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Linear representations of SU(2) described by using Kravchuk polynomials Nicolae Cotfas University of Bucharest, Faculty of Physics, P.O. Box MG-11, 077125 Bucharest, Romania∗ We show that a new unitary transform with characteristics almost similar to those of the finite Fourier transform can be definedin any finite-dimensional Hilbert space. It is definedby using the Kravchuk polynomials, and we call it Kravchuk transform. Some of its properties are investigated and used in order to obtain a simple alternative description for the irreducible representations of the Lie algebra su(2) and group SU(2). Our approach offers a deeper insight into the structure of thelinearrepresentationsofSU(2)andnewpossibilitiesofcomputation,veryusefulinapplications 6 in quantummechanics, quantuminformation, signal and image processing. 1 0 2 INTRODUCTION functions, the matrix elements of the irreducible repre- b sentations have simpler mathematical expressions. e The Hilbert space Cd, of dimension d=2j+1, can be In conclusion, we define a new unitary transform K, F regarded as the space of all the functions of the form similartothefiniteFouriertransformF.WehaveF4 =I, 7 respectivelyK3 =I, and in both cases,the definitions of 1 ψ: j, j+1, ...,j 1,j C, the direct and inverse transforms are almost identical. {− − − }−→ The new transform K allows a new description of the ] considered with the scalar product defined as h linear representations of su(2), SU(2), SO(3), and new p developments in all the mathematical models based on j - ϕ,ψ = ϕ(n)ψ(n). (1) these representations. New models in physics, quantum h h i t n=−j information, quantum finance [3], signal and image pro- a P m The finite Fourier transform F:Cd Cd :ψ F[ψ], cessing can be obtained by using K instead of F. −→ 7→ [ j 6 F[ψ](k)=√1d e−2dπiknψ(n), (2) KRAVCHUK POLYNOMIALS v n=−j P 4 plays a fundamental role in quantum mechanics, signal The functions 2 andimageprocessing. Itsinverseistheadjointtransform 4 6 K j, K j+1,...,Kj 1,Kj: j, j+1,...,j 1,j R 0 F+[ψ](k)= 1 j e2dπiknψ(n), (3) − − − {− − − }−→ . √d satisfying the polynomial relation [14] 1 n= j − 0 P 6 and F4 = I, where I is the identity operator Iψ = ψ. In j 1 the caseofaquantumsystemwith HilbertspaceCd,the (1 X)j+k(1+X)j−k = Km(k)Xj+m (7) − v: self-adjoint operator Q:Cd−→Cd :ψ 7→Qψ, mX=−j i X (Qψ)(n)=nψ(n), (4) arecalledKravchuk polynomials. The firstthree of them r are: K j(k)=1, K j+1(k)= 2k, K j+2(k)=2k2 j. a is usually regardedas a coordinate operator and By a−dmitting tha−t − − − P=F+QF (5) 1 =0 for n 0, 1, 2,... Γ(n) ∈{ − − } as a momentum operator [4, 5, 12, 15]. and using the binomial coefficients We show that a new remarkable unitary transform K:Cd Cd (6) Cmn=Γ(n+1Γ)(mΓ(+m1)n+1) −→ − m! for n 0,1,2,...,m , (8) ctraannsbfeordmefisnaetdisfibeysutshiengunthexepKecrtaevdchrueklatpioolnynKom3 =ialIs,.aOnudr =( n!