physreva.84.032109

7/30/2019 PhysRevA.84.032109

1/6

PHYSICAL REVIEW A 84, 032109 (2011)

Dirac oscillator interacting with a topological defect

J. CarvalhoUnidade Acad emica de Tecnologia de Alimentos, CCTA, Universidade Federal de Campina Grande, Pereiros, 58840-000,

Pombal, Paraba, Brazil

C. Furtado and F. MoraesDepartamento de Fsica, CCEN, Universidade Federal da Paraba, Cidade Universit aria, 58051-970 Jo ao Pessoa, Paraba, Brazil

(Received 11 April 2011; published 14 September 2011)

In this work we study the interaction problem of a Dirac oscillator with gravitational fields produced by

topological defects. The energy levels of the relativisticoscillator in the cosmic string and in the cosmic dislocation

space-times are sensible to curvature and torsion associated to these defects and are important evidence of the

influence of the topology on this system. In the presence of a localized magnetic field the energy levels acquire

a term associated with the Aharonov-Bohm effect. We obtain the eigenfunctions and eigenvalues and see that in

the nonrelativistic limit some results known in standard quantum mechanics are reached.

DOI: 10.1103/PhysRevA.84.032109 PACS number(s): 03.65.Ge, 03.65.Pm, 04.62.+v

I. INTRODUCTION

A relativistic generalization of the harmonic oscillator isfar from being a trivial topic and is certainly not unique.

Such a generalization depends on the choice of a definition

for the harmonic oscillator. The study of a relativistic electron

in the presence of a quadratic potential was investigated by

Nikolsky [1] and Postpska [2]. They obtained a Dirac equation

for a quadratic potential that results in a quartic equation in

the eigenvalues problem with no bound state solutions. They

have demonstrated that in the nonrelativistic limit the discrete

energy levels correspond in this oscillator to resonances [3].

In 1967, Ito et al. [4] investigated a problem of the spin-1/2particle in which the momentum p has the coupling

p p im x, (1)

where m and w are the mass and oscillator frequency,

respectively, is the standard matrix appearing in relativistic

quantum mechanics, and x is the position of the particle.Cook [5] demonstrated that this problem exhibits certain

peculiar accidental degeneracies. Moshinsky and Szczepaniak

[6] revived this problem and coined the term Dirac oscillator.

They did so due to the fact that in the nonrelativistic limit it

becomes a harmonic oscillator [6] with a very strong spin-orbit

coupling. In their study they obtained the eigenvalues and

eigenstates for this problem. Recently, the Dirac oscillator

was studied in a series of physical contexts: its covariance

properties were studied in [7], in the point of view of Lie

algebra in [8], hidden supersymmetry[7,911], using the shape

invariant method [12], conformal invariance properties [13],in

the presence of external magnetic fields, and in the presence of

Aharonov-Bohm flux [1417]. Furthermore, Dirac oscillator

seems to be related to superconductor systems, the quantum

Hall effect [17], and the Ramsey interferometry effect [18].

The study of the influence of topological defects in the

relativistic and nonrelativistic quantum dynamics is a topic

well investigated in recent years. In nonrelativistic quantum

mechanics a series of problems have been investigated:

scattering of particles by topological defects [19], bound states

of electrons and holes to a disclination [20,21], Landau levels

in the presence of topological defects [2224], and so on. The

harmonic oscillator interacting with topological defects was

investigated by two of us in [25]. In the relativistic point ofview, the quantum dynamics in the presence of topological

defects was investigated in various physical contexts: the

relativistic Hydrogen atom [26], space-times with topological

defects, quantum dynamics of a single particle in the presence

ofa series ofpotentials [27], andscattering of particles[28,29].

The aim in the present article is to investigate the Dirac

oscillator in the background produced by topological defects

such as cosmic strings. In this sense, our contribution fills a

lack on the physical properties of particles interacting with

gravitational fields due to topological defects.

In this contribution, we solve the Dirac equation for the

Dirac oscillator in the background space-times of a cosmic

string, of a magnetic cosmicstring, andof a cosmicdislocation.

Finally, to corroborate our results, a nonrelativistic study is

made for comparison with the known results in the literature.

