The Moyal representation of quantum mechanics and special function theory Joseph C. Várilly,1 José M. Gracia-Bondı́a,1 and Walter Schempp2 1 Escuela de Matemática, Universidad de Costa Rica, 11501 San José, Costa Rica 2 Lehrstuhl für Mathematik I, Universität Siegen, Siegen, Germany Acta Appl. Math. 18 (1990), 225–250 Abstract It is shown that the phase-space formulation of quantum mechanics is a rich source of special function identities. The Moyal formalism is reviewed for two phase spaces: the real plane and the sphere; and this is used to derive identities for Airy, Laguerre, Kummer and theta functions and for SU(2) rotation matrix elements, several of which are new. 1 Introduction Some forty years ago, Moyal [1] proposed an attractive description of quantum mechanical systems, taking place in phase space, i.e., the same arena as classical mechanics. Moyal noticed that Wigner’s recipe [1] associating a function on phase space (the so-called Wigner distribution) to a density operator on Hilbert space was essentially the inverse of the celebrated Weyl correspondence rule [3]. Thus the way was opened to represent Quantum Mechanics of spinless particles as a statistical theory in phase space. After a long period of neglect, Moyal’s formulation nowadays finds a wide range of applica- tion [4]. Also, mathematically very similar formalisms sprang up in apparently remote areas, such as signal theory [5], photometry, and (via the use of Gabor functions) in neurophysiology [6]. Recently, two of the present authors extended Moyal’s approach to cover spinning particles as well [7]. The purpose of this article is to outline a ‘mathematical application’ of Moyal’s theory, to wit, the derivation of properties of special functions. Such an endeavour has little precedent: indeed, we know only of Peetre’s paper [8] as a forerunner. More recently, there has been some work about symbolic calculus on the Heisenberg group, which bears a close relationship with our work [9]. Nevertheless, there the emphasis was put on the functional-analytic side and few explicit formulae were given. We write here, rather, for those who are fond of beautiful and surprising equalities. They will not be new for the most part, as Peetre’s formulae for Laguerre polynomials of course were not. What is new is the manner of their derivation. We hope to convince the reader that Moyal’s theory is indeed a rich mine for special function theory. 1 The approach sketched in this paper would seem to go against the grain, as the tide has run for many years now to putting order in the bewildering chaos of special functions, reinterpreting them as matrix elements of unitary irreducible representations of Lie groups [10]. But in fact, at a deeper level, this is not so. If we substitute another suitable group for the Heisenberg group in the ‘quantization’ procedure, we get again special function formulae after our fashion: and we do this for SU(2) in Section 4 of the paper. In fact, Moyal’s formulation of Quantum Mechanics is, mathematically speaking, a branch of Harmonic Analysis under a somewhat novel form, in much the same way as conventional Quantum Mechanics mirrors the theory of unitary representations of Lie groups. In order to avoid cumbersome arguments, we shall remain on a formal level and shall neglect most questions of convergence. We can put it in no better way than Peetre’s [8]: “In most instances it is obvious how to put things an a more rigorous status, but certainly then most of the present simplicity will be hidden. . . ” Section 2 of the paper contains a resumé of the Moyal representation of Quantum Mechanics in the simplest ordinary phase space ℝ2. Our presentation is based on the Grossmann–Royer reflection operators [11]. They are but the simplest instance of a Stratonovich–Weyl quantizer, which is the bridge between the Moyal representation and the usual brand of harmonic analysis (in this case, Fourier analysis on the Heisenberg group). In Section 3, two basic tricks are introduced: Wigner’s ‘transfer formula’ and the ‘functional calculus’ method. They are employed to (re)derive properties of Airy functions, Hermite and Laguerre polynomials and Kummer’s hypergeometric function. When one seeks to invert the functional calculus, three interesting families of polynomials appear. Then we exploit the connection between Moyal Quantum Mechanics and the holograph transform of signal theory. Using the Poisson formula, a discretization of the holograph transform identities, known as the radial pixel identity, is obtained. In this way, we derive several new identities for Jacobian theta-null values. Section 4 introduces the Moyal representation for spin and exploits it to give a new expression for the SU(2) representation matrix elements, in terms of canonical coordinates of the first kind on the group manifold. 