The relation between the symplectic group Sp(4, \mathbb{R}) and its Lie algebra: its application in polymer quantum mechanics
TThe relation between the symplectic group Sp (4 , R ) and its Lie algebra: itsapplication in polymer quantum mechanics Guillermo Chac´on-Acosta ∗ Departamento de Matem´aticas Aplicadas y Sistemas,Universidad Aut´onoma Metropolitana Cuajimalpa,Vasco de Quiroga 4871, Ciudad de M´exico 05348, MEXICO
Angel Garcia-Chung † Departamento de F´ısica, Universidad Aut´onoma Metropolitana - Iztapalapa,San Rafael Atlixco 186, Ciudad de M´exico 09340, M´exico andUniversidad Panamericana,Tecoyotitla 366. Col. Ex Hacienda Guadalupe Chimalistac,C.P. 01050 Ciudad de M´exico, M´exico
In this paper, we show the relation between sp (4 , R ), the Lie algebra of the symplecticgroup, and the elements of Sp (4 , R ). We use this result to obtain some special cases ofsymplectic matrices relevant to the study of squeezed states. In this regard, we provide someapplications in quantum mechanics and analyze the squeezed polymer states obtained fromthe polymer representation of the symplectic group. Remarkably, the polymer’s dispersionsare the same as those obtained for the squeezed states in the usual representation. Contents
I. Introduction II. Sp (4 , R ) group analysis and sp (4 , R ) Lie algebra III. Quantum relations and examples sp (4 , R ) and P (2 , R ) 9B. Examples 101. Case a , c (cid:54) = and b = . 102. Case c = a = diag( a , a ) and b =diag( b , b ). 113. Case a = c = and b (cid:54) = . 12 ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ m a t h - ph ] F e b IV. Quantum representation and its applications Sp (2 n, R ) 13B. Schr¨odinger representation of the squeeze operator for a bi-partite system. 14C. Covariance matrix for squeezed states 17 V. Squeezed states in polymer quantum mechanics VI. Conclusions VII. Acknowledgments VIII. Appendix: calculation of m n IX. Appendix: Series analysis X. Appendix: Covariance matrix coefficients References I. INTRODUCTION
Squeezed states are broadly used in many areas of physics [1–5]. Of particular interest is theuse of these squeezed states in cosmology [6–14], specifically when arguing for the emergence ofsemi-classical behavior in the early universe. Loop Quantum Cosmology (LQC) [15–19] is anotherscenario in which squeezed states are relevant. There, squeezed states for a single-mode showsome of the features of the quantum bounce and closely approximate solutions to the classicalEinstein equations [20–25]. The squeezed states used in LQC are constructed by hand, imposing theGaussian form of the states to obtain the squeezing nature of the dispersion relations. Moreover,the states describe systems with only one degree of freedom, i.e., single-mode squeezed states[26, 27].In quantum optics, squeezed states can be used to improve the sensitivity of measurementdevices beyond the usual quantum noise limits [2–5, 28–31]. They are defined by the squeezeoperators’ action on coherent states, or the vacuum state [2–4]. These operators are definedwithin the Fock representation using the annihilation and creation operators, or in the Wignerrepresentation, using the Wigner functional. A particular squeezed state used in quantum opticsis the two-mode squeezed state which plays a prominent role in the study of entanglement forbipartite systems. Particularly in the limit when the amount of squeezing is infinitely large, thestates become EPR-like states [32].Based on the relevance that squeezed states play in cosmology, LQC, and quantum optics, onemight ask whether there is a relation between them and whether it is possible to obtain squeezedstates in LQC the same way squeezed states are defined in quantum optics. Recall that theconstruction used in LQC for the squeezed states is somewhat artificial and does not correspondto any mechanism in the cosmological events. Hence, exploring whether LQC formulation admitsan operator similar to the squeeze operator and whose action on some state yields a squeezed statemight pave the way to construct such a mechanism in LQC.To do so, one must consider that in the LQC, the representation of the operators is not weaklycontinuous, hence the Fock representation is not suitable for the physical description. Instead, theSchr¨odinger representation, which is the scheme inherited from the quantization procedure, seemsto be the natural scheme to be considered [33–38]. Despite the Schr¨odinger representation of thesqueeze operator might be obtained using the representation of the infinitesimal squeeze operatorvia the exponential map [39], in LQC this cannot be done. In addition to the mathematicalchallenge that this operation requires in standard quantum mechanics, this is not possible inLQC because there is no infinitesimal representation of the squeeze operator. Therefore, in thepresent work, we will use the representation of the symplectic group Sp (2 n, R ) in polymer quantummechanics [38]. It is worth mentioning that Polymer Quantum Mechanics (PQM) can be consideredas a “toy model” of LQC because they share Hilbert spaces with the same mathematical structures.As a result, the representation of the symplectic group in PQM is mathematically the same as inLQC.To analyze the squeeze operator corresponding to the bipartite squeezed states in LQC, we willprovide the relation between the Lie algebra of the symplectic group sp (4 , R ) and the Lie group Sp (4 , R ). As far as the authors’ knowledge, this relation has not been reported before. Withthis result, we show some specific cases and then move to the analysis within polymer quantummechanics. Also, this relation allows us to describe the single-mode squeeze operator (specifically,the product of two single-mode operators) as a particular case of a symplectic matrix in Sp (4 , R ).We will show that the squeezed states derived in this way for LQC share the same featuresas those used in quantum optics. In particular, the correlations’ structure is the same for boththe single-mode and the two-mode squeezed states. However, there is no need for a Gaussian-likestructure for the initial states upon which the polymer squeeze operators act and such structure isabsent in the polymer squeezed states.This paper is organized as follows: in section (II) we calculate the relation between sp (4 , R )and Sp (4 , R ). In section (III) we discuss the isomorphism between sp (4 , R ) and the second-orderpolynomial operators P (2 , R ) and provide some examples. In Section (IV), we show some of theapplications of the results given in section III; in particular, we determine the covariance matrixfor the squeezed states in standard quantum mechanics. In Section (V), we analyze the squeezeoperators’ representation in polymer quantum mechanics and construct the polymer squeezed state.We also calculate the dispersion relations and show that they are equal to those obtained for thestandard squeezed states. We give our conclusion in Section (VI). II. Sp (4 , R ) GROUP ANALYSIS AND sp (4 , R ) LIE ALGEBRA
In this section we will detail the relation between an arbitrary element of sp (4 , R ) and itscorresponding element in the group Sp (4 , R ). This relation is the main result of this section andhas not been reported as far as we know. First, let us introduce some preliminary concepts andnotation, which we will use throughout the paper, to make the presentation self-contained.Let us begin by considering the Poisson manifold ( R n , { , } ) with Poisson bracket for the coor-dinates q j and momenta p j ( j = 1 , , , . . . , n ) given by { q j , q k } = 0 , { p j , p k } = 0 , { q j , p k } = δ jk . (1)These coordinates are collected using the array (cid:126)Y T = ( q , p , q , p , . . . , q n , p n ) for which thePoisson bracket (1) takes the form (cid:110) (cid:126)Y , (cid:126)Y T (cid:111) = J 0 · · ·
