- Email: [email protected]
Massless particle in 2d spacetime with constant curvature
25 February 1999
Physics Letters B 448 Ž1999. 203–208
Massless particle in 2d spacetime with constant curvature George Jorjadze a , Włodzimierz Piechocki
b,1
a
b
Razmadze Mathematical Institute, 380093 Tbilisi, Georgia Theory DiÕision (Zd.P8), Sołtan Institute for Nuclear Studies, 00-681 Warsaw, Poland Received 7 January 1999 Editor: P.V. Landshoff
Abstract We consider dynamics of massless particle in 2d spacetimes with constant curvature. We analyze different examples of spacetime. Dynamical integrals are constructed from spacetime symmetry related to sl Ž2.R. algebra. Mass-shell condition restricts dynamical integrals to a cone Žwithout vertex. which defines physical-phase space. We parametrize the cone by canonical coordinates. Canonical quantization with definite choice of operator ordering leads to unitary irreducible representations of SO ≠ Ž2.1. group. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 04.60.M; 02.20.S; 02.40.K Keywords: 2d spacetime with constant curvature; Dynamical integrals; Canonical quantization; Representations of SO ≠ Ž2.1. group
1. Introduction We analyze classical and quantum dynamics of a massless particle in two dimensional spacetime with constant curvature R 0 / 0. We consider the cases when spacetime is: Ži. hyperboloid with R 0 - 0, Žii. half-plane with R 0 - 0 and Žiii. stripe with R 0 ) 0. Presented results complete our analysis of dynamics of a particle with non-zero mass m 0 in 2d curved spacetime w1,2x. Taking formally m 0 ™ 0 leads to some singularities both at classical and quantum levels. Thus, the massless case needs separate treatment. As the result we get clear picture of the role played by topology and global symmetries of spacetime in the procedure of canonical quantization.
Dynamics of a massless particle in gravitational field gmn Ž x 0 , x 1 . is defined by the action w3x S s L Ž t . dt ,
H
L Ž t . :s y
1 2 lŽ t .
gmn Ž x 0 Ž t . , x 1 Ž t . .
=x˙ m Ž t . x˙ n Ž t . ,
where t is an evolution parameter, l plays the role of Lagrange multiplier and x˙ m :s dx mrdt . It is assumed that l ) 0 and x˙ 0 ) 0. The action Ž1.1. is invariant under reparametrization t ™ f Žt ., lŽt . ™ lŽt .rf˙Žt .. This gauge symmetry leads to the constrained dynamics in the Hamiltonian formulation w4x. The constraint reads
F :s g mn pm pn s 0,
Ž 1.2 . m
1
E-mail: [email protected]
Ž 1.1 .
where pm :s E LrE x˙ are canonical momenta Žwe use units with c s 1 s " ..
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 0 6 0 - X
G. Jorjadze, W. Piechockir Physics Letters B 448 (1999) 203–208
204
As in Refs. w1,2x we use the gauge invariant description. In the case of massive particle the set of spacetime trajectories can be considered as a physical phase-space of the system. This set has natural symplectic structure, which can be used for quantization. The spacetime trajectories of a massless particle has no symplectic structure Žthis set is only one dimensional, since particle velocity is fixed.. For the gauge invariant description we use the dynamical integrals constructed from the global symmetries of spacetime.
where the spacetime coordinates x 0 and x 1 are identified with the parameters r and u , respectively. The function Ž2.5. satisfies the Liouville Eq. Ž2.3. for R 0 s y2 m2 . The Lagrangian Ž1.1. in this case reads
2. Dynamics on hyperboloid
Ž r ,u . ™ Ž r ,u q a 0rm . ,
Let Ž y 0 , y 1 , y 2 . be the standard coordinates on 3d Minkowski space with the metric tensor ha b s diag Žq,y,y .. A one-sheet hyperboloid H is defined by
Ž r ,u . ™ Ž r y a 1rm sin m r sin m u ,
2
2
2
y Ž y 0 . q Ž y 1 . q Ž y 2 . s my2 ,
Lsy
r˙ 2 y u˙ 2 2 lsin2 m r
.
Ž 2.6 .
The hyperboloid Ž2.1. is invariant under the Lorentz transformations, i.e., SO ≠ Ž2,1. is the symmetry group of our system. The corresponding infinitesimal transformations Žrotation and two boosts. are
u q a 1rm cos m r cos mu . ,
Ž r ,u . ™ Ž r q a 2rm sin m r cos mu , u q a 2rm cos m r sin mu . .
Ž 2.1 .
Ž 2.7 .
The dynamical integrals for Ž2.7. read where m ) 0 is a fixed parameter. H has a constant curvature R s y2 m 2 Žsee w5x.. Any 2-dimensional Lorentzian manifold with constant curvature R 0 can be described Žlocally. by the conformal metric tensor w5x gmn Ž X . s exp w Ž X .
