A fuzzy bipolar celestial sphere
PPrepared for submission to JHEP
A fuzzy bipolar celestial sphere
Francesco Alessio, a Michele Arzano a a Dipartimento di Fisica “E. Pancini”, Universit`a di Napoli Federico II and INFN, Via Cinthia,80126 Fuorigrotta, Napoli, Italy
E-mail: [email protected] , [email protected] Abstract:
We introduce a non-commutative deformation of the algebra of bipolar spher-ical harmonics supporting the action of the full Lorentz algebra. Our construction is closein spirit to the one of the non-commutative spherical harmonics associated to the fuzzysphere and, as such, it leads to a maximal value of the angular momentum. We derivethe action of Lorentz boost generators on such non-commutative spherical harmonics andshow that it is compatible with the existence of a maximal angular momentum. a r X i v : . [ g r- q c ] J u l ontents The discovery of the connection between soft theorems in quantum field theory, the mem-ory effect and asymptotic symmetries has revealed an unexpected richness in the infraredstructure of gauge theories [1–3]. In gravity, the corner of this infrared triangle representedby the symmetries of asymptotically flat spacetimes has been subject of a revived interest,mainly due to the potential role of the BMS (Bondi-Metzner-Sachs) group [4] in the res-olution of the black hole information paradox [5, 6]. In this context, the existence of aninfinite number of conserved charges associated with BMS symmetries [7–9] can equip theblack hole with the soft hair [10, 11] needed to support correlations between the interiorof the black hole and the emitted Hawking quanta. To date, however, the exact mecha-nism from which the information can be recovered through the BMS charges is not known.Connected to this line of thought is the possibility that modes of a near-horizon BMSsymmetry might provide the degrees of freedom needed to microscopically reproduce theBekenstein-Hawking entropy [12, 13]. One of the obstacles in making such identificationconcrete is that the actual degrees of freedom which can be associated to BMS chargesare too many , in fact infinite, while the Bekenstein-Hawking entropy, albeit large, is finiteand proportional to the black hole area divided by the Planck length squared. This isalready evident in the simplest formulation of the BMS group as the semidirect productof the Lorentz group and the abelian group of supertranslations. The latter are indexed– 1 –y the angular momentum of spherical harmonics on the celestial sphere and are infinitein number since one can have infinite angular resolution on such sphere.In this note we explore the possibility of constructing a non-commutative deformationof the algebra of spherical harmonics supporting an action of the Lorentz algebra andexhibiting a maximal angular resolution. We show how such task cannot be accomplishedusing only one set of non-commutative spherical harmonics similar to the one used inthe literature to describe a non-commutative analogue of the two-sphere, the so-called fuzzy sphere [14]. We find, however, that a matrix generalization of the algebra of bipolarspherical harmonics [15] can be constructed, exhibiting a cut-off in the angular modes andcarrying a representation of the full Lorentz algebra.In the standard picture, Lorentz boosts acting on the celestial sphere do not commutewith the total angular momentum operator and hence they connect spherical harmonicswith different values of the angular momentum. One remarkable aspect of our constructionis that the action of Lorentz boosts on the algebra of non-commutative bipolar sphericalharmonics is found to be compatible with the existence of a maximal angular momentumand cannot produce harmonics labelled with an arbitrarily high angular momentum.In the next three Sections we recall some basic facts about the asymptotic structureof Minkowski spacetime, showing the action of the Poincar´e and the BMS algebra onthe celestial sphere. We then review the fuzzy sphere and in particular we will focus onthe mapping of ordinary spherical harmonics to the so-called fuzzy spherical harmonics ,characterized by a maximal angular momentum. Finally we extend such constructionin order to introduce an action of the full Lorentz algebra which is consistent with theexistence of a maximal value of the angular momentum. We close with a short summaryand an outline for future developments.
