Some Basics of Su(1,1)

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 introduçao˜ simples a` algebra´ su(1, 1) e´ apresentada, na qual discutimos a relaçao˜ com transformaçoes˜ canonicas,ˆ a realizaçao˜ em termos de modos quantizados do campo de radiaçao˜ e estados coerentes. Ao inves´ de entrar em detalhes a respeito desses topicos,´ preferimos enfatizar as conexoes˜ existentes entre eles. Discutimos duas parametrizaçoes˜ 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 espaço de Minkowski) e mostramos que se trata de um espaço de fase natural para sistemas quanticosˆ com simetria SU(1, 1). Palavras-chave: algebra´ su(1, 1), grupo SU(1, 1), transformaçoes˜ canonicas,ˆ estados coerentes, segunda quantizaçao,˜ 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 applications. 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) | 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) | i − | i √ 2E K k, m = (k + m) k, m , (8) − 0| i | 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 | 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) | i sm!Γ(2k + m) | i A system with n degrees of freedom, be it classical 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 manifold, 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 { } 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 [qˆi, pˆj] = 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) { i j} ij + + 2E R(r) = 0. (10) ˆ ˆ dr2 r dr − r − r2 [ξi, ξj] = i~Jij , (21) Some basics of su(1, 1) 353

T where ξ = (q1, ..., qN , p1, ..., pN ) , ξî 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 (a†) is easy to see that therefore n = 0 (28) | i √n! | i Sp(2N, R) = S SJST = J . (23) { | } 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(a†) = ∂ ∂ ∂ ∂ − q ∂p + p ∂q = 2iK0, q ∂p p ∂q = 2iK1, exp(ξ∗K ξK+) (29) {− − − − − ∂ ∂ q + p = 2iK2, . (24) − ∂q ∂p } 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 αa† 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 | i ˆ ˆ herent state ξj0 = Sijξi, (25) ˆ ˆ n such that [ξi0, ξj0 ] = i~Jij, are nonunitary (a 2 2 α 2/2 ∞ α × α = D(α) 0 = e−| | n . (31) nonunitary representation of su(1, 1) exists for exam- | i | i √n!| i i n=o ple in terms of Pauli matrices, K1 = 2 σ2, K2 = X i 1 ˆ ˆ σ1, K0 = σ3). However, since all ξi and all ξ0 are Action of S(ξ) on a coherent state gives a squeezed − 2 2 j hermitian and irreducible, by the Stone-von-Neumann coherent state, α, ξ = S(ξ) α . theorem [3, 5] there exists an operator U(S) that acts | i | i unitarily on the infinite dimensional Hilbert space of pure quantum states (Fock space). If we now see ξî 4.2. Two-mode realization ˆ and ξi0 as (infinite dimensional) matrices, then U(S) is It is also possible to introduce a two-mode realization ˆ ˆ 1 such that ξi0 = U(S)ξiU(S)− . Finding this unitary of the algebra su(1, 1). This is done by defining the operator in practice is in general a nontrivial task. generators

K+ = a†b†, K = ab, 4. Optics − 1 K0 = 2 (a†a + b†b + 1). (32) 4.1. One-mode realization We know the radiation field can be described by In this case the Casimir operator is given by C = 1 (a a b b)2 1 . If we introduce the usual two-mode bosonic operators a and a†. If we form the quadratic 4 † − † − 4 basis n, m then the states n + n , n with fixed n combinations | i | 0 i 0 form a basis for the representation of su(1, 1) in which 1 2 1 2 K+ = (a†) , K = a , 2 − 2 k = ( n0 + 1)/2. A charged particle in a magnetic 1 | | K0 = 4 (aa† + a†a) (26) field can also be described by this formalism [7]. 354 Novaes

The unitary operator From (33) we see that su(1, 1) coherent states are actually the result of a two-mode squeezing upon a S (ξ) = exp ξ ab ξa b = 2 ∗ † † Fock state of the kind n , 0 . On the other hand, from − | 0 i exp(ξ∗K ξK+) (33) the one-mode realization (29) they can be regarded as − − squeezed vacuum states. is called the two-mode squeeze operator [6], or down- converter. When we consider the other quadratic com- These states are not orthogonal, binations ( a b, (a )2, (b )2, a a b b and their her- { † † † † − † } mitian adjoint) we have the algebra sp(4, R), of which 2 k 2 k (1 z1 ) (1 z2 ) sp(2, R) su(1, 1) is a subalgebra. More detailed z1, k z2, k = − | | −2|k | (36) ∼ h | i (1 z∗z2) discussions about group theory and optics can be found − 1 for example in [3, 4, 8]. and they form an overcomplete set with resolution of 5. Coherent states unity given by