ž
1 0
0 , y1
/
Ž 2.2 . where the field w Ž X . satisfies the Liouville equation w6x
Ž 2.3 .
y sy where
cot m r m
,
1
y s
r g x 0,prm w ,
cos mu msin m r
,
2
y s
u g w 0,2 prm w
sin mu msin m r
,
Ž 2.4 .
pr m
, pr m
sin m r sin mu q
sin m r cos mu q
Ž 2.5 .
pu m
pu m
cos m r cos mu ,
cos m r sin mu ,
Ž 2.8 .
where pu :s E LrEu˙, pr :s E LrEr˙ are canonical momenta. Since J0 is connected with space translations Žsee Ž2.7.., it defines particle momentum pu s mJ0 . It is clear that the dynamical integrals Ž2.8. satisfy the commutation relations of sl Ž2.R. algebra
Ž 2.9 .
where h c d is the Minkowski metric tensor and ´ a b c is the anti-symmetric tensor with ´ 012 s 1. The mass shell condition Ž1.2. takes the form pr2 y pu2 s 0 and it leads to the relation J02 y J12 y J22 s 0.
we get the conformal form Ž2.2., with
w s ylog sin2 m r ,
m
Ja , J b 4 s ´ a b ch c d Jd ,
Making use of the parametrization 0
pu
J1 s y J2 s
X :s Ž x 0 , x 1 . ,
Ž E 02 y E 12 . w Ž X . q R 0 exp w Ž X . s 0.
J0 s
Ž 2.10 .
Eq. Ž2.10. defines two cones. The singular point of the cones J0 s 0 s J1 s J2 should be removed, since
G. Jorjadze, W. Piechockir Physics Letters B 448 (1999) 203–208
it corresponds to the massless particle with zero momentum Ž pr s 0 s pu ., which does not exist. Thus, the dynamical integrals Ž2.8. define the physical phase-space of the system and it consists of two disconnected cones Cq and Cy, for J0 ) 0 and J0 - 0, respectively. According to Ž2.8. and due to pr - 0 Žsince r˙ ) 0 and l ) 0. the trajectories satisfy the equations Ja y a s 0,
J1 y 2 y J 2 y 1 s
pr m
2
sy
< J0 < m
.
Ž 2.11 .
Each point Ž Ja . of the cone Cq or Cy defines uniquely the trajectory on the hyperboloid H. The trajectories Ž2.11. are straight lines in 3d Minkowski space. Hence, the set of trajectories is the set of generatrices of the hyperboloid Ž2.1.. For J0 ) 0 we get the ‘right’ moving particle with u˙) 0, while for J0 - 0 we have the ‘left’ moving one, with u˙- 0. Both cones, Cq and Cy, are invariant under SO ≠ Ž2.1. transformations. To quantize the system, we consider the cones Cq and Cy separately. We parametrize Cq as follows p J0 s 12 Ž p 2 q q 2 . , J1 s p2 q q2 , 2 q J2 s p2 q q2 , Ž 2.12 . 2 where Ž0,0. / Ž p,q . g R 2 . It is easy to see that Ž2.12. defines the one-to-one map from the plane without the origin to Cq. The canonical commutation relation p,q4 s 1 provides Ž2.9.. For quantization of Cq system we introduce the creation-annihilation operators a ":s Ž pˆ " iqˆ .r '2 and choose the definite operator ordering in Ž2.12.. This ordering is defined by the following requirements: Ža. the operators Jˆa are self-adjoint, Žb. they generate global SO ≠ Ž2.1. transformations, Žc. the spectrum of Jˆ0 is positive, ˆ Jˆ02 y Jˆ12 y Žd. the Casimir number operator C:s 2 ˆ J2 is zero. This leads to the following expressions
(
(
Jˆ0 s ay aq,
Jˆqs aq 'ay aqq 1 ,
Jˆys 'ay aqq 1 ay,
Ž 2.13 .
205
where Jˆ"s Jˆ1 " Jˆ2 . The creation and annihilation operators are defined in the Fock space in the standard way aq < n: s 'n q 1 < n q 1: ,
ay < n: s 'n < n y 1: ,
where the vectors < n:, Ž n G 0. form the basis of the corresponding Hilbert space Hq. The states < n: are the eigenstates of Jˆ0 Jˆ0 < n: s Ž n q 1 . < n: ,
Ž 2.14 .
and from Ž2.13. we get Jˆq < n: s Ž n q 1 . Ž n q 2 . < n q 1: ,
(
Jˆy < n: s n Ž n q 1 . < n y 1: .
(
Ž 2.15 .
Eqs. Ž2.13. – Ž2.15. define the unitary irreducible representation ŽUIR. of the group SO ≠ Ž2.1.. We identify this representation with the representation of Dq 1 of the discrete series of SLŽ2.R. group w7x. Quantization of the case Cy can be done in the same way. The corresponding Hilbert space Hy is again the Fock space, but we have to make the following replacements: Jˆ"™ Jˆ. and Jˆ0 ™ yJˆ0 . As a result we get the representation Dy 1. The Hilbert space of the whole system is H s Hq[ Hy and the corresponding representation Dq 1 [ Dy describes the SO ≠ Ž2.1. symmetry of the 1 quantum system. It is interesting to mention the following: At the classical level our system can be obtained from the massive case in the limit m 0 ™ 0 Žwhere m 0 is a particle mass. and by removing the singular point of the physical phase-space. The quantum theory of the massive case was considered recently in Ref. w1x. The corresponding representation is defined by the operators Jˆ0 cn s n cn ,
Jˆq cn s 'n 2 q n q a 2 cnq1 ,
Jˆy cn s 'n 2 y n q a 2 cny1 ,
Ž 2.16 .
