canonical transforms
Laurence BarkerCitation: Journal of Mathematical Physics 44, 1535 (2003); doi: 10.1063/1.1557331 View online: http://dx.doi.org/10.1063/1.1557331
View Table of Contents: http://aip.scitation.org/toc/jmp/44/4
Continuum quantum systems as limits of discrete
quantum systems. IV. Affine canonical transforms
Laurence Barkera)
Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey
共Received 4 June 2002; accepted 30 December 2002兲
Affine canonical transforms, complex-order Fourier transforms, and their associ-ated coherent states appear in two scenarios: finite-discrete and continuum. We examine the relationship between the two scenarios, making systematic use of inductive limits, which were developed in the preceding articles in this series. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1557331兴
I. INTRODUCTION
Inductive limits provide a clear and precise means whereby objects associated with a con-tinuum system can be realized as limits of objects associated with a sequence of discrete systems. Three preceding papers1–3 discuss inductive limits of vectors and operators. Another work4 con-cerns inductive limits of representations. In the present article, we illustrate the approach by applying it to a continuum scenario and a discrete scenario that lie in the core of quantum physics. Our main results are as follows. Theorem 6.1 realizes Glauber coherent states as inductive limits of spin coherent states. A practical version of the result goes back to Radcliffe5and Arecchi et al.6Theorem 5.3 realizes the group of continuum motion canonical transforms as an inductive limit of the group of discrete motion canonical transforms. A practical version was initiated in Ref. 6 and considerably developed by Atakishiyev et al.7,8 Theorem 5.1 and Corollary 5.2 realize parameter groups of continuum affine canonical transforms as inductive limits of single-parameter groups of discrete affine canonical transforms. Practical versions can be found in Do-brev et al.9Theorem 6.2 realizes continuum complex-order Fourier transforms as inductive limits of discrete complex-order Fourier transforms. From a practical point of view, that can be seen as a mild generalization of the fractional Fourier transforms in Ref. 10. In Ref. 4, Corollary 5.2 and Theorem 5.3 are expressed explicitly as inductive limits of representations but, in the present article, they are expressed simply as inductive limits of operators.
In using the adjective ‘‘practical,’’ rather than ‘‘heuristic,’’ we have erred towards understate-ment rather than overstateunderstate-ment. There is a vast body of literature on discrete to continuum corre-spondences that seem to be potential applications of inductive limits; see Sec. VII for a sample of further citations. Sometimes, in those works, the practical versions of the results have involved expressions of the formO⫽limnOn or On→O that do not conform to any evident definition of
limit. Sometimes, comparatively weak results have been stated and proved, yet with an apparently suggested meaning that goes beyond the literal interpretation; for instance, parallel discussion of continuum and discrete scenarios, the latter implicitly understood to be an approximation to the former. Actually, our use of inductive limits does have a practical intention, as we shall explain in Sec. VII.
Let us indicate the nature of the general kind of problem that concerns us. The limit equations in question are of the form O⫽limnOn, whereO is an object 共say, a vector, an operator or a
representation兲 associated with Hilbert space L, and each Onis an object associated with a Hilbert
space Ln. In this article, L⫽L2(R) and Ln is of finite dimension n. The problem is to select
appropriate definitions so as to make such limit equations potentially provable or refutable; or, at least, true or false. One approach is to embed the spaces Ln in the space L, and to replace
a兲Electronic mail: barker@fen.bilkent.edu.tr
1535
differential equations with corresponding difference equations. In general form, this is, of course, a numerical approximation technique that has been in widespread use ever since the emergence of statistical analysis in the 18th century. We must be very selective with our citations, since other-wise there would be no end to them. The convergence of eigenvectors examined in Ref. 11 may be applicable to the operators we consider below; this is significant, because convergence of spectral measures may be an interesting avenue for research into discrete to continuum limits of represen-tations共see Ref. 3, Sec. V兲. In Ref. 12, groups acting on Lnare embedded in groups acting onL,
and the discrete to continuum correspondence is characterized in terms of module induction. Another approach, proposed by Parthasarathy13,14and Lindsay–Parthasarathy,15is to collect all the spacesLntogether in a Fock space where limits can be examined without mentioning the spaceL.
Arguably, our approach is the most flexible of the three, since the definitions of inductive limits of vectors and operators do not require any constraints on the Hilbert spaces L and Ln 共except for
separability兲. However, it seems very probable that the particular limit equations in the present article can also be realized through the other two approaches.
Although some of the material below is in the nature of a review, this is a side-effect of a need to reformulate known results before presenting our own. We must also point out that although some of our limit formulas are unitary versions of accepted heuristic limits of Hermitian operators, the assertions that the formulas now express are new, since the kinds of limit involved had not previously been supplied with definitions.
II. CONTINUUM AFFINE CANONICAL TRANSFORMS
We shall introduce a six-dimensional connected real Lie group HSA⫽HSA(2,R) and an action of HSA as unitary operators on the continuum state space L2(R). As we shall see in the next section, HSA is a central extension of the special affine group SA on the plane; SA is also the Schoro¨dinger group with one space dimension and one time dimension. The group HSA, and its representation on L2(R), are discussed by Dobrev et al.,9 and Neiderer;16 for some other sources—oriented more towards the phase space picture—see Sec. III. Our main target, in this section, is to obtain explicit matrix representations for some generators of the Lie algebra of HSA. We shall also examine a subgroup HM of HSA. The group HM is a central extension of the Euclidian motion group.
The real Lie algebra hsa⫽hsa(2,R) has a basis兵iB,iC,iD,i P,iQ,iI其. The notation indicates that B, C, D, P, Q, I are elements of the complexification. The commutation relations are defined to be such that I is central,关Q,P兴⫽iI and, in the universal enveloping algebra,
B⫽12P
2, C⫽1
2Q
2, D⫽ 1
2共PQ⫹QP兲. It is not hard to show that the commutation relations involving B, C, D are
关B,P兴⫽0⫽关C,Q兴, 关C,P兴⫽iQ⫽⫺关D,Q兴, 关D,P兴⫽iP⫽⫺关B,Q兴,
共1兲 关B,C兴⫽⫺iD, 关B,D兴⫽⫺2iB, 关C,D兴⫽2iC.
For instance, 关B,C兴⫽ 1 4共P 2Q2⫺PQPQ⫹PQPQ⫺PQ2P⫹PQ2P⫺QPQP⫹QPQP⫺Q2P2兲 ⫽ 1 4共P关P,Q兴Q⫹PQ关P,Q兴⫹关P,Q兴QP⫹Q关P,Q兴P兲⫽⫺iD.
Let Bˆ , Cˆ, Dˆ, Pˆ, Qˆ, Iˆ be the Hermitian operators on L2(R) such that Iˆ is the identity operator and
Bˆ⫽12Pˆ
2, Cˆ⫽1
2Qˆ
2, Dˆ⫽1
2共PˆQˆ⫹QˆPˆ兲, 共3兲
wherebelongs to the Schwartz spaceS共R兲. The operators Pˆ and Qˆ are sometimes understood to correspond to momentum and position, respectively共or frequency and time, in signal processing, or frequency and position, in optics兲.
Letbe the anti-Hermitian representation of hsa on L2(R) such that the elements B, C, D, P, Q, I act as Bˆ , Cˆ, Dˆ, Pˆ, Qˆ, Iˆ, respectively. We introduce a real Lie group HSA⫽HSA共2,R兲 and a faithful unitary representationof HSA such that HSA has associated Lie algebra hsa and such that has differential representation. The elements of the group共HSA兲 are called continuum affine canonical transforms. Of course, there is no essential difference between the abstract Lie group HSA and the group of unitary operators共HSA兲. Each is isomorphic to the other via the isomorphism . Nevertheless, we do sometimes find it useful to distinguish between the two groups. Given real ,␥,␦,,,, we write
Hˆ共,␥,␦,,,兲⫽Bˆ⫹␥Cˆ⫹␦Dˆ⫹Pˆ⫹Qˆ⫹Iˆ, 共4兲 Uˆ共,␥,␦,,,兲⫽exp共⫺iHˆ共,␥,␦,,,兲兲. 共5兲 The continuum affine canonical transforms are the composites of operators that have the form Uˆ (, . . . ,).
Warning: some affine canonical transforms do not have the exponential form Uˆ (, . . . ,). We shall not be making use of this negative result, but we mention that it can be proved by considering the subquotient SL共2,R兲 of HSA, and using Eq. 共19兲.
