Generalised Jordan map, symplectic transformations and Dirac's representation of the 3 + 2 de Sitter group
Rabin Banerjee



  



Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences,
 Block-JD, Sector-III, Salt Lake City,
 Kolkata 700106, India






In his 1963 paper, Dirac <cit.> gave `A Remarkable Representation of the 3+2 de-Sitter Group'. We reproduce this representation using a generalised Jordan map which is motivated by the infinitesimal symplectic transformations related to the four dimensional symplectic 
group Sp(4). A physical picture of Dirac's representation is also discussed.  






Generalised Jordan map, symplectic transformations and Dirac's representation of the 3+2 de Sitter group
    Rabin Banerjee[
mailto:rabin@bose.res.inrabin@bose.res.in]
    
========================================================================================================







§ INTRODUCTION

Paul Dirac, in 1963, published a paper on A Remarkable Representation of the 3+2 de-Sitter Group <cit.>. He further showed that, among the infinitesimal generators of this group, there are four cyclic ones while the rest are hyperbolic. The representation is very simple with the wavefunctions depending on two variables. 
 
  The de-Sitter group is an important space-time symmetry in both particle physics and general relativity. It was initially introduced into physics for describing curved background <cit.>, leading to significant cosmological consequences. It has several interesting subgroups and is also the starting point for building representations of the Poincare group for relativistic particles <cit.>. The algebra presented in <cit.> has found many applications <cit.>, especially in providing an explicit example of connecting the standard Schroedinger approach to the quantum mechanics with Wigner's phase space approach <cit.>. Its mathematical construction has been used for building the two mode squeezed states in quantum optics <cit.>. Thus, it has an important role in contemporary physics.

   In this paper we give a simple derivation of Dirac's representation that is based on a generalisation of the well known Jordan-Schwinger map that yields a realisation of SU(2) generators. This generalisation is motivated by the fact that the algebraic structure of Dirac's representation may be understood by infinitesimal symplectic transformations related to the group Sp(4). It is possible to reproduce the complete structure of the representation <cit.> using these transformations. Finally, some other features are discussed that provide new insights into this representation. 
 
 In section II, we briefly review the salient features of Dirac's remarkable representation. This is followed by showing, in section III, that this representation may be obtained from the Sp(4) group using infinitesimal symplectic transformations. In section IV, we show that all results may be simply obtained by using a generalised Jordan-Schwinger type map. Section V discusses some physical aspects of Dirac's representation. Finally, Section VI contains our conclusion, including possible future directions.


§ THE REPRESENTATION
 
Dirac considered a pair of coupled harmonic oscillators. The ladder operators, denoted by a_i and a_i^† satisfy the basic commutation relations, 
    [a_i, a_j] = [a_i^†, a_j^†] = 0


    [a_i, a_j^†] = δ_ij                                  (i,j = 1,2).
 
Now sixteen quadratic operators can be constructed from these a_i and a_i^†. However, because of the relations (<ref>), only ten of these are independent. Appropriate linear combinations of these were used to define the following ten operators <cit.>,  
    L_1 = 12(a_1^†a_2 + a_2^†a_1)    ,    L_2 = i2(a_2^†a_1 - a_1^†a_2)
 
    L_3 = 12(a_1^†a_1 - a_2^†a_2)    ,    H = 12(a_1^†a_1 + a_2a_2^†)
 
    K_1 = -14(a_1^†a_1^† + a_1a_1 - a_2^†a_2^† - a_2a_2)
  
    K_2 = i4(a_1^†a_1^† +  a_2^†a_2^† - a_1a_1 - a_2a_2)
  
    K_3 = 12(a_1^†a_2^† + a_1a_2)

 
    B_1 = -i4(a_1^†a_1^† - a_1a_1 - a_2^†a_2^† + a_2a_2)
 
    B_2 = -14(a_1^†a_1^† +  a_2^†a_2^† + a_1a_1 + a_2a_2)
 

    B_3 = i2(a_1^†a_2^† - a_1a_2)


The first four (L_1, L_2, L_3, H) are cyclic rotations with H separate from the other three, since it commutes with all of them. The remaining six correspond to hyperbolic rotations. These operators provide a representation of the 3+2 de-Sitter group. This is the group of rotations of five real variables q_1, q_2, q_3, q_4, q_5 that keeps the quadratic form, 
    q_1^2+q_2^2+q_3^2-q_4^2-q_5^2
 invariant. Denoting the generators of this group by J_ij, we have, 