2 Moyal quantum mechanics on ℝ2 The Weyl correspondence between symbols (i.e., complex functions or distributions on phase space) and standard quantum-mechanical operators in modernly stated by means of the Grossmann–Royer reflection operators [11]. Namely, to each point 𝑢 := (𝑞, 𝑝) of phase space we associate an operator Π(𝑢) by: Π(𝑢)Ψ(Z) := exp [2𝑖 ℏ 𝑝(Z − 𝑞) ] Ψ(2𝑞 − Z) (1a) in configuration space, or Π′(𝑢)Φ(b) := exp [2𝑖 ℏ 𝑞(𝑝 − b) ] Φ(2𝑝 − b) (1b) in momentum space. In (1) ℏ denotes Planck’s constant and Ψ, Φ are wavefunctions belonging respectively to the Hilbert spaces 𝐿2(ℝ, 𝑑Z) =: HZ and 𝐿2(ℝ, 𝑑b) =: Hb . The operators Π and Π′ 2 are intertwined by the Fourier transform F: Π′ = FΠ F−1, where: FΨ(b) := 1 √ 2𝜋ℏ ∫ ℝ exp [ − 𝑖 ℏ bZ ] Ψ(Z) 𝑑Z connects position and momentum wavefunctions. The operators Π, Π′ are unitary and self-adjoint. An easy formal calculation gives the important formulas: trΠ(𝑢) = 1 2 , trΠ(𝑢)Π(𝑣) = 𝜋ℏ 2 𝛿(𝑢 − 𝑣), trΠ(𝑢)Π(𝑣)Π(𝑤) =: 1 8 𝐿 (𝑢, 𝑣, 𝑤) := 1 2 exp [ 2𝑖 ℏ ( 𝜎(𝑢, 𝑣) + 𝜎(𝑣, 𝑤) + 𝜎(𝑤, 𝑢) ) ] , (2) where 𝜎 is the standard symplectic form: namely, if 𝑢 = (𝑞, 𝑝) and 𝑣 = (𝑞′, 𝑝′), then 𝜎(𝑢, 𝑣) := 𝑞𝑝′ − 𝑞′𝑝. A symbol 𝑓 is mapped in a one-to-one linear way to an operator𝑊 𝑓 : 𝑊 𝑓 := 1 𝜋ℏ ∫ ℝ2 𝑓 (𝑢)Π(𝑢) 𝑑𝑢 (3) on the Hilbert space HZ , say. From (2) we infer that: (i) By using the same family of operators Π, formula (3) may be inverted: 𝑓 (𝑢) = 2 tr𝑊 𝑓Π(𝑢). (4) If 𝐴 is an operator on HZ , we shall write𝑊𝐴 (𝑢) := 2 tr 𝐴Π(𝑢), so𝑊𝑊𝐴 = 𝐴 and𝑊𝑊 𝑓 = 𝑓 . (ii) The constant function 1 corresponds to the identity operator. (iii) The product of two operators 𝑊 𝑓𝑊𝑔 is equal to 𝑊 𝑓×𝑔 where the twisted (quantum, Moyal) product of two symbols 𝑓 , 𝑔 is given by 𝑓 × 𝑔(𝑢) = 1 (2𝜋ℏ)2 ∫ ℝ2 ∫ ℝ2 𝑓 (𝑣) 𝑔(𝑤) 𝐿 (𝑢, 𝑣, 𝑤) 𝑑𝑣 𝑑𝑤 (5) with the property ∫ 𝑓 × 𝑔(𝑢) 𝑑𝑢 = ∫ 𝑓 (𝑢) 𝑔(𝑢) 𝑑𝑢. Clearly, the whole procedure works the same using Π′ on Hb instead. Once one has (5), one can forget about the Hilbert spaces HZ , Hb and their operatorial theory, and work exclusively with the symbols. This is, in short, Moyal Quantum Mechanics. We shall find it convenient to use Dirac’s notation |Ψ⟩ for wavefunctions Ψ(Z) seen as vectors of HZ . The inner product is written ⟨Ψ1 | Ψ2⟩HZ = ∫ ℝ Ψ1(Z) Ψ2(Z) 𝑑Z . We suppress the subscript HZ when there is no risk of confusion. Consider the rank one operator |Ψ1⟩⟨Ψ2 | associated to a couple Ψ1,Ψ2 of wavefunctions:( |Ψ1⟩⟨Ψ2 | ) Ψ(Z) = ⟨Ψ2 | Ψ⟩Ψ1(Z). 3 Then: 𝑊|Ψ1⟩⟨Ψ2 | (𝑢) = 2⟨Ψ2 | Π(𝑢)Ψ1⟩ = 2 ∫ ℝ Ψ2(𝑞 + 𝑦) Ψ1(𝑞 − 𝑦) exp [2𝑖 ℏ 𝑝𝑦 ] 𝑑𝑦. (6a) An analogous formula can be worked out in momentum space. One finds 𝑊|Φ1⟩⟨Φ2 | (𝑢) = 2 ∫ ℝ Φ2(𝑝 + 𝑦)Φ1(𝑝 − 𝑦) exp [ −2𝑖 ℏ 𝑞𝑦 ] 𝑑𝑦. (6b) When Ψ1 = Ψ2, the previous expressions are, but for a constant factor, the famous Wigner distributions [1]. It is however advisable, as argued by Dahl [12] on physical grounds, to treat states (i.e., the symbols corresponding to projectors |Ψ⟩⟨Ψ|) and transitions (i.e., the symbols of operators of the form |Ψ1⟩⟨Ψ2 | with Ψ1 ≠ Ψ2) on the same footing. Note that𝑊|Ψ2⟩⟨Ψ1 | = 𝑊 |Ψ1⟩⟨Ψ2 |. Because of (2) there holds in all generality: ⟨Ψ2 | 𝐴Ψ1⟩ = 1 2𝜋ℏ ∫ ℝ2 𝑊𝐴 (𝑢)𝑊|Ψ1⟩⟨Ψ2 | (𝑢) 𝑑𝑢. This is the formula which gives the aspect of a statistical theory in phase space to Moyal Quantum Mechanics. We denote by HZ the same space HZ , but with the multiplication law (𝛼,Ψ) ↦→ �̄�Ψ for 𝛼 ∈ ℂ and the inner product ⟨Ψ1 | Ψ2⟩HZ := ⟨Ψ1 | Ψ2⟩HZ . Then Ψ1 ⊕ Ψ2 ↦→ 𝑊|Ψ1⟩⟨Ψ2 | extends to a unitary isomorphism of the Hilbert space HZ ⊕ HZ onto 𝐿2(ℝ2, (2𝜋ℏ)−1𝑑𝑢). In practice, states are obtained, both in conventional and Moyal Quantum Mechanics, as eigen- states of given observables. Let us then look at dynamics in Moyal’s formulation. Suppose that 𝐻 is a real symbol such that𝑊𝐻 is selfadjoint. The basic equation: 𝑖ℏ 𝜕Ξ𝐻 𝜕𝑡 (𝑢; 𝑡) = 𝐻 × Ξ𝐻 (𝑢; 𝑡) (7) simply translates into the Moyal language the semigroup differential equation for the unitary operator 𝑈𝐻 (𝑡) := exp[−(𝑖/ℏ)𝑊𝐻𝑡]. Here 𝑈𝐻 = 𝑊Ξ𝐻 . We shall call Ξ𝐻 the Moyal propagator or evolution function corresponding to the Hamiltonian 𝐻. The formula (see Appendix B in the book by Taylor in [9]): 1 2𝜋ℏ ∫ ∞ −∞ 𝑈𝐻 (𝑡)𝑒𝑖𝑡𝐸/ℏ 𝑑𝑡 = 𝑃𝐻 (𝑑𝐸), for the projection-valued measure 𝑃𝐻 (‘resolution of the identity’) corresponding to the selfadjoint operator𝑊𝐻 by the spectral theorem, translates into: 1 2𝜋ℏ ∫ ∞ −∞ Ξ𝐻 (𝑢; 𝑡)𝑒𝑖𝑡𝐸/ℏ 𝑑𝑡 = 𝑊𝑃𝐻 (𝑑𝐸) =: Γ𝐻 (𝑢; 𝑑𝐸). (8) Here Γ𝐻 is a measure on the real line with distributional values in general. In all cases we shall meet, this measure is either discrete or absolutely continuous with respect to Lebesgue measure; so we simply write Γ𝐻 (𝑢; 𝑑𝐸) = Γ𝐻 (𝑢; 𝐸) 𝑑𝐸 . The corresponding ‘eigenvalue equation’ is clearly: 𝐻 × Γ𝐻 (𝑢; 𝐸) = Γ𝐻 (𝑢; 𝐸) × 𝐻 = 𝐸 Γ𝐻 (𝑢; 𝐸) (9) 4 and the spectrum sp𝐻 of𝑊𝐻 is the support of 𝑃𝐻 (𝑑𝐸), or equivalently, the support of Γ𝐻 (𝑢; 𝐸) in the 𝐸-variable. (For details of rigour on this matter, see Appendix A of [13]). The projection-valued measure 𝑃𝐻 is the cornerstone of the functional calculus for the self- adjoint operator𝑊𝐻 . We can now analogously build a twisted functional calculus with the symbols, with (for our purposes) an all-important difference: its elements are concrete functions (or distribu- tions) in phase space. Some important elements of a functional calculus are: (1) The aforesaid evolution function: Ξ𝐻 (𝑢; 𝑡) = ∫ sp𝐻 Γ𝐻 (𝑢; 𝐸) 𝑒−𝑖𝑡𝐸/ℏ 𝑑𝐸. (10) (2) The resolvent function: 𝑅𝐻 (𝑢;_) = ∫ sp𝐻 Γ𝐻 (𝑢; 𝐸) 𝐸 − _ 𝑑𝐸, (11) for _ ∈ ℂ, _ ∉ sp𝐻, which verifies 𝑅𝐻 (𝑢;_) × (𝐻 − _) = 1. (3) The twisted powers: 𝐻×𝑛 (𝑢) := 𝐻 × · · · × 𝐻 (𝑢) = ∫ sp𝐻 𝐸𝑛 Γ𝐻 (𝑢; 𝐸) 𝑑𝐸. (12) Now we reveal our strategy: one can try to solve equations (7) or (9) directly, or one can obtain Ξ𝐻 , Γ𝐻 as the transforms (4), through use of Green functions and Schrödinger wavefunctions of stan- dard Quantum Mechanics. Equating the results of both procedures, many interesting relationships emerge. A second device is direct use of the twisted functional calculus. Equations (7) and (9) are obviously very difficult to solve in general (this has deterred potential users of Moyal formalism for quantum physics). There is however an exception, to wit, when the symbol 𝐻 is some real quadratic polynomial in 𝑞, 𝑝. In that case the corresponding operators are