We start by recalling the notion of celestial sphere focusing for simplicity on Minkowskispacetime[16, 17], but keeping in mind that the same definition can be given for anyasymptotically flat spacetime, since it relies only on asymptotic properties. Given theMinkowski line element in cartesian coordinates ds = η µν dx µ dx ν = − ( dx ) + ( dx ) + ( dx ) + ( dx ) , (2.1)we first pass to ordinary spherical coordinates ( r, θ, φ ), x ± ix = re ± iφ sin θ, x = r cos θ, (2.2)and then switch from the inertial time coordinate x to the retarded time u = x − r .Consider now an observer emitting a light ray at x = 0 and r = 0 in a direction ( θ, φ ). Wecan assign to any point at finite distance along that ray the coordinates ( u, r, θ, φ ). Here r is just an affine parameter along the geodesic representing the null ray and can be thoughtas a measure of the distance between the emitter and the particular point considered, inthe frame of the emitter. Notice that u is constant along that ray (and is always equal to 0– 2 – igure 1 . The celestial sphere S + at u = 0 of an observer emitting a ray at x = 0 and r = 0,obtained as the intersection of the null cone N and I + . for the particular ray considered). The set ( u, r, θ, φ ) is called a retarded Bondi coordinatesystem.Future null infinity I + can be defined as the asymptotic null region obtained bysending r, x → ∞ while keeping the retarded time u = x − r constant. In such limit alight ray will intersect I + in a point, which we label by ( u, θ, ϕ ). By sending light raysin all possible directions one can cover the entire future null cone N . At null infinity thiscone will intersect I + on a sphere S + , spanned by the coordinates θ and φ . Similarly, itis possible to define a coordinate system on the past null cone, using ingoing null rays andthe advanced time v = x + r which is constant along them. The Minkowski conformaldiagram is represented in Figure 1.For any fixed value of the retarded time u , the points of I + are spheres S + of infiniteradius, called celestial spheres . They are the spheres of all directions towards which anobserver at r = 0 can look. Alternatively, it is possible to give a definition of celestialsphere [18], which does not rely on a particular choice of coordinates. It can be defined asthe set of future-directed null directions passing through a point, i.e. the complex projectiveline CP (cid:39) S . This will be very useful in the following. Notice that for any asymptoticallyflat spacetime future null infinity is always a 3-dimensional S × R manifold, whose S component is the celestial sphere S + . The connected component of the Lorentz group consists of transformations x (cid:48) µ = Λ µν x ν which relate the coordinates of two different inertial observers. They are isometries of theline element (2.1). In order to describe the action of such transformations on the celestialsphere, we start by introducing complex stereographic coordinates ( ζ, ¯ ζ ) for each point ofthe sphere. – 3 – igure 2 . The representation of a direction ( θ, φ ) in the sky S + as a stereographic coordinate ζ on the complex plane C ζ . ζ = e iφ tan θ x + ix r + x , ¯ ζ = ζ ∗ . (3.1)Then, it turns out [19] that any Lorentz transformation on I + for the stereographiccoordinates is given by a M¨obius map, ζ (cid:48) = aζ + bcζ + d , ad − bc = 1 , (3.2)while the retarded time transforms as u (cid:48) = 1 + | ζ | | aζ + b | + | cζ + d | u ≡ K ( ζ, ¯ ζ ) u, | ζ | ≡ ζ ¯ ζ, (3.3)i.e. u undergoes an angle-dependent rescaling. Equations (3.2) and (3.3) express therelations bewteen the coordinates on the celestial sphere associated to two different inertialobservers. Note that the M¨obius maps of (3.2) are just the group of complex projectivetransformations of the above mentioned complex projective line CP and that the inducedmetric on the celestial sphere undergoes a conformal transformation. Indeed, the lineelement of the unit sphere is ds = dθ + sin θdφ = 4 (cid:16) | ζ | (cid:17) dζd ¯ ζ, (3.4)under (3.2) transforms as ds (cid:48) = 4 (cid:16) | ζ (cid:48) | (cid:17) dζ (cid:48) d ¯ ζ (cid:48) = (cid:32) | ζ | | aζ + b | + | cζ + d | (cid:33) ds = K ( ζ, ¯ ζ ) ds , (3.5)– 4 – igure 3 . The point P on S + is obtained from P by means of a rotation about the x axis. Thecorresponding orbit in C ζ is a circle. i.e a conformal rescaling with conformal factor K ( ζ, ¯ ζ ). Equations (3.2),(3.3) tell us thatLorentz transformations on the celestial sphere are described by SL(2 , C ) / Z matrices A = ± (cid:32) a bc d (cid:33) , ad − bc = 1 . (3.6)In particular, a general rotation of an angle ϕ and of a boost of rapidity χ about an axisˆ n = (cos φ sin θ, sin φ sin θ, cos θ ) are described by the following SL(2 , C ) / Z matrices L ˆ n ( ϕ ) = ± cos ϕ − i cos θ sin ϕ − i sin θ sin ϕ e − iφ − i sin θ sin ϕ e iφ cos ϕ + i cos θ sin ϕ , (3.7) R ˆ n ( χ ) = ± cosh χ − cos θ sinh χ − sin θ sinh χ e − iφ − sin θ sinh χ e iφ cosh χ + cos θ sinh χ . (3.8)Notice that L ˆ n ( ϕ ) is an SU(2) transformation while R ˆ n ( χ ) is not. Indeed, for any rotationthe conformal factor is K ( ζ, ¯ ζ ) = 1, because rotations are pure isometries of the 2-sphere,while the boosts are only conformal symmetries.For example a rotation about the axis x of an angle ϕ is expressed