II. DIRAC OSCILLATOR IN THE COSMIC STRING

BACKGROUND

In this section, we proceed tosolve the Dirac oscillator in the

cosmic string background whose signature is (, + , + ,+).The cosmic string background is described by the following

metric written in cylindrical coordinates (t,,,z):

ds 2 = dt2 + d2 + 22d2 + dz2, (2)

with the following range on variables < (t,z) < , 0 < , and 0 2 . The parameter is the deficit angleassociated with the conical geometry obeying = 1 4 and is the linear mass density of the string in natural units.

The equation to be solved is the Dirac equation in the

background described by Eq. (2), which written in covariant

form is

[i(x) i(x)(x) m](t,x) = 0. (3)

Notice that this equation, when compared to theDirac equation

in flat space-time, has an extra (x)(x) term that takes into

account the correction introduced by the conical geometry

of the defect. The matrices (x) are the generalized Dirac

matrices for the background given by Eq. (2). They satisfy

032109-11050-2947/2011/84(3)/032109(6) 2011 American Physical Society
http://dx.doi.org/10.1103/PhysRevA.84.032109http://dx.doi.org/10.1103/PhysRevA.84.032109

7/30/2019 PhysRevA.84.032109

2/6

J. CARVALHO, C. FURTADO, AND F. MORAES PHYSICAL REVIEW A 84, 032109 (2011)

the anticommutation relation + = 2g and arewritten in terms of the standard Dirac matrices a(x) in

Minkowski space-time as

(x) = ea (x)a. (4)

The tetrad components ea(x) obey the relation

ea(x)eb (x)ab = 2g, (5)

where (,) = (0,1,2,3) are tensor indices and (a,b) =(0,1,2,3) are tetrad indices. We consider the following choice

for the tetrad inverse in the cosmic string space-time

ea (x) =

1 0 0 0

0 cos sin 0

0 sin

cos

0

0 0 0 1

. (6)

In this representation the Diracs matrices, (x), obey the

relations

0 = t, 3 = z, 1 = , = cos1 + sin2,(7)

2 =

, = sin1 + cos2.

The correction introduced by the geometry, (x)(x)

appearing in Eq. (3), can be calculated with the help of the

following one-form basis

e0 = dt, e3 = dz, e1 = cos d sin d,(8)

e2 = sin d + cos d.

Using the Maurer-Cartan structure equation de a + wab eb =0, we find the following nonnull components of the spin

connection to be w1 2 = w2 1 = (1 )d. Therefore, thespin connection matrix ab = w a bdx

is given by

ab (x) =

0 0 0 0

0 0 1 0

0 (1 ) 0 0

0 0 0 0

. (9)

Hence, we can calculate the spinorial connection using

(x) = i2

wabab

2with ab = i

2[a,b]. Using this infor-

mation and the results (7) and (9), we obtain that the curvature

correction is

(x)(x) =1

2 . (10)

In this point, we can solve the Dirac oscillator problem in the

cosmic string background. To include the Dirac oscillator term

imw into Eq. (3) we proceed with the following change in

the radial momentum component + mw. There-fore, Eq. (3), with the help of Eqs. (7) and (10) transforms

into itt + i

1 2

+ mw

+ i

+ izz m

= 0, (11)

This is the quantum equation for the Dirac oscillator of mass

m and frequency w in the background space-time described

by Eq. (2). A solution can be constructed assuming temporal

independence and rotational symmetry of the background

around z axis. For the positive energy solution we can choose

the following ansatz

= eiEt+i(l+1/23/2)+ikz

()

() . (12)In this way, Eq. (13) transforms into

2E + i

+

1

2+ mw

l + 1/2

zk m

= 0, (13)

where we use the property 3 = i imposed by relation (7). With the help of Eq. (7) we write the terms in the followingway

=cos

0 11 0

+sin

0 22 0

;

=cos

0 11 0

+sin

0 22 0

;

(14)

=

0

0

; = sin

0 1

1 0

+ cos

0 2

2 0

; z =

0 3

3 0

.

With these matrices Eq. (13) can be written as the two coupled equations below

(E m) +

i(cos1 + sin2)

+

1

2 mw

(sin1 + cos2) k3

= 0,

(15)

(E + m) +

i(cos1 + sin2)

+

1

2+ mw

(sin1 + cos2) k3

= 0.

