Ordering of operators is purely combinatorial task involving a number
of commutators shuffling components of operator expression to desired
form. Here we show how it can be illustrated by simple urn models in
which normal ordering procedure is equivalent to enumeration of urn
histories

Blasiak, Pawel

English

Repositorium für Naturwissenschaften und Technik  (TIB Hannover)

 Oberwolfach Preprints  Mathematisches Forschungsinstitut Oberwolfach gGmbH Oberwolfach Preprints (OWP)   ISSN 1864-7596 OWP 2008 - 06 PAWEL BLASIAK  Urn Models & Operator Ordering Procedures  Oberwolfach Preprints (OWP)  Starting in 2007, the MFO publishes a preprint series which mainly contains research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs-Programme (RiP) and the Oberwolfach-Leibniz-Fellows (OWLF), but this can also include an Oberwolfach Lecture, for example.   A preprint can have a size from 1 - 200 pages, and the MFO will publish it on its website as well as by hard copy. Every RiP group or Oberwolfach-Leibniz-Fellow may receive on request 30 free hard copies (DIN A4, black and white copy) by surface mail.   Of course, the full copy right is left to the authors. 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Imprint:   Mathematisches Forschungsinstitut Oberwolfach gGmbH (MFO)  Schwarzwaldstrasse 9-11  77709 Oberwolfach-Walke  Germany   Tel  +49 7834 979 50  Fax  +49 7834 979 55  @mfo.delf@mfo.dEmail   URL  www.mfo.de admin@mfo.d The Oberwolfach Preprints (OWP, ISSN 1864-7596) are published by the MFO.  Copyright of the content is hold by the authors.  Urn Models&Operator Ordering ProceduresP. BlasiakH. Niewodniczański Institute of Nuclear PhysicsPolish Academy of Sciencesul. Eliasza-Radzikowskiego 152PL 31342 Kraków, PolandAbstractOrdering of operators is purely combinatorial task involving a numberof commutators shuffling components of operator expression to desiredform. Here we show how it can be illustrated by simple urn models inwhich normal ordering procedure is equivalent to enumeration of urnhistories.11 IntroductionOperator algebras constitute mathematical framework within which manymodern theories are built. Probably the most spectacular one is QuantumMechanics with operator formalism at the very heart of the theory [1, 2]. Themost unexpected, yet unavoidable, characteristic that makes it so strangeand successful at the same time is non-commutativity. It has been real-ized very early since the advent of the theory non-commutativity has manyimportant consequences [3], such as Bose-Einstein condensation, supercon-ductivity, photon correlations, etc. At the same time, the new quality causedby the fact that the order of operators is relevant stimulated developmentof novel methods capable of tracing the order of components in operatorexpressions.A common realization of operator algebra in quantum theory is the occu-pation number representation in which the fundamental role is played by theannihilation a and creation a† operators acting in the infinite dimensionalHilbert space spanned by vectors |n〉 labeling some characteristic of a systemN |n〉 = n|n〉. Operators a and a† are interpreted as the operations shift-ing these characteristic by one, which is embodied in the algebraic relations[a, N ] = a and [a†, N ] = −a†. Conventionally, they are required to satisfythe canonical Heisenberg-Weyl commutation relation[a, a†] = 1 , (1)being the hallmark of non-commutativity in Quantum Theory. This causesambiguities in the representation of an operator as an expression in a and a†,e.g. aa† = a†a + 1, and one needs to standardize the notation by fixing thepreferred order of operators. An important practical example of operatorordering is the normally ordered form in which all annihilation operatorsstand to the right of the creation operators. Shuffling operators into thisform comes down to a number of commutations of type Eq. (1) which, ingeneral, is a highly nontrivial problem of genuine combinatorial origin [4, 5].Here we construct an elementary urn model illustrating the normal order-ing procedure for operators satisfying Eq. (1). It’s convenience comes fromintuitive concept of enumeration of urn histories providing a simple pictureof this, after all, abstract mathematical construction. We also develop anelegant resolution to the problem based on generating functions methodol-ogy [6]. The primary interest of this paper is