Fuzzy spheres from inequivalent coherent states quantizations

Fuzzy spheres from inequivalent coherent states quantizations J.-P. Gazeau, E. Huguet, M. Lachièze-Rey, J. Renaud To cite this version: J.-P. Gazeau, E. Huguet, M. Lachièze-Rey, J. Renaud. Fuzzy spheres from inequivalent coherent states quantizations. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2007, 40, pp.10225-10249. <10.1088/1751-8113/40/33/018>. <hal-00105288> HAL Id: hal-00105288 https://hal.archives-ouvertes.fr/hal-00105288 Submitted on 10 Oct 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Fuzzy spheres from inequivalent coherent states quantizations Jean Pierre Gazeau, Eric Huguet, Marc Lachièze-Rey and Jacques Renaud Laboratoire Astroparticules et Cosmologie‡ Boite 7020, Université Paris 7-Denis Diderot F-75251 Paris Cedex 05, France gazeau@ccr.jussieu.fr, huguet@ccr.jussieu.fr, marclr@cea.fr, renaud@ccr.jussieu.fr ‡ UMR 7164 (CNRS,Université Paris 7, CEA, Observatoire de Paris) July 4, 2006 ccsd-00105288, version 1 - 10 Oct 2006 Abstract We present a new procedure which allows a coherent state (CS) quantization of any set with a measure. It is manifest through the replacement of classical observables by CS quantum observables, which acts on a Hilbert space of prescribed dimension N . The algebra of CS quantum observables has the finite dimension N 2 . The application to the 2-sphere provides a family of inequivalent CS quantizations, based on the spin spherical harmonics (the CS quantization from usual spherical harmonics appears to give a trivial issue for the cartesian coordinates). We compare these CS quantizations to the usual (Madore) construction of the fuzzy sphere. The difference allows us to consider our procedures as the constructions of new type of fuzzy spheres. The very general character of our method suggests applications to construct fuzzy versions of a variety of sets. 1 Some ideas on quantization A classical description of a set of data, say X, is usually carried out by considering sets of real or complex functions on X. Depending on the context (data handling, signal analysis, mechanics. . . ) the set X will be equipped with a definite structure (topological space, measure space, symplectic manifold. . . ) and the set of functions on X which will be considered as classical observables must be restricted with regard to the structure on X; for instance, signals should be square integrable with respect to the measure assigned to set X. How to provide instead a “quantum description” of the same set X? As a first characteristic, the latter replaces - this is a definition - the classical observables by quantum observables, which do not commute in general. As usual, these quantum observables will be realized as operators acting on some Hilbert space H, whose projective version will be considered as the set of quantum states. This Hilbert space will be constructed as a subset in the set of functions on X. The advantage of the coherent states (CS) quantization procedure, in a standard sense [1, 2, 3] as in recent generalizations [4] and applications [5] is that it requires a minimal significant structure on X, namely the only existence of a measure µ(dx), together with a σ-algebra of measurable subsets. As a measure space, X will be given the name of an observation set in the present context, and the existence of a measure provides us with a statistical reading of the set of measurable real or complex valued functions on X: computing for instance average values on subsets with bounded measure. The quantum states will correspond to measurable and square integrable functions on the set X, but not all square integrable functions are eligible as quantum states. The construction of H is equivalent to the choice of a class of eligible quantum states, together with a technical condition of continuity. This provides a correspondence between classical and quantum observables by defining a generalization of the so-called coherent states. Although the procedure appears mathematically as a quantization, it may also be considered as a change of point of view for looking at the system, not necessarily a path to quantum physics. In this 1 sense, it could be called a discretization or a regularization [6]. It shows a certain resemblance with standard procedures pertaining to signal processing, for instance those involving wavelets, which are coherent states for the affine group transforming the half-plane time-scale into itself [7, 8]. In many respects, the choice of a quantization appears here as the choice of a resolution to look at the system. As is well known, some aspects of (ordinary) quantum mechanics may be seen as a non commutative version of the geometry of the phase space, where position and momentum operators do not commute. It appears as a general fact that the quantization of a “set of data” makes a fuzzy (non commutative) geometry to emerge [9]. We will show explicitly how the CS quantization of the ordinary sphere leads to its fuzzy geometry. In Section 2 we present a construction of coherent states which is very general and encompasses most of the known constructions, and we derive from the existence of a CS family what we call CS quantization. The latter extends to various situations the well-known Klauder-Berezin quantization. The formalism is illustrated with the standard Glauber-Klauder-Sudarshan coherent states and the related canonical quantization of the classical phase space of the motion on the real line. In Section 3, we apply the formalism to the sphere S 2 by using orthonormal families of spin spherical harmonics (σ Yjm )−j6m6j [10, 11, 12]. For a given σ such that 2σ ∈ Z and j such that 2|σ| 6 2j ∈ N there corresponds a continuous family of coherent states and the subsequent 2j + 1-dimensional quantization of the 2-sphere. For a given j, we thus get 2j + 1 inequivalent quantizations, corresponding to the possible values of σ. Note that the classical Gilmore-Perelomov-Radcliffe case [13, 14, 15] correspond to the particular value σ = j. On the other hand, the case σ = 0 is proved to be singular in the sense that it leads to a null quantization of the cartesian coordinates of the 2-sphere. The section 4 establishes the link between the CS quantization approach to the 2-sphere and the Madore construction [9, 16] of the fuzzy sphere. We examine there the question of equivalence between the two procedures. Note that a construction of the fuzzy sphere based on Perelomov coherent states has already been carried out by Grosse and Pres̆najder [17]. They proceed to a covariant symbol calculus à la Berezin with its corresponding ⋆-product. However, their approach is different of ours. The appendices give an exhaustive set of formulas, particularly concerning the spin spherical harmonics, needed for a complete description of our CS approach to the 2-sphere. 