- Email: [email protected]
Electromagnetic duality for anyons
1 December
1997
PHYSICS
ElSEVlER
Physics Letters A 236
LETTERS
A
( 1997) 5-7
Electromagnetic duality for anyons A. Maghakian I, A. Sissakian, V. Ter-Antonyan2 Bogoliubov
Laboratov
of
Dubna.
Moscow
Received 7 July 1997; revised manuscript
Theoretical
Physics, Joint Institute
Region, RU-141980,
for
Nuclear
Research,
Russian Federation
received 17 August 1997; accepted for publication 19 September 1997 Communicated by P.R. Holland
Abstract Electromagnetic duality was established for anyons: the 3D oscillator with coordinates confined by the 2D half-up cone for an angle of 7r/6 is dual to the 2D charge-dyon bound system obeying fractional statistics. @ 1997 Published by Elsevier Science B.V. PACS: 02.40.+m; Keywords:
Duality;
03.65.Ge; Anyons;
12.4O.Lk; 14.80.H~ Dyons
In modern field theory a central role is attributed to the idea of electromagnetic duality [ 11, according to which a strongly coupled theory can be formulated in terms of weakly coupled magnetic monopoles. Electromagnetic duality has a beautiful analog in non-relativistic quantum mechanics [ 2-41: oscillators in R*, @ and R2 can be transformed into the chargedyon bound systems in RS, R3 and R2, respectively. These magic dimensions follow from the Hurwitz theorem [ 51. In particular, the reduction of the circular oscillator by the Z2 group action results in two systems, a two-dimensional hydrogen atom and a twodimensional spin-l /2 charge-dyon bound system [ 41. In two space dimensions, however, we not only have bosons and fermions but also anyons, i.e. objects with fractional statistics [ 61. These possibilities arise from the peculiar topological properties of the configuration space of the collection of two-dimensional iden-
[7]. Hence, the question arises as to whether it is possible to construct an oscillator model which can be transformed into a charge-dyon bound system with fractional statistics, i.e. a system continuously interpolating between the bosonic and fermionic systems considered in Ref. [ 41. The answer we have found in this paper is positive. We show that the role of the desired oscillator model is played by the Cc oscillator, i.e. the 3D oscillator with coordinates confined by the 2D half-up cone C$ for an angle of n-16. 3 Thus, we have an important extension of electromagnetic duality for anyons. At the same time, our result opens up a wide range of applications for oscillator physics because anyons are important for massive gauge fields and soluble quantum gravity [ IO], the fractional quantum Hall effect [ 111 and high-ir, superconductivity [ 121.
’ Present address: The Rockefeller University, Box 238, New York, NY 10021, USA. ? E-mail: [email protected].
3The analogy between anyon (z Aharonov-Bohm) problems gravity) has and free motion on a cone (Z 2 + I-dimensional been previously noted in Refs. [ 8,9].
03759601/97/$17.00 @ 1997 Published PII SO375-9601(97)00758-5
1230 York Avenue.
tical particles
by Elsevier Science B.V. All rights reserved
6
A. Maghakian
et al/Physics
Let ufi be the Cartesian coordinates in R3 and Cc c R3. By definition, u!, E Cz if 3( uf +uz) - ui = 0, and us > 0. It is convenient to describe the cone Cl by the parametrization ut + iuf = ue2’4/2, us = &u/2, where 4 E [ 0, ~1. It should be emphasized that u E (0, oo), i.e. the tip of the cone is not included. As a consequence, the space C; is infinitely connected, i.e. loops with different winding numbers are homotopically inequivalent. After each winding, the wavefunction !P acquires the phase eiT” with v E [ 0, 11. Thus, the function P” = e-‘“&?P satisfies the usual condition ?P((u,& + rr) = e’““P(u,qS). Since d12 = du: + duz + du: = du2 + u2d42, we have
Letters A 236 (1997) 5-7
MJ E=-21i2(nl+Im+sl+~)2’
It follows from (4) that for s = 0 and s = l/2, E describes the (2n, + 2(ml + l)- and (2n, + 21rnl + l)fold energy levels of the two-dimensional hydrogen atom and spin-l /2 charge-dyon system, respectively. For s E (0,1/2), the energy levels (4) are decomposedintosublevelswith(n,+(ml+l)and(n,+Im() degrees of degeneracy 4 . It follows from r = u2 and the accepted parametrization that Cc =+ R2(x) can be interpreted as the mapping of the half-up cone Cc on the paraboloid R( r = u2) with the following projection R + R2(x). This composite transformation can be written as XI + ix2 = 4(uf + ui) ‘/2(ut + iu2).
