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Free-field potentials in electrodynamics
PHYSICS LE1”I’ERS A
Volume 142, number 4,5
11 December 1989
FREE-FIELD POTENTIAlS IN ELECTRODYNAMICS V.M. DUBOVIK and S.V. SHABANOV’ Joint Inst itutefor Nuclear Research, Laboratory of Theoretical Physics, P.O. Box 79, 101000 Moscow, USSR Received 12 April 1989; revised manuscript received 28 June 1989; accepted for publication 6 October 1989 Communicated by J.P. Vigier
A physical (gauge invariant) degree of freedom of the vector potential generating no electromagnetic fields is singled out. The problem of observing this degreeof freedom is discussed.
1. About 60 years ago Fock [1] and Weyl [2] have proposed a gauge method for describing interactions of quantum charged particles with an external electromagnetic field. It consists in the requirement for the Schrodinger equation to be invariant for a charged particle under local phase transformations of the wave function. For this purpose, the vector potential A should be introduced so that the equation (l) ihä,yi={(l/2m)[ —ihV— (e/c)A]2+e9,}~v, where e is the particle charge and q’ is the scalar ~ tential, does not change under gauge transformations ~v—exp(iew/~c)~v, A—~A+Vw,
~—~—
-L9~w.
(2)
C
This invariance was called the U (1) gauge symmetry. Further, this method of introducing an interaction was generalised by Yang and Mills to non-Abehan symmetry groups [3]. The experience of classical physics suggests that only the elecirornagnetic fields E= —VQ~—(1 /c)8~A and B=rot A act on charged particles. However, in contrast with the Lorentz equation for a classical particle in E and B fields, eq. (1) shows that a quanturn charged particle interacts with an electromagnetic field through the potentials A and This mitiated the discussion of the vector potential. IsA just ~.
Novosibirsk State University, Pirogova 2, 630090 Novosibirsk, USSR.
a convenient form describing interactions of partides with an electromagnetic field or is it a more fundamental physical concept than a field itself? In other words, is there any A which does not correspond to any electromagnetic field? A reason for a negative answer to this question is usually ambiguity in determing A (see (2)) since the observed quantities, like E and B, cannot depend on gauge arbitrariness. In the present paper we will show a vector potential to have such a physical (gauge invariant) degree of freedom which is locally associated with zero values of strengths. This degree of freedom, as any physical quantity, should manifest itself in nature. It turns out to act only on quantum systems; therefore, its macroscopic manifestation is possible only as a collective quantum effect, which is observed in superconductivity. 2. Gauge symmetry in electrodynamics leads to constraints on dynamic variables in the theory [4], i.e. between generahised coordinates and momenta of a dynamic system there are some relations contaming no derivatives with respect to time. Canonical coordinates in electrodynamics are the components of the vector potential A~(x) (n = 1, 2, 3) and ~(x) at each point of the space. Canonical momenta conjugate to them correspond to components of the electric field —E~(x)with an opposite sign; the momentum conjugate to ~(x) equals zero E 0(x) =0 (this is the primary constraint in electrodynamics). For the quantities A~,q’ and E~,E0 the Poisson brackets 211
Volume 142, number 4,5
PHYSICS LETrERS A
{A~(x),E,(x)}=—o,~o3(x_y), {ço(x), E 3(x—y) (3) 0(y)}= —~ are determined. The total set of constraints in electrodynamics is [4]
11 December 1989
substitute (7) into (6); them,into the integration in the operator A~ decays two parts Vregion and R3\V. The integral over R3\V transforms by the Gauss theorem and then using the equality 41 x— y I = 4itô3 (x—y) we get —‘
—
A ~ = VXPh, E 0(x)=0,
0nEn(x)=p(x),
(4)
where p(x) is the density of charges generating an electric field. Every observed quantity F~h[A, E, E0] depend-
x~~h[Al=o,,
~,
r v~(y)~(y) —on ~ 4Ex—yI dy,
ing on canonical variables should be a gauge invariant and thus satisfy the equations {F~h,E0}=0,
A”(y) dy J 4~~—yl
where x~8V,V and (5)
v~,is the
(8) external normal to the
as the constraints (4) are generators of gauge transformations [4]. It follows from the first equation of (5) and (4) that FPh can depend only on A and E. The general sohution of the second equation of (5) is F~h[A, E] =F[APh, E], where [5] A [Al =A~ —o n8k4 1Ak, 0 nA ~h 0. (6)
surface OV. Functional (8) is a gauge invariant (Av_~Av+Vw, x—~x+w);moreover, by definition XPh is a harmonic function (see (6)) in the region ~ \ V. In fact, XPh is the solution of the external Dirichlet problem for the Laplace equation in the multiply connected In the simplest cases when 3\Vregion. is doubly connected~”(8) has a the region P simple form
The operator A~is determined in a standard way,
XPh[A]
{FPh, OnEn_P}m{Fph, OnEn}0,
~h
—
4’f(x)=— Jd3x(4RIx_YIY’f(y),
~,
~‘~‘
xeV, rot Av=B,
A (x) = V~(x), x~V, B= 0.