2 Coherent states 2.1 The construction The (classical) system to be quantized is considered as a set of data, X = {x ∈ X}, assumed to be equipped with a measure µ defined on a σ-field B. We consider the Hilbert spaces L2K (X, µ) (K = R or C) of real or complex functions, with the usual Hermitian inner product hf | gi. The quantization is defined by the choice of a closed subspace H of L2K (X, dµ). The only requirements on H, in addition to be an Hilbert space, amount to the following technical conditions: • For all ψ ∈ H and all x, ψ(x) is well defined (this is of course the case whenever X is a topological space and the elements of H are continuous functions) • the linear map (“evaluation map”) δx : H → K (1) ψ 7→ ψ(x) is continuous with respect to the topology of H, for almost all x. The last condition is realized as soon as the space H is finite dimensional since all the linear forms are continuous in this case. We see below that some other examples can be found. As a consequence, using the Riesz theorem, there exists, for almost all x, an unique element px ∈ H (a function) such that hpx | ψi = ψ(x). (2) We define the coherent states as the normalized vectors corresponding to px written in Dirac notation: | xi ≡ | px i 1 [N (x)] 2 where N (x) ≡ hpx | px i. (3) One can see at once that, for any ψ ∈ H: 1 ψ(x) = [N (x)] 2 hx | ψi. 2 (4) As a consequence, one obtains the following resolution of the identity of H which is at the basis of the whole construction: Z IdH = | x ih x | N (x) µ(dx). (5) Note that φ(x) = Z p X N (x) N (x′ ) hx|x′ i φ(x′ ) µ(dx′ ), ∀φ ∈ H. Hence, H is a reproducing Hilbert space with kernel p K(x, x′ ) = N (x) N (x′ ) hx |x′ i, (6) (7) and the latter assumes finite diagonal values (a.e.), K(x, x) = N (x), by construction. Note that this construction yields an embedding of X into H and one could interpret | x i as a state localized at x once a notion of localization has been properly defined on X. In view of (5) the set {| x i} is called a frame for H. This frame is said to be overcomplete when the the vectors {| x i} are not linearly independent [18, 19]. We define a classical observable over X in a loose way as a function f : X 7→ K (R or C). As a matter of fact we will not retain a priori the usual requirements on f like to be real valued and smooth with respect to some topology defined on X. To any such function f , we associate the quantum observable over H through the map: Z f 7→ Af ≡ N (x) µ(dx) f (x) | x ih x | . (8) X The operator corresponding to a real function is Hermitian by construction. Hereafter, we will also use the notation f˜ for Af . The existence of the continuous frame {| x i} enables us to carry out a symbolic calculus à la BerezinLieb [2, 20]. To each linear, self-adjoint operator (observable) O acting on H, one associates the lower (or covariant) symbol Ǒ(x) ≡ h x | O | x i, (9) b such that and the upper (or contravariant) symbol (not necessarily unique) O Z b O = N (x) µ(dx) O(x) | x ih x | . (10) X Note that f is an upper symbol of Af . The technical conditions and the definition of coherent states can be easily expressed when we have a Hilbertian basis of H. Let (φn )n∈I such a basis, the technical condition is equivalent to X |φn (x)|2 < ∞ a.e. (11) n The coherent state is then defined by |xi = 1 (N (x)) 1 2 X n φ∗n (x) φn with N (x) = X n |φn (x)|2 . To a certain extent, the quantization scheme exposed here consists in adopting a certain point of view in dealing with X, determined by the choice of the space H. This choice specifies the admissible quantum states and the correspondence “classical observables versus quantum observables” follows. 2.2 The standard coherent states Let us illustrate the above construction for the dynamics of a particle moving on the real line. This leads to the well-known Klauder-Glauber-Sudarshan coherent states [21] and the subsequent so-called canonical quantization (with a slight difference of notation). The construction can be easily extended to the dynamics of the particle in a flat higher dimensional spacetime. The observation set X is the classical phase space R2 ≃ C = {z = √12 (q + ip)} (in complex notations) of a particle with one degree of freedom. The symplectic form identifies with 2i dz ∧ dz̄ ≡ d2 z, the Lebesgue measure of the plane. Here we adopt the Gaussian measure on X, µ(dz) = 1 π 2 e−|z| d2 z. 3 The quantization of X is hence achieved by a choice of polarization (in the language of geometric quantization): the selection, in L2 (X, dµ), of the Hilbert subspace H defined as the so-called FockBargmann space of all antiholomorphic entire functions that are square integrable with respect to the Gaussian measure. n The Hilbertian basis is given by the functions φn (z) ≡ √z̄ n! , the normalized powers of the conjugate 2 P 2 = e|z| , the coherent states read of the complex variable z. Thus, since n |z| n! |zi = e− |z|2 2 X zn √ |ni, n! n (12) where |ni stands for ϕn , and one easily checks the normalization and unity resolution: Z 1 hz |zi = 1, |zihz| d2 z = IdH . π C (13) ′ Note that the reproducing kernel is simply given by K(z, z ′ ) = ez z̄ . Quantum operators acting on H are yielded by using (8). We thus have for the most basic one, Z X√ 1 n + 1 |nihn + 1|, (14) a ≡ Az = z |zihz| d2 z = π C n √ which appears as the lowering operator, a|ni = n |n − 1i. Its adjoint a† is obtained by replacing z by z̄ † in (14), and we get the factorization N = a a for the number operator, together with the commutation rule [a, a† ] = IdH . Also note that a† and a realize on H as multiplication operator and derivation operator respectively, a† f (z) = zf (z), af = df /dz. From q = √12 (z + z̄) et p = √12i (z − z̄), one easily infers by linearity that q and p are upper symbols for √12 (a + a† ) ≡ Q and √12i (a − a† ) ≡ P respectively. In consequence, the (essentially) self-adjoint operators Q and P obey the canonical commutation rule [Q, P ] = iIdH , and for this reason fully deserve the name of position and momentum operators of the usual (Galilean) quantum mechanics, together with all localization properties specific to the latter. 