(1) The change of variables r = u2, p = 24 transforms the Cl oscillator into the system described by the equation 1
c p + eA CL
-i/i-&
2M
(2)
Here XI + ix2 = re’+‘, Aj = -(g/2r)EjkXk/r2, h/e, h = fi/27r, P( r, cp) = P’(u, 4) and e=--
Mm2 8
’
E cr=--, 4
s = v/2 E [O, l/2].
g =
(3)
Since B = V A A = g8c2) (x), Eq. (2) describes the motion of a particle of mass M and electric charge e = -e/2 in the fields of the dyon of electric charge Q = +a/2 and magnetic charge g, localized at the origin. It follows from (2) that s = v/2 has the meaning of the spin of the dyon. Since s E [ 0,1/2], the Cc oscillator is dual to the fractional charge-dyon bound system which continuously interpolates between the bosonic (s = 0) and fermionic (s = l/2) systems considered in Ref. [6]. E$. (2) leads to the bounded solution
x’F(-n,,2]j/
+ I,z)eimP/J2?T
with z = 2Ar/r0, ro = fi2/pcq h = ( -2efi2/~ury2) ‘i2, n,=0,1,2 ,..., m E Z, j = m+s, and the eigenvalues
(4)
(5)
Eq. (5) together with the ansatz (3) forms the duality transformation mapping the CT oscillator onto the fractional charge-dyon system. Let us stress what we mean by the term duality. Consider the system S with energy E and coupling constant w. The system Sd is called dual to S if ,Y?is governed by the same dynamical equation as S but with fixed E and quantized w. The transformation Td which maps S onto the system sd with energy E and coupling constant (Y, such that E = E(O) and (Y = a(E), is called the transformation of duality. In our case, S, sd and Td mean the Cz oscillator, fractional charge-dyon bound system and paraboloidal projection, respectively. The electromagnetic duality for anyons is conserved even after the arbitrary scalar potential V( u2) is excluded. The dual partner of the C; oscillator modified in this way is the fractional charge-dyon system placed in the potential V( r)/4r. Finally we explain why we considered the half-up cone for an angle of 0 = 7r/6 only, but not for arbitrary 8. The answer is that only for 8 = 7r/6 the paraboloidal projection transforms C$ into the Euclidean plane. For other values of 8, after such a projection, one obtains a plane with non-Euclidean metric d12 = dr2 + 4r2 sin2 Bdp2. It is to be emphasized that for 8 = 7r/6 Cc has the topology of the manifold describing the relative coordinate of two identical parti-
4 The dynamical symmetry of the magnetic by Jackiw in his beautiful article I 13 1.
vortex was established
A. Mqhakian et al./Physics Letters A 236 (1997) S-7
cles placed in the Euclidean plane [ 71, and therefore, the CT oscillator is also a prototype of the anyon. We would like to thank Yeranuhy Hakobyan, Levon Mardoyan, Armen Nersessian and George Pogosyan for discussions. One of us (V.T.-A.) thanks Simon Ter-Antonian for hospitality at the University of Minnesota and acquaintance with the subject of anyons.
A 10 (1995)
] [4 I [3
A. Nersessian,
[7 ]
Introduction 1960) p. 52.
1 I 1 N. Seiberg, E. Witten, V.
Ter-Antonyan.
Nucl. Phys. B 431 ( 1994) A.
Nersessian.
Mod.
484.
Phys.
191 P Gerber% R. Jackiw, Comm. R. Jackiw,
( 1982)
Math.
Phys. Rev. Lett. 52
Ann.
Mod.
Phys.
E. Witten.
(McGraw-Hill,
957. 237 B
Phys. II8
Math.
Nucl. Phys. B (Proc.
Y.-H. Chen. F. Wilczek.
Analysis
Nuovo Cimento
181 S. Deser. R. Jackiw. Comm.
1131 R. Jackiw, Lett.
to Matrix
Phys. Rev. Len. 49
J.M. Leinaas. J. Myrheim.
B. Halperin,
M. Tsulaja,
160.5.
New York,
Phys. B 3 (1989)
[ 1I
J. Geom. and Phys. ( 1996) 250.
V.M. Ter-Antonyan,
R. Bellman.
161 F. Wilczek.
[ IO] II I I 1121
References
2633.
T. Iwai. T. Sunako, Lett. A 11 (1996)
[5 )
I
Phys. 124
Suppl.)
18 A
( 1989)
1.583.
B. Halperin.
1001. Phys. (NY)
201
( 1990)
83.
( 1977) I.
( 1988)
495.