    [J_ij, J_kl] = i(J_ikη_jl - J_ilη_jk + J_jlη_ik - J_jkη_il)       i,j = 1, 2, 3, 4, 5   ,  η_ij = (+ + + - -)

The correspondance with the operators (<ref>) is given by, 
    J_ij = ε_ijkL_k     for   i, j, k = 1, 2, 3
 
    J_i4 = K_i        for    i = 1,2,3
 
    J_i5 = B_i        for    i = 1,2,3
  

    J_45 = H



There are three spacial rotation operators (L_1, L_2, L_3), two boosts (K_i, B_i) corresponding to two times and a hamiltonian (H). It may be observed that this H in (<ref>) is half the energy of two harmonic oscillators, as also mentioned in <cit.>. In section V we shall give an explanation for this half factor. The individual algebra satisfied by these operators is given by, 
    [L_i, L_j] = iε_ijkL_k  ,   [L_i, H] = 0
  
    [L_i, K_j] = iε_ijkK_k ,   [K_i, K_j] = -iε_ijkL_k
  
    [L_i, B_j] = iε_ijkB_k  ,   [B_i, B_j] = -iε_ijkL_k


    [K_i, H] = iB_i  ,   [B_i, H] = -iK_i  ,   [K_i, B_j] = iδ_ijH


The operators L_i satisfy the SU(2) algebra. Also, there are three sets of operators (K_1, B_1, H), (K_2, B_2, H) and (K_3, B_3, H) that satisfy the hyperbolic SU(1, 1) algebra. We will return to these issues later.



§ SYMPLECTIC TRANSFORMATIONS AND DIRAC'S REPRESENTATION

Since the 3+2 de-Sitter group is isomorphic to the four dimensional symplectic group Sp(4), the algebra (<ref>) also serves to characterise the latter. This fact was utilised in <cit.> to illuminate the nature of the operators given in (<ref>). It was shown that the unitary transformations generated by these operators are translated into linear canonical transformations of the Wigner function for a pair of coupled oscillators. The corresponding group in this case is just Sp(4). 

 In this section we provide an alternative approach that does not require the introduction of the Wigner function. The structure of the generators (<ref>) is deduced solely from properties of symplectic transformations. Starting from the matrix representation of the ten generators of Sp(4), we construct the corresponding vector fields. From these fields, quadratic homogeneous polynomials involving four phase space variables are obtained. Finally, replacing the phase space variables by the oscillator variables, the ten generators in (<ref>) are reproduced. 
 
 A symplectic matrix M is defined by the condition, 

    MJM^T = J


where M^T is the transpose of M and J has the canonical form, 

    J = 
    [  0  I; -I  0 ]


with I being the identity matrix. For two pairs of canonical variables relevant for Sp(4), J takes the antisymmetric structure, 

    J = 
    [  0  0  1  0;  0  0  0  1; -1  0  0  0;  0 -1  0  0;    ]

Introducing the generators G_i of Sp(4) we have, 

    M = e^-iα_iG_i

where G_i is a set of ten pure imaginary 4×4 matrices. The symplectic condition (<ref>) now implies that G be either antisymmetric and commute with J (first set) or, be symmetric and anticommute with J (second set) <cit.>. 
  
 Using the Pauli spin matrices and the 2×2 identity matrix it is possible to construct the relevant generators. The first set is given by, 

    L_1 = i2[    0  σ_1; -σ_1    0 ]  ,   L_2 = 12[ σ_2   0;   0 σ_2 ]


    L_3 = i2[    0  σ_3; -σ_3    0 ]  ,   H = i2[  0  I; -I  0 ]

Likewise, the second set is given by, 

    K_1 = i2[   0 σ_3; σ_3   0 ]  ,   K_2 = i2[  I  0;  0 -I ]  ,  K_3 = -i2[   0 σ_1; σ_1   0 ]


    B_1 = -12[  σ_3    0;    0 -σ_3 ]  ,   B_2 = i2[ 0 I; I 0 ]  ,  B_3 = i2[  σ_1    0;    0 -σ_1 ]

Using the result, 

    [σ_i, σ_j] = 2iε_ijkσ_k

it is possible to show that the matrices (<ref>, <ref>) satisfy the algebra (<ref>). 

