Energy cutoff, effective theories, noncommutativity, fuzzyness: the case of O(D)-covariant fuzzy spheres
EEnergy cutoff, effective theories, noncommutativity,fuzzyness: the case of O ( D ) -covariant fuzzy spheres Gaetano Fiore ∗ , Francesco Pisacane Dip. di Matematica e Applicazioni, Università di Napoli “Federico II”,& INFN, Sezione di Napoli,Complesso Universitario M. S. Angelo, Via Cintia, 80126 Napoli, ItalyE-mail: [email protected] , [email protected] Projecting a quantum theory onto the Hilbert subspace of states with energies below a cut-off E may lead to an effective theory with modified observables, including a noncommutativespace(time). Adding a confining potential well V with a very sharp minimum on a submanifold N of the original space(time) M may induce a dimensional reduction to a noncommutative quantumtheory on N . Here in particular we briefly report on our application [1, 2, 3, 4, 5] of this procedureto spheres S d ⊂ R D of radius r = D = d + > E and the depth of the well dependon (and diverge with) Λ ∈ N we obtain new fuzzy spheres S d Λ covariant under the full orthogonalgroups O ( D ) ; the commutators of the coordinates depend only on the angular momentum, as inSnyder noncommutative spaces. Focusing on d = ,
2, we also discuss uncertainty relations, lo-calization of states, diagonalization of the space coordinates and construction of coherent states.As Λ → ∞ the Hilbert space dimension diverges, S d Λ → S d , and we recover ordinary quantummechanics on S d . These models might be suggestive for effective models in quantum field theory,quantum gravity or condensed matter physics. Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity"(CORFU2019), 31 August - 25 September 2019, Corfu, Greece ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ m a t h - ph ] A ug nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore
1. Introduction
The first example of noncommutative spacetime was proposed in 1947 by Snyder [6] with thehope that nontrivial (but Poincaré covariant) commutation relations among the coordinates couldcure ultraviolet (UV) divergences in quantum field theory (QFT) . Shortly afterwards the regular-ization of UV divergences based on an energy cutoff, although not Poincaré covariant, allowedthe renormalization of quantum electrodynamics; in the following decades this and other regu-larization methods within the renormalization program have allowed the extraction of physicallyaccurate predictions from quantum electrodynamics, chromodynamics, and the Standard Model ofelementary particle physics. Therefore Snyder’s model was almost forgotten for long time. On theother hand, there is general consensus that any merging of quantum theory and general relativityin an acceptable quantum gravity theory should lead to a cutoff (upper bound) on the local concen-tration of energy and to an associated lower bound (the Planck length l p = (cid:112) ¯ hG / c ∼ − cm)on the localizability of events. In fact, by Heisenberg uncertainty relations, to reduce the uncer-tainty ∆ x of the coordinate x of an event one must increase the uncertainty ∆ p x of the conjugatedmomentum component by use of higher energy probes; but by general relativity the associatedconcentration of energy in a small region would produce a trapping surface (event horizon of ablack hole) if it were too large; hence the size of this region, and ∆ x itself, cannot be lower than theassociated Schwarzschild radius, i.e. l p . This heuristic argument [8] was made made more preciseby Doplicher, Fredenhagen, Roberts [9], who also proposed that the latter bound could follow fromappropriate noncommuting coordinates (for a review of more recent developments see [10]).We begin this paper observing that in fact all these facts may stem from the same (energy cut-off) mechanism: introducing an energy cutoff E in a quantum theory on a commutative space(time) M , i.e. projecting the theory on the Hilbert subspace with energy below E , directly induces a non-commutative deformation of the latter and lower bounds for the space(time) localizability. More-over, adding a confining potential well V with a very sharp minimum on a submanifold N of M mayinduce a dimensional reduction to a noncommutative quantum theory on N . In [1, 2, 5] we haveapplied this idea to obtain new fuzzy spheres S d Λ of any dimension d starting from quantum me-chanics on ordinary Euclidean spaces; while the seminal Madore-Hoppe fuzzy sphere (FS) [17, 18]is covariant only under the rotation group, our S d Λ are covariant under the whole orthogonal groups.After the mentioned general arguments, here we summarize how the S d Λ are constructed and theirmain features, including uncertainty relations, localization of states, diagonalization of the spacecoordinates and construction of coherent states [3, 4] for d = , M is a sequence { A n } n ∈ N of finite-dimensional algebras such that A n n → ∞ −→ A ≡ algebra of regular functions on M . Since their intro-duction fuzzy spaces have raised a keen interest among mathematical and high-energy physicistsas a non-perturbative technique in QFT (or string, or M-, theory) based on a finite-discretizationof space(time) alternative to the lattice one; one main advantage is that the algebras A n can carryrepresentations of Lie groups (not only of discrete ones). In a QFT on a fuzzy space the “cutoff” n works as a parameter regularizing UV divergences, because integration over fields amounts tointegration over matrices of a finite size, growing with n (see e.g. [19, 20] for the first QFT on The idea had originated in the ’30s from Heisenberg, who proposed it in a letter to Peierls [7]; the idea propagatedvia Pauli to Oppenheimer, who asked his student Snyder to develop it. nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore the FS [17, 18], and [21, 22, 23, 24] for examples of QFT on fuzzy spheres of higher dimensions).If spacetime M is enlarged to a higher-dimensional one M (cid:48) = M × S n - where S n is a fuzzy space,instead of a compact manifold S - it reduces the number of massive Kaluza-Klein modes of a fieldtheory on M (cid:48) to a finite value [25, 26] (the extra dimensions can be used to describe internal degreesof freedom). In the matrix model formulations of M -theory [27, 28] and string theory [29] fuzzyspaces may arise as subalgebras giving the leading contribution to the path-integrals over largermatrix algebras; they respectively lead to quantized branes in a 11- or 10- dimensional spacetime.Consider a quantum theory T ; we denote the Hilbert space of the system S by H , thealgebra of observables on H (or with a domain dense in H ) by A ≡ Lin ( H ) , the Hamiltonianby H ∈ A . For a generic subspace H ⊂ H let P : H (cid:55)→ H be the associated projection and A ≡ Lin (cid:0) H (cid:1) = { A ≡ PAP | A ∈ A } (cid:54) = A . Assume now H is a subspace such that: i) PH = HP ; ii) A contains all the observablescorresponding to measurements that we can really perform with the experimental apparati at ourdisposal. If the initial state of the system belongs to H , then neither the dynamical evolutionruled by H , nor any measurement can map it out of H , and we can describe S by the effectivetheory T based on the projected Hilbert space H , algebra of observables A and Hamiltonian H = H | H . If H , H are invariant under some group G , then P , A , H , T will be as well.As a particular consequence, if the theory T is based on commuting coordinates x i (com-mutative space) this will be in general no longer true for T : [ x i , x j ] (cid:54) = . A physically relevant instance of the above projection mechanism occurs when H is thesubspace of H characterized by energies E below a certain cutoff, E ≤ E ; then T is a low-energy effective approximation of T . The prototypical example is Peierls projection [11] (see also[12, 13]) applied to the Landau model of a charged particle in a plane subject to a perpendicularmagnetic field B : choosing E equal to the ground state energy E implies [ x , x ] = i ¯ hcieB (here e is the electric charge of the particle, c is the speed of light, x , x are the Cartesian coordinatesof the particle on the plane), so that the effective theory is on a noncommutative space. E isa deformation parameter, in the sense T → T as E → ∞ . If H is G -invariant then also H andtherefore P , A , H , T automatically are. Given an observable A (e.g. A = x , in the Landau model), A will measure the same physical quantitity (the x coordinate of the particle, in the mentionedexample) with an uncertainty compatible with E ≤ E ; in other words, the measurement processcannot make the system jump out of H , i.e. in states of energy E > E .Imposing an energy cutoff E ≤ E on theory T may be useful at least for the following reasons(which may co-exist): • If H ⊥ is practically not accessible in preparing the initial state, nor through the dynami-cal evolution (which may include interactions with the environment, encoded in the possi-bly time-dependent Hamiltonian), nor through the measurement processes, then T on thesmaller Hibert space H is in principle sufficient for determining all physical predictions andin fact simpler to work with. If the state is not pure, but described by a density matrix ρ , the condition becomes "if ρ ∈ A ". nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore • If at E > E we expect new physics not accountable by T , then T may also help to figureout a new theory T (cid:48) valid for all E . • As a regularization procedure of a QFT T , an energy cutoff E may allow to make senseof T if this is originally ill-defined due to UV divergences - e.g. divergent contributions(loop integrals) to transition amplitudes - for generic finite values of (a finite number of) bareparameters µ I (e.g. masses, coupling constants,...) present in the Hamiltonian/Action. Thesedivergent contributions are due to virtual intermediate states of arbitrarily high energy E thatcan be assumed by the system during the interaction. Imposing E ≤ E (or some other regu-larization scheme) allows to make the (unknown) µ I well-defined (at least in a perturbativesense) functions µ I ( Q , E ) of a small number of observable quantities Q i (e.g. masses ofasymptotic states, large distance coupling constants,...) and of E (or the other regularizationparameter). Replacing these functions in the dependences O A ( µ ) of all the observables O A (e.g. cross sections in scattering processes, decay times of unstable particles, etc.; here A stands for a collective index which allows to distinguish not only the type of observable, butalso the involved initial and final data, e.g. the initial and final type, number, momenta of theparticles involved in a scattering process) on the µ I yields functions O A ( Q , E ) . If the latteradmit E → ∞ limits the theory is said to be UV renormalizable, and these limits tipically givea physically accurate relation between Q and the observed O A .As a T consider now quantum mechanics (QM) of a zero spin particle on R D with a Hamil-tonian H ( xxx , ppp ) . If H is the subspace with energies E ≤ E then its dimension is approximatelythe phase space volume of the classical region B E determined by the inequality H (cid:0) xxx c , ppp c (cid:1) ≤ E , inPlanck constant units: dim ( H ) (cid:39) Vol ( B E ) / h D . This is still infinite if e.g. H reduces to the kinetic term ppp (upper part of fig. 1), while it is finiteif H contains a sufficiently strong binding potential V ( xxx ) (lower part of fig. 1); consequently T will be a fuzzy approximation of QM approximately confined in the (configuration space) region R ⊂ R D determined by the inequality V ( xxx ) ≤ E . We can obtain a
NC, fuzzy approximation ofQM on a submanifold N of R D adding a ‘dimensional reduction’ mechanism , more precisely a V ( xxx ) with a sharp minimum on N . In the rest of the paper we report on our application [1, 2, 3, 4, 5] of the mechanism for N equal to the d = ( D − ) -dimensional sphere S d of radius r = r : = xxx is the square distance fromthe origin) and on the study of the resulting fuzzy spheres for d = , R (a thin spherical shell of radius (cid:39)
1) in the d = S d Λ for generic d ≥
1. In sections 4, 5 we respectively review the mainfeatures of S Λ , S Λ , the eigenvalues and eigenvectors of the associated coordinate operators x i (for Of course, one can obtain a fuzzy noncommutative approximation of QM in a region R also imposing an energycutoff on a pre-existing noncommutative deformation of QM on R , see e.g. the fuzzy disc of [14]. In passing, we note that defining submanifolds of noncommutative spaces is delicate problem [15];[16] proposes arather general procedure in the framework of Drinfel’d twist deformations of differential geometry. nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore
Figure 1:
Up: Classically allowed phase space region H ( xxx c , ppp c ) ≤ E if H = ppp . Down: Classically allowedconfiguration space region if H = ppp c + V (cid:0) xxx c (cid:1) , with the potential of the form V (cid:0) xxx c (cid:1) = a ( x c ) + b ( x c ) (left)or V (cid:0) xxx c (cid:1) = k ( r c − ) (right), where r c = √ xxx c . the latter we prove slightly stronger results than in [3]); then we present various systems of coherentstates (SCS) on them and discuss their localization both in configuration and (angular) momentumspace. Finally, in section 6 we draw the conclusions and add final remarks, while comparing our S d Λ with other fuzzy spheres, in particular the celebrated Madore-Hoppe Fuzzy sphere (FS) [17, 18].