by ζ (cid:48) = e − iϕ ζ , i.e.a rotation on C ζ (see Figure 3). Two observers that are rotated about the axis x see thesame celestial sphere, but their coordinates are rotated. Rotations of the celestial spheremap into rotations of the complex plane.On the other hand, a boost of rapidity χ along the x axis is given by ζ (cid:48) = e − χ ζ .In this case, the two inertial observers still see the same celestial sphere, but the pointsof the celestial sphere of the boosted observer are dragged away from the south pole and– 5 – igure 4 . The action of a boost along the x direction on points on the celestial sphere, and thecorresponding action on C ζ . The two radii are such that | ζ (cid:48) i | / | ζ i | = e − χ . come closer to the north pole as χ increases. On the complex plane this corresponds to acontraction, as shown in Figure 4.Furthermore, it is possible to show [17] that the conformal factor of boosts is relatedto the Lorentz factor γ through the following relation K ( ζ, ¯ ζ ) = 1[ γ (1 − v · ˆ r c )] . (3.9)If all the stars in one observer’s sky are thought of as projected onto its celestial sphere,two boosted observer see a different night sky [20]. This is the classical phenomenon ofstellar aberration.The infinitesimal transformations of (3.2),(3.3) are described by the following vectorfields on on I + , L x = i (cid:18) sin ϕ ∂∂θ + cot θ cos ϕ ∂∂φ (cid:19) , (3.10) L y = i (cid:18) − cos ϕ ∂∂θ + cot θ sin ϕ ∂∂φ (cid:19) , (3.11) L z = − i ∂∂φ , (3.12) R x = − i (cid:18) cos θ cos ϕ ∂∂θ − sin ϕ sin θ ∂∂φ − u sin θ cos ϕ ∂∂u (cid:19) , (3.13) R y = − i (cid:18) cos θ sin ϕ ∂∂θ + cos ϕ sin θ ∂∂φ − u sin θ sin ϕ ∂∂u (cid:19) , (3.14) R z = i (cid:18) sin θ ∂∂θ + u cos θ ∂∂u (cid:19) , (3.15)and it is easy to prove [19] that they are a representation of the Lorentz algebra on I + – 6 –nd on the celestial sphere, having fixed the value of u :[ L i , L j ] = i(cid:15) ijk L k , [ R i , R j ] = − i(cid:15) ijk L k , [ L i , R j ] = i(cid:15) ijk R k . (3.16)The celestial sphere, as a smooth manifold, is a 2-sphere S and the commutative algebraof smooth functions defined on it, which will be denoted by C ( S ), is generated by thespherical harmonics { Y lm ( θ, φ ) } , which provide an orthonormal and complete basis withinner product given by (cid:90) d Ω Y ∗ l m ( θ, ϕ ) Y l m ( θ, ϕ ) = δ l l δ m m , (3.17)Thus, any smooth function f ( θ, φ ) ∈ C ( S ) can be expanded as f ( θ, φ ) = ∞ (cid:88) l =0 l (cid:88) m = − l f lm Y lm ( θ, φ ) , (3.18)with the components of the expansion given by f lm = (cid:90) d Ω Y ∗ lm ( θ, ϕ ) f ( θ, φ ) . (3.19)The product of two spherical harmonics can expressed in terms of a linear combination ofspherical harmonics using the Clebsch-Gordan coefficients: Y l m Y l m = l + l (cid:88) l = | l − l | l (cid:88) m = − l (cid:115) (2 l + 1)(2 l + 1)4 π (2 l + 1) C l l l C lml m l m Y lm . (3.20)Note that such product is commutative, since C l l l C lml m l m = C l l l C lml m l m and thatthe maximum value of the angular momentum l is given by l max = l + l .Let us now consider the ladder operators L + = L x + iL y = e iϕ (cid:18) ∂∂θ + i cot θ ∂∂φ (cid:19) , (3.21) L − = L x − iL y = − e − iϕ (cid:18) ∂∂θ − i cot θ ∂∂φ (cid:19) , (3.22) L z = − i ∂∂φ , (3.23) R + = R x + iR y = − ie iϕ (cid:18) cos θ ∂∂θ + i sin θ ∂∂φ − u sin θ ∂∂u (cid:19) , (3.24) R − = R x − iR y = − ie − iϕ (cid:18) cos θ ∂∂θ − i sin θ ∂∂φ − u sin θ ∂∂u (cid:19) , (3.25) R z = i (cid:18) sin θ ∂∂θ + u cos θ ∂∂u (cid:19) . (3.26)– 7 –heir action on spherical harmonics Y lm ( θ, φ ) is given by [15]: L + ( Y lm ) = (cid:112) l ( l + 1) − m ( m + 1) Y l,m +1 , (3.27) L − ( Y lm ) = (cid:112) l ( l + 1) − m ( m − Y l,m − , (3.28) L z ( Y lm ) = mY lm , (3.29) R + ( Y lm ) = − il (cid:115) ( l + m + 1)( l + m + 2)(2 l + 1)(2 l + 3) Y l +1 ,m +1 − i ( l + 1) (cid:114) ( l − m − l − m )4 l − Y l − ,m +1 , (3.30) R − ( Y lm ) = il (cid:115) ( l − m + 1)( l − m + 2)(2 l + 1)(2 l + 3) Y l +1 ,m − + i ( l + 1) (cid:114) ( l + m − l + m )4 l − Y l − ,m − , (3.31) R z ( Y lm ) = il (cid:115) ( l − m + 1)( l + m + 1)(2 l + 1)(2 l + 3) Y l +1 ,m − i ( l + 1) (cid:114) ( l + m )( l − m )4 l − Y l − ,m . (3.32)Notice that since the total angular momentum L does not commute with the boosts R i , theaction of a boost on a spherical harmonic changes in general its total angular momentum l . So far, we have discussed what is the effect of Lorentz transformations on the celestialsphere. The isometries of Minkowski space however comprise also the four-translations x (cid:48) µ = x µ + δx µ . In this section we describe their effect on the celestial sphere. Anyinfinitesimal time translation x (cid:48) = x + δx clearly maps u into u (cid:48) = u + δx . It meansthat the first observer will see the same celestial sphere of the second after a proper timeinterval δx . The two