 We now show that the above matrices can be translated into oscillator variables that precisely reproduce the representation (<ref>). Let us consider the space X =  IR^2n =  IR^n +  IR^n  which has the symplectic form w,

    w = Σ dq_i dp_i

Let ξ be the vector field on X and i(ξ) denote the interior product by ξ. Then there is a smooth function H determined up to a local constant such that <cit.>, 

    i(ξ)w = dH
 
Hence, for w defined in (<ref>), the vector field is given by, 

    ξ_H = Σ (∂ H∂ p_i∂∂ q_i -∂ H∂ q_i∂∂ p_i)


The function H is the hamiltonian and determines the evolution of the vector field (<ref>) from the differential equations, 

    dq_idt = ∂ H∂ p_i  ,  dp_idt = -∂ H∂ q_i

The poisson bracket of two functions f_1 and f_2 is defined as, 

    {f_1, f_2} = -ξ_f_1f_2

so that the familiar Hamilton's equations are obtained from (<ref>),

    dq_idt = {q_i, H}  ,  dp_idt = {p_i, H}

Using the group property it can be shown that <cit.>, 

    [ξ_f_1, ξ_f_2] = -ξ_{f_1, f_2}

This is illustrated by taking a simple example. Consider the functions f_1 and f_2 to be the hamiltonians for the harmonic oscillator and free particle. respectively, 

    f_1 = 12(p^2 + q^2)   ,   f_2 = p^22

Then the corresponding vector fields are obtained from (<ref>),

    ξ_f_1 = p∂∂ q - q∂∂ p  ,  ξ_f_2 = p∂∂ q

The vector field corresponding to the Poisson bracket {f_1, f_2} (defined in (<ref>)) turns out to be, 

    ξ_{f_1, f_2} = ξ_qp = q∂∂ q - p∂∂ p

It is now easy to prove (<ref>) using (<ref>) and (<ref>). This shows an isomorphism between the algebra of quadratic homogeneous polynomials (in two variables) and that of their corresponding vector fields. The isomorphism can be pushed further to include the symplectic algebra. To do this we first write the appropriate matrices.  

  For the function f_1 the equations of motion (<ref>) are given by, 

    dqdt = ∂ f_1∂ p = p   ,  dpdt = -∂ f_1∂ q = -q
 
 In matrix notation, 
 
    ddt[ q; p ]
    = 
    [  0  1; -1  0 ][ q; p ]

 Thus the function f_1 corresponds to the vector field ξ_f_1 and the matrix [  0  1; -1  0 ] belonging to the Lie algebra of the symplectic group Sp(2). Likewise we can find the matrices corresponding to f_2 and {f_1, f_2}. Collecting all results, 

    M_f_1 = 
    [  0  1; -1  0 ]  ,   
    M_f_2 = 
    [ 0 1; 0 0 ]  ,   
    M_{f_1, f_2} = 
    [  1  0;  0 -1 ]

It is easy to see that the algebra closes,

    [M_f_1, M_f_2] = M_{f_1, f_2}

Associating a self adjoint operator f_q to each homogeneous quadratic polynomial f, it is possible to construct the map f ⟶ -if_q such that Poisson brackets get replaced by commutators. We then have an isomorphim between the commutator algebra of the quadratic polynomials and the Lie algebra of the symplectic group. Finally, the quadratic polynomials may be equivalently represented in terms of the oscillator variables. Following this approach Dirac's representation (<ref>) is derived starting from the generators (<ref>, <ref>) of the symplectic group Sp(4). 

 Let us explicitly work out one specific example. Take L_1 from the set (<ref>).