attached to intriguing analogybetween combinatorial urn models used in description of various discrete2phenomena [7, 8] and ordering procedures forced by operator formalism ofquantum physics.2 Urn model2.1 Urns and processesWe consider here urns U containing balls which are strictly distinguishablebetween each other, e.g. numbered with different integers. For short, Undenotes urn with n balls. One can modify contents of the urn by elementaryoperations of removing a ball out or adding one into the urn, see Fig. 1.We assume here that these operations are done one-by-one, i.e. only oneball is taken out or put in at a time, and denote them by D and X re-spectively. A basic process is just the composition of the elementary ones,e.g. X 2D3X 4D ≡ XXDDDXXXXD represents composite process mean-ing: “take a ball out, put four balls in, take three balls out, and then puttwo balls in”. Each process may be realized in many ways since there aredifferent choices of the balls in the urn. Note also that the order in whichelementary operations are made is crucial for the number of possible historiesthat may occur – there is one more possibility for DX than for XD – beinga sign of non-commutativity in the model.We shall further need to capture indeterminacy of the process itself. Inother words, we assume that each time it is chosen at random from a givenrepertoire denoted by H. To account for different probabilities with whichbasic processes may occur we allow for their copies in H. The numbers hkcounting copies of the same basic process Hk in H describe their relativeprobabilities, shortly denoted as H = ∑k hkHk. For example, in processH = 2X 3D + 5XDX probability that occurs X 3D or XDX is as 2 : 5.2.2 Urn historiesClearly, applying process H to urn U (possibly many times) one ends withoutcome which has some history behind, i.e. the record of events whichhappened in a series of steps. We shall be interested in enumerating allpossible histories with given input and output, i.e.G(n)l→k :={number of historiesin n steps from urn Ul to Uk}. (2)3234512345123456XDFigure 1: Urn containing 5 balls and elementary operations of removal Dand addition X of a ball. There are 5 possibilities to take a ball out, andonly 1 to put a ball in.Enumeration of histories is a nontrivial task, especially if one needs to do itfor general n. An elegant and efficient way of sorting and tackling informationabout sequences is attained through their generating functions. Hence, foreach n we define the multivatiate generating functionsG(n)(x, y) =∑k,lG(n)l→k xk yll!, (3)and the exponential generating functionG(x, y, z) =∑nG(n)(x, y)znn!. (4)In practice one often faces the problem of explicit calculation or at leaststudying properties of these objects. A typical issue addressed in this contextconcerns finding distribution of probabilities P(n)l→k of ending with urn Uk ifstarted from Ul as the results of n iterations of a given process H, defined byP(n)l→k =G(n)l→k∑k G(n)l→k. (5)4Probability generating function is then simply expressed trough G(x, y, z) as∑k,lP(n)l→k xk yll!=[zn]G(x, y, z)[zn]G(1, y, z)(6)=∂nz G(x, y, z)|z=0∂nz G(1, y, z)|z=0, (7)and all statistical properties, such as moments, asymptotic, etc., can be con-veniently studied with the methods of generating functions [6].In Section 3 we shall provide a simple scheme of calculating generatingfunctions of Eqs. (3) and (4) deriving from the operator ordering methodol-ogy.2.3 Operator representationThis simple urn model can be conveniently described in terms of polynomialsand differential operators. Let us represent an urn Un containing n balls bymonomial xn, and elementary operations X and D by multiplication X andderivative D operators respectively, i.e.Un ←→ xn,D ←→ D,X ←→ X.Observe that acting with thus obtained representation of a basic process onxn we get monomial corresponding to the resulting urn multiplied by thenumber enumerating all possible histories in which the process could haveoccurred, e.g. X2D3X3D xn = n(n + 2)(n + 1)n xn+1. Following up thisremark one finds out that correspondenceH ←→ H(X, D)holds true for any process H. Accordingly, applying operator H(X, D) to xnwe get polynomial indicating possible results with coefficients enumeratingall histories in which the process could have occurred, e.g. for H = 2X 3D+5XDDX , one gets H(X, D) xn ≡ (X3D + 5 XDDX) xn = 2n xn+2 + 5(n +1)n xn.5These