QUATUM MECHAICS O DISCRETE SPACE AD TIME Mguel Lorente Departamento de Físca, Facultad de Cencas Unversdad de Ovedo E-33007, Ovedo, Span We propose the assumpton of quantum mechancs on a dscrete space and tme, whch mples the modfcaton of mathematcal expressons for some postulates of quantum mechancs. In partcular we have a Hlbert space where the vectors are complex functons of dscrete varable. As a concrete example we develop a dscrete analog of the one-dmensonal quantum harmonc oscllator, usng the dependence of the Wgner functons n terms of Kravchuk polynomals. In ths model the poston operator has a dscrete spectrum gven by one ndex of the Wgner functons, n the same way that the energy egenvalues are gven by the other matrcal ndex. Smlar pcture can be made for other models where the dfferental equaton and ther solutons correspond to the contnuous lmt of some dfference operator and orthogonal polynomal of dscrete varable.. ITRODUCTIO In the last Symposum on Fundamental Problems n Quantum Physcs [] I explored the hypothess of a realstc nterpretaton of lattce theores based on some ontologcal model that presupposes some fundamental network prevous to the concept of space and tme. Accordng to ths model the structure of space-tme s a consequence of the relatons among these fundamental enttes, and gves rase to a dscrete character of the space and tme varables. Even the standard nterpretaton of quantum mechancs n our model s conserved the assumpton of a dscrete space and tme ntroduces some drastc changes n the mathematcal formulaton of quantum mechancs. These consequences have been also shared by recent authors who use lattce models as a mathematcal tool. In partcular non-commutatve geometry
Lorente leads n a natural way to dfference operators []. Quantum groups and q- analss mples a deformaton of space-tme groups wth a q-casmr defned on a space or tme lattce [3]. Lattce feld theores are wdely spread n an attempt to overcome nfntes n perturbaton theores [4]. Specal functons and orthogonal polynomals on contnuous varable are studed together wth those of dscrete varable [5]. on-eucldean crystallography requres the use of dscrete groups of hyperbole type that has been recently developped [6]. Recent lterature advocates for the applcaton of dscrete models to unfy general relatvty and quantum mechancs as Wheeler, Ponzano and Regge and others have proposed [7]. For ths purpose some modern tools are used, such as dscrete topology and partton functons defned over symplcal networks.. THE POSTULATES OF QUATUM MECHAICS O Z 4 The assumpton of dscrete space and tme mposses some restrctons on the mathematcal expressons for the postulates of Quantum Mechancs. The queston s now whether ths assumpton keeps the analogy wth standard formulaton n the contnuum case n the lmt both formulatons concde and at the same tme avods the unwanted nfntes. The answer s, generally speakng, n the affrmatve. In order to be more explct, we start wth the Hlbert space: ths must be defned over the gaussan numbers complex of nteger components wth scalar products constructed wth summaton nstead of ntegrals b fj g j j=a where we use the notaton f j f jε, ε beng the fundamental length of the one-dmensonal lattce, j nteger numbers. As an example, we take the followng orthonormal bass where f j k m = + εk j m εk, j =0,, m k m ε tg πm, m =0,, satsfyng j =0 f j k m f j k m =δ mm