2. Further preliminaries O ( D ) -covariance of the theory means that for any orthogonal matrix g ∈ O ( D ) there is a unitarytransformation U ( g ) of the Hilbert space H (or H ) such that g i j v j = U † ( g ) v i U ( g ) for all vectors vvv , and similarly for other O ( D ) -tensors. Fixed a vvv , we can split H = (cid:91) uuu ∈ S d H uuu , H uuu : = (cid:110) ψ ∈ H | (cid:104) vvv (cid:105) ψ = (cid:12)(cid:12) (cid:104) vvv (cid:105) ψ (cid:12)(cid:12) uuu (cid:111) . (2.1)For each (unit vector) uuu ∈ S d consider a g ∈ O ( D ) such that guuu = eee , where eee : = ( , ,..., ) , anddefine vvv (cid:48) : = gvvv , so that vvv · uuu = v (cid:48) . For all ψ ∈ H uuu we find (cid:104) vvv (cid:48) (cid:105) ψ = |(cid:104) vvv (cid:105) ψ | eee ; moreover, H uuu = U † ( g ) H eee (of course one obtains the same result replacing eee by any other eee i ).4 nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore
Figure 2:
Left: the vectors xxx , uuu ≡ (cid:104) xxx (cid:105) , xxx − (cid:104) xxx (cid:105) , the region σ and the tangent plane T uuu S d at uuu . Right:perpendicularity of xxx and LLL . R D , S d and S d Λ A good measure of the localization of a state in configuration space R D is its spacial disper-sion , i.e. the O ( D ) -invariant (and therefore reference-frame-independent) expectation value ( ∆ xxx ) ≡ D ∑ i = ( ∆ x i ) ≡ (cid:68) ( xxx −(cid:104) xxx (cid:105) ) (cid:69) = (cid:10) xxx (cid:11) − (cid:104) xxx (cid:105) (2.2)on the state. Here xxx ≡ ( x , ..., x n ) is the vector position observable of the particle in the ambientEuclidean space R D , the vector (cid:104) xxx (cid:105) ≡ ( (cid:104) x (cid:105) , ..., (cid:104) x n (cid:105) ) pinpoints the average position, the scalarobservable xxx : = ∑ Di = x i x i measures the square distance from the origin, the vector observable xxx −(cid:104) xxx (cid:105) measures the displacement from (cid:104) xxx (cid:105) ; (2.2) is the expectation value of the square of the latter.We adopt ( ∆ xxx ) also on S d , S d Λ : in fact, if the state is localized in a small region σ ⊂ S d around apoint uuu ≡ (cid:104) xxx (cid:105) ∈ S d then ( ∆ xxx ) essentially reduces to the average square displacement in the tangentplane at uuu (see fig. 2, left), as wished. If xxx ≡ H (this occurs strictlyif H = L ( S d ) and also on Madore’s FS S n , only approximately on our S d Λ ), then (cid:10) xxx (cid:11) is state-independent, and (2.2) is minimal on the states with maximal (cid:104) xxx (cid:105) . By (2.1) with vvv = xxx , in each H uuu (cid:104) xxx (cid:105) is maximized on the eigenvector(s) ψ of x (cid:48) = xxx · uuu with the highest (in absolute value)eigenvalue (the latter exists on the Madore’s FS, while on S d it exists as a generalized eigenvector). x i , and most localized states For x i to approximate well and O ( D ) -covariantly a coordinate of a quantum particle forced tostay on the commutative sphere S d , its spectrum Σ x i should fulfill at least the following properties:1. Σ x i is the same for all i = , ..., D and choices of the reference frame. In particular, it isinvariant under inversion x i (cid:55)→ − x i .2. In the commutative limit Σ x i becomes uniformly dense in [ − , ] , in particular the maximaland the minimal eigenvalues converge to 1 and −
1, respectively.These properties are fulfilled by both the Madore FS and (at least for d = ,
2) our S d Λ . As explainedin the previous subsection, the eigenstates with maximal eigenvalue (in absolute value) have alsomaximal localization on S d , S n ; this also approximately ture on our S d Λ .5 nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore
We recall that the canonical SCS { φ z } z ∈ Ω ⊂ H on R D can be defined in three equivalent ways:1. As the set of states saturating Heisenberg uncertainty relations (HUR) ∆ x i ∆ p i ≥ / a i with set of joint eigenvalues z ∈ Ω .3. As the set of states generated by the group G acting on the vacuum state φ .Here H = L ( R D ) , Ω ≡ C D , all variables have been made dimensionless, a i = x i + ip i , and G isthe Heisenberg-Weyl group. Characterizations 1 (for D = Strong continuity of φ z as a function of z ∈ Ω ;b) Resolution of the identity: id = (cid:82) Ω d µ ( z ) P z , P z = φ z (cid:104) φ z , ·(cid:105) ≡ | φ z (cid:105)(cid:104) φ z | ;c) Completeness : Span { φ z | z ∈ Ω } = H .where d µ ( z ) = d ℜ ( z ) d ℑ ( z ) , and the resolution b) is in the weak sense. These properties are oftenused [37] for defining SCS in general: a set { φ z } z ∈ Ω ⊂ H , where Ω is a topological label space,is a strong SCS if it fulfills a), b) with a suitable integration measure d µ ( z ) on Ω ; a weak SCS ifit fulfills a), c). As b) implies c), a strong SCS is also weak. Perelomov and Gilmore develop[30, 32] the concept of SCS through approach 3 choosing Ω either a generic Lie group G , or moregenerally a coset G / H thereof, acting on H via an irreducible unitary representation T (see e.g.