where cn :s exp in f Ž n g Z . form the basis of the Hilbert space L2 Ž S 1 . and a s m 0rm. For a s 0 Ž m 0 y s 0. this representation terns into Dq 1 [ A [ D1 , where A is a one-dimensional trivial representation
G. Jorjadze, W. Piechockir Physics Letters B 448 (1999) 203–208
206
on the vector c 0 . By removing A we get the quantum theory of the massless case. The momentum of a quantum particle pˆu s mJˆ0 can take only discrete values Pn s mn, where n is a nonzero integer.
The mass-shell condition Ž1.2. leads to pqs 0 for P ) 0 and pys 0 for P - 0. Due to Ž3.4. and the mass-shell condition, we have K 2 y PM s 0
Ž 3.6 .
and the trajectories read x1 y e Ž P . x 0 s
3. Dynamics on half-plane Let us consider the Liouville field w s y2log m < x 0 <, given on a plane Ž x 0 , x 1 .. This field defines constant spacetime curvature R 0 s y2 m2 Žsee Ž2.3... The Lagrangian Ž1.1. in this case reads Lsy
2 x˙qx˙y
l m2 Ž xqq xy .
2
,
Ž 3.1 .
where x ":s x 0 " x 1. It is assumed that x˙ 0 ) 0, which leads to pqq py- 0. Formally, Ž3.1. is invariant under the fractionallinear transformations xq™
axqq b cxqq d
xy™
,
axyy b ycxyq d
,
ad y bc s 1.
Ž 3.2 . Thus, formally, SLŽ2.R.rZ 2 Žwhich is isomorphic to SO ≠ Ž2.1.. is the symmetry of our system. The transformations Ž3.2. are well defined on the plane only for c s 0. The corresponding transformations with c s 0 form the group of dilatations and translations Žalong x 1 ., which is a global symmetry of the considered spacetime. The infinitesimal transformations for Ž3.2. are x "™ x "" a 0 ,
where
eŽ P. s
P
.
Ž 3.7 .
It is clear that x 0 s 0 is the singularity line in the spacetime. In the massive case this singularity leads to the dynamical ambiguities w2,8x. However, the dynamics of the massless particle is defined uniquely due to Ž3.7.. The physical phase-space is defined by two cones Ž3.6. without the line corresponding to P s 0. Thus, we have two disconnected parts: Pq with P ) 0 and Py with P - 0. Both cones Ž Pq and Py . are invariant under dilatations and translations generated by the dynamical integrals K and P. But, the transformations generated by M are not defined globally. Thus, the physical phase-space has the same symmetry as the spacetime. Let us quantize the system corresponding to Pq case Žthe case Py can be done in the same way.. We parametrize Pq as follows w9x P s p,
K s pq,
M s pq 2 ,
Ž 3.8 .
where Ž p,q . are the coordinates on half-plane with p ) 0. The canonical commutation relation p,q4 s 1 provides the commutation relations of sl Ž2.R. algebra
P,K 4 sP, P,M 4 s2 K , K ,M 4 sM . Ž 3.9 .
2
Ž 3.3 .
and the corresponding dynamical integrals read K s pq xqq py xy,
2
2
M s pq Ž xq . y py Ž xy . ,
Ž 3.4 .
"
where p "s E LrE x˙ . The dynamical integrals Ž3.4. satisfy again the commutation relations Ž2.9. with 1 2
P
,
x "™ x "q a 1 x ",
x "™ x "" a 2 Ž x " . P s pqy py ,
K
J0 s Ž P q M . ,
1 2
J1 s Ž P y M . ,
J2 s K .
Ž 3.5 .
In the ‘p-representation’ Ž qˆ s i Ep . we choose the operator ordering for Kˆ and Mˆ by the following requirements: Ža. Kˆ and Mˆ are self-adjoint, ˆ Kˆ and Mˆ satisfy the commutation relaŽb. P, tions corresponding to Ž3.9., ˆ ˆq Žc. the Casimir number operator Cˆ s 1r2Ž PM ˆ ˆ . y Kˆ 2 equals zero. MP As a result we get Pˆ s p,
Kˆ s i Ž pEp q 12 . ,
Mˆ s ypEp2 y Ep q
1
. 4p Ž 3.10 .
G. Jorjadze, W. Piechockir Physics Letters B 448 (1999) 203–208
The operators defined by Ž3.10. have continuous spectrum. The operator Jˆ0 s Ž Pˆ q Mˆ .r2 has the discrete spectrum Jˆ0 cn Ž p . s l n cn Ž p . ,
l n s n q 1, where by w10x
L1nŽ x .