Lorente 3 wth respect to whch a fnte Fourer transform equvalent to a Fourer seres can be defned F j = m =0 a m f j k m, a m = j =0 f j k m F j otce that although the space-tme varables are dscrete the functons can be contnuous. If we consder observables, the correspondng operators must be self adjont wth respect to the scalar product mentoned before, and the spectrum s always dscrete. As an example, we gve the egenvalues and egenfunctons of the poston and momentum operators. X : jδ jl = lδ jl, l fx ; j =0,, P : ε jf j k m = εk f j k m m where δ jl s the Kronecker functon and f j k m s defned as before. Wth the help of the scalar product we can defned espectaton values, projecton operators, densty matrx, uncertantes or mean values of some operators. Suppose now that a physcal system s represented by a state vector ψ t dependng on dscrete tme t = nτ. If H s the operator correspondng to the hamltonan of the system for smplcty we take H constant n tme the Schrödnger equaton for that system s the soluton of whch s k m τ nψ n = H n ψ n ψ n = τh n + τh ψ 0 wth the ntal condton ψ 0. Here n s the forward dfference operator n ψ n = ψ n+ ψ n and n the mean operator n ψ n = ψ n+ + ψ n As n the contnuous case we can defne an untary evoluton operator U n τh n + τh whch s untary because H s self adjont and satsfes the dfference equaton τ nu n = H n U n
Lorente 4 If we use the Hesenberg pcture the evoluton of some operators n tme s gven by A n = U n + A 0 U n where A n s some operators dependng on dscrete tme and U n s the evoluton operator defned before. The Hesenberg equaton now reads [8] τ A n = τh [A n,h n ] + τh the soluton of whch s. An other scheme may be used f we take the symmetrc dfference operator then Also we have τ δ na n = τ A n+ A n = δ n A n A n+ + 4 τ H A n [A n,h n ], + 4 τ H + 4 τ H τ [A n,h n ] 4 + 4 τ H Some smplfcaton can be acheved f we take the partcular case H =. It can be easly proved the followng equaton for the operator n the Hesenberg pcture: τ na n = τh [A + τh n,h] + τ 4 τ na n = τ + τ 4 n n A n = τ δ na n = + τ 4 + τh [A τh n,h] + τ 4 [A n,h] τ A n+ A n = [[A n,h],h] τ 4 + τ 4 [A n,h]
Lorente 5 In the last three equatons the dependence on the Hamltonan operator H s lnear as n the contnuous case. The realzaton of operators n the coordnate or poston representaton s gven by the substtuton X jε, P ε j The substtuton s not unque. We wll dscuss n the next secton some dfferent realzaton for the poston and momentum operator of he harmonc oscllator. 3. QUATUM HARMOIC OSCILLATOR OF DISCRETE VARIABLE The quantum harmonc oscllator s descrbed by the Schrödnger equaton hω [ ] d dξ + ξ ψ ξ =λψ ξ wth ω k M, ξ = αs, α Mω h, λ = E hω For smplcty, we take α = The normalzed solutons are ψ n s = π n n! e s / H n s, n =0,,, where H n s are the Hermte polynomals. The ψ n s are egenfunctons correspondng to the egenvalues λ = n +, and they satsfy the followng recurrence relatons: sψ n s = n +ψ n+ s+ nψ n s 3 d ds ψ n s = n +ψ n+ s+ nψ n s 4 From these two relatons one defnes the creaton and annhlaton operators: a ψ n s s d ψ n s = n +ψ n+ s 5 ds aψ n s s + d ψ n s = nψ n s 6 ds
Lorente 6 It s well known that the Hermte polynomals are the contnuous lmt of the Kravchuk polynomals of dscrete varables k n x andtheweght functon of the Hermte polynomals s the contnuous lmt of the bnomal dstrbuton ρ x whch n turns s the weght functon of the Kravchuk polynomal [9]. But the product of the Kravchuk polynomals tme ther weght functon s proportonal to the Wgner functons d j mm β thatappearn the generalzed sphercal functons namely, D j mm α, β, γ =exp mα d j mm βexp m γ, 7 d j ρ x mm β = m m k n p x, d n 8 where d n s some normalzaton constant, and n = j m, x= j m, p = sn β/. Ths connecton betwen the functons of dscrete and contnuous varable suggests that