observations are a simple consequence of intentional choice of therepresentation of urns by monomials and basic processes by multiplicationand derivation operators respectively, dictated by the relationsD xn = n xn−1,X xn = xn+1, (8)and reflecting the facts that• there are n ways of removing a ball from urn containing n distinguish-able balls,• a ball can be added to any urn only in one way.This gives surprisingly simple combinatorial insight into the commutator[D, X] = 1. (9)Thus established correspondence translating model of urn processes intothe language of polynomials and differential operators will establish equiva-lence with operator ordering procedures. We will find it useful in enumeratingurn histories in Section 3.3.3 Normal ordering procedure3.1 Normal orderMotivation for operator ordering procedures has strong grounds in quantumphysics where reshuffling of operator expressions to desired form introducesboth calculational and interpretative advantages. A typical example is theoccupation number representation where the annihilation a and creation a†operators, satisfying the relation of Eq. (1), are best handled if all annihila-tors stand to the right of creators. For a gentle introduction to the normalordering methods as well as physical motivation see [5].For the purpose at hand we observe that algebra of Eq. (1) may be realizedin the space of polynomials (formal power series) as followsa† ←→ Xa ←→ D (10)6where X and D ≡ ∂x are the multiplication and derivative operators definedin Eq. (8). We note that choice of this representation is not accidental as weare interested in combinatorics underpinning the Heisenberg-Weyl commu-tation relation of Eqs. (1) or (9).Accordingly, we standardize the notation of any operator H given asexpression in operators X and D to its normally ordered form in whichall multiplication operators X stand to the left of derivative operators D.Clearly, each H can be unambiguously put in this form, i.e.H(X, D) =∑k,l≥0hkl XkDl. (11)By the normal ordering of an operator we mean procedure of moving all thederivatives D to the right using commutation relation of Eq. (9). Below weshall be interested in normal ordering of powers and exponential of a givenoperator.3.2 Formal resolutionSuppose we are given operator H(X, D) and look for the normally orderedform of its n-th power, i.e.(H(X, D))n =∑k,l≥0h(n)kl XkDl. (12)(If H(X, D) is as in Eq. (11) one has h(0)kl = δk0δ0l and h(1)kl = hkl.)The normally ordered form of operator in Eq.(12) can be conveniently en-coded into polynomialBn(x, y) =∑k,l≥0h(n)kl xkyl, (13)Observe that it satisfies the identityBn(x, y) = e−xyH(X, D)nexy. (14)Now we are ready to derive the recurrence for the polynomials Bn(x, y). ByEq. (14) we haveBn+1(x, y) = e−xyH(X, D)exyBn(x, y),7and using property Dlexy = exy(D + y)l, we arrive at the recurrenceBn+1(x, y) = H(X, D + y) Bn(x, y). (15)The exponential generating function of polynomials Bn(x, y) is defined byB(x, y, λ) =∞∑n=0Bn(x, y)λnn!. (16)Note that, similarly to Eq. (14), one hasB(x, y, λ) = e−xyeλH(X,D)exy. (17)Differentiating B(x, y, λ) with respect to λ yields∂λ B(x, y, λ) =∞∑n=0Bn+1(x, y)λnn!=∞∑n=0H(X, D + y) Bn(x, y)λnn!,which gives the following partial differential equation∂λ B(x, y, λ) = H(X, D + y) B(x, y, λ), (18)with initial condition B(x, y, 0) = 1.In this way we have obtained formal resolution to the problem. Observethat the right hand sides of equations(H(X, D))n = Bn(X, D), (19)eλH(X,D) = B(X, D, λ) (20)are in the normally ordered form. We note that in many cases Eq. (18) canbe solved providing explicit analytic formulas.3.3 Enumeration of urn historiesHere we will show utility of ordering methods in calculating generating func-tion of urn histories of Eqs. (3) and (4). Recall from Section 2.3 that for agiven process H we haveHnxl =∑kG(n)l→k xk, (21)8which if multiplied by yl/l! and summed over l yields generating functions ofurn historiesG(n)(x, y) = Hn exy (22)andG(x, y, z) = ezH exy. (23)For efficient use of Eqs. (22) and (23) one needs to know the action of oper-ators (H(X, D))n and ezH(X,D) on functions (in this particular case on exy).It becomes trivial when the normally ordered form of operator is known.Hence, from Eqs. (19) and (20) we haveG(n)(x, y) = Bn(x, y) exy (24)andG(x, y, z) = B(x, y, z) exy (25)proving that calculation of both functions G(x, y, z) and B(x, y, z) is equiva-lent. In this way we