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To a gauge invariant model of superconductors
Physics Letters A 171 (1992) 220-222 North-Holland
PHYSICS LETTERS A
To a gauge invariant model of superconductors I.E. Bulyzhenkov Institute of Spectroscopy, Russian Academy of Sciences, Troitsk~ 142092 Moscow Region, Russian Federation Received 15 August 1992; accepted for publication 11 September 1992 Communicated by V.M. Agranovich
A scalar self-interacting charged field for the Lagrangian invariant under local gauge transformations is used for the relativistic description of superconductivity. The unique boundary of a charge density and a relativistic chemical potential of superconducting carriers makes it possible to consider the scalar field with a real mass which differs from the usual Goldstone-I-Iil~ model. The magnetic flux quantization in three-dimensional superconducting loops is independent of stationary gravity or inertia fields while superconducting phases are coherent in the four-dimensional space.
The self-interacting charged field with an imaginary mass could give rise to a lot of analogies with superconductors due to the Goldstone-Higgs mechanism of the spontaneous symmetry-breaking because this approach leads to a nonzero vacuum-expectation field value [ 1,2 ]. But in reality, superconducting particles are formed from paired normal carriers with a real mass m which enforce the two-fluid concept of superconductivity. This way it would be natural to start with a Lagrangian density .Y for the scalar charged field ~= r(2M)-~/2exp (ix/he) with a real mass M = 2m and consequently a monotone self-interacting potential
~ = ( _g)l/2[ (Du~0)+ (DU~0) _M2c2~+ ~ _ 2pM2(~+ ~)2_ ( 1/16~r)F~F ~ ] ,
( 1)
where Du = hVu+ iq(Au/c) ,
q= 2e,
Fu, = VuA~ - V~A u .
Notice that the Lagrangian density ( 1 ) is invariant for the local gauge transformations
~exp(-iO/hc)~,
~+--.exp(iO/hc)~ +,
1
Au--,A u - qVuO.
The relativistic chemical potential/h of superconducting particles is fixed by the one, P-n, of normal carriers in the equilibrium states of two fluids,/A = 2/A~.It strictly determines the superconducting charge density p, = qr21A/ Mc 2 [ 3 ] and leads to a minimum of the effective self-interacting potential of the scalar field in ( 1 ) at a nonzero value r. Varying the action with the density ( 1 ) by the parameter r and the electromagnetic field A u one obtains the main equations h2
2M VuVUr+
½Mc2r (1 - (V~x+ qA~(~Ux + qAU) ) + ,r3 =O ,
1 V~'Fu"=-g uP~q (V~z+qA~)r 2 gU~'J'~'=-4-~
(2)
(3)
The covariant differentiation of the superconducting four-current J,,-qno,u,,,c from eq. (3) leads to the relativistic Bernoulli equation 7uPp--VvP u for superconducting carders in external fields, where P u = - V u x / c 220
0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 171,number3,4
PHYSICSLETTERSA
7 December1992
= (p.o,U=u+qA~,)/c is the generalized four-momentum of the superconducting component and the relativistic chemical potential/~ in the proper frame is proportional to the proper superconducting density Po, [ 3 ]. If the reference frame metric is plane (g0o= 1; g # = - 1; g~,~=0, #~z,;/z=0, 1, 2, 3; i=1, 2, 3), one could easily represent eq. (2) through components of the superconducting four-current J=~= ( p c , - j ~ ) , h2~"
Mj2 2----M- d- 2 ~
~[/4~2-]
~2~2 r
2Mc 2
d- ½Mc 21_ - ~-'~c2)
(4)
r + flr3 =O ,
where
qr2 L=p,v,= Mcc(Vz-qA),
(5)
~" - 2 / c - q O = M c 2 P---~-~ /4= (1 - v 2 / c 2) a/2 qr 2"
(6)
The value of the relativistic chemical potential/~ of normal carders in the laboratory frame is practically independent of a movement of a small superfluid part of charged carders in the vicinity of the superconducting transition temperature. In this case the value 2/~ can be considered as a constant as well as the equal value /4. Thus the Ginzburg-Landau coefficient a for the linear terra with respect to r in eq. (4) takes the form Ol--
(Me2) 2 - (2/~)2 2Mc2
(7)
This result corresponds to a conclusion that superconductivity (or < 0) could be realized if the paired carder's rest energy M c 2 is less than the relativistic energy 2/~ of a couple of normal carders. This seems natural and leads in the nonrelativistic limit (c-~oo) to the exact Ginzburg-Landan equation with a = - ( 2 ~ - M c ' ) . The gravity and inertia phenomena in superconductors can be considered with the assistance ofeqs. (2) and (3) when both the velocity three-vector V~ =dx~/ds and the impulse three-vector P~ depend on the gravity three-vector g,.= - goi/ goo while ds = ~ ( d x ° - g~ d x i)/c. For example, if world lines of two superconducting carders coincide at one point A= (x °, XA) and then intersect again at another point B= (x °, xB), the coherence of the phases )fi/c~ and Z2/ch will be conserved at any macroscopic path of every carder in the four-dimensional space. It leads to the following rule of the path integrals for generalized impulses Pu= - V u z / c in both electromagnetic and gravity fields, B
B
f Pu(1) dxU- f .~(2) d x U - 2 ~ h . = 0 , A
(8)
A
where each four-vector Pu takes the following components, c x / l _ V ~ i V ~ / c 2 + - - c, -
d x / 1 - 1,',, v ~ / d
-
.