032109-2

7/30/2019 PhysRevA.84.032109

3/6

DIRAC OSCILLATOR INTERACTING WITH A . . . PHYSICAL REVIEW A 84, 032109 (2011)

We can multiply the first equation of Eq. (15) by (E + m) and using the second equation we obtain

(E2 m2)

i(cos1 + sin2)

+

1

2 mw

(sin1 + cos2) k3

i(cos1 + sin2)

+

1

2+ mw

(sin1 + cos2) k3

= 0. (16)

An analog equation for can be written as

(E2 m2)

i(cos1 + sin2)

+

1

2+ mw

(sin1 + cos2) k3

i(cos1 + sin2)

+

1

2 mw

(sin1 + cos2) k3

= 0. (17)

After simple algebraic manipulations we arrive at

() + ()

1

2

1

4+ i12 + 2

m2w22 ()

+ (E2 m2 + mw k2) () 2mw[i12 + ik( cos 13 + sin 23)] () = 0. (18)

The third term in this expression can be written as a perfect

square. To the last term, we use the definition of the scalar

product S L in cylindrical coordinates. We get then

[i12 + ik ( cos 13 + sin 23)] = L.

(19)

As S = 2

the above term becomes

[i12 + ik ( cos 13 + sin 23)] = 2 S L.

(20)

We can simplify our problem by assuming a tridimensional

planar system in which the term S L has an eigenvalue (l +1/2)/2. With this we can write Eq. (18) in a compact form

d2

d2+

1

d

d

1

2

l + 1/2

1

2

2+ m2w22

= 0,

(21)

with the term given by

= E2 m2 + 2mwl + 1/2

+1

2 k2. (22)

An analog equation can be found for . In summary, we have

the set of two decoupled differential equations to solved2

d2+

1

d

d

2

2+ m2w22

= 0,

(23)

with

=l + 1/2

1

2,

(24)

e = E2 m2 + 2mw

l + 1/2

1

2 k2.

The aim now is to solve Eq. (23). The equation for the

component can be transformed by the variable change = mw2 resulting in

() +1

()

2+

42+

1

4

4mw

() = 0, (25)

with + = (l + 1/2)/ + 1/2. The requirements of finitenessat the origin and at infinity are obtained by studying the

appropriate limits. This leads us to the following ansatz

() = |+|

2 e

2 F(). (26)

With this, Eq. (25) is transformed into the equation

F() + (|+| + 1 )F()

|+|

2+

1

2

4mw

F() = 0, (27)

which is of the confluent hypergeometric function type

F() + (c + 1 )F() aF() = 0. To obtain normal-ized solutions to this equation we require that the polynomial

series terminates. This requirement is satisfied by equating the

independent term to a negative integer or zero (i.e, a = n).By simple manipulation we find the following energy levels:

E2 = m2 + 4mw

n +

l + 1/2(1 )

2

l + 1/2(1 )2

+ 1 s

2

+ k2, (28)

with n = 0,1,2, . . . , l = 0,1,2, . . . , and s = 1. Theeigenfunctions to our problem are described by

= eiEt+i(l+1/23/2)+ik z

()

()

, (29)

with the up (+) and down () spinor components written interms of hypergeometric confluent functions as below

()

()

= (mw)

||2 ||e

mw2

2 F

n, || + 1, mw2

.

(30)

032109-3

7/30/2019 PhysRevA.84.032109

4/6


Note that taking the limit 1 into Eq. (28) leads tothe energy levels of the particle in flat space added by its

rest energy. Moreover, for integer values of , the infinite

degeneracy of the eigenvalues is present, in accordance with

the degenerate states of the nonrelativistic case. For noninteger values the degeneracies are absent.

III. DIRAC OSCILLATOR IN THE MAGNETIC COSMIC

STRING BACKGROUND

In this section, the Dirac equation in the background of a

cosmic string with an internal magnetic field will be solved.

We choose the magnetic vector potential to be A = iB

2e.

This choice gives a flux tube coinciding with the cosmic string

and the z axis. We assume that the Hamiltonian is minimally

coupled to this magnetic field. In this field configuration, the

Dirac oscillator coupled to the cosmic string background obeysitt + i

1

2+ mw

+ i

+ i e

B

2

+ izz m

= 0.

(31)

Using Eq. (12) the above equation transforms into2E + i

+

1

2+ mw

l + 1/2 + eB /2

zk m

= 0.