have shown the one-to-one correspondence between theproblems of enumeration of urn histories of iterated process H and normalordering of powers of the associated operator H(X, D). From one side or-dering methods provide efficient tools capable of tracing of urn histories, andvice versa – the urn model illustrates the procedure itself.For illustration consider urn process H = XD + gX + gD. It corre-sponds to the Hamiltonian H = a†a + g (a† + a) describing oscillator drivenby external force – a simple model of a system coupled to the environment.Particularly interesting is the picture of dynamics generated by H in termsof urn histories. It translates into the process H in which at each step oneball can be either inspected, added or removed with relative probabilities 1,g and g respectively. Inspecting XD means just drawing a ball, looking atand then putting it back into the urn – hence in no way affecting number ofthe balls in the urn – representing free (undisturbed) evolution of a system.The remaining two terms introduce disturbance into the scheme by randomaddition X or removal D of a ball, interpreted as the effect of external fac-tor. Consequently, number of balls in the urn may change and urn historiesproliferate as conveniently described by generating function of Eq. (4)G(x, y, z) = e(x+g)(y+g)(ez−1)e−g2zexy, (26)obtained from Eqs. (18) and (25). Statistical properties of the model arestraightforward application of combinatorial analysis [6].94 OutlookWe have pointed out that abstract Heisenberg-Weyl commutation relationhave purely combinatorial underpinning. It allowed for construction of ele-mentary urn model illustrating the operator normal ordering procedure whichwas proved equivalent to enumeration of urn histories. In this way, theseboth seemingly unrelated procedures gain new perspective and may draw onmethods taken one from another. From one side combinatorial enumerationof histories is facilitated if the normally ordered form of involved operators isknown. On the other hand abstract mathematical construction of operatorordering gains straightforward interpretation as enumeration of histories insimple combinatorial models. The latter particularly interesting in descrip-tion of quantum systems, and pointing at efficiency of combinatorial methodsbased on generating functions approach.Anticipating further analogies we observe that multi-mode systems canbe modeled by urns containing balls of different type, e.g. each mode havingdifferent colour. Mixing terms in the Hamiltonian, introducing entanglementinto quantum system, may be then interpreted as processes swapping betweendifferent types of balls in the urn. We expect that at least some of strangequantum phenomena can be made more intuitive if looked from this simplecombinatorial perspective.AcknowledgmentsI would like to appreciate warm hospitality and support of MathematischesForschungsinstitut Oberwolfach where much of this research was carried outunder the Oberwolfach Leibniz Fellowships Programme. The author wassupported by the Polish Ministry of Science and Higher Education GrantsNo. N202 061434 and N202 107 32/2832.10References[1] P. A. M. Dirac. The Principles of Quantum Mechanics. Oxford UniversityPress, New York, 4th edition, 1982.[2] C. J. Isham. Lectures on Quantum Theory: Mathematical and StructuralFoundations. Imperial College Press, London, 1995.[3] L. E. Ballentine. Quantum Mechanics: A Modern Development. WorldScientific, Singapore, 1998.[4] W. H. Louisell. Quantum Statistical Properties of Radiation. John Wiley& Sons, New York, 1990.[5] P. Blasiak, A. Horzela, K. A. Penson, A. I. Solomon, and G. H. E.Duchamp. Combinatorics and Boson normal ordering: A gentle intro-duction. Am. J. Phys., 75:639–646, 2007. arXiv:0704.3116 [quant-ph].[6] P. Flajolet and R. Sedgewick. Analytic Combinatorics. To be publishedby Cambridge University Press, 2008. Web edition available from authors’websites.[7] P. Flajolet, P. Dumas, and V. Puyhaubert. Some exactly solvable modelsof urn process theory. In Discrete Mathematics & Theoretical ComputerScience AG, pages 59–118, 2006.[8] N. L. Johnson and S. Kotz. Urn Models and Their Applications. JohnWiley & Sons, New York, 1977.11

Urn models & operator ordering procedures

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Urn models & operator ordering procedures

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