(9)
where the contour C encircles once the region V, i.e., 0 is the magnetic field flux through the contour C, and the function f(x) depends on the geometry of the region V. Since XI~h[A] is a gauge invariant, one can pass from (8) to (9) in any appropriate gauge. For instance, for an infinite solenoid directed along the Oz axis f(x) = O/27L, where 0 is the angle of the cylindrical system of coordinates. Note that in spite ofthe “pure” gauge form of the vector potential (7) for x~V it cannot be reduced to zero in the P3 \ V gauge transformation since x is a multi-valued function. Gauge transformations with multivalued functions are inadmissible as they change the flux of the magnetic field in the region B: ~ (VX, dI) 0 is the gauge invariant (observed quantity) where C is the contour that cannot be con~‘
(7)
Moreover, at the boundary OV there is a condition of continuity A”=VX, xeOV, i.e., B=0, xeOV. We 212
(A, dI)=f(x)Ø, C
and integration is performed over the whole threedimensional space P3. It is to be noted that from the viewpoint of Hamihtonian dynamics A ~“ is the canonical variable describing two transverse physical degrees of freedom of the electromagnetic field (the conjugatemomenta are the components ofthe transverse part of the electric field [5]). This variable should naturally be introduced while quantizing electrodynamics as a system with constraints [5]. Let us find the value of the functional A 1” [A] at the space points where E=B=0. Since E= 0, we considerthe stationary problem. Then, from B= rot A = 0 it follows A =V~.If B=0 at all the points xeP then from (6) we have the trivial equality A [VX]= 0. A different situation arises when there is a region V where Bs~0.In this case we have A(x) =Av(x),
=f(x)
Multiconnectedness of the region R3\V follows form the equation div 8=0. If B~0only in V, then there always exists a contour that is not in V the magnetic flux through which is nonzero. For instance, this contour can be chosen on ÔV (8=0, xet9V see (7)) as the lines of the field B are always closed: divB=’O.
Volume 142, number 4,5
PHYSICS LETTERS A
11 December 1989
tracted at a point without intersecting the boundary OV. From this point of view the existence of XPh [A] seems to be quite natural. Thus, around the regions occupied by the magnetic field there is a real (gauge invariant) field. However, the definition “real” needs to be refined, namely, one should point out the way of its observing. This is just the purpose of the next section.
mations L—+L+dQ/dt are admissible including those with multiple-valued functions £2. Using the rules of canonical quantization one can formulate a quantum theory corresponding to the Lagrangian (11). The obtained SchrOdinger equation will contain only gauge-invariant quantities. The same result can be obtained from eq. (1) by introducing the gauge-invariant wave function
3. To solve the problem of physical manifestation of the field Xt~I~, let us consider the interaction of a charged particle with an external electromagnetic field. The Lagrangian is
~
—
—
(10)
component of an electromagnetic field are linear combinations of one physical and one nonphysical degree of freedom [5]: Using the Schrodinger equa-
Under gauge transformations (2) the Lagrangian (10) is added by a total derivative
tion in the gauge invariant variables we can show that the field XPh affects a charged particle. These are wellknown results [7,8]. However, it is to be noted that
L—~L+e d C dt
our consideration is by construction independent of the choice of gauge in contrast with refs. [7,8].