celestial spheres are just shifted in time by δx . A displacement byan infinitesimal spatial vector δ(cid:126)x induces the transformation u (cid:48) = u + (cid:126)x · δ(cid:126)xr and thus we canwrite an infinitesimal four-translation δx µ of the retarded time using spherical harmonics– 8 –s u (cid:48) = u + δx + δx cos φ sin θ + δx sin φ sin θ + δx cos θ ≡ u + (cid:88) l ∈{ , } l (cid:88) m = − l α lm Y lm ( θ, φ ) , (4.1)where α = √ πδx , α = − (cid:114) π δx ,α , − = − (cid:114) π δx + iδx ) , α = − (cid:114) π − δx + iδx ) . (4.2)While SL(2 , C ) / Z is a symmetry group at null infinity both for Minkowski spacetimeand for asymptotically flat spacetimes, the picture for the four-translations is dramaticallydifferent in the two cases. In fact, the boundary conditions of asymptotically flat spacetimesallow a larger class of transformations, known as supertranslations , which generalize (4.1)to arbitrary values of l : u (cid:48) = u + ∞ (cid:88) l =0 l (cid:88) m = − l α lm Y lm ( θ, φ ) , (4.3)with α lm complex numbers satisfying α lm = ( − m α ∗ l, − m . The generators of these trans-formations are the vector fields P lm = Y lm ( θ, φ ) ∂∂u , (4.4)which span the abelian algebra of supertranslations. The vector fields (4.4), together with(3.10)-(3.15), form the BMS algebra found by Sachs [4], which contains the Poincar´e algebraas a subalgebra. This shows that the asymptotic symmetry group of asymptotically flatspacetimes at null infinity is not the Poincar´e group, but the BMS group [21, 22] which isinfinite dimensional instead, and it is the semi-direct product SL(2 , C ) / Z (cid:110) S , where S isthe abelian group of supertanslations. The first step in order to obtain a non-commutative deformation of the celestial spherewill be to deform the algebra of spherical harmonics (3.20). This essentially boils downto the introduction of fuzzy spherical harmonics [23–26] which can be thought of as thealgebra of functions on a non-commutative space known as the fuzzy sphere [14, 27–37].This deformation of the algebra of spherical harmonics is concretely realized in terms ofa “quantization map” between the commutative algebra of functions on the two-sphere C ( S ) and the algebra of N × N complex matrices M N ( C ),Ω N : C ( S ) → M N ( C ) ; Ω N [ Y lm ( θ, ϕ )] = (cid:40) ˆ Y ( N ) lm l < N l ≥ N (5.1)– 9 –here the mapping between the spherical harmonics Y lm ( θ, φ ) and the matrices ˆ Y ( N ) lm isexplicitly realized as:ˆ Y ( N ) lm = 2 l l ! (cid:20) N ( N − − l )!( N + l )! (cid:21) ( J ( N ) · ∇ ) l (cid:16) r l Y lm ( θ, ϕ ) (cid:17) , (5.2)with J ( N ) = ( J ( N ) x , J ( N ) y , J ( N ) z ) and J ( N ) i are the N -dimensional spin matrices with spin j N [ J ( N ) i , J ( N ) j ] = i(cid:15) ijk J ( N ) k , J ( N )2 = j N ( j N + 1) I ( N ) , j N + 1 = N . (5.3)The fuzzy spherical harmonics are irreducible tensor operators of rank l and are propor-tional to the polarization tensors ˆ Y ( N ) lm = (cid:113) N π T ( N ) lm . We thus have that, given the ladderoperators J ( N ) ± = J ( N ) x ± iJ ( N ) y , their adjoint action (cid:46) on the fuzzy spherical harmonics isgiven by J ( N ) ± (cid:46) ˆ Y ( N ) lm ≡ [ J ( N ) ± , ˆ Y ( N ) lm ] = (cid:112) ( l ∓ m )( l ± m + 1) ˆ Y ( N ) l,m ± , (5.4) J ( N ) z (cid:46) ˆ Y ( N ) lm ≡ [ J ( N ) z , ˆ Y ( N ) lm ] = m ˆ Y ( N ) lm . (5.5)Furthermore, J ( N )2 (cid:46) ˆ Y ( N ) lm = (cid:104) J ( N )+ , (cid:104) J ( N ) − , ˆ Y ( N ) lm (cid:105)(cid:105) + (cid:104) J ( N ) z , (cid:104) J ( N ) z , ˆ Y ( N ) lm (cid:105)(cid:105) (5.6)(5.7) − (cid:104) J ( N ) z , ˆ Y ( N ) lm (cid:105) = l ( l + 1) ˆ Y ( N ) lm ≡ (cid:52) ˆ Y ( N ) lm , (5.8)where we have introduced the fuzzy Laplacian (cid:52) . This is the non-commutative analogue ofthe ordinary angular Laplacian and its eigenmatrices are the fuzzy harmonics. Its spectrumis truncated at l = l max = 2 j N = N −
1. Note that the operation (cid:46) is a derivation, that isthe non-commutative analogue of a vector field.The product of ˆ Y ( N ) l m and ˆ Y ( N ) l m can be expanded as a linear combination of ˆ Y ( N ) lm using6j-symbols [15] ˆ Y ( N ) l m ˆ Y ( N ) l m = j N (cid:88) l =0 ( − j N + l (cid:114) (2 l + 1)(2 l + 1)(2 j N + 1)4 π (5.9) × (cid:40) l l lj N j N j N (cid:41) C lml m l m ˆ Y ( N ) lm . (5.10)Notice that the 6j-symbols of (5.9) automatically vanish if the triangular conditions | l − l | 1, in contrast to what happens in the product of ordinary spherical– 10 –armonics (3.20). From the product above we can write the commutator (cid:104) ˆ Y ( N ) l ,m , ˆ Y ( N ) l m (cid:105) = j N (cid:88) l =0 ( − j N + l (cid:114) (2 l + 1)(2 l + 1)(2 j N + 1)4 π (5.11) × (cid:40) l l lj N j N j N (cid:41) C lml m l m ˆ Y ( N ) lm [1 − ( − l + l − l ] . (5.12)Using the product rule (5.9) and the asymptotic behaviour of the 6j symbols [38] for largevalues of N (cid:40) l l lj N j N j N (cid:41) ≈ ( − j + l (cid:112) (2 l + 1)(2 j N + 1) C l l l , (5.13)we have that lim N →∞ Ω − N (cid:16) ˆ Y ( N ) l m ˆ Y ( N ) l m (cid:17) = Y l m ( θ, φ ) Y l m ( θ, φ ) . and thus the commutator (5.11) vanishes in the large- N limit leading to the the usualcommutative algebra of spherical harmonics. On M N ( C ) we can introduce the followingscalar product (cid:16) ˆ Y ( N ) l m , ˆ Y ( N ) l m (cid:17) ( N ) = 4 πN Tr (cid:16) ˆ Y ( N ) † l m ˆ Y ( N ) l m (cid:17) = δ l l δ m m . (5.14)Since there are (cid:80) j N l =0 (2 l + 1) = N independent fuzzy spherical harmonics the set (cid:110) ˆ Y ( N ) lm (cid:111) ,equipped with (5.14) is a orthonormal basis in M N ( C ). Any element ˆ f ( N ) ∈ M N ( C ) canthus be expanded as ˆ f ( N ) = j N (cid:88) l =0 l (cid:88) m = − l (cid:16) ˆ Y ( N ) † lm , ˆ f ( N ) (cid:17) ( N ) ˆ Y ( N ) lm . (5.15)Again, note that this expansion is truncated at l max , in contrast to what happens in (3.18).The quantization map (5.1) can be extended by linearity to arbitrary functions of ( θ, φ )Ω N : f ( θ, φ ) = ∞ (cid:88) l =0 l (cid:88) m = − l f lm Y lm ( θ, φ ) → ˆ f ( N ) = N − (cid:88) l =0 l (cid:88) m = − l f lm ˆ Y ( N ) lm . (5.16)The set C N ( S ) ⊂ C ( S ) of truncated functions on the 2-sphere, i.e. the set of functionswhose expansion in terms of the spherical harmonics includes only terms with l < N as f ( N ) ( θ, φ ) = (cid:80) j N l =0 (cid:80) lm = − l f lm Y lm ( θ, φ ) is a vector space, but not an algebra with thestandard definition of pointwise product of two functions, since the product of two sphericalharmonics of order say N − N − 1, asremarked before. However, we can equip this vector space with a non-commutative (cid:63) -product via the Weyl-Wigner map: (cid:16) f ( N ) (cid:63) g ( N ) (cid:17) ( θ, φ ) = j N (cid:88) l =0 l (cid:88) m = − l (cid:16) ˆ Y ( N ) † lm , ˆ f ( N ) ˆ g ( N ) (cid:17) ( N ) Y lm ( θ, φ ) , (5.17)– 11 –urning C N ( S ) into a non-commutative algebra. This non-commutative algebra of func-tions can be interpreted as functions on the fuzzy sphere. An important feature introducedby the non-commutativity is that we now have a cut-off on the allowed values of the an-gular momentum l max in a way which is compatible with the multiplicative structure onthe space of non-commutative spherical harmonics. In what follows we will see how thenon-commutative deformation of spherical harmonics we just presented can be extended inorder to include an action of the Lorentz algebra which, together with the new multiplica-tive structure, is compatible with the presence of a maximal allowed value of the angularmomentum. In order to construct a non-commutative generalization of angular mode functions whichsupports an action of the full Lorentz algebra we look at the finite dimensional represen-tations of the latter. Every finite-dimensional irreducible representation of the Lorentzalgebra with dimension N = N N can be constructed in terms of spin matrices as L ( N ) i = J ( N ) i ⊗ I ( N ) + I ( N ) ⊗ J ( N ) i , (6.1) R ( N ) i = i (cid:16) J ( N ) i ⊗ I ( N ) − I ( N ) ⊗ J ( N ) i (cid:17) . (6.2)It is easy to check that these matrices close the Lorentz Lie algebra (3.16). For both sets ofspin matrices J ( N ) i and J ( N ) i we can construct their associated fuzzy spherical harmonicsˆ Y ( N ) lm and ˆ Y ( N ) lm which are N × N and N × N matrices, respectively, that satisfy all theproperties discussed in the previous Section. In particular, using (5.1), one can construct,for any fixed l < N i the complete set of fuzzy harmonics asˆ Y ( N i ) ll ∝ (cid:16) J ( N i )+ (cid:17) l , (6.3)which implies ˆ Y ( N i ) lm ∝ (cid:16) J ( N i ) − (cid:17) l − m (cid:46) ˆ Y ( N i ) ll (6.4)up to normalization factors. This is shown in Appendix A. This procedure automaticallystops when l = N i since (cid:16) J ( N i )+ (cid:17) N i = 0 for the spin matrices. The most straightforwardattempt at generalizing this procedure for the representation (6.1) would be thus to usethe the generator L + in place of J + . However, the matrices constructed using this strategydo not provide a basis for M N ( C ). Indeed writing the n -th power of the generator L + as (cid:16) L ( N )+ (cid:17) n = n (cid:88) k =0 (cid:32) nk (cid:33) (cid:16) J ( N )+ (cid:17) n − k ⊗ (cid:16) J ( N )+ (cid:17) k . (6.5)Setting n = N + h in the above sum the terms with k ≤ h are always 0 because (cid:16) J ( N )+ (cid:17) N =0. The term with k = h + 1 is (cid:16) J ( N )+ (cid:17) N − ⊗ (cid:16) J ( N )+ (cid:17) h +1 – 12 –hen h + 1 = N and hence n = N + N − (cid:16) L ( N )+ (cid:17) n = 0. We can thus onlyconstruct (cid:80) N + N − l =0 (2 l + 1) = ( N + N − independent matrices. But for N , N (cid:54) = 1we always have N + N − < N N , and hence we cannot construct a basis