L1n Ž x . s
Ž 3.12 .
are the Laguerre polynomials defined
Ý Ž y.
n! Ž n q 1 . PPP Ž k q 2 . k k! Ž n y k . !
ks0 n
J0 s y J1 s y
n G 0,
ny1
lead to the dynamical integrals
Ž 3.11 .
cn Ž p . s 'p exp Ž yp . L1n Ž 2 p . ,
n
q Ž y. x .
207
J2 s y
pt m pt m pt m
, cos mx cos mt q cos mx sin mt y
px m px m
sin mx sin mt , sin mx cos mt ,
which satisfy the commutation relations Ž2.9.. The mass-shell condition pt2 s p x2 provides
xk
J02 y J12 y J22 s 0,
Ž 3.13 .
Eqs. Ž3.11.-Ž3.13. show that our representation is Ž Ž . unitarily equivalent to Dq 1 representation see 2.14 and Ž2.15... The scheme presented here can be generalized to include other representations of the discrete series of SLŽ2.R. group w9x.
4. Dynamics on stripe Let us consider the spacetime to be a stripe
Ž 4.1 .
with the conformal metric tensor Ž2.2. and the Liouville field
w Ž t , x . s ylog sin2 mx.
Ž 4.2 .
It defines the spacetime with constant positive curvature R s 2 m 2 . The Lagrangian Ž1.1. in this case reads Lsy
t˙2 y x˙ 2 2 lsin2 mx
Ž 4.7 .
Due to Ž4.6., the trajectories of massless particle Ž4.7. are zigzag lines with the slope x˙ s "1. A particle with the velocity equal to one reaches the ‘edge’ of space where it ‘changes’ the direction. Then, it reaches another ‘edge’, again changes the direction, etc. This motion is periodic with the period 2prm. Since the physical phase-space is the upper-cone without the vertex, the system can be quantized as in the case of Cq system, presented in Section 2. The corresponding quantum theory is defined by the Dq 1 representation.
Ž 4.3 .
and it is invariant under the action of the universal & covering group SLŽ2.R.. The corresponding infinitesimal transformations
Ž t , x . ™ Ž t y a 0rm, x . , Ž t , x . ™ Ž t y a 1rm cos mx cos mt , x q a 1rm sin mx sin mt . ,
Ž t , x . ™ Ž t y a 2rm cos mx sin mt , x y a 2rm sin mx cos mt .
Ž 4.6 .
which defines two cones. The physical conditions t˙) 0 and l ) 0 give pt - 0. Thus, the upper-cone Ž J0 ) 0. without the vertex J0 s 0 s J1 s J2 is the physical phase-space of our system. By Ž4.5. we get J0 cos mx s J1cos mt y J2 sin mt.
S :s Ž t , x . < t g R, x g x 0,prm w 4 ,
Ž 4.5 .
Ž 4.4 .
5. Discussion Any two Lorentzian 2d manifolds with the same constant curvature R 0 are Žlocally. isometric. Therefore, the considered systems give the exhaustive picture for the dynamics of massless particle in 2d spacetime with constant curvature. It is interesting to note that these mechanical systems are related to the model of 2d gravity with the dilaton field w11x. In the conformal gauge this 2d gravity model is described by the Liouville and free
208
G. Jorjadze, W. Piechockir Physics Letters B 448 (1999) 203–208
fields, which have equal traceless energy-momentum tensors. One can check that the massless field, which has the same traceless energy-momentum tensor as the Liouville field Ž4.2., is singular and the singularity lines coincide with the trajectories of the massless particle given by Ž4.7.. A similar picture exists for the hyperboloid Ž2.1. described by the Liouville field Ž2.5.. The Hamiltonian reduction to the physical variables eliminates all field degrees of freedom of the indicated model of 2d gravity and the physical phase-space of the model is finite dimensional. The character of the reduced system and its relation to the considered dynamics of massless particle depends on the global properties of spacetime. Details concerning this relation will be discussed elsewhere.
Acknowledgements This work was supported by the grants from: INTAS Ž96-0482., RFBR Ž96-01-00344., the Georgian Academy of Sciences and the Sołtan Institute for Nuclear Studies.
We wish to thank Mikhail Plyushchay for correspondence.
References w1x G. Jorjadze, W. Piechocki, Geometry of 2d spacetime and quantization of particle dynamics, gr-qcr9811094. w2x G. Jorjadze, W. Piechocki, hep-thr9709059, Theor. Math. Phys. 118 Ž1999., in press. w3x L. Brink, P. Di Vecchia, P. Howe, Nucl. Phys. B 118 Ž1977. 76. w4x P.A.M. Dirac, Lectures on quantum mechanics, Yeshiva University Press, New York, 1964. w5x J.A. Wolf, Spaces of constant curvature, Publish or Perish, INC, 1974. w6x J. Liouville, J. Math. Pures Appl. 18 Ž1853. 71. w7x D.P. Zhelobenko, A.I. Stern, Representations of Lie groups, Nauka, Moscow, 1983. w8x G. Jorjadze, W. Piechocki, Class. Quantum Gravity 15 Ž1998. L41. w9x M. Plyushchay, J. Math. Phys. 34 Ž1993. 3954. w10x A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and series, vol. 2, Nauka, Moscow, 1983. w11x N. Seiberg, in: O. Alvarez et al. ŽEds.., Rundom surfaces and quantum gravity, Plenum Press, New York, 1991, p. 363.
