ЯДЕРНАЯ ФИЗИКА, 2010, том 73, № 2, с. 276-284
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
SUPERINTEGRABLE POTENTIALS ON 3D RIEMANNIAN AND LORENTZIAN SPACES WITH NONCONSTANT CURVATURE
© 2010 A. Ballesteros1^, A. Enciso2)**, F. J. Herranz3)***, O. Ragnisco4)****
Received April 17,2009
A quantum sl(2, R) coalgebra (with deformation parameter z) is shown to underly the construction of a large class of superintegrable potentials on 3D curved spaces, that include the nonconstant curvature analog of the spherical, hyperbolic, and (anti-)de Sitter spaces. The connection and curvature tensors for these "deformed" spaces are fully studied by working on two different phase spaces. The former directly comes from a 3D symplectic realization of the deformed coalgebra, while the latter is obtained through a map leading to a spherical-type phase space. In this framework, the nondeformed limit z ^ 0 is identified with the flat contraction leading to the Euclidean and Minkowskian spaces/potentials. The resulting Hamiltonians always admit, at least, three functionally independent constants of motion coming from the coalgebra structure. Furthermore, the intrinsic oscillator and Kepler potentials on such Riemannian and Lorentzian spaces of nonconstant curvature are identified, and several examples of them are explicitly presented.
1. INTRODUCTION
In the context of Hamiltonian systems with an arbitrary finite number of degrees of freedom, a deep connection between the coalgebra symmetry of a given system and its Liouville integrability was firmly established in [1]. Moreover, the intrinsic superin-tegrability properties of the coalgebra construction were further explored in [2]. Since then, this framework has lead to the coalgebra interpretation of the integrability properties of many well-known systems, as well as to the construction of many new super-integrable systems by using both Lie and g-Poisson coalgebras (see [1—4] and references therein).
In particular, by making use of the Poisson coalgebra given by the nonstandard quantum deformation of sl(2, R), the construction of integrable 2D geodesic flows corresponding to 2D Riemannian and Lorentzian spaces with nonconstant curvature was presented in [5]. Furthermore, these systems revealed a geometric interpretation of the quantum deformation, since the (in general, nonconstant) curvature
1)Departamento de Física, Facultad de Ciencias, Universidad de Burgos, Spain.
2)Departamento de Física Teórica II, Universidad Complutense, Madrid, Spain.
3)Departamento de Física, Escuela Politécnica Superior, Universidad de Burgos, Spain.
4)Dipartimento di Fisica, Universita di Roma Tre and Istituto Nazionale di Fisica Nucleare sezione di Roma Tre, Italy. E-mail: angelb@ubu.es
E-mail: aenciso@fis.ucm.es E-mail: fjherranz@ubu.es E-mail: ragnisco@fis.uniroma3.it
of these spaces was just a smooth function of the deformation parameter. Later, integrable potentials on such 2D "quantum deformed" spaces were introduced by preserving the underlying deformed coalgebra symmetry [6]. In this context, the search for the appropriate Kepler—Coulomb (KC) and oscillator potentials on such 2D curved spaces was posed as an interesting problem, and some hints were proposed.
In this contribution we present the generalization of all these results to the 3D case and we fully solve the question concerning the generic form of the intrinsic oscillator and KC potentials on all the corresponding "quantum deformed" 3D spaces, thus opening the path for the generalization of this construction to N dimensions. In the next section we make use of the quantum sl(2, R) coalgebra symmetry in order to construct the family of super-integrable 3D geodesic flows that define the spaces of hyperbolic type, whose sectional and scalar curvatures are also obtained. In Section 3 an analytic continuation procedure is introduced through a set of appropriate spherical-type coordinates, thus leading to the Lorentzian counterparts of the previous spaces. Moreover, it is shown that these coordinates allow the separability of the geodesic flow Hamiltonians in all the cases. Finally, Section 5 is devoted to the characterization of those potentials that will preserve the superintegrability properties of the free Hamiltonian. In particular, by applying the prescription given in [7— 10], the intrinsic oscillator and KC potentials on all the previous spaces are explicitly constructed, and some particular examples are analysed.
SUPERINTEGRABLE POTENTIALS ON 3D RIEMANNIAN 2. SUPERINTEGRABLE HAMILTONIANS
Let us consider the Poisson coalgebra version of the nonstandard quantum deformation of sl(2, R), hereafter denoted (slz(2,R), A) = slz(2), where z is a real deformation parameter (q = ez). Its deformed Poisson brackets, coproduct A and Casimir C are given by [3]:
[J3,J+ } = 2J+ cosh zJ-, (1)
{Ja ,J-} = -2
sinh zJ-
{J-,J+} = 4 Ja,
A(J-) = J- 0 1 + 10 J-,
A(Ji) = Ji 0 ezJ- + e-zJ- 0 Ji, l = +, 3,
(2)
С
sinh zJ-
z
J+ - J3 •
(3)
A one-particle symplectic realization of (1) reads
J^ = q2,
J
(i) _ sinhzq\ 2 zbi
zq1
Pi +
sinh zqi'
(4)
J
(i) sinh zqi
zqi
qiPi,
where b1 is a real parameter that labels the representation through C(1) = bi. Hence dimensions of the deformation parameter are [z] = [J_] 1 = [q1 ] 2.
Starting from (4), the coproduct (2) determines the corresponding two-particle realization of (1) defined on slz (2) 0 slz(2) that depends on two real parameters b1, b2:
J(2) = q2 + q2,
J
(2) _ sinh zqj
zq2
qiPiezq2 +
2 sinh zq?
zq22
q?P?e zq2,
J
(2)
+
zbi
zq2
+ (^ßti +
sinh zq2
zb?
zq?
ezq22
-zq\
+
sinh zq|,
Then the two-particle Casimir is given by
zq2
zq?
