A HOLOMORPHIC REPRESENTATION OF THE SEMIDIRECT SUM OF SYMPLECTIC AND HEISENBERG LIE ALGEBRAS

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1 JGSP 4 (005 9 JOURNAL OF Geometry and Symmetry in Physics A HOLOMORPHIC REPRESENTATION OF THE SEMIDIRECT SUM OF SYMPLECTIC AND HEISENBERG LIE ALGEBRAS STEFAN BERCEANU March 4, 006 Communicated by S.T. Ali Abstract. A representation of the Jacobi algebra by first order differential operators with polynomial coefficients on a Kähler manifold which as set is the product of the complex multidimensional plane times the Siegel ball is presented.. Introduction In this paper we construct a holomorphic polynomial first order differential representation of the Lie algebra which is the semidirect sum h n sp(n, R, on the manifold C n D n, different from the extended metaplectic representation [6]. The case n = corresponding to the Lie algebra h su(, was considered in [3]. The natural framework of such an approach is furnished by the so called coherent state (CS-groups, and the semi-direct product of the Heisenberg-Weyl group with the symplectic group is an important example of a mixed group of this type []. We use Perelomov s coherent state aproach []. Previous results concern the hermitian symmetric spaces [] and semisimple Lie groups which admit CS-orbits [4]. The case of the symplectic group was previously investigated in [], [6], [5],[0], []. Due to lack of space we do not give here the proofs, but in general the technique is the same as in [3], where also more references are given. More details and the connection of the present results with the squeezed states [3] will be discussed elsewhere.. The differential action of the Jacobi algebra The Heisenberg-Weyl (HW group is the nilpotent group with the n+-dimensional real Lie algebra h n =< is + n i= (x ia + i x i a i > s R,xi C, where a + i (a i are the boson creation (respectively, annihilation operators.

2 Stefan Berceanu Table : The generators of the symplectic group: operators, matrices, and bifermion operators ( K + ij K ij + 0 = i eij + e ji ( 0 0 K ij Kij = i 0 0 ( e ij + e ji 0 K 0 ij Kij 0 = eij 0 0 e ji a+ i a+ j a ia j 4 (a+ i a j + a j a + i We consider the realization of the Lie algebra of the group Sp(n, R [], [6]: sp(n, R =< n (a ij Kij 0 + b ijk ij + b ij Kij >, a = a, b t = b. ( i,j= With the notation: X := dπ(x, we have the correspondence: X sp(n, R X, where the real symplectic Lie algebra sp(n, R is realized as sp(n, C u(n, n X = ( a b b ā X = n i,j= (a ij K 0 ij + z ij K + ij z ijk ij, b = iz. ( The Jacobi algebra is the the semi-direct sum g J := h n sp(n, R, where h n is an ideal in g J, i.e. [h n, g J ] = h n, determined by the commutation relations: [a i, a + j ] = δ ij; [a i, a j ] = [a + i, a+ j ] = 0 (3a [K ij, K kl ] = [K+ ij, K+ kl ] = 0; [K0 ji, K 0 kl ] = K0 jl δ ki K 0 ki δ lj (3b [K ij, K+ kl ] = K0 kj δ li + K 0 lj δ ki + K 0 ki δ lj + K 0 li δ kj (3c [K ij, K0 kl ] = K il δ kj + K jl δ ki; [K + ij, K0 kl ] = K+ ik δ jl K + jk δ li (3d [a i, K + kj ] = δ ika + j + δ ija + k : [K kj, a+ i ] = δ ika j + δ ij a k (3e [K 0 ij, a + k ] = δ jka + i ; [a k, K 0 ij] = δ ik a j ; [a + k, K+ ij ] = [a k, K ij ] = 0.(3f Perelomov s coherent state vectors associated to the group G J with Lie algebra the Jacobi algebra, based on the complex N-dimensional manifold, N = n(n+3, M := HW/R Sp(n, R/U(n; M = D := C n D n (4

3 A holomorphic representation of the semidirect sum of symplectic and Heisenberg Lie algebras 3 are defined as e z,w = exp(xe 0, X := i z i a + i + ij w ij K + ij, z Cn ; W D n. (5 The non-compact hermitian symmetric space X n = Sp(n, R/U(n admits a realization as a bounded homogeneous domain, precisely the Siegel ball [7],[8]: D n := { W M(n, C; W = W t, W W > 0 }. (6 The extremal weight vector e 0 verify the equations a i e o = 0, i =,, n (7a K + ij e 0 0; K ij e 0 = 0; K 0 ij e 0 = k 4 δ ije 0. (7b Proposition The differential action of the generators of the Jacobi algebra is: a = z ; a+ = z + W z K = W ; K0 = k 4 + z z + W W K + = k W + z z + (W z z + z z W + W W W. (8a (8b (8c Proof. The calculation is an application of the formula Ad (exp X = exp(ad X. We have used the convention: [( W W f(w ] f(w := kl w ki w il, W = (w ij. 3. The group action The displacement operator, i.e. D(α := exp(αa + ᾱa, has the addition property D(α D(α = e iθ h(α,α D(α + α, θ h (α, α := Im (α ᾱ. Concerning the real symplectic group, we extract from [], [6] Remark ( To every g Sp(n, R, g g c Sp(n, C U(n, n, or denoted just a b g, g =, where aa b bb = ; ab t = ba t ; a a b t b = ; a t b = b a. ā