(d/dt= O,+ (A, V) is the material derivative). Therefore, the Lagrangian equations of motion are invariant under gauge transformations. To introduce XPh into L we must find its gauge-invariant form. For this purpose we add dQ/dt into the right-hand side of(10)whereQ=(e/c)8,,A’A~.Asaresult,wehave obtained the gauge-invariant Lagrangian
4. A superconductor is the simplest system in which one can observe the action of the field XPh on a quanturn system. Its free energy in the external electromagnetic field is
e h )—e4 L(x,x)=~mx——(x,A”
+(l/4m)~ft[—i*V+(2e/C)A]2W}, (13) where V is the volume occupied by a superconduc-
L (x, x) =
e
= exp [ i (e/hC)4 ‘O nAn] t~, (12) then substituting i,v into (1) and expressing ço from constraint (4). Formula (12) reflects the known fact that the phase of a charged field and the transverse
~
—
—
(x, A) + eq’.
C
.
p,
(11)
C
where we have used the equality E= —Vço— (1 /C)A and constraint (4). The Lagrangian (11) is independent of the choice of gauge. Note that the Lagrangian (11) differs from the gauge invariant Lagrangian, containing path-dependent integrals on A, in the Mandelstam formulation [6].Now, to introduce an interaction with ~Ph, it is sufficient to put p = 0 and A “ VX~.Then, the Lagrangian (11) differs from the free one only by the total derivative (e/c)d~~/dl.Hence, a classical particle is insensitive to XPh. This was to be expected as at E=B= 0 the Lorentz strength is absent. From the viewpoint of the Lagrangian formalism this means that the freedom in choosing the L.agrangian by given equations of motion is greater than gauge arbitrariness in —
electrodynamics: Thus, in the first case transfor-
2+~bIwI4
Fs=j dx{—aI~I v
and ~visthe complex parameter of the Landau— Ginzburg order or the wave function of the Cooper pair. Using transformation (12) (e—~2e) expression (13) can be rewritten in terms of gauge-invariant variables ~ and A ~ so that by varying them one can obtain the gauge-invariant Landau—Ginzburg equations. Our further analysis will be qualitative; therefore, in (13) we have omitted the magnetic field energy, i.e., disregarded the depth of penetration of the magnetic field into the superconductor. The basic state of the superconductor is specified by the minimum ofF,. IfA~’=0,then the minimum ofF, is achieved at ~u=y,~= const as the kinetic energy vanishes (this is an absolute minimum ofF,). Usually, ~v 0is normalised as Wo 12 = n,/2 where n, is the density of paired electrons. tor,
213
Volume 142, number 4,5
PHYSICS LETTERS A
Consider now a superconducting ring put on the sohenoid. If the ring temperature is T> T~,where T~ is the critical temperature, the current in the ring dies out. Let us stabilize the flux of the magnetic field for T> T~.Then, upon cooling the ring to T< T~it transforms into a superconducting state. After substitution ofA = VXPl~for an infinite long solenoid in eq. (13) we find that the only difference of F, in cornparison with the case XPh = 0 consists in the change of the rotation energy of the condensate L ~ (L~ h0/00 2, where L~= ihO/8O is the operator ~
-+
—
—
of the angular momentum projection onto the solenoid axis, 0 is the polar angle, and 0o = lthC/e is the magnetic flux quantum (fluxon). For the internal quantisation of the angular momentum m0 = [0/0o1, (14) where rn0 is chosen from the condition of minimal F,, integer, and [0/00] rounding of 0/00 nearest if themeans integral quantisation ofto thethe angular momentum is valid, then in the superconducting ring there arises a current proportional to the difference m 0Ø0 _~*2 Note that there is no current in the superconductor if the angular moment is quantised with an arbitrary number [101 which however disagrees with experiment (see ref. [11]and references therein), Based on the simplest idea of a short-range interaction, i.e., transfer of interaction by a field, one should point out what physical (i.e., independent of gauge) field forced the Cooper pairs to move to create a current. Just the field XPh plays the role of a transmitter of interaction. In this connection, we should like to recall the paper by Wu and Yang [12] in which they have introduced the “minimal” description of electromagnetic interactions in terms of a nonintegrable phase factor. Again, keeping the idea of a short range interaction and locality, one should point out what field is responsible for the appearance in the wave function of this phase factor. As above, Wph
=
Wo exp (iOm0),
~,
S2
1*3
The exact value of the current can be calculated by the formula I,=c(m0Ø,—0) IL, where L is the contour inductance that is exact for superconductors [9]. The authors of ref. [10] have discussed the behaviour of a superconductor in the field of a solenoid but they were not correct in their reasonings, by, for example, incorrect use of the Coulomb gauge and the use ofsingular gauge transformations ElI].