(m0−n)! for n∈Z{ 0,1,2,...,}m . ∈ \{ } iQ, iK+QK, iKQK+ the Kravchuk polynomials can be defined as − − − form a basis of a Lie algebra isomorphic to su(2). This j+m arellsoewntsautisontos oofbtSaUin(2a)naenwddSeOsc(r3i)p.tiIonntfeorrmtsheofliKneraavrcrheupk- Km(k)= (−1)nCjn+kCjj−+km−n. (9) n=0 X 2 The hypergeometric function Theorem 1. Kravchuk polynomials satisfy the relation 2F1(cid:18) a,cb(cid:12)(cid:12)z(cid:19)=Xk∞=0(a)(kc)(kb)k zkk!, (10) kX=j−j(−i)kKm(k)Kk(n)=2jij+mij+nKm(n) (16) (cid:12) where (cid:12) for any m,n j, j+1,...,j 1,j . ∈{− − − } Γ(α+k) (α) =α(α+1)...(α+k 1)= , k Proof. Direct consequence of the polynomial relation − Γ(α) satisfies the relation [10] j j ( i)j+kK (k)K (n) Xj+m m k m= j k= j − ! n, β − − 2F1 −γ z = ΓΓ((γγ+)nΓ)(γΓ−(βγ+βn)) P = Pj ( i)j+k(1 X)j+k(1+X)j kK (n) (cid:18) (cid:12) (cid:19) − (11) − − − k (cid:12) n, β k= j (cid:12)(cid:12) ×2F1(cid:18)β−−γ−n+1(cid:12)1−z(cid:19). =(1P+−X)2j j Kk(n) XX+−11i j+k. (cid:12) k= j Since (−α)k=(−1)kΓ(α+1)/Γ(α−k+1)(cid:12)(cid:12),the relation =(1+X)2j P1− X 1i (cid:16)j+n 1(cid:17)+ X 1i j−n − X+−1 X+−1 2F1(cid:18)−j1−−mk,−−mj−k(cid:12)−1(cid:19)= ΓΓ((j1−−kk+−1m)Γ)(Γj(−2mj++11)) =(1+i+(1(cid:16)−i)X)j+n(cid:17)(1−i(cid:16)+(1+i)X)(cid:17)j−n (cid:12)(cid:12)(cid:12) ×2F1 −j−m,2j−j−k 2 =(1+i)j+n(1−i)j−n(1−iX)j+n(1+iX)j−n (cid:18) − (cid:12) (cid:19) (cid:12) j can be written as (cid:12)(cid:12) =2j(−i)jij+n Km(n)(iX)j+m. (cid:3) m= j − j m, j k P Km(k)=C2jj+m2F1 − − 2j− − 2 . (12) By using (12), the relation (16) can be written as (cid:18) − (cid:12) (cid:19) (cid:12) From the polynomial relation (cid:12)(cid:12) j (−i)k (j+k()2!j()j! k)! 2F1 −j−m,2j−j−k 2 k= j − (cid:18) − (cid:12) (cid:19) j j P− (cid:12) 1 Cj+kK (k)K (k) Xj+mYj+n j k, j n(cid:12) m,n= j 22j k= j 2j m n ! ×2F1 − − 2−j − (cid:12)2 (17) P− j P− j j (cid:18) − (cid:12) (cid:19) ===2212112jjk[=Pk(j=1P−−jCjX2jCj+)2jk(j+1(k1−mYP=X)−+)jj+K(1km+(1(Xk+))XX(1)j+j+−Ymk()1n]2−=Pj−Y=j)K(j1+nk+((k1X)+YYYj)+)2jnj−k and is a s=pe2cjiaijl+cmasiej+fnor2F(112(cid:18))−frjo−mm−[6,2]j−anj−dn(5(cid:12)(cid:12)(cid:12)(cid:12).5(cid:12)(cid:12)(cid:12)2)(cid:19)fr,om [8]. 22j − − j = Cj+mXj+mYj+m. KRAVCHUK FUNCTIONS 2j m= j − P The space Cd can be regarded as a subspace of the it follows the well-known relation [10, 14] space of all the functions of the form ψ : Z C by −→ j identifying it either with the space 1 Cj+kK (k)K (k)=Cj+mδ . (13) 22j k=P−j 2j m n 2j mn ℓ2(Zd)={ ψ:Z−→C | ψ(n+d)=ψ(n) for any n∈Z } (18) We extend the set K j by admitting that { m}m=−j of periodic functions of period d or with the space Km =0 for m Z j, j+1,...,j 1,j . ℓ2[ j,j]= ψ:Z C ψ(n)=0 for n >j (19) ∈ \{− − − } − { −→ | | | } With this convention, by differentiating (7) we get of all the functions null outside j, j+1,...,j 1,j . The use of ℓ2(Z ) or ℓ2[ j,j] a{l−lows−us to defin−e ne}w d (j+m+1)K (k) − m+1 mathematical objects and to obtain new results. The (14) +(j−m+1)Km−1(k)=−2kKm(k). functions Km:Z−→R, defined as For the half-integer powers we use the definiton 1 Cj+k K (k)= 2j K (k) (20) zk = z keikarg(z). (15) m 2jvuC2jj+m m | | u t 3 canbeexpressedintermsofthehypergeometricfunction The transform K :ℓ2[ j,j] ℓ2[ j,j], − −→ − 1 j m, j k j Km(k)= 2jqC2jj+mC2jj+k 2F1(cid:18)− −−2j− − (cid:12)(cid:12)2(cid:19), (21) K =(−1)2jn,mX=−jinK−n(m) |j;nihj;m|, (29) belong