3 3.1 Quantizations of the 2-sphere The 2-sphere We now apply our method to the quantization of the observation set X = S 2 , the unit 2-sphere. This is not to be confused with the quantization of the phase space for the motion on the two-sphere (i.e.quantum mechanics on the two-sphere, see for instance [22], [23], [24]). A point of X is denoted by its spherical coordinates, x = (θ, φ). Through the usual embedding in R3 , we may see x as a point (xi ) ∈ R3 obeying P3 i 2 (x ) = 1. We adopt on S 2 the normalized measure µ(dx) = sin θ dθ dφ/4π, proportional to the i=1 SO(3)-invariant measure, which is also the surface element. We know that µ is a symplectic form, with the canonical coordinates q = φ, p = − cos θ. This allows to see S 2 itself as the phase space for the theory of (classical) angular momentum. In this spirit, we will be able to interpret our procedure as the construction of families of spin coherent states including the Gilmore-Perelomov-Radcliffe (hereafter, GPR) ones [15]. Also, our construction will take advantage of the group action of SO(3) on S 2 embedded in R3 . This three-dimensional group acts as isometries in R3 , as rotations in S 2 . However, we emphasize again that our quantization procedure is based on the only existence of a measure, and may be used in the absence of metric or symplectic structure. 3.2 3.2.1 The CS quantization of the 2-sphere The Hilbert space and the coherent states At the basis of the CS quantization procedure is the choice of a finite dimensional Hilbert space, which is a subspace of L2 (S 2 ), and which carries a UIR of the group SU(2). We write its dimension 2j + 1, with j integer or semi-integer. Although it could have appeared natural to choose this space as V j , the linear span of ordinary spherical harmonics Yjm , this choice would not allow to consider half-integer values of j. Moreover, it happens that the quantization so obtained gives trivial results for the cartesian coordinates. Namely, the quantum counterparts of the cartesian coordinates (or, equivalently, the spherical harmonics Y1m ) are identically zero. Thus we are led to define H on a general setting as the linear span of spin spherical harmonics (hereafter SSH’s). 4 3.2.2 The spin spherical harmonics We define H = Hσj as the vector space spanned by the spin spherical harmonics σ Yjµ ∈ L2 (S 2 ), where −j ≤ σ, µ ≤ j, and σ is fixed in this range. Note that σ and j are both integer or semi-integer. The spin spherical harmonics (SSH’s) were first introduced in [10] (see also [12] and [11] for their main properties). In view of their importance in the context of the present work, they are comprehensively described in Appendix A. The special case σ = 0 corresponds to the ordinary spherical harmonics 0 Yjm = Yjm . A CS quantization is defined after a choice of values for j and σ, that we consider as fixed in the sequel. With the usual inner product of L2 (S 2 ), the SSH’s provide an ON basis (σ Yjµ )µ=−j...j of Hσj (hereafter the SSH basis). The Hilbert space Hσj carries the 2j + 1-dimensional UIR of SU(2) (see Appendix A). The generators of SU(2) in this representation can be taken as those corresponding to the three rotations around the orthogonal axes of x1 , x2 , x3 . They are called the “spin” angular momentum operators (SAMOs, to be distinguished from the usual angular momentum operators Ji ), and will be written as Λσj a . Hereafter, the index a = 1, 2, 3 will refer to the three spatial directions. We have Λ0j a = Ja , the usual angular σj σj momentum operators. As usual, we define Λσj ǫ = Λ1 + ǫ i Λ2 , ǫ = ±1. All these generators obey the usual commutation relations of the group SU(2). They act on the ON basis as σj Λσj 3 σ Yjµ = µ σ Yjµ , Λǫ σ Yjµ = aǫ (j, µ) σ Yjµ+ǫ , (15) where the aǫ (j, µ), given in (71,72), are the same as for the usual angular momentum operators Ja . The SSH basis allows to identify Hσj with C2j+1 : σ Yjµ ❀ | µ i ֒→ (0, . . . , 0, 1, 0, . . . , 0)t with µ = −j, −j + 1, . . . , j , (16) where the 1 is at position µ and the superscript t denotes the transpose. By construction we have the Hilbertian orthonormality relations: Z ∗ hµ | ν i ≡ (x) σ Yjν (x) = δµν . (17) µ(dx) σ Yjµ X The CS construction presented in Sect. (2.1) leads to the following class of coherent states | x i =| θ, φ i = p with N (x) = j X 1 N (x) µ=−j j X µ=−j ∗ σ Yjµ (x) |σ Yjµ (x) |2 = | µ i; | x i ∈ H, (18) 2j + 1 . 4π For σ = ±j, they reduce to the spin coherent states [13, 14, 15]. 3.2.3 Operators We call Oσj ≡ End(Hσj ) the space of linear operators (endomorphisms) acting on Hσj . This is a complex vector space of dimension (2j + 1)2 and an algebra for the natural composition of endomorphisms. The SSH basis allows to write a linear endomorphism of Hσj (i.e., an element of Oσj ) in a matrix form. This provides the algebra isomorphism Oσj ❀ Mat2j+1 , the algebra of complex matrices of order 2j + 1, equipped with the matrix product. The projector | x ih x | is a particular linear endomorphism of Hσj , i.e., an element of Oσj . Being Hermitian by construction, it may be seen as an Hermitian matrix of order 2j + 1, i.e., an element of Herm2j+1 ⊂ Mat2j+1 . Note that Herm2j+1 and Mat2j+1 have respective (complex) dimensions (j + 1) (2j + 1) and (2j + 1)2 . We have resolution of identity and normalization by construction: Z µ(dx) N (x) | x ih x | = Id, h x | x i = 1. S2 5 3.2.4 Observables According to the prescription (8), the CS quantization associates to the classical observable f : S 2 7→ C the quantum observable Z f˜ ≡ Af = µ(dx) f (x) N (x) | x ih x | = j X µ,ν=−j Z µ(dx) f (x) [σ Yjµ (x)]∗ σ Yjν (x) | µ ih ν | . (19) This operator is an element of Oσj ∼ End(Hσj ) ❀ Mat(2j+1) . Of course its existence is submitted to the convergence of (19) in the weak sense as an operator integral. The expression above gives directly its expression as a matrix in the SSH basis, with matrix elements f˜µν : f˜ = j X µ,ν=−j f˜µν | µ ih ν | with f˜µν = Z ∗ µ(dx) f (x) σ Yjµ (x) σ Yjν (x). (20) When f is real-valued, the corresponding matrix belongs to Herm(2j+1) . Also, we have ff∗ = (f˜)† (matrix transconjugate), where we have used the same notation for the operator and the associated matrix. 3.2.5 The usual spherical harmonics as classical observables An usual spherical harmonics Yℓm is a particular classical observable and, as such, may be quantized. The quantization procedure associates to Yℓm the operator Yg ℓm . The details of the computation are given in Appendix A and the result is given in Subsection 7.13, Eq. (91). We hence obtain the matrix elements of Yg ℓm in the SSH basis: r    h i (2ℓ + 1) j j ℓ j j ℓ σ−µ Yg = (−1) (2j + 1) , (21) ℓm −σ σ 0 −µ ν m 4π µν in terms of the 3j-symbols. This generalizes the formula (2.7) of [25]. This expression is a real quantity. Any function f on the 2-sphere with reasonable properties (continuity, integrability...) may be expanded in spherical harmonics as ∞ ℓ X X fℓm Yℓm , (22) f= ℓ=0 m=−ℓ from which results the corresponding expansion of f˜. However, the 3j-symbols are non zero only when a triangular inequality is satisfied. This implies that the expansion is cut at a finite value, giving f˜ = 2j ℓ X X ℓ=0 m=−ℓ g fℓm Y ℓm . (23) σj This relation means that the (2j + 1)2 observables (Yg ℓm )ℓ62j, −ℓ6m6ℓ provide a second (SH) basis of O . σj ˜ The fℓm are the components of the matrix f ∈ O in this basis. 