the soluton of the quantum harmonc oscllator are the contnuous lmt, up to a factor, of the Wgner functons. In order to prove ths Ansatz we compare the recurrence relatons of the two types of functons as t s done for the orthogonal polynomals of dscrete varable. In our case we take the dfferental equaton for the Wgner functon [0] ± d dβ dj mm β+ m m cos β sn β d j mm β = = j mj ± m +d j m±,m β 90 From ths we deduce two recurrence relatones: m mcos β d j sn β mm β = = j mj + m +d j m+,m β+ + j + mj m +d j m,m β j + m j m +d j m,m β j m j + m +d j m,m + β = = j mj + m +d j m+,m j + mj m +d j m,m
Lorente 7 ote that the last expresson has been obtaned wth the help of the well known property of Wgner functons We suppose that d j mm β = m m d j m m β lm C n d j mm β =ψ n s 3 where we take m = j n, m = j x, =j, x = p + pq s,and C n some normalzaton constant to be determned. We compare the recurrence relatons that s to say, formulas 3 and. We dvde the second one by j and substtute d j mm β by vnx C, n wth v n x C n d j j n,j x β. Theresults: or j x j ncos β v n x j sn β C n = + n n + j nn + j v n x C n x + v n+ x C n+ x p n vn x s + pq C n = n + n v n+ x C n+ + + n n v n x C n In the lmt ths expresson goes to the recurrence relaton 3 provded Cn C n+ = Cn C n =,orc n =const=. The recurrence relatons formules 4 and can be compared by the same method. We substtute n and dvde both sdes by. Theresults v n x d j j n, j x β 4 x x + x +v n x + x v n x = n + n = n v n x n +v n+ x
Lorente 8 Substtutng x = p+ pqs, and extractng p h sde, we obtan n the left h x + q + p s = n n q p s + v n x + p x v n x = v n x n + In the lmt, h 0 ths expresson goes to lm h 0 n v n+ x h {ψ n s + h ψ n s h} = n +ψ n s+ nψ n s that concdes wth 4 We can use these results to construct creaton and annhlaton operators for the Wgner functons. We defne Ad j mm β { m mcos β d j j sn β mm β+ j + m + j m + d j 4j m,m β } j m j + m + d j m,m 4j + β = j mj + m + = d j m+,m β 5 j A d j mm β { m mcos β d j j sn β mm β j + m j m + d j m,m 4j β+ } j m + j + m + d j m,m 4j + β = = j + mj m + d j m,m β j 6 Usng the lmt of the recurrence relatons we obtan
Lorente 9 lm Adj mm β = s + d ψ n s aψ n s 7 ds lm A d j mm β = s d ds ψ n s a ψ n s 8 Relatons 5 and 6 suggest that the creaton and annhlaton operators are connected wth the rasng and lowerng operators for the sphercal harmoncs Y jm. In fact, multplyng 5 and 6 by Y jm and addng for m we have or A m d j mm β Y jm = j mj + m + d j m+,m β Y j jm m AY jm = j j mj + m +Y j,m+ = j J + Y jm 9 smlarly, or A d j mm β Y jm = j + mj m + d j m,m β Y j jm m A Y jm = j j + mj m +Y j,m = j J Y jm 0 In order to make more transparent the connecton between the creaton and annhlaton operators wth the rasng and lowerng operators of the sphercal harmoncs, we take the commutaton and antcommutaton relatons of the former operators. AA A A Y jm = j J +J J J + Y jm = = j J zy jm = m j Y jm = n j Y jm Substtutng Y jm = d j m mm β Y jm we get [ A, A ] d j mm β = n d j mm β j whch n the lmt j goes to
Lorente 0 Smlarly [ a, a ] ψ n s =ψ n s AA + A A Y jm = j J +J + J J + Y jm = j = j j j + m Y jm = J Jz Y jm = { } n + n j or { } AA + A A d j mm β = n + n d j mm β j whch n the lmt j goes to aa + a a ψ n s =n +ψ n s If we multply both sdes by hω/ we obtan the egenvalue equaton for the hamltonan. The nterpretaton of ths model can be taken from the quantum harmonc oscllator of contnuous varable. The energy levels are equally dstant bytheamount hω and are labelled by n =0,,,. In the quantum harmonc oscllator of dscrete varable we have also the dscrete egenvalues of