As an element of the Lie algebra hsa, we define
N⫽B⫹C⫺I/2 . The corresponding Hermitian operator on L2(R) is
Nˆ⫽共N兲⫽Bˆ⫹Cˆ⫹Iˆ/2.
Let hm⫽hm共2,R兲 be the subalgebra of hsa with basis兵iI,iN,i P,iQ其and let HM⫽HM共2,R兲 be the subgroup of HSA with associated Lie algebra hm. We call HM the group of Heisenberg motions, and we call the elements of the group 共HM兲 the continuum motion canonical transforms. Again, there is no essential difference between the two isomorphic groups HM and共HM兲. The commutation relations for HM are given by Eq.共1兲 together with
关N,I兴⫽0, 关N,P兴⫽iQ, 关N,Q兴⫽⫺iP. 共6兲
The continuum共and discrete兲 motion canonical transforms will be of particular importance to us, and it is worth introducing some special notation for them. Given,,,苸R, we define
Eˆ共,,,兲⫽exp共⫺i共Iˆ⫹Nˆ⫹Pˆ⫹Qˆ兲兲. 共7兲 By passing to the quotient group HM/Z(HM)⬵EM 共see Sec. III兲, it can easily be shown that the operators having the form Eˆ (,,,) are closed under composition. In other words, the con-tinuum motion canonical transforms are precisely the operators having the form Eˆ (,,,).
We shall give some explicit matrix equations for the infinitesmal generators Bˆ , Cˆ, Dˆ, Pˆ, Qˆ, Iˆ of the continuum affine canonical transforms. For that, we need to specify a complete orthonor-mal set. Recall that, for s苸N, the s-th Hermite polynomial Hs and the s-th Hermite function
共⫺1兲sexp共q2/2兲 d
s
dqsexp共⫺q 2兲⫽H
s共q兲⫽
冑
s!2s冑
exp共q2/2兲hs共q兲. 共8兲Switching to Dirac notation, we write兩s
典
⫽hs. Note that the zeroth Hermite function兩0典
⫽h0 is the Gaussian functionh0共q兲⫽⫺1/4exp共⫺q2/2兲. 共9兲
Recall that 兵兩s
典
:s苸N其 is a complete orthonormal set in L2(R). Also recall that the annihilation operator Aˆ⫽(Qˆ⫹iPˆ)/& and its Hermitian conjugate, the creation operator Aˆ†⫽(Qˆ⫺iPˆ)/&, act byAˆ兩s
典
⫽冑
s兩s⫺1典
, Aˆ†兩s典
⫽冑
s⫹1兩s⫹1典
. 共10兲 By direct calculation using Eq.共10兲, we obtainBˆ 兩s
典
⫽⫺1 4冑
s共s⫺1兲兩s⫺2典
⫹ 2s⫹1 4 兩s典
⫹ ⫺1 4冑共s⫹1兲共s⫹2兲 兩s⫹2
典
, 共11兲 Cˆ 兩s典
⫽1 4冑
s共s⫺1兲 兩s⫺2典
⫹ 2s⫹1 4 兩s典
⫹ 1 4冑共s⫹1兲共s⫹2兲 兩s⫹2
典
, 共12兲 Dˆ 兩s典
⫽⫺i 2冑
s共s⫺1兲 兩s⫺2典
⫹ i 2冑共s⫹1兲共s⫹2兲 兩s⫹2
典
, 共13兲 Pˆ 兩s典
⫽⫺i冑
s 2兩s⫺1典
⫹i冑
s⫹1 2 兩s⫹1典
, 共14兲 Qˆ 兩s典
⫽冑
s 2兩s⫺1典
⫹冑
s⫹1 2 兩s⫹1典
, 共15兲 Iˆ兩s典
⫽兩s典
, 共16兲 Nˆ 兩s典
⫽s 兩s典
. 共17兲In Sec. IV, we shall find discrete analogues of these seven matrix equations.
Let us end this section with an example. Recall that the continuum Fourier transform is the unitary operator Fˆ on L2(R) such that Fˆ 兩s
典
⫽is兩s典
. More generally, after Namias,17 the con-tinuum fractional Fourier transform of order t苸R is the unitary operator Fˆt on L2(R) such that Fˆt兩s典
⫽exp(2ist)兩s典
. In other words,Fˆt⫽exp共2itNˆ兲⫽e⫺itexp共2i共Bˆ⫹Cˆ兲兲. 共18兲 III. THE CONTINUUM PHASE SPACE PICTURE
This section has two purposes. One of them is to fulfill the promise, made above, to explain how the groups HSA and HM are central extensions of the groups SA and EM, which act on the real plane. The other purpose is to clarify the relationship between the Hermitian operators and their corresponding unitary operators. In Refs. 6 –9 and 12, and many other works, limits are described mainly in terms of Hermitian operators. But inductive limits are defined for unitary operators; they are not defined for unbounded Hermitian operators. So we do need to be able to move freely from Hermitian operators to unitary operators, and in reverse.
The phase space picture provides much insight into these matters. There is a vast literature on phase space, and much attention has been paid to affine canonical transforms, especially special linear canonical transforms. See, for instance work by Folland,18 Hillery et al.,19 Littlejohn,20 Ozorio de Almeida;21 we also mention two collections of papers edited by Forbes et al.22 共on applications to optics兲 and Mecklenbra¨uker–Hlawatsch23 共on applications to signal processing兲. The relevant material, though, is not easy to extract from the literature. Let us give a brief self-contained account of it.
The phase space plane, denotedP, is defined to be a copy of R2. We regardP as a Euclidean plane equipped with a fixed coordinate system; the vectors are written as coordinate vectors ( p,q) where p and q are formal variables.
Recall that the group of special linear transforms of the real plane, denoted SL⫽SL共2,R兲, has Lie algebra sl⫽sl共2,R兲 with basis兵iB¯ ,iC¯ ,iD¯其 where
B ¯⫽
冉
0 ⫺i 0 0冊
, C ¯⫽冉
0 0 i 0冊
, D ¯⫽冉
i 0 0 ⫺i冊
. Thus, SL is generated by the elements having the form冉
a bc d
冊
⫽exp共⫺i共B¯⫹␥C¯⫹␦D¯兲兲⫽exp冉
␦ ⫺ ␥ ⫺␦
冊
,where,␥,␦苸R. Diagonalizing, a straightforward calculation shows that
冉
a b c d冊
⫽冉
cos␣⫹␦␣⫺1sin␣ ⫺␣⫺1sin␣
␥␣⫺1sin␣ cos␣⫺␦␣⫺1sin␣
冊
, 共19兲where␣is the real or imaginary number such that␣2⫽␥⫺␦2 and, for imaginary␣, we under-stand that cos␣⫽cosh i␣ and sin␣⫽i sinh i␣. Note that, for given real a, b, c, d satisfying ad ⫺bc⫽1, Eq. 共19兲 has a solution in reals,␥,␦if and only if a⫹d⭓⫺2. The natural action of SL on the real plane is given by
exp共⫺i共B¯⫹␥C¯⫹␦D¯兲兲
冉
x y冊
⫽冉
a b c d冊
冉
x y冊
⫽冉
ax⫹by cx⫹dy冊
. 共20兲The Euclidean special affine group SA⫽SA共2,R兲 共which coincides with the Schro¨dinger group with one space and one time dimension兲 is generated by SL and the plane translates. The associated Lie algebra sa⫽sa共2,R兲 has basis兵iB¯ ,iC¯ ,iD¯ ,iP¯ ,iQ¯其, where
exp共⫺i共P¯⫹Q¯兲兲
冉
x y冊
⫽冉
x⫹
y⫹
冊
. 共21兲Evidently,关P¯,Q¯兴⫽0. It is easy to check that the 14 other commutation relations are as in Eq. 共1兲. We allow SA to act onP via the identification (p,q)⫽(⫺y,x). Thus
exp共⫺i共B¯⫹␥C¯⫹␦D¯兲兲
冉
p q冊
⫽冉
d ⫺c b a冊
冉
p q冊
, exp共⫺i共P¯⫹Q¯兲兲冉
p q冊
⫽冉
p⫺ q⫹冊
.By comparing commutation relations, we see that there is a Lie algebra epimorphism hsa→sa mapping B, C, D, P, Q, I to B¯ , C¯ , D¯ , P¯ , Q¯ , 0, respectively. The group epimorphism HSA→SA has kernel
Ker共HSA→SA兲⫽Z共HSA兲⫽兵exp共⫺itI兲:t苸R其. We allow HSA to act onP by inflation from SA. Thus
exp共⫺i共B⫹␥C⫹␦D兲兲
冉
p q冊
⫽冉
d ⫺c b a冊
冉
p q冊
, 共22兲 exp共⫺i共P⫹Q⫹I兲兲冉
p q冊
⫽冉
p⫺ q⫹冊
. 共23兲The state space L2(R) and the phase space plane P are related to each other via the continuous function :L2共R兲苹哫关兴苸L R 2共P兲, 关兴共p,q兲⫽1
冕
⫺⬁ ⬁ dt共q⫹t兲共q⫺t兲 exp共2ipt兲.The functionis essentially a specialization of the famous Weyl–Wigner correspondence; see the references at the beginning of this section, especially Refs. 19 and 18. Given g苸HSA and 苸L2(R), then
关共g兲兴共g共p,q兲兲⫽关兴共p,q兲.
