ = = ωµ
in which the plasma frequency pω is defined by
2 2
0/p e en e mω ≡ ε (at T = 0 sn goes to the electron density
en ).
Combining the first London equation (3) with the
Maxwell equation
2
2
02 2
1
s
c t
∂
∇ − = −µ
∂
AA I ,
the London’s obtain [3]
2
2
2 2 2
1 ,
Lc t
∂
∇ − =
∂ λ
A AA (6)
2
2
2 2 2
1 ,
Lc t
∂
∇ − =
∂ λ
B BB (7)
2
2
2 2 2
1 .
Lc t
∂
∇ − =
∂ λ
E EE (8)
In a static situation Eq. (7) leads to 2 2/ L∇ = λB B which
explains the Meissner effect.
The differential Eqs. (6)–(8) contain, respectively, only
A, B and E and its spatial and time differentials of second
order separately and the constants 2
Lλ and 2c .
However, the brilliant observation of Anderson [2]
(1963) was that the Eqs. (6), (7) and (8) of the work of the
London`s is also applicable to the photon field inside the
superconductor with massive photons presented by the
terms 2/ LλA , 2/ LλB and 2/ LλE .
In this phenomenological description is for comparison
written down the relativistic Schrödinger wave equation
[6] for a free particle which shows the same structure and
describes Bose particles, hence also photons with mass m
and wave function ψ :
2 2 2
2
2 2 2 2
1
c
m c
c t
∂ ψ ψ
∇ ψ − = ψ ≡
∂ λ
(9)
in which /c mcλ ≡ is equals to the Compton wave length
/h mc divided by 2π for a photon with mass m .
The Eqs. (6)–(9) are relativistic equations and have the
same form, are mathematically identical and describe ex-
actly the same phenomenon [1]. The squared lengths on
the right hand sides of the Eqs. (6)–(8) 2
Lλ , and in Eq. (9),
2 2 2 2/c m cλ ≡ should be equal to each other
2 2 2 2 2/L c m cλ = λ = ,
or
2
L
mc c
=
λ
, (10)
or
/ Lm c= λ . (11)
Mass (11) is the mass of the photon inside the supercon-
ductor. This implies when penetrating the superconductor
from outside into the bulk, superconductivity and photon
mass arises in the same way. At T = 0, sn goes to the elec-
tron density en so that the plasma frequency is equals to
2
(0)
p
L
c mc
ω = =
λ
. (12)
If we use (0)Lλ = 500 Å = 5⋅10–8 m we find
34
36
8 8
10/ 7 10 kg
3 10 5 10
Lm c
−
−
−
= λ = = ⋅
⋅ ⋅ ⋅
.
For comparison 319,1 10em −= ⋅ kg and we find for
15/ (0) 6 10 Hzp Lcω = λ = ⋅ . For comparison the gap fre-
quency 11 12
gap 2 (0)/ 10 –10 Hzω ≈ ∆ ≈ .
The very small photon mass m implies a very large ze-
ro-point motion inside the superconductor.
These macroscopic phenomenological considerations
are very academic since free space between the atoms in
the superconductor is very limited for investigation. A pho-
ton in empty spaces moves at the speed of light, v = c, and
has two transverse field components (E and B) perpendicu-
lar to each other and perpendicular to the direction of wave
propagation. From the Eqs. (7) and (8) follows that a pho-
ton in motion inside a superconductor acquires also a third
degree of freedom forward and back in the direction of
motion (left and right, up and down and forward and back
oscillations) leading to a particle with mass.
A massless photon moves at the speed of light in vacu-
um and moves into a bulk superconductor through a medi-
um, a field inside the superconductor (the Ginzburg–
Landau free energy density) of which the symmetry is bro-
ken and the photon acquires mass. We start by investigat-
ing the penetration depth in the Ginzburg–Landau theory
(1950) [4]. We write down a modern version of the second
Ginzburg–Landau equation, an equation also present in the
theory of F. and H. London (1935) [3]:
1110 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7
On massive photons inside a superconductor as follows from London and Ginzburg–Landau theory
2
2**
2 2
(2 ) (2 )( )
2(2 ) 2
2(2 ) (2 )
2
s s s s s s
e e
s s s
e
i e e
m m
ee e
m
= ψ ∇ψ −ψ ∇ψ − ψ =
= ψ ∇φ− = ψ
I A
A v
(13)
from which follows Eq. (1). Taking the curl of Eqs. (1)
and (2) is obtained. This Eq. (13) follows in form from
both the relativistic [8] and the nonrelativistic [6,8] ex-
pression, in which 2 /2s snψ = is the pair density in
modern language. In the GL theory the difference in the
free energy density 2 ,sF T ψ
can be written as
2 2 41( , ) ( ) ( ) ...