3.3 3.3.1 The spin angular momentum operators Action on functions The Hilbert space Hσj carries a unitary irreducible representation of the group SU(2) with generators σj Λσj a (the SAMOs), which belong to O . Their action is given in (70-71-72). Explicit calculations shown in the appendix (see 98) give the crucial relations: fa = K Λσj x a , with K ≡ σ . j(j + 1) (24) We see here the peculiarity of the ordinary spherical harmonics (σ = 0) as an ON basis for the quantization procedure: they would lead to a trivial result for the quantized version of the cartesian coordinates! On the other hand, the quantization based on the GPR spin coherent states yields the maximal value: K = 1/(j + 1). Hereafter we assume σ 6= 0. 6 3.3.2 Action on operators The SU(2) action on Hσj induces the following canonical (infinitesimal) action on Oσj = End(Hσj ): σj σj Lσj a : 7→ La A ≡ [Λa , A] (the commutator) (25) here expressed through the generators. g ^ We prove in Appendix A, (104), that Lσj a Yℓm = Ja Yℓm , from which it results: σj 2 g g g g Lσj 3 Yℓm = m Yℓm and (L ) Yℓm = ℓ (ℓ + 1) Yℓm . σj g We recall that the (Yg ℓm )ℓ62j form a basis of O . The relations above make Yℓm appear as the unique σj σj (up to a constant) element of O that is common eigenvector to L3 and (Lσj )2 , with eigenvalues m and ℓ (ℓ + 1) respectively. This implies by linearity that for all f such that fe makes sense σj 2 e g 2 e g Lσj a f = Ja f and (L ) f = J f . 4 Link with the fuzzy sphere 4.1 The construction of the fuzzy sphere Let us first recall an usual construction of the fuzzy sphere (see for instance [9] p.148), that we slightly modify to make the correspondence with the CS quantization. It starts from the decomposition of any smooth function f ∈ C ∞ (S 2 ) in spherical harmonics, f= ∞ X ℓ=0 ℓ X fℓm Yℓm . (26) m=−ℓ Let us denote by V ℓ the (2ℓ + 1)-dimensional vector space generated by the Yℓm , at fixed ℓ. Through the embedding of S 2 in R3 , any function in S 2 can be seen as the restriction of a function on R3 (that we write with the same notation), and under some mild conditions such functions are generated by the homogeneous polynomials in R3 . This allows us to express (26) in a polynomial form in R3 : X X (27) f(i1 i2 ...iℓ ) xi1 xi2 ...xiℓ + ..., f(i) xi + ... + f (x) = f(0) + (i1 ) (i1 i2 ...iℓ ) where each sum subtends a V ℓ and involves all symmetric combinations of the ik indices, each varying from 1 to 3. This gives, for each fixed value of ℓ, 2ℓ + 1 coefficients f(i1 i2 ...iℓ ) (ℓ fixed), which are those of a symmetric traceless 3 × 3 × .... × 3 (ℓ times) tensor. The fuzzy sphere with 2j + 1 cells is usually written Sfuzzy,j , with j an integer or semi-integer. Here, our slightly modified procedure leads to a different fuzzy sphere that we write σ Sfuzzy,j . We detail the steps of its standard definition. 1. We consider a 2j + 1 dimensional irreducible unitary representation (UIR ) of SU(2). The standard construction considers the vector space V j of dimension 2j + 1, on which the three generators of SU(2) are expressed as the usual (2j + 1) × (2j + 1) Hermitian matrices Ja . Here we will make a different choice, namely the three SAMOs Λj , which correspond to the choice of the representation space Hσj (instead of V j in the usual construction). Since they obey the commutation relations of SU(2), σj σj (28) [Λσj a , Λb ] = i ǫabc Λc , the usual procedure may be applied. As we have seen, Hσj can be realized as the Hilbert space spanned by the spin spherical harmonics {σ Yjµ }µ=−j...j , with the usual inner product. The latter provide the SSH (ON) basis. Since the standard derivation of all properties of the fuzzy sphere rest only upon the abstract commutation rules (28), nothing but the representation space changes if we adopt the representation space H instead of V . σj 2. The operators Λσj a belong to O , and have a Lie algebra structure, through the skew products defined by the commutators. But the symmetrized products of operators provide a second algebra structure, that we write Oσj , at the basis of the construction of the fuzzy sphere: these symmetrized σj products of the Λσj (of dimension (2j + 1)2 ) of all linear a , up to power 2j, generate the algebra O σj endomorphisms of H , exactly like the ordinary Ja do in the original Madore construction. This σj is the standard construction of the fuzzy sphere, with the Ja and V j replaced by Λσj a and H . 7 3. The construction of the fuzzy sphere (of radius r) is defined by associating an operator fˆ in Oσj to any function f . Explicitly, this is done by first replacing each coordinate xi by the operator ca ≡ κ Λσj x a ≡ p r Λσj a , j(j + 1) (29) in the above expansion (27) of f (in the usual construction, this would be Ja instead of Λσj a ). Next, we replace in (27) the usual product by the symmetrized product of operators, and we truncate the sum at index ℓ = 2j. This associates to any function f an operator fˆ ∈ Oσj . 4. The vector space Mat2j+1 of (2j + 1) × (2j + 1) matrices is linearly generated by a number (2j + 1)2 of independent matrices. According to the above construction, a basis of Mat2j+1 can be taken as all the products of the Λσj a up to power 2j + 1 (which is necessary and sufficient to close the algebra). 5. The commutative algebra limit is restored by letting j go to the infinity while parameter κ goes to zero and κj is fixed to κj = r. The geometry of the fuzzy sphere Sfuzzy,j is thus constructed after making the choice of the algebra of the matrices of the representation, with their matrix product. It is taken as the algebra of operators, which generalize the functions. The rank (2j + 1) of the matrices invites us to view them as acting as endomorphisms in an Hilbert space of dimension (2j + 1). This is exactly what allows the coherent states quantization introduced in the previous section. 