(9)
Equation (8) describes also the well-known magnetic flux quantization in superconducting loops when carders with closed paths in three-dimensional space are compared with motionless carders,
We consider stationary fields when V.)'o=0 and the superconducting electrochemical potential cPo is the same constant for all supercarder paths. The value of the chemical potential/4 could also be considered as a constant for the usual equilibrium superconductors without scalar electric fields ( ~ - 0 ) , 221
Volume 171, number 3,4
/Zo,~ vvv
=
PHYSICS LETTERS A
-- const.
7 December 1992
( 11 )
By using this relation in eq. ( I 0) we find
The second equality in eqs. (12) follows from the clock synchronization rule along a closed three=dimensional path [4]. The net result is that the unique quantization of a magnetic flux in superconducting loops when using fourdimensional paths is independent of stationary gravity or inertia fields, and it is decribed by the usual threedimensional rule
~
qA,
c
clx~=2 ~ n ,
(13)
while V.~=0 in the bulk of superconductors. This makes it useless to apply the magnetic flux quantization to measurements of stationary gravity fields or a reference frame rotation rate [ 5 ]. For nonstationary gravity or inertia fields the problem should be calculated by simultaneous use of eq. (10) and the relativistic Bernoulli equation V,.P0+ VoP~=0.
References [ 1] [2] [3] [4] [5]
222
D.A. Kirzhnits, Usp. Fiz. Nauk 125 (1978) 169. B.J. Harrington and N.IC Shepard, Nucl. Phys. B 105 (1976) 527. I.E. Bulyzhenkov, Phys. Lett. A 158 (1991) 483. L.D. Landau and E.M. Lifshits, Theory of fields (Nauka, Moscow, 1973) ch. 10. G. Papini, Phys. Lett. A 24 (1967) 32.
PHYSICS LETTERS A
To a gauge invariant model of superconductors I.E. Bulyzhenkov Institute of Spectroscopy, Russian Academy of Sciences, Troitsk~ 142092 Moscow Region, Russian Federation Received 15 August 1992; accepted for publication 11 September 1992 Communicated by V.M. Agranovich
A scalar self-interacting charged field for the Lagrangian invariant under local gauge transformations is used for the relativistic description of superconductivity. The unique boundary of a charge density and a relativistic chemical potential of superconducting carriers makes it possible to consider the scalar field with a real mass which differs from the usual Goldstone-I-Iil~ model. The magnetic flux quantization in three-dimensional superconducting loops is independent of stationary gravity or inertia fields while superconducting phases are coherent in the four-dimensional space.
The self-interacting charged field with an imaginary mass could give rise to a lot of analogies with superconductors due to the Goldstone-Higgs mechanism of the spontaneous symmetry-breaking because this approach leads to a nonzero vacuum-expectation field value [ 1,2 ]. But in reality, superconducting particles are formed from paired normal carriers with a real mass m which enforce the two-fluid concept of superconductivity. This way it would be natural to start with a Lagrangian density .Y for the scalar charged field ~= r(2M)-~/2exp (ix/he) with a real mass M = 2m and consequently a monotone self-interacting potential
~ = ( _g)l/2[ (Du~0)+ (DU~0) _M2c2~+ ~ _ 2pM2(~+ ~)2_ ( 1/16~r)F~F ~ ] ,
( 1)
where Du = hVu+ iq(Au/c) ,
q= 2e,
Fu, = VuA~ - V~A u .
Notice that the Lagrangian density ( 1 ) is invariant for the local gauge transformations
~exp(-iO/hc)~,
~+--.exp(iO/hc)~ +,
1
Au--,A u - qVuO.