(32)

With Eq. (14), we obtaind2

d2+

1

d

d

a2

2+ 22

= 0, (33)

with = mw. Now, the terms a and assume the form

a =l + 1/2 + eB /2

1

2,

= E2 m2 k2 (34)

+ 2mw

l + 1/2 + eB /2

1

2

.

In this way the angular term is added to the Aharonov-Bohm

contribution eB /2 . A mathematical procedure to solve the

equation for the component begins with the substitution = 2. At the origin and at infinity, normalized solutionsare found by choosing

() = |a+|

2 e

2 F(), (35)

and a+ = (l + 1/2 + eB /2 )/ + 1/2. By direct substitu-tion of this equation into Eq. (33) we get

F() + (|a+| + 1 )F()

|a+|

2+

1

2

4m

F() = 0, (36)

which is, again, a confluent hypergeometric equation similar

to the one in the previous section. Here the angular momentum

part is added by the magnetic flux contribution eB /2 .

Normalized solutions to this equation are guaranteed by

imposing thatthe last term should assume only negative integer

or zero values. This results in

|a+|

2+

1

2

4mw= n. (37)

This procedure allows us to determine the energy eigenvaluesassociated with the Dirac oscillator as

E2 = m2 + 4mw

n +

lAB + 1/2(1 )2

lAB + 1/2(1 )

2

+

1 s

2

+ k2, (38)

with lAB = l + eB /2 and which has two contributions.The first is related to the nonlocal or topological features

of the background space-times ascribed by the parameter,

although the background space-time is locally flat. The second

features the electromagnetic contribution to these levels, even

though the magnetic field is restricted to the symmetry axisof the string, a forbidden region to our particle. We can

summarize these observations as a gravitational analog of the

well-known electromagnetic Aharonov-Bohm effect. More-

over, the analysis has important consequences: Information

on the background space-time or on the electromagnetic

fields present can be found by a critical analysis of the

eigenvalues and eigenfunctions. From this problem of Dirac

particles interacting with a conical singularity we can obtain

the Minkowski space-time result taking the limit 1.Independent of , assuming integer or noninteger values, the

degenerate states of the energy levels are absent, unlike the

previous case. To finalize this section, we see that in the limit

of weak magnetic field B 0 the results of the previoussection are reached.

IV. DIRAC OSCILLATOR IN COSMIC DISLOCATION

SPACE-TIME

From now on we proceed to consider the Dirac oscillator

in a cosmic dislocation space-time. This geometry was found

by Galtsov and Letelier [30] as

ds 2 = dt2 + d2 + 22d2 + (dz + Jzd)2, (39)

in cylindrical coordinates, with 0 and 0 2 . The

parameter Jz is related to the torsion source and = 1 4,

where is the linear mass density of the string, as before.We observe that, when Jz = 0, the metric of the cosmic stringspace-time is recovered. Therefore, we are adding a torsion

source, represented by Jz, to our cosmic string. Observe that

this metric is locally flat, as it is easily seen by the coordinate

change t T , = ,z Z = z + Jz.To write the Dirac equation in this background, let us

consider the following inverse tetrad:

ea (x) =

1 0 0 0

0 cos sin 0

0 sin

cos

0

0Jzsin

J

z cos

1

. (40)

032109-4

7/30/2019 PhysRevA.84.032109

5/6

DIRAC OSCILLATOR INTERACTING WITH A . . . PHYSICAL REVIEW A 84, 032109 (2011)

The Dirac matrices (x) (4) assume the form

0 = t, 1 = , 2 =

, 3 = z

Jz

,

(41)

where again, = sin1 + cos2, = cos1 +

sin2

. The Dirac equation in the cosmic dislocation space-time, with the Dirac oscillator and with a magnetic field flux

B , along its symmetry axis, becomesitt + i

+ mw +

1

2

+ i

J

zz + ieB

2

+ izz m

(t,r) = 0.

(42)

Using the ansatz (12) and the relation (14) added to the

properties of the Pauli matrix, Eq. (42) becomes

d2d2 +

1

d

d 2

2 + m2

w2

2

= 0,

(43)

with

=l + 1/2 kJz + eB /2

1

2,

(44) = E

2 m2 + 2mw[(l + 1/2

+ e/2 kJz)/ 1/2] k2.