In other words,is covariant with the actions of HSA on the signal space L2(R) and on the phase space P. The result is proved in, for instance, Ref. 20 共Equations 6.18, 6.23, 6.27兲, and Ref. 18 共Proposition 2.13, Theorem 2.15兲. The rationale for our terminology should now be apparent: the ‘‘Heisenberg’’ groups HSA and HM are central extensions 共or quantized versions兲 of the groups SA and EM.
The special linear canonical transforms are usually understood to be unitary actions of SL on the state space L2(R). For an element of SL as in Eq. 共19兲, the action on state space is taken to be the unitary operator
共a,b,c,d兲Uˆ共,␥,␦,0,0,0兲⫽共a,b,c,d兲exp共⫺i共Bˆ⫹␥Cˆ⫹␦Dˆ兲兲,
where(a,b,c,d) is a phase. The phases (a,b,c,d) cannot be chosen so as to yield a unitary representation of SL. True enough, they can be chosen so as to preserve composition up to ⫾ signs, thus determining a unitary representation of the metaplectic group Mp共2,R兲, which is the double cover of SL. But that observation has limited practical use, since the description of the metaplectic group is very complicated; see Ref. 18, Chap. 4. For practical purposes, the special linear canonical transforms comprise a four-dimensional group, one of the degrees of freedom being the multiplications by phases. In fact, to establish a clear correspondence with the discrete scenario, we have no choice but to include the momentum and position translates, as well as the multiplications by phases. Thus, even if one is primarily concerned with the three-parameter group SL, the connection with the discrete scenario demands that we consider all six degrees of freedom in the group HSA.
IV. DISCRETE AFFINE CANONICAL TRANSFORMS
We shall introduce some discrete affine canonical transforms whose infinitesmal generators satisfy matrix equations analogous to Eqs. 共11兲–共17兲. First, we need to look at the Kravchuk functions, which are discrete analogs of the Hermite functions. We closely follow the representation-theoretic discussion of the Kravchuk functions in Ref. 24共Chap. 6兲 and, to a lesser extent, Ref. 25 共Chap. 8兲. For parallel discussions of the Kravchuk and Hermite functions in
connection with discrete and continuum oscillator algebras, see Refs. 10 and 26. An alternative approach to the comparison of Kravchuk and Hermite functions, making systematic use of cre-ation and annihilcre-ation operators, can be found in Ref. 27.
All lemmas that we state without proof can be obtained from the earlier lemmas together with routine calculations as in Ref. 24. There is only one argument that is not straightforward, namely, the proof of Lemma 4.5. For this, Ref. 24 invokes the theory of hypergeometric functions, and that requires some delicate analysis, the Kravchuk functions being specializations of hypergeometric functions at singular points. Our more direct argument is purely algebraic. The results proved below concerning Kravchuk functions and Kravchuk polynomials are summarized in Appendix B. Let n be a positive integer. Write n⫽2ᐉ⫹1. Let 关n兴 denote the set of k such that ᐉ⫹k and ᐉ⫺k are natural numbers. Thus, 关n兴 consists of n integers or n halves of odd integers. Let Ln be
the n-dimensional Hilbert space of functions关n兴→C, the inner product being
具
兩典⫽兺
k⫽⫺ᐉ
ᐉ
共k兲共k兲, where,苸Ln, and the bar denotes complex conjugation. Let兩k
典
nZ
denote the vector inLnsuch
that, given苸Ln, then(k)⫽
具
兩k典
n Z. The set 兵兩k
典
nZ:k苸关n兴其 is an orthonormal basis forLn.Via the equation
兩k
典
nZ⫽ u
ᐉ⫹kvᐉ⫺k
冑共ᐉ⫹k兲!共ᐉ⫺k兲!
共24兲we identifyLn with the space of homogenous polynomials of degree 2ᐉ in variables u and v.
Later, we shall be realizing Ln as a representation space of the Lie group U共2兲. For the
following three preliminary results, though, we may as well consider, more generally, the Lie group GL共2,C兲. We define a group representationn of GL共2,C兲 on Ln such that
共n共g兲 F兲共u,v兲⫽F共au⫹cv,bu⫹dv兲, g⫽
冉
a b
c d
冊
. 共25兲Lemma 4.1: Let j,k苸关n兴. Put max⫽max(0,j⫹k) and min⫽min(ᐉ⫹j,ᐉ⫹k). Then, with re-spect to the orthonormal basis兵兩k
典
nZ
:k苸关n兴其, the ( j ,k) entry of the matrix representingn(g) is
n Z
具
j兩n共g兲兩k典
n Z⫽冑
共ᐉ⫹ j兲!共ᐉ⫺ j兲! 共ᐉ⫹k兲!共ᐉ⫺k兲!r⫽max兺
min冉
ᐉ⫹k r冊冉
ᐉ⫺k ᐉ⫹ j⫺r冊
arbᐉ⫹ j⫺rcᐉ⫹k⫺rdr⫺ j⫺k. Henceforth, we work directly from Lemma 4.1, and we can forget about the characterization of Ln as a space of polynomials.Lemma 4.2: Now suppose that g苸SL(2,C), and that the matrix entries b, c, d are nonzero. Given j,k苸关n兴, then n Z
具
j兩 n共g兲兩k典
n Z⫽b ᐉ⫹ jcᐉ⫹k dj⫹k冑
共ᐉ⫹ j兲!共ᐉ⫹k兲! 共ᐉ⫺ j兲!共ᐉ⫺k兲! r兺
⫽0 min(ᐉ⫹ j,ᐉ⫹k) 共2ᐉ⫺r兲!共bc兲⫺r !共ᐉ⫹ j⫺r兲!共ᐉ⫹k⫺r兲!. Let cr⫽ᐉ(ᐉ⫹1)⫺2⫹1 4 for 2苸Z. Thus ck⫹1/2⫽共ᐉ⫺k兲共ᐉ⫹k⫹1兲, ck⫺1/2⫽共ᐉ⫹k兲共ᐉ⫺k⫹1兲. Letn be the differential representation ofn.Lemma 4.3: Given an element H⫽(CA DB) of gl共2,C兲 and an element k苸关n兴, then
n共H兲 兩k
典
n Z⫽冑
ck⫺1/2C兩k⫺1典
n Z ⫹共共ᐉ⫹k兲A⫹共ᐉ⫺k兲D兲 兩k典
n Z⫹冑
ck⫹1/2B兩k⫹1典
n Z .The real Lie algebra u共2兲 and its subalgebra su共2兲 have bases兵⫺iW,⫺iX,⫺iY,⫺iZ其 and
兵⫺iX,⫺iY,⫺iZ其, respectively, where W⫽1 2
冉
1 0 0 1冊
, X⫽ 1 2冉
0 1 1 0冊
, Y⫽ 1 2冉
0 ⫺i i 0冊
, Z⫽ 1 2冉
1 0 0 ⫺1冊
. Note that W commutes with X, Y , Z, and the other commutation relations are 关X,Y 兴⫽iZ and 关Y,Z兴⫽iX and 关Z,X兴⫽iY. LetWˆn⫽n共W兲, Xˆn⫽n共X兲, Yˆn⫽n共Y 兲, Zˆn⫽n共Zn兲.