4.2 Operators We have defined the action on Oσj : σj Lσj a A ≡ [Λa , A]. The formula (27) expresses any function f of V ℓ as the reduction to S 2 of an homogeneous polynomials homogeneous of order ℓ: X fα,β,γ (x1 )α (x2 )β (x3 )γ ; α + β + γ = ℓ. f= α,β,γ The action of the ordinary momentum operators J3 and J 2 is straightforward. Namely, h i X fα,β,γ (−i) β(x1 )α+1 (x2 )β−1 (x3 )γ − α(x1 )α−1 (x2 )β+1 (x3 )γ , J3 f = α,β,γ and similarly for J1 and J2 . On the other hand, we have by definition   X c1 )α (x c2 )β (x c3 )γ , fα,β,γ S (x fˆ = (30) α,β,γ σj ca = κ Λσj where S(·) means symmetrization. Recalling x a , and using (28), we apply the operator L3 to this expression: h  α β γ i X σj ˆ ˆ1 xˆ2 xˆ3 ˆ fα,β,γ Λσj Lσj . (31) 3 f ≡ [Λ3 , f ] = 3 ,S x α,β,γ We prove in appendix B that the commutator of the symmetrized is the symmetrized of the commutator. Then, using the identity [J, AB · · · M ] = [J, A] B · · · M + A [J, B] · · · M + · · · + AB · · · [J, M ], which results easily (by induction) from [J, AB] = [J, A] B + A [J, B], it follows that   X α−1 β+1 γ α+1 β−1 γ σj ˆ ˆ fα,β,γ iα xˆ1 Lσj xˆ2 xˆ3 − iβ xˆ1 xˆ2 xˆ3 . 3 f ≡ [Λ3 , f ] = α,β,γ We thus have proven Similar identities hold for Lσj 1 , Lσj 2 ˆ d Lσj 3 f = J3 f . and thus for (Lσj )2 . 8 (32) coherent states Madore-like fuzzy sphere fuzzy sphere H = Hσj = span(σ Yjµ ) ⊂ L2 (S 2 ) Hilbert space O = Oσj = EndHσj endomorphisms spin angular momentum operators Λσj a ∈ O fa = K Λσj fe ∈ Oσj ; x a observables action of angular momentum σj e e g Lσj a f ≡ [Λa , f ] = Ja f ca = κ Λσj fb ∈ Oσj ; x a σj b b d Lσj a f ≡ [Λa , f ] = Ja f d Yg ℓm = C(ℓ) Yℓm correspondence Table 1: Coherent state quantization of the sphere is compared to the standard construction of the fuzzy sphere through correspondence formula. σj It results that Yd which is a common eigenvector of Lσj ℓm appears as an element of O 3 , with value m, and of (Lσj )2 , with value ℓ(ℓ + 1). Since we have proved above that such an element is unique (up to j g d a constant), it results that each Yd ℓm ∝ Yℓm . Thus, the Yℓm ’s, for ℓ ≤ j, −j ≤ m ≤ j form a basis of A . g d Then, the Wigner-Eckart theorem (see 7.15) implies that Yℓm = C(ℓ) Yℓm , where the proportionality constant C(ℓ) does not depend on m (what can also be checked directly). These coefficients can be calculated directly, after remarking that ℓ c1 c2 ℓ Yc ℓℓ ∝ (Λ+ ) ∝ (x + i x ) . In fact, c1 c2 ℓ Yc ℓℓ = a(ℓ) (x + i x ) ; a(ℓ) = We obtain (−1)j+σ−2 ℓ (2 j + 1) C(ℓ) = 2 κℓ ℓ 5 s p (2ℓ + 1)! √ . 2ℓ+1 π ℓ! (2j − ℓ)! (2j + ℓ + 1)!  j −σ j σ  ℓ . 0 Discussion We thus have two families of quantization of the sphere. • The usual construction of the fuzzy sphere, which depends on the parameter j. This parameter defines the “size” of the discrete cell. • The present construction coherent states which makes use of coherent states and which depends on two parameters, j and σ 6= 0. These two quantizations may be formulated as involving the same algebra of operators (quantum observables) O, acting on the same Hilbert space H (see Table 1). Note that H and O are not the Hilbert space and algebra usually involved in the usual expression of the fuzzy sphere (when we consider them as embedded in the space of functions of the spheres, and of operators acting on them), but they are isomorphic to them, and nothing is changed. The difference lies in the fact that the quantum counterparts, f˜ and fˆ of a given classical observable f differ in both approaches. Thus, the CS quantization really differs from the usual fuzzy sphere quantization. This raises the question iof whether the CS quantization is or is not a construction of a new type offuzzy sphere. It results from the calculations above that all properties of the usual fuzzy sphere are shared by the CS quantized version. The only point to be checked is if it gives the sphere manifold in some classical limit. The answer is positive as far as the classical limit is correctly defined. Simple 9 calculations show that it is obtained as the limit j 7→ ∞, σ 7→ ∞, provided that the ratio σ/j tends to a finite value. Thus, one may consider that the CS quantization leads to a one parameter family of fuzzy spheres if we impose relations of the type σ = j − σ0 , for fixed σ0 > 0 (for instance). 6 Conclusion We have proposed a general quantization procedure which applies to any measurable set X. It proceeds from the choice of an Hilbert space H of prescribed dimension. We have presented in details an implementation of this procedure (non necessarily unique) from an explicit family of coherent states, which realizes a natural embedding of X into H. We have applied this CS procedure to the sphere S 2 . We started from a natural basis linked to the UIR’s of the group SU(2): for any value of j and σ, we chose the Hilbert space Hσj , which carries a UIR of SU(2). Our CS construction associates, to any classical observable f ∈ L2 , a quantum observable fe, which belong to the algebra of endomorphisms Oσj ≡ End(Hσj ). On the other hand, we also followed the usual fuzzy sphere construction (with 2j + 1 cells), by replacing the coordinates by operators acting on the same Hilbert space. This allowed us to associate a fuzzy observable fb to any classical observable f . Those form the algebra of operators acting on the fuzzy sphere. For the particular classical observables provided by the ordinary spherical harmonics, we have shown g that the CS quantum observable and the fuzzy observable coincide up to a constant, Yd ℓm = C(ℓ) Yℓm , e b and the explicit value of this constant has been given. However, in general, f differs from f , although the correspondence is easy established from the relation above, through a development in the usual spherical harmonics. Thus, the CS quantization procedure really differs from the construction of the usual fuzzy sphere. Although they share the same algebra of quantum observables, acting on the same Hilbert space, the CS quantum observables fe and the fuzzy one, fb, associated to the same classical observable f differ. And there is no way to make them coincide, since the CS quantization with σ = 0 leads to trivial results. Our discussion in (5) allows us to consider our CS quantization procedure as a construction of a new type of fuzzy sphere, with properties differing from the standard one. It shares most of the properties of the usual fuzzy sphere, but appears more economic in the sense that - it does not require a group action on the space to be quantized; - it does not require an initial expansion of the functions into spherical harmonics. Applications of procedures of this type to the sphere have appeared in different contexts. For instance, a similar procedure is carried out in [6] in order to achieve a regularization of a membrane, with surface S 2 , by a mapping of functions to matrices, similar to the one presented here. Despite analog mathematics, the procedure there is not seen as a quantization and, according to the author, the regularized theory still