[31]). The steps are as follows: • For all φ ∈ H , let φ g ≡ T ( g ) φ for all g ∈ G , H ≡ { h ∈ G | φ h = exp [ i α ( h )] φ } . • Then | φ g (cid:105)(cid:104) φ g | = | φ gh (cid:105)(cid:104) φ gh | ≡ P z , i.e. depends only on z ≡ [ g ] ∈ G / H ≡ Ω . • If φ is admissible , i.e. (cid:82) G |(cid:104) φ , T ( g ) φ (cid:105)| dg < ∞ , where dg is the left-invariant Haar measureon G , then b) holds with d µ ( z ) the normalized measure induced by dg on Ω .Clearly, if G is compact all φ ∈ H are admissible. Following Perelomov, the CS that are closest toclassical states are obtained from a φ that maximizes H , or better the isotropy subalgebra b in the complex hull of the Lie algebra of G ; φ is annihilated by some element(s) in b , the corresponding φ g are eigenvectors of the latter (property 2) and minimize the G -invariant uncertainty associated tothe quadratic Casimir [ ( ∆ LLL ) = ∑ i < j ∆ L i j in the case G = SO ( D ) ]. For G = SO ( ) it is H = SO ( ) , ( ∆ LLL ) = (cid:10) LLL (cid:11) − (cid:104) LLL (cid:105) (with L i ≡ ε i jk L jk / ( ∆ LLL ) amounts to saturating a specificUR [4] (hence also property 1 holds); this SCS consists of the socalled coherent spin or Bloch states.In introducing SCS on S d Λ ( d = ,
2) we follow in spirit Perelomov’s approach, with G theisometry group O ( D ) of S d (a compact group). However, our Hilbert space H Λ will in generalcarry a reducible representation of O ( D ) ; moreover, we study the localization properties of theseSCS both in configuration and (angular) momentum space.6 nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore
Figure 3:
Three-dimensional plot of V ( r ) Figure 4:
Two-dimensional plot of V ( r ) includingthe energy-cutoff and allowed energy levels (black).
3. Construction of the S d Λ for general d ≥ The main steps of the costructions are as follows: • We adopt a O ( D ) invariant Hamiltonian H = − ∆ + V ( r ) , (3.1)where the confining potential V ( r ) has a very sharp minimum V = V ( ) at r =
1. Moreprecisely, we assume that V ( r ) (cid:39) V + k ( r − ) if V ( r ) ≤ E , (3.2)so that V ( r ) has a harmonic behavior for | r − | ≤ (cid:113) E − V k , and that V (cid:48)(cid:48) ( ) ≡ k (cid:29) k thusparametrizes the sharpness of the minimum); we fix V so that the ground state has energy E =
0. Using polar coordinates we can decompose ∆ = ∂ r + dr ∂ r − r LLL , where LLL : = L i j L i j / L i j : = i ( x j ∂∂ x i − x i ∂∂ x j ) are the angular momentumcomponents], i.e. the Hamiltonian of free motions (the Laplacian) on S d . Looking for theeigenfunctions ψ of H in the form ψ = f ( r ) Y ( ϕ , ... ) , where ϕ , ... are the angular coordinates,we reduce the eigenvalue equation H ψ = E ψ to a 1-dimensional Schrödinger equation in theform of an ordinary differential equation with respect to r . The eigenvalues are parametrizedby integers l , n ≥
0; they respectively determine the eigenvalue E l ≡ l ( l + d − ) of LLL andthe radial excitation, which at least for small n are approximately of harmonic type, (cid:39) √ kn . • We choose E low enough, e.g. E (cid:46) √ k , to constrain n to be zero, namely to eliminate radialexcitations from the spectrum Σ H of H , so that the latter reduces to that of LLL , Σ H = { E l } .Then we also find that the x i generate the whole algebra of observables A , and [ x i , x j ] (cid:39) nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore − iL i j / k , i.e. we find Snyder-type commutation relations among the coordinates . There isa residual freedom in the choice of V ( r ) (the higher order terms in the Taylor expansion of V ( r ) around r = [ x i , x j ] = − iL i j / k (up to terms thatact non-trivially only on the highest energy states). • To obtain a sequence of finite-dimensional models going to QM on S d we make E grow anddiverge with a natural number Λ ; so must also k do, in order that the above inequality keepsholding. We choose E ≡ E Λ = Λ ( Λ + d − ) and V depending on Λ so that k ( Λ ) ≥ Λ ( Λ + ) ;correspondingly, Σ H = { E l } Λ l = , and replacing everywhere the bar by the subscript Λ we find ( H Λ , A Λ ) Λ → ∞ −→ ( H , A ) ≡ (cid:16) L ( S d ) , Lin (cid:16) L ( S d ) (cid:17)(cid:17) (3.4)in a suitable sense [1]. { S d Λ } Λ ∈ N ≡ { ( H Λ , A Λ ) } Λ ∈ N is our d -dimensional, O ( D ) -covariantfuzzy sphere , i.e. a sequence of finite-dimensional approximations of ordinary QM on S d .It turns out that (at least for D = ,