of ( N N ) matrices for the space of complex matrices M N ( C ).A resolution of this problem is found if we notice that the first equation in (6.1) is just thestatement that L is the sum of two angular momenta. From angular momentum theory, ifwe construct the matrices l l ˆ Y ( N ) LM = (cid:88) m m C LMl m l m ˆ Y ( N ) l m ⊗ ˆ Y ( N ) l m , (6.6)where C LMl m l m are the Clebsh-Gordan coefficients, we automatically have that L ( N ) ± (cid:46) l l ˆ Y ( N ) LM = (cid:112) ( L ∓ M )( L ± M + 1) l l ˆ Y ( N ) L,M ± , (6.7) L ( N ) z (cid:46) l l ˆ Y ( N ) LM = M l l ˆ Y ( N ) LM , (6.8) L ( N )2 (cid:46) l l ˆ Y ( N ) LM = L ( L + 1) l l ˆ Y ( N ) LM . (6.9)The matrices l l ˆ Y ( N ) LM are irreducible tensors of rank L and are eigenmatrices of J ( N )2 ⊗ I ( N ) and I ( N ) ⊗ J ( N )2 with eigenvalues l ( l + 1) and l ( l + 1), respectively. The allowedvalues of the total angular momentum are L = l min , ..., l max , and M = m + m = − L, ..., L with l min = | l − l | and l max = l + l as follows from the rules for the addition of twoangular momenta. Note that, since l = N − l = N − L isnever greater than L max = N + N − 2. The set (cid:110) l l ˆ Y ( N ) LM (cid:111) is an orthonormal basis in M N ( C ) with a scalar product analogous to the one of the fuzzy spherical harmonics, givenby (cid:16) l l ˆ Y ( N ) L M , l (cid:48) l (cid:48) ˆ Y ( N ) L M (cid:17) ( N ) = (4 π ) N Tr (cid:16) l l ˆ Y ( N ) † L M l (cid:48) l (cid:48) ˆ Y ( N ) L M (cid:17) = δ L L δ M M δ l l (cid:48) δ l l (cid:48) , . We would now like to obtain the explicit form for the action of boost generators on l l ˆ Y ( N ) LM .Using the expression for R ( N )+ (6.2) and l l ˆ Y ( N ) LM (6.6) R ( N )+ (cid:46) l l ˆ Y ( N ) LM = i (cid:88) m m C LMl m l m (cid:16)(cid:112) ( l − m )( l + m + 1) ˆ Y ( N ) l ,m +1 ⊗ ˆ Y ( N ) l m − (cid:112) ( l − m )( l + m + 1) ˆ Y ( N ) l m ⊗ ˆ Y ( N ) l ,m +1 (cid:17) . – 13 –ur goal is to express the right hand side of the action above as a linear combination ofthe basis matrices (cid:110) l l ˆ Y ( N ) LM (cid:111) . In order to do so one can evaluate the matrix elements (cid:16) ˆ l (cid:48) l (cid:48) Y ( N ) L (cid:48) M (cid:48) , R ( N )+ (cid:46) l l ˆ Y ( N ) LM (cid:17) ( N ) = i (4 π ) N (cid:88) m m (cid:48) m m (cid:48) C LMl m l m C L (cid:48) M (cid:48) l (cid:48) m (cid:48) l (cid:48) m (cid:48) × (cid:104) µ + ( l , m )Tr (cid:16) ˆ Y ( N ) † l (cid:48) m (cid:48) ˆ Y ( N ) l ,m +1 (cid:17) Tr (cid:16) ˆ Y ( N ) † l (cid:48) m (cid:48) ˆ Y ( N ) l m (cid:17) − µ + ( l , m )Tr (cid:16) ˆ Y ( N ) † l (cid:48) m (cid:48) ˆ Y ( N ) l ,m (cid:17) Tr (cid:16) ˆ Y ( N ) † l (cid:48) m (cid:48) ˆ Y ( N ) l ,m +1 (cid:17)(cid:105) = i (cid:88) m m C LMl m l m (cid:16) µ + ( l , m ) C L (cid:48) M (cid:48) l (cid:48) m +1 l (cid:48) m − µ + ( l , m ) C L (cid:48) M (cid:48) l (cid:48) m l (cid:48) m +1 (cid:17) δ l l (cid:48) δ l l (cid:48) , (6.10)where we used the shorthand notation µ ± ( l, m ) = (cid:112) ( l ∓ m )( l ± m + 1) . Notice that these matrix elements are non-vanishing only if l i = l (cid:48) i . The reader will find thedetails of the calculation in Appendix B. The final expression for the action of the boost R ( N )+ on our fuzzy harmonics is R ( N )+ (cid:46) l l ˆ Y ( N ) LM = iL (cid:115) ( L − M )( L − M − L − ( l min ) ][( l max + 1) − L )](4 L − l l ˆ Y ( N ) L − ,M +1 , (6.11)+ i l min ( l max + 1) L ( L + 1) (cid:112) ( L − M )( L + M + 1) l l ˆ Y ( N ) L,M +1 , − i ( L + 1) (cid:115) ( L + M + 1)( L + M + 2)[( L + 1) − ( l min ) ][( l max + 1) − ( L + 1) ](2 L + 1)(2 L + 3) l l ˆ Y ( N ) L +1 ,M +1 . The action of R ( N ) − and R ( N ) z can be calculated similarly and are given by R ( N ) − (cid:46) l l ˆ Y ( N ) LM = − iL (cid:115) ( L + M )( L + M − L − ( l min ) ][( l max + 1) − L )](4 L − l l ˆ Y ( N ) L − ,M − + i l min ( l max + 1) L ( L + 1) (cid:112) ( L + M )( L − M + 1) l l ˆ Y ( N ) L,M − + i ( L + 1) (cid:115) ( L − M + 1)( L − M + 2)[( L + 1) − ( l min ) ][( l max + 1) − ( L + 1) ](2 L + 1)(2 L + 3) l l ˆ Y ( N ) L +1 ,M − , (6.12)– 14 –nd R ( N ) z (cid:46) l l ˆ Y ( N ) LM = iL (cid:115) ( L + M )( L − M )[ L − ( l min ) ][( l max + 1) − L )](4 L − l l ˆ Y ( N ) L − ,M + iM l min ( l max + 1) L ( L + 1) l l ˆ Y ( N ) LM + i ( L + 1) (cid:115) ( L + M + 1)( L − M + 1)[( L + 1) − ( l min ) ][( l max + 1) − ( L + 1) ](2 L + 1)(2 L + 3) l l ˆ Y ( N ) L +1 ,M . (6.13)While these results might appear at first sight not very illuminating they are in fact re-markable. Indeed, unlike the case of commutative spherical harmonics on the celestialsphere, we now have a maximum value of the angular momentum L . Moreover the coeffi-cients of the l l ˆ Y ( N ) L +1 M + q terms automatically vanish if L equals l max and thus the actionof boosts, which in the standard case always maps the harmonic with given l