requires a further quantization. Similar regularization exists for surfaces of arbitrary genius, and it would be interesting to apply the CS procedure in these cases. Also, it should not be difficult to explore cases with more dimensions, and in particular S 3 . This offers possibilities to construct new fuzzy versions of these spaces. Moreover, authors in [25] have given a description of the fuzzy sphere in terms of SU(2) spin networks. Since the latter play an important role in the canonical quantization of general relativity, this suggests that the application of the CS procedure to the quantization of gravity or to various geometries, compact or non-compact [26] could be fruitful, a program that we start to explore. Furthermore, the universality of the CS procedure would allow explicit constructions of spin networks associated to different groups, in particular SU(3). Since it has claimed that the latter could be of importance for quantum gravity, this reveals to be a promising field of research also. 7 7.1 Appendix A: Spin spherical harmonics SU(2)-parameterization SU (2) ∋ ξ =  ξ0 + iξ3 ξ2 + iξ1 −ξ2 + iξ1 ξ0 − iξ3  . (33) ξ0 + iξ3 = cos ωeiψ1 , ξ1 + iξ2 = sin ωeiψ2 π 0 6 ω 6 , 0 6 ψ1 , ψ2 < 2π. 2 (34) In bicomplex angular coordinates, 10 (35) and so SU (2) ∋ ξ = in agreement with Talman [27].  cos ωeiψ1 i sin ωe−iψ2 i sin ωeiψ2 cos ωe−iψ1  , (36) Matrix elements of SU(2)-UIR 7.2 j Dm (ξ) = (−1)m1 −m2 [(j + m1 )!(j − m1 )!(j + m2 )!(j − m2 )!]1/2 × 1 m2 X (ξ0 + iξ3 )j−m2 −t (ξ0 − iξ3 )j+m1 −t (−ξ2 + iξ1 )t+m2 −m1 (ξ2 + iξ1 )t × , (j − m2 − t)! (j + m1 − t)! (t + m2 − m1 )! t! t (37) in agreement with Talman. With angular parameters the matrix elements of the UIR of SU (2) are given in terms of Jacobi polynomials [28] by: s (j − m1 )!(j + m1 )! j −im1 (ψ1 +ψ2 ) −im2 (ψ1 −ψ2 ) m2 −m1 × Dm1 m2 (ξ) = e e i (j − m2 )!(j + m2 )! × m1 +m2 m1 −m2 1 (m1 −m2 ,m1 +m2 ) (1 + cos 2ω) 2 (1 − cos 2ω) 2 Pj−m (cos 2ω), 1 2m1 (38) in agreement with Edmonds [29] (up to an irrelevant phase factor). Orthogonality relations and 3j-symbols 7.3 Let us equip the SU (2) group with its Haar measure : µ(dξ) = sin 2ω dω dψ1 dψ2 , (39) in terms of the bicomplex angular parametrization. Note that the volume of SU (2) with this choice of j normalization is 8π 2 . The orthogonality relations satisfied by the matrix elements Dm (ξ) reads as: 1 m2 Z  ′ ∗ 8π 2 j j Dm (ξ) Dm µ(dξ) = (40) δjj ′ δm1 m′1 δm2 m′2 . ′ m′ (ξ) 1 m2 1 2 2j +1 SU (2) in connection with the reduction of the tensor product of two UIR’s of SU (2), we have the following equivalent formula involving the so-called 3 − j symbols (proportional to Clebsch-Gordan coefficients), in the Talman notations :    ∗ X j j′ j ′′ j j′ j ′′  j ′′ ′′ j j′ (2j + 1) Dm1 m2 (ξ) Dm′ m′ (ξ) = Dm′′ m′′ (ξ) , (41) ′ ′′ ′ ′′ m1 m1 m1 m2 m2 m2 1 2 1 2 ′′ ′′ ′′ j m1 m2 Z ′ SU (2) ′′ j j j 2 Dm (ξ) Dm ′ m′ (ξ) Dm′′ m′′ (ξ) µ(dξ) = 8π 1 m2 1 2 1 2  j m1 j′ m′1 j ′′ m′′1  j m2 j′ m′2  j ′′ . m′′2 (42) One of the multiple expressions of the 3 − j symbols (in the convention that there are all real) is given by:  j m j′ m′ × j ′′ m′′ X s  =(−1)j−j (−1)s ′ −m′′  (j + j ′ − j ′′ )!(j − j ′ + j ′′ )!(−j + j ′ + j ′′ )! (j + j ′ + j ′′ + 1)! 1/2 1/2 [(j + m)!(j − m)!(j ′ + m′ )!(j ′ − m′ )!(j ′′ + m′′ )!(j ′′ − m′′ )!] s!(j ′ + m′ − s)!(j − m − s)!(j ′′ − j ′ + m + s)!(j ′′ − j − m′ + s)!(j + j ′ − j ′′ − s)! (43) 11 7.4 Spin spherical harmonics The spin spherical harmonics, as functions on the 2-sphere S 2 are defined as follows: r r i∗ 2j + 1 h j 2j + 1 j µ−σ Dµσ (ξ (Rr̂ )) = (−1) D−µ−σ (ξ (Rr̂ )) σ Yjµ (r̂) = 4π 4π r  2j + 1 j  † = Dσµ ξ (Rr̂ ) , 4π (44) (45) where ξ (Rr̂ ) is a (nonunique) element of SU (2) which corresponds to the space rotation Rr̂ which brings c3 to the unit vector b the unit vector e r with polar coordinates :  1  x = sin θ cos φ, b x2 = sin θ sin φ, (46) r=  3 x = cos θ. We immediately infer from the definition (44) the following properties: (σ Yjµ (r̂))⋆ = (−1)σ−µ −σ Yj−µ (r̂), µ=j X µ=−j |σ Yjµ (r̂)|2 = 2j + 1 . 4π (47) (48) Let us recall here the correspondence (homomorphism) ξ = ξ(R) ∈ SU (2) ↔ R ∈ S0(3) ≃ SU (2)/Z2 :  ix′3 x′2 + ix′1 b r′ = (x′1 , x′2 , x′3 ) = R · b r ←→    ′ ′ ix3 −x2 + ix1 −x2 + ix1 ξ† . =ξ ′ x2 + ix1 −ix3 −ix3 (49) (50) In the particular case of (44) the angular coordinates ω, ψ1 , ψ2 of the SU (2)-element ξ (Rr̂ ) are constrained by cos 2ω = cos θ, sin 2ω = sin θ, e i(ψ1 +ψ2 ) =ie iφ so so 2ω = θ, π ψ1 + ψ2 = φ + . 2 (51) (52) Here we should pay a special attention to the range of values for the angle φ, depending on whether j and consequently σ and m are half-integer or not. If j is half-integer, then angle φ should be defined mod (4π) whereas if j is integer, it should be defined mod (2π). We still have one degree of freedom concerning the pair of angles ψ1 , ψ2 . We leave open the option concerning the σ-dependent phase factor by putting def i−σ eiσ(ψ1 −ψ2 ) = eiσψ , (53) where ψ is arbitrary. With this choice and considering (37) we get the expression of the spin spherical harmonics in terms of φ, θ/2 and ψ: s r 2j + 1 (j + µ)!(j − µ)! σ iσψ iµφ e × σ Yjµ (r̂) = (−1) e 4π (j + σ)!(j − σ)!    2j X 2t+σ−µ  θ θ j+σ j−σ tan (−1)t , (54) × cos t+σ−µ t 2 2 t s r 2j + 1 (j + µ)!(j − µ)! σ iσψ iµφ = (−1) e e × 4π (j + σ)!(j − σ)!     2j X 2t+σ−µ θ θ j+σ j−σ × sin cot (−1)j−t+µ−σ , (55) t+σ t−µ 2 2 t which are not in agreement with the definitions of Newman and Penrose [10], Campbell [12] (note there is a mistake in the expression given by Campbell, in which a cos θ2 should read cot θ2 ), and Hu and White 12 [30]. Besides presence of different phase factors, the disagreement is certainly due to a different relation between the polar angle θ and the Euler angle. Now, considering (38), we get the expression of the spin spherical harmonics in terms of the Jacobi polynomials, valid in the case in which µ ± σ > −1: s r 2j + 1 (j − µ)!(j + µ)! µ iσψ × σ Yjµ (r̂) = (−1) e 4π (j − σ)!(j + σ)! µ+σ µ−σ 1 (µ−σ,µ+σ) (cos θ) eiµφ . (1 + cos θ) 2 (1 − cos θ) 2 Pj−µ 2µ For other cases, it is necessary to use alternate expressions based on the relations [28]:
l n+β  x−1 (l,β) (α,β) Pn(−l,β) (x) = nl
Pn−l (x), P0 (x) = 1. 2 l × Note that with σ = 0 we recover the expression of the normalized spherical harmonics : r 1 2j + 1 p (m,m) m (j − m)!(j + m)! (sin θ)m Pj−m (cos θ) eimφ 0 Yjm (r̂) = Yjm (r̂) = (−1) 4π j! 2m s r 2j + 1 (j − m)! m = Pj (cos θ)eimφ 4π (j + m)! (56) (57) (58) since we have the following relation between associated Legendre polynomials and Jacobi polynomials m (m,m) Pj−m (z) = (−1)m 2m (1 − z 2 )− 2 j! Pjm (z), (j + m)! (59) for m > 0. We recall also the symmetry formula Pj−m (z) = (−1)m (j − m)! m Pj (z). (j + m)! (60) Our expression of spherical harmonics is rather standard, in agreement with Arkfen [31, 32]1 7.5 Transformation laws We consider here the transformation law of the spin spherical harmonics under the rotation group. From the relation RRt Rr̂ = Rr̂ (61) ′ ′ for any R ∈ SO(3), and from the homomorphism ξ(RR ) = ξ(R)ξ(R ) between SO(3) and SU (2), we deduce from the definition (44) of the spin spherical harmonics the transformation law r  r 2j + 1 