214
II December 1989
this is provided by the interaction
with XPh.
5. In conclusion we should like to remind ofan old but very physical paper [13]. Based on ref. [7]its author raised two questions. The first is the possibility ofobserving the effect on an electron in the region where E=B= 0 and the potential is nonzero. We give a positive answer to this question. The second question is: does there exist a specific action of the potential in quantum mechanics, differing from that in classical physics, which would enable one to treat the potential as a more fundamental quantity than the field intensity and would require reformulation of the basic assumptions of the theory? The author ofref. [13]gives a negative answer. The literature on this problem is quite voluminous. Our point of view is the following. In electrodynamics, the vector potential really has a physical degree of freedom that is not3\V associated (i.e., at this locally is important) each any pointelectromagnetic of the space Pfield. Thus, the vector powith tentiah is a fundamental object with respect to the field. The observed degree of freedom manifests it—
self only in quantum theory; it possesses the basic attribute of a physical field ability to act on a physical system. Moreover, as has been shown, the fundamentals of neither classical electrodynamics nor quantum theory need be reformulated. —
References [1] V.A. Fock, Z. Phys. 39 (1927) 226. [2] H. Weyl, Z. Phys. 56 (1929) 330. [3] P.A.M. C.N. Yang and Lectures R.L. Mills,onPhys. Rev. 96 (1954) 191. [4] Dirac, quantum mechanics (Yeshiva University, New York, 1964). [5] L.V. Prohorov, Yad. Fiz. 35 (1982) 229. [6] S. Mandelstam, Ann. Phys. (NY) 19 (1962)1; E. Kazes et al., Phys. Rev. D 27 (1982) 1388. [7] W. Ehrenberg and R.E. Disay, Proc. Phys. Soc. B 62 (1949) 8; Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [8]M. Peshkin et al., Ann. Phys. (NY) 12 (1961) 426; 16 (1961) 177; M. Peshkin, Phys. Rep. 80 (1981) 375. [9] V.A. Fock, Phys. Z. Sowjetunion 1 (1932) 215. [10] J.Q. Liang and X.X. Ding, Phys. Rev. Lett. 60 (1988) 836. [11] A. Tonomura and A. Fukuhara, Phys. Rev. Lett. 62 (1989) 113. [12] T.T. Wu and C.N. Yang, Preprint ITP-SB 751331, New York (1975). [13] E.A. Feinberg, Usp. Fiz. Nauk. 78 (1962) 53.
Volume 142, number 4,5
11 December 1989
FREE-FIELD POTENTIAlS IN ELECTRODYNAMICS V.M. DUBOVIK and S.V. SHABANOV’ Joint Inst itutefor Nuclear Research, Laboratory of Theoretical Physics, P.O. Box 79, 101000 Moscow, USSR Received 12 April 1989; revised manuscript received 28 June 1989; accepted for publication 6 October 1989 Communicated by J.P. Vigier
A physical (gauge invariant) degree of freedom of the vector potential generating no electromagnetic fields is singled out. The problem of observing this degreeof freedom is discussed.
1. About 60 years ago Fock [1] and Weyl [2] have proposed a gauge method for describing interactions of quantum charged particles with an external electromagnetic field. It consists in the requirement for the Schrodinger equation to be invariant for a charged particle under local phase transformations of the wave function. For this purpose, the vector potential A should be introduced so that the equation (l) ihä,yi={(l/2m)[ —ihV— (e/c)A]2+e9,}~v, where e is the particle charge and q’ is the scalar ~ tential, does not change under gauge transformations ~v—exp(iew/~c)~v, A—~A+Vw,
~—~—
-L9~w.
