ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
SUPERINTEGRABILITY ON N-DIMENSIONAL SPACES OF CONSTANT CURVATURE FROM so(N +1) AND ITS CONTRACTIONS
© 2008 F. J. Herranz1)*, A. Ballesteros2)**
Received August 20, 2007
The Lie—Poisson algebra so(N + 1) and some of its contractions are used to construct a family of superintegrable Hamiltonians on the N-dimensional spherical, Euclidean, hyperbolic, Minkowskian, and (anti-)de Sitter spaces. We firstly present a Hamiltonian which is a superposition of an arbitrary central potential with N arbitrary centrifugal terms. Such a system is quasi-maximally superintegrable since this is endowed with 2N — 3 functionally independent constants of the motion (plus the Hamiltonian). Secondly, we identify two maximally superintegrable Hamiltonians by choosing a specific central potential and finding at the same time the remaining integral. The former is the generalization of the Smorodinsky—Winternitz system to the above six spaces, while the latter is a generalization of the Kepler—Coulomb potential, for which the Laplace—Runge—Lenz N-vector is also given. All the systems and constants of the motion are explicitly expressed in a unified form in terms of ambient and polar coordinates as they are parametrized by two contraction parameters (curvature and signature of the metric).
PACS:02.30.Ik; 02.20.Sv; 02.40.Ky
1. INTRODUCTION Let us consider the following potential on the N-dimensional (ND) Euclidean space [ 1]:
N
и
F(r) + E
i=1
fk X2 '
(1)
where 3i are arbitrary real constants, xi are Cartesian coordinates, and F(r) is an arbitrary smooth function depending on the Euclidean distance r =
n a1'2
A=l xi I
This potential is known to be su-
perintegrable and can be interpreted as the superposition of a central term with N centrifugal barriers associated to the j3i's. Furthermore, two particular choices of F(r) provide well-known maximally su-perintegrable (MS) Euclidean Hamiltonians:
If F(r) = u2r2, we obtain an isotropic harmonic oscillator with angular frequency u. The N arbitrary centrifugal terms can be added to the oscillator potential keeping maximal superintegrability; the resulting potential is the Smorodinsky—Winternitz (SW) system [2-5].
If F(r) = -k/r, where k is a real constant, we find the Kepler-Coulomb (KC) potential. In this
'■'Departamento de Fisica, Escuela Politecnica Superior, Universidad de Burgos, Spain.
2)Departamento de Fi sica, Facultad de Ciencias, Universidad de Burgos, Spain. E-mail: fjherranz@ubu.es E-mail: angelb@ubu.es
case, only a maximum number of (N — 1) centrifugal terms can be considered in order to preserve maximal superintegrability [1,6, 7].
Our aim in this paper is to present a unified generalization of the superintegrable potential (1) and its two particular MS cases to the ND spherical, Euclidean, hyperbolic, Minkowskian, and both de Sitter spaces. The approach we shall make use of is based on the Lie—Poisson algebras associated to the Lie groups and subgroups involved in the construction of the above spaces as symmetrical homogeneous ones. Thus, in the next section we introduce the basics on the Lie groups of isometries on these six ND spaces together with the two coordinate systems we shall deal with: (N + 1) ambient coordinates in an auxiliary linear space and N intrinsic geodesic polar (spherical) coordinates. The kinetic energy which gives rise to the geodesic motion is studied in Section 3 by starting from the metric. The generalization of the potential (1) is addressed in Section 4 in such a manner that general and global expressions for the Hamiltonian and its 2N — 3 functionally independent integrals of motion are explicitly given. Finally, the last section is devoted to the study of the two MS Hamiltonians arising in the above family by choosing in an adequate way the radial function and by finding at the same time the remaining constant of the motion. Consequently, we obtain the generalization of the SW and generalized KC potentials for any value of the curvature and signature of the metric.
931
9
*
We remark that these results generalized to arbitrary contractions of so(N + 1), where m and K2 are two dimension N the 3D case recently studied in [8].
real contraction parameters. The non-vanishing Lie
2. RIEMANNIAN SPACES AND RELATIVISTIC SPACE-TIMES brackets of so^(N + 1) in the basis spanned by
Let us consider a set of real Lie algebras soK1, K2 (N + 1) which come from ZfN graded } (p,v = 0,1,...,N; f < v ) read [9]:
[Jij, Jik ] - — Jjk, [Jij ,Jjk] = Jik, [Jik, Jjk\ = Jij,
[J1 j, J1k] = K2Jjk, [J1j ,Jjk] = - J1 k , [J1k, Jjk] = J1j,
[J01, J0k] = K1 J1k, [J01, J1k] = -J0k, [J0k, J1k] = K2 J01,
[J0j, J0k] = K1K2Jjk, [J0j ,Jjk] = -J0k, [J0k, Jjkk] = J0j,
(2)
where i,j,k = 2,...,N and i < j < k. Both contraction parameters k (l = 1,2) can take any real value. By scaling the Lie generators, each k can be reduced to either +1, 0, or —1; the limit k ^ 0 is equivalent to apply an InonU—Wigner contraction.
The quadratic Casimir for soK1,K2(N + 1), associated to the Killing—Cartan form, is given by
N
N
N
C = K2 Jqi + J02j + Jfj + Kl K2 J,
j=2
j=2
i,j=2
S[Ki]K2 = soki,K2 (N + 1)/soK2 (N ),
can be written as k1 = ±1/R2, where R is the radius of the space (R for the Euclidean case).