Given k苸关n兴, then, by Lemma 4.3,
Wˆn兩k
典
n Z⫽ᐉ 兩k典
n Z , 共26兲 Xˆn兩k典
n Z⫽1 2共冑ck⫺1/2兩k⫺1典
n Z⫹冑
c k⫹1/2兩k⫹1典
n Z), 共27兲 Yˆn兩k典
n Z⫽i 2共冑
ck⫺1/2兩k⫺1典
n Z⫺冑
ck⫹1/2兩k⫹1典
n Z ), 共28兲 Zˆn兩k典
n Z⫽k兩k典
n Z . 共29兲Thus, the algebra representationn of gl共2,C兲 restricts to anti-Hermitian representations of
u共2兲 and isu共2兲. In other words, the group representationn of GL共2,C兲 restricts to unitary
repre-sentations of U共2兲 and SU共2兲. It is well-known 共by an easy ladder argument兲 that the two restricted representations are irreducible.
For each k苸关n兴, we define a vector 兩k
典
nX
⫽exp共⫺iYˆn/2兲 兩k
典
n Z. 共30兲
To rewrite Eqs.共26兲–共29兲 with respect to the orthonormal basis兵兩k
典
nX:k苸关n兴其, let us first deter-mine the exponentials of iW, iX, iY , iZ. By evaluating derivatives at t⫽0, or by appealing to Eq. 共19兲 共with complex values of,␥,␦兲, we haveexp共⫺itW兲⫽
冉
e⫺it/2 0
0 e⫺it/2
冊
, exp共⫺itX兲⫽冉
cos t/2 ⫺i sin t/2 ⫺i sin t/2 cos t/2
冊
,共31兲 exp共⫺itY 兲⫽
冉
cos t/2 ⫺sin t/2sin t/2 cos t/2
冊
, exp共⫺itZ兲⫽冉
e⫺it/2 0 0 eit/2
冊
. By direct calculation, e⫺itYZeitY⫽Z cos t⫹X sint for all t苸R. Soexp共⫺iYˆn/2兲Zˆnexp共iYˆn/2兲⫽Xˆn, exp共⫺iYˆn/2兲Xˆnexp共iYˆn/2兲⫽⫺Zˆn.
We can now rewrite Eqs.共26兲–共29兲 as
Wˆn兩k
典
nX⫽ᐉ兩k典
nX , 共32兲 Xˆn兩k典
n X⫽k兩k典
n X , 共33兲 Yˆn兩k典
n X⫽ i 2共冑ck⫺1/2兩k⫺1典
n X⫺冑
c k⫹1/2兩k⫹1典
n X), 共34兲Zˆn兩k
典
n X⫽⫺1 2 共冑ck⫺1/2兩k⫺1典
n X⫹冑
c k⫹1/2兩k⫹1典
n X). 共35兲Lemmas 4.1 and 4.2 now yield the following result. Lemma 4.4: Given j,k苸关n兴, then
(1) nZ
具
j兩k典
nX⫽共⫺1兲 ᐉ⫹ j 2ᐉ冑
共ᐉ⫹ j兲!共ᐉ⫺ j兲! 共ᐉ⫹k兲!共ᐉ⫺k兲!兺
r冉
ᐉ⫹k r冊冉
ᐉ⫺k ᐉ⫹ j⫺r冊
共⫺1兲r, (2) nZ具
j兩k典
nX⫽共⫺1兲 ᐉ⫹ j 2ᐉ冑
共ᐉ⫹ j兲!共ᐉ⫹k兲! 共ᐉ⫺ j兲!共ᐉ⫺k兲!兺
s 共2ᐉ⫺s兲!共⫺2兲s s!共ᐉ⫹ j⫺兲!共ᐉ⫹k⫺s兲!,where the indices of the sums run over the values for which the terms are defined, namely, max(0,j⫹k)⭐r⭐min(ᐉ⫹j,ᐉ⫹k) and 0⭐s⭐min(ᐉ⫹j,ᐉ⫹k).
Lemma 4.5: Given j,k苸关n兴, then
n Z
具
k兩 j
典
nX⫽共⫺1兲j⫺k Zn具
j兩k典
nX⫽共⫺1兲ᐉ⫹k nZ具
k兩⫺ j典
nX⫽共⫺1兲ᐉ⫺ j nZ具
k兩 j典
nX.Proof: Throughout the argument, when multiplying powers of⫺1, we must bear in mind that j,k,ᐉ are all integers or all halves of odd integers. By Lemma 4.4共2兲,
共⫺1兲ᐉ⫹ j
n Z
具
j兩k
典
nX⫽共⫺1兲ᐉ⫹k nZ具
k兩 j典
nX . The first asserted equality follows.Since the eigenvalues of Xˆn are distinct, the eigenvector equations Xˆn兩 j
典
Xn⫽兩j典
nX and Xˆn兩 ⫺ j典
nX⫽⫺ j 兩j
典
n Xdetermine the unit vectors兩 j
典
nX and兩⫺ j典
nX up to phase factors. By Eqs.共27兲 and 共33兲, the matrix entry nZ
具
j兩Xˆ n兩k典
nX is zero unless兩 j⫺k兩⫽1. Therefore, fixing j, there is a phase
such that, for all k, we have
n Z
具
k兩⫺ j典
n X⫽共⫺1兲ᐉ⫹k n Z具
k兩 j典
n X . 共In other words, if we multiply the Z-coordinates of 兩j典
nX
by an alternating⫾1, then we get a multiple of兩⫺ j
典
nX .) Putting k⫽⫺ᐉ, and noting that, by Lemma 4.4共1兲,n Z
具
⫺ᐉ兩⫺ j典
n X⫽1 2冑
冉
2ᐉ ᐉ⫹ j冊
n Z具
⫺ᐉ兩j典
n X ,we deduce that⫽1. The second asserted equality follows and, hence, the third. 䊐 Lemma 4.6: Given j,k苸关n兴, then
(1)
冑
ck⫺1/2 nZ具
k⫺1兩j典
n X⫺2 j n Z具
k兩 j典
n X⫹冑
c k⫹1/2 nZ具
k⫹1兩j典
n X⫽0, (2)冑
cj⫺1/2 nZ具
k兩 j⫺1典
n X⫺2k n Z具
k兩 j典
n X⫹冑c j⫹1/2 nZ具
k兩 j⫹1典
n X⫽0.LetNn denote the set of natural numbers less than n. For each s苸Nn, we define the
Krav-chuk polynomial Ks,n:Nn→C and the Kravchuk function hs,n:关n兴→C such that
共⫺1兲ᐉ⫹ j 2ᐉ
冑
冉
2ᐉ ᐉ⫹ j冊冉
2ᐉ ᐉ⫹k冊
Kᐉ⫹ j,n共ᐉ⫹k兲⫽hᐉ⫹ j,n共k兲⫽n Z具
j兩k典
nX for j,k苸关n兴. The formulas in Appendix B are precisely Lemmas 4.4–4.6.Proposition 4.7: The set of Kravchuk functions兵hs,n:s苸Nn其is an orthonormal basis forLn.
Proof: The values of the Kravchuk functions are the overlaps of two orthonormal bases. 䊐 We now rewrite the Kravchuk functions as兩s
典
n⫽hs,n.Proposition 4.8: Given s苸关n兴, then 兩s
典
n⫽(⫺1)s兩ᐉ⫺s典
n X.Proof: Apply Lemma 4.5. 䊐
Via Proposition 4.8, we can rewrite Eqs.共32兲–共35兲 as
Wˆn兩s
典
n⫽ᐉ兩s典
n , 共36兲 Xˆn兩s典
n⫽共ᐉ⫺s兲兩s典
n , 共37兲 Yˆn兩s典
n⫽ i 2共⫺冑s共2ᐉ⫹1⫺s兲 兩s⫺1典
n⫹冑共s⫹1兲共2ᐉ⫺s兲 兩s⫹1
典
n), 共38兲 Zˆn兩s典
n⫽ 1 2共冑
s共2ᐉ⫹1⫺s兲 兩s⫺1典
n⫹冑共s⫹1兲共2ᐉ⫺s兲 兩s⫹1
典
n). 共39兲We define Hermitian operators
Iˆn⫽Xˆn/ᐉ, Pˆn⫽⫺Yˆn/
冑ᐉ,
Qˆn⫽Zˆn/冑ᐉ,
2Bˆn⫽Pˆn 2, 2Cˆ n⫽Qˆn 2, 2Dˆ n⫽PˆnQˆn⫹QˆnPˆn.We can understand Pˆnas discrete momentum共or frequency兲 and Qˆnas discrete position共or time兲.