3) there exist O ( D ) -covariant ∗ -algebra isomorphisms A Λ (cid:39) π Λ [ U so ( D + )] , where ( π Λ , H Λ ) is a suitable irreducible unitary representation of U so ( D + ) . More precisely, in terms of the canonical basis { L IJ | ≤ I < J ≤ D + } of so ( D + ) , L i j = π Λ ( L i j ) , x h = π Λ (cid:2) f (cid:0) LLL (cid:1) L h ( D + ) f (cid:0) LLL (cid:1)(cid:3) , ≤ i , j , h ≤ D , i < j , (3.5)where and f ( s ) , f ( s ) are suitable analytic functions.To simplify the notation, below we shall remove the bar and denote the generic A ∈ A Λ as A . D = : O ( ) -covariant fuzzy circle In a suitable orthonormal basis B Λ : = { ψ Λ , ψ Λ − , ..., ψ − Λ } of the Hilbert space H Λ consistingof eigenvectors of the angular momentum L ≡ L , L ψ n = n ψ n , (4.1)the action of the noncommutative coordinates x ± : = x ± ix of the fuzzy circle S Λ read x ± ψ n = (cid:20) + n ( n ± ) k (cid:21) ψ n ± if − Λ ≤ ± n ≤ Λ − , k = k ( Λ ) ≥ Λ ( Λ + ) . In the Λ = ∞ limit x ± = e ± i ϕ , ψ n = e in ϕ (up to a phase); ϕ is theangle along S . L , x + , x − and xxx : = x + x = ( x + x − + x − x + ) fulfill the O ( ) -equivariant relations [ L , x ± ] = ± x ± , x + † = x − , L † = L , (4.3) Snyder’s quantized spacetime algebra is generated by 4 hermitean Cartesian coordinate operators { x µ } µ = , , , ,and 4 hermitean momentum operators (cid:8) p µ (cid:9) µ = , , , fulfilling (here α is a suitable constant) [ p µ , p ν ] = , [ x µ , p ν ] = i ¯ h ( δ µν − α p µ p ν ) , [ x µ , x ν ] = − i ¯ h α L µν , µ , ν = , , , L µν = x µ p ν − x ν p µ and v µ = η µν v ν , with η = diag ( , − , − , − ) = η − the Minkowski metric matrix. Here we use the conventions of [3, 4], rather than those of [1]. nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore [ x + , x − ] = − Lk + (cid:20) + Λ ( Λ + ) k (cid:21)(cid:16) (cid:101) P Λ − (cid:101) P − Λ (cid:17) ≡ L (cid:48) , (4.4) xxx = + L k − (cid:20) + Λ ( Λ + ) k (cid:21) (cid:101) P Λ + (cid:101) P − Λ , (4.5) Λ ∏ n = − Λ ( L − n I ) = , ( x ± ) Λ + = . (4.6)Here (cid:101) P n is the projection onto the 1-dim subspace C ψ n . Terms marked red are absent in the com-mutative case. In the Λ → ∞ limit also the non-vanishing ones will play no role at any fixed energy E , as they are proportional to the projections (cid:101) P ± Λ onto the states with highest energy E Λ → ∞ ;(4.6a) gives back Σ L = Z , whereas (4.6b) looses meaning and must be dropped. We point out that: • xxx (cid:54) =
1, but it is a function of L , hence the ψ n are its eigenvectors; its eigenvalues (except on ψ ± Λ ) are close to 1, slightly grow with | n | and collapse to 1 as Λ → ∞ . • The ordered monomials x h + L l x n − [with degrees h , l , n bounded by (4.3)-4.6] make up a basisof the ( Λ + ) -dim vector space underlying the algebra of observables A Λ : = End ( H Λ ) (the (cid:101) P n themselves can be expressed as polynomials in L ). • x + , x − generate the whole ∗ -algebra A Λ , because also L can be expressed as a non-orderedpolynomial in x + , x − . • As anticipated in (3.5), actually there are O ( ) -equivariant ∗ -algebra isomorphisms A Λ A Λ (cid:39) M N ( C ) (cid:39) π Λ [ U so ( )] , N = Λ + , (4.7)where π Λ is the N -dimensional unitary irreducible representation of U so ( ) . The latter ischaracterized by the condition π Λ ( C ) = Λ ( Λ + ) , where C = E a E − a is the Casimir (sumover a ∈ { + , , −} ), and E a make up the Cartan-Weyl basis of so ( ) , [ E + , E − ] = E , [ E , E ± ] = ± E ± , E † a = E − a . (4.8)In fact we can realize L , x + , x − by setting [1] (we simplify the notation dropping π Λ ) L = E , x ± = f ± ( E ) E ± , f + ( s ) = (cid:115) + s ( s − ) / k Λ ( Λ + ) − s ( s − ) = f − ( s − ) , (4.9)i.e. in a sense the x ± are E ± (which play the role of x ± in Madore FS) squeezed in the E direction; one can easily check (4.3-4.6) using (5.2), with L a , l , m resp. replaced by E a , Λ , n .Hence π Λ ( E + ) , π Λ ( E − ) are generators of A Λ alternative to x + , x − . • The group Y Λ (cid:39) SU ( Λ + ) of ∗ -automorphisms of A Λ is inner and includes a subgroup SO ( ) independent of Λ (acting irreducibly via π Λ ) and a subgroup O ( ) ⊂ SO ( ) corre-sponding to orthogonal transformations (in particular, rotations) of the coordinates x i , whichplays the role of isometry group of S Λ .As in the commutative case we define (cid:104) xxx (cid:105) : = (cid:104) x (cid:105) + (cid:104) x (cid:105) and find (cid:104) xxx (cid:105) = (cid:104) x + (cid:105)(cid:104) x − (cid:105) = |(cid:104) x + (cid:105)| .9 nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore x i on S Λ As said, by O ( ) -covariance Σ x i ( Λ ) = Σ x ( Λ ) for all i , so we can study just the spectrum Σ x ( Λ ) . L is invariant under 2-dimensional rotations, whereas L → − L under x - or x -inversion. On thebasis B Λ the operator x is represented by the ( Λ + ) × ( Λ + ) symmetric tridiagonal matrix X ( Λ ) = b Λ . . . b Λ b Λ − . . . b Λ − b Λ − . . . · · · b − Λ b − Λ · · · b − Λ = X ( Λ ) + O (cid:18) Λ (cid:19) . Here b n ≡ (cid:112) + n ( n − ) / k , and X is the k → ∞ limit of X , i.e. is obtained replacing all b n by 1.The eigenvectors and eigenvalues of Toeplitz matrices such as X are known (see e.g. [47] p. 2-3)and are good approximations of those of x ; in [3] we have studied the latter estimating the neededcorrections. The spectrum of X ( Λ ) arranged in descending order is Σ X : = { (cid:101) α h ( Λ ) } Nh = , where (cid:101) α h = cos (cid:18) h π N + (cid:19) , (4.10)and N = Λ +
1. We shall arrange also the spectrum Σ X ( Λ ) = { α h ( Λ ) } Λ + h = of x (cid:39) X in decreasingorder. Improving the results of Theorem 3.1 in [3], here we prove Theorem 4.1.
For all Λ ∈ N
1. If α belongs to Σ X , then also − α does.2. Σ X ( Λ ) , Σ X ( Λ + ) interlace, i.e. between any two consecutive eigenvalues of X ( Λ + ) there isexactly one of X ( Λ ) (see fig. 5): α ( Λ + ) > α ( Λ ) > α ( Λ + ) > α ( Λ ) > ... > α Λ ( Λ ) > α Λ + ( Λ + ) ; Σ X becomes uniformly dense in [ − , ] as Λ → ∞ , in particular α ( Λ ) ≥ − π ( Λ + ) .Proof. Items 1., 3. were proved in Theorem 3.1 in [3]. Item 1. follows also from Proposition7.1, after the inversion X m (cid:55)→ − X m . We prove item 2.with the help of Proposition 7.1 (?) COM-PLETARE.Note that item 2. implies in particular that all eigenvalues are simple.In the Λ → ∞ the eigenvectors of x become generalized eigenvectors, as expected.10 nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore O ( ) -covariant uncertainty relations and O ( ) -invariant strong SCS systems From (4.3) one can derive [4] for both S , S Λ the O ( ) -covariant ‘Heisenberg’ uncertainty relations ∆ L ∆ x ≥ |(cid:104) x (cid:105)| , ∆ L ∆ x ≥ |(cid:104) x (cid:105)| , ∆ L ( ∆ xxx ) ≥ (cid:104) xxx (cid:105) ψ n ( ∆ L = ∆ x , ∆ x may vanish separately,but not simultaneously, because ( ∆ xxx ) ≥ ( ∆ xxx ) min ∼ Λ . (4.12) Theorem (section 3.1 in [4])
The system S β ≡ (cid:26) ωωω βα ≡ Λ ∑ n = − Λ e i ( α n + β n ) √ Λ + ψ n (cid:27) α ∈ R / π Z is a strong SCS, Λ + π (cid:90) π d α P βα = id , P βα ≡ ωωω βα (cid:104) ωωω βα , ·(cid:105) , (4.13) for all β ∈ ( R / π Z ) Λ + (the label space is R / π Z (cid:39) S ≡ Ω ). It is fully O ( ) -covariant if β − n = β n . On all ωωω βα it is (cid:104) L (cid:105) = , ( ∆ L ) = Λ ( Λ + ) , whereas ( ∆ xxx ) is minimized by the φφφ α ≡ ωωω α , with ( ∆ xxx ) < Λ + (cid:18) + Λ (cid:19) . (4.14)Within the class of strong SCS, the φφφ α are closest to classical states(=points) of S , and in one-to-one correspondence with them: S ↔ S ≡ { φφφ α } α ∈ R / π Z (cid:39) S ≡ Ω . O ( ) -invariant weak SCS minimizing ( ∆ xxx ) Since ( ∆ xxx ) is O ( ) -invariant, so is the set W of states minimizing it; W is a weak SCS.We can recover the whole set from any element χχχ through rotations, W = (cid:110) χχχ α ≡ e i α L χχχ (cid:111) α ∈ [ , π [ .Choosing χχχ so that (cid:104) x (cid:105) χχχ =
0, by (2.1) we find (cid:104) xxx (cid:105) χχχ α = (cid:12)(cid:12)(cid:12) (cid:104) xxx (cid:105) χχχ (cid:12)(cid:12)(cid:12) uuu α , where uuu α = ( cos α , sin α ) . Wehave shown that 0 < ( ∆ xxx ) min = ( ∆ xxx ) χχχ α < . ( Λ + ) . (4.15)The (rays associated to) χχχ α are closest to classical states(=points) of S , and in one-to-one corre-spondence with them: S ↔ S ≡ { φφφ α } α ∈ R / π Z (cid:39) S .