(2)
C
This invariance was called the U (1) gauge symmetry. Further, this method of introducing an interaction was generalised by Yang and Mills to non-Abehan symmetry groups [3]. The experience of classical physics suggests that only the elecirornagnetic fields E= —VQ~—(1 /c)8~A and B=rot A act on charged particles. However, in contrast with the Lorentz equation for a classical particle in E and B fields, eq. (1) shows that a quanturn charged particle interacts with an electromagnetic field through the potentials A and This mitiated the discussion of the vector potential. IsA just ~.
Novosibirsk State University, Pirogova 2, 630090 Novosibirsk, USSR.
a convenient form describing interactions of partides with an electromagnetic field or is it a more fundamental physical concept than a field itself? In other words, is there any A which does not correspond to any electromagnetic field? A reason for a negative answer to this question is usually ambiguity in determing A (see (2)) since the observed quantities, like E and B, cannot depend on gauge arbitrariness. In the present paper we will show a vector potential to have such a physical (gauge invariant) degree of freedom which is locally associated with zero values of strengths. This degree of freedom, as any physical quantity, should manifest itself in nature. It turns out to act only on quantum systems; therefore, its macroscopic manifestation is possible only as a collective quantum effect, which is observed in superconductivity. 2. Gauge symmetry in electrodynamics leads to constraints on dynamic variables in the theory [4], i.e. between generahised coordinates and momenta of a dynamic system there are some relations contaming no derivatives with respect to time. Canonical coordinates in electrodynamics are the components of the vector potential A~(x) (n = 1, 2, 3) and ~(x) at each point of the space. Canonical momenta conjugate to them correspond to components of the electric field —E~(x)with an opposite sign; the momentum conjugate to ~(x) equals zero E 0(x) =0 (this is the primary constraint in electrodynamics). For the quantities A~,q’ and E~,E0 the Poisson brackets 211
Volume 142, number 4,5
PHYSICS LETrERS A
{A~(x),E,(x)}=—o,~o3(x_y), {ço(x), E 3(x—y) (3) 0(y)}= —~ are determined. The total set of constraints in electrodynamics is [4]
11 December 1989
substitute (7) into (6); them,into the integration in the operator A~ decays two parts Vregion and R3\V. The integral over R3\V transforms by the Gauss theorem and then using the equality 41 x— y I = 4itô3 (x—y) we get —‘
—
A ~ = VXPh, E 0(x)=0,
0nEn(x)=p(x),
(4)
where p(x) is the density of charges generating an electric field. Every observed quantity F~h[A, E, E0] depend-
x~~h[Al=o,,
~,
r v~(y)~(y) —on ~ 4Ex—yI dy,
ing on canonical variables should be a gauge invariant and thus satisfy the equations {F~h,E0}=0,
A”(y) dy J 4~~—yl
where x~8V,V and (5)
v~,is the
(8) external normal to the
as the constraints (4) are generators of gauge transformations [4]. It follows from the first equation of (5) and (4) that FPh can depend only on A and E. The general sohution of the second equation of (5) is F~h[A, E] =F[APh, E], where [5] A [Al =A~ —o n8k4 1Ak, 0 nA ~h 0. (6)
surface OV. Functional (8) is a gauge invariant (Av_~Av+Vw, x—~x+w);moreover, by definition XPh is a harmonic function (see (6)) in the region ~ \ V. In fact, XPh is the solution of the external Dirichlet problem for the Laplace equation in the multiply connected In the simplest cases when 3\Vregion. is doubly connected~”(8) has a the region P simple form
The operator A~is determined in a standard way,
XPh[A]
{FPh, OnEn_P}m{Fph, OnEn}0,
~h
—
4’f(x)=— Jd3x(4RIx_YIY’f(y),
~,
~‘~‘
xeV, rot Av=B,
A (x) = V~(x), x~V, B= 0.
