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Some Integrable Quantum Systems on the Lattice Miguel Lorente 4 0 Departamento de F´ısica (Facultad de Ciencias) 0 Universidad de Oviedo 2 33007 Oviedo, Spain n a J 4 1 1 Abstract v 7 TheWeyl relations, theharmonicoscillator, the hydrogen atom, the Dirac equa- 7 tion on the lattice are presented with the help of the difference equations and the 0 1 orthogonal polynomials of discretevariable. Thisarea of research isattracting more 0 interest due to the lattice field theories and the hypothesis of a finite space. 4 0 / h p - t n a 1 Weyl group on finite space u q : We defined the position space of dimension N with orthonormal basis v i X 1 0 0 r 0 1 0 a       0 = 0 , 1 = 0 , , N 1 = 0 , i j = δ | i  ...  | i  ...  ··· | − i  ...  h | i ij             0 0 1             Similarly we construct an N-dimensional momentum space with ortonormal basis 1 0 1 1 1 ω ω2 ωN 1        −  0 = 1 , 1 = ω2 , 2 = ω4 , N 1 = ω2N 2 − | i  ...  | i  ...  | i  ...  | − i  ...          1 ωN−1 ω2N−2 ω(N−1)2                 where ωN = 1, and l k = δ lk h | i Two operators acting on these spaces are defined as 0 1 0 0 1 ··· 0 0 1 0 ω    ··· A = 0 0 0 1 , B = ω2 , a,b Z  ···  ...  ∈ ·1·· ·0·· ·0·· ·0·· ······  ωN−1     from which we construct U Aa, V Bb, a,b Z a b ≡ ≡ ∈ On the position space, we have U j = j a , V j = ωbj j , (mod. N) and in the a b | i | − i | i | i momentum space U k = ωak k , V k = k +b , (mod. N) a b | i | i | i | i We can define the representation of vectors and operators as follows. From the expansion in the momentum space 1 N 1 N 1 − − F = k k F = b k k | i N | ih | i | i k=0 k=0 X X we obtain the representation in position space F (j) j F = b f (j) (1) k k ≡ h | i wheref (j) j k = 1 ωjk X k ≡ h | i √N Similarly, from the expansion in momentum space, 1 N 1 N 1 − − G = j j G = a j j | i N | ih | i | i k=0 j=0 X X we obtain the representation in momentum space G(k) k G = a f (k) (2) j j ≡ h | i X For the operators V , V we have in position space a b (U F)(j) = j U F = F (j +a) a a h | | i (V F)(j) = j V F = ω bjF (j) b b − h | | i and in momentum space (U G)(k) = k U G = ω akG(k) a a − h | | i (V G)(k) = k V G = G(k +b) b b h | | i From (1) and (2) we construct a Finite Fourier transform 1 N 1 Fˆ(k) = − F ωjk, F = j F j j √N h | i j=0 X 1 N 1 F (j) = − Fˆ(k)ω jk, Fˆ(k) = k F − √N h | i k=0 X The Weyl approach to Quantum Mechanics[1] is based in the properties of the operators A,B when the N-dimensional space becomes infinite. Postulate I. There exist two parameter abelian groupin anN-dimensional space whose elements A and B satisfy AB = ωBA , ωN = 1 As j = j +s Bt j = ωjt j | i | i | i | i Postulate II. In the continuous limit N we can identify → ∞ ω ei2Nπ eiξη, ξ << 1, η << 1 ≡ −→ s As eiξP eiσP, ξs σ ≡ −→ → (cid:16) (cid:17)t Bt eiηQ eiτQ, ηt τ ≡ −→ → (cid:16) (cid:17) ωst eisξtη eiστ ≡ −→ AsBt = ωstBtAs eiσPeiτQ = eiστeiτQeiσP −→ The justification of Postulate II lyes in the fact that in the continuous case the action of the translation and multiplication operators is U f (x) eiσPf (x) = f (x+σ) σ ≡ V f (x) eiτQf (x) = eiτqf (x) τ ≡ which are equivalent to the relations of Postulate I. From these equations the interpretation of the operators P, Q is derived as the generators of the translations and multiplica- tions operators. 