 2j + 1 j  † t j Dσµ ξ (Rt R·r̂ ) = Dσµ ξ † t RRr̂ σ Yjµ ( R · r̂) = 4π 4π r   r 2j + 1 X  2j + 1 j  † j j Dσµ ξ (Rr̂ ) ξ (R) = Dσν ξ † (Rr̂ ) Dνµ (ξ (R)) = 4π 4π ν X j (62) = σ Yjν (r̂) Dνµ (ξ (R)) , ν as expected if we think to the special case (σ = 0) of the spherical harmonics. Given a function f (x) on the sphere S 2 belonging to the 2j + 1-dimensional Hilbert space Hσj and a rotation R ∈ SO(3), we define the rotation operator Dσj (R) for that representation by   Dσj (R)f (x) = f (R−1 · x) = f (t R · x). (63) Thus, in particular,   Dσj (R) σ Yjµ (r̂) = σ Yjµ ( t R · r̂). (64) (a) The generators of the three rotations R , a = 1, 2, 3, around the three usual axes, are the angular momentum operator in the representation. When σ = 0, we recover the usual SHs, and these generators (j) are the usual angular momentum operators J i (short notation for Ji ) for that representation. In the (σj) general case σ 6= 0, we call them Λa . We study their properties below. 1 Sometimes (e.g., Arfken 1985 [31]), the Condon-Shortley phase (−1)m is prepended to the definition of the spherical harmonics. Talman adopted this convention. 13 7.6 Infinitesimal transformation laws Recalling that the components Ja = −i ǫabc xb ∂c of the ordinary angular momentum operator are given in spherical coordinates by: J3 = −i∂φ , (65) J+ = J1 + iJ2 = e iφ J− = J1 − iJ2 = −e (∂θ + i cot θ partialφ) , −iφ (∂θ − i cot θ partialφ ) . We have introduced the “spin” angular momentum operators: Λσj 3 = J3 = −i∂φ , Λσj + Λσj − = Λσj 1 = Λσj 1 +i Λσj 2 −i Λσj 2 (66) iφ = J+ + σ csc θ e , = J− + σ csc θ e −iφ (67) . (68) They obey the expected commutation rules, σj σj [Λσj + , Λ− ] = 2Λ3 . σj σj [Λσj 3 , Λ± ] = ±Λ± , (69) These operators are the infinitesimal generators of the action of SU (2) on the spin spherical harmonics: 7.7 Λσj 3 σ Yjµ = µ σ Yjµ p σj Λ+ σ Yjµ = (j − µ)(j + µ + 1) σ Yjµ+1 p Λσj (j + µ)(j − µ + 1) σ Yjµ−1 . − σ Yjµ = (70) (71) (72) Integrals and 3j-symbols Specifying the equation (40) to the spin spherical harmonics lead to the following orthogonality relations which are valid for j integer (and consequently σ integer). Z ∗ (73) σ Yjµ (r̂) (σ Yj ′ ν (r̂)) µ(dr̂) = δjj ′ δµν , S2 We recall that in the integer case, the range of values assumed by the angle φ is 0 6 φ < 2π. Now, if we consider half-integer j (and consequently σ), the range of values assumed by the angle φ becomes 0 6 φ < 4π. The integral above has to be carried out on the “doubled” sphere Se2 and an extra 1 normalization factor equal to √ is needed in the expression of the spin spherical harmonics. 2 For a given integer σ the set { σ Yjµ , −∞ 6 µ 6 ∞, j > max (0, σ, m)} form an orthonormal basis of the Hilbert space L2 (S 2 ). Indeed, at µ fixed so that µ ± σ > 0, the set s ) (r µ+σ µ−σ 2j + 1 (j − µ)!(j + µ)! 1 (µ−σ,µ+σ) 2 2 P (1 + cos θ) (1 − cos θ) (cos θ), j > µ j−µ 4π (j − σ)!(j + σ)! 2µ is an orthonormal basis of the Hilbert space L2 ([−π, π], sin θ dθ). The same holds for other ranges of values of µ by using alternate expressions like (57) N for Jacobi polynomials. Then it suffices to view L2 (S 2 ) as the tensor product L2 ([−π, π], sin θ dθ) L2 (S 1 ). Similar reasoning is valid for half-integer σ. Then, the Hilbert space to be considered is the space of “fermionic” functions on the doubled sphere Se2 , i.e. such that f (θ, φ + 2π) = −f (θ, φ). Specifying the equation (41) to the spin spherical harmonics leads to r X (2j + 1)(2j ′ + 1)(2j ′′ + 1) × σ Yjµ (r̂) σ ′ Yj ′ µ′ (r̂) = 4π j ′′ µ′′ σ ′′    j j ′ j ′′ j j ′ j ′′ × (74) (σ ′′ Yj ′′ µ′′ (r̂))∗ . ′ ′′ ′ ′′ µ µ µ σ σ σ 14 We easily deduce from (74) the following integral involving the product of three spherical spin harmonics (in the integer case, but analog formula exists in the half-integer case) and with the constraint that σ + σ ′ + σ ′′ = 0: r Z (2j + 1)(2j ′ + 1)(2j ′′ + 1) × σ Yjµ (r̂) σ ′ Yj ′ µ′ (r̂) σ ′′ Yj ′′ µ′′ (r̂) µ(dr̂) = 4π S2     j j ′ j ′′ j j ′ j ′′ × . (75) ′ ′′ µ µ µ σ σ ′ σ ′′ Note that this formula is independent of the presence of a constant phase factor of the type eiσψ in the definition of the spin spherical harmonics because of the a priori constraint σ + σ ′ + σ ′′ = 0. On the other hand, we have to be careful in applying Eq. (75) because of this constraint, i.e. since it has been derived from Eq. (74) on the ground that σ ′′ was already fixed at the value σ ′′ = −σ − σ ′ . Therefore, the computation of Z S2 σ Yjµ (r̂) σ ′ Yj ′ µ′ (r̂) σ ′′ Yj ′′ µ′′ (r̂) µ(dr̂) for an arbitrary triplet (σ, σ ′ , σ ′′ ) should be carried out independently. 7.8 Important particular case : j = 1 In the particular case j = 1, we get the following expressions for the spin spherical harmonics: r σ  1 3 θ iσψ p cos θ, cot σ Y10 (r̂) = e 4π (1 + σ)!(1 − σ)! 2 r  σ θ 1 3 iσψ p cot sin θ eiφ , σ Y11 (r̂) = −e 4π 2(1 + σ)!(1 − σ)! 2 r  σ 3 θ 1 σ −iσψ p tan sin θ e−iφ . Y (r̂) = (−1) e σ 1−1 4π 2(1 + σ)!(1 − σ)! 2 (76) (77) (78) For σ = 0, we recover familiar formula connecting spherical harmonics to components of vector on the unit sphere: r r 3 3 Y10 (r̂) = cos θ = z, (79) 4π 4π r r 3 1 3 x + iy √ sin θeiφ = − √ , Y11 (r̂) = − (80) 4π 2 4π 2 r r 3 1 3 x − iy √ sin θe−iφ = √ . Y1−1 (r̂) = (81) 4π 2 4π 2 7.9 Another important case : σ = j For σ = j, due to the relations (57), the spin spherical harmonics reduce to their simplest expressions : v ! r u j+µ  j−µ u 2j θ θ 2j + 1 j ijψ t cos sin eiµφ . (82) j Yjµ (r̂) = (−1) e j+µ 4π 2 2 They are precisely the states which appear in the construction of the Perelomov coherent states. Otherwise said, the Perelomov CS [15] and related quantization are just particular cases of our approach. 7.10 Spin coherent states For a given pair (j, σ), we define the family of coherent states in the 2j + 1-dimensional Hilbert space Hσj : j X 1 ∗ (83) | x i =| θ, φ i = p σ Yjµ (x) | σjµ i; | x i ∈ Hσj , N (x) µ=−j 15 with N (x) = j X µ=−j | σ Yjµ (x) |2 = 2j + 1 . 4π For σ = j, these coherent states identify to the so-called spin or atomic or Bloch coherent states [15]. But, for a given j and two different σ 6= σ ′ , the corresponding families are distinct because they live in different Hilbert spaces of same dimension 2j + 1. This is due to the fact that the map between the two orthonormal sets is not unitary, since we should deal with expansions like: X (84) Mj ′ µ′ ,jµ (σ ′ , σ) σ ′ Yj ′ µ′ , σ Yjµ = j ′ µ′ where Mj ′ µ′ ,jµ (σ ′ , σ) = Z S2 (σ ′ Yj ′ µ′ (r̂))∗ σ Yjµ (r̂) µ(dr̂) = [j ′ jσ ′ σµ] δµµ′ , (85) the (non-trivial!) coefficient [j ′ jσ ′ σµ] being to be determined and forcing the sum to run on values of j ′ different of j. 7.11 Covariance properties of spin CS The definition of the rotation operator Dσj (R) was given in (63). Starting from a CS | x i, let us consider the coherent state with rotated parameter R · x. Due to the transformation property (62), the invariance of N (x) and the unitarity of Dj , we find: |R · xi = p = p = p j X 1 N (x) µ=−j ∗ t σ Yjµ ( R j X 1 N (x) µ,µ′ =−j j X 1 N (x) µ′ =−j · x) | σjµ i ∗ σ Yjµ′ (x) ∗ σ Yjµ′ (x)  Dµj ′ µ ξ R−1 j X µ=−j = Dσj (R) | x i,