4 4 Stefan Berceanu We consider a particular case of the positive discrete series representation [9] of Sp(n, R and let us denote S(Z = S(W. The vacuum is chosen such that equation (7b is satisfied. Here: S(Z = exp( z ij K + ij z ijk ij, Z = (z ij (9a S(W = exp(w K + exp(ηk 0 exp( W K (9b Z W = Z tanh Z (9c Z Z Z = arctanh W W W = W W log + W W W W (9d W W η = log( W W = log cosh ZZ. (9e Perelomov s un-normalized CS-vectors for Sp(n, R are: e Z := exp( ( z ij K + iz ij e 0 = π e 0 0, Z = (z ij ; Z = Z t. (0 Remark 3 For g Sp(n, R, the following relations between the normalized and un-normalized Perelomov s CS-vectors hold: e g := π(ge 0 = (det ā k/ e Z = S(Ze 0 = det( W W k/4 e W ( ( k det a 4 S(Ze0, Z = det ā i bā ( S(ge W/i = det(w b + a k/ e Y/i (3 where W D n, and Z C n in ( are related by equations (9c, (9d, and the linear-fractional action of the group Sp(n, R on the unit ball D n in (3 is Y := g W = (a W + b( b W + ā = (W b + a (b t + W a t. (4 Let us introduce the notation à := ( Ā A and D(Z = e X = cosh Z Z sinh Z Z Z Z Z sinh ZZ ZZ cosh ZZ ZZ, X := ( 0 Z Z 0. (5 Remark 4 The following (Holstein-Primakoff-Bogoliubov equation is true: S (ZãS(Z = D(Zã.

5 A holomorphic representation of the semidirect sum of symplectic and Heisenberg Lie algebras 5 Remark 5 If D is the displacement operator and S(Z is defined by (9a, then D(αS(Z = S(ZD(β, β = D( Z α; α = D(Z β. (6 Let us introduce the notation S(g = S(Z, A := exp( a ij K 0 ij + z ijk + ij z ijk ij. (7 Remark 6 If S denotes the representation of Sp(n, R, in the matrix realization of Table, we have S (g ã S(g = g ã, and S(gD(αS (g = D(α g, α g = a α + b ᾱ. (8 Lemma 7 The normalized and un-normalized Perelomov s coherent state vectors Ψ α,w := D(αS(W e 0 ; e z,w := exp(za + + W K + e 0 are related by the relation Ψ α,w = det( W W k/4 exp( ᾱ ze z,w, z = α W ᾱ. (9 Comment 8 Starting from (9, we obtain the expression of the reproducing kernel K = K( x, V ; y, W (e x,v, e y,w = det(u k/ exp [ < x, Uy > (0 + < V ȳ, Uy > + < x, UW x >], U = ( W V. From the following proposition we can see the holomorphic action of the Jacobi group G J := HW Sp(n, R on the manifold (4: Proposition 9 Let us consider the action S(gD(αe z,w, where g Sp(n, R, and the coherent state vector is defined in (5. Then we have: S(gD(αe z,w = λe z,w, λ = λ(g, α; z, W ( z = (W b + a (z + α W ᾱ ( W = g W = (aw + b( bw + ā = (W b + a (b t + W a t (3 λ = det(w b + a k/ exp( x z ȳ z exp iθ h (α, x (4 x = ( W W (z + W z; y = a(α + x + b(ᾱ + x. (5 Corollary 0 The action of the Jacobi group G J on the manifold (4 is given by (, (. The composition law in G J is (g, α, t (g, α, t = (g g, g α +α, t +t +Im (g α ᾱ. (6 The proof of Proposition 9 is based on the previous assertions of this section.

6 6 Stefan Berceanu 4. The scalar product Following the general prescription for CS-groups [4], we calculate the Kähler potential f as the logarithm of the reproducing kernel K, and the Kähler twoform: f = k log det( W W + z i ( W W ij z j + (7 [z i[ W ( W W ] ij z j + z i [( W W W ] ij z j ] iω = k Tr[( W W dw ( W W d W ] (8 + Tr[dz t ( W W d z] Tr[d z t ( W W dw x] + cc + Tr[ x t dw ( W W d W x]. Applying the technique of Ch. IV in [8] and a property extracted from the first reference [5] p. 398, we find out for the density of the volume form: Q = det( W W (n+. (9 Now we determine the scalar product. If f ψ (z := (e z, ψ, then (φ, ψ = Λ fφ (z, W f ψ (z, W QK dzdw (30 z C n ; W W >0;W =W t dz = n dre z i dim z i ; dw = i= i j n dre w ij dim w ij. (3 We take in (30 φ, ψ =, we change the variable z = ( W W / x, we apply equations (A, (A in Bargmann [] and Theorem.3. p. 46 in [8] W W >0,W =W t det( W W λ dw = J n (λ and we find for Λ in (30 (below p := (k 3/ n > : Λ = π n Jn (p, J n (p = n π n(n+ n i= Γ(p + i Γ(p + n + i +. (3