2 The harmonic oscillator on the lattice In the discrete case we take the Kravchuk polynomials k(p)(x,N) with x = 0,1,2, n ··· N 1, [2] and for the normalized functions the Wigner functions, that appear in the − representation of the rotation group, dj (β) mm′ (−1)m−m′djmm′ (β) = d−n1 ρ(x)kn(p)(x,N) q β β with N = 2j m = j n m = j x p = sin2 , q = cos2 ′ − − 2 2 After substitution in the fundamental formulas for the orthogonal polinomials we get [3] for the creation and annihilation operators pq(n+1)(N n)dj (β) = − j−n−1, j−x q = p(N x n)dj (β)+ pqx(N x+1)dj − − j−n, j−x − j−n, j−x+1 q pqn(N n+1)dj (β) = − j−n+1, j−x q = p(N x n)dj (β)+ pq(x+1)(N x)dj (β) − − j−n, j−x − j−n, j−x−1 q The last equations can be written down in terms of the new parameters 1 sinβ (j +m)(j m+1)dj (β) = sin2β (m+m)dj (β)+ 2 − m 1,m′ 2 ′ m,m′ − q +1 sin β (j m)(j +m +1)dj (β) 2 − ′ ′ m,m′+1 q 1 sin β (j m)(j +m+1)dj (β) = sin2β (m+m)dj (β)+ 2 − m+1,m′ 2 ′ m,m′ q +1 sin β (j +m)(j m +1)dj (β) 2 ′ − ′ m,m′ 1 − q The creation and annihilation operators are connected with the raising and lowering operators for the spherical harmonics Y . In fact, from the connection between dj and jm mm′ Y , we get jm 1 1 AY = (j m)(j +m+1)Y = J Y jm j,m+1 + jm √2j − √2j q 1 1 A Y = (j +m)(j +m+1)Y = J Y † jm j,m 1 jm √2j − √2j − q In order to make more transparent the connection between the creation and annihilation operators with the raising and lowering operators of the spherical harmonics, we take the commutation and anticommutation relations of the former operators. n j j A,A d (β) = 1 d (β) † mm′ − j! mm′ h i wich in the limit j goes to → ∞ a,a ψ (s) = ψ (s) † n n h i Similarly n2 AA +A A dj (β) = (2n+1) dj (β) † † mm′ ( − j ) mm′ (cid:16) (cid:17) which in the limit j goes to → ∞ aa +a a ψ (s) = (2n+1)ψ (s) † † n n (cid:16) (cid:17) If we multiply both sides by h¯ω/2 we obtain the eigenvalue equation for the hamiltonian. The interpretation of this model can be taken from the quantum harmonic oscillator. The energy levels are equally distant by the amount h¯ω and are labelled by n = 0,1,2, . In the quantum harmonic oscillator of discrete variable we have also the ···∞ discrete eigenvalues of the hamiltonian connected with the index m = j n of the Wigner − function dj (β). mm′ These values are equally separeted but finite (m = j, +j). Similarly the eigenval- − ··· ues of theposition operatorA+A+ arealso discrete and connected to the index m = j x ′ − of the Wigner functions but finite (m = j, ,+j). ′ − ··· The integer numbers x = 0,1, 2j are related to the quantity x = αs where s is the ··· continuousvariableandα = Mω/h¯. Sincexisapurenumberandshasthedimensionof a length, the spacing of the oqne-dimensional lattice is equal to 1/α = h¯/Mω. Therefore the Planck’s constant h¯ play role with respect to discrete space similqar to the role with respect to discrete energy values. 3 The Hidrogen atom in the lattice We start from the difference equation for the Meixner polynomials, the limit of which goes to the Laguerre polynomials in the continuous case. For the normalized Meixner polynomilas M(γ)(x) = d 1 ρ(x)mγ (x) we get the following difference equation [3] n −n n q µ(γ +x)(x+1)M (x+1)+ µx(x+γ 1)M (x 1) n n − − − q [µ(x+n+γ) n+x]Mq(x) = 0 n − − For the Hidrogen atom in the continuous case one takes as the solution of the reduced radical equation the wave function ψ2l+1(s) = ρ (s)L2l+1 (s), ρ (s) = sρ(s) n 1 n l 1 1 −− q where ρ(s) is the weight function. In the discrete case we take the wave