selfadjoint (see Chapter 1 of the book by Taylor in [9], and [13]), and moreover (7), (9) degenerate into linear second-order differential equations which can, in fact, be easier to solve than the corresponding equations in the conventional formalism. This quantization scheme can be generalized to phase spaces other than ℝ2. The starting point is to realize that ℝ2, with its natural symplectic structure, may be considered as a coadjoint orbit of the 3-dimensional Heisenberg group ℍ3; and that the Grossmann–Royer operators Π(𝑢) transform covariantly under the coadjoint action of ℍ3 on ℝ2 and conjugation by the associated unitary irreducible representation 𝑈 of ℍ3 on the Hilbert space HZ . Let 𝑓 be a function defined on the underlying manifold of ℍ3 and set, for _ ∈ ℝ∗ indexing the set of generic coadjoint orbits or the nontrivial unitary irreducible representations of ℍ3: 𝑓 (_) := ∫ ℍ3 𝑓 (𝑔)𝑈_ (𝑔) 𝑑𝑔, where 𝑑𝑔 is the Haar measure on ℍ3. This is the (operatorial) Fourier transform of 𝑓 . The abstract Plancherel formula: 𝑓 (0) = 𝑐 ∫ ∞ −∞ |_ | tr[ 𝑓 (_)] 𝑑_ (for a suitable constant 𝑐) 5 is well known. As indicated above, one has 𝑈_ (𝑔) Π_ (𝑢)𝑈_ (𝑔−1) = Π_ (𝑔 · 𝑢) where 𝑔 · 𝑢 denotes the coadjoint action and Π_ is always given in the Schrödinger representation by (1a) with the formal identification _ = ℏ. Now we apply the inverse Weyl correspondence to the representation operators themselves: 𝐸 (𝑔, 𝑢;_) := 2 tr [ 𝑈_ (𝑔) Π_ (𝑢) ] , and think of 𝐸 as a (scalar) Fourier kernel. Then it is clear from the foregoing that: 𝑓 (𝑢;_) := ∫ ℍ3 𝐸 (𝑔, 𝑢;_) 𝑓 (𝑔) 𝑑𝑔 = 𝑊 𝑓 (_) (𝑢). By direct computation, 𝐸 is seen to be simply the ordinary Fourier kernel in three dimensions: that is to say, the ordinary Plancherel theorem and the Plancherel theorem for the Heisenberg group are equivalent, as has been pointed out by Howe [14] and by Taylor [9] from slightly different points of view. The Weyl correspondence ferries back and forth between 𝑓 and 𝑓 . The important fact is that one can duplicate this scheme for several other Lie groups. The seminal paper by Stratonovich [15] pointed this out for the group SU(2), and this approach to spin quantization has been explored in detail by two of us in [7]. In that general context the equivalent of 𝐸 may be highly nontrivial. We shall then speak of a Stratonovich–Weyl correspondence. Before we get down to business, we shall be mindful of Dahl’s contention and take advantage of the remark following (6): we state the spectral theorem for Moyal Quantum Mechanics in a fairly general form. (a) The solutions of the system: 𝐻 × Γ𝐻 (𝑢; 𝐸, 𝐸′) = 𝐸 Γ𝐻 (𝑢; 𝐸, 𝐸′), Γ𝐻 × 𝐻 (𝑢; 𝐸, 𝐸′) = 𝐸′ Γ𝐻 (𝑢; 𝐸, 𝐸′), (13) span the space 𝐿2(ℝ2, (2𝜋ℏ)−1 𝑑𝑢). The permissible values of 𝐸 or 𝐸′ give the spectrum. Omitting the 𝑢-variable for simplicity of notation, we have Γ𝐻 (𝐸, 𝐸′) = Γ𝐻 (𝐸′, 𝐸) and Γ𝐻 (𝐸, 𝐸) = Γ𝐻 (𝐸) as defined in (8). (b) The Γ𝐻 can be appropriately normalized so that the following relations hold: Γ𝐻 (𝐸, 𝐸′) × Γ𝐻 (𝐸′′, 𝐸′′′) = 𝛿𝐸 ′,𝐸 ′′ Γ𝐻 (𝐸, 𝐸′′′). (14a) (c) If we denote by (· | ·) the inner product on 𝐿2(ℝ2, (2𝜋ℏ)−1𝑑𝑢), then we have the orthogonality condition: (Γ𝐻 (𝐸, 𝐸′) | Γ𝐻 (𝐸′′, 𝐸′′′)) = 𝛿𝐸,𝐸 ′′ 𝛿𝐸 ′,𝐸 ′′′ . (15) Note that this implies: ∫ ℝ2 Γ𝐻 (𝐸, 𝐸′) 𝑑𝑢 2𝜋ℏ = 𝛿𝐸𝐸 ′ . 6 (d) The functional calculus gives in general: 𝑔×(𝐻) = ∑︁ 𝐸∈sp𝐻 𝑔(𝐸) Γ𝐻 (𝐸) (16) with an obvious notation for the ‘twisted’ function 𝑔 of the Hamiltonian 𝐻. The equations (10)–(12) are already instances of (16). In particular: ∑︁ sp𝐻 Γ𝐻 (𝐸) = 1; ∑︁ sp𝐻 𝐸 Γ𝐻 (𝐸) = 𝐻. (e) In view of (a), (b) and (c), the following closure relations hold:∑︁ 𝐸,𝐸 ′∈sp𝐻 Γ𝐻 (𝑢; 𝐸, 𝐸′) Γ𝐻 (𝑣; 𝐸′, 𝐸) = 2𝜋ℏ 𝛿(𝑢 − 𝑣),∑︁ 𝐸,𝐸 ′,𝐸 ′′∈sp𝐻 Γ𝐻 (𝑢; 𝐸, 𝐸′) Γ𝐻 (𝑣; 𝐸′, 𝐸′′) Γ𝐻 (𝑤; 𝐸′′, 𝐸) = 𝐿 (𝑢, 𝑣, 𝑤). (17) Of course, these are but (2) under another guise. We have assumed in the notation that 𝐻 has a simple pure point spectrum. Examples of continuous spectra will shortly appear. Then the transitions Γ𝐻 (𝐸, 𝐸′) are generalized functions outside the Hilbert space; formula (13) applies in a weak sense, and (14a) then reads: Γ𝐻 (𝐸, 𝐸′) × Γ𝐻 (𝐸′′, 𝐸′′′) = 𝛿(𝐸′ − 𝐸′′) Γ(𝐸, 𝐸′′′). (14b) The new rendition of the other formulas (15), (16) and (17), wherein integrals substitute for the Fourier–Dirichlet series, is also straightforward. 3 Special functions To reach our conclusions, we shall apply the foregoing theory to three illustrative examples, which are also of great physical significance: the free-fall Hamiltonian 𝐻 = (𝑝2/2𝑚) + 𝐹𝑞; the harmonic oscillator 𝐻 = (𝑝2/2𝑚) + 𝑘𝑞2 with 𝑘 > 0; and the harmonic barrier (same expression with 𝑘 < 0). It is known [13, 16] that for any quadratic Hamiltonian 𝐻 = 1 2𝑢 𝑡𝐵𝑢 + 𝑐𝑡𝑢, where 𝑢 = (𝑞 𝑝 ) , 𝑢𝑡 = (𝑞, 𝑝), 𝐵 is a nonsingular 2×2 symmetric matrix, and 𝑐 is a constant vector, the solution of (7) is Ξ𝐻 (𝑢; 𝑡) = 2𝑛𝑒−𝑖𝛽(𝑡)/ℏ√︁ det(1 + 𝑀 (𝑡)) exp [ − 𝑖 ℏ 𝑔𝐻 (𝑢; 𝑡) ] , (18) where 𝑀 (𝑡) is the symplectic matrix exp(−𝐽𝐵𝑡) with 𝐽 = ( 0 1 −1 0 ) ; 𝑔𝐻 (𝑢; 𝑡) = (𝑢 + 1 2𝑎(𝑡)) 𝑡𝐽 (𝑀 (𝑡) − 1) (𝑀 (𝑡) + 1)−1(𝑢 + 1 2𝑎(𝑡)) − 𝑢 𝑡𝐽𝑎(𝑡); 𝑎(𝑡) = ∫ 𝑡 0 𝑀 (𝑠)𝐽𝑐 𝑑𝑠; 𝛽(𝑡) = 1 2 ∫ 𝑡 0 ∫ 𝑠 0 𝑐𝑡𝑀 (𝑟 − 𝑠)𝐽𝑐 𝑑𝑟 𝑑𝑠. By employing this result, one can get at the spectrum of 𝐻 and, through (8), the ‘resolution of the identity’ Γ𝐻 (𝐸), in a single stroke. The ‘eigenvalue equations’ (13) remain useful, however, to find the nondiagonal Γ𝐻 . 7 3.1 Free-fall Hamiltonian and Airy functions We take 𝐹 = 𝑚 = 1 to avoid encumbering our notation unnecessarily. Writing 𝐻 = 1 2 𝑝 2 + 𝑞, from (18) we obtain Ξ𝐻 (𝑢; 𝑡) = exp [ − 𝑖 ℏ ( 𝐻𝑡 + 𝑡3 24 )] . (19) Recalling Ai(𝑥) := 1 2𝜋 ∫ ∞ −∞ exp(𝑖a𝑥 + 1 3𝑖a 3) 𝑑a, we get: Γ𝐻 (𝑢; 𝐸) = 3 √︂ 8 ℏ2 Ai ( 3 √︂ 8 ℏ2 (𝐻 − 𝐸) ) . (20) The twisted product (5) has the following asymptotic expansion [17]: 𝑓 × 𝑔 ∼ ∞∑︁ |𝛼 |=0 ( 𝑖ℏ 2 ) |𝛼 | 1 𝛼! 