For real,␥,␦,,,, we introduce a Hermitian operator
Hˆn共,␥,␦,,,兲⫽Bˆn⫹␥Cˆn⫹␦Dˆn⫹Pˆn⫹Qˆn⫹Iˆn. 共40兲
We define a discrete affine canonical transform to be a unitary operator having the form
Uˆn共,␥,␦,,,兲⫽exp共⫺iHˆn共,␥,␦,,,兲兲. 共41兲 Recall that, in the continuum scenario, we defined the continuum affine canonical transforms to be the composites of the unitary operators having the form Uˆ (, . . . ,). Our reason for not defining the discrete affine canonical transforms in the same way is that the infinitesmal generators Hˆn(, . . . ,) do not span a Lie algebra. We can work with single-parameter groups of discrete
affine canonical transforms—including fractional Fourier transforms, chirps and dilations—and these single-parameter groups, of course, have the index-additivity property UˆsUˆt⫽Uˆs⫹t. In gen-eral, though, we do not retain any tractible closure property if we compose elements of distinct single-parameter groups.
However, in the continuum scenario, we defined the motion canonical transforms to be pre-cisely the unitary operators having the form Eˆ (,,,), these operators being closed under composition. That feature can be retained in the discrete scenario. Let
Nˆn⫽Wˆn⫺Xˆn⫽ᐉ共1ˆ⫺Iˆn兲.
The operators Iˆn, Nˆn, Pˆn, Qˆn are closed under commutators. We define a discrete motion
canonical transform to be a unitary operator having the form
Eˆn共,,,兲⫽exp共⫺i共Iˆn⫹Nˆn⫹Pˆn⫹Qˆn兲兲⫽共En共,,,兲兲, 共42兲
where ,,,苸R. Let us put it in the language of representations. The Lie group u共2兲 has a basis兵In,Nn, Pn,Qn其 where
In⫽X/ᐉ , Nn⫽W⫺X, Pn⫽⫺Y/冑ᐉ , Qn⫽Z/冑ᐉ .
关In,Nn兴⫽0, 关In, Pn兴⫽⫺iQn/ᐉ , 关In,Qn兴⫽iPn/ᐉ ,
共43兲 关Nn, Pn兴⫽iQn, 关Nn,Qn兴⫽⫺iPn, 关Pn,Qn兴⫽iIn.
The algebra representation n maps In, Nn, Pn, Qn to Iˆn, Nˆn, Pˆn, Qˆn, respectively. Observe
that, asᐉ→⬁, the structural constants for In, Nn, Pn, Qnconverge to those given in Sec. 2 for
the basis elements I, N, P, Q of hm. The algebra iu共2兲 and the group U共2兲 are to serve as the discrete analogs of the algebra hm and the group HM.
Now let us write down the matrices for Bˆn, Cˆn, Dˆn, Pˆn, Qˆn, Nˆn, Iˆnwith respect to the basis of Kravchuk functions. For 2r⫹1苸N, let
tn共r兲⫽冑共2r⫹1兲共4ᐉ⫺2r⫹1兲/16ᐉ. Given s苸Nn, then tn
冉
s⫹ 1 2冊
⫽冑
s⫹1 2冉
1⫺ s 2ᐉ冊
, tn冉
s⫺ 1 2冊
⫽冑
s 2冉
1⫺ s⫺1 2ᐉ冊
. By Eqs.共36兲–共39兲, Bˆn兩s典
n⫽⫺ 1 2tn冉
s⫺ 1 2冊
tn冉
s⫺ 3 2冊
兩s⫺2典
n⫹冉
s 2冉
1⫺ s 2ᐉ冊
⫹ 1 4冊
兩s典
n ⫺1 2tn冉
s⫹ 1 2冊
tn冉
s⫹ 3 2冊
兩s⫹2典
n , 共44兲 Cˆn兩s典
n⫽ 1 2tn冉
s⫺ 1 2冊
tn冉
s⫺ 3 2冊
兩s⫺2典
n⫹冉
s 2冉
1⫺ s 2ᐉ冊
⫹ 1 4冊
兩s典
n⫹ 1 2tn冉
s⫹ 1 2冊
tn冉
s⫹ 3 2冊
兩s⫹2典
n , 共45兲 Dˆn兩s典
n⫽⫺itn共s⫺ 1 2兲tn共s⫺ 3 2兲 兩s⫺2典
n⫹itn共s⫹ 1 2兲tn共s⫹ 3 2兲 兩s⫹2典
n , 共46兲 Pˆn兩s典
n⫽⫺itn共s⫺1 2兲 兩s⫺1典
n⫹itn共s⫹ 1 2兲 兩s⫹1典
n , 共47兲 Qˆn兩s典
n⫽tn共s⫺ 1 2兲 兩s⫺1典
n⫹tn共s⫹ 1 2兲 兩s⫹1典
n , 共48兲 Iˆn兩s典
n⫽共1⫺s/ᐉ兲 兩s典
n , 共49兲 Nˆn兩s典
n⫽s 兩s典
n . 共50兲Again, we observe a suggestive connection with the continuum scenario. As ᐉ→⬁, the matrix entries in Eqs.共44兲–共50兲 converge to the matrix entries in Eqs. 共11兲–共17兲.
In Sec. II, we ended with an example. Let us end the present section with the analogous example. The discrete Fourier transform of Atakishiyev–Wolf10 is the unitary operator Fˆn onLn
such that Fˆn兩s
典
n⫽is兩s典
n. More generally, their discrete fractional Fourier transform of order t 苸R is the unitary operator Fˆnt onL
n such that Fˆn t 兩s
典
n⫽exp(2ist)兩s
典
n. In other words,Fˆn t
⫽exp共2itNˆn兲. 共51兲
V. CONVERGENCE OF UNITARY TRANSFORMS
We wish to say that the continuum affine canonical transforms are limits of discrete affine canonical transforms. The whole problem lies in making the assertion absolutely unambiguous; then the proof will follow purely by deductive reasoning. Parts of the proof are deferred to Ref. 4.
Usually, when one writes an equation of the form x⫽limn→⬁xn, the object x and the objects xnall
belong to the same space共or category兲. Such is not the case in our situation. We need to specify an interface between the continuum scenario and the discrete scenario. Let us describe the inter-face in two different ways, the first one clear and precise, the second one more illuminating from a practical perspective.
The clear description of the interface makes use of inductive limits, which are introduced in Refs. 1–3. A summary is given in Ref. 4, Sec. 2. LetS共R兲 be the Schwartz subspace of L2(R). For each positive integer n, let resn be the linear map S(R)→Ln such that, given 苸S(R), and writingn⫽resn(), then
n共k兲⫽ᐉ⫺1/4共ᐉ⫺1/2k兲, 共52兲
where k苸关n兴. The linear maps resncomprise an inductive resolution of L2(R). We are now in a position to realize vectors in L2(R) as limits ⫽limnn, where each n is a vector in the n-dimensional spaceLn. We can do the same for bounded operators and, in particular, for unitary
operators.
The following alternative description is rather more intuitive. Let be a continuous and well-behaved complex-valued function with one real variable. For each n, let n be a vector in Ln. We regardn as a good approximation toprovided
n共k兲⬇ᐉ⫺1/4共ᐉ⫺1/2k兲
for almost all k苸关n兴. As the number of sample points n⫽2ᐉ⫹1 increases, the mesh ᐉ⫺1/2 decreases and the width of the sample window 2ᐉ1/2increases. Ifnbecomes an arbitrarily good
approximation to in a certain manner that preserves everything involving inner products, then we say thatnconverges to, and we write⫽limnn. Limits of unitary operators are required
to preserve limits of vectors.
For example, Ref. 2, Theorem 5.1, says that 兩s
典
⫽limn
兩s
典
n, 共53兲for all natural numbers s. In other words, the Hermite functions are the inductive limits of the Kravchuk functions.
Theorem 5.1: Let ⫽limnn, ␥⫽limn␥n, ␦⫽limn␦n, ⫽limnn, ⫽limnn,
⫽limnn as limits of real sequences. Then
Uˆ共,␥,␦,,,兲⫽lim
n
Uˆn共n,␥n,␦n,n,n,n兲.