Physics Letters B 448 Ž1999. 203–208
Massless particle in 2d spacetime with constant curvature George Jorjadze a , Włodzimierz Piechocki
b,1
a
b
Razmadze Mathematical Institute, 380093 Tbilisi, Georgia Theory DiÕision (Zd.P8), Sołtan Institute for Nuclear Studies, 00-681 Warsaw, Poland Received 7 January 1999 Editor: P.V. Landshoff
Abstract We consider dynamics of massless particle in 2d spacetimes with constant curvature. We analyze different examples of spacetime. Dynamical integrals are constructed from spacetime symmetry related to sl Ž2.R. algebra. Mass-shell condition restricts dynamical integrals to a cone Žwithout vertex. which defines physical-phase space. We parametrize the cone by canonical coordinates. Canonical quantization with definite choice of operator ordering leads to unitary irreducible representations of SO ≠ Ž2.1. group. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 04.60.M; 02.20.S; 02.40.K Keywords: 2d spacetime with constant curvature; Dynamical integrals; Canonical quantization; Representations of SO ≠ Ž2.1. group
1. Introduction We analyze classical and quantum dynamics of a massless particle in two dimensional spacetime with constant curvature R 0 / 0. We consider the cases when spacetime is: Ži. hyperboloid with R 0 - 0, Žii. half-plane with R 0 - 0 and Žiii. stripe with R 0 ) 0. Presented results complete our analysis of dynamics of a particle with non-zero mass m 0 in 2d curved spacetime w1,2x. Taking formally m 0 ™ 0 leads to some singularities both at classical and quantum levels. Thus, the massless case needs separate treatment. As the result we get clear picture of the role played by topology and global symmetries of spacetime in the procedure of canonical quantization.
Dynamics of a massless particle in gravitational field gmn Ž x 0 , x 1 . is defined by the action w3x S s L Ž t . dt ,
H
L Ž t . :s y
1 2 lŽ t .
gmn Ž x 0 Ž t . , x 1 Ž t . .
=x˙ m Ž t . x˙ n Ž t . ,
where t is an evolution parameter, l plays the role of Lagrange multiplier and x˙ m :s dx mrdt . It is assumed that l ) 0 and x˙ 0 ) 0. The action Ž1.1. is invariant under reparametrization t ™ f Žt ., lŽt . ™ lŽt .rf˙Žt .. This gauge symmetry leads to the constrained dynamics in the Hamiltonian formulation w4x. The constraint reads
F :s g mn pm pn s 0,
Ž 1.2 . m
1
E-mail: [email protected]
Ž 1.1 .
where pm :s E LrE x˙ are canonical momenta Žwe use units with c s 1 s " ..
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 0 6 0 - X
G. Jorjadze, W. Piechockir Physics Letters B 448 (1999) 203–208
204
As in Refs. w1,2x we use the gauge invariant description. In the case of massive particle the set of spacetime trajectories can be considered as a physical phase-space of the system. This set has natural symplectic structure, which can be used for quantization. The spacetime trajectories of a massless particle has no symplectic structure Žthis set is only one dimensional, since particle velocity is fixed.. For the gauge invariant description we use the dynamical integrals constructed from the global symmetries of spacetime.
where the spacetime coordinates x 0 and x 1 are identified with the parameters r and u , respectively. The function Ž2.5. satisfies the Liouville Eq. Ž2.3. for R 0 s y2 m2 . The Lagrangian Ž1.1. in this case reads
2. Dynamics on hyperboloid
Ž r ,u . ™ Ž r ,u q a 0rm . ,
Let Ž y 0 , y 1 , y 2 . be the standard coordinates on 3d Minkowski space with the metric tensor ha b s diag Žq,y,y .. A one-sheet hyperboloid H is defined by
Ž r ,u . ™ Ž r y a 1rm sin m r sin m u ,
2
2
2
y Ž y 0 . q Ž y 1 . q Ž y 2 . s my2 ,
Lsy
r˙ 2 y u˙ 2 2 lsin2 m r
.
Ž 2.6 .
The hyperboloid Ž2.1. is invariant under the Lorentz transformations, i.e., SO ≠ Ž2,1. is the symmetry group of our system. The corresponding infinitesimal transformations Žrotation and two boosts. are
u q a 1rm cos m r cos mu . ,
Ž r ,u . ™ Ž r q a 2rm sin m r cos mu , u q a 2rm cos m r sin mu . .
Ž 2.1 .
Ž 2.7 .
The dynamical integrals for Ž2.7. read where m ) 0 is a fixed parameter. H has a constant curvature R s y2 m 2 Žsee w5x.. Any 2-dimensional Lorentzian manifold with constant curvature R 0 can be described Žlocally. by the conformal metric tensor w5x gmn Ž X . s exp w Ž X .