00 J · · · ... ... ... ... · · · J = n × n ⊗ J , (2)where the is the 2 × n × n is the identity matrix and the matrix J is given by J = − . (3)The group action over the manifold R n is Sp (2 n, R ) × R n → R n ; (cid:16) M , (cid:126)Y (cid:17) (cid:55)→ (cid:126)Y (cid:48) T = M (cid:126)Y T , (4)provided that the matrix M satisfies the condition( n × n ⊗ J ) = M ( n × n ⊗ J ) M T , (5)where M T is the transpose matrix. That is, the symplectic group Sp (2 n, R ) can be defined asthe set of 2 n × n real matrices satisfying (5) and, additionally, its group action on the Poissonmanifold ( R n , { , } ) is given by (4). Note that a “coordinatization” of ( R n , { , } ) different from (cid:126)Y yields a condition for the symplectic group matrices different to that in (5). To show this, considernow the array (cid:126)X T = ( (cid:126)q T , (cid:126)p T ) where (cid:126)q T = ( q , q , . . . , q n ) and (cid:126)p T = ( p , p , . . . , p n ) are thecoordinates on the space R n . The Poisson bracket for this array is given by (cid:110) (cid:126)X, (cid:126)X T (cid:111) = n × n − n × n = J ⊗ n × n . (6)The group action is now given by Sp (2 n, R ) × R n → R n ; (cid:16)(cid:102) M , (cid:126)X (cid:17) (cid:55)→ (cid:126)X (cid:48) where (cid:126)X (cid:48) is (cid:126)X (cid:48) T = (cid:102) M (cid:126)X T , (7)and the matrix (cid:102) M satisfies ( J ⊗ n × n ) = (cid:102) M ( J ⊗ n × n ) (cid:102) M T . (8)Hence, both conditions (5) and (8), can be considered as definitions for the symplectic group indifferent “coordinatizations” of the phase space R n . Naturally, both group actions (cid:102) M and M arerelated via the similarity transformation Γ ( n ) [4] as (cid:102) M = Γ ( n ) M Γ − ( n ) , (9)where Γ ( n ) is given by (cid:126)X T = Γ ( n ) (cid:126)Y T , (10)and is such that Γ T ( n ) = Γ − ( n ). Since the present work concerns the case where n = 2, it isworth showing the explicit form of Γ (2) which is Γ (2) = . (11)Having provided the two group actions over the manifold R n using different “coordinatizations”and their relation for arbitrary n , let us now focus on the symplectic group Sp (4 , R ). According to(5) this group is given by 4 × M for which the following condition holds J J = M J J M T , (12)The matrix M can be written in block form as M := A BC D , (13)where the 2 × A , B , C and D satisfy the conditions J = AJA T + BJB T = CJC T + DJD T , = AJC T + BJD T , (14)which result from (12).The Lie algebra of Sp (4 , R ), denoted as sp (4 , R ), is given by 4 × m such that theexponential map [39] of the Lie algebra element m yields symplectic matrices M close to theidentity, i.e., M = e m := + m + 12 m + · · · + 1 n ! m n + . . . (15)It can be shown that the matrices in sp (4 , R ) can be written as the product m = J J L , (16)where L is a real symmetric matrix written in block form as L = a bb T c , (17)and where b is a 2 × a and c are also real but 2 × M can be written as in (15), then its inverse M − , its transpose M T and the n -powermatrix ( M ) n , can be written respectively as follows M − = exp − J J L , (18) M T = − J J M − J J , (19)( M ) n = exp J J ( n L ) . (20)Thus, the Lie algebra multiplication in sp (4 , R ) is given by the matrix commutator [ , ] m . Whenthis multiplication acts on two arbitrary elements m and m gives the element m defined as m := [ m , m ] m = J J L , J J L = J J L , (21)where the matrix L is also a real symmetric matrix with components of the form L = a Ja + b Jb T − a Ja − b Jb T a Jb + b Jc − a Jb − b Jc b T Ja + c Jb T − b T Ja − c Jb T c Jc + b T Jb − c Jc − b T Jb , (22)hence, m is clearly an element in sp (4 , R ).Up to this point, we introduced the main concepts and notations required to derive the relationbetween sp (4 , R ) and its corresponding Lie group Sp (4 , R ). Let us proceed then to obtain theexplicit relation between the block matrices A , B , C and D and the Lie algebra element L . Itis worth noting that the following procedure can be applied to higher-order symplectic groups Sp (2 n, R ) for n ≥
3, being this the main reason for its exposition in this section.Let us collect the even and odd terms of the expansion in (15) as follows M = (cid:20) + 12! m + · · · + 1(2 n )! m n + . . . (cid:21) + m (cid:20) + 13! m + · · · + 1(2 n + 1)! m n + . . . (cid:21) , (23)where m takes the form m = − (det a + det b ) × Jd − Jd T − (det b + det c ) × , (24)and the matrix d is defined as d = aJb + bJc . (25)As can be seen from the expansion (23), to obtain the expression for M we need first todetermine m n . In Appendix VIII we obtain the expression for m n given in Eq. (111). Let usreplace this result in the series expansion (23), which, after collecting the even and odd terms,gives the following A = α ( e ) + ( α ( o ) − β ( o ) det b ) J a + β ( o ) J b J c J b T , (26) B = ( γ ( o ) − β ( o ) det a ) J b + β ( e ) ( J a J b + J b J c ) + β ( o ) J a J b J c , (27) C = ( α ( o ) − β ( o ) det c ) J b T + β ( e ) ( J b T J a + J c J b T ) + β ( o ) J c J b T J a , (28) D = γ ( e ) + ( γ ( o ) − β ( o ) det b ) J c + β ( o ) J b T J a J b . (29)The coefficients α ( e ) , α ( o ) , β ( e ) , β ( o ) , γ ( e ) and γ ( o ) were defined in the appendix IX.These expressions for the matrices A , B , C and D link the components of the Lie algebraelement L with the corresponding symplectic matrix M and constitute the main result of thissection. Note also the “non-linear matrix relation” between the Lie algebra elements and thegroup elements, particularly the role of the block matrix b .A remarkable and direct application of this result is that it allows us to compute the symplecticeigenvalues of the matrix M . To do so, recall that the characteristic polynomial for a 4 × M with det( M ) = 1 can be written in terms of the trace of its first three powers by the expressionΛ − (Tr( M ))Λ + (cid:2) (Tr( M )) − (Tr( M )) (cid:3) + (cid:20) − (Tr( M )) M ))(Tr( M ))2 + − (Tr( M ))3 (cid:21) Λ + 1 = 0 . (30)where Λ are the eigenvalues of the arbitrary matrix M and Tr( M ) is the trace of the matrix.Using the relations (26)–(29) we obtain that the trace Tr( M ) is given byTr( M ) = 2( α ( e ) + γ ( e ) ) = cosh (cid:16)(cid:112) λ + (cid:17) + cosh (cid:16)(cid:112) λ − (cid:17) , (31)where α ( e ) and γ ( e ) are given in (129) and (133), respectively, and eigenvalues λ ± are given in(120). Moreover, due to the linearity of the trace and the relation (20), we can verify that Tr( M n )is given by Tr( M n ) = cosh (cid:16) n (cid:112) λ + (cid:17) + cosh (cid:16) n (cid:112) λ − (cid:17) , (32)from which we obtain the expressions for Tr( M ) and Tr( M ).Inserting all these expressions in (30) for a symplectic matrix of the form (15), and whencalculating the roots of the polynomial, we obtain that their eigenvalues are given byΛ = cosh (cid:16)(cid:112) λ + (cid:17) − sinh (cid:16)(cid:112) λ + (cid:17) = e − √ λ + , (33)Λ = cosh (cid:16)(cid:112) λ − (cid:17) − sinh (cid:16)(cid:112) λ − (cid:17) = e − √ λ − , (34)Λ = sinh (cid:16)(cid:112) λ − (cid:17) + cosh (cid:16)(cid:112) λ − (cid:17) = e √ λ − , (35)Λ = sinh (cid:16)(cid:112) λ + (cid:17) + cosh (cid:16)(cid:112) λ + (cid:17) = e √ λ + , (36)Note that if λ + > λ − , then the eigenvalues are ordered as Λ < Λ < Λ < Λ which coincideswith the results of Williamson’s theorem [40, 41].In [26, 42] an alternative (and different) formulation for each of the symplectic group genera-tors was provided. Our approach, however, not only reproduces to the same expressions for thegenerators but also provides a direct relation with the Lie algebra matrix generators a , b and c apoint that is absent in [26, 42]. As a result, we can relate a broader range of Lie algebra elementswith their corresponding group elements.Let us now show some of the relevant matrices and examples in the next section in which thisresult can be applied. III. QUANTUM RELATIONS AND EXAMPLES
This section provides three examples where the relation between the Lie algebra element andthe group element is explicit. However, before proceeding, let us introduce additional concepts andnotations (see [4, 39] for more details), which will be relevant for the quantum description.
