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Relativistic strings and electromagnetic flux tubes
Volume 82B, number 2
PHYSICS LETTERS
26 March 1979
RELATIVISTIC STRINGS AND ELECTROMAGNETIC FLUX TUBES H.A. KASTRUP Institut fiir Theoretische Physik, R WTH Aachen, 51 Aachen, FR Germany Received 17 January 1979
The physical significance of electromagnetic fields with rank 2 - i.e. for which E'B = 0 - and which are obtained from the relativistic string via Plficker's coordinates, is discussed, especially the relation of the resulting flux tubes to electric and magnetic confinement.
In a recent paper [1] it was shown, how the motion
x 0 =t=r, x 2 =A(o-
X 1 =A(o-½~r)coscor, ½7r)sincor,
x 3 = 0,
Aco =2/7r,
(1)
of the relativistic string can be mapped onto a rank-2 solution o f Maxwell's equations. Such solutions are characterized by the property that the invariant *FuvFUV = 4 E ' B , *Fuv = ½euvoaFOa, vanishes, or since det(Fuv ) = (E.B)2, that det(Fuv ) = 0, without Fur vanishing identically. Since the matrix Fur defines a closed 2-form (curvature form) F = dA = d(A u dxU), then, according to Darboux's theorem [2,3], there is a coordinate system in which it takes the form
(2)
if and only if it has rank 2! The functions a(~)(x), a = 1,2, may be interpreted as scalar potentials. The representation (2) has the following interesting interpretation: Consider a closed curve C(r), r -~ xU(z), 0 <~ "c <~ 1, in Minkowski space M 4. This curve induces a closed curve ~ ( r ) in the two-dimensional "a-space", with points (a (1), a(2)). The curve C(r) constitutes the boundary o f a two-dimensional area S (2) in a-space. Consider now in M4 the line integral fc(r) Au dxU. Because of eqs. (2) Stoke's theorem
gives
(3)
where V(S (2)) ~ 0 is the volume of the two-dimensional area S (2). The sign in eq. (3) depends on the induced orientation of C(7-). Thus, rank-2 electromagnetic fields fulfill a modified Wilson criterium [4] for confinement ! In ref. [ l ] I considered only the rigid motion (1) of the string: By using Pliicker's coordinates our = ~cUxv' - fcVx u', fcu =- ~rx u, xU' = a axU , it can be mapped onto the electromagnetic field
E(x, t) = XA(cos cot, sin cot, 0), B(x, t) = -X(0, 0, (2/Tr)p), /9 = +[(X1) 2 + (X2) 2] 1/2,
F = dA = da(1)(x) A da(2)(x), A = ( 1 + a (1)) da (2),
f A u dx u = + f da (1)/x da (2) = +V(S(2)), C(r) S(Z)
(4)
which can be derived from the four-potential [1] A0=0,
AI=
(2/37r)~ox2-X(A/co)sincot,
A 2 = (2/3zr)X/gx 1 + ?~(A/co)cos cot,
A 3 = O.
Actually, it can be shown [5] that any string solution can be mapped onto a rank-2 solution o f Maxwell's equations! Here, I would like to make some more comments on the physical interpretation of the solutions (4) and their dual counterpart ,1 : The fields (4) q-1 To consider the dual configuration of eqs. (4) was suggested to me by F. Englert. 237
Volume 82B, number 2
PHYSICS LETTERS
represent a magnetic flux tube in a three-direction, on which a periodic electric field is superposed. The time average (E) t of the electric field vanishes and we have E 2 - B 2 = X2(A 2 - (4/zr2)02)/> 0, for 0 ~< p ~< lzrA inside the tube, with the equality holding at the surface p = ½zrA. The equations curl E = -Ot B,
divE = 0,
divB = 0,
(5)
curlB = Ot E +], imply a continuous electric field and a continuous normal component B n, but the tangential component B t may have a discontinuity, due to surface currents, at any boundary. However, we may assume that our magnetic flux tube is imbedded in a (type II) superconductor and that B t goes to zero for p > i nA in a thin layer [1 ]. The magnetic flux inside the tube has the value 1 ~m = -g)k/r3A 3 = - 4 ~.co- 3 .
