Revista Brasileira de Ensino de F´ısica, v. 26, n. 4, p. 351 - 357, (2004) www.sbfisica.org.br Some basics of su(1; 1) (Introduc¸ao˜ a su(1; 1)) Marcel Novaes1 Instituto de F´ısica “Gleb Wataghin”, Universidade Estadual de Campinas, Campinas, SP, Brasil Recebido em 22/07/2004; Aceito em 19/08/2004 A basic introduction to the su(1; 1) algebra is presented, in which we discuss the relation with canonical transformations, the realization in terms of quantized radiation field modes and coherent states. Instead of go- ing into details of these topics, we rather emphasize the existing connections between them. We discuss two parametrizations of the coherent states manifold SU(1; 1)=U(1): as the Poincare´ disk in the complex plane and as the pseudosphere (a sphere in a Minkowskian space), and show that it is a natural phase space for quantum systems with SU(1; 1) symmetry. Keywords: su(1; 1) algebra, SU(1; 1) group, canonical transformations, coherent states, second quantization, pseudosphere. Uma introduc¸ao˜ simples a` algebra´ su(1; 1) e´ apresentada, na qual discutimos a relac¸ao˜ com transformac¸oes˜ canonicas,ˆ a realizac¸ao˜ em termos de modos quantizados do campo de radiac¸ao˜ e estados coerentes. Ao inves´ de entrar em detalhes a respeito desses topicos,´ preferimos enfatizar as conexoes˜ existentes entre eles. Discutimos duas parametrizac¸oes˜ da variedade dos estados coerentes SU(1; 1)=U(1): como o disco de Poincare´ no plano complexo e como a pseudoesfera (uma esfera em um espac¸o de Minkowski) e mostramos que se trata de um espac¸o de fase natural para sistemas quanticosˆ com simetria SU(1; 1). Palavras-chave: algebra´ su(1; 1), grupo SU(1; 1), transformac¸oes˜ canonicas,ˆ estados coerentes, segunda quantizac¸ao,˜ pseudoesfera. 1. Introduction and appears naturally in a wide variety of physical problems. A realization in terms of one-variable dif- What I wish to present here is a very basic and accessi- ferential operators, ble introduction to the su(1; 1) algebra and its applica- tions. I present for example how to obtain the energy d2 a y2 i d 1 K1 = 2 + 2 + ; K2 = − y + ; spectrum of hydrogen without solving the Schrodinger¨ dy y 16 2 dy 2 2 y2 equation. I also present the relation with the symplectic K = d + a ; (2) 0 dy2 y2 − 16 algebra and canonical transformations. But the main focus is on coherent states and the geometry of the quo- for example, allows any ODE of the kind tient space SU(1; 1)=U(1). Taking into account the d2 a realization of this algebra by creation and annihilation + + by2 + c f(y) = 0 (3) dy2 y2 operators, I hope these geometrical considerations may have some importance to the field of quantum optics. to be expressed in terms of a su(1; 1) element [1]. The The su(1; 1) sp(2; R) so(2; 1) algebra is de- ∼ ∼ radial part of the hydrogen atom and of the 3D har- fined by the commutation relations monic oscillator, and also the Morse potential fall into this category, and the analytical solution of these sys- [K ; K ] = iK ; [K ; K ] = iK ; 1 2 − 0 0 1 2 tems is actually due to their high degree of symme- [K2; K0] = iK1; (1) try. In fact, the close relation between the concepts of 1Enviar correspondenciaˆ para Marcel Novaes. E-mail: mnovaes@ifi.unicamp.br. Copyright by the Sociedade Brasileira de F´ısica. Printed in Brazil. 352 Novaes symmetry, invariance, degeneracy and integrability is 2 3=2 of great importance to all areas of physics [2]. If we make r = y and R(r) = y− Y (y) we Just like for su(2), we can choose a different basis have d2 4l(l + 1) 3=4 − + 8Ey2 8Z Y (y) = 0; (11) dy2 − y2 − K = (K1 iK2); (4) „ « and, as already noted in the introduction, this can be in which case the commutation relations become written in terms of the su(1; 1) generators (2). A little algebra gives [K0; K ] = K ; [K+; K ] = 2K0: (5) − − 1 1 ( 64E)K + ( + 64E)K 8Z Y (y) = 0; (12) 2 − 0 2 1 − Note the difference in sign with respect to su(2). The Casimir operator, the analog of total angular mo- and the Casimir reduces to C = l(l + 1), which gives mentum, is given by k = l + 1. Using the transformation equations C = K2 K2 K2 = K2 0 − 1 − 2 0 − 1 iθK2 iθK2 (K+K + K K+): (6) e− K0e = K0 cosh