A. Relation between sp (4 , R ) and P (2 , R ) Consider the Lie algebra formed by second-order (operator) polynomials, denoted by P (2 , R ).An arbitrary element (cid:98) s is given as (cid:98) s = − i (cid:126) (cid:104) a (cid:98) q + a (cid:98) q (cid:98) p + (cid:98) p (cid:98) q ) + a (cid:98) p + b (cid:98) q (cid:98) q + b (cid:98) q (cid:98) p + b (cid:98) p (cid:98) q + b (cid:98) p (cid:98) p ++ c (cid:98) q + c (cid:98) q (cid:98) p + (cid:98) p (cid:98) q ) + c (cid:98) p (cid:105) . (37)Here, (cid:98) q j and (cid:98) p j , with j = 1 ,
2, are the position and momenta operators satisfying the canonicalcommutation relations [ (cid:98) q j , (cid:98) p k ] = i (cid:126) δ j,k , and a ij , b ij and c ij are all real coefficients. The reason forthis notation is that formally i (cid:126) (cid:98) s is a self-adjoint operator to be represented in a Hilbert space H , hence the exponential map e (cid:98) s gives rise to a unitary operator in H . Thus, in this sense, thisnotation smoothes the way to the quantum representation analysis in section IV.It is easy to check that (cid:98) s can be written in the following form (cid:98) s = − i (cid:126) (cid:98) R T a a b b a a b b b b c c b b c c (cid:98) R , (38)that is a symmetric matrix, and where (cid:98) R T = (cid:16) (cid:98) q (cid:98) p (cid:98) q (cid:98) p (cid:17) . (39)Instead of the matrix commutator, the Lie algebra multiplication in P (2 , R ) is given by theoperator commutator [ , ]. Therefore, the Lie algebra multiplication of two elements (cid:98) s and (cid:98) s givesa third element (cid:98) s of the form (cid:98) s = [ (cid:98) s , (cid:98) s ] = − i (cid:126) (cid:98) R T L (cid:98) R , (40)where the matrix L is given by (22). Due to L is a symmetric matrix, the operator (cid:98) s is clearlyin P (2 , R ). Naturally, this result provides the isomorphism between sp (4 , R ) and P (2 , R ), i.e., the0map ι : sp (4 , R ) → P (2 , R ); m (cid:55)→ (cid:98) s = ι ( m ) = − i (cid:126) (cid:98) R T L (cid:98) R , (41)and this map preserves the linear properties of both Lie algebras, i.e., it is a Lie algebra isomor-phism.An implication of this isomorphism is that due to P (2 , R ) is a Lie algebra isomorphic to sp (4 , R ),then the exponential map of its elements ( (cid:98) s (cid:55)→ e (cid:98) s ) gives a (quantum) unitary operator ( e (cid:98) s ) whichcan be seen as the (quantum) unitary representation of Sp (4 , R ) as showed in the following diagram sp (4 , R ) ι ⇐ == ⇒ P (2 , R ) (cid:121) (cid:121) Sp (4 , R ) −→ (cid:99) Sp (4 , R ) (42)We can expect that if a representation of P (2 , R ) in a Hilbert space H is known, then there is alsoa representation of (cid:99) Sp (4 , R ) in H . However, in some scenarios like in polymer quantum mechanicsand LQC, it is not possible to obtain the representation of (cid:98) S out of the representation of (cid:98) s in H .The reason is that some elements of P (2 , R ) cannot be represented in the corresponding Hilbertspace. This difficulty can be overcome if we can represent directly the exponential e (cid:98) s instead ofits infinitesimal generator (cid:98) s . This approach was done for the case of polymer quantum mechanicsin [38]. Consequently, due to the operators in (37) can be used to describe the dynamics of manyphysical systems ranging from two decoupled quantum harmonic oscillators to the bipartite squeezeoperators, a polymer representation of these operators is possible, as we will show in section V.More details about these aspects will be provided in section V.In the next subsection, we show some of the explicit forms of M . B. Examples
1. Case a , c (cid:54) = and b = . Let us consider the Lie algebra element with b = and a , c (cid:54) = 0, which, according to theexpression (38), implies that there is no interaction between the subsystems, that is, (cid:98) s is of theform (cid:98) s = − i (cid:126) (cid:2) a (cid:98) q + a ( (cid:98) q (cid:98) p + (cid:98) p (cid:98) q ) + a (cid:98) p + c (cid:98) q + c ( (cid:98) q (cid:98) p + (cid:98) p (cid:98) q ) + c (cid:98) p (cid:3) . (43)1In this case, d = and λ + = − det a and λ − = − det c . After inserting b = 0 and theexpressions for λ ± in (129)-(134) we obtain the following symplectic matrix M = cosh (cid:0) √− det a (cid:1) + sinh ( √− det a ) √− det a J a 00 cosh (cid:0) √− det c (cid:1) + sinh ( √− det c ) √− det c J c . (44)As can be seen, both block matrices in (44) are elements of Sp (2 , R ) hence, the Lie algebra elementsgiven by the parameters a and c can be considered as the Lie algebra generators of Sp (2 , R ) ⊗ Sp (2 , R ) ⊂ Sp (4 , R ). Moreover, the matrix M is diagonal if and only if a and c are anti-diagonalmatrices, i.e., only when there are no squared terms in (37).An important symplectic matrix of this type is J J , (45)which is often used to derive the transpose matrix as in (19). One can check that this matrix canbe obtained from (44) when a = c = diag( π , π ), i.e., J J = exp π J J . (46)
2. Case c = a = diag ( a , a ) and b = diag ( b , b ) . In this case the operator (cid:98) s is of the form (cid:98) s = − i (cid:126) (cid:20) a (cid:18) (cid:98) p + 12 (cid:98) p (cid:19) + a (cid:18) (cid:98) q + 12 (cid:98) q (cid:19) + b (cid:98) q (cid:98) q + b (cid:98) p (cid:98) p (cid:21) , (47)i.e., the sub-systems interact via the matrix b but only with couplings between coordinates (cid:98) q (cid:98) q and momenta operators (cid:98) p (cid:98) p . According to (120), the expression for λ ± for this case is λ ± = − ( a a + b b ) ± ( a b + a b ) = − ( a ∓ b )( a ∓ b ) . (48)Note that when b = ± a or b = ± a the eigenvalues are null and two particular systemsemerge with their operators given by b = ± a → (cid:98) s = − i (cid:126) (cid:20) a (cid:18) (cid:98) p + 12 (cid:98) p (cid:19) + a (cid:98) q ± (cid:98) q ) + b (cid:98) p (cid:98) p (cid:21) , (49) b = ± a → (cid:98) s = − i (cid:126) (cid:20) a (cid:98) p + (cid:98) p ) + a (cid:18) (cid:98) q + 12 (cid:98) q (cid:19) + b (cid:98) q (cid:98) q (cid:21) . (50)Both systems represent two interacting harmonic oscillators with a coupling term in the momentaand the coordinates, respectively.2The symplectic matrix, denoted in this case as M , is given by M = cosh (cid:16) √ λ − (cid:17) +cosh (cid:16) √ λ + (cid:17) a + b ) S − +( a − b ) S + (cid:16) √ λ − (cid:17) − cosh (cid:16) √ λ + (cid:17) a + b ) S − +( b − a ) S + b − a ) S + − ( a + b ) S − (cid:16) √ λ − (cid:17) +cosh (cid:16) √ λ + (cid:17) a − b ) S + − ( a + b ) S − (cid:16) √ λ − (cid:17) − cosh (cid:16) √ λ + (cid:17) (cid:16) √ λ − (cid:17) − cosh (cid:16) √ λ + (cid:17) a + b ) S − +( b − a ) S + (cid:16) √ λ − (cid:17) +cosh (cid:16) √ λ + (cid:17) a + b ) S − +( a − b ) S + a − b ) S + − ( a + b ) S − (cid:16) √ λ − (cid:17) − cosh (cid:16) √ λ + (cid:17) b − a ) S + − ( a + b ) S − (cid:16) √ λ − (cid:17) +cosh (cid:16) √ λ + (cid:17) , (51) where we introduce the parameters S ± as S ± := sinh (cid:16)(cid:112) λ ± (cid:17)(cid:112) λ ± . (52)
3. Case a = c = and b (cid:54) = . In this case, the operator (cid:98) s is of the form (cid:98) s = − i (cid:126) [ b (cid:98) q (cid:98) q + b (cid:98) q (cid:98) p + b (cid:98) p (cid:98) q + b (cid:98) p (cid:98) p ] , (53)and this system corresponds, as we will see in the next section, to the general case of the squeezeoperator for a bi-partite system [4].Note that in this case, not only the matrices a = c are null, but also the matrix d , which impliesthat λ + = λ − = − det b . Once we replace these expressions in (129)-(134) the symplectic matrixtakes the form M = cosh √− det b sinh √− det b √− det b J b sinh √− det b √− det b J b T cosh √− det b , (54)where the block matrices A and D are diagonal matrices. Clearly, when det b < M will be given by hyperbolic functions. In case det b > IV. QUANTUM REPRESENTATION AND ITS APPLICATIONS