To the wave equation we input the change of variables = mw2, and the result is

d2

d2 +1

d2

d 2

+42 +

14

4mw

= 0, (45)

with + = (l + 1/2 + eB /2 kJz)/ + 1/2. By study-

ing the asymptotic limits of Eq. (45), we assume a solution of

the type

() = |+|

2 e

2 R(). (46)

Substituting the above ansatz into Eq. (45) that equation

transforms into a confluent hypergeometric equation for

R(n,|+| + 1,mw2), with the following eigenvalues

E2 = m2 + 4mwn + leff + 1/2(1 )

2

leff + 1/2(1 )

2

+

1 s

2

+ k2, (47)

with leff = l + eB /2 kJz. In this way an analog equa-

tion to the eigenfunction (12) can be constructed with

the help of the hypergeometric series. Comparatively to

the results in Sec. II, the spinor for the Dirac oscil-

lator on the cosmic dislocation space-time assumes the

form ()

()

= (mw)

||2 ||e

mw2

2 F(n, || + 1, mw2).

(48)

Again we observe a Landau structure in the above energy

levels. Comparatively to the previous cases, the energy levels

now have a torsional contribution, besides the curvature

and electromagnetic parts. For Jz = 0, all previous resultsare recovered. Contrary to the nonrelativistic case [25], the

degeneracy of the Landau levels is not removed by choosing

integer. The results (47) and (48) yield all the previously

obtained results in the appropriate limits. Notice that torsionas well as magnetic flux give rise to an effective angular

momentum.

V. NONRELATIVISTIC LIMIT

To analyze the nonrelativistic limit we assume E = m + such that m and Eq. (43) assumes the form

=

1

2m

d2

d2+

1

d

d

2

2 k2

+1

2m22

2 2 S L

. (49)

We note that this equation is the Schrodinger equation in thedispiration background space-time with an harmonicoscillator

term and the spin-orbit interaction. To analyze the nonrela-

tivistic limit of the energy, we write Eq. (47) in the appropriate

form

E = m

1 +

4

m

n + +

1 s

2

+

k2

m2, (50)

with

=|l + eB /2 kJ

z + 1/2(1 )|

2

[l + eB /2 kJ

z + 1/2(1 )]

2. (51)

Using the Taylor expansion up to second order in the energy

expression results in

m +k2

2m+ 2

n + +

1 s

2

22

m

n + +

1 s

2

2. (52)

The first two terms in this expression are, respectively, the rest

energy added to the kinetic energy of the particle along the

string, the second term is the energy of the nonrelativistic

harmonic oscillator, and the last one is associated to the

relativistic correction term.

VI. CONCLUSION

Line defects like the cosmic string and its variations

(magnetic cosmic string and cosmic dislocation) even though

they correspond to locally flat geometries and have global

properties that affect quantum systems like the harmonic

oscillator. The analysis of the emission spectra of spacebound

atomic species from a region where there is gravitational

lensing is an indication of a cosmic string [31] and might

give further evidence for the presence of the defect. In this

article we analyzed the energy spectrum of one of the possible

relativistic generalizations of the harmonic oscillator in the

032109-5

7/30/2019 PhysRevA.84.032109

6/6


background space-times of the cosmic string, the magnetic

cosmic string, and the cosmic dislocation. In all cases we

recover the known nonrelativistic limits. The eigenvalues

and eigenfunctions found depend explicitly on the nonlocal

parameters of the space-time in consideration even though it

is locally flat. This result may be thought of as a variation

on the well-known electromagnetic Aharonov-Bohm effect.

Also, differently from the case of Minkowski space-time, the

presence of the defects breaks the degeneracy of the energy

levels for noninteger values of the parameter .

ACKNOWLEDGMENTS

We thank CAPES/PROCAD, NANOBIOTEC/CAPES,

INCT-FCx, and CNPq for financial support.

[1] K. Nikolski, Z. Phys. 62, 677 (1930).

[2] I. Posepska, Acta Phys. Pol. 4, 269 (1935).

[3] J. Karwoski and G. Pestka, Theor. Chem. Acc. 118, 519 (2007).

[4] D. Ito, K. Mori, and E. Carrieri, Nuovo Cimento A 51, 1119

(1967).