θ + K1 sinh θ (13) 2 − − iθK2 iθK2 e− K1e = K0 sinh θ + K1 cosh θ (14) This operator commutes with all of the K's. Since the group SU(1; 1) is non-compact, all we can choose its unitary irreducible representations are infinite- 64E + 1=2 dimensional. Basis vectors k; m in the space where tanh θ = (15) j i 64E 1=2 the representation acts are taken as simultaneous eigen- − vectors of K0 and C: in order to obtain Z C k; m = k(k 1) k; m ; (7) K0Y~ (y) = Y~ (y); (16) j i − j i p 2E K k; m = (k + m) k; m ; (8) − 0j i j i iθK2 where Y~ (y) = e− Y (y). Since we know the spec- where the real number k > 0 is called the Bargmann trum of K0 from (8) we can conclude that the energy index and m can be any nonnegative integer (we con- levels are given by sider only the positive discrete series). All states can be obtained from the lowest state k; 0 by the action Z j i En = 2 ; n = m + l + 1: (17) of the ”raising” operator K+ according to −2n Γ(2k) m 3. Relation with Sp(2; R) k; m = (K+) k; 0 : (9) j i sm!Γ(2k + m) j i A system with n degrees of freedom, be it classi- cal or quantum, always has Sp(2n; R) as a symmetry 2. Energy levels of the hydrogen atom group. Classical mechanics takes place in a real mani- fold, and the kinematics are given by Poisson brackets The hydrogen atom, as well as the Kepler problem, (i; j = 1::N) has a high degree of symmetry, related to the partic- qi; pj = δij: (18) ular form of the potential. This symmetry is reflected f g in the conservation of the Laplace-Runge-Lenz vector, Quantum mechanics takes place in a complex Hilbert and leads to a large symmetry group, SO(4; 2). Here space, and the kinematics determined by the canonical we restrict ourselves to the radial part of this problem, commutation relations (i; j = 1::N) as an example of the applicability of group theory to [q^i; p^j] = i~δij: (19) quantum mechanics and of su(1; 1) in particular. For more complete treatments see [1, 2]. The radial part of These relations can also be written in the form (now the Schrodinger¨ equation for the hydrogen atom is i; j = 1::2N) ξ ; ξ = J ; (20) d2 2 d 2Z l(l + 1) f i jg ij + + 2E R(r) = 0: (10) ^ ^ dr2 r dr − r − r2 [ξi; ξj] = i~Jij ; (21) Some basics of su(1; 1) 353 T where ξ = (q1; :::; qN ; p1; :::; pN ) , ξ^i is the hermitian we obtain a realization of the su(1; 1) algebra. In this operator corresponding to ξ and J is the 2N 2N case the Casimir operator reduces identically to i × matrix given by 3 0 1 C = k(k 1) = ; (27) J = : (22) − −16 1 0 − k = 1=4 k = 3=4 The symplectic group Sp(2N; R) (in its defining which corresponds to or . It is not representation) is composed by all real linear transfor- difficult to see that the states mations that preserve the structure of relations (20). It n (ay) is easy to see that therefore n = 0 (28) j i pn! j i Sp(2N; R) = S SJST = J : (23) f j g with even n form a basis for the unitary representation For a far more extended and detailed discussion, see with k = 1=4, while the states with odd n form a basis [3] for the case k = 3=4. For a classical system with only one degree of free- The unitary operator dom, such canonical transformations are generated by the vector fields [4] 1 2 1 2 S(ξ) = exp 2 ξ∗a 2 ξ(ay) = @ @ @ @ − q @p + p @q = 2iK0; q @p p @q = 2iK1; exp(ξ∗K ξK+) (29) {− − − − − @ @ q + p = 2iK2; : (24) − @q @p g is called the squeeze operator in quantum optics, and It is easy to see that these operators have the same is associated with degenerate parametric amplification commutation relations as the su(1; 1) algebra (1). [6]. There is also the displacement operator Note that the symplectic groups Sp(2n; R) are non-compact, and therefore any finite dimensional rep- D(α) = exp αay α∗a ; (30) resentation must be nonunitary. In the quantum case, − that means that the matrices S implementing the trans- which acts on the vacuum state 0 to generate the co- formations j i ^ ^ herent state ξj0 = Sijξi; (25) ^ ^ n such that [ξi0; ξj0 ] = i~Jij, are nonunitary (a 2 2 α 2=2 1 α × α = D(α) 0 = e−| j n : (31) nonunitary representation of su(1; 1) exists for exam- j i j i pn!j i i n=o ple in terms of Pauli matrices, K1 = 2 σ2; K2 = X i 1 ^ ^ σ1; K0 = σ3).