𝜕𝛼 𝑓 𝜕𝛼𝑔, (21) where 𝛼 = (𝛼1, 𝛼2) ∈ ℕ2; |𝛼 | := 𝛼1 + 𝛼2; 𝜕𝛼 = 𝜕𝛼1 𝜕𝑞𝛼1 𝜕𝛼2 𝜕𝑝𝛼2 ; 𝜕𝛼 = (−1)𝛼2 𝜕𝛼1 𝜕𝑝𝛼1 𝜕𝛼2 𝜕𝑞𝛼2 . This expansion is exact whenever one of the factors is a polynomial. Using this, and separating real and imaginary parts in (13), we get:( 𝑝2 2 + 𝑞 − ℏ2 8 𝜕2 𝜕𝑞2 ) Γ𝐻 (𝑞, 𝑝; 𝐸, 𝐸′) = 𝐸 + 𝐸′ 2 Γ𝐻 (𝑞, 𝑝; 𝐸, 𝐸′),( −𝑝 𝜕 𝜕𝑞 + 𝜕 𝜕𝑝 ) Γ𝐻 (𝑞, 𝑝; 𝐸, 𝐸′) = − 𝑖 ℏ (𝐸 − 𝐸′) Γ𝐻 (𝑞, 𝑝; 𝐸, 𝐸′). In the new canonical variables (𝐻, 𝑝), this reads:[ ℏ2 8 𝜕2 𝜕𝐻2 − ( 𝐻 − 𝐸 + 𝐸′ 2 )] Γ𝐻 (𝐸, 𝐸′) = 0, 𝜕Γ𝐻 (𝐸, 𝐸′) 𝜕𝑝 = − 𝑖 ℏ (𝐸 − 𝐸′) Γ𝐻 (𝐸, 𝐸′), whose (suitably normalized) regular solution is: Γ𝐻 (𝐸, 𝐸′) = 3 √︂ 8 ℏ2 Ai ( 3 √︂ 8 ℏ2 ( 𝐻 − 𝐸 + 𝐸′ 2 )) 𝑒𝑖𝑝(𝐸 ′−𝐸)/ℏ. In particular, we recover (20). Now the solution of the standard Schrödinger equation for the free-fall is again an Airy function! We have, in fact, the equation: −ℏ 2 2 𝜕2Ψ𝐻,𝐸 𝜕𝑞2 + 𝑞Ψ𝐻,𝐸 = 𝐸 Ψ𝐻,𝐸 8 whose regular solution [normalized by ⟨Ψ𝐻,𝐸 | Ψ𝐻,𝐸 ′⟩ = 𝛿(𝐸 − 𝐸′)] is Ψ𝐻,𝐸 = 3 √︂ 2 ℏ2 Ai ( 3 √︂ 2 ℏ2 (𝑞 − 𝐸) ) . Now we write down equation (6a) for this case and see what happens. After eliminating constants irrelevant to the matter at hand, we get Ai [ 3√4(𝑎2 + 1 2 (𝑏 + 𝑏 ′)) ] = 3√2 𝑒𝑖(𝑏 ′−𝑏)𝑎 ∫ ∞ −∞ Ai(𝑏′ + 𝑐) Ai(𝑏 − 𝑐) 𝑒2𝑖𝑎𝑐 𝑑𝑐. (22a) Moreover, from the general relation: Ψ𝐻,𝐸 (𝑞) Ψ𝐻,𝐸 ′ (𝑞) = 1 2𝜋ℏ ∫ ℝ Γ𝐻 (𝑞, 𝑝; 𝐸, 𝐸′) 𝑑𝑝, which follows from (6a), or working directly on (22a), we obtain Ai(𝑏) Ai(𝑏′) = 1 𝜋 3√2 ∫ ∞ −∞ Ai ( 3√4(𝑎2 + 1 2 (𝑏 + 𝑏 ′)) ) 𝑒𝑖𝑎(𝑏−𝑏 ′) 𝑑𝑎. (22b) These curious nonlinear relations among Airy functions seem to be new. Special cases have appeared in [18] and [19]. Many properties of Airy functions can be derived through (20) from the twisted functional calculus. We shall not go into that; instead we note only that the family of polynomials 𝐺1 𝑛 (𝑥) defined by 𝐺1 𝑛 (𝐻) := 𝐻×𝑛 (with ℏ = 1) obeys, by (21), the following recurrence relation: 𝐺1 𝑛 (𝑥) = 𝑥 𝐺1 𝑛−1(𝑥) − 1 8 𝑑𝐺1 𝑛−1 𝑑𝑥2 (𝑥) for 𝑛 ⩾ 1; 𝐺1 0(𝑥) = 1. Then, from (12) and (20): 𝐺1 𝑛 (𝑥) = ∫ (𝑥 − 1 2 𝑦) 𝑛 Ai(𝑦) 𝑑𝑦. In particular, the moments of the Airy function are calculated to be `𝑛 (Ai) = (−1)𝑛2𝑛𝐺1 𝑛 (0). The possibility of inverting the functional calculus is interesting. Note that Ξ𝐻 (𝑢; 𝑡) is formally an exponential: Ξ𝐻 (𝑢; 𝑡) = ∞∑︁ 𝑛=0 (−𝑖ℏ𝑡)𝑛 𝑛! 𝐻×𝑛. We write Exp(−𝑖𝐻𝑡/ℏ) := Ξ𝐻 (𝑢; 𝑡). From (19), with ℏ = 1, we get: Exp [ −𝑖𝑡 ( 𝐻 − 𝑡2 24 )] = 𝑒−𝑖𝑡𝐻 . Formally, 𝑒𝑎𝑥 = Exp [ 𝑎 ( 𝑥 + 𝑎 2 24 )] =: 𝑔1(𝑥; 𝑎) =: ∞∑︁ 𝑛=0 𝑇× 𝑛 (𝑥) 𝑎𝑛. 9 We are treating 𝑔1 as a generating function; from the differential equation for 𝑔1: 𝜕𝑔1 𝜕𝑎 = (𝑥 + 1 8𝑎 2) × 𝑔1, one gets the recurrence relation for the polynomials 𝑇𝑛: 𝑛𝑇𝑛 (𝑥) − 𝑥 𝑇𝑛−1(𝑥) − 1 8𝑇𝑛−3(𝑥) = 0 for 𝑛 ⩾ 3; 𝑇0(𝑥) = 1; 𝑇1(𝑥) = 𝑥; 𝑇2(𝑥) = 1 2𝑥 2. If 𝑓 denotes an ordinary function of 𝐻 such that 𝑓 (𝐻) = ∑∞ 𝑛=0 𝑎𝑛𝐻 𝑛, it will follow that 𝑓 (𝐻) = 𝑔×(𝐻) with 𝑔(𝐻) = ∑∞ 𝑛=0 𝑎𝑛 𝑛!𝑇𝑛 (𝐻). Proceeding in the same way, we shall get two more families of polynomials respectively associ- ated to the harmonic oscillator and the harmonic barrier Hamiltonians. The last one turns out to be quite interesting (see Subsection 3.3 below). From (14b) and the analogues of (15, 17), there issue several integral formulas. We shall not bother to write them explicitly, except for:∭ Ai(𝑝2 + 2𝑞 − 𝑎) Ai(𝑝′2 + 2𝑞′ − 𝑎′) Ai(𝑝′′2 + 2𝑞′′ − 𝑎′′) 𝑒𝑖[𝑝(𝑎 ′−𝑎′′)+𝑝′ (𝑎′′−𝑎)+𝑝′′ (𝑎−𝑎′)] 𝑑𝑎 𝑑𝑎′ 𝑑𝑎′′ = 1 8 exp [ 2𝑖(𝑞𝑝′ − 𝑞′𝑝 + 𝑞′𝑝′′ − 𝑞′′𝑝′ + 𝑞′′𝑝 − 𝑞𝑝′′) ] . 3.2 Harmonic oscillator Hamiltonian and Laguerre functions Writing 𝐻 = 1 2 (𝑝 2 + 𝑞2), we obtain from (18): Ξ𝐻 (𝑢; 𝑡) = sec 𝑡 2 exp ( −2𝑖 ℏ 𝐻 tan 𝑡 2 ) . (23) The eigenvalue equations (13), on the other hand, turn out to be 𝑝2 + 𝑞2 2 − ℏ2 8 ( 𝜕2 𝜕𝑞2 + 𝜕2 𝜕𝑝2 ) Γ𝐻 (𝑞, 𝑝; 𝐸, 𝐸′) = 𝐸 + 𝐸′ 2 Γ𝐻 (𝑞, 𝑝; 𝐸, 𝐸′),( 𝑞 𝜕 𝜕𝑝 − 𝑝 𝜕 𝜕𝑞 ) Γ𝐻 (𝑞, 𝑝; 𝐸, 𝐸′) = − 𝑖 ℏ (𝐸 − 𝐸′) Γ𝐻 (𝑞, 𝑝; 𝐸, 𝐸′). (24) If we introduce the new canonical variables (𝐻, 𝜏), where 𝜏 = arctan(𝑝/𝑞), the (appropriately normalized) regular solutions of (24) are: 𝑓𝑚𝑛 (𝐻, 𝜏) = 2(−1)𝑛 √︂ 𝑛! 𝑚! ( 4𝐻 ℏ ) (𝑚−𝑛)/2 𝑒−𝑖𝜏(𝑚−𝑛)𝐿𝑚−𝑛𝑛 ( 4𝐻 ℏ ) 𝑒−2𝐻/ℏ (25a) if 𝑚 ⩾ 𝑛; 𝑓𝑚𝑛 = 𝑓𝑛𝑚 if 𝑛 < 𝑚. Here 𝐿𝑚−𝑛𝑛 denotes the associated Laguerre polynomial of type 𝑚−𝑛 and degree 𝑛, with 𝑚, 𝑛 ∈ ℕ; the allowed values of 𝐸, 𝐸′ are (2𝑚 + 1)ℏ/2, (2𝑛 + 1)ℏ/2. 10 In what follows we take ℏ = 2 for convenience. In terms of the original variables, (25a) then reads: 𝑓𝑚𝑛 (𝑞, 𝑝) = 2(−1)𝑛 √︂ 𝑛! 𝑚! (𝑞2 + 𝑝2) (𝑚−𝑛)/2𝑒−𝑖(𝑚−𝑛) arctan(𝑝/𝑞)𝐿𝑚−𝑛𝑛 (𝑞2 + 𝑝2) 𝑒−(𝑞2+𝑝2)/2 (25b) for 𝑚 ⩾ 𝑛, say. From (16) one sees that the following equation holds: sec 𝑡 2 𝑒−𝑖𝐻 tan(𝑡/2) = ∞∑︁ 𝑛=0 𝑓𝑛𝑛 (𝐻)𝑒−𝑖(2𝑛+1)𝑡 . (26) As observed by Peetre [8], this equation gives the generating function formula for the Laguerre polynomials. A number of interesting consequences can be extracted from (26). If 𝛿0 denotes the Dirac measure concentrated at the origin of phase space, by taking limits in (26) as 𝑡 → 𝜋/2, we get 𝛿0 = 1 2𝜋 ∞∑︁ 𝑛=0 (−1)𝑛 𝑓𝑛𝑛. (27) The functions 𝑓𝑚𝑛, with 𝑚 ≠ 𝑛 in general, were introduced first by Groenewold [20] and by Bartlett and Moyal [21]. They have been rediscovered many times [16, 22], which surely is a measure of usefulness. All their basic properties, such as: 1 4𝜋 ∫ ℝ2 𝑓𝑚𝑛 (𝑞, 𝑝) 𝑑𝑞 𝑑𝑝 = 𝛿𝑚𝑛, (28a) 1 4𝜋 ∫ ℝ2 𝑓𝑚𝑛 (𝑞, 𝑝) 𝑓𝑟𝑠 (𝑞, 𝑝) 𝑑𝑞 𝑑𝑝 = 𝛿𝑛𝑟 𝛿𝑚𝑠, (28b) F( 𝑓𝑚𝑛) = (−𝑖)𝑚+𝑛 𝑓𝑚𝑛, (28c) follow easily from our apparatus. For the last one, for instance, note thatF 𝑓𝑚𝑛 (𝑞, 𝑝) = 𝐹 𝑓𝑚𝑛 (−𝑝, 𝑞), where 𝐹 𝑓 := 𝑓 ×𝛿0 is the ‘symplectic Fourier transform’. Using (27) we obtain 𝐹 ( 𝑓𝑚𝑛) = (−1)𝑛 𝑓𝑚𝑛 and F( 𝑓𝑚𝑛) = (−𝑖)𝑚+𝑛 𝑓𝑚𝑛. The orthonormal basis { 𝑓𝑚𝑛}𝑚,𝑛∈ℕ has theoretical importance. It has been employed by two of us [23] to give a rather complete account of duality and functional-analytic properties of twisted product theory. The 𝑓𝑚𝑛 could be employed to derive orthogonality and other properties of the associated Laguerre polynomials, in much the same way as Peetre exploited the ‘diagonal’ 𝑓𝑛𝑛 to reobtain properties of the ordinary Laguerre polynomials. We shall not go into that straightforward matter. Instead, we note that they are related to Hermite polynomials. It is well known that the solutions of the harmonic oscillator problem in the conventional formulation are: Ψ𝐻,𝐸𝑛 (𝑞) = (2𝜋)−1/4(2𝑛𝑛!)