(6)
The flux tube leads to magnetic confinement [6,7] if we place magnetic charges g = + qbm at the two ends of a flux tube with length L. Notice that we get "infrared slavery" in our case, because g(co) = co- 3 which goes to infinity if co ~ 0 and to zero if co -+ oo (asymptotic freedom!). In addition, we have "flux quantization" in the following sense: Since rr2A 2 = 2 4 ( a ' ) J , where a ' is a constant and J the spin of the system, we have q~2 m= 4.(2/3)4X2(a')3J 3. Thus, the flux ~m becomes quantized if the spin J takes only values which are integer multiples of Planck's constant! In order to obtain electric confinement, we merely have to pass to the dual [ 8 - 1 0 ] of the fields (4): Fuu ~ *Fu~ means E - + - B , B-~ E, Pe ~ - P m ,Je -+ - ] m , yielding curlB = OtE, divE = 0, divB = 0, curlE = 3 t B +ira, i.e. we have no magnetic charge density - in the rest system of the tube - but a magnetic current. We now have an electric flux tube with E 2 B 2 < 0 inside and E 2 = B 2 at the boundary. The electric flux inside the tube has the value q5e = _ 4 Xco-3. We obtain charge confinement, if we place two charges q = -+ q5e at the ends of the tube with length L. Again we have infrared "slavery" as above. We see that the rank-2 solutions of Maxwell's equations obtained froln the string motion lead to a classical model for confinement in a straightforward way! 238
26 March 1979
The best known example for a rank-2 solution of Maxwell's equations is the radiation field. If we place a charge q in such a field, again interesting things happen: The corresponding classical equations of motion and the Dirac equation can be solved exactly [ 1 1 - 1 3 ] , with the following results: Classically the point charge moves along a figure eight or a circle, depending on the polarization of the electromagnetic wave, i.e. the charge gets trapped in the radiation field! In the case of a Dirac particle in an external radiation field the result may be interpreted [13] as a change of the electron mass squared by the amount e2(A u A~)t! However, this term looks very similar to a mass term of the electromagnetic fields and resembles very much the dynamical vector field mass term in two-dimensional QED [14,15]. This is no coincidence, because in the case of a radiation field the potentials Au = AU(t, n . x ) , n 2 = 1, depend only on two variables t and n . x , i.e. we have essentially a twodimensional problem! All this shows that rank-2 gauge fields are interesting candidates for solving the confinement problem in four dimensions! I am very much indebted to F. Englert for stimulating discussions concerning the confinement problem in classical gauge theories and I thank M. Rinke for discussions on the string problem. [1] H.A. Kastrup, Phys. Lett. 78B (1978) 433. [2] C1. Godbillon, G~om&rie diff~rentielle et m~chanique analytique (Hermann, Paris, 1969) pp. 118-120. [3} R. Abraham, Foundations of mechanics (Benjamin, New York, 1967) p. 95. [4] K.G. Wilson, Phys. Rev. D10 (1974) 2445. [5] M. Rinke, to be published. [6] Y. Nambu, Phys. Rev. D10 (1974) 4262. [7] F. Englert, Electric and magnetic confinement schemes, Lectures given at the Carg~se Summer School 1977; here, further references can be found. [8] S. Mandelstam, Phys. Rep. 23 (1976) 245. [9] Y. Nambu, Phys. Rep. 23 (1976) 250. [10] F. Englert and P. Windy, Nucl. Phys. B135 (1978) 529. [11] L.D. Landau and E.M. Lifshitz, The classical theory of fields, 3rd rev. ed. (Pergamon, Oxford, 1971) pp. 112114,118. [12] D.M. Wolkow, Z. Phys. 94 (1935) 250. [ 13] V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Relativistic quantum theory (Pergamon, Oxford, 1971) pp. 122-124. [14] J.H. Lowenstein and J.A. Swieca, Ann. Phys. (NY) 68 (1971) 172. [15] 1. Kogut and L. Susskind, Phys. Rev. Dll (1975) 3594.