⋆ | σjµ i j Dµµ ′ (ξ (R)) | σjµ i (86) where the Dσj have been defined in (63). Hence, we get the (standard) covariance property of the spin CS: Dσj (R)|R−1 · xi =| x i. 7.12 (87) Spin CS quantization A classical observable on X is a function f : X 7→ C. To any such function f , we associate the operator Af in Hσj through the map: Z f 7→ Af ≡ f (x) | x ih x | N (x) µ(dx). (88) X Occasionally we might use the notation f˜ for Af . In terms of its matrix elements in the basis of spin harmonics, this operator reads: Af = j X µ,µ′ =−j 7.13 Z X ∗ f (x) σ Yjµ (x) σ Yjµ′ (x) | σjµ ih σjµ′ | µ(dx) ≡ j X µ,µ′ =−j [Af ]µµ′ | σjµ ih σjµ′ | . (89) Spin CS quantization of spin spherical harmonics The quantization of an arbitrary spin harmonics ν Ykn yields an operator in Hσj whose (2j + 1) × (2j + 1) matrix elements are given by the following integral resulting from (89): 16 h e ν Ykn i µµ′ = = Z Z ∗ σ Yjµ (x) σ Yjµ′ (x) ν Ykn (x) µ(dx) X X (−1)σ−µ −σ Yj−µ (x) σ Yjµ′ (x) ν Ykn (x) µ(dx). (90) As asserted above, it is only when ν − σ + σ = 0, i.e. when ν = 0, that the integral (90) is given in terms of a product of two 3j-symbols as follows: h Yekn i µµ′ = = Z Z ∗ σ Yjµ (x) (−1)σ−µ X = (−1) 7.14 σ Yjµ′ (x) Ykn (x) µ(dx) X σ−µ −σ Yj−µ (x) σ Yjµ′ (x)Ykn (x) µ(dx) (2j + 1) r (2k + 1) 4π  j −µ j µ′ k n  j −σ j σ  k . 0 (91) Checking quantization in the simplest case : j = 1 With the notations of the text, we find for the matrix elements of the CS quantized versions of the above spherical harmonics: r h i 1 3 e mδmn , (92) Y10 =σ 4π j(j + 1) mn r r h i 1 3 (j − n)(j + n + 1) Ye11 = −σ δmn+1 , (93) 4π j(j + 1) 2 mn r r h i 3 (j + n)(j − n + 1) 1 =σ Ye1−1 δmn−1 . (94) 4π j(j + 1) 2 mn Comparing with the actions (70), (71), (72) of the spin angular momentum on the spin-σ spherical harmonics, we have the identification: Ye11 Ye1−1 r 1 3 Λ3 , 4π j(j + 1) r 1 3 = −σ Λ+ , 8π j(j + 1) r 3 1 =σ Λ− . 8π j(j + 1) Ye10 = σ (95) (96) (97) Hence, we can conclude on the following identification between quantized versions of the components of the vector on the unit sphere and the components of the spin angular momentum operator: σ Λ1 , j(j + 1) σ Λ2 , ye = j(j + 1) σ ze = Λ3 . j(j + 1) x e= 7.15 (98) (99) (100) Rotational covariance properties of operators By construction, the operators ν] Ykn acting on Hσj are tensorial irreducible. Indeed, under the action σj of the representation operator D (R) in Hσj , due to (87), the rotational invariance of the measure and N (x), and (62), they transform as: 17 Dσj (R) ν] Ykn Dj (R−1 ) = = Z ν Ykn (x) ZX X = X n′ = X | R · x ih R · x | N (x) µ(dx) −1 ν Ykn (R · x) | x ih x | N (x) µ(dx) Z Dnk ′ n (ξ (R)) ν Ykn′ (x) | x ih x | N (x) µ(dx) X ^ k ν Ykn′ Dn′ n (ξ (R)) . (101) n′ Therefore, the Wigner-Eckart [29] tells us that the matrix elements of the operator ^ ν Ykn n theorem o with respect to the SSH basis σ Yejm are given by:   h i j j k e K(ν, σ, j, k). (102) = (−1)j−m ν Ykn ′ −m m n mm′ Note that the presence of the 3j symbol in (102) implies the selection rules n+m′ = m and the triangular rule 0 6 k 6 2j. The proportionality coefficient K can be computed directly from (90) by choosing therein suitable values of m, m′ . On the other hand, we have by definition (62,64) X k νk ν Ykn′ Dn′ n (ξ (R)) = D (R) ν Ykn . n′ Thus, from the formula above, Dσj (R) ν] Ykn Dj (R−1 ) = Dνk ^ (R) ν Ykn . In the special case ν = 0, j −1 g Dσj (R) Y ) = D0k^ (R) Ykn . kn D (R (103) This has the infinitesimal version (see xxx), for the three rotations Ri , (σj) [Λi ^ (k) g , Y Ykn . kn ] = Ji (104) Appendix B: Symmetrization of the commutator One intends to show that S ([J3 , J1α1 J2α2 J3α3 ]) = [J3 , S (J1α1 J2α2 J3α3 )], where Ji is a representation of so(3). Let us make a first comment on the symmetrization : S(J1α1 J2α2 J3α3 ) = 1 X Ji . . . Jiσ(l) , l! σ∈S σ(1) l where l = α1 + α2 + α3 . The terms of the sum are not all distinct, since the exchange of, e.g., two J1 gives the same term: each term appears in fact α1 !α2 !α3 ! times, so that there are l!/(α1 !α2 !α3 !) distinct terms. This is the number of sequences of length l, with values in {1, 2, 3}, where there are αi occurrences of the value i (for i = 1, 2, 3). One denotes this set as Uα1 ,α2 ,α3 . After grouping of identical terms, one obtains : X α1 !α2 !α3 ! S(J1α1 J2α2 J3α3 ) = Ju1 . . . Jul , l! u∈U α1 ,α2 ,α3 where all the terms of the summation are now different. Let us now calculate S ([J3 , J1α1 J2α2 J3α3 ]). First, we write [J3 , J1α1 J2α2 J3α3 ] = [J3 , J1α1 ]J2α2 J3α3 + J1α1 [J3 , J2α2 ]J3α3 , {z } | {z } | A 18 B with α1 X J . . . J J J . . . J J α2 J α3 . | 1 {z 1} 2 | 1 {z 1} 2 3 k=1 k−1 terms α1 −k terms The different terms in A give the same symmetrized. Thus,
S(A) = α1 S J1α1 −1 J2α2 +1 J3α3 X (α1 − 1)!(α2 + 1)!α3 ! = α1 Ju1 . . . Jul . l! u∈U A= α1 −1,α2 +1,α3 Similarly, for B, S(B) = −α2 (α1 + 1)!(α2 − 1)!α3 ! l! X Ju1 . . . Jul . u∈Uα1 +1,α2 −1,α3 Now we calculate I = [J3 , S(J1α1 J2α2 J3α3 )] = α1 !α2 !α3 ! l! X l X Ju1 . . . Juk−1 [J3 , Juk ]Juk+1 . . . Jul . u∈Uα1 ,α2 ,α3 k=1 The sum splits in two parts, according to the value of uk = 1 or 2. I = A′ + B ′ , with A′ = α1 !α2 !α3 ! l! and B′ = − X X Ju1 . . . Juk−1 J2 Juk+1 . . . Jul , u∈Uα1 ,α2 ,α3 k|uk =1 α1 !α2 !α3 ! l! X X Ju1 . . . Juk−1 J1 Juk+1 . . . Jul . u∈Uα1 ,α2 ,α3 k|uk =2 Let us examine the constituents of A′ . There are of the form Ju1 . . . Jul with u ∈ Uα1 −1,α2 +1,α3 . Their number is l!/(α1 !α2 !α3 !) × α1 , but they are not all different. Each monomial is issued from a term where a J1 has been transformed into a J2 . Since there are α2 + 1 occurrences of J2 in each term, each monomial appears α2 + 1 times. We now group these identical terms : A′ = X α1 !α2 !α3 ! 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