−1/2𝐻𝑛 ( 𝑞 √ 2 ) 𝑒−𝑞 2/4 with 𝐸𝑛 = 2𝑛 + 1 (recall that ℏ = 2). 11 Use of (6a) gives then:∫ ℝ 𝑒−𝑥 2/2𝐻𝑛 (𝑥 − 𝑧) 𝐻𝑚 (𝑥 + 𝑧) 𝑑𝑥 = 2𝑚 √ 𝜋 𝑛! 𝑧𝑚−𝑛𝐿𝑚−𝑛𝑛 (2|𝑧 |2) (𝑚 ⩾ 𝑛), which is a well-known integral [24, 7.377]. Now, the 𝑓𝑚𝑛 themselves are eigenfunctions of an harmonic-oscillator-type Schrödinger equation in two variables (see (24) and also (28c)). Accordingly, 𝑓𝑚𝑛 (𝑞, 𝑝) = ∑︁ 𝑘+𝑙=𝑚+𝑛 𝑐𝑘𝑙𝑚𝑛ℎ𝑘 (𝑞) ℎ𝑙 (𝑝), where ℎ𝑘 (𝑞) := (2𝑘−1𝑘!)−1/2𝐻𝑘 (𝑞)𝑒−𝑞 2/2, and we have computed that 𝑐𝑘𝑙𝑚𝑛 = 2(𝑚−𝑛)/2𝑖2𝑚+𝑙 ( 𝑚 + 𝑛 𝑙 )1/2 ( 𝑚 + 𝑛 𝑚 )−1/2 𝑃𝑙−𝑚,𝑘−𝑚𝑚 (0), where 𝑃𝑙−𝑚,𝑘−𝑚𝑚 denotes the usual Jacobi polynomial. Finally, from (17), we obtain ∞∑︁ 𝑚,𝑛,𝑘=0 𝑓𝑚𝑛 (𝑞1, 𝑝1) 𝑓𝑛𝑘 (𝑞2, 𝑝2) 𝑓𝑘𝑚 (𝑞3, 𝑝3) = 4 exp [ 𝑖(𝑞1𝑝2 − 𝑞2𝑝1 + 𝑞2𝑝3 − 𝑞3𝑝2 + 𝑞3𝑝1 − 𝑞1𝑝3) ] which no doubt holds in quite a strong sense. Surprisingly, no one seems to have remarked this before. Now we turn to the functional calculus, with makes use only of the diagonal elements. Our primary interest is to find an explicit formula for the twisted resolvent function for the harmonic oscillator. Here we cannot follow Peetre, as he uses (−1)𝑛 𝑓𝑛𝑛 for his diagonal elements, so in fact he essentially calculates the Hankel transform of our resolvent and other twisted functions. (This discrepancy can be traced back to the use of the twisted convolution in [8] instead of the twisted product.) For calculating the resolvent we have at our disposal at least two different methods. Besides 𝑅𝐻 (𝑞, 𝑝;_) = 𝑒−𝐻 ∞∑︁ 𝑛=0 (−1)𝑛𝐿𝑛 (2𝐻) 𝑛 + 1 2 (1 − _) , (29) which is (11), we note that the following Laplace-transform formula is available: 𝑅𝐻 (𝑞, 𝑝;_) = ∫ ∞ 0 exp(−𝐻 tanh 𝛽 + _𝛽) cosh 𝛽 𝑑𝛽, (30) the integral being valid for Re_ < 1. From the last formula, by a change of variables, we get 𝑅𝐻 (𝑞, 𝑝;_) = ∫ 1 0 𝑒−𝐻𝑥 (1 + 𝑥) (_−1)/2(1 − 𝑥) (−_−1)/2 𝑑𝑥 = ∫ 𝜋/2 0 𝑒−𝐻 cos \ ( tan \ 2 )−_ 𝑑\ = 2 1 − _Φ1 ( 1, 1 − _ 2 , 3 − _ 2 ;−1,−𝐻 ) , 12 where Φ1 denotes a confluent hypergeometric function in two variables [25]. From (29) and (30) we extract ∞∑︁ 𝑛=0 (−1)𝑛 𝐿𝑛 (𝑥) 𝑛 + 𝑎 = 𝑒𝑥/2 𝑎 Φ1 ( 1, 𝑎, 1 + 𝑎;−1,−𝑥 2 ) . This summation formula appears to be new. In particular, the ‘twisted inverse’ 𝐻×−1 of the harmonic oscillator Hamiltonian is 𝐻×−1(𝑞, 𝑝) = 𝑅𝐻 (𝑞, 𝑝; 0) = ∫ 𝜋/2 0 𝑒−𝐻 cos \ 𝑑\ = 𝜋 2 (𝐼0(𝐻) − 𝐿0(𝐻)). (31) Here 𝐼0 is the modified Bessel function of order 0 and 𝐿0 is the modified Struve function of order 0. It is amusing to see how, as 𝐼0(0) = 1 and 𝐿0(0) = 0, the series (29) for 𝐻 = 0 reduces to Gregory’s series: 𝜋 2 = 2 ( 1 − 1 3 + 1 5 − 1 7 + · · · ) . Note that 𝐻 × 𝑓 (𝐻) = 𝐻 𝑓 (𝐻) − 𝑓 ′(𝐻) − 𝐻 𝑓 ′′(𝐻) so 𝐻×−1 could also be sought as a series solution of the differential equation: 𝐻 𝑓 ′′(𝐻) + 𝑓 ′(𝐻) − 𝐻 𝑓 (𝐻) = −1. One obtains:∗ 𝑓 (𝐻) = ∞∑︁ 𝑛=0 𝑎𝑛𝐻 𝑛, with 𝑎2𝑛 = 𝜋 2 · 4𝑛 (𝑛!)2 , 𝑎2𝑛+1 = − 1 (2𝑛 + 1)!! . This coincides with (31) – consult, for instance, [26]. In contrast, the ‘parametrix’ method for twisted inversion suggested in [27] leads right away to very clumsy expressions. We can associate two families of polynomials to the harmonic oscillator problem, in the same way as before. The family 𝐻×𝑛 =: 𝐺2 𝑛 (𝐻) does not seem very interesting. On the other hand, if Exp denotes the quantum oscillator exponential, we get by obvious manipulations of (23): 𝑒𝑥𝑎 = Exp(𝑥 arctanh 𝑎) √ 1 − 𝑎2 =: 𝑔2(𝑥; 𝑎). We again treat 𝑔2(𝑥; 𝑎) as a generating function: 𝑔2(𝑥; 𝑎) = ∞∑︁ 𝑛=0 𝑄× 𝑛 (𝑥)𝑎𝑛. From the differential equation for 𝑔2: (1 − 𝑎2) 𝜕𝑔2 𝜕𝑎 = (𝑥 + 𝑎) × 𝑔2, ∗We are indebted to J. F. Ávila for performing this calculation. 13 one quickly gets the recurrence relation for the 𝑄 polynomials: 𝑛𝑄𝑛 (𝑥) − 𝑥 𝑄𝑛−1(𝑥) − (𝑛 − 1)𝑄𝑛−2(𝑥) = 0 if 𝑛 ⩾ 2, 𝑄0(𝑥) = 1, 𝑄1(𝑥) = 𝑥. If 𝑓 (𝐻) = ∑∞ 𝑛=0 𝑎𝑛𝐻 𝑛 denotes an ordinary function of the harmonic oscillator Hamiltonian, then 𝑓 (𝐻) = 𝑔×(𝐻), where 𝑔(𝐻) = ∑∞ 𝑛=0 𝑎𝑛𝑛!𝑄𝑛 (𝐻). The 𝑄𝑛 are apparently new; nevertheless, they are very similar on the surface to the so-called ‘continuous Hahn polynomials’ that pop out by the same trick in Subsection 3.3. 3.3 Harmonic barrier Hamiltonian and Kummer functions We take 𝐻 = 1 2 (𝑝 2 − 𝑞2) in this subsection. Formula (18) yields at once: Ξ𝐻 (𝑢; 𝑡) = sech 𝑡 2 exp ( −2𝑖 ℏ 𝐻 tanh 𝑡 2 ) . On the other hand, (13) can be solved using the same techniques as before and gives (with the appropriate normalization): Γ𝐻 (𝑢; 𝐸, 𝐸′) = 1 ℏ sech 𝜋(𝐸 + 𝐸′) 2ℏ 𝑒−2𝑖𝐻/ℏ 1𝐹1 ( 1 2 − 𝑖(𝐸 + 𝐸′) 2ℏ , 1; 4𝑖𝐻 ℏ ) exp [ 𝑖 ℏ (𝐸 − 𝐸′) arctanh 𝑝 𝑞 ] . It is amusing that the reality of the Γ𝐻 for 𝐸 = 𝐸′ entails a particular case of Kummer’s transforma- tion: 𝑒−𝑧/2 1𝐹1(𝑎, 1, 𝑧) = 𝑒𝑧/2 1𝐹1(1 − 𝑎, 1,−𝑧). Another very elementary fact in our context will give the integral representation of Kummer’s function: we must have (with ℏ = 2): Γ𝐻 (𝑢; 𝐸) = ∫ ∞ −∞ sech 𝑡 2 𝑒−𝑖(𝐻 tanh(𝑡/2)−𝐸𝑡/2) 𝑑𝑡 = 2𝑒−𝑖𝐻 ∫ 1 0 𝑒2𝑖𝐻𝑢𝑢−(1+𝑖𝐸)/2(1 − 𝑢)−(1−𝑖𝐸)/2 𝑑𝑢 using the change of variables tanh 𝑡/2 = 1 − 2𝑢. If we put 2𝑖𝐻 = 𝑧, 1 − 𝑖𝐸 = 2𝑎, and take into