( 1989) 229. ( 1991 ) 107. Int. J. Mod.
1997
PHYSICS
ElSEVlER
Physics Letters A 236
LETTERS
A
( 1997) 5-7
Electromagnetic duality for anyons A. Maghakian I, A. Sissakian, V. Ter-Antonyan2 Bogoliubov
Laboratov
of
Dubna.
Moscow
Received 7 July 1997; revised manuscript
Theoretical
Physics, Joint Institute
Region, RU-141980,
for
Nuclear
Research,
Russian Federation
received 17 August 1997; accepted for publication 19 September 1997 Communicated by P.R. Holland
Abstract Electromagnetic duality was established for anyons: the 3D oscillator with coordinates confined by the 2D half-up cone for an angle of 7r/6 is dual to the 2D charge-dyon bound system obeying fractional statistics. @ 1997 Published by Elsevier Science B.V. PACS: 02.40.+m; Keywords:
Duality;
03.65.Ge; Anyons;
12.4O.Lk; 14.80.H~ Dyons
In modern field theory a central role is attributed to the idea of electromagnetic duality [ 11, according to which a strongly coupled theory can be formulated in terms of weakly coupled magnetic monopoles. Electromagnetic duality has a beautiful analog in non-relativistic quantum mechanics [ 2-41: oscillators in R*, @ and R2 can be transformed into the chargedyon bound systems in RS, R3 and R2, respectively. These magic dimensions follow from the Hurwitz theorem [ 51. In particular, the reduction of the circular oscillator by the Z2 group action results in two systems, a two-dimensional hydrogen atom and a twodimensional spin-l /2 charge-dyon bound system [ 41. In two space dimensions, however, we not only have bosons and fermions but also anyons, i.e. objects with fractional statistics [ 61. These possibilities arise from the peculiar topological properties of the configuration space of the collection of two-dimensional iden-
[7]. Hence, the question arises as to whether it is possible to construct an oscillator model which can be transformed into a charge-dyon bound system with fractional statistics, i.e. a system continuously interpolating between the bosonic and fermionic systems considered in Ref. [ 41. The answer we have found in this paper is positive. We show that the role of the desired oscillator model is played by the Cc oscillator, i.e. the 3D oscillator with coordinates confined by the 2D half-up cone C$ for an angle of n-16. 3 Thus, we have an important extension of electromagnetic duality for anyons. At the same time, our result opens up a wide range of applications for oscillator physics because anyons are important for massive gauge fields and soluble quantum gravity [ IO], the fractional quantum Hall effect [ 111 and high-ir, superconductivity [ 121.
’ Present address: The Rockefeller University, Box 238, New York, NY 10021, USA. ? E-mail: [email protected].
3The analogy between anyon (z Aharonov-Bohm) problems gravity) has and free motion on a cone (Z 2 + I-dimensional been previously noted in Refs. [ 8,9].
03759601/97/$17.00 @ 1997 Published PII SO375-9601(97)00758-5
1230 York Avenue.
tical particles
by Elsevier Science B.V. All rights reserved
6
A. Maghakian
et al/Physics
Let ufi be the Cartesian coordinates in R3 and Cc c R3. By definition, u!, E Cz if 3( uf +uz) - ui = 0, and us > 0. It is convenient to describe the cone Cl by the parametrization ut + iuf = ue2’4/2, us = &u/2, where 4 E [ 0, ~1. It should be emphasized that u E (0, oo), i.e. the tip of the cone is not included. As a consequence, the space C; is infinitely connected, i.e. loops with different winding numbers are homotopically inequivalent. After each winding, the wavefunction !P acquires the phase eiT” with v E [ 0, 11. Thus, the function P” = e-‘“&?P satisfies the usual condition ?P((u,& + rr) = e’““P(u,qS). Since d12 = du: + duz + du: = du2 + u2d42, we have
Letters A 236 (1997) 5-7
MJ E=-21i2(nl+Im+sl+~)2’
It follows from (4) that for s = 0 and s = l/2, E describes the (2n, + 2(ml + l)- and (2n, + 21rnl + l)fold energy levels of the two-dimensional hydrogen atom and spin-l /2 charge-dyon system, respectively. For s E (0,1/2), the energy levels (4) are decomposedintosublevelswith(n,+(ml+l)and(n,+Im() degrees of degeneracy 4 . It follows from r = u2 and the accepted parametrization that Cc =+ R2(x) can be interpreted as the mapping of the half-up cone Cc on the paraboloid R( r = u2) with the following projection R + R2(x). This composite transformation can be written as XI + ix2 = 4(uf + ui) ‘/2(ut + iu2).