[5] P. A. Cook, Lett. Nuovo Cimento 1, 419 (1971).

[6] M. Moshinsky and A. Szczepaniak, J. Phys. A: Math. Gen 22,

L817 (1989).[7] M. Moreno and A. Zanttella, J. Phys. A : Math. Gen. 22, L821

(1989).

[8] C. Quesne and M. Moshinsky, J. Phys. A: Math. Gen. 23, 2263

(1990).

[9] J. Bentez, R. P. Martnez y Romero, H. N. Nunez-Yepez, and

A. L. Salas-Brito, Phys. Rev. Lett. 64, 1643 (1990); 65, 2085

(1990).

[10] J. Beckers and N. Debergh, Phys. Rev. D 42, 1255 (1990).

[11] C. Quesne, Int. J. Mod. Phys. A 6, 1567 (1991).

[12] C. Quesne and V. M. Tkachuk, J. Phys A: Math. Gen. 38, 1747

(2005).

[13] R. P. Martnez y Romero and A. L. Salas Brito, J. Math. Phys.

33, 1831 (1992).

[14] L. Gonzalez-Diaz and V. M. Villalba, Phys. Lett. A 352, 202

(2006).

[15] V. M. Villalba and Clara Rojas, Int. J. Mod. Phys. A 21, 313

(2006).

[16] C. Rojas andV. M. Villalba, Mod. Phys. Lett. A 20, 2245 (2005).

[17] N. Ferkous and A. Bounames, Phys. Lett. A 325, 21 (2004).

[18] A. Bermudez, M. A. Martin-Delgado, and A. Luis, Phys. Rev.

A 77, 033832 (2008).

[19] C. Furtado, F. Moraes, and V. B. Bezerra, Phys. Lett. A 289, 160

(2001).

[20] C. A. de Lima Ribeiro, C. Furtado, and F. Moraes, Europhys.

Lett. A 62, 306 (2003).[21] C. Furtado and F. Moraes, Phys. Lett. A 188, 394 (1994).

[22] C. Furtado and F. Moraes, Phys. Lett. A 195, 90 (1994).

[23] C. Furtado and F. Moraes, Europhys. Lett. 45, 179 (1999).

[24] G. de A. Marques, C. Furtado, V. B. Bezerra, and F. Moraes,

J. Phys. A: Math. Gen. 31, 5945 (2001).

[25] C. Furtado and F. Moraes, J. Phys. A: Math. Gen. 33, 5513

(2000).

[26] G. A. Marques and V. B. Bezerra, Phys. Rev. D 66, 105011

(2002).

[27] G. A. Marques and V. B. Bezerra, Classical Quantum Gravity

19, 985 (2002).

[28] S. Deser and R. Jackiw, Commun. Math. Phys. 118, 495

(1988).

[29] P. de Sousa Gerbert and R. Jackiw, Commun. Math. Phys. 124,

229 (1989).

[30] D. V. Galtsov and P. S. Letelier, Phys. Rev. D 47, 4273 (1993).

[31] M. Sazhin et al., Mon. Not. R. Astron. Soc. 343, 353 (2003).