account Γ(𝛼)Γ(1 − 𝛼) = 𝜋 csc 𝜋𝛼, this gives 1𝐹1(𝑎, 1, 𝑧) = 1 Γ(𝑎)Γ(1 − 𝑎) ∫ 1 0 𝑒𝑧𝑡𝑡𝑎−1(1 − 𝑡)−𝑎 𝑑𝑡. Now we investigate again the two associated families of polynomials. We obtain, from (21) once more 𝐻×𝑛 = 𝐺3 𝑛 (𝐻), where 𝐺3 𝑛 (𝐻) = 𝐻 𝐺3 𝑛−1(𝐻) + 𝑑𝐺3 𝑛−1 𝑑𝐻 (𝐻) + 𝐻 𝑑2𝐺3 𝑛−1(𝐻) 𝑑𝐻2 for 𝑛 ⩾ 1; 𝐺3 0(𝐻) = 1. The functional calculus then gives 2𝑒𝑖𝑥𝐺3 𝑛 (𝑥) = ∫ ∞ −∞ 𝑦𝑛 cosh 1 2𝜋𝑦 1𝐹1( 1 2 (1 − 𝑖𝑦), 1, 2𝑖𝑥) 𝑑𝑦. 14 In particular: 𝐺3 𝑛 (0) = { 0 if 𝑛 is odd, |𝐸𝑛 | if 𝑛 is even, (32) where 𝐸𝑛 are the Euler numbers. Proceeding as before for inverting the functional calculus, we arrive at a new generating function: 𝑒𝑎𝑥 = Exp(𝑥 arctan 𝑎) √ 1 + 𝑎2 =: 𝑔3(𝑥; 𝑎) =: ∞∑︁ 𝑛=0 𝑆×𝑛 (𝑥) 𝑎𝑛. From the differential equation for 𝑔3: (1 + 𝑎2) 𝜕𝑔3 𝜕𝑎 = (𝑥 − 𝑎) × 𝑔3, we derive the recurrence relation for the 𝑆𝑛: 𝑛𝑆𝑛 (𝑥) − 𝑥 𝑆𝑛−1(𝑥) + (𝑛 − 1)𝑆𝑛−2(𝑥) = 0 for 𝑛 ⩾ 2; 𝑆0(𝑥) = 1, 𝑆1(𝑥) = 𝑥. It is clear that the 𝑆𝑛 exhibit parity. Suppose that they satisfy an orthonormality relation, with weight function 𝑤(𝑥). We note that we must then have ∫ ∞ −∞ 𝑥 2𝑛𝑤(𝑥) 𝑑𝑥 = 𝐺3 2𝑛 (0). From (32), it is thus obvious that these polynomials are indeed orthogonal on the real line, with positive weight function 𝑤(𝑥) = 1 2 sech(𝜋𝑥/2). The 𝑆𝑛 are none other than the classical Meixner–Pollaczek polynomials, which have been recently discussed under the name ‘continuous Hahn polynomials’ [28]. These appeared in an investigation about a discrete version of quantum mechanics; but we find our approach to be simpler and more direct. Again, several integral relations can be derived from the spectral theorem. We write down only:∭ 𝑑𝑎 𝑑𝑎′ 𝑑𝑎′′ sech 𝜋𝑎 2 sech 𝜋𝑎′ 2 sech 𝜋𝑎′′ 2 1𝐹1( 1 2 (1 − 𝑖𝑎), 1; 𝑖(𝑝2 − 𝑞2)) 1𝐹1( 1 2 (1 − 𝑖𝑎′), 1; 𝑖(𝑝′2 − 𝑞′2)) 1𝐹1( 1 2 (1 − 𝑖𝑎′′), 1; 𝑖(𝑝′′2 − 𝑞′′2)) exp [ −𝑖((𝑎′ − 𝑎′′) arctanh(𝑝/𝑞) + (𝑎′′ − 𝑎) arctanh(𝑝′/𝑞′) + (𝑎 − 𝑎′) arctanh(𝑝′′/𝑞′′)) ] = 8 exp [ 𝑖 2 (𝑝2 + 𝑝′2 + 𝑝′′2 − 𝑞2 − 𝑞′2 − 𝑞′′2 + 2𝑞𝑝′ − 2𝑞′𝑝 + 2𝑞′𝑝′′ − 2𝑞′′𝑝′ + 2𝑞′′𝑝 − 2𝑞𝑝′′) ] . 3.4 Steps towards identities for theta-null values As Howe has well said [14], the “rather Hinduish multiplicity-in-one is characteristic of the theory of [the Heisenberg group] and adds greatly to its richness.” Substituting Weil’s form of the representations for Schrödinger’s (see, for instance, [29]) leads directly to the use of the Poisson formula and the theory of theta functions. It was C. G. J. Jacobi who invented the theta functions in the 1820’s. The classical first-order theta function is defined by the Fourier series \ (𝑧, 𝜏) := ∑̀︁ ∈ℤ 𝑒−𝜋` 2𝜏𝑒2𝜋𝑖`𝑧 15 which is normally convergent within the domain { (𝑧, 𝜏) ∈ ℂ2 : Re 𝜏 > 0 }. In the usual reduced characteristics notation, we have \ (𝑧, 𝜏) = \ [ 0 0 ] (𝑧, 𝜏); see, for instance, the monograph by Rauch and Lebowitz [30]. The function 𝜏 ↦→ \ (0, 𝜏) = \ [ 0 0 ] (0, 𝜏) = ∑̀︁ ∈ℤ 𝑒−𝜋` 2𝜏 is holomorphic in the open right half-plane { 𝜏 ∈ ℂ : Re 𝜏 > 0 } and is called the theta-null value (of characteristic [0 0 ] ). Now let Planck’s constant be normalized by setting ℎ = 1. Then ℏ = 1/2𝜋 and the real line ℝ carries Lebesgue measure 𝑑𝑡. For wavefunctions Ψ,Φ ∈ 𝐿2(ℝ), the Wigner function (6a) in position space takes the form 𝑊|Ψ⟩⟨Φ| (𝑞, 𝑝) = ∫ ℝ Ψ(𝑞 + 1 2 𝑡)Φ(𝑞 − 1 2 𝑡) exp[−2𝜋𝑖𝑝𝑡] 𝑑𝑡. The holographic transformation H is defined [31, 31] by the identity H(Ψ,Φ; 𝑥, 𝑦) := ∫ ℝ Ψ(𝑡 + 1 2𝑥)Φ(𝑡 − 1 2𝑥) exp[2𝜋𝑖𝑦𝑡] 𝑑𝑡 (33) for all pairs (𝑥, 𝑦) ∈ ℝ2. If Φ̌ denotes the reflection 𝑡 ↦→ Φ(−𝑡) of Φ, we have the identity H(Ψ,Φ; 𝑥, 𝑦) = 1 2𝑊|Ψ⟩⟨Φ̌| ( 1 2𝑥,− 1 2 𝑦). In signal theory, the holographic transform provides a mixing of coherent waves with amplitudes Ψ and Φ; this relates to the interpretation of Wigner functions as transitions. The symmetry between the position and momentum representations (6a) and (6b) of the Wigner functions translates into a corresponding symmetry of the holographic transforms. It is readily seen from (6), or more directly from (33), that H(Ψ̂, Φ̂; 𝑥, 𝑦) = H(Ψ,Φ;−𝑦, 𝑥) (34) where Ψ̂ = FℝΨ is the Fourier transform of Ψ. In fact, this is a simple special case of a more general identity H(Ψ𝜎,Φ𝜎;𝜎(𝑎, 𝑏)) = H(Ψ,Φ; 𝑎, 𝑏) where 𝜎 is a symplectic transformation of ℝ2 and Ψ𝜎 is the image of Ψ under the corresponding element of the metaplectic representation of the symplectic group Sp(2,ℝ): see [31], for example. From (33) one can verify directly that Fℝ2 ( H(Ψ,Ψ; ·, ·)H(Φ,Φ; ·, ·) ) (𝑥, 𝑦) = ��H(Ψ̂, Φ̂; 𝑥, 𝑦) ��2 which, in view of (34), yields Fℝ2 ( H(Ψ,Ψ; ·, ·)H(Φ,Φ; ·, ·) ) (𝑥, 𝑦) = ��H(Ψ,Φ;−𝑦, 𝑥) ��2. 16 Now an application of the Poisson summation formula yields the general pixel identity∑︁ (`,a)∈ℤ2 H(Ψ,Ψ; `, a)H(Φ,Φ; `, a) = ∑︁ (`,a)∈ℤ2 ��H(Ψ,Φ; `, a) ��2 for all wavefunctions Ψ,Φ ∈ 𝐿2(ℝ). Due to the radial isotropy of the harmonic oscillator wavefunctions and (25a), we get from the last identity the ‘radial pixel identity’ for 𝑚 ⩾ 𝑛 ⩾ 0:∑︁ (`,a)∈ℤ2 𝑙0𝑚 (𝜋(`2 + a2)) 𝑙0𝑛 (𝜋(`2 + a2)) = 𝑛! 𝑚! 𝜋𝑚−𝑛 ∑︁ (`,a)∈ℤ2 (`2 + a2)𝑚−𝑛 [ 𝑙𝑚−𝑛𝑛 (𝜋(`2 + a2)) ]2 , (35) where 𝑙𝛼𝑛 (𝑥) := 𝑒−𝑥/2𝐿𝛼𝑛 (𝑥). As special cases of the identity (35) we may relate the odd powers of 𝜋 to the theta-null value and its derivatives at the point 𝜏 = 1: • Case 𝑚 = 1, 𝑛 = 0: 𝜋 = ∑̀︁ ∈ℤ 𝑒−𝜋` 2 / 4 ∑̀︁ ∈ℤ `2𝑒−𝜋` 2 . (36) To see this, notice that (35) reduces to∑︁ (`,a)∈ℤ2 (1 − 𝜋(`2 + a2))𝑒−𝜋(`2+a2) = ∑︁ (`,a)∈ℤ2 𝜋(`2 + a2)𝑒−𝜋(`2+a2) which becomes ∑̀︁ ∈ℤ 𝑒−𝜋` 2 − 2𝜋 ∑̀︁ ∈ℤ `2𝑒−𝜋` 2 = 2𝜋 ∑̀︁ ∈ℤ `2𝑒−𝜋` 2 , yielding (36). Similar elementary manipulations yield the formulas: • Case 𝑚 = 2, 𝑛 = 1: 𝜋3 = 15 ∑̀︁ ∈ℤ (8𝜋2`4 − 1)𝑒−𝜋`2 / 32 ∑̀︁ ∈ℤ `6𝑒−𝜋` 2 . • Case 𝑚 = 3, 𝑛 = 2: 𝜋5 = 45 ∑̀︁ ∈ℤ (16𝜋4`8 − 140𝜋2`4 + 21)𝑒−𝜋`2 / 64 ∑̀︁ ∈ℤ `10𝑒−𝜋` 2 . • Case 𝑚 = 4, 𝑛 = 3: 𝜋7 = 91 ∑ `∈ℤ(256𝜋6`12 − 15840𝜋4`8 + 166320𝜋2`4 − 25245)𝑒−𝜋`2 1024 ∑ `∈ℤ `14𝑒−𝜋`2 . 