(5)
sinh zqf sinh zq2 J
+ b1e2z?2 + b2e_2zq2.
Next, the 3-sites coproduct, A(3) = (A 0 id) o A = = (id 0 A) o A, gives rise to a three-particle symplec-tic realization of (1) in terms of three real parameters
bi which is defined on slz(2) 0 slz(2) 0 slz(2) [1, 3]; namely
J(3) = q? + q? + q? = q2
(6)
J
(3) _ sinhzgf
zq?
q1p1ezq2 ezq2 +
+ S-^q2P2e~zqhzq 1 +
zq?
+
sinh zq?
q3p3e-zq2 e-zq2,
J
(3) _ ( sinh zq\ 2
zbi
+
^ zqf ^ ^ sinh zqf -.2
ezq2 ezq2 +
+ ( SiDh^2p2 + Zb2 \ e-ZQleZq¡ + V zq? sinh zq? J
+ / sinhz9|^2 + zb3
V zq2 sinh zq2
e-zq2 e-zq2 •
Hence if we denote the three sites on slz (2) 0 0 slz(2) 0 slz(2) by 1 0 2 0 3 the coalgebra approach [1, 11] provides three "relevant" functions, coming from the two- and three-sites coproduct of the Casimir (3): (i) the two-particle Casimir which is defined on 1 0 2; (ii) another two-particle Casimir C(2) but defined on 2 0 3; and (iii) the three-
particle Casimir C(3) defined on 1 0 2 0 3. These are given by (5) and
/sinh zq? sinh zq? ?
C(2) = -2--2 (92P3 - Q3P2)
zq?
zq3?
+ (7)
sinh zq? sinh zq? J
+ b?e?zq2 + b3e-?zq22,
zq2
zq??
+
sinh zq? sinh
sinh zq? /
/ sinh zq? sinh zq? , ,2
+ --^^ (qm ~ qsPl) +
zq3
sinh zq? sinh zq? /
( sinh zq? sinh zq? , + ( „„2 »„2 - 93P2) +
zq??
zq?
+ 62^4 + e-2zq?e"zqiezqi +
sinh zq? sinh zq? /
+ b1e?zq22 e2zq2 + b2e-2zq2 e2zq2 + b3e-2zq2 e-2zq22 •
z
All the above expressions give rise to the (nonde-formed) sl(2, R) coalgebra [4] under the limit z — 0, that is, the Poisson brackets and Casimir are non-deformed, the coproduct is primitive, A(X) = X ® ® 1 + 1 ® X, and the symplectic realization reads
J-3) = q2, J+> = P2 + £3=1 bi/q2, J we set all the bi = 0, the three Casimir functions reduce to the components of the angular momentum
lij = QiVj - qjPi +12
7(3)
(3)
q • p.
If
с(2) = &,
c,
(2)
= l¡3, and с(3) =
_ l2 + 12 — l12 ' ll
13
23
In this way a large family of superintegrable Hamiltonians can be constructed through the following statement.
Proposition 1. (i) The three-particle generators (6) fulfil the commutation rules (1) with respect to the canonical Poisson bracket {qi,pj } = = Ôij.
(ii) These generators Poisson commute with the three functions C(2), C(2), and C(3).
(iii) Any function defined on (6), i.e., H = H(JÍ3),J^3),J:(3)),
(8)
3. CURVED SPACES FROM GEODESIC FLOWS
As a byproduct of Proposition 1 we can obtain an infinite family of superintegrable free Hamiltonians by setting the three bi = 0 and by choosing, amongst the family (8), the following expression for H:
H=l-J+J{zJ_),
(9)
where f is any smooth function such that lim^o f(zJ-) = 1; hence lim^o'W = -gP2 gives the kinetic energy on the 3D Euclidean space. For the sake of simplicity, from now on we drop the index "(3)" in the generators. Thus by writing the Hamiltonian (9) as a free Lagrangian,
If zqj
^ = • Г1 ^~zq2^~zqkï +
2 \ sinh zq2
+
zq|
sinh zq2
ezqi e-zq2 q2 +
2
sinh zq2> )
we find the geodesic flow on a 3D space with a definite positive metric given by
provides a completely integrable Hamiltonian since either {C(2), C(3), H} or {C(2), C(3), H} are three functionally independent functions in involution.
(iv) The four functions {C(2), C(2), C(3), H} are functionally independent.
We remark that in the 2D case the generic function H = H(J— ,J++2) ,j32) ) determines, in principle, an integrable (but not superintegrable!) Hamiltonian as it is only endowed with a single constant of the motion c(2) [6]. On the contrary, in the 3D case any Hamiltonian H (8) is (at least) a weakly-superintegrable one [12], since one more constant of the motion is lacking in order to ensure the maximal superintegrability of the system. It is well known [1,2] that in the ND generic case the coalgebra approach would provide 2N — 2 constants of the motion, thus leading to the construction of quasi-maximally superintegrable systems. Obviously, in 3D, quasi-maximal superintegrability is equivalent to weak superintegrability, but this is no longer true in higher dimensions. Therefore the 3D case can be considered as the cornerstone for the generalization of all the results we shall present here to arbitrary dimension N.
ds2 _
( 2zq\ „-zq2 -
+
\sinhzgf 2zq2
e-zq2 e-zq3 dq2 +
(10)
+
sinh zq2 2
ezqi e-zq3 dq2 +
2zq2
sinhzg2 J f(zq2)
The
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