function U (x) = d 1 ρ (x)Mγ (x), ρ = µ(x+γ)ρ(x) n −n 1 n 1 q The difference equation now reads: µ(x+1) µx µ(x+γ)+x U (x) n U (x+1)+ U (x 1) U (x) = (µ 1)n n n n sx+γ +1 sx+γ − − x+γ − x+γ This equation is of Sturn-Liouville type, from which an orthogonality relation can be derived: U (x)U (x) ∞ m n = 0, if n = m x+γ 6 x=0 X We can construct also raising and lowering operators for the normalized Meixner functions. We get L+U (x) = µ(γ +n)(n 1)U (x) = µ(x+n+γ)U (x) µx(x+γ)U (x 1) n n+1 n n − − − q q x+1 L U (x) = µn(n+γ 1)U = µ(x+γ +n)U (x) µ(x+γ) U (x+1) − n n 1 n n − − − sx+γ +1 q The action of these operators is to create or annihilate a new state the eigenvalue of which (with respect to the energy operator) is increased or decreased by unity. 4 Dirac and Klein-Gordon equation on the lattice From the Dirac equation in momentum space (γ p m c)ψ(p) = 0 we can construct µ µ 0 − the wave equation in position space with the help of the Fourier transform. We define the following difference operators 1 ∆f (j) = f (j +1) f (j), ∆˜f (j) = f (j +1)+f (j) , − 2 { } 1 ˜ f (j) = f (j) f (j 1), f (j) = f (j)+f (j 1) , ∇ − − ∇ 2 { − } and the partial difference operators with respect to a function of several discrete variables ∆ f (j ) = f (j +δ ) f (j ), ν µ µ µν µ − 1 ˜ ∆ f (j ) = f (j +δ )+f (j ) , ν µ µ µν µ 2 { } and similarly f (j ) and ˜ f (j ). ν µ ν µ ∇ ∇ From these operators we construct 1 1 δ+ ∆ ∆˜ , δ ˜ , µ ≡ ε µ ν µ− ≡ ε∇µ ∇ν ν=µ ν=µ Y6 Y6 3 3 η+ ∆˜ , η ˜ . µ − µ ≡ ≡ ∇ µ=0 µ=0 Y Y From the Fourier transform we can derive the wave equation in lattice space. The kernel of the transform satisfies: 1 2 ∆ exp (2πi(k.j)ε) = i tan (πk ε)∆˜ exp (2πi(k.j)ε) µ µ µ ε ε We could apply the Fourier transform to the Dirac equation in momentum space and we would obtain the discrete wave equation. Instead we postulate a difference equation that in the limit goes to the continuous differential equation, namely, iγµδ+ m cη+ ψ(j ) = 0. µ − 0 µ The kernel of the Fourier (cid:16)transform or “pl(cid:17)ane wave” is a particular solution of this equation if it satisfies 2 γµ tanπk ε m c η+ exp 2πi(k j)ε = 0. µ 0 ε − · (cid:18) (cid:19) Applying the operator γµ2 tanπk ε+m c from the left to the last equation we obtain ε µ 0 4 (tan πkµε)(tan πk ε) m2c2 = 0, ε2 µ − 0 which is the integrability condition for the solution of the wave equation. Applying to the wave equation the operator iγµδ + m cη from the left we obtain µ− 0 − the discrete version of the Klein-Gordon equation in the lattice space δ+δµ m2c2η+η ψ(j ) = 0 µ − − 0 − µ a particular solution of which(cid:16)is again the “plane(cid:17)wave” provided the integrability condi- tions is satisfied. Acknowledgment(s) The autor expresses his gratitude to Prof. Doebner, Dobrev, Hennig and Luecke for the invitations to the Symposium. This work has been partially supported by D.G.I.C.Y.T. (Spain) # Pb 96 0538. References [1] H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, New York 1950. [2] A.F. Nikiforov, S.K. Suslov, V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin 1990. [3] M. Lorente, “Creation and Annihilation operators for orthogonal Polynomials of continuous and Discrete Variables”, Electronic Transactions on Numerical Analysis, Vol. 9 (1999), pp. 102-111. [4] M. Lorente, “Representations of the Discrete Inhomogeneous Lorentz Group and Dirac Wave equation on the lattice”, J.Phys.A: Math. Gen. 32 (1999) 2.481-2.497.