(9)
where the contour C encircles once the region V, i.e., 0 is the magnetic field flux through the contour C, and the function f(x) depends on the geometry of the region V. Since XI~h[A] is a gauge invariant, one can pass from (8) to (9) in any appropriate gauge. For instance, for an infinite solenoid directed along the Oz axis f(x) = O/27L, where 0 is the angle of the cylindrical system of coordinates. Note that in spite ofthe “pure” gauge form of the vector potential (7) for x~V it cannot be reduced to zero in the P3 \ V gauge transformation since x is a multi-valued function. Gauge transformations with multivalued functions are inadmissible as they change the flux of the magnetic field in the region B: ~ (VX, dI) 0 is the gauge invariant (observed quantity) where C is the contour that cannot be con~‘
(7)
Moreover, at the boundary OV there is a condition of continuity A”=VX, xeOV, i.e., B=0, xeOV. We 212
(A, dI)=f(x)Ø, C
and integration is performed over the whole threedimensional space P3. It is to be noted that from the viewpoint of Hamihtonian dynamics A ~“ is the canonical variable describing two transverse physical degrees of freedom of the electromagnetic field (the conjugatemomenta are the components ofthe transverse part of the electric field [5]). This variable should naturally be introduced while quantizing electrodynamics as a system with constraints [5]. Let us find the value of the functional A 1” [A] at the space points where E=B=0. Since E= 0, we considerthe stationary problem. Then, from B= rot A = 0 it follows A =V~.If B=0 at all the points xeP then from (6) we have the trivial equality A [VX]= 0. A different situation arises when there is a region V where Bs~0.In this case we have A(x) =Av(x),
=f(x)
Multiconnectedness of the region R3\V follows form the equation div 8=0. If B~0only in V, then there always exists a contour that is not in V the magnetic flux through which is nonzero. For instance, this contour can be chosen on ÔV (8=0, xet9V see (7)) as the lines of the field B are always closed: divB=’O.
Volume 142, number 4,5
PHYSICS LETTERS A
11 December 1989
tracted at a point without intersecting the boundary OV. From this point of view the existence of XPh [A] seems to be quite natural. Thus, around the regions occupied by the magnetic field there is a real (gauge invariant) field. However, the definition “real” needs to be refined, namely, one should point out the way of its observing. This is just the purpose of the next section.
mations L—+L+dQ/dt are admissible including those with multiple-valued functions £2. Using the rules of canonical quantization one can formulate a quantum theory corresponding to the Lagrangian (11). The obtained SchrOdinger equation will contain only gauge-invariant quantities. The same result can be obtained from eq. (1) by introducing the gauge-invariant wave function
3. To solve the problem of physical manifestation of the field Xt~I~, let us consider the interaction of a charged particle with an external electromagnetic field. The Lagrangian is
~
—
—
(10)
component of an electromagnetic field are linear combinations of one physical and one nonphysical degree of freedom [5]: Using the Schrodinger equa-
Under gauge transformations (2) the Lagrangian (10) is added by a total derivative
tion in the gauge invariant variables we can show that the field XPh affects a charged particle. These are wellknown results [7,8]. However, it is to be noted that
L—~L+e d C dt
our consideration is by construction independent of the choice of gauge in contrast with refs. [7,8].
(d/dt= O,+ (A, V) is the material derivative). Therefore, the Lagrangian equations of motion are invariant under gauge transformations. To introduce XPh into L we must find its gauge-invariant form. For this purpose we add dQ/dt into the right-hand side of(10)whereQ=(e/c)8,,A’A~.Asaresult,wehave obtained the gauge-invariant Lagrangian
4. A superconductor is the simplest system in which one can observe the action of the field XPh on a quanturn system. Its free energy in the external electromagnetic field is
e h )—e4 L(x,x)=~mx——(x,A”
+(l/4m)~ft[—i*V+(2e/C)A]2W}, (13) where V is the volume occupied by a superconduc-
L (x, x) =
e
= exp [ i (e/hC)4 ‘O nAn] t~, (12) then substituting i,v into (1) and expressing ço from constraint (4). Formula (12) reflects the known fact that the phase of a charged field and the transverse
~
—
—
(x, A) + eq’.
C
.