(1) The change of variables r = u2, p = 24 transforms the Cl oscillator into the system described by the equation 1
c p + eA CL
-i/i-&
2M
(2)
Here XI + ix2 = re’+‘, Aj = -(g/2r)EjkXk/r2, h/e, h = fi/27r, P( r, cp) = P’(u, 4) and e=--
Mm2 8
’
E cr=--, 4
s = v/2 E [O, l/2].
g =
(3)
Since B = V A A = g8c2) (x), Eq. (2) describes the motion of a particle of mass M and electric charge e = -e/2 in the fields of the dyon of electric charge Q = +a/2 and magnetic charge g, localized at the origin. It follows from (2) that s = v/2 has the meaning of the spin of the dyon. Since s E [ 0,1/2], the Cc oscillator is dual to the fractional charge-dyon bound system which continuously interpolates between the bosonic (s = 0) and fermionic (s = l/2) systems considered in Ref. [6]. E$. (2) leads to the bounded solution
x’F(-n,,2]j/
+ I,z)eimP/J2?T
with z = 2Ar/r0, ro = fi2/pcq h = ( -2efi2/~ury2) ‘i2, n,=0,1,2 ,..., m E Z, j = m+s, and the eigenvalues
(4)
(5)
Eq. (5) together with the ansatz (3) forms the duality transformation mapping the CT oscillator onto the fractional charge-dyon system. Let us stress what we mean by the term duality. Consider the system S with energy E and coupling constant w. The system Sd is called dual to S if ,Y?is governed by the same dynamical equation as S but with fixed E and quantized w. The transformation Td which maps S onto the system sd with energy E and coupling constant (Y, such that E = E(O) and (Y = a(E), is called the transformation of duality. In our case, S, sd and Td mean the Cz oscillator, fractional charge-dyon bound system and paraboloidal projection, respectively. The electromagnetic duality for anyons is conserved even after the arbitrary scalar potential V( u2) is excluded. The dual partner of the C; oscillator modified in this way is the fractional charge-dyon system placed in the potential V( r)/4r. Finally we explain why we considered the half-up cone for an angle of 0 = 7r/6 only, but not for arbitrary 8. The answer is that only for 8 = 7r/6 the paraboloidal projection transforms C$ into the Euclidean plane. For other values of 8, after such a projection, one obtains a plane with non-Euclidean metric d12 = dr2 + 4r2 sin2 Bdp2. It is to be emphasized that for 8 = 7r/6 Cc has the topology of the manifold describing the relative coordinate of two identical parti-
4 The dynamical symmetry of the magnetic by Jackiw in his beautiful article I 13 1.
vortex was established
A. Mqhakian et al./Physics Letters A 236 (1997) S-7
cles placed in the Euclidean plane [ 71, and therefore, the CT oscillator is also a prototype of the anyon. We would like to thank Yeranuhy Hakobyan, Levon Mardoyan, Armen Nersessian and George Pogosyan for discussions. One of us (V.T.-A.) thanks Simon Ter-Antonian for hospitality at the University of Minnesota and acquaintance with the subject of anyons.
A 10 (1995)
] [4 I [3
A. Nersessian,
[7 ]
Introduction 1960) p. 52.
1 I 1 N. Seiberg, E. Witten, V.
Ter-Antonyan.
Nucl. Phys. B 431 ( 1994) A.
Nersessian.
Mod.
484.
Phys.
191 P Gerber% R. Jackiw, Comm. R. Jackiw,
( 1982)
Math.
Phys. Rev. Lett. 52
Ann.
Mod.
Phys.
E. Witten.
(McGraw-Hill,
957. 237 B
Phys. II8
Math.
Nucl. Phys. B (Proc.
Y.-H. Chen. F. Wilczek.
Analysis
Nuovo Cimento
181 S. Deser. R. Jackiw. Comm.
1131 R. Jackiw, Lett.
to Matrix
Phys. Rev. Len. 49
J.M. Leinaas. J. Myrheim.
B. Halperin,
M. Tsulaja,
160.5.
New York,
Phys. B 3 (1989)
[ 1I
J. Geom. and Phys. ( 1996) 250.
V.M. Ter-Antonyan,
R. Bellman.
161 F. Wilczek.
[ IO] II I I 1121
References
2633.
T. Iwai. T. Sunako, Lett. A 11 (1996)
[5 )
I
Phys. 124
Suppl.)
18 A
( 1989)
1.583.
B. Halperin.
1001. Phys. (NY)
201
( 1990)
83.
( 1977) I.
( 1988)
495.
( 1989) 229. ( 1991 ) 107. Int. J. Mod.