Normalized coherent states can be deﬁned for a general unitary irreducible representation of su(1, 1) as [9] 2k 1 dz dz − ∧ ∗ z, k z, k = z, k = (1 z 2)k π (1 z 2)2 | ih | | i − | | ZD − | | ∞ 1 k, m k, m = 1 (k > ). (37) ∞ Γ(2k + m) m | ih | 2 z k, m , (34) m=0 s m!Γ(2k) | i X mX=0 where z is a complex number inside the unit disk, D = z, z < 1 . Similar to the usual coherent states, From the integration measure we see that the co- { | | } they can be obtained from the lowest state by the action herent states are parametrized by points in the Poincare´ of a displacement operator: disk (or Bolyai-Lobachevsky plane), which we discuss in the next section. The expectation value for a product z, k = exp(ζK+ ζ∗K ) k, 0 , p q | i − − | i of algebra generators like K K Kr was presented in ζ 0 + z = tanh ζ . (35) [10] and is given by − ζ | | | |

p q r 2 2k p r ∞ Γ(m + p + 1)Γ(m + p + 2k) q 2m z, k K K0 K+ z, k = (1 z ) z − (m + p + k) z . (38) h | − | i − | | m!Γ(m + p + 1 r)Γ(2k) | | mX=0 −

Simple particular cases of this expression are 1 + z 2 2z | | z, k z, k . (40) z, k K z, k = k , 1 z 2 | ih | h | −| i 1 z 2 − | | − | | 2 1 + z Just as usual spin coherent states are parametrized z, k K0 z, k = k | |2 . (39) h | | i 1 z by points on the space SU(2)/U(1) S2, the − | | ∼ two-dimensional spherical surface, su(1, 1) coher- Moreover, for k > 1/2 the operator K has a diag- 0 ent states are parametrized by points on the space onal representation as SU(1, 1)/U(1), which corresponds to the Poincare´ disk. This space can also be seen as the two- 2k 1 d2z dimensional upper sheet of a two-sheet hyperboloid, K = − (k 1) 0 4π (1 z 2)2 − ∗ also known as the pseudosphere. ZD − | | Some basics of su(1, 1) 355