The unitary representation of the group Sp (2 n, R ) was given by Moshinsky and Quesne in [43].A review and a historical analysis can be found in [44, 45]. However, to be self-contained, we will3show the main aspects of this group’s quantum representation in standard quantum mechanics inthe next subsection. A. Schr¨odinger representation of Sp (2 n, R ) The symplectic group is a non-compact group which implies an infinite-dimensional Hilbertspace for its unitary representation. Consider the Hilbert space H = L ( R n , d(cid:126)x ) where d(cid:126)x is thestandard Lebesgue measure. The unitary representation of Sp (2 n, R ) is the map (cid:98) C : Sp (2 n, R ) → U ( H ); (cid:102) M (cid:55)→ (cid:98) C (cid:102) M , (55)where (cid:98) C (cid:102) M is a unitary operator over H , i.e., formally (cid:98) C † (cid:102) M = (cid:98) C − (cid:102) M . Note that the group action con-sidered in this map is (cid:102) M instead of M , i.e., we used the “coordinatization” given by (cid:126)X introducedin section II. Hence, in order to obtain a quantum (unitary) representation of a given symplecticmatrix M we first have to transform it into the other group action (cid:102) M using Eq. (9) with thecorresponding matrix Γ ( n ) given by (10) or Γ (2) for Sp (4 , R ) given in (11).The map (cid:98) C is given by the integral operator (cid:98) C (cid:102) M Ψ( (cid:126)x ) = (cid:90) d(cid:126)x (cid:48) C (cid:102) M ( (cid:126)x, (cid:126)x (cid:48) )Ψ( (cid:126)x (cid:48) ) , Ψ( (cid:126)x ) ∈ H , (56)and the kernel C (cid:102) M ( (cid:126)x, (cid:126)x (cid:48) ) of this integral is C (cid:102) M ( (cid:126)x, (cid:126)x (cid:48) ) = e i (cid:126) [ (cid:126)x T (cid:101) D (cid:101) B − (cid:126)x − (cid:126)x (cid:48) T (cid:101) B − (cid:126)x + (cid:126)x (cid:48) T (cid:101) B − (cid:101) A (cid:126)x (cid:48) ] (cid:113) (2 πi (cid:126) ) n det (cid:101) B . (57)According to [43], this representation results from imposing two conditions on the operators (cid:98) C (cid:102) M . The first one is given by (cid:98) C (cid:102) M (cid:126) (cid:98) q T (cid:126) (cid:98) p T (cid:98) C − (cid:102) M = (cid:102) M − (cid:126) (cid:98) q T (cid:126) (cid:98) p T , (58)and relates the symplectic group elements (cid:102) M with the operators (cid:98) C (cid:102) M . Here, (cid:126) (cid:98) q := ( (cid:98) q , (cid:98) q , . . . , (cid:98) q n )and (cid:126) (cid:98) p := ( (cid:98) p , (cid:98) p , . . . , (cid:98) p n ) are the coordinate and momenta operators associated to the HeisenbergLie algebra of the system. The second condition is that (cid:98) C (cid:102) M · (cid:16) (cid:98) C (cid:102) M (cid:17) † = (cid:98) , (59)where (cid:98) (cid:98) C (cid:102) M .4The factor det B in (57) gives rise to a well define operator even in the case where the matrix B is singular (for more details see [43, 44]). Finally, it is worth mentioning that this representation(56) is valid for the entire symplectic group and not just for those elements close to the groupidentity.Since the fundamental operators are unbounded the condition (58) only holds in a subspacegiven by the domain of the operators (cid:98) q j and (cid:98) p j in H . To obtain a condition valid in the full Hilbertspace, we are forced to introduce the exponentiated version of (cid:98) q j and (cid:98) p j , that is to say, the Weylalgebra. Briefly, the Weyl algebra is a C ∗ -unital algebra whose generators, denoted by (cid:99) W ( (cid:126)a,(cid:126)b ), arerelated with (cid:98) q j and (cid:98) p j with the following relation (cid:99) W ( (cid:126)a,(cid:126)b ) := e i (cid:126) (cid:16) (cid:126)a (cid:126) (cid:98) q T + (cid:126)b (cid:126) (cid:98) p T (cid:17) , (60)and such that the real arrays (cid:126)a = ( a , a , . . . , a n ) and (cid:126)b = ( b , b , . . . , b n ), which have dimensions[ a j ] = momentum and [ b j ] = position, label the Weyl algebra generators.The standard Schr¨odinger representation of (cid:98) q j and (cid:98) p j is now used to obtain a representationfor the generators (cid:99) W ( (cid:126)a,(cid:126)b ) in H given by (cid:99) W ( (cid:126)a,(cid:126)b )Ψ( (cid:126)x ) = e i (cid:126) (cid:126)a (cid:126)b T e i (cid:126) (cid:126)a (cid:126)x T Ψ( (cid:126)x + (cid:126)b ) , (61)and such that the canonical commutation relations give rise to the Weyl algebra mutiplication (cid:99) W ( (cid:126)a ,(cid:126)b ) (cid:99) W ( (cid:126)a ,(cid:126)b ) = e − i (cid:126) ( (cid:126)a (cid:126)b T − (cid:126)b (cid:126)a T ) (cid:99) W ( (cid:126)a + (cid:126)a ,(cid:126)b + (cid:126)b ) . (62)Combining (58) and (60) to obtain the exponentiated version of (58) yields (cid:98) C (cid:102) M (cid:99) W ( (cid:126)a,(cid:126)b ) ( (cid:98) C (cid:102) M ) − = (cid:99) W ( (cid:126)a (cid:101) D T − (cid:126)b (cid:101) C T , − (cid:126)a (cid:101) B T + (cid:126)b (cid:101) A T ) , (63)where (cid:101) A , (cid:101) B , (cid:101) C and (cid:101) D are the block matrices in (cid:102) M . This relation allows us to obtain a representationof the symplectic group in the Hilbert space used in polymer quantum mechanics and in loopquantum cosmology [38].We are now ready to show, in the next subsections, some of the applications of the representationof Sp (4 , R ) given by (56) and (57). B. Schr¨odinger representation of the squeeze operator for a bi-partite system.
The squeeze operator (cid:98) S ( ζ ) for a bi-partite system is given by the exponential map (cid:98) S ( ζ ) = e (cid:98) s ζ , (64)5where the operator (cid:98) s ζ , is given by (cid:98) s ζ := 12 (cid:16) ζ ∗ (cid:98) a (cid:98) a − ζ (cid:98) a † (cid:98) a † (cid:17) . (65)Here, (cid:98) a and (cid:98) a are the annihilation operators for each of the sub-systems, say, 1 and 2, of thebi-partite system, (cid:98) a † and (cid:98) a † are their adjoint operators respectively and ζ is a complex numberlabelling the amount of squeezing. The operator (cid:98) S ( ζ ), when acting on the vacuum state of thebi-partite quantum harmonic oscillators, gives a family of squeezed states labelled by ζ .The operators in (65) are in the Fock representation, hence, let us transform (65) to theSchr¨odinger representation described with operators (cid:98) q , (cid:98) q , (cid:98) p and (cid:98) p . The relation between theserepresentations is given by (cid:98) a j = 1 √ (cid:98) q j l j + i √ l j (cid:98) p j (cid:126) , (cid:98) a † j = 1 √ (cid:98) q j l j − i √ l j (cid:98) p j (cid:126) , (66)for j = 1 , l j := (cid:113) (cid:126) m j ω j where m j and ω j stand for the masses and the frequencies of theoscillators. Inserting these expressions for (cid:98) a j and (cid:98) a † j in (65) the operator (cid:98) s ζ takes the followingform (cid:98) s ζ = 12 i (cid:126) (cid:20) (cid:126) ζ y l l (cid:98) q (cid:98) q − l ζ x l (cid:98) q (cid:98) p − l ζ x l (cid:98) p (cid:98) q − l l ζ y (cid:126) (cid:98) p (cid:98) p (cid:21) , (67)where ζ x and ζ y are the real and imaginary parts of ζ .We now rewrite this operator in the form (cid:98) s ζ = − i (cid:126) ( (cid:126) (cid:98) R T , (cid:126) (cid:98) R T ) bb T (cid:126) (cid:98) R (cid:126) (cid:98) R , (68)where the matrix b is the following b = (cid:126) ζ y l l − l ζ x l − l ζ x l − l l ζ y (cid:126) . (69)Using the isomorphism ι − defined in (41) we obtain that the corresponding Lie algebra element m ζ = ι − ( (cid:98) s ζ ) is given by m ζ = J J bb T . (70)Note that the Lie algebra matrix m ζ isomorphic to the squeeze operator (cid:98) s ζ , is of the type given inthe third case (III B 3).6To obtain the symplectic matrix associated to this Lie algebra element, we insert m ζ and itsexpressions for a , b and c in (54). This results in the following symplectic matrix M s ( r, φ ) = cosh( r ) 0 − sinh( r ) cos( φ ) l l − sinh( r ) sin( φ ) l l (cid:126) r ) − sinh( r ) sin( φ ) (cid:126) l l sinh( r ) cos( φ ) l l − sinh( r ) cos( φ ) l l − sinh( r ) sin( φ ) l l (cid:126) cosh( r ) 0 − sinh( r ) sin( φ ) (cid:126) l l sinh( r ) cos( φ ) l l r ) , (71)where r and φ are defined as ζ = re iφ . Matrix M s ( r, φ ) can be considered as the classical symplectictransformation such that when represented in L ( R , d (cid:126)x ), gives rise to the quantum operator (cid:98) S ( ζ ).Naturally, this means also that the unitary representation of (cid:98) S ( ζ ) in the Schr¨odinger representationis given by (cid:98) C M s , i.e., (cid:98) C M s = e (cid:98) s ζ .It is worth to mention that although the expression (71) depends on the proper lengths l and l , the matrix M s ( r, φ ) is (cid:126) − independent, i.e., it is entirely a classical object. Also, matrix M s produces classical squeezing but of course, adapted to the classical phase space, which in thiscase is ( R , { , } ). To illustrate the squeezing and the rotation properties of the matrix M s as acanonical transformation for different values of r and φ we consider its action on a circular trajectory( q ( t ) , p ( t ) , q ( t ) , p ( t )) where, q j ( t ) = cos( t ) q j + sin( t ) p j and p j ( t ) = − sin( t ) q j + cos( t ) p j , for j = 1 , - - - q - - p (a) Squeezing a circular trajectory. - - q - - p (b) Rotating a squeezed trajectory with r = 0 . FIG. 1:
In both figures, the solid, the dashed and the dotted lines correspond to: (a) r = 0, r = 0 . r = 0 . φ = 0, φ = π/ φ = π/
2, respectively. M s on the trajectories is explicitly of the form q (cid:48) ( t ) p (cid:48) ( t ) q (cid:48) ( t ) p (cid:48) ( t ) = M s ( r, φ ) q ( t ) p ( t ) q ( t ) p ( t ) . (72)In figure 1 we showed the plot of ( q (cid:48) ( t ) , p (cid:48) ( t )). As expected, we note in Fig. (1a), that theamount of squeezing r squeezes the circular trajectory. Recall that symplectic transformations alsopreserve the area, hence the trajectories are squeezed but the area is preserved. On the other hand,the rotation angle φ , as showed in (1b), rotates the trajectories and also preserves the area.Finally, observe that M s is given in the (cid:126)Y “coordinatization” which is not suitable for itsquantum representation. To make it suitable, let us provide the expression for the matrix (cid:102) M s ,which is given by (cid:102) M s = cosh( r ) − l sinh( r ) cos(2 φ ) l − l l sinh( r ) sin(2 φ ) (cid:126) − l sinh( r ) cos(2 φ ) l cosh( r ) − l l sinh( r ) sin(2 φ ) (cid:126) − (cid:126) sinh( r ) sin(2 φ ) l l cosh( r ) l sinh( r ) cos(2 φ ) l − (cid:126) sinh( r ) sin(2 φ ) l l l sinh( r ) cos(2 φ ) l cosh( r ) . (73)This expression will be used to explore the analog of the bipartite squeeze operator in polymerquantum mechanics in section V. C. Covariance matrix for squeezed states
Now we will show the relation between the covariance matrix, denoted by V (2) , and the sym-plectic matrix (cid:102) M . Let us consider the state | Ψ (cid:102) M (cid:105) ∈ L ( R n , d(cid:126)x ) related with the symplectic matrix (cid:102) M as | Ψ (cid:102) M (cid:105) = (cid:98) C (cid:102) M | (cid:105) , (74)where | (cid:105) = (cid:82) d(cid:126)x Ψ ( (cid:126)x ) | (cid:126)x (cid:105) is the state | (cid:105) = | (cid:105) ⊗ | (cid:105) . . . | (cid:105) n , and the ket | (cid:105) j is the vacuum stateof the j -th quantum harmonic oscillator. Note that this construction can be extended to otherstates in L ( R n , d(cid:126)x ) and not only for | (cid:105) . However, for simplicity in our exposition, let us considerthe simplest example of the covariance matrix for (cid:98) C (cid:102) M | (cid:105) .To obtain the covariance matrix we first calculate the following amplitude (cid:104) Ψ (cid:102) M | (cid:99) W ( (cid:126)a,(cid:126)b ) | Ψ (cid:102) M (cid:105) = (cid:104) | (cid:98) C † (cid:102) M (cid:99) W ( (cid:126)a,(cid:126)b ) (cid:98) C (cid:102) M | (cid:105) , (75)8where (cid:99) W ( (cid:126)a,(cid:126)b ) is the Weyl-algebra generator introduced in (60). Combining (56), (61) and theGaussian form of the vacuum state of the system given by n -decoupled harmonic oscillators, weobtain the following expression for the amplitude in (75) (cid:104) Ψ (cid:102) M | (cid:99) W ( (cid:126)a,(cid:126)b ) | Ψ (cid:102) M (cid:105) = exp − (cid:16) (cid:126)a (cid:126)b (cid:17) T Λ (cid:126)a(cid:126)b , (76)where the matrix Λ is given by Λ := (cid:102) M (cid:126) L
00 L − (cid:102) M T , (77)and L = diag( l , l , . . . , l n ), where l j was defined earlier (66).The covariance matrix V (2) has components given by V (2) = (cid:104) Ψ (cid:102) M | (cid:98) x j (cid:98) x k | Ψ (cid:102) M (cid:105) (cid:104) Ψ (cid:102) M | { (cid:98) x j , (cid:98) p k } | Ψ (cid:102) M (cid:105) (cid:104) Ψ (cid:102) M | { (cid:98) p j , (cid:98) x k } | Ψ (cid:102) M (cid:105) (cid:104) Ψ (cid:102) M | (cid:98) p j (cid:98) p k | Ψ (cid:102) M (cid:105) , (78)and these components can be obtained from (76) using the following relations (cid:104) Ψ (cid:102) M | (cid:98) x j (cid:98) x k | Ψ (cid:102) M (cid:105) = − (cid:126) ∂ a j a k (cid:104) Ψ (cid:102) M | (cid:99) W ( (cid:126)a,(cid:126)b ) | Ψ (cid:102) M (cid:105)| (cid:126)a,(cid:126)b =0 , (79)12 (cid:104) Ψ (cid:102) M | { (cid:98) x j , (cid:98) p k } | Ψ (cid:102) M (cid:105) = − (cid:126) ∂ a j b k (cid:104) Ψ (cid:102) M | (cid:99) W ( (cid:126)a,(cid:126)b ) | Ψ (cid:102) M (cid:105)| (cid:126)a,(cid:126)b =0 , (80)12 (cid:104) Ψ (cid:102) M | { (cid:98) p j , (cid:98) x k } | Ψ (cid:102) M (cid:105) = − (cid:126) ∂ b j a k (cid:104) Ψ (cid:102) M | (cid:99) W ( (cid:126)a,(cid:126)b ) | Ψ (cid:102) M (cid:105)| (cid:126)a,(cid:126)b =0 , (81) (cid:104) Ψ (cid:102) M | (cid:98) p j (cid:98) p k | Ψ (cid:102) M (cid:105) = − (cid:126) ∂ b j b k (cid:104) Ψ (cid:102) M | (cid:99) W ( (cid:126)a,(cid:126)b ) | Ψ (cid:102) M (cid:105)| (cid:126)a,(cid:126)b =0 . (82)Remarkably, the resulting expression for V (2) in terms of the symplectic matrix (cid:102) M is V (2) = 12 (cid:102) M L (cid:126) L − (cid:102) M T , (83)and this shows the direct relation between the covariance matrix V (2) for the state | Ψ (cid:102) M (cid:105) and thesymplectic matrix (cid:102) M associated with the unitary operator (cid:98) C (cid:102) M . Moreover, if we now consider thedefinition (8), it can be shown that (cid:126) V (2) is actually a symplectic matrix. Let us apply this formulato some of the systems considered before.Consider the matrix M given in (44). Using (9) we obtain the expression for (cid:102) M which thenis replaced in (83) giving rise to the following covariance matrix V (2)1 = 12 V (2)11 V (2)13 V (2)22 V (2)24 V (2)13 V (2)33 V (2)24 V (2)44 . (84)9Its components are given in the appendix (X) and in the particular case where a = a = c = c = 0, the covariance matrix V (2)1 reduces to V (2)1 = 12 l e a l e c (cid:126) l e − a
00 0 0 (cid:126) l e − c . (85)We use this result to derive the uncertainties in the coordinates ∆ x j for j = 1 , x j := (cid:113) (cid:104) Ψ (cid:102) M | (cid:98) x j | Ψ (cid:102) M (cid:105) − (cid:104) Ψ (cid:102) M | (cid:98) x j | Ψ (cid:102) M (cid:105) = l j e α j √ , (86)where due to the symmetry of the vacuum wavefunction we have (cid:104) Ψ (cid:102) M | (cid:98) x j | Ψ (cid:102) M (cid:105) = 0. This canbe verified calculating the first derivatives in (76). Here, for simplicity we make α = a and α = c . Note the remarkably property of the squeezed states like in (74) which is that ∆ x j canbe smaller than the proper length of the vacuum state l j when α j < p j , are∆ p j := (cid:113) (cid:104) Ψ (cid:102) M | (cid:98) p j | Ψ (cid:102) M (cid:105) − (cid:104) Ψ (cid:102) M | (cid:98) p j | Ψ (cid:102) M (cid:105) = (cid:126) √ l j e α j , (87)which can also be smaller than (cid:126) l j when α j > (cid:104) Ψ (cid:102) M | (cid:98) p j | Ψ (cid:102) M (cid:105) = 0 as the previous case.Nevertheless, both uncertainties satisfy Heisenberg’s uncertainty principle:∆ x j ∆ p j = (cid:18) l j e α j √ (cid:19) (cid:32) (cid:126) √ l j e α j (cid:33) = (cid:126) / . (88)Another interesting covariance matrix is the one related with the bipartite squeeze operator(73) derived in the previous subsection. Inserting (73) in (83) yields V (2) ( r, φ ) = l cosh(2 r )2 − l l sinh(2 r ) cos(2 φ )2 − l (cid:126) sinh(2 r ) sin(2 φ )2 l − l l sinh(2 r ) cos(2 φ )2 l cosh(2 r )2 − l (cid:126) sinh(2 r ) sin(2 φ )2 l − l (cid:126) sinh(2 r ) sin(2 φ )2 l (cid:126) cosh(2 r )2 l (cid:126) sinh(2 r ) cos(2 φ )2 l l − l (cid:126) sinh(2 r ) sin(2 φ )2 l (cid:126) sinh(2 r ) cos(2 φ )2 l l (cid:126) cosh(2 r )2 l , (89)and this allows us to determine the correlation between the second moments of the subsystem 1and the subsystem 2 (cid:104) Ψ (cid:102) M | ( (cid:98) x ± (cid:98) x ) | Ψ (cid:102) M (cid:105) = e r (cid:2) l + l ∓ l l cos(2 φ ) (cid:3) + e − r (cid:2) l + l ± l l cos(2 φ ) (cid:3) , (90) (cid:104) Ψ (cid:102) M | ( (cid:98) p ± (cid:98) p ) | Ψ (cid:102) M (cid:105) = (cid:126) e r l l (cid:2) l + l ± l l cos(2 φ ) (cid:3) + (cid:126) e − r l l (cid:2) l + l ∓ l l cos(2 φ ) (cid:3) . (91)0In the particular case where l = l = l and φ = π , the uncertainties (∆ x ) (cid:101) Ψ and (∆ x ) (cid:101) Ψ forthe state (cid:101) Ψ are correlated as follows(∆ x ) (cid:101) Ψ + (∆ x ) (cid:101) Ψ = l cosh(2 r ) . (92)These are the main results, at the standard quantum mechanics level, which we want to showregarding the representation of the symplectic group in quantum mechanics. There are others ap-plications like the analysis of the Bohmian trajectories for bipartite squeezed states, the analysis ofthe fidelity for bipartite or tripartite squeezed states, and others which are currently in preparation.For now, let us move to the analysis of the squeezed states in polymer quantum mechanics givenin the next section. V. SQUEEZED STATES IN POLYMER QUANTUM MECHANICS
Polymer quantum mechanics [33, 34, 36, 37], is a quantization scheme which can be considered asa “toy model” looming from loop quantum cosmology. Hence, exploring the nature and propertiesof squeezed states in polymer quantum mechanics will help study those scenarios in loop quantumcosmology where such states might play a significant role.For this example, we will consider a system with two degrees of freedom and both will bepolymer quantized. Therefore, the Hilbert space of the entire system is given by H poly = H (1) poly × H (2) poly , (93)where the Hilbert spaces H ( j ) poly with j = 1 , H ( j ) poly = L ( R , dp ( j ) Bohr ) , (94)where R is the Bohr compactification of real line and dp ( j ) Bohr is the Bohr measure (see [36] formore details). This Hilbert space resembles the momentum representation used in the standardquantum mechanics.An arbitrary state in this Hilbert space H poly is given byΨ( p , p ) = (cid:88) { (cid:126)x j } Ψ (cid:126)x j e i (cid:126) (cid:126)x Tj (cid:126)p , (95)where { (cid:126)x j } is a shorthand notation for the graph { ( x (1) j , x (2) j ) } j = nj =1 associated with the state Ψ( (cid:126)p ).In this notation, the array (cid:126)p = ( p , p ) denotes the momentum variables for the system 1 and 2,1respectively. The coefficients Ψ (cid:126)x j provide the value for the norm of the state which is given by || Ψ( (cid:126)p ) || = (cid:88) { (cid:126)x j } | Ψ (cid:126)x j | , (96)hence these coefficients are different from zero and the sum converges (they are non-null overcountable number of points in the graph { (cid:126)x j } ). This norm arises from the inner product (cid:104) Ψ | Φ (cid:105) = lim L ,L →∞ L L (cid:90) L − L (cid:90) L − L Ψ ∗ ( (cid:126)p ) Φ( (cid:126)p ) dp dp , (97)which for the specific case of the plane waves takes the form of the Kronecker delta (cid:104) e i (cid:126) (cid:126)x T (cid:126)p | e i (cid:126) (cid:126)x (cid:48) T (cid:126)p (cid:105) = δ (cid:126)x,(cid:126)x (cid:48) . (98)This inner product is the main signature of the polymer quantization as it violates the Stone-von Neumann theorem. Consequently, polymer quantum mechanics is not unitarily equivalent tothe standard Schr¨odinger representation. Moreover, in polymer quantum mechanics there is nomomentum operator hence infinitesimal spatial translations cannot be implemented. Nevertheless,we can obtain a representation for the position operator, which in the present case is given by (cid:98) q Ψ( p , p ) = i (cid:126) ∂∂p Ψ( p , p ) = − (cid:88) { (cid:126)x j } Ψ (cid:126)x j x (1) j e i (cid:126) (cid:126)x Tj (cid:126)p , (99) (cid:98) q Ψ( p , p ) = i (cid:126) ∂∂p Ψ( p , p ) = − (cid:88) { (cid:126)x j } Ψ (cid:126)x j x (2) j e i (cid:126) (cid:126)x Tj (cid:126)p . (100)Despite these peculiarities with the non-regularity of the polymer representation, the represen-tation of the symplectic group Sp (2 n, R ) on the Hilbert space of polymer quantum mechanics wasprovided recently by one of the authors in Ref. [38]. There, the representation is given by the map (cid:98) C ( poly ) : Sp (2 n, R ) → L ( H poly ) , (cid:102) M (cid:55)→ (cid:98) C ( poly ) (cid:102) M , where the linear operator (cid:98) C ( poly ) (cid:102) M acts on H poly as (cid:98) C ( poly ) (cid:102) M Ψ( (cid:126)p ) = lim L ,L →∞ L L (cid:90) L − L (cid:90) L − L C ( poly ) (cid:102) M ( (cid:126)p, (cid:126)p (cid:48) )Ψ( (cid:126)p (cid:48) ) d(cid:126)p (cid:48) . (101)The polymer kernel C ( poly ) (cid:102) M ( (cid:126)p, (cid:126)p (cid:48) ) is given by C ( poly ) (cid:102) M ( (cid:126)p, (cid:126)p (cid:48) ) = det( (cid:101) D (cid:101) A T ) − e − i (cid:126) (cid:126)p T (cid:101) B (cid:101) D − (cid:126)p (cid:88) (cid:126)x e i (cid:126) (cid:126)p T (cid:126)x − i (cid:126) (cid:126)p (cid:48) T (cid:101) D T (cid:126)x + i (cid:126) (cid:126)x T (cid:101) D (cid:101) C T (cid:126)x , (102)and note that when the factor det( (cid:101) D (cid:101) A T ) − (cid:54) = 1 it implies that this representation is not unitary[38].Recall that one of the main features of loop quantum cosmology is its intrinsic length scale givenby the Planck length. As a toy model, polymer quantum mechanics does not have an intrinsic2length scale. However, it admits a length scale that mimics some of the features of loop quantumcosmology. This length scale is introduced at hand and is called polymer scale , usually denoted by µ . This polymer scale constitutes the analog of minimum length for polymer quantum mechanicsmodels, and therefore, it can be considered as a lower bound for the uncertainties. In the presentanalysis, each system admits a polymer scale µ and µ when the dynamics is considered.Let us now consider the following questions: (1) is it possible to have polymer states such thattheir uncertainties are lower than the polymer scale? Furthermore, (2) do the correlations foundin (90) have an analog in polymer quantum mechanics?To answer these questions let us consider the matrix (cid:102) M with a = a = c = c = 0, usedto calculate the covariance matrix V (2)1 in (85), but now with a = − r and c = − r . Theexplicit form is (cid:102) M (cid:102) M = e − r e − r e r
00 0 0 e r . (103)The action of the group element (cid:98) C ( poly ) (cid:102) M on an arbitrary polymer state (95) gives the followingstate (cid:101) Ψ (cid:102) M ( p , p ) = (cid:98) C ( poly ) (cid:102) M Ψ( p , p ) = (cid:88) { (cid:126)x j } Ψ (cid:126)x j e i (cid:126) (cid:16) e − r p x (1) j + e − r p x (2) j (cid:17) . (104)Using the representation of the position operators (cid:98) q and (cid:98) q given in (99) and (100) we obtainthe dispersion relations (∆ x ) (cid:101) Ψ = e − r (∆ x ) Ψ , (∆ x ) (cid:101) Ψ = e − r (∆ x ) Ψ , (105)which show that the squeezed polymer state is indeed squeezed by a factor e − r or e − r . Con-sequently, if the initial dispersion of the polymer state is given by (∆ x ) Ψ or (∆ x ) Ψ , then thesqueeze operator (cid:98) C ( poly ) (cid:102) M gives rise to a polymer state (cid:101) Ψ (104) whose dispersion is smaller than thatof the initial polymer state Ψ. Moreover, due to there is no upper bound for the parameter r , thesedispersion relations can be smaller than the corresponding polymer scales µ and µ .Let us now consider the analog of the correlations (92) but for polymer states. To do so,consider the polymer representation (cid:98) C ( poly ) (cid:102) M s of the symplectic matrix (73) corresponding to a bi-partite system. The action of (cid:98) C ( poly ) (cid:102) M s on an arbitrary polymer state (95) is given by (cid:101) Ψ s = (cid:98) C ( poly ) (cid:102) M s Ψ( p , p ) = (cid:88) { (cid:126)x j } Ψ (cid:126)x j e i (cid:126) (cid:104)(cid:16) cosh( r ) x (1) j +sinh( r ) x (2) j (cid:17) p + (cid:16) cosh( r ) x (2) j +sinh( r ) x (1) j (cid:17) p (cid:105) , (106)3where again the representation in (101) was used.We combine this result with the representation of the position operators in (99) and (100) andobtain the following relations(∆ x ) (cid:101) Ψ s − (∆ x ) (cid:101) Ψ s = (∆ x ) − (∆ x ) , ∀ Ψ ∈ H poly , (107)where the conditions l = l and φ = { , π , π } were imposed. Remarkably, this result not onlyis independent of the parameter r (which labels the amount of squeezing) but also applies to anypolymer state Ψ ∈ H poly . As can be seen, the difference of dispersions squared is conserved,regardless of the amount of squeezing. Also, note that l and l are considered group parametersand have no relation to the dynamics, i.e., we are considering general states in H poly . The sameapplies for φ .Let us now consider pure and symmetric polymeric states. The pure states are those that canbe written as the following productΨ ( p ) ( p , p ) = (cid:88) { x (1) j } Ψ (1) x (1) j e i (cid:126) p x (1) j (cid:88) { x (2) j } Ψ (2) x (2) j e i (cid:126) p x (2) j . (108)Secondly, both lattices { x (1) j } and { x (2) j } are symmetric, i.e., for every positive point 0 < x ( s ) j ∈{ x ( s ) j } there exist a negative point 0 > x ( s ) j (cid:48) ∈ { x ( s ) j } , such that x ( s ) j + x ( s ) j (cid:48) = 0, and the statesare also symmetric which implies that Ψ ( s ) x ( s ) j = Ψ ( s ) x ( s ) j (cid:48) . These states are the analog of the statesdescribed with even functions in the standard quantum mechanics.The dispersion relation for squeezed pure symmetric polymer states is given by(∆ x ) (cid:101) Ψ s + (∆ x ) (cid:101) Ψ s = cosh(2 r ) (cid:0) (∆ x ) ( p ) + (∆ x ) ( p ) (cid:1) , (109)which takes the form (∆ x ) (cid:101) Ψ s + (∆ x ) (cid:101) Ψ s = l cosh(2 r ) , (110)when the dispersion of the pure states are ((∆ x ) Ψ ( p ) = (∆ x ) Ψ ( p ) ) = l/ √
2. Notably, Eq. (110)is the same as that obtained in (92) for the Schr¨odinger representation. This shows that thecorrelations present in the standard quantum mechanics using the (cid:98) C (cid:102) M operator for both symplecticmatrices (cid:102) M and (cid:102) M s are the same to those obtained in polymer quantum mechanics using theoperator (cid:98) C ( poly ) (cid:102) M .4 VI. CONCLUSIONS
In this paper we provided the direct relation between the Lie algebra sp (4 , R ) and the symplecticgroup Sp (4 , R ). The expression shows the link between the block matrices A , B , C and D withthose of the Lie algebra a , b and c given in the Eqs. (26)-(29). This result has not been reportedbefore and applies to the full Lie algebra sp (4 , R ) of the symplectic group Sp (4 , R ).Such relation allows us to obtain some important symplectic matrices that were used in subse-quent sections. In the first example for a , c (cid:54) = and b = 0, we show that the corresponding symplec-tic matrix M , given in (44), can be written as ( M (cid:48) ⊗ ) · ( ⊗ M (cid:48) ), where , M (cid:48) , M (cid:48) ∈ Sp (2 , R ).Here, M (cid:48) and M (cid:48) are symplectic matrices acting over each of the sub-systems with coordinates( q , p ) and ( q , p ) respectively. In the case a = c = diag( a , a ) and b = diag( b , b ) thesymplectic matrix M in (51), describes two coupled harmonic oscillators with interaction termslabeled by the coefficients of the matrix b . Finally, in the equation (54) we showed the symplecticmatrix M for the case in which a = c = and a general form of the matrix b .In section (IV) we analyzed the classical description of squeeze operators. We showed that thesymplectic matrix M s is the classical analog of the squeeze operator (cid:98) S ( ζ ) = e (cid:98) s ζ = (cid:98) C (cid:102) M s . Also, weremarked the isomorphism between the Lie algebra sp (4 , R ) and P (2 , R ). Additionally, the generalform of the covariance matrix V (2) for the squeezed vacuum state | Ψ (cid:102) M (cid:105) was derived using theWeyl algebra representation and the symplectic matrix M . The components of this covariantmatrix were used to calculate the dispersion relations (86) and (87) for the particular case where a = a = c = c = 0. As is already known, these dispersions can be smaller than the vacuumcharacteristic length for the harmonic oscillators. They also satisfy the Heisenberg uncertaintyprinciple as was shown in (88). We then calculated the covariance matrix for the symplectic matrix M s corresponding to the classical analog of the bipartite squeeze operator. With this matrix wedetermined the correlation (92). We also provided the general expressions for these correlations inequations (90) and (91).Applying the previous results, it is also possible to represent operators in non-regular Hilbertspaces that are non-unitarily equivalent to the Fock-Schr¨odinger representation, so in section (V)we analyze polymer quantized systems. We calculated the dispersion relation for an arbitrarypolymer state using the representation of the symplectic group in polymer quantum mechanics.We obtained that the polymer representation of the squeeze operator given by (cid:98) C ( poly ) (cid:102) M , yields adispersion relation Eq. (105), which can be smaller than those of the initial state. This impliesthat (cid:98) C ( poly ) (cid:102) M is indeed a polymer squeeze operator and (104) describes a polymer squeezed state.5On the other hand, the polymer representation of the bipartite squeeze operator given by (cid:98) C ( poly ) (cid:102) M s was used to derive the polymer correlations (107) and (110). The first correlation (107) shows thatthe difference of the dispersions’ square is preserved and is independent of the initial polymer state.Clearly, this result only holds for (cid:98) C ( poly ) (cid:102) M s so a symplectic matrix different than the one used in (103)will produce a different result. In the case of (110), the result has the same form as the standardcorrelation in Eq. (92), hence, the polymer representation of (cid:98) C ( poly ) (cid:102) M s can be used to constructcorrelated squeezed states for bipartite polymer systems. Naturally, this brings some questions likewhether there is any mechanism in nature, say loop quantum cosmology or the interior of a blackhole, from which a polymer squeezed state can be created.Moreover, establishing that squeezing is a property also present in non-regular representationsquestions its role in the classicality of some cosmological models. As we showed, it is possibleto construct entangled polymer states using (cid:98) C ( poly ) (cid:102) M s . Such polymer entangled states correlationssatisfy a relation identical to the one obtained in the standard quantum mechanics. In this case,the states are polymer bipartite squeezed states, similar to those used in the quantum descriptionof the inflaton field.Finally, it is worth to mention that the polymer squeezed states obtained as a result of therepresentation of Sp (4 , R ) in H poly , given in (104, 106), differ from those reported in the LQCliterature [20–25]. In these references, a Gaussian form of the states is considered, whereas inour case, the polymer state (104) is a general polymer state. In LQC, the squeezed states areconstructed by hand due after imposing some conditions to achieve the squeezed nature of thedispersion relations. In our case, the squeezed state results from the action of the squeeze operator.The results of this work open the doors for studying the entangled states of matter and geometryand the role that their correlations might play in some physical scenarios. VII. ACKNOWLEDGMENTS
I thank Academia de Matem´aticas and Colegio de F´ısica, UP, for the support and enthusiasm.