Proof: This is part of Ref. 4, Theorem 7.2. 䊐
A comparison of Eqs. 共11兲–共16兲 with Eqs. 共44兲–共49兲 provides a heuristic justification for Theorem 5.1, but not a proof. Convergence of matrix entries of infinitesmal generators does not, in general, imply convergence of the corresponding unitary operators.
Although arbitrary pairs of discrete affine canonical transforms do not compose in a tractible way, let us draw attention to the index-additivity property of single-parameter groups of discrete affine canonical transforms. Fix reals,␥,␦,,,. Theorem 5.1 tells us that
Uˆ共t,t␥,t␦,t,t,t兲⫽lim
n
Uˆ n共tn,t␥n,t␦n,tn,tn,tn兲 共54兲
for all t苸R. Since Bˆn,Cˆn,Dˆn, Pˆn,Qˆn,Iˆn are linearly independent for n⭓3, we have the
follow-ing.
Corollary 5.2: For fixed n⭓3, Eq. (54) describes a bijective correspondence between the single-parameter groups of continuum affine canonical transforms and the single-parameter
groups of discrete affine canonical transforms on Ln. Now let n vary. The elements of a
single-parameter group of continuum affine canonical transforms are inductive limits of sequences of elements of the corresponding single-parameter groups of discrete affine canonical transforms.
We now turn to motion canonical transforms.
Theorem 5.3: Let ⫽limnn, ⫽limnn, ⫽limnn, ⫽limnn as limits of real
se-quences. Then
Eˆ共,,,␥兲⫽lim
n
Eˆn共n,n,n,␦n兲.
Proof: The limit of representations in Ref. 4, Theorem 10.2, is a stronger result. 䊐 Warning: Theorem 5.3 is not a special case of Theorem 5.1. Not all of the discrete motion canonical transforms are discrete affine canonical transforms.
Comparing Eqs.共18兲 and 共51兲, we see that Theorem 5.3 recovers the convergence of fractional Fourier transforms Fˆt⫽lim n F ˆ n t . 共55兲
A more direct proof of Eq.共55兲 is given in Ref. 3, Example 4.F. The equation 共not expressed in the form of an inductive limit兲 is due to Atakishiyev–Wolf.10
VI. COMPLEX-ORDER FOURIER TRANSFORMS AND COHERENT STATES
We introduce two more objects to the continuum scenario: the system of Glauber coherent states共Gabor functions兲 and the continuum Hermite semigroup 共the semigroup of complex-order Fourier transforms兲. Then we introduce the analogous objects to the discrete scenario: the system of spin coherent states and the discrete Hermite semigroup共discrete complex-order Fourier trans-forms兲. As in the previous section, the analogy between the discrete and continuum objects is plain enough; our purpose is to express the analogy precisely using inductive limits.
For an introduction to the Glauber and spin coherent states, see Ref. 28, Chap. 1 or Ref. 29. To fix notation, we shall recall the relevant definitions, but we shall not discuss the measures on the label spaces. The Glauber coherent state兩z
典
C with label z苸C can be defined as兩z
典
C⫽exp共⫺兩z兩2/2兲exp共zAˆ†兲兩0典
⫽exp共⫺兩z兩2/2兲兺
s⫽0
⬁
zs
冑
s!兩s典
. 共56兲 Writing gz to denote兩z典
C regarded as a 共rapidly decreasing兲 function R→C, we have1/4g z共q兲⫽exp
冉
⫺ q2 2 ⫹&zq⫺ z2 2 ⫺ 兩z兩2 2冊
⫽exp冉
⫺ q2 2 ⫺共u⫹iv兲q⫺ u2 2 ⫺ iuv 2冊
, 共57兲 where&z⫽u⫹iv with u,v苸R. We note one other useful characterization:兩z
典
C⫽exp共⫺iuPˆ⫹ivQˆ兲)兩0典
. 共58兲In electrical enginnering and signal processing, Glauber coherent states are usually called Gabor functions, and are usually expressed in the form of Eq. 共57兲. The other two equations are more normally used in quantum physics. As a gesture of mediation between the two disciplines, let us give a quick proof that the three equations are mutually equivalent. From Eq.共58兲, rewritten as
gz⫽exp共⫺iuPˆ⫹ivQˆ兲h0, it is easy to obtain Eq.共57兲 using the identities
exp共⫺i共uPˆ⫹vQˆ兲兲⫽exp共iuv/2兲exp共ivQˆ兲exp共⫺iuPˆ兲, exp共⫺iuPˆ兲共q兲⫽共q⫺u兲, exp共⫺ivQˆ兲共q兲⫽exp共ivq兲共q兲, where苸S(R). Using the generating function
exp共2qt⫺t2兲⫽
兺
s⫽0
⬁ ts
s!Hs共q兲 together with Eq.共57兲, straightforward manipulation yields
兺
s⫽0 ⬁ ts s!冕
⫺⬁ ⬁dqHs共q兲exp共⫺q2/2兲gz共q兲⫽1/4exp共⫺兩z兩2/2兲exp共&zt兲.
Comparing coefficients of powers of t, we obtain
具
s兩z典
C⫽exp(⫺兩z兩2/2)/冑
s!. The equivalence of Eqs.共56兲–共58兲 is now established.For苸C with 兩兩⭐1, the continuum complex-order Fourier transform Fˆ() is defined to be the bounded operator on L2(R) such that
Fˆ共兲 兩s
典
⫽s兩s典
. 共59兲The integral kernel for Fˆ () may be found in Ref. 30. An optical realization of Fˆ () is discussed in Ref. 31. We have an obvious composition law
Fˆ共兲Fˆ共⬘兲⫽Fˆ共⬘兲. 共60兲
The commutative semigroup兵Fˆ ():兩兩⭐1其, called the continuum Hermite semigroup, is evi-dently isomorphic to the semigroup兵苸C:兩兩⭐1其. Writing
⫽exp共2it兲, 共61兲
we say that Fˆ () has order t. Given Fˆ (), the real part of t is well-defined up to congruence modulo 1. The condition兩兩⭐1 is precisely the condition that t lies in the closed upper half of the complex plane. By Eq. 共17兲,
Fˆ共兲⫽exp共2itNˆ兲.
The continuum fractional Fourier transforms are precisely the unitary continuum complex-order Fourier transforms. By Eqs. 共56兲 and 共59兲, the continuum Hermite semigroup permutes the Glauber coherent states共up to scalar factors兲 according to the equation
Fˆ共兲 兩z
典
C⫽exp共兩z兩2/2⫺兩z兩2/2兲 兩z典
C. 共62兲 Now let us look at the discrete scenario. The discrete annihilation operator Aˆn and itsHermitian conjugate, the discrete creation operator Aˆn†, are defined to be
Aˆn⫽共Qˆn⫹iPˆn兲/& , Aˆn
†⫽共Qˆ
n⫺iPˆn兲/& .
From Eqs.共47兲 and 共48兲 we have Aˆn兩s
典
n⫽冑
s冉
1⫺ s⫺1 2ᐉ冊
兩s⫺1典
n , Aˆn †兩s典
n⫽冑
共s⫹1兲冉
1⫺ s 2ᐉ冊
兩s⫹1典
n .The spin coherent state兩z
典
nC with label z苸C is defined by冉
1⫹兩z兩 2 2ᐉ冊
ᐉ 兩z典
n C ⫽exp共zAˆn †兲兩0典
n⫽兺
s⫽0 2ᐉ冑
冉
2ᐉ s冊
冉
z冑
2ᐉ冊
s 兩s典
n . 共63兲We also allow a spin coherent state
兩⬁
典
n C⫽ lim z→⬁ 兩z典
n C⫽兩2ᐉ典
n C .For arbitrary苸C, the discrete complex-order Fourier transform Fˆn() is defined to be the operator on L2(R) such that
Fˆn共兲 兩s
典
n⫽s兩s典
n . 共64兲Using Eqs.共30兲 and 共31兲, followed by Lemma 4.1 and Proposition 4.8, it can be shown that Fˆn共兲⫽n共K共兲兲, K共兲⫽
1 2
冉
1⫹ 1⫺
1⫺ 1⫹
冊
. 共65兲 Evidently, we have a composition lawFˆn共兲Fˆn共⬘兲⫽Fˆn共
⬘
兲. 共66兲The semigroup 兵Fˆ ():苸C其 is called the discrete Hermite semigroup. Letting t be as in Eq. 共61兲, we say that Fˆn() has order t. The real part of t is still well-defined only up to congruence
modulo 1, but there are now no constraints on the range of t. By Eq.共50兲, Fˆn共兲⫽exp共2itNˆn兲.