ž
1 0
0 , y1
/
Ž 2.2 . where the field w Ž X . satisfies the Liouville equation w6x
Ž 2.3 .
y sy where
cot m r m
,
1
y s
r g x 0,prm w ,
cos mu msin m r
,
2
y s
u g w 0,2 prm w
sin mu msin m r
,
Ž 2.4 .
pr m
, pr m
sin m r sin mu q
sin m r cos mu q
Ž 2.5 .
pu m
pu m
cos m r cos mu ,
cos m r sin mu ,
Ž 2.8 .
where pu :s E LrEu˙, pr :s E LrEr˙ are canonical momenta. Since J0 is connected with space translations Žsee Ž2.7.., it defines particle momentum pu s mJ0 . It is clear that the dynamical integrals Ž2.8. satisfy the commutation relations of sl Ž2.R. algebra
Ž 2.9 .
where h c d is the Minkowski metric tensor and ´ a b c is the anti-symmetric tensor with ´ 012 s 1. The mass shell condition Ž1.2. takes the form pr2 y pu2 s 0 and it leads to the relation J02 y J12 y J22 s 0.
we get the conformal form Ž2.2., with
w s ylog sin2 m r ,
m
Ja , J b 4 s ´ a b ch c d Jd ,
Making use of the parametrization 0
pu
J1 s y J2 s
X :s Ž x 0 , x 1 . ,
Ž E 02 y E 12 . w Ž X . q R 0 exp w Ž X . s 0.
J0 s
Ž 2.10 .
Eq. Ž2.10. defines two cones. The singular point of the cones J0 s 0 s J1 s J2 should be removed, since
G. Jorjadze, W. Piechockir Physics Letters B 448 (1999) 203–208
it corresponds to the massless particle with zero momentum Ž pr s 0 s pu ., which does not exist. Thus, the dynamical integrals Ž2.8. define the physical phase-space of the system and it consists of two disconnected cones Cq and Cy, for J0 ) 0 and J0 - 0, respectively. According to Ž2.8. and due to pr - 0 Žsince r˙ ) 0 and l ) 0. the trajectories satisfy the equations Ja y a s 0,
J1 y 2 y J 2 y 1 s
pr m
2
sy
< J0 < m
.
Ž 2.11 .
Each point Ž Ja . of the cone Cq or Cy defines uniquely the trajectory on the hyperboloid H. The trajectories Ž2.11. are straight lines in 3d Minkowski space. Hence, the set of trajectories is the set of generatrices of the hyperboloid Ž2.1.. For J0 ) 0 we get the ‘right’ moving particle with u˙) 0, while for J0 - 0 we have the ‘left’ moving one, with u˙- 0. Both cones, Cq and Cy, are invariant under SO ≠ Ž2.1. transformations. To quantize the system, we consider the cones Cq and Cy separately. We parametrize Cq as follows p J0 s 12 Ž p 2 q q 2 . , J1 s p2 q q2 , 2 q J2 s p2 q q2 , Ž 2.12 . 2 where Ž0,0. / Ž p,q . g R 2 . It is easy to see that Ž2.12. defines the one-to-one map from the plane without the origin to Cq. The canonical commutation relation p,q4 s 1 provides Ž2.9.. For quantization of Cq system we introduce the creation-annihilation operators a ":s Ž pˆ " iqˆ .r '2 and choose the definite operator ordering in Ž2.12.. This ordering is defined by the following requirements: Ža. the operators Jˆa are self-adjoint, Žb. they generate global SO ≠ Ž2.1. transformations, Žc. the spectrum of Jˆ0 is positive, ˆ Jˆ02 y Jˆ12 y Žd. the Casimir number operator C:s 2 ˆ J2 is zero. This leads to the following expressions
(
(
Jˆ0 s ay aq,
Jˆqs aq 'ay aqq 1 ,
Jˆys 'ay aqq 1 ay,
Ž 2.13 .
205
where Jˆ"s Jˆ1 " Jˆ2 . The creation and annihilation operators are defined in the Fock space in the standard way aq < n: s 'n q 1 < n q 1: ,
ay < n: s 'n < n y 1: ,
where the vectors < n:, Ž n G 0. form the basis of the corresponding Hilbert space Hq. The states < n: are the eigenstates of Jˆ0 Jˆ0 < n: s Ž n q 1 . < n: ,
Ž 2.14 .
and from Ž2.13. we get Jˆq < n: s Ž n q 1 . Ž n q 2 . < n q 1: ,
(
Jˆy < n: s n Ž n q 1 . < n y 1: .
(
Ž 2.15 .
Eqs. Ž2.13. – Ž2.15. define the unitary irreducible representation ŽUIR. of the group SO ≠ Ž2.1.. We identify this representation with the representation of Dq 1 of the discrete series of SLŽ2.R. group w7x. Quantization of the case Cy can be done in the same way. The corresponding Hilbert space Hy is again the Fock space, but we have to make the following replacements: Jˆ"™ Jˆ. and Jˆ0 ™ yJˆ0 . As a result we get the representation Dy 1. The Hilbert space of the whole system is H s Hq[ Hy and the corresponding representation Dq 1 [ Dy describes the SO ≠ Ž2.1. symmetry of the 1 quantum system. It is interesting to mention the following: At the classical level our system can be obtained from the massive case in the limit m 0 ™ 0 Žwhere m 0 is a particle mass. and by removing the singular point of the physical phase-space. The quantum theory of the massive case was considered recently in Ref. w1x. The corresponding representation is defined by the operators Jˆ0 cn s n cn ,
Jˆq cn s 'n 2 q n q a 2 cnq1 ,
Jˆy cn s 'n 2 y n q a 2 cny1 ,
Ž 2.16 .