to one with l + 1, is now compatible with the existence of a cut-off in the value of L . Thus the actions(6.7),(6.8),(6.9) and (6.11)(6.12),(6.13) could be thought as the non-commutative analogueof (3.27)-(3.32).As a final step let us write explicitly the algebra of the matrices (6.6). Using the summationrule [15] (cid:88) βγ(cid:15)ϕ C aαbβcγ C dδe(cid:15)fϕ C gηe(cid:15)bβ C jµfϕcγ = Π adgj (cid:88) ρσ C ρσgηjµ C ρσdδaα c b af e dj g k , where Π ab...c = (cid:112) (2 a + 1)(2 b + 1) ... (2 c + 1) and c b af e dj g k are 9j-symbols one finds thatsuch product is given by l (cid:48) l (cid:48) ˆ Y ( N ) L (cid:48) M (cid:48) l (cid:48)(cid:48) l (cid:48)(cid:48) ˆ Y ( N ) L (cid:48)(cid:48) M (cid:48)(cid:48) = (cid:88) LMl l √ N π (cid:113) (2 l + 1)(2 l (cid:48) + 1)(2 l (cid:48)(cid:48) + 1)(2 L (cid:48) + 1)(2 l + 1)(2 l (cid:48) + 1)(2 l (cid:48)(cid:48) + 1)(2 L (cid:48)(cid:48) + 1) × (cid:40) l (cid:48) l (cid:48)(cid:48) l j N j N j N (cid:41) (cid:40) l (cid:48) l (cid:48)(cid:48) l j N j N j N (cid:41) ( − j N +2 j N + l + l C LML (cid:48) M (cid:48) L (cid:48)(cid:48) M (cid:48)(cid:48) l (cid:48) l (cid:48) L (cid:48) l (cid:48)(cid:48) l (cid:48)(cid:48) L (cid:48)(cid:48) l l L l l ˆ Y ( N ) LM . (6.14)– 15 –or large values of N = N N we have (cid:40) l (cid:48) l (cid:48)(cid:48) l j N j N j N (cid:41) (cid:40) l (cid:48) l (cid:48)(cid:48) l j N j N j N (cid:41) ≈ ( − j N + j N + l + l (cid:112) N (2 l + 1)(2 l + 1) C l l (cid:48) l (cid:48)(cid:48) C l l (cid:48) l (cid:48)(cid:48) , (6.15)so that the algebra becomes l (cid:48) l (cid:48) ˆ Y ( N ) L (cid:48) M (cid:48) l (cid:48)(cid:48) l (cid:48)(cid:48) ˆ Y ( N ) L (cid:48)(cid:48) M (cid:48)(cid:48) ≈ (cid:88) LMl l π (cid:113) (2 l (cid:48) + 1)(2 l (cid:48)(cid:48) + 1)(2 L (cid:48) + 1)(2 l (cid:48) + 1)(2 l (cid:48)(cid:48) + 1)(2 L (cid:48)(cid:48) + 1) × C l l (cid:48) l (cid:48)(cid:48) C l l (cid:48) l (cid:48)(cid:48) C LML (cid:48) M (cid:48) L (cid:48)(cid:48) M (cid:48)(cid:48) l (cid:48) l (cid:48) L (cid:48) l (cid:48)(cid:48) l (cid:48)(cid:48) L (cid:48)(cid:48) l l L l l ˆ Y ( N ) LM , (6.16)which is exactly the algebra closed by the bipolar spherical harmonics (see e.g. [15]), asone would expect. For these reason, the matrices of (6.7) can be thought of as fuzzybipolar spherical harmonics . The ordinary bipolar spherical harmonics form a basis inthe algebra of functions on the manifold S × S and hence the fuzzy bipolar sphericalharmonics of (6.7) can be thought as a realization of a non-commutative S × S space.Their commutator is given by (cid:104) l (cid:48) l (cid:48) ˆ Y ( N ) L (cid:48) M (cid:48) , l (cid:48)(cid:48) l (cid:48)(cid:48) ˆ Y ( N ) L (cid:48)(cid:48) M (cid:48)(cid:48) (cid:105) = (cid:88) LMl l √ N π (cid:113) (2 l + 1)(2 l (cid:48) + 1)(2 l (cid:48)(cid:48) + 1)(2 L (cid:48) + 1)(2 l + 1)(2 l (cid:48) + 1)(2 l (cid:48)(cid:48) + 1)(2 L (cid:48)(cid:48) + 1) × (cid:40) l (cid:48) l (cid:48)(cid:48) l j N j N j N (cid:41) (cid:40) l (cid:48) l (cid:48)(cid:48) l j N j N j N (cid:41) ( − j N +2 j N + l + l C LML (cid:48) M (cid:48) L (cid:48)(cid:48) M (cid:48)(cid:48) l (cid:48) l (cid:48) L (cid:48) l (cid:48)(cid:48) l (cid:48)(cid:48) L (cid:48)(cid:48) l l L × [1 − ( − l + l + l (cid:48) + l (cid:48) + l (cid:48)(cid:48) + l (cid:48)(cid:48) ] l l ˆ Y ( N ) LM . (6.17)These equations define our non-commutative algebra of fuzzy bipolar spherical harmonics. We have shown how the algebra of spherical harmonics on the celestial sphere can be gen-eralized to a non-commutative algebra in order to accommodate a maximal value of theangular momentum. In particular, we derived an action of Lorentz boosts which is consis-tent with the existence of a maximal angular momentum. Our construction is based on amatrix realization of angular mode functions and uses basic techniques of non-commutative– 16 –eometry. These results suggest that, since the generators of supertranslations of the BMSgroup are proportional to the spherical harmonics on the celestial sphere, it could be pos-sible to construct a generalization of the BMS algebra characterized by a non-abelian sub-algebra of supertranslations having a finite number of generators. These would givea finite number of conserved supertranslation charges and thus non-commutativity, or thefuzziness of the angular mode functions, could be the ingredient needed to provide a consis-tent cut-off mechanism for soft modes. It is tempting to speculate that a similar mechanismcould be used to provide the missing link between soft hair and the Bekenstein-Hawkingentropy for black holes. Acknowledgements We would like to thank Patrizia Vitale and Alessandro Zampini for very useful discussionson various aspects of fuzzy geometries. A Construction of the fuzzy spherical harmonics An explicit way to construct the fuzzy