The discrete fractional Fourier transforms are precisely the unitary discrete complex-order Fourier transforms. By Eqs. 共63兲 and 共59兲, the discrete Hermite semigroup permutes the spin coherent states共up to scalar factors兲 according to the equation
Fˆn共兲 兩z
典
n C⫽冉
2ᐉ⫹兩z兩 2 2ᐉ⫹兩z兩2冊
ᐉ 兩z典
nC. 共67兲Theorem 6.1: Given z苸C, then 兩z
典
C⫽limn兩z典
n C.
Proof: Consider a vector 苸L2(R) and vectors n苸Ln such that the set 兵储n储:n苸N其 is
bounded. By Eq.共53兲 and Ref. 1, Theorem 3.4,⫽limnn if and only if
具
s兩典
⫽limn
n
具
s兩n典
for all s苸N. These two equivalent conditions hold when⫽兩z
典
C andn⫽兩z典
n C because exp共⫺兩z兩2/2兲冑
s! ⫽ limᐉ→⬁冉
1⫹ 兩z兩1 2ᐉ冊
⫺ᐉ冑
冉
2ᐉ s冊
冉
1冑
2ᐉ冊
s . 䊐 Theorem 6.2: Given苸C with 兩兩⭐1, then Fˆ()⫽limnFˆn().Proof: Let苸L2(R) andn苸Lnsuch that⫽limnn. Using the criterion for limits noted
具
s兩Fˆ共兲典⫽s具
s兩典⫽lim n s n具
s兩n典
⫽lim n n具
s兩Fˆn共兲n典
, and Fˆ ()⫽limnFˆn()n. 䊐 VII. CONCLUSIONSWe have used inductive limits to express the way in which the discrete scenario and the continuum scenario are related to each other. From a procedural point of view 共oriented, say, towards implementation of numerical calculations兲, the relationships between the two scenarios has two significant aspects: approximation and analogy. Not only do the discrete objects serve as approximations to their corresponding continuum objects, but they are also analogs in the sense that the algebraic structures in the discrete scenario mirror the algebraic structures in the con-tinuum scenario. For the purpose of numerical calculation, that feature is important, because it ensures that errors due to inaccuracy of the approximating formulas are not compounded under repeated composition. Our approach provides some rationale for both of those aspects: inductive limits serve as approximations; they also preserve algebraic structures, specifically, they preserve inner products, operator-vector compositions, and operator-operator compositions.
We propose inductive limits as a way of providing theoretical justification for discrete ap-proximations in cases where precise error analysis would be too difficult. As concrete examples become more complicated, intuition may become unreliable, and a precise criterion for the limits may become increasingly useful. Inductive limits of representations, as in Sec. V and Ref. 4, appears to be applicable to various other limits of representations. See, for instance, Refs. 8 and 32–35. It is to be expected that, through moderately complicated but routine exercises in epsilon-ics, the limits of operators in those works can be shown to be inductive limits.
However, to plough through such calculations would be to overlook a more interesting line of study. Limits of representations are more subtle than limits of individual operators. The result 共Ref. 4, Theorem 9.4兲 on convergence of structural constants requires, in addition to convergence of individual operators, an analytic convergence hypothesis. The hypothesis is potentially verifi-able, in practice, for concrete examples, but some simplifications may be possible; perhaps it suffices to check the uniformity condition in Ref. 4, Sec. 8 only for a spanning set of infinitesmal generators. Thus, at the time of writing, the criterion for inductive limits of representations should be regarded as subject to simplification or modification.
Besides, in order to be of significant practical use, the theory of inductive limits of represen-tations is in need of general theorems. For a limit of represenrepresen-tations⫽limnn 共Ref. 4. Proposi-tion 9兲 asserts that, ifis faithful, thennis faithful for large n. That result is unlikely to be useful
in application to concrete examples, since faithfulness is usually obvious to start with. However, the result may point the way forwards: ifis irreducible, mustn be irreducible for large n? To
prove theorems, of course, it is sometimes necessary to tinker with definitions. So, again, we conclude that the present criterion for inductive limits of representations should be regarded as subject to change.
It appears that inductive limits can also be used to describe a correspondence between a finite-discrete periodic scenario based on the integers modulo pmand a continuous periodic sce-nario based on the p-adic integers. Here, p is a rational prime. For the discrete context, see Refs. 36 and 37; for the continuous context, see Refs. 38, and 39. In this p-adic scenario, purely intuitive arguments are to be distrusted, so the use of some or another precise notion of limit is essential. Discrete versus continuum correspondences of operators and representations arise frequently. Without attempting to classify the various directions of study, let us list some papers on the topic where the term limit is used explicitly and is probably interpretable as inductive limit: Refs. 40, 6, 7, 8, 32, 10, 41, 42, 5, 43, 44, 35, and 45– 47. We have given a broad spread of citations so as to provide evidence that an intuitive equivalent of the notion of an inductive limit is in widespread use. The list could be extended considerably. The author has come across only one paper共citation
omitted兲 in which the limits of operators are nonsensical 共arbitrary SL共2,R兲 canonical transforms written as ‘‘limits’’ of SL(2,p) canonical transforms, where p runs over the rational primes兲.
There are also a great many works where the term limit is not used for our purpose, but inductive limits seem to be involved implicitly. This point pertains, in particular, to many single-parameter discrete systems used as approximations to continuum systems in signal processing. For some examples, see Refs. 48 and 49 and references therein.
APPENDIX A: A COUNTER-EXAMPLE
By the definition of inductive limits of operators, Theorem 6.2 can be expressed as follows. Theorem A.1: Given a vector苸L2(R) and vectorsn苸Lnsuch that⫽limnn, then, for
all苸C with 兩兩⭐1, we have Fˆ()⫽limnFˆn()n.
For arbitrary苸C⫻, we can still define Fˆ () to be the operator on L2(R) satisfying Eq. 共59兲. If兩兩⬎1, then Fˆ() is unbounded. The domain of Fˆ (), in this case, has been studied by Byun.50 Plainly, for arbitrary, the conclusion of Theorem A.1 still holds when is a Glauber cat state 共linear combination of coherent states兲 in L2(R) andis the corresponding spin cat state inL
n.
However, for arbitrary and arbitrary in the domain of Fˆ (), the conclusion of Theorem A.1 can fail. A counter-example is⫽0 andn⫽2⫺2ᐉ兩2ᐉ
典
with⫽3.It is difficult to imagine how the mainstream techniques共formal manipulation兲 could be used to ‘‘derive’’ Theorem A.1 without also ‘‘deriving’’ the fallacy refuted in the previous paragraph.