17 These special cases have also been proved in [33] and numerically checked by Martin Schmidt. Remark 1. The three cases 3.1–3.3 give all the interesting mathematics that can be extracted using the ‘transfer formula’ and ‘functional calculus’ methods from Moyal theory in ℝ2, because all the nontrivial Hamiltonians are canonically equivalent to one of these [13]. One can go to higher-dimensional spaces, but the spectrum will be in general degenerate, which entails additional complications. An important exception, which warrants further study, is the generic Hamiltonian 𝐻 = 1 2𝑢 𝑡𝐵𝑢, where 𝑢 ∈ ℝ4 and 𝐵 is a 4 × 4 symmetric matrix such that the eigenvalues ±𝛼 ± 𝑖𝛽 of 𝐽𝐵 are all distinct. Then (with ℏ = 2) one can compute: Ξ𝐻 (𝑢; 𝑡) = 2 cosh𝛼𝑡 + cos 𝛽𝑡 exp [ −𝑖 𝛼 −1𝐻1 sinh𝛼𝑡 + 𝛽−1𝐻2 sin 𝛽𝑡 cosh𝛼𝑡 + cos 𝛽𝑡 ] with 𝐻1 + 𝐻2 = 𝐻. The associated functional calculus will give interesting properties for new special functions in two variables. It is no longer true that a twisted function of 𝐻 is an (ordinary) function of 𝐻 alone, as we consistently found in ℝ2. Remark 2. Another useful device is the use of limiting processes. For instance, a Hamiltonian of the form 1 2 (𝑝 2 + 𝑘𝑞2) + 𝑞 is of the harmonic oscillator type [respectively, of the harmonic barrier type] as long as 𝑘 > 0 [respectively, 𝑘 < 0]. Reworking our formulas to include 𝑘 explicitly and taking the limit 𝑘 → 0 should give interesting limit formulas for Laguerre and Kummer functions in terms of Airy functions. Remark 3. The Laguerre polynomials 𝐿𝛼𝑛 can be expressed in terms of the Poisson–Charlier poly- nomials [34]: 𝑐𝑛 (𝑥; 𝑎) := 2𝐹0(−𝑛,−𝑥;−𝑎−1) = 𝑛!(−𝑎)−𝑛𝐿𝑥−𝑛𝑛 (𝑎). (37) Notice that the argument 𝑥 of the Poisson–Charlier polynomial appears as a parameter to the Laguerre polynomial and, vice versa, the parameter 𝑎 of 𝑐𝑛 is the argument of 𝐿𝑥−𝑛𝑛 . The importance of the Poisson–Charlier polynomials stems from their discrete orthogonality relation [34]: ∞∑︁ 𝑥=0 𝑐𝑛 (𝑥; 𝑎) 𝑐𝑚 (𝑥; 𝑎) 𝑎 𝑥 𝑥! = 𝑒𝑎𝑎𝑛𝑛! 𝛿𝑚𝑛. (38) It is also possible to express the Laguerre functions in terms of the generalized Bateman functions 𝑘𝑚𝑛 (see [35]). On using (35) and (37), we obtain a type of trace formula for the integer values of the Poisson– Charlier polynomials:∑︁ (`,a)∈ℤ2 (−1)𝑚+𝑛 (` 2 + a2)𝑚+𝑛 (𝑚 + 𝑛)! 𝑒−𝜋(` 2+a2)𝑐𝑚 (𝑚; 𝜋(`2 + a2)) 𝑐𝑛 (𝑛; 𝜋(`2 + a2)) = ∑︁ (`,a)∈ℤ2 (`2 + a2)𝑚+𝑛 (𝑚 + 𝑛)! 𝑒−𝜋(` 2+a2) [𝑐𝑛 (𝑚; 𝜋(`2 + a2)) ]2 whenever 𝑚 ⩾ 𝑛 ⩾ 0. This can be thought of as a two-dimensional discrete overcompleteness relation corresponding to the one-dimensional discrete orthogonality relation (38). 18 Remark 4. A new insight into the deep links between Moyal theory and harmonic analysis has been revealed in the discovery of the mysterious role played by ordinary (not twisted) convolution in the theory of Wigner functions. By this means, two of us have shown in [36] that Hudson’s theorem [37] fails for mixed states (for a thorough exposition of Hudson’s theorem, see [38]; also, [39] is relevant in this context). 4 Moyal representation for spin and special functions We now show how the Moyal approach to spin quantization yields more information about special functions. The appropriate phase space is the sphere 𝕊2, which is a coadjoint orbit for SU(2). One chooses a unitary irreducible representation 𝜋 𝑗 of the compact Lie group SU(2) on ℂ2 𝑗+1, for some positive half-integer 𝑗 . The role of the operators Π(𝑢) is played by the SU(2) Stratonovich–Weyl operator kernel Δ 𝑗 (𝒏) which associates to each 𝒏 on 𝕊2 a (2 𝑗 + 1) × (2 𝑗 + 1) selfadjoint matrix, satisfying appropriate conditions. If 𝑅 ∈ SU(2) and if 𝑅 ∈ SO(3) denotes its image under the covering homomorphism SU(2) → SO(3), then SU(2)-covariance of Δ 𝑗 means that 𝜋 𝑗 (𝑅) Δ 𝑗 (𝒏) 𝜋 𝑗 (𝑅)−1 = Δ 𝑗 (𝑅𝒏) (39) for 𝑅 ∈ SU(2), 𝒏 ∈ 𝕊2. The matrix elements of 𝜋 𝑗 (𝑅) in the standard presentation are customarily written D 𝑗 𝑚𝑛 (𝑅). Let 𝑍 𝑗𝑠𝑟 (𝒏) denote the (𝑟, 𝑠)-element of the matrix Δ 𝑗 (𝒏), with 𝑟, 𝑠 = − 𝑗 , . . . , 𝑗 − 1, 𝑗 . It is shown in [7] that these matrix elements are: 𝑍 𝑗 𝑠𝑟 (𝒏) := √ 4𝜋 2 𝑗 + 1 2 𝑗∑︁ 𝑙=0 √ 2𝑙 + 1 ⟨ 𝑗 𝑙 𝑟 (𝑠 − 𝑟) | 𝑗 𝑠⟩𝑌𝑙,𝑠−𝑟 (𝒏), (40) where𝑌𝑙𝑚 denotes the usual spherical harmonics and ⟨ 𝑗 𝑙 𝑟 (𝑠−𝑟) | 𝑗 𝑠⟩ is a Clebsch–Gordan coefficient. (In [7], the indices 𝑟, 𝑠 are erroneously switched.) Using the well-known formula [40] for transforming spherical harmonics: 𝑌𝑙𝑚 (𝑅𝒏) = 𝑙∑︁ 𝑛=−𝑙 D𝑙∗ 𝑚𝑛 (𝑅)𝑌𝑙𝑛 (𝒏), (41) from (40) and (41) one derives [7]: 𝑍 𝑗 𝑠𝑟 (𝑅𝒏) = 𝑗∑︁ 𝑝,𝑞=− 𝑗 D 𝑗 𝑟 𝑝 (𝑅)D 𝑗∗ 𝑠𝑞 (𝑅) 𝑍 𝑗𝑞𝑝 (𝒏), (42) which is just (39) in explicit form. A property directly analogous to (2) is valid for the SU(2) case: trΔ 𝑗 (𝒏) = 1, tr ( Δ 𝑗 (𝒎) Δ 𝑗 (𝒏) ) = 4𝜋 2 𝑗 + 1 𝐾 𝑗 (𝒎, 𝒏), tr ( Δ 𝑗 (𝒎) Δ 𝑗 (𝒏) Δ 𝑗 (𝒌) ) =: 𝐿 𝑗 (𝒎, 𝒏, 𝒌), (43) 19 where 𝐾 𝑗 (𝒎, 𝒏) = 2 𝑗∑︁ 𝑙=0 𝑙∑︁ 𝑠=−𝑙 𝑌𝑙𝑠 (𝒎)𝑌 ∗ 𝑙𝑠 (𝒏) is the reproducing kernel of the space H2 𝑗 of spherical harmonics of degree ⩽ 2 𝑗 . Now one can define the Stratonovich–Weyl correspondence between ‘symbols’ – functions on the phase space 𝕊2 belonging to the function space H2 𝑗 – and operators on ℂ2 𝑗+1 by the formulas 𝑊 𝑓 = 2 𝑗 + 1 4𝜋 ∫ 𝕊2 𝑓 (𝒏) Δ 𝑗 (𝒏) 𝑑𝒏; 𝑊𝐴 (𝒏) = tr(𝐴Δ 𝑗 (𝒏)) (with 𝑑𝒏 = sin \ 𝑑\ 𝑑𝜙 if 𝒏 = (\, 𝜙) in spherical coordinates). Once more, we have 𝑊𝑊𝐴 = 𝐴 and 𝑊𝑊 𝑓 = 𝑓 , and the constant function 1 corresponds to the identity operator. From (43), one verifies that the product of two operators 𝑊 𝑓 , 𝑊𝑔 is equal to 𝑊 𝑓×𝑔 where the SU(2) twisted product in H2 𝑗 is given by 𝑓 × 