the hamltonan connected wth the ndex m = j n of the Wgner functon d j mm β. These values are equally separeted but fnte m = j, + j. Smlarly the egenvalue of the poston operator A+A + are also dscrete and connected to the ndex m = j x of the Wgner functons but fnte m = j,, +j. The nteger numbers x =0,, j are related to the quantty x = αs where s s the contnuous varable and α = Mω/ h. Snce x s a pure number and s has the dmenson of a length, the spacng of the onedmensonal lattce s equal to /α = h/mω. Therefore the Planck s constant h play an role wth respect to dscrete space smlar to the role wth respect to dscrete energy values. Y jm 4. COCLUDIG REMARKS The analyss we have made for the quantum harmonc oscllator of dscrete varable can be appled to other cases, where the functons nvolved are orthogonal polynomals of contnuous varable the lmt of whch are some orthogonal polynomal of dscrete varables; we gve some examples:
Lorente. The functon f j k m descrbed n secton are polynomals of dscrete varable the contnuous lmt of whch are the exponental functon. We have developed a new scheme for the Klen-Gordon and Drac feld equaton that can be extended to lattce gauge theores [].. The soluton of the Schrödnger equaton for the hdrogen atom s gven n term of the orthogonal Laguerre polynomals and the sphercal harmoncs. The radcal equaton can be translated nto the dfference equaton for the Mexner polynomal of dscrete varable. 3. The quantfcaton of the electromagnetc fels leads to the D Alambert equaton the soluton of whch are gven n terms of the Bessel sphercal functons that are related to the trgonometrc functons. These functons suggest the connecton wth the orthogonal polynomals of dscrete varable, that are solutons of dfference equatons of the hypergeometrc type. General speakng a parallel study of dscrete and contnuous model can be made smlar to that made by the russan school of mathematcans wth respect to orthogonal polynomals. ACKOWLEDGEMETS I would lke to thank the organzers for the nvtaton to ths meetng. I want to thank also to Prof. André Ronveaux for very llumnatng suggestons about the connecton between orthogonal polynomals of dscrete and contnuous varable and Prof. A. Galndo, A. Fz. Rañada, R. Alvarez Estrada for very nterestng comments about ths model. Ths work has been supported partally by D.G.I.C.Y.T. grant PB94-38. References [] M. Lorente, A realstc nterpretaton of lattce gauge theores, n Fundamental Problems n Quantum Physcs M. Ferrero and Alwyn van der Merwe ed. Kluwer Academc. Dordrecht 995, p. 77-86. [] A. Connes, Geometre non-commutatve, Interedtons, Pars 990. [3] A. Ballesteros, F.J. Herranz, M.A. Olmo, M. Santander, Deformaton of space-tme symmetres and fundamental scales, n Problems n Quantum Physcs M. Ferrero and A. van der Merwe ed. Kluwer Academc Dordrecht 995, p. 9-35. [4] A. I. Montvay, G. Münster, Quantum Felds on a Lattce, Cambrdge U. Press, 994. [5] A.F. kforov, V.B. Uvarov, Specal functons of mathematcal physcs, Brkhäuser 988.
Lorente [6] M. Lorente, P. Kramer, on-eucldean crstallography, Symmetres n Scence VII, B. Gruber, ed. Plenum ew York 995. [7] A. Kheyfets,.J. Lafave, W.A. Mler, ule-strut calculus, Phys. Rev.D4, 368, 3637 990 [8] M. Lorente, The method of fnte dfferences for some operator feld equatons, Lett. Math. Phys. 3 987 9-36. [9] A.F. kforov, S.K. Suslov, V.B. Uvarov, Classcal orthogonal polynomals of a dscrete varable. Sprnger Berln 99, p. 48. [0] See Ref.9 p. 33. [] M. Lorente, A ew Scheme for the Klen-Gordon and Drac Felds on the Lattce wth Axal Anomaly. J. of Group Theory n Physcs 993 p. 05-.