to ℓ2[ j,j], and satisfy the relations (cid:12)(cid:12) we call Kravchuk transform, is a unitary operator. − The direct and inverse transforms are quite identical: K (n)=K (m), K ( n)=( 1)j+mK (n). m n m m − − j The functions K j , called Kravchuk functions, K[ψ](n) = ( 1)2j in( 1)j+m Kn(m)ψ(m) form an orthono{rmma}lmb=as−isj in ℓ2[ j,j], that is, we have − mPj=−j − − K+[ψ](n)=( 1)2j ( i)m( 1)j+nK (m)ψ(m). n − − − j m= j − K K =δ , K K =I. (22) P h m| ni mn | mih m| The operator Q:ℓ2[ j,j] ℓ2[ j,j], (Qψ)(n)=nψ(n), m= j − → − X− admitting the spectral decomposition From (14) we get the relation j Q= n j;n j;n, (30) | ih | (j−m)(j+m+1) Km+1(k) n=P−j (23) can be regarded as a coordinate operator in ℓ2[ j,j]. p + (j+m)(j−m+1) Km−1(k)=−2kKm(k) Theorem 3. Kravchuk transform satisfies the −relations p and its direct consequence K3 =I (31) (j+m)(j m+1) for n=m 1 j − − K+QK+QK+ KQKQK =iQ. (32) 2 kKm(k)Kn(k)=p(j m)(j+m+1) for n=m+1 − − kX=−j p0 − for n=m 1.Proof. A consequence of the equality (26) is the relation 6 ± Therelation(23)canalsobewrittenintheform[1,2,(2144)] K2 = j j im+kK m(k)K k(n) j;m j;n n,m= jk= j − − | ih | j P− P− j (j m)(j+m+1)K (m+1) = ( 1)j+nim j( i)j ( i)kK (k)K (n) j;m j;n k − m k − (25) n,m= −j − k= j− | ih | p+ (j+m)(j m+1)K (m 1)= 2kK (m). Pj− P− − k − − k = ( 1)j+m+nij+n( i)jKm(n) j;m j;n − − | ih | Theorpem2. TheKravchukfunctionssatisfytherelation n,mPj=−j = ( 1)nij+n( i)jK (m) j;m j;n =K+ n j n,m= j − − − | ih | − ( i)kK (k)K (n)=ij+mij+nK (n) (26) P − m k m which shows that K3 =I. From (24) and k= j X− j for any m,n j, j+1,...,j 1,j . QK+QK j;m = ( 1)m+nn ∈{− − − } | i − n= j P− j Proof. Direct consequence of (16) and (20). (cid:3) kK (k)K (k) j;n m n × | i k= j j P− K+QKQj;m = ( 1)m+nm KRAVCHUK TRANSFORM | i − n= j P− j kK (k)K (k) j;n The functions {δm}jm=−j, defined by the relation j ×k=P−j m n | i KQK+ j;m = ( i)min δ :Z C, δ (k)=δ = 1 for k=m, (27) | i n= j − m −→ m km ( 0 for k6=m, P− j kK (k)K (k) j;n m n × | i k= j formanorthonormalbasisintheHilbertspaceCd. With P− the traditional notation j;m instead of δ , we have it follows the relation m | i QK+QK K+QKQ=iKQK+ j − j;mj;n =δ and j;m j;m =I. (28) h | i mn | ih | equivalent to (32). (cid:3) m= j − P 4 THE IRREDUCIBLE REPRESENTATIONS OF QUANTUM SYSTEMS WITH FINITE THE LIE ALGEBRA su(2) AND GROUP SU(2) DIMENSIONAL HILBERT SPACE Theorem 4. The self-adjoint operators In the continuous case, the coordinate operator (qˆψ)(q)=qψ(q) and the momentum operator pˆ= i d Jz =Q, Jx =K+QK, Jy =KQK+ (33) − dq satisfy the relation pˆ= +qˆ , where satisfy the relations F F [Jx,Jy]=iJz, [Jy,Jz]=iJx, [Jz,Jx]=iJy (34) F[ψ](p)= √12π ∞ e−ipqψ(q) (37) and define a linear representation of su(2) in ℓ2[ j,j]. Z−∞ − is the Fourier transform. In the finite-dimensional case, Proof. Eachof the relations (34) is equivalent to (32). (cid:3) P =F+QF definedbyusingthefiniteFouriertransform, is usually regarded as a momentum operator [5, 12, 15]. The operators J =Jx iJy verify the relations ByfollowingAtakishyiev,Wolfet al. [1,2,16],we can ± ± consider P˜ = K+QK as a second candidate for the role [J ,J ]= J , [J ,J ]= 