When k2 is negative, we find a Lorentzian metric corresponding to relativistic space-times; namely, the anti-de Sitter (ki > 0), Minkowskian (ki = 0), and de Sitter spaces (ki < 0): Sj+i =
[+]-
1) SN]- =
(3)
Next from the Lie group SOK1,K2 (N + 1) with Lie algebra (2) we construct the following ND symmetrical homogeneous space: j
(4)
SOK2 (N ) = J ; i,j = 1,...,N).
The parameter m turns out to be the constant sectional curvature of the space, while k2 determines the signature of the metric as diag(+1, k2,...,k2).In this way, we find that S|K1 comprises well-known spaces of constant curvature:
When k2 is positive, say k2 = +1, we recover the three classical Riemannian spaces. These are the spherical (ki > 0), Euclidean (ki = 0), and hyperbolic spaces (ki < 0): S|+]+ = SO(N +
+ 1)/SO(N), Sj0]+ = ISO(N)/SO(N), and = = SO(N, 1)/SO( N ). The first two rows of (2) s pan the rotation Lie subalgebra so(N), while the N generators joî (i = 1,...,N) appearing in the last two rows play the role of translations. The curvature
= SO(N — 1, 2)/SO(N — 1, = ISO(N — 1, 1)/SO(N — 1, 1), and Sjj_ = = SO(N, 1)/SO(N — 1,1). The generators J01, J0j, Jij and Jij (i,j = 2,..., N) are identified with time translation, space translations, boosts and spatial rotations, respectively. The first row of (2) is a rotation subalgebra so(N — 1) and the two first rows span the Lorentz subalgebra so(N — 1,1). The two contraction parameters can be expressed as k1 = ±1/t2, where t is the (time) universe radius, and k2 = —1/c2, where c is the speed of light.
Finally, the contraction k2 = 0 gives rise to Newtonian (nonrelativistic) space-times with a degenerate metric. Since we shall construct superintegrable systems on Sj]K2, for which the kinetic energy is provided by the metric, hereafter we assume k2 = 0.
In what follows, we introduce an explicit model of the space S^]K2 in terms of (N + 1) ambient coordinates and also of N intrinsic geodesic quantities.
The vector representation of soKlK2(N + 1) is given by (N + 1) x (N + 1) real matrices [9]:
J01 = —Kieoi + eio, Joj = —KiK2eoj + j (5) J1j = —K2e1j + ej1, Jjk = ejk + ekj,
j,k = 2,
,N,
where eij is the matrix with entries (eij)lm = Si5j™. Any generator X e soK1>K2 (N + 1) fulfils
XT Ik + IkX = 0, (6)
Ik = diag(+1, m, Ki K2, ..., Kl K2),
so that any element G e SOKlK2(N + 1) verifies GTIkG = Ik. Then SOkuk2(N + 1) is a group of isometries of IK acting on a linear ambient space Rn+1 = (xo,xi ,...,xN) through matrix multiplication. The origin O in §NKl]K2 has ambient coordinates O = (1,0,..., 0) which is invariant under the (Lorentz) rotation subgroup SOK2 (N) (4). The orbit of O corresponds to the homogeneous space S^]K2 which is contained in the "sphere" provided by IK:
N
N
£ = x0 + k1x2 + k1 k2 ^ x2 = 1.
(7)
j=2
The (N + 1) ambient coordinates x = = (x0,x1,... ,xN), subjected to (7), are also called Weierstrass coordinates. The metric on §N ,
[Kl]K2
follows from the flat ambient metric in MN+1 in the form
1
N
dsz = — I dxo + n\dx\ + H\H2 7 dx K1\ j=
(8)
A differential realization of soK1, K2 (N + 1) (2), coming directly from the vector representation (5), is given by
J01 = k 1 x 1 do — xod1, (9)
Joj = K1K2xj do — xodj,
J1j = K2xj d1 — x1dj, Jjk = xk dj — xj dk,
where j,k = 2,...,N and d^ = d/dx
Next we parametrize the (N + 1) ambient coordinates x of a generic point P in terms of N intrinsic quantities (r, 0,fa3,..., ) on the space §N]K2 through the following action of N one-parametric subgroups of SOK(N + 1) on the origin O:
x = exp(^N Jn-1 n) x
(10)
x exp(0N-1 Jn-2 n-1)... exp(03 J23) x x exp(6>J12)exp(rJo1 )O. This gives (i = 2,...,N — 1):
xo = Cki (r), (11)
x1 = Ski (r)CK2 (0), i
xi = Skl (r)SK2 (0)n sin $s cos 0i+1,
s=3
xn = Ski (r)SK2 (0) sin fa,
s=3
where hereafter a product nis such that s > i is assumed to be equal to 1. The K-dependent trigonometric functions CK(x) and SK(x) are defined by [10, 11]:
{cos t/kx, k > 0,
1, K = 0, (12)
cosh ^/^nx, n < 0;
sin y/nx, n > 0, Sk(x) = <( x, k = 0,
7== sinh J—nx, n < 0.
Notice that here k e {k1 , k2}. The K-tangent is defined by TK(x) = SK(x)/CK(x) and its contraction k = 0 gives To(x) = x.
The canonical parameters (r,0,fa3,..., ) dual to (Jo1,J12,J23,...,JN-1N) are called geodesic polar coordinates. In order to explain their (physical) geometrical role, let us consider a (time-like) geodesic l1 and other (N — 1) (space-like) geodesics lj (j = 2,... ,N) in Sj^]K2 which are
orthogonal at the origin O (each translation Joi moves O along li). Then [8, 12]:
The radial coordinate r is the distance between the point P and the origin O measured along the geodesic l that joins both points. In the Rieman-nian spaces with k1 = ±
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