6. The pseudosphere 6.1. Action of the group

The sphere S2 is the set of points equidistant from the The symmetry group of the pseudosphere is the group that preserves the relation y2 + y2 y2 = R2, the origin in a Euclidian space: 1 2 − 0 − Lorentz-like group SO(2, 1). The so(2, 1) algebra as- 2 2 2 2 2 sociated with this group is isomorphic to the su(1, 1) S = (x1, x2, x3) x1 + x2 + x3 = R . (41) { | } algebra we are studying. All isometries can be repre- sented by 3 3 matrices Λ that are orthogonal with The pseudosphere H2 plays a similar role in a × respect to the Minkowski metric Q = diag(1, 1, 1) Minkovskian space, that is, take the space defined by − 2 2 2 2 (actually we must also impose Λ00 > 0 so that we (y1, y2, y0) y + y y = R , which is a two- { | 1 2 − 0 − } are restricted to the upper sheet of the hyperboloid), sheet hyperboloid that crosses the y0 axis at two points, and they can be generated by 3 basic types: A) Eu- R, called poles. The pseudosphere, which is a Rie- clidian rotations, by an angle φ0, on the (y1, y2) plane; mannian space, is the upper sheet, y0 > 0. The pseu- B) Boosts of rapidity τ along some direction in the dosphere is related to the Poincare´ disk by a stereo- 0 (y1, y2) plane; C) Reflections through a plane contain- graphic projection in the plane (y1, y2), using the point ing the y axis. As examples, we show a rotation, a (0, 0, R) as base point. The relation between the pa- 0 − boost in the y direction and a reflection through the rameters is 2 plane (y1, y0): y = R cosh τ, y = R sinh τ cos φ, 0 1 cos φ sin φ 0 0 − 0 y2 = R sinh τ sin φ, (42) A) sin φ cos φ 0 ,  0 0  0 0 1 and  1 0 0  iφ τ y1 + iy2 z = e tanh = . (43) B) 0 cosh τ0 sinh τ0 , 2 R + y0   0 sinh τ0 cosh τ0 The distance ds2 = dy2 + dy2 dy2 and the area  1 0 0  1 2 − 0 dµ = sinh τdτ dφ become C) 0 1 0 (47) ∧  −  0 0 1 dz dz ds2 = dτ 2 + sinh τdφ2 = · ∗ , (44)   (1 z 2)2 − | | Incidentally, the geometrical character of the pre- dz dz∗ (τ, φ) dµ = ∧ . (45) viously used parameters becomes clear. (1 z 2)2 − | | Using the complex coordinates of the Poincare´ disk we have Note that the metric is conformal, so the actual angles R (z) = eiφ0 z (48) coincide with Euclidian angles. Geodesics, which are φ0 intersections of the pseudosphere with planes through for rotations, the origin, become circular arcs (or diameters) orthog- iφ0 onal to the disk boundary (the non-Euclidian character (cosh τ0/2)z + e sinh τ0/2 Tτ ,φ (z) = (49) of the Poincare´ disk appears in some beautiful draw- 0 0 iφ0 (e− sinh τ0/2)z + cosh τ0/2 ings of M.C. Escher, the “Circle Limit” series [11]). A very good discussion about the geometry of the pseu- for boosts of rapidity τ0 in the φ0 direction and S(z) = dosphere can be found in [12], and we follow this pre- z∗ for reflections through the (y1, y0) plane. We see sentation. that, except for reflections, all isometries can be writ- In the pseudosphere coordinates the average values ten as of the su(1, 1) generators are very simple: αz + β 2 2 z0 = , with α β = 1, (50) z, k K z, k = k y , z, k K z, k = k y , β∗z + α∗ | | − | | h | 1| i R 1 h | 2| i R 2 z, k K z, k = k y . (46) h | 0| i R 0 and if, as usual, we represent these transformations by α β matrices ∗ there is a realization of the trans- From now on we set R = k = 1. β α ∗ 356 Novaes formation group by 2 2 matrices, in which this property was examined for example in [13]. This × phase space can also be used to define path integrals eiφ0/2 0 Rφ0 = iφ /2 , for SU(1, 1) (see [14, 15] and references therein), and 0 e− 0 obtain a semiclassical approximation to this class of cosh τ /2 e iφ0 sinh τ /2 T = 0 − 0 .(51) quantum systems. τ0,φ0 eiφ0 sinh τ /2 cosh τ /2 0 0 This is the basic representation of the group 7. Summary SU(1, 1). For other parametrizations of the pseudosphere, see [12]. We have presented a very basic introduction to the su(1, 1) algebra, discussing the connection with 6.2. Canonical coordinates canonical transformations, the realization in terms of quantized radiation field modes and coherent states. We present one last set of coordinates, one that has an We have not explored these subjects in their full de- important physical property. Let us first note that if we tail, but instead we emphasized how they can be re- define i = z, k Ki z, k , then there exists an opera- lated. The coherent states, for example, can be re- K h | | i tion , such that the commutation relations garded as one-mode vacuum squeezed states or as two- {· ·} mode number squeezed states. The coherent states [K , K ] = iK , [K , K ] = iK , 1 2 − 0 0 1 2 manifold SU(1, 1)/U(1) was treated as the Poincare´ [K2, K0] = iK1 (52) disk and as the pseudosphere, and shown to be a natural phase space for quantum systems with SU(1, 1) are exactly mapped to symmetry. , = , , = , {K1 K2} K0 {K0 K1} −K2 , = , (53) {K2 K0} −K1 Acknowledgments in agreement with the usual quantization condition After completion of this manuscript I became aware of , i[ , ]. This Poisson Bracket is written in terms {· ·} → · · reference [16], which presents a very long account of of the Poincare´ disk coordinates as the group SO(2, 1), its geometrical properties and ap- (1 z 2)2 ∂f ∂g ∂f ∂g plications to quantum optics. I acknowledge financial f, g = − | | . (54) { } 2ik ∂z ∂z − ∂z ∂z support from Fapesp (Fundaçao˜ de Amparo a` Pesquisa ∗ ∗ do Estado de Sao˜ Paulo). It is possible to define new coordinates (q, p) that are canonical in the sense that References ∂f ∂g ∂f ∂g f, g = . (55) { } ∂q ∂p − ∂p ∂q [1] B.G. Wybourne, Classical Groups for Physicists (Wi- ley, New York, 1974). These coordinates are given by [2] R. Gilmore, Lie Groups, Lie Algebras and Some of q + ip z Their Applications (Krieger, Malabar, 1994). = (56) √4k 1 z 2 − | | [3] Arvind et al., Pramana J. Physics 45, 471 (1995). and the classical functions arep written in terms of them [4] A. Wunsche, J. Opt. B: Quantum Semiclass. Opt. 2, 73 as (2000).

q 2 2 [5] T.F. Jordan, Linear Operators for Quantum Mechanics 1 = 4k + q + p , K 2 (John Wiley, New York, 1974). p 2 2 2 = p 4k + q + p , K 2 [6] M.O. Scully and M.S. Zubairy, Quantum Optics (Cam- q2+p2 =pk + . (57) bridge University Press, Cambridge, 1999). K0 2 We thus see that there is a natural phase space for [7] M. Novaes and J.-P. Gazeau, J. Phys. A: Math. Gen. quantum systems that admit SU(1, 1) as a symme- 36, 199 (2003). try group. Dynamics of time-dependent systems with [8] S.L. Braunstein, quant-ph/9904002. Some basics of su(1, 1) 357

[9] A. Perelomov, Generalized Coherent States and Their [14] C.C. Gerry, Phys. Rev. A 39, 971 (1989). Applications (Springer, Berlin, 1986). [10] T. Lisowski, J. Phys. A: Math. Gen. 25, L1295 (1992). [15] C. Grosche and F. Steiner, Handbook of Feynman Path Integrals (Springer, Berlin, 1998). [11] D. Schattschneider, M.C. Escher: Visions of Symmetry (Harry N Abrams, New York, 2004), 2nd ed. [16] H. Kastrup, Fortschritte d. Physik 51, 975 (2003). Also [12] N.L. Balasz and A. Voros, Phys. Rep. 143, 109 (1986). available as quant-ph/0307069. [13] A. Bechler, J. Phys. A: Math. Gen. 34, 8081 (2001).