2J , (35) z ± ± ± − + − z of momentum operator. In this case [Q,P˜]=iJy, and and, by using (24), one can prove that 1 1 j(j+1) 1 Jz|j;mi=m|j;mi H˜=2P˜2+ 2Q2 = 2 − 2Jy2 (38) J j;m = (j m)(j+m+1) j;m+1 (36) + corresponds to the quantum harmonic oscillator. | i − | i J j;m =p(j+m)(j m+1) j;m 1 . The function K j;m is an eigenstate of H˜ for any m. −| i − | − i | i The author is grateful to J. Van der Jeugt for letting In certain cases, pit is useful to describe the structure of him know that the relation (16) from Theorem 1 is a J , J , J by using only two operators. Beside z x y special case of a known identity concerning the hyperge- 1 1 J , J = (J +J+), J = (J J+) ometric function. z x 2 + + y 2i +− + we have now the alternative representation (33), more advantageous when we pass form su(2) to SU(2) and SO(3). The operator J admits the decomposition x ∗ [email protected];http://fpcm5.fizica.unibuc.ro/ñcotfas/ j j [1] M. K.Atakishyieva,N.M. Atakishyiev,and K.B. Wolf, Jx=K+ k j;k j;k K = k K k K k . J. Phys: Conf. Ser. 512, 012031 (2014). | ih | | − ih − | k= j k= j [2] N.M.AtakishyievandK.B.Wolf,1997J.Opt.Soc.Am. X− X− A 14, 1467 (1997). and consequently [3] L.-A. Cotfas, Physica A 392, 371 (2013). j j [4] N. Cotfas and D. Dragoman, J. Phys. A: Math. and e−iβJx = e−iβk K k K k = eiβk Kk Kk Theor. 46, 355301 (2013). k=P−jj j| − ih − | k=P−j | ih | [5] N. Cotfas, J.-P. Gazeau and A. Vourdas, J. Phys. A: = eiβkK (m)K (n) j;m j;n. Math. Theor. 44 175303 (2011). k k | ih | [6] A. Erdélyi, Higher Transcedental Functions, Vol. 1, pag m,n= jk= j P− P− 85 (McGraw-Hill, New York,1953). In the case of the representation of SU(2) in ℓ2[ j,j], [7] J.-P. Gazeau, Coherent States in Quantum Physics − the element with Euler angles α, β, γ corresponds to (Berlin: Wiley-VCH,Berlin, 2009). [8] J.VanderJeugtandR.Jagannathan,J.Math.Phys.39 j e−iαJze−iβJxe−iγJz= e−i(αm+γn) 5062 (1998). m,n= j [9] W. Miller, Jr., Lie Theory and Special Functions (Aca- Pj − demic Press, New York,1968). eiβkKm(k)Kn(k) j;m j;n. [10] A.F.Nikiforov,S.K.SuslovandV.B.Uvarov,Classical × | ih | k=−j OrthogonalPolynomialsofaDiscreteVariable(Springer, P We think that, in certain applications, our description Berlin, 1991). of the linear representations of su(2) and SU(2) is more [11] A. M. Perelomov, Generalized Coherent States and their Applications (Springer, Berlin, 1986). advantageous than other known descriptions [9, 10, 13, [12] J. Schwinger, Proc. Natl. Acad. Sci. (USA) 46 570 14]. The spin coherent states [7, 11], that is, the orbit of (1960). SU(2) passing through j; j , is formed by the states [13] J. D. Talman, Special Functions. A Group Theoretical | − i α,β =e−iαJze−iβJxe−iγJz j; j Approach (Benjamin, New York,1968). | i | − i [14] N. Ja. Vilenkin, Fonctions Spéciales et Théorie de la =eijγ j e iαm j eiβk Cj+k K (k) j;m . Représentation des Groupes (Dunod,Paris, 1969). 2j − 2j m | i [15] A. Vourdas, Rep.Prog. Phys.67 267 (2004). m=−j k=−j q [16] K.B. WolfandG. Krötzsch,J.Opt.Soc.Am.A24651 P P Evidently, we can neglect the unimodular factor eijγ. (2007). 