PHYSICS LETTERS
26 March 1979
RELATIVISTIC STRINGS AND ELECTROMAGNETIC FLUX TUBES H.A. KASTRUP Institut fiir Theoretische Physik, R WTH Aachen, 51 Aachen, FR Germany Received 17 January 1979
The physical significance of electromagnetic fields with rank 2 - i.e. for which E'B = 0 - and which are obtained from the relativistic string via Plficker's coordinates, is discussed, especially the relation of the resulting flux tubes to electric and magnetic confinement.
In a recent paper [1] it was shown, how the motion
x 0 =t=r, x 2 =A(o-
X 1 =A(o-½~r)coscor, ½7r)sincor,
x 3 = 0,
Aco =2/7r,
(1)
of the relativistic string can be mapped onto a rank-2 solution o f Maxwell's equations. Such solutions are characterized by the property that the invariant *FuvFUV = 4 E ' B , *Fuv = ½euvoaFOa, vanishes, or since det(Fuv ) = (E.B)2, that det(Fuv ) = 0, without Fur vanishing identically. Since the matrix Fur defines a closed 2-form (curvature form) F = dA = d(A u dxU), then, according to Darboux's theorem [2,3], there is a coordinate system in which it takes the form
(2)
if and only if it has rank 2! The functions a(~)(x), a = 1,2, may be interpreted as scalar potentials. The representation (2) has the following interesting interpretation: Consider a closed curve C(r), r -~ xU(z), 0 <~ "c <~ 1, in Minkowski space M 4. This curve induces a closed curve ~ ( r ) in the two-dimensional "a-space", with points (a (1), a(2)). The curve C(r) constitutes the boundary o f a two-dimensional area S (2) in a-space. Consider now in M4 the line integral fc(r) Au dxU. Because of eqs. (2) Stoke's theorem
gives
(3)
where V(S (2)) ~ 0 is the volume of the two-dimensional area S (2). The sign in eq. (3) depends on the induced orientation of C(7-). Thus, rank-2 electromagnetic fields fulfill a modified Wilson criterium [4] for confinement ! In ref. [ l ] I considered only the rigid motion (1) of the string: By using Pliicker's coordinates our = ~cUxv' - fcVx u', fcu =- ~rx u, xU' = a axU , it can be mapped onto the electromagnetic field
E(x, t) = XA(cos cot, sin cot, 0), B(x, t) = -X(0, 0, (2/Tr)p), /9 = +[(X1) 2 + (X2) 2] 1/2,
F = dA = da(1)(x) A da(2)(x), A = ( 1 + a (1)) da (2),
f A u dx u = + f da (1)/x da (2) = +V(S(2)), C(r) S(Z)
(4)
which can be derived from the four-potential [1] A0=0,
AI=
(2/37r)~ox2-X(A/co)sincot,
A 2 = (2/3zr)X/gx 1 + ?~(A/co)cos cot,
A 3 = O.
Actually, it can be shown [5] that any string solution can be mapped onto a rank-2 solution o f Maxwell's equations! Here, I would like to make some more comments on the physical interpretation of the solutions (4) and their dual counterpart ,1 : The fields (4) q-1 To consider the dual configuration of eqs. (4) was suggested to me by F. Englert. 237
Volume 82B, number 2
PHYSICS LETTERS
represent a magnetic flux tube in a three-direction, on which a periodic electric field is superposed. The time average (E) t of the electric field vanishes and we have E 2 - B 2 = X2(A 2 - (4/zr2)02)/> 0, for 0 ~< p ~< lzrA inside the tube, with the equality holding at the surface p = ½zrA. The equations curl E = -Ot B,
divE = 0,
divB = 0,
(5)
curlB = Ot E +], imply a continuous electric field and a continuous normal component B n, but the tangential component B t may have a discontinuity, due to surface currents, at any boundary. However, we may assume that our magnetic flux tube is imbedded in a (type II) superconductor and that B t goes to zero for p > i nA in a thin layer [1 ]. The magnetic flux inside the tube has the value 1 ~m = -g)k/r3A 3 = - 4 ~.co- 3 .