    L_1 = i2[    0  σ_1; -σ_1    0 ]
    = i2[  0  0  0  1;  0  0  1  0;  0 -1  0  0; -1  0  0  0 ]

The corresponding vector field is given by, 

    ξ_L_1 = i2(p_2∂∂ q_1 + p_1∂∂ q_2 - q_2∂∂ p_1 - q_1∂∂ p_2)
 
 The homogeneous quadratic polynomial leading to (<ref>) is obtained from (<ref>), 
 
    L_1 = i2(p_1p_2 + q_1q_2)

 As a cross check we reproduce (<ref>) from (<ref>). The equations of motion (<ref>) are,
 
    dq_1dt = ∂ L_1∂ p_1 = i2p_2 , dq_2dt = i2p_1 , dp_1dt = -i2q_2 , dp_2dt = -i2q_1

 so that, 
 
    ddt[ q_1; q_2; p_1; p_2 ]
     = i2[  0  0  0  1;  0  0  1  0;  0 -1  0  0; -1  0  0  0 ][ q_1; q_2; p_1; p_2 ]

thereby yielding the matrix (<ref>). Next, replacing L_1 by -iL_1 and substituting q, p by corresponding operators, we obtain the operator version of (<ref>), 

    L̂_̂1̂ = 12(p̂_̂1̂p̂_̂2̂ + q̂_̂1̂q̂_̂2̂)
 
which is already hermitian since, 

    [q̂_̂î, p̂_̂ĵ] = iδ_ij  ,   [q̂_̂î, q̂_̂ĵ] = [p̂_̂î, p̂_̂ĵ] = 0
 
Introducing the ladder (creation and destruction) operators for oscillators, 

    q̂_̂î = 1√(2)(a_i + a^†_i)   ,  p̂_̂î = -i√(2)(a_i - a^†_i)
 
 we finally obtain from (<ref>), 
 
    L̂_̂1̂ = 12(a^†_1a_2 + a^†_2a_1)

 which reproduces the first relation in (<ref>). 

  In cases, unlike (<ref>), which are not hermitian, one has to just add the hermitian conjugate. As an example, consider K_2 from (<ref>),
 
    K_2 = i2[  1  0  0  0;  0  1  0  0;  0  0 -1  0;  0  0  0 -1 ]

 The vector field is, 
 
    ξ_K_2 = i2(q_1∂∂ q_1 + q_2∂∂ q_2 - p_1∂∂ p_1 - p_1∂∂ p_2)
 
 and the corresponding polynomial follows from (<ref>),
 
    K_2 = i2(q_1p_1 + q_2p_2)

 The operator version is given by, 
 
    K̂_̂2̂ = 14(q̂_̂1̂p̂_̂1̂ + p̂_̂1̂q̂_̂1̂ + q̂_̂2̂p̂_̂2̂ + p̂_̂2̂q̂_̂2̂)

 where we have included the hermitian conjugate so that K̂_̂2̂ is hermitian. Using the basic brackets (<ref>), this reduces to, 
 
    K̂_̂2̂ = 12(q̂_̂1̂p̂_̂1̂ + p̂_̂2̂q̂_̂2̂)

 Translating into the oscillator variables (<ref>), this reproduces the corresponding result in (<ref>). 
 
  In this manner it is possible to derive the complete representation (<ref>) starting from (<ref>, <ref>). 
 

§ GENERALISED JORDAN-TYPE MAP

It is useful to note that the matrices (<ref>, <ref>) which eventually led to the representation (<ref>) are all composed of the Pauli matrices and the identity matrix. This suggests the possibility of obtaining (<ref>) directly from the Pauli and identity matrices. Here we show how this can be done. 
 
 Before coming to the actual solution, let us first show that there are other possible representations of the de-Sitter group in terms of oscillator variables, which are different from (<ref>). This is based on Jordan's map,

    M̂_̂î = 12a^†M_ia

which states that the algebra of the operators M̂_̂î is isomorphic to that of the matrices M_i, where a and a^† are the usual ladder operators. In other words if the M_i satisfy, 

    [M_i, M_j] = ω_ijkM_k
 
where ω_ijk are some structure constants, then, 

    [M̂_̂î, M̂_̂ĵ] = ω_ijkM̂_̂k̂

so that M̂_̂î may be interpreted as an operator representation of the matrices M_i. Since we have earlier provided the ten generators of Sp(4) in matrix form (<ref>, <ref>), it is straightforward to compute their corresponding operator versions from (<ref>). Thus for L_1 (<ref>), we have, 

    L̂_̂1̂ = i4(a_1^†a_4+a_2^†a_3-a_3^†a_2-a_4^†a_1)
 
Likewise, it is possible to obtain the relevant operators for all the matrices (<ref>, <ref>). Clearly, these will satisfy the de-Sitter algebra (<ref>).