5 SUPPLEMENTARY INFORMATION The reader can directly check the relation (16) by using the program in Mathematica • j=10 K[m_,n_]:=Sum[(-1)^k Binomial[j+n, k] Binomial[j-n,j+m-k], {k,0,j+m}] MatrixForm[Table[Sum[(-I)^k K[m,k] K[k,n], {k,-j,j}] - (-2)^j I^(m+n) K[m,n], {m,-j,j}, {n,-j,j}]] for j 0,1,2,3,... , and the program ∈{ } j=3/2 K[m_,n_]:=Sum[(-1)^k Binomial[j+n, k] Binomial[j-n,j+m-k], {k,0,j+m}] N[MatrixForm[Table[Sum[(-I)^k K[m,k] K[k,n], {k,-j,j}] - (-2)^j I^(m+n) K[m,n], {m,-j,j}, {n,-j,j}]]] for j 1,3,5,... . ∈ 2 2 2 (cid:8) (cid:9) By using the program • j = 10 KP[m_, n_] := Sum[(-1)^k Binomial[j + n, k] Binomial[j - n, j + m - k], {k, 0, j + m}] KF[m_, n_] := (1/2^j) Sqrt[Binomial[2 j, j + n]/Binomial[2 j, j - m]] KP[m, n] K := Table[(-1)^(2 j) I^m KF[-m, n], {m, -j, j}, {n, -j, j}]; Jz := Table[m DiscreteDelta[m - n], {m, -j, j}, {n, -j, j}] Jx := K.K.Jz.K Jy := K.Jz.K.K MatrixForm[MatrixPower[K, 3]] MatrixForm[Jz] MatrixForm[Jx] MatrixForm[Jy] MatrixForm[Jx.Jy - Jy.Jx - I Jz] MatrixForm[Jy.Jz - Jz.Jy - I Jx] MatrixForm[Jz.Jx - Jx.Jz - I Jy] for j 0,1,2,3,... , and ∈{ } j = 3/2 KP[m_, n_] := Sum[(-1)^k Binomial[j + n, k] Binomial[j - n, j + m - k], {k, 0, j + m}] KF[m_, n_] := (1/2^j) Sqrt[Binomial[2 j, j + n]/Binomial[2 j, j - m]] KP[m, n] K := Table[(-1)^(2 j) I^m KF[-m, n], {m, -j, j}, {n, -j, j}]; Jz := Table[m DiscreteDelta[m - n], {m, -j, j}, {n, -j, j}] Jx := K.K.Jz.K Jy := K.Jz.K.K N[MatrixForm[MatrixPower[K, 3]]] N[MatrixForm[Jz]] N[MatrixForm[Jx]] N[MatrixForm[Jy]] N[MatrixForm[Jx.Jy - Jy.Jx - I Jz]] N[MatrixForm[Jy.Jz - Jz.Jy - I Jx]] N[MatrixForm[Jz.Jx - Jx.Jz - I Jy]] for j 1,3,5,... , one can directly verify the relation ∈ 2 2 2 (cid:8) (cid:9) K3 =I and to see that the operators J =K+QK, J =KQK+ and J =Q x y z satisfy the relations [J ,J ]=iJ , [J ,J ]=iJ , [J ,J ]=iJ . x y z y z x z x y 6 Itisknownthatsomeidentitiesforpowersfailforcomplexnumbers,nomatterhowcomplexpowersaredefined • as single-valued functions. For example: ( 1)21 =eπ2i − i21 =eπ4i and consequently ( 1)21 i21 =( i)21; ( i)21 =e−π4i − 6 − − (−1)12 =e1π212i==1i and consequently −11 21 6= ( 1112)12. (cid:16) (cid:17) − So, we have to be careful in computations involving half-integer powers of complex non-positive numbers. In our computations involving half-integer powers of 1, i, i, we use: − − – the definition zk = z keikarg(z); | | – the identity zk zm=zk+m for integer as well as half-integer k and m; – the identities zk zk=(z z )k and z1k= z1 k only for integer k. 1 2 1 2 z2k z2 (cid:16) (cid:17) The proofs of our theorems are based on the relations: • (1 i)2j =(√2)2je−2j4πi =2je−j2πi =2j( i)j; − − j+n (1+i)j+n(1 i)j n= 1+i (1 i)2j =ij+n(1 i)2j =2j( i)jij+n; − − 1 i − − − − (cid:16) (cid:17) im+k( 1)j+k( 1)j+n=( 1)j+nim jij+k( 1)j+k − − − − − =( 1)j+nim j( i)j+k =( 1)j+nim j( i)j( i)k; − − − − − − − ( 1)j+nim j( i)jij+mij+n=( 1)j+ni2mij+n( i)j =( 1)j+m+nij+n( i)j − − − − − − − =( 1)nij+n( i)j( 1)j+m − − − ( 1)nij+n( i)j= 1 ( 1)j+nij+n( i)j = 1 ( i)j+n( i)j = 1 ( i)2j( i)n − − ( 1)j − − ( 1)j − − ( 1)j − − − − − = ((−11))2jj (−i)2j(−i)n =(−1)j ((−1i))22jj (−i)n =(−1)j −1i 2j (−i)n =(−1)2j(−i)n. − − − (cid:16) (cid:17)