p,
(11)
C
where we have used the equality E= —Vço— (1 /C)A and constraint (4). The Lagrangian (11) is independent of the choice of gauge. Note that the Lagrangian (11) differs from the gauge invariant Lagrangian, containing path-dependent integrals on A, in the Mandelstam formulation [6].Now, to introduce an interaction with ~Ph, it is sufficient to put p = 0 and A “ VX~.Then, the Lagrangian (11) differs from the free one only by the total derivative (e/c)d~~/dl.Hence, a classical particle is insensitive to XPh. This was to be expected as at E=B= 0 the Lorentz strength is absent. From the viewpoint of the Lagrangian formalism this means that the freedom in choosing the L.agrangian by given equations of motion is greater than gauge arbitrariness in —
electrodynamics: Thus, in the first case transfor-
2+~bIwI4
Fs=j dx{—aI~I v
and ~visthe complex parameter of the Landau— Ginzburg order or the wave function of the Cooper pair. Using transformation (12) (e—~2e) expression (13) can be rewritten in terms of gauge-invariant variables ~ and A ~ so that by varying them one can obtain the gauge-invariant Landau—Ginzburg equations. Our further analysis will be qualitative; therefore, in (13) we have omitted the magnetic field energy, i.e., disregarded the depth of penetration of the magnetic field into the superconductor. The basic state of the superconductor is specified by the minimum ofF,. IfA~’=0,then the minimum ofF, is achieved at ~u=y,~= const as the kinetic energy vanishes (this is an absolute minimum ofF,). Usually, ~v 0is normalised as Wo 12 = n,/2 where n, is the density of paired electrons. tor,
213
Volume 142, number 4,5
PHYSICS LETTERS A
Consider now a superconducting ring put on the sohenoid. If the ring temperature is T> T~,where T~ is the critical temperature, the current in the ring dies out. Let us stabilize the flux of the magnetic field for T> T~.Then, upon cooling the ring to T< T~it transforms into a superconducting state. After substitution ofA = VXPl~for an infinite long solenoid in eq. (13) we find that the only difference of F, in cornparison with the case XPh = 0 consists in the change of the rotation energy of the condensate L ~ (L~ h0/00 2, where L~= ihO/8O is the operator ~
-+
—
—
of the angular momentum projection onto the solenoid axis, 0 is the polar angle, and 0o = lthC/e is the magnetic flux quantum (fluxon). For the internal quantisation of the angular momentum m0 = [0/0o1, (14) where rn0 is chosen from the condition of minimal F,, integer, and [0/00] rounding of 0/00 nearest if themeans integral quantisation ofto thethe angular momentum is valid, then in the superconducting ring there arises a current proportional to the difference m 0Ø0 _~*2 Note that there is no current in the superconductor if the angular moment is quantised with an arbitrary number [101 which however disagrees with experiment (see ref. [11]and references therein), Based on the simplest idea of a short-range interaction, i.e., transfer of interaction by a field, one should point out what physical (i.e., independent of gauge) field forced the Cooper pairs to move to create a current. Just the field XPh plays the role of a transmitter of interaction. In this connection, we should like to recall the paper by Wu and Yang [12] in which they have introduced the “minimal” description of electromagnetic interactions in terms of a nonintegrable phase factor. Again, keeping the idea of a short range interaction and locality, one should point out what field is responsible for the appearance in the wave function of this phase factor. As above, Wph
=
Wo exp (iOm0),
~,
S2
1*3
The exact value of the current can be calculated by the formula I,=c(m0Ø,—0) IL, where L is the contour inductance that is exact for superconductors [9]. The authors of ref. [10] have discussed the behaviour of a superconductor in the field of a solenoid but they were not correct in their reasonings, by, for example, incorrect use of the Coulomb gauge and the use ofsingular gauge transformations ElI].
214
II December 1989
this is provided by the interaction
with XPh.
5. In conclusion we should like to remind ofan old but very physical paper [13]. Based on ref. [7]its author raised two questions. The first is the possibility ofobserving the effect on an electron in the region where E=B= 0 and the potential is nonzero. We give a positive answer to this question. The second question is: does there exist a specific action of the potential in quantum mechanics, differing from that in classical physics, which would enable one to treat the potential as a more fundamental quantity than the field intensity and would require reformulation of the basic assumptions of the theory? The author ofref. [13]gives a negative answer. The literature on this problem is quite voluminous. Our point of view is the following. In electrodynamics, the vector potential really has a physical degree of freedom that is not3\V associated (i.e., at this locally is important) each any pointelectromagnetic of the space Pfield. Thus, the vector powith tentiah is a fundamental object with respect to the field. The observed degree of freedom manifests it—
self only in quantum theory; it possesses the basic attribute of a physical field ability to act on a physical system. Moreover, as has been shown, the fundamentals of neither classical electrodynamics nor quantum theory need be reformulated. —
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