 The structure found in this way is obviously different from the representation (<ref>). It now involves four coupled oscillators instead of two. Among other things, the representation (<ref>) is minimal in the sense that, to construct ten independent quadratic operators in the ladder variables, the minimum number of the oscillators needed is two. It is now clear that to reproduce (<ref>) in this fashion, we have  to work with (2×2) matrices instead of (4×4)ones. As indicated, the obvious choice would be the Pauli matrices and the (2×2) identity matrix.

 There is a well known representation of the SU(2) generators in terms of the oscillator (ladder) operators. This is called the Jordan-Schwinger map. It is a special version of (<ref>) with M_i replaced by the Pauli matrices and M̂_̂î by L_i, the angular momentum operator,

    L_i = 12 a^†σ_ia
  
from which follows, 

    [L_i, L_j] = iε_ijkL_k

 which is the first relation in (<ref>). Indeed, using the explicit structure of the Pauli matrices, L_1, L_2 and L_3 exactly reproduce the corresponding relations in (<ref>). 
 
  The point is that (<ref>) is not the only possible way of expressing a hermitian operator in terms of a product of  the Pauli matrices and quadratic functions of the oscillator variables. For instance, we can have the form, 
 
    W_i = 14(a^†σ_ia^† + aσ_ia)

 which, like (<ref>), is also hermitian. Inserting the explicit form for the σ-matrices, we find, 
 
    W_1 = K_3  ,   W_3 = -K_1

Note that W_2 = 0 due to reasons of symmetry. Another hermitian construction, similar to (<ref>), is given by, 

    Z_i = i4(a^†σ_ia^† - aσ_ia)
  
 We find, 
 
    Z_1 = B_3  ,   Z_3 = -B_1

while Z_2 = 0. This exhausts possible (quadratic) hermitian combinations involving the Pauli matrices. Now we consider the identity matrix. 

 The analogues of (<ref>) and (<ref>) involving the identity matrix are given by,

    W = 14(a^†_ia^†_i + a_ia_i)
  
 and, 

    Z = i4(a^†_ia^†_i - a_ia_i)
 
 which yield, 
 
    W = -B_2  ,   Z = K_2

 Finally, we have the structure,
 
    H = 14(a^†_ia_i + a_ia^†_i)

 which reproduces the corresponding expression in (<ref>). Thus the complete representation (<ref>) may be deduced by using a generalisation of the Jordan map to include all possible quadratic combinations of the oscillator operators sandwitching the Pauli and identity matrices. These combinations have to be hermitian.
 

§ PHYSICAL INTERPRETATION

 In this section we discuss certain features of the representation (<ref>) that manifest a physical structure. In the cyclic sector comprising L_1, L_2, L_3 and H, while L_1 and L_2 have coupled expressions, L_3 and H correspond to decoupled oscillators. Indeed it is possible to reproduce these expressions by considering a two dimensional oscillator as a pair of chiral oscillators, one rotating in the clockwise direction while the other in the anticlockwise direction. 
 
  The lagrangian for a one-dimensional harmonic oscillator is given by, 
 
    L = 12(q̇^2 - q^2)
  
 By introducing an additional variable it can be converted into its first order form <cit.>, 
 
    L_± = 12(± ε_αβq_αq̇_̇β̇ - q^2_α)

 where α = 1, 2 is an internal index with ϵ_12 = 1. Eliminating either q_1 or q_2 in favour of the other yields (<ref>) with q = q_1 or q_2. The lagrangians L_± are a pair of chiral oscillators with the sign of the first term determining their chirality. 

  Although the lagrangians (<ref>) each seem to have two degrees of freedom, only one of them is independent due to the presence of constraints which imply the symplectic structure,
 
    {q_α, q_β} = ∓ ε_αβ

 for L_±. The corresponding hamiltonians are identical due to the first order nature of (<ref>), 
 
    H_± = 12(q_1^2 + q_2^2) = H̃

 The angular momentum is given by, 
 
    J_± = ε_αβq_αp_β

 where the canonical momenta p_β is, 
 
    p_β = ∂ L_±∂q̇_̇β̇ = ± 12ε_ρβq_ρ

 Thus, 
 
    J_± = ± 12q_α^2 = ± H̃

 so that the angular momenta have the same magnitude, but differ in sign, a consequence of chirality. 