where cn :s exp in f Ž n g Z . form the basis of the Hilbert space L2 Ž S 1 . and a s m 0rm. For a s 0 Ž m 0 y s 0. this representation terns into Dq 1 [ A [ D1 , where A is a one-dimensional trivial representation
G. Jorjadze, W. Piechockir Physics Letters B 448 (1999) 203–208
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on the vector c 0 . By removing A we get the quantum theory of the massless case. The momentum of a quantum particle pˆu s mJˆ0 can take only discrete values Pn s mn, where n is a nonzero integer.
The mass-shell condition Ž1.2. leads to pqs 0 for P ) 0 and pys 0 for P - 0. Due to Ž3.4. and the mass-shell condition, we have K 2 y PM s 0
Ž 3.6 .
and the trajectories read x1 y e Ž P . x 0 s
3. Dynamics on half-plane Let us consider the Liouville field w s y2log m < x 0 <, given on a plane Ž x 0 , x 1 .. This field defines constant spacetime curvature R 0 s y2 m2 Žsee Ž2.3... The Lagrangian Ž1.1. in this case reads Lsy
2 x˙qx˙y
l m2 Ž xqq xy .
2
,
Ž 3.1 .
where x ":s x 0 " x 1. It is assumed that x˙ 0 ) 0, which leads to pqq py- 0. Formally, Ž3.1. is invariant under the fractionallinear transformations xq™
axqq b cxqq d
xy™
,
axyy b ycxyq d
,
ad y bc s 1.
Ž 3.2 . Thus, formally, SLŽ2.R.rZ 2 Žwhich is isomorphic to SO ≠ Ž2.1.. is the symmetry of our system. The transformations Ž3.2. are well defined on the plane only for c s 0. The corresponding transformations with c s 0 form the group of dilatations and translations Žalong x 1 ., which is a global symmetry of the considered spacetime. The infinitesimal transformations for Ž3.2. are x "™ x "" a 0 ,
where
eŽ P. s
P
.
Ž 3.7 .
It is clear that x 0 s 0 is the singularity line in the spacetime. In the massive case this singularity leads to the dynamical ambiguities w2,8x. However, the dynamics of the massless particle is defined uniquely due to Ž3.7.. The physical phase-space is defined by two cones Ž3.6. without the line corresponding to P s 0. Thus, we have two disconnected parts: Pq with P ) 0 and Py with P - 0. Both cones Ž Pq and Py . are invariant under dilatations and translations generated by the dynamical integrals K and P. But, the transformations generated by M are not defined globally. Thus, the physical phase-space has the same symmetry as the spacetime. Let us quantize the system corresponding to Pq case Žthe case Py can be done in the same way.. We parametrize Pq as follows w9x P s p,
K s pq,
M s pq 2 ,
Ž 3.8 .
where Ž p,q . are the coordinates on half-plane with p ) 0. The canonical commutation relation p,q4 s 1 provides the commutation relations of sl Ž2.R. algebra
P,K 4 sP, P,M 4 s2 K , K ,M 4 sM . Ž 3.9 .
2
Ž 3.3 .
and the corresponding dynamical integrals read K s pq xqq py xy,
2
2
M s pq Ž xq . y py Ž xy . ,
Ž 3.4 .
"
where p "s E LrE x˙ . The dynamical integrals Ž3.4. satisfy again the commutation relations Ž2.9. with 1 2
P
,
x "™ x "q a 1 x ",
x "™ x "" a 2 Ž x " . P s pqy py ,
K
J0 s Ž P q M . ,
1 2
J1 s Ž P y M . ,
J2 s K .
Ž 3.5 .
In the ‘p-representation’ Ž qˆ s i Ep . we choose the operator ordering for Kˆ and Mˆ by the following requirements: Ža. Kˆ and Mˆ are self-adjoint, ˆ Kˆ and Mˆ satisfy the commutation relaŽb. P, tions corresponding to Ž3.9., ˆ ˆq Žc. the Casimir number operator Cˆ s 1r2Ž PM ˆ ˆ . y Kˆ 2 equals zero. MP As a result we get Pˆ s p,
Kˆ s i Ž pEp q 12 . ,
Mˆ s ypEp2 y Ep q
1
. 4p Ž 3.10 .
G. Jorjadze, W. Piechockir Physics Letters B 448 (1999) 203–208
The operators defined by Ž3.10. have continuous spectrum. The operator Jˆ0 s Ž Pˆ q Mˆ .r2 has the discrete spectrum Jˆ0 cn Ž p . s l n cn Ž p . ,
l n s n q 1, where by w10x
L1nŽ x .