2s s n s sF F T F T T∆ = ψ − = α ψ + β ψ +
(14)
in which β is positive at all temperatures and ( ) 0Tα < for
cT T< . The equilibrium superconductive state is
2/ 0sF∂∆ ∂ ψ = , hence 2 2/ 0s sF∂∆ ∂ ψ = α +β ψ = , or
2
equil / /sψ = −α β = α β and 2
equil equil
1
2 sF∆ = − α ψ . We
find the GL penetration depth GLλ by
2
2
0 02 2
1 2
2
s
s s
n
e
c t
∂
∇ − = −µ = −µ =
∂
AA I v
2
0 2
(2 )
2 e GL
e
m
α
= µ =
β λ
AA (15)
in which
2
22
0
2
.
(2 )
e
GL
s
m
e
λ =
µ ψ
If 2
equil /2s snψ = this equation is identical with Eq. (4),
the London penetration depth, and
2 4 2 2 2
2
02 2
(2 ) .
2 s
eL
m c c e c
m
= = µ ψ
λ
(16)
In the spirit of the “Higgs mechanism” and the principle of
broken symmetry which starts when the temperature is low-
ered from cT T> to cT T< we plot the relevant Ginzburg–
Landau free energy density F versus 2
sψ , which is often
called the “Higgs field”, for both cT T> and cT T< when the
symmetry of the “Higgs field” is broken (Fig. 1).
The lowest potential energy density for cT T< corre-
sponds to a finite displacement of a non-zero value of
2
sψ . There is a small bump in the bottom of the curve,
the presence of this bump forces the symmetry to break as
the “field” cools from cT T> to cT T< and a valley appears
in the curve. The lowest point in the curve corresponds to a
non-zero value of the scalar “field”. The photon in the su-
perconductor does interact with the Ginzburg–Landau or
“Higgs field”, it interacts with this field, gains energy,
slows down, the “field” dragged on the photon and the
interaction with the particle photon and the field is mani-
fested as a resistance of the photon particle acceleration.
When the photon particle moves at constant velocity it is
not affected by the “field” and 2/ 0sF∂∆ ∂ ψ = . The
Ginzburg–Landau “field” is a scalar field with no direc-
tions. During the cooling of the superconductor from
cT T> to cT T< each photon inside the superconductor
acquires an energy 2mc .
We now consider the solution of the Eqs. (6)–(9) inside
the bulk superconductor
2 4 2
2 2 2 2
2 1 L
E m c mccω = = + = + λk k k
(17)
in which 2E = ω = πνk is the relativistic energy
2 2 4 2 2( )pE m c p c= + and 2 /p k= = π λ is the relativistic
momentum. We have plotted ω versus k for the photon
mass m inside the bulk superconductor and for a photon in
empty space /c k= ω (Fig. 2) for two cases: a type I and a
type II superconductors. Inside the bulk superconductor the
massive photon particle has a group velocity /g k= ∂ω ∂v of
the associated wave.
We remarked already that if 2 / 1/ GL∇φ = π λ < ξ super-
conductivity exists [10], and if 2 / 1/ GL∇φ = π λ > ξ super-
conductivity ceases to exist [10]. For k values smaller than
1/ GLξ the wave-packet of the photon behaves massive,
contrary to the opposite case for k values larger than 1/ .GLξ
It should be possible in principle to observe this transition
from the superconductive state 1/ GL∇φ < ξ to the state of
anomalous conductivity 1/ GL∇φ > ξ with a quantum foam-
like structure (hence from the m to the /k cω = state of
photons) in a superconducting layer in an external radia-
tion field 2 /∇φ = π λ of which the frequency increases.