  Let us now show explicitly how L_+ and L_- may be combined to yield a two dimensional oscillator. Consider the following combination of L_+(q_α) and L_-(r_α),
 
    L = L_+(q_α) + L_-(r_α) = 12ε_αβq_αq̇_̇β̇ - 12q_α^2 - 12ε_αβr_αṙ_̇β̇ - 12r_α^2

 Introducing new variables, 
 
    q_α + r_α = Q_α  ,   q_α - r_α = R_α

 we get,

    L = 12ε_αβR_αQ̇_̇β̇ - 14(Q_α^2 + R_α^2)
 
where a total time derivative term has been dropped. Eliminating either R or Q in favour of the other yields, 

    L = 14(Ż_̇α̇^2 - Z_α^2)    ;   Z_α = R_α  or   Q_α
 
 which is the standard lagrangian for the two dimensional oscillator but with an averall normalisation of 12. 
 
  From (<ref>, <ref>) we find that the hamiltonians for the chiral oscillators have identical expressions as normal oscillators.footnote^1 Hence, accounting for the 12 factor mentioned above, the final hamiltonian has the form 

 [b]1
                   

 15mm0.005pt

footnote^1Note that the canonical pairs (q, p) for L_+ and L_- are (q_2, q_1) and (q_1, q_2), respectively.
 

 
    H = 12(H_+ + H_-) = 12((a^†_1a_1 + 12) + (a^†_2a_2 + 12))   = 12(a^†_1a_1 + a^†_2a_2 + 1)

 while the angular momentum, which is the difference between H_+ and H_-, becomes, 
 
    J = 12((a^†_1a_1 + 12) - (a^†_2a_2 + 12)) = 12(a^†_1a_1 - a^†_2a_2)

These expressions match with the corresponding ones in (<ref>) (i.e. H and L_z). Thus two chiral oscillators rotating in th x-y plane in opposite (clockwise and anticlockwise) directions effectively yield the hamiltonian (H) and angular momentum (L_z or L_3) given in (<ref>). From (<ref>) it is seen that all eigenvalues of the hamiltonian are positive with the minimum being 12, corresponding to the zero point energy of the two chiral oscillators. If they had been usual oscillators then this minimum would have been 1. Likeiwse, the z-component of the angular momentum (<ref>) can have both integral and half integral values with the sign determined by the difference in the number of clockwise and anticlockwise rotating chiral oscillators. 

 A similar analysis may be done to understand the physical content of the remaining generators containing only uncoupled terms (like K_1, K_2, B_1, B_2). Incidentaly, (K_1, B_1, H) and (K_2, B_2, H) satisfy an algebra isomorphic to SU(1,1), as easily checked from (<ref>). Thus the starting point is to consider the basic matrices defining this algebra.
 
Consider the following 2×2 matrices,

    S_1 = i2[  1  0;  0 -1 ]  ,   
    S_2 = i2[  0  1; -1  0 ]  ,   
    S_3 = i2[ 0 1; 1 0 ]

 which satisfy the SU(1,1) algebra,
 
    [S_1, S_2] = iS_3  ,   [S_2, S_3] = iS_1  ,   [S_3, S_1] = -iS_2

Following the technique discussed in sction III, the vector field corresponding to S_1 is given by,

    ξ_S_1 = i2(q∂∂ q - p∂∂ p)
 
Using (<ref>), the relevant polynomial generating this vector is found to be, 

    S_1 = i2(qp)
 
and the corresponding hermitian operator is,  
 
    Ŝ_̂1̂ = -14(q̂p̂ + p̂q̂)

 Translating into oscillator variables, 
  
    Ŝ_̂1̂ = i4(a^†2 - a^2)

 Similarly S_2 and S_3 translate into,
 
    Ŝ_̂2̂ = 14(a^†a + aa^†)   ,  Ŝ_̂3̂ = -14(a^†2 + a^2)

 The operators Ŝ_̂1̂,Ŝ_̂2̂ and Ŝ_̂3̂ satisfy the same algebra as (<ref>). 
 