L1n Ž x . s
Ž 3.12 .
are the Laguerre polynomials defined
Ý Ž y.
n! Ž n q 1 . PPP Ž k q 2 . k k! Ž n y k . !
ks0 n
J0 s y J1 s y
n G 0,
ny1
lead to the dynamical integrals
Ž 3.11 .
cn Ž p . s 'p exp Ž yp . L1n Ž 2 p . ,
n
q Ž y. x .
207
J2 s y
pt m pt m pt m
, cos mx cos mt q cos mx sin mt y
px m px m
sin mx sin mt , sin mx cos mt ,
which satisfy the commutation relations Ž2.9.. The mass-shell condition pt2 s p x2 provides
xk
J02 y J12 y J22 s 0,
Ž 3.13 .
Eqs. Ž3.11.-Ž3.13. show that our representation is Ž Ž . unitarily equivalent to Dq 1 representation see 2.14 and Ž2.15... The scheme presented here can be generalized to include other representations of the discrete series of SLŽ2.R. group w9x.
4. Dynamics on stripe Let us consider the spacetime to be a stripe
Ž 4.1 .
with the conformal metric tensor Ž2.2. and the Liouville field
w Ž t , x . s ylog sin2 mx.
Ž 4.2 .
It defines the spacetime with constant positive curvature R s 2 m 2 . The Lagrangian Ž1.1. in this case reads Lsy
t˙2 y x˙ 2 2 lsin2 mx
Ž 4.7 .
Due to Ž4.6., the trajectories of massless particle Ž4.7. are zigzag lines with the slope x˙ s "1. A particle with the velocity equal to one reaches the ‘edge’ of space where it ‘changes’ the direction. Then, it reaches another ‘edge’, again changes the direction, etc. This motion is periodic with the period 2prm. Since the physical phase-space is the upper-cone without the vertex, the system can be quantized as in the case of Cq system, presented in Section 2. The corresponding quantum theory is defined by the Dq 1 representation.
Ž 4.3 .
and it is invariant under the action of the universal & covering group SLŽ2.R.. The corresponding infinitesimal transformations
Ž t , x . ™ Ž t y a 0rm, x . , Ž t , x . ™ Ž t y a 1rm cos mx cos mt , x q a 1rm sin mx sin mt . ,
Ž t , x . ™ Ž t y a 2rm cos mx sin mt , x y a 2rm sin mx cos mt .
Ž 4.6 .
which defines two cones. The physical conditions t˙) 0 and l ) 0 give pt - 0. Thus, the upper-cone Ž J0 ) 0. without the vertex J0 s 0 s J1 s J2 is the physical phase-space of our system. By Ž4.5. we get J0 cos mx s J1cos mt y J2 sin mt.
S :s Ž t , x . < t g R, x g x 0,prm w 4 ,
Ž 4.5 .
Ž 4.4 .
5. Discussion Any two Lorentzian 2d manifolds with the same constant curvature R 0 are Žlocally. isometric. Therefore, the considered systems give the exhaustive picture for the dynamics of massless particle in 2d spacetime with constant curvature. It is interesting to note that these mechanical systems are related to the model of 2d gravity with the dilaton field w11x. In the conformal gauge this 2d gravity model is described by the Liouville and free
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fields, which have equal traceless energy-momentum tensors. One can check that the massless field, which has the same traceless energy-momentum tensor as the Liouville field Ž4.2., is singular and the singularity lines coincide with the trajectories of the massless particle given by Ž4.7.. A similar picture exists for the hyperboloid Ž2.1. described by the Liouville field Ž2.5.. The Hamiltonian reduction to the physical variables eliminates all field degrees of freedom of the indicated model of 2d gravity and the physical phase-space of the model is finite dimensional. The character of the reduced system and its relation to the considered dynamics of massless particle depends on the global properties of spacetime. Details concerning this relation will be discussed elsewhere.
Acknowledgements This work was supported by the grants from: INTAS Ž96-0482., RFBR Ž96-01-00344., the Georgian Academy of Sciences and the Sołtan Institute for Nuclear Studies.
We wish to thank Mikhail Plyushchay for correspondence.
References w1x G. Jorjadze, W. Piechocki, Geometry of 2d spacetime and quantization of particle dynamics, gr-qcr9811094. w2x G. Jorjadze, W. Piechocki, hep-thr9709059, Theor. Math. Phys. 118 Ž1999., in press. w3x L. Brink, P. Di Vecchia, P. Howe, Nucl. Phys. B 118 Ž1977. 76. w4x P.A.M. Dirac, Lectures on quantum mechanics, Yeshiva University Press, New York, 1964. w5x J.A. Wolf, Spaces of constant curvature, Publish or Perish, INC, 1974. w6x J. Liouville, J. Math. Pures Appl. 18 Ž1853. 71. w7x D.P. Zhelobenko, A.I. Stern, Representations of Lie groups, Nauka, Moscow, 1983. w8x G. Jorjadze, W. Piechocki, Class. Quantum Gravity 15 Ž1998. L41. w9x M. Plyushchay, J. Math. Phys. 34 Ž1993. 3954. w10x A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and series, vol. 2, Nauka, Moscow, 1983. w11x N. Seiberg, in: O. Alvarez et al. ŽEds.., Rundom surfaces and quantum gravity, Plenum Press, New York, 1991, p. 363.