7 A holomorphic representation of the semidirect sum of symplectic and Heisenberg Lie algebras 7 Proposition Let us consider the Jacobi group G J with the composition rule (6, acting on the coherent state manifold (4 via ( (5. The manifold M has the Kähler potential (7 and the G J -invariant Kähler two-form ω given by (8. The Hilbert space of holomorphic functions F K associated to the holomorphic kernel K : M M C given by (0 is endowed with the scalar product (30, where the normalization constant Λ is given by (3 and the density of volume given by (9. Proposition Let h = (g, α G J, and we consider the representation π(h = S(gD(α, g Sp(n, R, α C n, and let the notation x = (z, W D. Then the continuous unitary representation (π K, H K attached to the positive definite holomorphic kernel K defined by (0 is (π K (h.f(x = J(h, x f(h.x, where the cocycle J(h, x = λ(h, x with λ defined by equations (- (5 and the function f belongs to the Hilbert space of holomorphic functions H K F K endowed with the scalar product (30. Comment 3 The value of Λ given by (3 corresponds to the one given in (7.6 in [3], taking above n =, k 4k. Note that p defying the normalization constant Λ in (3 for the Jacobi group is related with q = k n in (φ, ψ FH = Λ W W >0;W =W t fφ (W f ψ (W det( W W q dw (33 defining the normalization constant Λ = Jn (q for the group Sp(n, R by the relation p = q. It is well known [5], [8] that the admissible set for k for the space of functions F H endowed with the scalar product (33 is the set Σ = {0,,, n } ((n,. The integral (33 deals with a non-negative scalar product if k n, in which the domain of convergence k n is included, and the separate points k = 0,,..., n. Acknowledgements The author is grateful to Anatol Odzijewicz for the possibility to attend the Białowieża, Poland meetings in 004 and 005 and to Tudor Ratiu for the hospitality at the Bernoulli Center, EPFL Lausanne, Switzerland, where this investigation was started in November December 004.

8 8 Stefan Berceanu References [] Bargmann V., Group Representations on Hilbert Spaces of Analytic Functions, In: Analytic Methods in Mathematical Physics, R. Gilbert and R. Newton (Eds, Gordon and Breach, New York, London, Paris, 970, pp [] Berceanu S. and Gheorghe A., On Equations of Motion on Hermitian Symmetric Spaces, J. Math. Phys. 33 ( ; Berceanu S. and Boutet de Monvel L., Linear Dynamical Systems, Coherent State Manifolds, Flows and Matrix Riccati Equation, J. Math. Phys. 34 ( [3] Berceanu S., A Holomorphic Representation of the Jacobi Algebra, arxiv: math.dg/04089, v 4 Mar006; to appear in Rev. Math. Phys. [4] Berceanu S., Realization of Coherent State Algebras by Differential Operators, In: Advances in Operator Algebras and Mathematical Physics (Sinaia, 003, F. Boca, O. Bratteli, R. Longo, H. Siedentop (Eds., The Theta Foundation, Bucharest 005, pp -4; arxiv: math.dg/ [5] Berezin F., Quantization in Complex Symmetric Spaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (975, , 47; Berezin F., Models of Gross-Neveu Type are Quantization of a Classical Mechanics with a Nonlinear Phase Space, Commun. Math. Phys. 63 (978, [6] Folland G., Harmonic Analysis in Phase Space, Princeton University Press, Princeton, New Jersey, 989. [7] Helgason S., Differential Geometry, Lie Groups and Symmetric Spaces, Academic, New York, 978. [8] Hua L. K., Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. AMS, Providence, R.I., 963. [9] Knapp A., Representations Theory of Semisimple Lie groups- An overview Based on Examples, Princeton University, Princeton, NJ, 986. [0] Monastyrsky M. and Perelomov A., Coherent States and Bounded Homogeneous Domains, Reports Math. Phys. 6 ( [] Neeb K. H., Holomorphy and Convexity in Lie Theory, de Gruyter Expositions in Mathematics 8, Walter de Gruyter, Berlin-New York, 000. [] Perelomov A., Generalized Coherent States and their Applications, Springer, Berlin, 986. [3] Stoler P., Equivalence Classes of Minimum Uncertainty Packets, Phys. Rev. D ( ;, II, Phys. Rev D 4 (

9 A holomorphic representation of the semidirect sum of symplectic and Heisenberg Lie algebras 9 Stefan Berceanu Institute for Physics and Nuclear Engineering Department of Theoretical Physics PO BOX MG-6, Bucharest-Magurele Romania address: Berceanu@theor.theory.nipne.ro