  We may now identify, with appropriate modifications, the set Ŝ_̂1̂, Ŝ_̂2̂, Ŝ_̂3̂ with K_2, H and B_2, respectively, as defined in (<ref>). One has to just consider two independent oscillators instead of a single one. Then the transition of Ŝ_̂1̂ and Ŝ_̂3̂ to K_2 and B_2 is straightforward. For Ŝ_̂2̂, taking two independent oscillators yields, 
 
    Ŝ_̂2̂|_1 +  Ŝ_̂2̂|_2 = 14(a^†_1a_1 + a_1a^†_1 + a^†_2a_2 + a_2a^†_2 )

 
    = 12(a^†_1a_1 + a_2a^†_2)

 obtained on using basic commutation relations. The final form is just H in (<ref>). Incidently, the distinctive roles of SU(2) and SU(1) groups in interferometry has been discussed in <cit.>. Here we see another example in the case of the de-Sitter representation (<ref>).
 

§ CONCLUSION

In this paper we have looked at the remarkable representation (<ref>) of the 3+2 de-Sitter group found by Dirac <cit.>. Although this work was done in 1963, its importance can be gauged from the fact that it is still relevant in contemporary physics. Among other applications <cit.>, this construction became the basic mathematical language for discussing two mode squeezed states in quantum optics <cit.>. 

Since the algebra of the 3+2 de-Sitter group is isomorphic to the four dimensional symplectic group Sp(4), it is usually customary to analyse the representation (<ref>) through this group. This is because the symplectic group, being the group of canonical transformations, provides a natural connection between physics and mathematics.

 The matrices representing the generators of Sp(4) were translated into homogeneous quadratic polynomials involving a pair of phase space variables (q_1, p_1, q_2, p_2). The step by step sequence of this construction was discussed. Once the polynomial was obtained, it was straightforward to express these in terms of the oscillator (ladder) operators using the standard expressions. In this way the complete representation (<ref>) was obtained.

 It may be recalled that, in refs.<cit.>, the set (<ref>) was derived by using the ten matrices of Sp(4) and performing canonical transformations on Wigner's phase space function involving a pair of canonical variables. Our derivation is more geometrical and does not need the Wigner function. While the connection of this function with density matrices is well known [ ], the present analysis reveals its geometrical origin. Indeed the vector fields found here by symplectic considerations agree with those found by Wigner's phase space approach <cit.>.

 Motivated by the use of Pauli and identity matrices in the building of the Sp(4) representation, we have given a simple derivation of (<ref>) that generalises the well known Jordan-Schwinger map for SU(2) matrices. By constructing the most general quadratic hermitian combinations of the ladder operators with the Pauli and identity matrices, this result was obtained. Effectively, the set (<ref>) may be interpreted as this generalised Jordan type map for the de-Sitter group. Since SU(2) is a subgroup of the de-Sitter group, the usual Jordan map is just a subset of this map that is restricted to the three operators L_1, L_2 and L_3 in (<ref>).

 A physical interpretation of the results has also been provided. It is easy to see from (<ref>) that there are some operators that involve a coupling of the pair of oscillators, but some are uncoupled. The uncoupled set L_3 and H was shown to fit in the interpretation of regarding (a_1, a_1^†) and (a_2, a_2^†) as the ladder operators corresponding to chiral oscillators rotating in clockwise and anticlockwise directions. Furthermore, this interpretation explains the appearance of the half factor in the hamiltonian H in (<ref>). Also, some other operators of (<ref>) were shown to be a simple addition of the corresponding operators for two independent oscillators whose ladder operators are used to define the SU(1,1) representation. 

  The techniques developed here may be generalised in a systematic way to higher dimensions. For example, knowing the twenty one generators of Sp(6) (Sp(2n) has 2n^2+n generators) it would be possible to build a similar representation as (<ref>) comprising three coupled oscillators. Like (<ref>), this would also be the optimal or minimal representation since the number of independent quadratic functions of a_i, a_i^† (i = 1, 2, 3) is exactly twentyone. Similarly, with more operators at hand, other generalised Jordan type maps analogous to (<ref>) and (<ref>) can be constructed. It is known that the Wigner phase space approach gets connected to other formalisms (like the Schoedinger approach) through density matrices. Our symplectic analysis provides another route, which is complimentary to the convensional group theoretic approach <cit.>. This aspect may be studied further by looking at specific problems in quantum mechanics.

                                        

                                

Acknowledgements: I thank Ankur Srivastav for typing the manuscript. 

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