032109-6
http://dx.doi.org/10.1007/BF01843485http://dx.doi.org/10.1007/BF01843485http://dx.doi.org/10.1007/BF01843485http://dx.doi.org/10.1007/s00214-007-0362-yhttp://dx.doi.org/10.1007/s00214-007-0362-yhttp://dx.doi.org/10.1007/s00214-007-0362-yhttp://dx.doi.org/10.1007/BF02721775http://dx.doi.org/10.1007/BF02721775http://dx.doi.org/10.1007/BF02721775http://dx.doi.org/10.1007/BF02721775http://dx.doi.org/10.1007/BF02785170http://dx.doi.org/10.1007/BF02785170http://dx.doi.org/10.1007/BF02785170http://dx.doi.org/10.1088/0305-4470/22/17/002http://dx.doi.org/10.1088/0305-4470/22/17/002http://dx.doi.org/10.1088/0305-4470/22/17/002http://dx.doi.org/10.1088/0305-4470/22/17/002http://dx.doi.org/10.1088/0305-4470/22/17/003http://dx.doi.org/10.1088/0305-4470/22/17/003http://dx.doi.org/10.1088/0305-4470/22/17/003http://dx.doi.org/10.1088/0305-4470/22/17/003http://dx.doi.org/10.1088/0305-4470/23/12/011http://dx.doi.org/10.1088/0305-4470/23/12/011http://dx.doi.org/10.1088/0305-4470/23/12/011http://dx.doi.org/10.1088/0305-4470/23/12/011http://dx.doi.org/10.1103/PhysRevLett.64.1643http://dx.doi.org/10.1103/PhysRevLett.64.1643http://dx.doi.org/10.1103/PhysRevLett.64.1643http://dx.doi.org/10.1103/PhysRevLett.65.2085http://dx.doi.org/10.1103/PhysRevLett.65.2085http://dx.doi.org/10.1103/PhysRevLett.65.2085http://dx.doi.org/10.1103/PhysRevD.42.1255http://dx.doi.org/10.1103/PhysRevD.42.1255http://dx.doi.org/10.1103/PhysRevD.42.1255http://dx.doi.org/10.1142/S0217751X91000836http://dx.doi.org/10.1142/S0217751X91000836http://dx.doi.org/10.1142/S0217751X91000836http://dx.doi.org/10.1088/0305-4470/38/8/011http://dx.doi.org/10.1088/0305-4470/38/8/011http://dx.doi.org/10.1088/0305-4470/38/8/011http://dx.doi.org/10.1088/0305-4470/38/8/011http://dx.doi.org/10.1063/1.529660http://dx.doi.org/10.1063/1.529660http://dx.doi.org/10.1063/1.529660http://dx.doi.org/10.1016/j.physleta.2005.12.003http://dx.doi.org/10.1016/j.physleta.2005.12.003http://dx.doi.org/10.1016/j.physleta.2005.12.003http://dx.doi.org/10.1016/j.physleta.2005.12.003http://dx.doi.org/10.1142/S0217751X06025158http://dx.doi.org/10.1142/S0217751X06025158http://dx.doi.org/10.1142/S0217751X06025158http://dx.doi.org/10.1142/S0217751X06025158http://dx.doi.org/10.1142/S0217732305017524http://dx.doi.org/10.1142/S0217732305017524http://dx.doi.org/10.1142/S0217732305017524http://dx.doi.org/10.1016/j.physleta.2004.03.033http://dx.doi.org/10.1016/j.physleta.2004.03.033http://dx.doi.org/10.1016/j.physleta.2004.03.033http://dx.doi.org/10.1103/PhysRevA.77.033832http://dx.doi.org/10.1103/PhysRevA.77.033832http://dx.doi.org/10.1103/PhysRevA.77.033832http://dx.doi.org/10.1103/PhysRevA.77.033832http://dx.doi.org/10.1016/S0375-9601(01)00615-6http://dx.doi.org/10.1016/S0375-9601(01)00615-6http://dx.doi.org/10.1016/S0375-9601(01)00615-6http://dx.doi.org/10.1016/S0375-9601(01)00615-6http://dx.doi.org/10.1209/epl/i2003-00396-3http://dx.doi.org/10.1209/epl/i2003-00396-3http://dx.doi.org/10.1209/epl/i2003-00396-3http://dx.doi.org/10.1209/epl/i2003-00396-3http://dx.doi.org/10.1016/0375-9601(94)90482-0http://dx.doi.org/10.1016/0375-9601(94)90482-0http://dx.doi.org/10.1016/0375-9601(94)90482-0http://dx.doi.org/10.1016/0375-9601(94)90432-4http://dx.doi.org/10.1016/0375-9601(94)90432-4http://dx.doi.org/10.1016/0375-9601(94)90432-4http://dx.doi.org/10.1209/epl/i1999-00159-8http://dx.doi