APPENDIX B: IDENTITIES FOR THE KRAVCHUK FUNCTIONS
Let n be a positive integer. As in Sec. IV, we write n⫽2ᐉ⫹1 and 关n兴⫽兵⫺ᐉ,1⫺ᐉ, . . . ,ᐉ ⫺1,ᐉ其 and we define ck⫹1/2⫽(ᐉ⫺k)(ᐉ⫹k⫹1), equivalently, ck⫺1/2⫽(ᐉ⫹k)(ᐉ⫺k⫹1), where 2k苸Z. The Kravchuk polynomials K0,n,K1,n, . . . ,K2ᐉ,n are the functions 兵0,1, . . . ,2ᐉ其→C given by Kᐉ⫹ j,n共ᐉ⫹k兲 ⫽
冉
2ᐉ l⫹ j冊
⫺1兺
⫽max(0,j⫹k) min(ᐉ⫹ j,ᐉ⫹k)冉
ᐉ⫹k 冊冉
ᐉ⫹ j⫺ᐉ⫺k冊
共⫺1兲 ⫽兺
⫽0 min(ᐉ⫹ j,ᐉ⫹k)冉
2ᐉ 冊
⫺1冉
ᐉ⫹ j 冊冉
ᐉ⫹k冊
共⫺2兲,where j,k苸关n兴. Note that, in each of the two formulas, the indexorruns over all values for which the terms are defined. In Sec. IV, it is shown that the two formulas are equivalent to each other. It is also shown that the Kravchuk polynomials satisfy
Kᐉ⫹k,n共ᐉ⫹ j兲⫽Kᐉ⫹ j,n共ᐉ⫹k兲, 共B1兲
Kᐉ⫹ j,n共ᐉ⫺k兲⫽共⫺1兲ᐉ⫹ jKᐉ⫹ j,n共ᐉ⫹k兲, 共B2兲
Kᐉ⫺ j,n共ᐉ⫹k兲⫽共⫺1兲ᐉ⫹kKᐉ⫹ j,n共ᐉ⫹k兲, 共B3兲
共ᐉ⫺k兲Kᐉ⫹ j,n共ᐉ⫹k⫹1兲⫹2 jKᐉ⫹ j,n共ᐉ⫹k兲⫹共ᐉ⫹k兲Kᐉ⫹ j,n共ᐉ⫹k⫺1兲⫽0, 共B4兲
共ᐉ⫺ j兲Kᐉ⫹ j⫹1,n共ᐉ⫹k兲⫹2kKᐉ⫹ j,n共ᐉ⫹k兲⫹共ᐉ⫹ j兲Kᐉ⫹ j⫺1,n共ᐉ⫹k兲⫽0. 共B5兲
The Kravchuk functions h0,n,h1,n, . . . ,h2ᐉ,nare the functions 关n兴→C given by hs,n共k兲⫽ 共⫺1兲s 2ᐉ
冑
冉
2ᐉ s冊冉
2ᐉ ᐉ⫹k冊
Ks,n共ᐉ⫹k兲. 共B6兲 In other words,hᐉ⫹ j,n共k兲⫽共⫺1兲 ᐉ⫹ j 2ᐉ
冑
共ᐉ⫹ j兲!共ᐉ⫺ j兲! 共ᐉ⫹k兲!共ᐉ⫺k兲!兺
冉
ᐉ⫹k 冊冉
ᐉ⫹ j⫺ᐉ⫺k冊
共⫺1兲 ⫽共⫺1兲ᐉ⫹ j 2ᐉ冑
共ᐉ⫹ j兲!共ᐉ⫹k兲! 共ᐉ⫺ j兲!共ᐉ⫹k兲!兺
共2ᐉ⫺兲!共⫺2兲 !共ᐉ⫹ j⫺兲!共ᐉ⫹k⫺兲!. 共B7兲 Equations共B1–B5兲 can be rewritten ashᐉ⫹k,n共 j兲⫽共⫺1兲k⫺ jhᐉ⫹ j,n共k兲, 共B8兲
hᐉ⫹ j,n共⫺k兲⫽共⫺1兲ᐉ⫹ jhᐉ⫹ j,n共k兲, 共B9兲
hᐉ⫺ j,n共k兲⫽共⫺1兲ᐉ⫺khᐉ⫹ j,n共k兲, 共B10兲
冑
ck⫹1/2hᐉ⫹ j,n共k⫹1兲⫹2 jhᐉ⫹ j,n共k兲⫹冑ck⫺1/2hᐉ⫹ j,n共k⫺1兲⫽0, 共B11兲冑
cj⫹1/2hᐉ⫹ j⫹1,n共k兲⫺2khᐉ⫹ j,n共k兲⫹冑cj⫺1/2hᐉ⫹ j⫺1,n共k兲⫽0. 共B12兲Proposition 4.7 says that Kravchuk functions comprise an orthonormal basis for the space of functions关n兴→C.
1L. Barker, J. Funct. Anal. 186, 153共2001兲. 2L. Barker, J. Phys. A 22, 4673共2001兲. 3L. Barker, J. Math. Phys. 42, 4653共2001兲. 4
L. Barker, Limits of representations of Lie groups and Lie algebras共submitted兲. 5J. M. Radcliffe, J. Phys. A 4, 313共1971兲.
6F. T. Arecchi, E. Courtans, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211共1972兲. 7N. Atakishiyev, S. M. Chumakov, and K. B. Wolf, J. Math. Phys. 39, 6247共1998兲. 8
N. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, J. Phys. A 34, 9381共2001兲. 9
V. K. Dobrev, H.-D. Doebner, and C. Mrugalla, Mod. Phys. Lett. A 14, 1113共1999兲. 10N. M. Atakishiyev and K. B. Wolf, J. Opt. Soc. Am. A 14, 1467共1997兲.
11T. Digernes, V. S. Varadara´jan, and S. R. S. Varadhan, Rev. Math. Phys. 6, 612共1994兲.
12P. Kramer and M. Lorente, ‘‘Discrete and continuous symmetry via induction and duality,’’ in Symmetries in Science, X
(Bregenz, 1997)共Plenum, New York, 1998兲, pp. 165–177.
13K. R. Parthasarathy, J. Appl. Probab. Special Vol. 25A, 151共1988兲.
14K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus共Birkha¨user, Basel, 1992兲. 15J. M. Lindsay and K. R. Parthasarathy, Sankhya, Ser. A 50, 151共1988兲.
16
U. Neiderer, Helv. Phys. Acta 45, 802共1972兲. 17
V. Namias, J. Inst. Math. Appl. 25, 214共1980兲.
18G. B. Folland, Harmonic Analysis in Phase Space共Princeton University Press, Princeton, NJ, 1989兲. 19M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121共1984兲.
20R. G. Littlejohn, Phys. Rep. 138, 193共1986兲. 21
A. M. Ozorio de Almeida, Phys. Rep. 295, 265共1998兲.
22G. W. Forbes, V. I. Man’ko, H. M. Ozaktas, R. Simon, and K. B. Wolf共eds.兲, ‘‘Wigner Distributions and Phase Space in Optics,’’ J. Opt. Soc. Am. A共feature issue兲 17共12兲, 2274 共2000兲.
23The Wigner Distribution, Theory and Applications in Signal-Processing, edited by W. Mecklenbra¨uker and F. Hlawatsch 共Elsevier, Amsterdam, 1996兲.
24
N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie Groups and Special Functions, Vol. 1共Kluwer, Dordrecht, 1993兲.
25A. Wawrzyn´czyk, Group Representations and Special Functions共Kluwer, Dordrecht, 1978, 1984兲. 26M. Lorente, Phys. Lett. A 285, 119共2001兲.
27
M. Lorente, El. Trans. Numer. Anal. 9, 102共1999兲.
28J. R. Klauder and B-S. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics共World Scien-tific, Singapore, 1985兲.
29W-M. Zhang, D. H. Feng, and R. Gilmore, Rev. Mod. Phys. 62, 867共1990兲. 30F. B. Weissler, J. Funct. Anal. 32, 102共1979兲.
31
C-C. Shih, Opt. Lett. 20, 1178共1995兲.
32N. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, J. Phys. A 34, 9399共2001兲.
33M. Lorente and P. Kramer, ‘‘Induced representations of the Poincare´ group on the lattice: spin 1/2 and spin 1 case,’’ in
Symmetries in Science. X (Bregenz, 1997)共Plenum, New York, 1998兲, pp. 179–195.
34
35M. Ruzzi and D. Galetti, J. Phys. A 33, 1065共2000兲. 36
M. S. Richman and T. W. Parks, ‘‘Understanding discrete rotations,’’ in Proc. 1997 IEEE Int. Conf. Acoustics, Speech
and Signal Processing共IEEE, Piscataway, NJ 1997兲.
37M. S. Richman, T. W. Parks, and R. G. Shenoy, IEEE Trans. Signal Process. 46, 1517共1998兲. 38A. Bechata, C. R. Acad. Sci., Ser. I: Math. 326, 157共1998兲.
39V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics共World Scientific, Singapore, 1994兲.
40R. Aldrovandi and D. Galetti, J. Math. Phys. 31, 2978共1990兲. 41
D. B. Fairlie and C. K. Zachos, Phys. Lett. B 224, 101共1989兲. 42
D. Galetti and A. F. R. de Toledo Piza, Physica A 214, 207共1995兲. 43A. M. F. Rivas and A. M. Ozorio de Almeida, Ann. Phys. 276, 223共1999兲. 44M. Ruzzi, J. Phys. A 35, 1763共2002兲.
45M. Ruzzi and D. Galetti, J. Phys. A 35, 4633共2002兲. 46
J. A. Vaccaro and D. T. Pegg, Phys. Rev. A 41, 5156共1990兲. 47V. R. Vieira and P. D. Sacramento, Ann. Phys. 242, 188共1995兲. 48S-C. Pei and M-H. Yeh, IEEE Trans. Signal Process. 49, 1198共2001兲. 49C. Richard, IEEE Trans. Signal Process. 49, 2536共2001兲.
50