spherical harmonics can done by using directly theWeyl-Wigner map of (5.2) and equation (5.4). The scalar product J ( N ) · ∇ in sphericalcomponents reads as J ( N ) · ∇ = J ( N ) i ∇ i = − J ( N )+1 ∇ − + J ( N )0 ∇ − J ( N ) − ∇ + , (A.1)where the spherical components of a vector A are defined as usual, A ± = ∓ √ ( A x ± iA y )and A = A z . Hence, the contact with the notation we used previously is J ( N )+1 = − √ J ( N )+ , J ( N ) − = 1 √ J ( N ) − , J ( N )0 = J ( N ) z . (A.2)Furthermore the followig identities hold [15] ∇ [ r l Y lm ( θ, φ )] = (cid:115) l − m (2 l + 1)(2 l − 1) (2 l + 1) r l − Y l − m ( θ, φ ) , (A.3) ∇ ± [ r l Y lm ( θ, φ )] = − (cid:115) ( l ∓ m − l ∓ m )2(2 l − l + 1) (2 l + 1) r l − Y l − m − ( θ, φ ) . (A.4)Suppose we want to construct ˆ Y ( N ) ll . We must apply (A.1) l times to Y lm ( θ, φ ). Thefirst time we apply it, only the term proportional to ∇ − in (A.1) contributes, producing − (cid:113) l (2 l +1)2 r l − Y l − m − ( θ, φ ). In general, everytime we apply the operator (A.1) only theterm proportional to ∇ − will contribute. Acting n times we have( − J ( N )+1 ∇ − ) n ( r l Y lm ( θ, φ )) = (cid:16) J ( N )+1 (cid:17) n (cid:114) (2 l + 1)(2 l ) ... (2 l − n + 2)2 n r l − n Y l − n,m − n ( θ, φ ) . (A.5) For a recent attempt at generalizing the BMS algebra using quantum group techniques see [39]. – 17 –or n = l we get, for ˆ Y ( N ) ll ˆ Y ( N ) ll = 2 l l ! (cid:20) N ( N − − l )!( N + l )! (2 l + 1)2 l (2 l − ... l (cid:21) √ π (cid:16) J ( N )+1 (cid:17) l ∝ (cid:16) J ( N )+ (cid:17) l , (A.6)as claimed in (6.3). By acting on Y ( N ) lm with the lowering operator J ( N ) − it is possible toconstruct all the 2 l + 1 fuzzy spherical harmonics at fixed l . B Derivation of the action of Lorentz boosts on bipolar fuzzy sphericalharmonics From the following recursion formula for the Clebsch-Gordan coefficients µ − ( L (cid:48) , M (cid:48) ) C L (cid:48) M (cid:48) − l m l m (cid:48) = µ − ( l , m + 1) C L (cid:48) M (cid:48) l m +1 l m + µ − ( l , m + 1) C L (cid:48) M (cid:48) l m l m +1 , we have that C L (cid:48) M (cid:48) l m +1 l m = µ − ( L (cid:48) , M (cid:48) ) µ − ( l , m + 1) C L (cid:48) M (cid:48) − l m l m (cid:48) − µ − ( l , m + 1) µ − ( l , m + 1) C L (cid:48) M (cid:48) l m l m +1 . Plugging this expression in (6.10) we obtain (cid:16) ˆ l l Y ( N ) L (cid:48) M (cid:48) , R ( N )+ (cid:46) l l ˆ Y ( N ) LM (cid:17) ( N ) = i (cid:88) m m C LMl m l m (cid:20) µ + ( l , m ) µ − ( L (cid:48) , M (cid:48) ) µ − ( l , m + 1) C L (cid:48) M (cid:48) − l m l m (cid:48) − (cid:18) µ + ( l , m ) µ − ( l , m + 1) µ − ( l , m + 1) + µ + ( l , m ) (cid:19) C L (cid:48) M (cid:48) l m l m +1 (cid:21) = i (cid:88) m m (cid:104) µ − ( L (cid:48) , M (cid:48) ) C LMl m l m C L (cid:48) M (cid:48) − l m l m (cid:48) − µ + ( l , m ) C LMl m l m C L (cid:48) M (cid:48) l m l m +1 (cid:105) , (B.1)where we have used µ + ( l, m ) = µ − ( l, m + 1) . From the orthogonality of the Clebsh-Gordan coefficients (cid:88) m m C LMl m l m C L (cid:48) M (cid:48) l m l m = δ LL (cid:48) δ MM (cid:48) , we have that the first term of the matrix element (B.1) is iµ − ( L (cid:48) , M (cid:48) ) δ LL (cid:48) δ M (cid:48) M +1 . (B.2)For the second term − i (cid:88) m m (cid:112) ( l − m )( l + m + 1) C LMl m l m C L (cid:48) M (cid:48) l m l m +1 , (B.3)– 18 –e use the following recursion relation [15] (cid:112) ( l − m )( l + m + 1) C LMl m l m = − L (cid:115) ( L − M )( L − M − L − ( l − l ) ][( l + l + 1) − L )](4 L − C L − M +1 l m l m +1 + 12 L ( L + 1) [( l ( l + 1) − l ( l + 1) + L ( L + 1)] (cid:112) ( L − M )( L + M + 1) C LM +1 l m l m +1 + 12( L + 1) (cid:115) ( L + M + 1)( L + M + 2)[( L + 1) − ( l − l ) ][( l + l ) − L + 2( l + l − L )](2 L + 1)(2 L + 3) × C L +1 M +1 l m l m +1 . Substituting and using again the orthogonality condition for the Clebsh-Gordan coefficientswe have that the second term in (B.3) can be written as iL (cid:115) ( L − M )( L − M − L − ( l − l ) ][( l + l + 1) − L )](4 L − δ L (cid:48) ,L − δ M (cid:48) ,M +1 − iL ( L + 1) [( l ( l + 1) − l ( l + 1) + L ( L + 1)] (cid:112) ( L − M )( L + M + 1) δ L (cid:48) ,L δ M (cid:48) ,M +1 − i ( L + 1) (cid:115) ( L + M + 1)( L + M + 2)[( L + 1) − ( l − l ) ][( l + l ) − L + 2( l + l − L )](2 L + 1)(2 L + 3) × δ L (cid:48) ,L +1 δ M (cid:48) ,M +1 . The term proportional to δ L (cid:48) ,L δ M (cid:48) ,M +1 in the previous expression, together with (B.2), canbe written as i [ l ( l + 1) − l ( l + 1)] L ( L + 1) (cid:112) ( L − M )( L + M + 1) = i ( l − l )( l + l + 1) L ( L + 1) (cid:112) ( L − M )( L + M + 1) = ± i l min ( l max + 1) L ( L + 1) (cid:112) ( L − M )( L + M + 1) . (B.4)Using similar procedures it is possible to obtain the action of R ( N ) − and of R ( N ) z on l l ˆ Y ( N ) LM . 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