(6)
The flux tube leads to magnetic confinement [6,7] if we place magnetic charges g = + qbm at the two ends of a flux tube with length L. Notice that we get "infrared slavery" in our case, because g(co) = co- 3 which goes to infinity if co ~ 0 and to zero if co -+ oo (asymptotic freedom!). In addition, we have "flux quantization" in the following sense: Since rr2A 2 = 2 4 ( a ' ) J , where a ' is a constant and J the spin of the system, we have q~2 m= 4.(2/3)4X2(a')3J 3. Thus, the flux ~m becomes quantized if the spin J takes only values which are integer multiples of Planck's constant! In order to obtain electric confinement, we merely have to pass to the dual [ 8 - 1 0 ] of the fields (4): Fuu ~ *Fu~ means E - + - B , B-~ E, Pe ~ - P m ,Je -+ - ] m , yielding curlB = OtE, divE = 0, divB = 0, curlE = 3 t B +ira, i.e. we have no magnetic charge density - in the rest system of the tube - but a magnetic current. We now have an electric flux tube with E 2 B 2 < 0 inside and E 2 = B 2 at the boundary. The electric flux inside the tube has the value q5e = _ 4 Xco-3. We obtain charge confinement, if we place two charges q = -+ q5e at the ends of the tube with length L. Again we have infrared "slavery" as above. We see that the rank-2 solutions of Maxwell's equations obtained froln the string motion lead to a classical model for confinement in a straightforward way! 238
26 March 1979
The best known example for a rank-2 solution of Maxwell's equations is the radiation field. If we place a charge q in such a field, again interesting things happen: The corresponding classical equations of motion and the Dirac equation can be solved exactly [ 1 1 - 1 3 ] , with the following results: Classically the point charge moves along a figure eight or a circle, depending on the polarization of the electromagnetic wave, i.e. the charge gets trapped in the radiation field! In the case of a Dirac particle in an external radiation field the result may be interpreted [13] as a change of the electron mass squared by the amount e2(A u A~)t! However, this term looks very similar to a mass term of the electromagnetic fields and resembles very much the dynamical vector field mass term in two-dimensional QED [14,15]. This is no coincidence, because in the case of a radiation field the potentials Au = AU(t, n . x ) , n 2 = 1, depend only on two variables t and n . x , i.e. we have essentially a twodimensional problem! All this shows that rank-2 gauge fields are interesting candidates for solving the confinement problem in four dimensions! I am very much indebted to F. Englert for stimulating discussions concerning the confinement problem in classical gauge theories and I thank M. Rinke for discussions on the string problem. [1] H.A. Kastrup, Phys. Lett. 78B (1978) 433. [2] C1. Godbillon, G~om&rie diff~rentielle et m~chanique analytique (Hermann, Paris, 1969) pp. 118-120. [3} R. Abraham, Foundations of mechanics (Benjamin, New York, 1967) p. 95. [4] K.G. Wilson, Phys. Rev. D10 (1974) 2445. [5] M. Rinke, to be published. [6] Y. Nambu, Phys. Rev. D10 (1974) 4262. [7] F. Englert, Electric and magnetic confinement schemes, Lectures given at the Carg~se Summer School 1977; here, further references can be found. [8] S. Mandelstam, Phys. Rep. 23 (1976) 245. [9] Y. Nambu, Phys. Rep. 23 (1976) 250. [10] F. Englert and P. Windy, Nucl. Phys. B135 (1978) 529. [11] L.D. Landau and E.M. Lifshitz, The classical theory of fields, 3rd rev. ed. (Pergamon, Oxford, 1971) pp. 112114,118. [12] D.M. Wolkow, Z. Phys. 94 (1935) 250. [ 13] V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Relativistic quantum theory (Pergamon, Oxford, 1971) pp. 122-124. [14] J.H. Lowenstein and J.A. Swieca, Ann. Phys. (NY) 68 (1971) 172. [15] 1. Kogut and L. Susskind, Phys. Rev. Dll (1975) 3594.