Fig. 1. The relevant GL free energy density F versus 2
sψ for
cT T> and cT T< .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 1111
R. de Bruyn Ouboter and A.N. Omelyanchouk
1. F. Wilczek, A Beautiful Question — Finding Nature’s Deep
Design, Penguin Books, New York (2016).
2. P.W. Anderson, Phys. Rev. 130, 439 (1963).
3. F. London and H. London, Proc. Roy. Sec. (London) A 149
(1937); F. London, Superfluids. Macroscopic Theory of
Superconductivity, Vol. I (1950), Dover Publications, New
York (1961).
4. V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20,
1064 (1950).
5. L.N. Cooper, Phys. Rev. 104, 1189 (1956); J. Bardeen, L.N.
Cooper, and J.R. Schrieffer, Phys. Rev. 108, 1175 (1957).
6. W. Pauli, Wave Mechanics, Pauli Lectures on Physics,
Vol. 5, The M.I.T. Press, Cambridge, Massachsetts (1973).
7. J.R. Waldram, Superconductivity of Metals and Cuprates,
Institute of Physics Publishing (1996); M. Tinkham,
Introduction to Superconductivity, McGraw-Hill (1996).
8. H.A. Kramers, Quantum Mechanics (1933, 1937), North
Holland Publ. Comp., Amsterdam (1957); L.I. Schiff,
Quantum Mechanics, 2nd ed., McGraw-Hill Publ. Comp.,
New York (1955).
9. A.J. Leggett, Quantum Liquids, Bose Condensation and
Cooper Pairing in Condensed Matter Systems, Oxford
University Press (2006).
10. P. Noziers, Superfluidity in Bose and Fermi Liquids, in D.F.
Brewer, Quantum Fluids, North Holland Publ. Comp.,
Amsterdam (1966).
Fig. 2. ω versus k for the photon mass m inside the bulk su-
perconductor and for a photon in empty space /c k= ω for two
cases: a type I and a type II superconductors.
1112 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7
|
| id | nasplib_isofts_kiev_ua-123456789-129539 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-11-27T10:27:55Z |
| publishDate | 2017 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | de Bruyn Ouboter, R. Omelyanchouk, A.N. 2018-01-19T20:52:03Z 2018-01-19T20:52:03Z 2017 On massive photons inside a superconductor as follows from London and Ginzburg–Landau theory / R. de Bruyn Ouboter, A.N. Omelyanchouk // Физика низких температур. — 2017. — Т. 43, № 7. — С. 1109-1112. — Бібліогр.: 10 назв. — англ. 0132-6414 PACS: 03.50.De, 31.30.J– https://nasplib.isofts.kiev.ua/handle/123456789/129539 A phenomenological derivation in the frame of London's and Ginzburg-Landau theories is given that photons behave inside a superconductor as if they have mass. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Сверхпроводящие и мезоскопические структуры. К 70-летию со дня рождения А.Н. Омельянчука On massive photons inside a superconductor as follows from London and Ginzburg–Landau theory Article published earlier |
| spellingShingle | On massive photons inside a superconductor as follows from London and Ginzburg–Landau theory de Bruyn Ouboter, R. Omelyanchouk, A.N. Сверхпроводящие и мезоскопические структуры. К 70-летию со дня рождения А.Н. Омельянчука |
| title | On massive photons inside a superconductor as follows from London and Ginzburg–Landau theory |
| title_full | On massive photons inside a superconductor as follows from London and Ginzburg–Landau theory |
| title_fullStr | On massive photons inside a superconductor as follows from London and Ginzburg–Landau theory |
| title_full_unstemmed | On massive photons inside a superconductor as follows from London and Ginzburg–Landau theory |
| title_short | On massive photons inside a superconductor as follows from London and Ginzburg–Landau theory |
| title_sort | on massive photons inside a superconductor as follows from london and ginzburg–landau theory |
| topic | Сверхпроводящие и мезоскопические структуры. К 70-летию со дня рождения А.Н. Омельянчука |
| topic_facet | Сверхпроводящие и мезоскопические структуры. К 70-летию со дня рождения А.Н. Омельянчука |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/129539 |
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