The relativistic chemical potential/h of superconducting particles is fixed by the one, P-n, of normal carriers in the equilibrium states of two fluids,/A = 2/A~.It strictly determines the superconducting charge density p, = qr21A/ Mc 2 [ 3 ] and leads to a minimum of the effective self-interacting potential of the scalar field in ( 1 ) at a nonzero value r. Varying the action with the density ( 1 ) by the parameter r and the electromagnetic field A u one obtains the main equations h2
2M VuVUr+
½Mc2r (1 - (V~x+ qA~(~Ux + qAU) ) + ,r3 =O ,
1 V~'Fu"=-g uP~q (V~z+qA~)r 2 gU~'J'~'=-4-~
(2)
(3)
The covariant differentiation of the superconducting four-current J,,-qno,u,,,c from eq. (3) leads to the relativistic Bernoulli equation 7uPp--VvP u for superconducting carders in external fields, where P u = - V u x / c 220
0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 171,number3,4
PHYSICSLETTERSA
7 December1992
= (p.o,U=u+qA~,)/c is the generalized four-momentum of the superconducting component and the relativistic chemical potential/~ in the proper frame is proportional to the proper superconducting density Po, [ 3 ]. If the reference frame metric is plane (g0o= 1; g # = - 1; g~,~=0, #~z,;/z=0, 1, 2, 3; i=1, 2, 3), one could easily represent eq. (2) through components of the superconducting four-current J=~= ( p c , - j ~ ) , h2~"
Mj2 2----M- d- 2 ~
~[/4~2-]
~2~2 r
2Mc 2
d- ½Mc 21_ - ~-'~c2)
(4)
r + flr3 =O ,
where
qr2 L=p,v,= Mcc(Vz-qA),
(5)
~" - 2 / c - q O = M c 2 P---~-~ /4= (1 - v 2 / c 2) a/2 qr 2"
(6)
The value of the relativistic chemical potential/~ of normal carders in the laboratory frame is practically independent of a movement of a small superfluid part of charged carders in the vicinity of the superconducting transition temperature. In this case the value 2/~ can be considered as a constant as well as the equal value /4. Thus the Ginzburg-Landau coefficient a for the linear terra with respect to r in eq. (4) takes the form Ol--
(Me2) 2 - (2/~)2 2Mc2
(7)
This result corresponds to a conclusion that superconductivity (or < 0) could be realized if the paired carder's rest energy M c 2 is less than the relativistic energy 2/~ of a couple of normal carders. This seems natural and leads in the nonrelativistic limit (c-~oo) to the exact Ginzburg-Landan equation with a = - ( 2 ~ - M c ' ) . The gravity and inertia phenomena in superconductors can be considered with the assistance ofeqs. (2) and (3) when both the velocity three-vector V~ =dx~/ds and the impulse three-vector P~ depend on the gravity three-vector g,.= - goi/ goo while ds = ~ ( d x ° - g~ d x i)/c. For example, if world lines of two superconducting carders coincide at one point A= (x °, XA) and then intersect again at another point B= (x °, xB), the coherence of the phases )fi/c~ and Z2/ch will be conserved at any macroscopic path of every carder in the four-dimensional space. It leads to the following rule of the path integrals for generalized impulses Pu= - V u z / c in both electromagnetic and gravity fields, B
B
f Pu(1) dxU- f .~(2) d x U - 2 ~ h . = 0 , A
(8)
A
where each four-vector Pu takes the following components, c x / l _ V ~ i V ~ / c 2 + - - c, -
d x / 1 - 1,',, v ~ / d
-
.
(9)
Equation (8) describes also the well-known magnetic flux quantization in superconducting loops when carders with closed paths in three-dimensional space are compared with motionless carders,
We consider stationary fields when V.)'o=0 and the superconducting electrochemical potential cPo is the same constant for all supercarder paths. The value of the chemical potential/4 could also be considered as a constant for the usual equilibrium superconductors without scalar electric fields ( ~ - 0 ) , 221
Volume 171, number 3,4
/Zo,~ vvv
=
PHYSICS LETTERS A
-- const.
7 December 1992
( 11 )
By using this relation in eq. ( I 0) we find
The second equality in eqs. (12) follows from the clock synchronization rule along a closed three=dimensional path [4]. The net result is that the unique quantization of a magnetic flux in superconducting loops when using fourdimensional paths is independent of stationary gravity or inertia fields, and it is decribed by the usual threedimensional rule
~
qA,
c
clx~=2 ~ n ,
(13)
while V.~=0 in the bulk of superconductors. This makes it useless to apply the magnetic flux quantization to measurements of stationary gravity fields or a reference frame rotation rate [ 5 ]. For nonstationary gravity or inertia fields the problem should be calculated by simultaneous use of eq. (10) and the relativistic Bernoulli equation V,.P0+ VoP~=0.
References [ 1] [2] [3] [4] [5]
222
D.A. Kirzhnits, Usp. Fiz. Nauk 125 (1978) 169. B.J. Harrington and N.IC Shepard, Nucl. Phys. B 105 (1976) 527. I.E. Bulyzhenkov, Phys. Lett. A 158 (1991) 483. L.D. Landau and E.M. Lifshits, Theory of fields (Nauka, Moscow, 1973) ch. 10. G. Papini, Phys. Lett. A 24 (1967) 32.