VIII. APPENDIX: CALCULATION OF m n In this appendix the expression for the matrix m n is obtained. To do so, recall that the matrix m is formed by four 2 × × . The upper right block is the matrix Jd whereas the lower6left is − Jd T . Notably, we found that this block structure is preserved after exponentiating thematrix m an integer number of times. That is to say, the n-power of matrix m yields a newmatrix ( m ) n given by ( m ) n = α n × β n Jd − β n Jd T γ n × . (111)It is this pattern the one to be considered when this procedure is applied to higher order symplecticgroups.The coefficients α n , β n and γ n , are to be determined and depend on the values of the matrices a , b , c and d . For n = 1, these coefficients are given by the factors in the block matrices of m given in (24) and can be directly defined as α := − (det a + det b ) , β := +1 , γ := − (det c + det b ) . (112)To calculate α n , β n and γ n for arbitrary n , first note that they can be generated with the( n − T as α n β n γ n = T n − α β γ , (113)where the matrix T is given by T = α β det d β γ β det d γ . (114)The calculation shows that T n − is a matrix of the form T n − = U n − (cid:126) T (cid:126)u T γ n − (cid:80) n − j =0 γ − j U j γ n − , (115)where (cid:126) ,
0) and (cid:126)u = (0 , β det d ) and matrix U is given by U = α β det d β γ . (116)Then, using (113) we have the following relation for the coefficients α n β n = U n − α β , γ n = γ n + (cid:126)u T γ n − n − (cid:88) j =0 γ − j U j α β . (117)7In order to calculate U n − we need to diagonalize the matrix U . Let P be the matrix diago-nalizing U , then U = P D P − , (118)where the matrix P is P = ( λ + − γ ) β k λ − − γ ) β k k k . (119)The real arbitrary parameters k and k result from the diagonalization procedure. Its valueswill be automatically cancelled as part of the calculation of U n − further below. The eigenvaluesof U , denoted by λ ± , have the following expression λ ± = α + γ ± (cid:113) ( α − γ ) + 4 β det d , = − det a + det c + 2 det b ± (cid:112) (det a − det c ) + 4 det d . (120)and the diagonal matrix D is D = λ + λ − . (121)We now take the n − U given in (118) to obtain the following result U n − = ( λ + − γ ) β k λ − − γ ) β k k k λ n − λ n − − ( λ + − γ ) β k λ − − γ ) β k k k − , (122)which, when combined with the result in (117) together with the expression for (cid:126)u , gives α n = ( λ + − det b − det c ) λ n + − ( λ − − det b − det c ) λ n − (cid:112) (det a − det c ) + 4 β det d , (123) β n = λ n + − λ n − (cid:112) (det a − det c ) + 4 β det d , (124) γ n = (cid:2) ( λ + − det b − det c ) λ n − − ( λ − − det b − det c ) λ n + (cid:3)(cid:112) (det a − det c ) + 4 β det d . (125)These are the final expressions for the coefficients in ( m ) n . IX. APPENDIX: SERIES ANALYSIS
In this appendix we calculate the series expansion terms. To do so, note that once the expressionfor ( m ) n is inserted the expansion (23) and the even and odd terms are collected, the matrix8 M ( a , b , c ) takes the form M ( a , b , c ) = α ( e ) × β ( e ) Jd − β ( e ) Jd T γ ( e ) × + m α ( o ) × β ( o ) Jd − β ( o ) Jd T γ ( o ) × , (126)where the following coefficients are given by α ( e ) := 1 + + ∞ (cid:88) n =1 n )! α n , β ( e ) := + ∞ (cid:88) n =1 n )! β n , γ ( e ) := 1 + + ∞ (cid:88) n =1 n )! γ n , (127) α ( o ) := 1 + + ∞ (cid:88) n =1 n + 1)! α n , β ( o ) := + ∞ (cid:88) n =1 n + 1)! β n , γ ( o ) := 1 + + ∞ (cid:88) n =1 n + 1)! γ n . (128)We now insert (123), (124) and (125) in the relations (127) - (128) to obtain α ( e ) = 12 (cid:104) cosh (cid:16)(cid:112) λ + (cid:17) + cosh (cid:16)(cid:112) λ − (cid:17)(cid:105) + (det c − det a ) (cid:104) cosh (cid:16)(cid:112) λ + (cid:17) − cosh (cid:16)(cid:112) λ − (cid:17)(cid:105) (cid:112) (det a − det c ) + 4 det d , (129) α ( o ) = 12 sinh (cid:16)(cid:112) λ + (cid:17)(cid:112) λ + + sinh (cid:16)(cid:112) λ − (cid:17)(cid:112) λ − + (det c − det a )2 (cid:112) (det a − det c ) + 4 det d sinh (cid:16)(cid:112) λ + (cid:17)(cid:112) λ + − sinh (cid:16)(cid:112) λ − (cid:17)(cid:112) λ − , (130) β ( e ) = 1 (cid:112) (det a − det c ) + 4 det d (cid:104) cosh (cid:16)(cid:112) λ + (cid:17) − cosh (cid:16)(cid:112) λ − (cid:17)(cid:105) , (131) β ( o ) = 1 (cid:112) (det a − det c ) + 4 det d sinh (cid:16)(cid:112) λ + (cid:17)(cid:112) λ + − sinh (cid:16)(cid:112) λ − (cid:17)(cid:112) λ − , (132) γ ( e ) = 12 (cid:104) cosh (cid:16)(cid:112) λ + (cid:17) + cosh (cid:16)(cid:112) λ − (cid:17)(cid:105) − (det c − det a ) (cid:104) cosh (cid:16)(cid:112) λ + (cid:17) − cosh (cid:16)(cid:112) λ − (cid:17)(cid:105) (cid:112) (det a − det c ) + 4 det d , (133) γ ( o ) = 12 sinh (cid:16)(cid:112) λ + (cid:17)(cid:112) λ + + sinh (cid:16)(cid:112) λ − (cid:17)(cid:112) λ − − (det c − det a )2 (cid:112) (det a − det c ) + 4 det d sinh (cid:16)(cid:112) λ + (cid:17)(cid:112) λ + − sinh (cid:16)(cid:112) λ − (cid:17)(cid:112) λ − , (134)where we have to recall the expression for the eigenvalues λ ± in (120).9 X. APPENDIX: COVARIANCE MATRIX COEFFICIENTS
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