.org/10.1209/epl/i1999-00159-8http://dx.doi.org/10.1209/epl/i1999-00159-8http://dx.doi.org/10.1088/0305-4470/34/30/306http://dx.doi.org/10.1088/0305-4470/34/30/306http://dx.doi.org/10.1088/0305-4470/34/30/306http://dx.doi.org/10.1088/0305-4470/33/31/306http://dx.doi.org/10.1088/0305-4470/33/31/306http://dx.doi.org/10.1088/0305-4470/33/31/306http://dx.doi.org/10.1088/0305-4470/33/31/306http://dx.doi.org/10.1103/PhysRevD.66.105011http://dx.doi.org/10.1103/PhysRevD.66.105011http://dx.doi.org/10.1103/PhysRevD.66.105011http://dx.doi.org/10.1103/PhysRevD.66.105011http://dx.doi.org/10.1088/0264-9381/19/5/310http://dx.doi.org/10.1088/0264-9381/19/5/310http://dx.doi.org/10.1088/0264-9381/19/5/310http://dx.doi.org/10.1007/BF01466729http://dx.doi.org/10.1007/BF01466729http://dx.doi.org/10.1007/BF01466729http://dx.doi.org/10.1007/BF01466729http://dx.doi.org/10.1007/BF01219196http://dx.doi.org/10.1007/BF01219196http://dx.doi.org/10.1007/BF01219196http://dx.doi.org/10.1007/BF01219196http://dx.doi.org/10.1103/PhysRevD.47.4273http://dx.doi.org/10.1103/PhysRevD.47.4273http://dx.doi.org/10.1103/PhysRevD.47.4273http://dx.doi.org/10.1046/j.1365-8711.2003.06568.xhttp://dx.doi.org/10.1046/j.1365-8711.2003.06568.xhttp://dx.doi.org/10.1046/j.1365-8711.2003.06568.xhttp://dx.doi.org/10.1046/j.1365-8711.2003.06568.xhttp://dx.doi.org/10.1103/PhysRevD.47.4273http://dx.doi.org/10.1007/BF01219196http://dx.doi.org/10.1007/BF01219196http://dx.doi.org/10.1007/BF01466729http://dx.doi.org/10.1007/BF01466729http://dx.doi.org/10.1088/0264-9381/19/5/310http://dx.doi.org/10.1088/0264-9381/19/5/310http://dx.doi.org/10.1103/PhysRevD.66.105011http://dx.doi.org/10.1103/PhysRevD.66.105011http://dx.doi.org/10.1088/0305-4470/33/31/306http://dx.doi.org/10.1088/0305-4470/33/31/306http://dx.doi.org/10.1088/0305-4470/34/30/306http://dx.doi.org/10.1209/epl/i1999-00159-8http://dx.doi.org/10.1016/0375-9601(94)90432-4http://dx.doi.org/10.1016/0375-9601(94)90482-0http://dx.doi.org/10.1209/epl/i2003-00396-3http://dx.doi.org/10.1209/epl/i2003-00396-3http://dx.doi.org/10.1016/S0375-9601(01)00615-6http://dx.doi.org/10.1016/S0375-9601(01)00615-6http://dx.doi.org/10.1103/PhysRevA.77.033832http://dx.doi.org/10.1103/PhysRevA.77.033832http://dx.doi.org/10.1016/j.physleta.2004.03.033http://dx.doi.org/10.1142/S0217732305017524http://dx.doi.org/10.1142/S0217751X06025158http://dx.doi.org/10.1142/S0217751X06025158http://dx.doi.org/10.1016/j.physleta.2005.12.003http://dx.doi.org/10.1016/j.physleta.2005.12.003http://dx.doi.org/10.1063/1.529660http://dx.doi.org/10.1063/1.529660http://dx.doi.org/10.1088/0305-4470/38/8/011http://dx.doi.org/10.1088/0305-4470/38/8/011http://dx.doi.org/10.1142/S0217751X91000836http://dx.doi.org/10.1103/PhysRevD.42.1255http://dx.doi.org/10.1103/PhysRevLett.65.2085http://dx.doi.org/10.1103/PhysRevLett.65.2085http://dx.doi.org/10.1103/PhysRevLett.64.1643http://dx.doi.org/10.1088/0305-4470/23/12/011http://dx.doi.org/10.1088/0305-4470/23/12/011http://dx.doi.org/10.1088/0305-4470/22/17/003http://dx.doi.org/10.1088/0305-4470/22/17/003http://dx.doi.org/10.1088/0305-4470/22/17/002http://dx.doi.org/10.1088/0305-4470/22/17/002http://dx.doi.org/10.1007/BF02785170http://dx.doi.org/10.1007/BF02721775http://dx.doi.org/10.1007/BF02721775http://dx.doi.org/10.1007/s00214-007-0362-yhttp://dx.doi.org/10.1007/BF01843485

physreva.84.032109

Documents