𝑔(𝒏) = (2 𝑗 + 1 4𝜋 )2 ∫ 𝕊2 ∫ 𝕊2 𝑓 (𝒎) 𝑔(𝒌) 𝐿 𝑗 (𝒏,𝒎, 𝒌) 𝑑𝒎 𝑑𝒌 with the property that ∫ 𝕊2 𝑓 × 𝑔(𝒏) 𝑑𝒏 = ∫ 𝕊2 𝑓 (𝒏) 𝑔(𝒏) 𝑑𝒏. With these ingredients, one can carry out the quantization scheme outlined in Section 2 for the SU(2) case, using the formal analogues of equations (7) and (9): this is developed more fully in [7]. The special functions 𝑍 𝑗𝑟𝑠 (𝒏) are the primary computational tools, since they have the orthogonality and product properties: 2 𝑗 + 1 4𝜋 ∫ 𝕊2 𝑍 𝑗 𝑟𝑠 (𝒏) 𝑍 𝑗𝑡𝑢 (𝒏) 𝑑𝒏 = 𝛿𝑟𝑢 𝛿𝑠𝑡 ; 𝑍 𝑗 𝑟𝑠 × 𝑍 𝑗𝑡𝑢 = 𝛿𝑠𝑡 𝑍 𝑗 𝑟𝑢; (44) as is evident from the Stratonovich–Weyl correspondence, since 𝑍 𝑗𝑟𝑠 = 𝑊| 𝑗 ,𝑟⟩⟨ 𝑗 ,𝑠 | where | 𝑗 , 𝑟⟩ denotes a spin eigenvector. They constitute an interesting subject in themselves. From (40), we get in particular the spin eigenstates: 𝑍 𝑗 𝑚𝑚 (𝒏) = 2 𝑗∑︁ 𝑙=0 2𝑙 + 1 2 𝑗 + 1 ⟨ 𝑗 𝑙 𝑚0 | 𝑗𝑚⟩ 𝑃𝑙 (cos \), (45) where the 𝑃𝑙 are the Legendre polynomials. If𝑊 𝑗 𝑧 := ∑ 𝑗 𝑚=− 𝑗 𝑚𝑍 𝑗 𝑚𝑚 is the symbol associated to the 𝐽𝑧 spin operator, then 𝑊 𝑗 𝑧 (𝒏) = 𝑗∑︁ 𝑚=− 𝑗 2 𝑗∑︁ 𝑙=0 𝑚 2𝑙 + 1 2 𝑗 + 1 ⟨ 𝑗 𝑙 𝑚0 | 𝑗𝑚⟩ 𝑃𝑙 (cos \) = √︁ 𝑗 ( 𝑗 + 1) cos \. Let us proceed as indicated at the end of Section 2. The Fourier kernel 𝐸 defined by 𝐸 (𝑔; 𝑗 , 𝒏) := tr ( 𝜋 𝑗 (𝑔) Δ 𝑗 (𝒏) ) for 𝑔 ∈ SU(2), 𝒏 ∈ 𝕊2, is of central importance for the harmonic analysis of SU(2). Note that 𝐸 (𝑔; 𝑗 , 𝒏) is the Stratonovich–Weyl symbol of 𝜋 𝑗 (𝑔). Explicitly, 𝐸 (𝑔; 𝑗 , 𝒏) = 𝑗∑︁ 𝑟,𝑠=− 𝑗 𝑍 𝑗 𝑠𝑟 (𝒏)D 𝑗 𝑠𝑟 (𝑔). (46) 20 From the orthogonality properties (44) of the 𝑍 𝑗𝑟𝑠 functions, and the orthogonality properties of the matrix elements D 𝑗 𝑚𝑛 with respect to the normalized Haar measure 𝑑𝑔 on SU(2), from (46) we derive at once: D 𝑗 𝑚𝑛 (𝑔) = 2 𝑗 + 1 4𝜋 ∫ 𝕊2 𝐸 (𝑔; 𝑗 , 𝒏) 𝑍 𝑗𝑛𝑚 (𝒏) 𝑑𝒏, 𝑍 𝑗 𝑟𝑠 (𝒏) = (2 𝑗 + 1) ∫ SU(2) 𝐸 (𝑔; 𝑗 , 𝒏)D 𝑗 𝑠𝑟 (𝑔) 𝑑𝑔. (47) We now show, by a specific example, how this machinery may be used to generate identities for special functions associated to the representation theory of SU(2). The representative functions D 𝑗 𝑚𝑛 are most often written explicity in terms of the Eulerian angle parametrization of SU(2), although some other parametrizations are occasionally employed [40]. However, an SU(2)-element is most naturally written in the angle-axis parametrization: 𝑔 = 𝑔(𝜓,𝒎) := exp(− 𝑖 2𝜓𝒎 · 𝝈) with |𝒎 |2 = 1. We thus require a formula which expresses D 𝑗 𝑚𝑛 (𝑔(𝜓,𝒎)) directly in terms of 𝜓 and 𝒎. Let us write 𝒏0 for the north-pointing vector in 𝕊2, so that the one-parameter subgroup of SU(2) generated by − 𝑖 2𝜎𝑧 is 𝑡 ↦→ 𝑔(𝑡, 𝒏0); applying the homomorphism and the Stratonovich–Weyl correspondence, this yields the evolution equation: 𝑖 𝜕𝐸 𝜕𝑡 (𝑔(𝑡, 𝒏0); 𝑗 , 𝒏) = 𝑊 𝑗 𝑧 × 𝐸 (𝑔(𝑡, 𝒏0); 𝑗 , 𝒏) whose solution is clearly 𝐸 (𝑔(𝑡, 𝒏0); 𝑗 , 𝒏) = 𝑗∑︁ 𝑘=− 𝑗 𝑒−𝑖𝑘𝑡 𝑍 𝑗 𝑘 𝑘 (𝒏). (48) A general expression for the Fourier kernel may now be found from its covariance properties. Let us, by an abus de notation, write 𝑍 𝑗 𝑘 𝑘 (cos \) instead of 𝑍 𝑗 𝑘 𝑘 (𝒏) for the spin states, in view of their cylindrical symmetry (45). Then since 𝑔(𝜓,𝒎) is conjugate within SU(2) to 𝑔(𝜓, 𝒏0) by an element whose associated rotation takes 𝒏0 to 𝒎, we obtain from (42), (46) and (48): 𝐸 (𝑔(𝜓,𝒎); 𝑗 , 𝒏) = 𝑗∑︁ 𝑘=− 𝑗 𝑒−𝑖𝑘𝜓𝑍 𝑗 𝑘 𝑘 (𝒎 · 𝒏). (49) From (47) and (49) we now derive D 𝑗 𝑚𝑛 (𝑔(𝜓,𝒎)) = 𝑗∑︁ 𝑘=− 𝑗 𝑒−𝑖𝑘𝜓 2 𝑗 + 1 4𝜋 ∫ 𝕊2 𝑍 𝑗 𝑘 𝑘 (𝒎 · 𝒏) 𝑍 𝑗𝑛𝑚 (𝒏) 𝑑𝒏 = 2 𝑗∑︁ 𝑙,𝑙′=0 𝑗∑︁ 𝑘=− 𝑗 (2𝑙 + 1) √ 2𝑙′ + 1 (2 𝑗 + 1) √ 4𝜋 𝑒−𝑖𝑘𝜓 ⟨ 𝑗 𝑙 𝑘0 | 𝑗 𝑘⟩ ⟨ 𝑗 𝑙′𝑚(𝑛 − 𝑚) | 𝑗𝑛⟩∫ 𝕊2 𝑃𝑙 (𝒎 · 𝒏)𝑌𝑙′,𝑛−𝑚 (𝒏) 𝑑𝒏 = 2 𝑗∑︁ 𝑙=0 √ 2𝑙 + 1 2 𝑗 + 1 ⟨ 𝑗 𝑙 𝑚(𝑛 − 𝑚) | 𝑗𝑛⟩ 𝑗∑︁ 𝑘=− 𝑗 ⟨ 𝑗 𝑙 𝑘0 | 𝑗 𝑘⟩ 𝑒−𝑖𝑘𝜓 √ 4𝜋𝑌𝑙′,𝑛−𝑚 (𝒎). (50) 21 (The integration relies on the known expansion 𝑃𝑙 (𝒎 · 𝒏) = 𝑙∑︁ 𝑠=−𝑙 𝑌𝑙𝑠 (𝒎)𝑌 𝑙𝑠 (𝒏) and the orthonormality of the spherical harmonics over 𝕊2.) This explicit development of the representative functions in terms of Clebsch–Gordan coefficients and spherical harmonics only, in the angle-axis parametrization, appears to be new. Writing 𝒎 = (\, 𝜙) in spherical polar coordinates, we can, for instance, write down the following special cases of (50) for spin one: D1 00(𝑔(𝜓,𝒎)) = cos𝜓 + (1 − cos𝜓) cos2 \, D1 11(𝑔(𝜓,𝒎)) = 1 2 (1 + cos𝜓) − 𝑖 sin𝜓 cos \ − 1 2 (1 − cos𝜓) cos2 \, D1 01(𝑔(𝜓,𝒎)) = − 1 √ 2 𝑒𝑖𝜙 sin \ [ 𝑖 sin𝜓 + (1 − cos𝜓) cos \ ] , D1 −1,1(𝑔(𝜓,𝒎)) = −1 2 (1 − cos𝜓)𝑒2𝑖𝜙 sin2 \. 5 Conclusion The general scheme outlined in this paper can in principle be applied to other Lie groups having no evident connection with Quantum Mechanics; the point of contact is the theory of Kirillov, Kostant and Souriau which allows one to define a natural symplectic structure on the coadjoint orbits of the group: see [41, 42], for example. This is merely Classical Mechanics, in principle; but whenever a ‘Stratonovich–Weyl correspondence’ can be found, a bridge is built to operator theory on the representation spaces and standard noncommutative harmonic analysis, a twisted product appears, and a systematic comparison of the twisted product formalism with the conventional operator theory yields many identities between special functions. Some of these identities will be old, and are merely derived in a fresh and illuminating manner; and several will be new. Thus the apparatus of Moyal Quantum Mechanics provides a means of demonstrating unexpected connections among many parts of mathematics. Acknowledgements Two of the authors (JCV and JMG-B) are grateful for support from the Deutsche Akademische Austauschdienst (DAAD), which gave them the opportunity to visit the University of Siegen, where part of this work was done. WS thanks the University of Costa Rica for its hospitality. 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