5. D=3: O ( ) -covariant fuzzy sphere We use two related sets of angular momentum and space coordinate operators: the hermiteanones { L i } i = (with L i ≡ ε i jk L jk /
2) and { x i } i = , and the partly hermitean conjugate ones { L a } , { x a } (here a = , + , − ), which are obtained from the former as follows : L ± : = L ± iL , L : = L , x ± : = x ± ix , x : = x . Again, here we use the conventions of [3, 4], rather than those of [1]. nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore
The square distance from the origin can be expressed as xxx : = x i x i = x + ( x + x − + x − x + ) /
2. Asa preferred orthonormal basis of the carrier Hilbert space H Λ we adopt one B Λ consisting ofeigenvectors of L , LLL = L i L i = L + ( L + L − + L − L + ) / B Λ : = { ψ ml } l = , ,..., Λ ; m = − l ,..., l , LLL ψ ml = l ( l + ) ψ ml , L ψ ml = m ψ ml . (5.1)On the ψ ml the L a , x a act as follows: L ψ ml = m ψ ml , L ± ψ ml = (cid:112) ( l ∓ m )( l ± m + ) ψ m ± l , (5.2) x a ψ ml = c l A a , ml ψ m + al − + c l + B a , ml ψ m + al + if l < Λ , c l A a , ml ψ m + a Λ − if l = Λ , A , ml : = (cid:115) ( l + m )( l − m )( l + )( l − ) , A ± , ml : = ± (cid:115) ( l ∓ m )( l ∓ m − )( l − )( l + ) , B a , ml = A − a , m + al + , c l : = (cid:114) + l k ≤ l ≤ Λ , c = c Λ + = , (5.4)and k ( Λ ) fulfills k ( Λ ) ≥ Λ ( Λ + ) . The L i , x i fulfill the following O ( ) -covariant relations: x † i = x i , L † i = L i , [ L i , x j ] = i ε i jh x h , [ L i , L j ] = i ε i jh L h , x i L i = , (5.5) [ x i , x j ] = i ε i jh L h (cid:18) − k + K (cid:101) P Λ (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) Snyder − like , xxx = + LLL + k − (cid:20) + ( Λ + ) k (cid:21) Λ + Λ + (cid:101) P Λ , (5.6) Λ ∏ l = (cid:2) LLL − l ( l + ) I (cid:3) = , l ∏ m = − l ( L − mI ) (cid:101) P l = , ( x ± ) Λ + =
0; (5.7)here K = k + + Λ k Λ + , (cid:101) P l is the projection on the LLL = l ( l + ) eigenspace. Again, terms marked redare absent in the commutative case. In the Λ → ∞ limit also the non-vanishing ones will play norole at any fixed energy E , as they are proportional to the projection (cid:101) P Λ onto the states with highestenergy E Λ → ∞ ; (5.7a,b) give back the spectra of LLL , L on L ( S ) , L ( R ) , whereas (5.7c) loosesmeaning and must be dropped. We point out that: • xxx (cid:54) =
1; but it is a function of
LLL , hence the ψ ml are its eigenvectors; its eigenvalues (exceptwhen l = Λ ) are close to 1, slightly grow with l and collapse to 1 as Λ → ∞ . • The ordered monomials in x i , L i [with degrees bounded by (5.5-5.7)] make up a basis of the ( Λ + ) -dim vector space A Λ : = End ( H Λ ) (cid:39) M ( Λ + ) ( C ) , because the (cid:101) P l themselves can beexpressed as polynomials in LLL . • The x i generate the ∗ -algebra A Λ , because also the L i can be expressed as non-ordered poly-nomials in the x i . 12 nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore • As anticipated in (3.5), actually there are O ( ) -covariant ∗ -algebra isomorphisms A Λ (cid:39) M N ( C ) (cid:39) πππ Λ [ U so ( )] , N : = ( Λ + ) . (5.8)where πππ Λ is the N -dimensional unitary vector (and irreducible) representation of U so ( ) onthe Hilbert space V Λ characterized by the conditions πππ Λ ( C ) = Λ ( Λ + ) , πππ Λ ( C (cid:48) ) = { L HI } ( H , I ∈ { , , , } ) of so ( ) , [ L HI , L JK ] = i ( δ HJ L IK − δ HK L IJ − δ IJ L HK + δ IK L HJ ) , L † HI = L HI = − L IH , (5.9) C = L IJ L IJ , C (cid:48) = ε HIJK L HI L JK (sum over repeated indices). To simplify the notation we drop πππ Λ . In fact one can realize L i , x i , i ∈ { , , } , by setting [1] L i = ε i jk L jk , x i = g ∗ ( λ ) L i g ( λ ) , g ( l ) = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) Γ (cid:0) Λ + l + (cid:1) Γ (cid:0) Λ − l + (cid:1) Γ (cid:0) Λ + + l + (cid:1) Γ (cid:0) Λ − l + (cid:1) Γ (cid:16) l + + i √ k (cid:17) Γ (cid:16) l + − i √ k (cid:17) √ k Γ (cid:16) l + + i √ k (cid:17) Γ (cid:16) l + − i √ k (cid:17) = (cid:118)(cid:117)(cid:117)(cid:116) ∏ l − h = ( Λ + l − h ) ∏ lh = ( Λ + l + − h ) [ l − ] ∏ j = + ( l − j ) k + ( l − − j ) k ; (5.10)here we have introduced the operator λ : = [ √ L i L i + − ] / l ∈{ , , ..., Λ } ), Γ is Euler gamma function, the last equality holds only if l ∈ N , and [ b ] stands for the integer part of b . Therefore the L HI in the πππ Λ -representation make up also analternative set of generators of A Λ (in [1] L i is denoted by X i ). • The group Y Λ (cid:39) SU ( N ) of ∗ -automorphisms of A Λ is inner and includes a subgroup SO ( ) independent of Λ (acting irreducibly via πππ Λ ) and a subgroup O ( ) ⊂ SO ( ) correspondingto orthogonal transformations (in particular, rotations) of the coordinates x i , which play therole of isometries of S Λ . x i on S Λ Again, by O ( ) -covariance all x i have the same spectrum, so we study the one Σ x of x ≡ x .Since [ x , L ] =
0, and Σ L is known from (5.1), we look for simultaneous eigenvectors of L , x L χχχ m α = m χχχ m α , x χχχ m α = α χχχ m α , m = − Λ , − Λ , ..., Λ (5.11)in the form χχχ m α = ∑ Λ l = | m | χ m α , l ψ ml . The second equation can be rewritten in the matrix form X m ( Λ ) χ = α χ , where χ = (cid:16) χ m α , | m | , χ m α , | m | + , . . . , χ m α , Λ (cid:17) T and X m ( Λ ) is the following N ( Λ ; m ) × nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore N ( Λ ; m ) [with N ( Λ ; m ) : = Λ −| m | + X m ( Λ ) = c | m | + A , m | m | + c | m | + A , m | m | + c | m | + A , m | m | + c | m | + A , m | m | + c | m | + A , m | m | + c Λ − A , m Λ − c Λ A , m Λ c Λ A , m Λ . Since X m ( Λ ) ≡ X − m ( Λ ) , we can stick to m ∈ { , , ..., Λ } . We shall arrange the spectrum of X m Σ X m = { α h ( Λ ; m ) } N ( Λ ; m ) h = in descending order. Improving the results of Theorem 4.1 in [3] we prove Theorem 5.1.
For all Λ ∈ N
1. If α belongs to Σ X m , then also − α does.2. α ( Λ ; 0 ) > α ( Λ ; 1 ) > · · · > α ( Λ ; Λ ) Σ X m ( Λ ) , Σ X m ( Λ + ) interlace, i.e. between any two consecutive eigenvalues of X m ( Λ + ) thereis exactly one of X m ( Λ ) (see fig. 5): α ( Λ + m ) > α ( Λ ; m ) > α ( Λ + m ) > α ( Λ ; m ) > ... > α Λ ( Λ ; m ) > α Λ + ( Λ + m ) ; Σ X becomes uniformly dense in [ − , ] as Λ → ∞ , with α ( Λ ; 0 ) ≥ − π ( Λ + ) if Λ ≥ .Proof. Items 1., 2., 4. were proved in Theorem 4.1 in [3]. In particular 4. is based on the fact thatmost the highest α h ( Λ ; m ) are well approximated by the eigenvalues of the Toeplitz matrix X m thatis obtained from X m replacing all nonzero elements by 1 /
2, although in this case c l A , ml (cid:57) /
2, and c l A , ml (cid:39) / | m | (cid:28) l ; the eigenvalues of X m are given by (4.10) with N = N ( Λ ; m ) .Item 2. is a direct consequence of Proposition 7.1. Also item 1., after the inversion X m (cid:55)→ − X m .Item 2. implies in particular that all eigenvalues are simple.As Λ → ∞ the eigenvectors of x become generalized eigenvectors, as expected; in particular, theone with the highest eigenvalue α ( Λ ; 0 ) becomes a Dirac delta concentrated in the North pole. O ( ) -invariant UR and strong SCS on S Λ Theorem 4.1 in [4].
The uncertainty relation ( ∆ LLL ) ≥ |(cid:104) LLL (cid:105)| ⇔ (cid:104)
LLL (cid:105) ≥ |(cid:104) LLL (cid:105)| ( |(cid:104) LLL (cid:105)| + ) (5.12) holds on H Λ = ⊕ Λ l = V l and is saturated by the spin coherent states φφφ l , g : = πππ Λ ( g ) ψ ll ∈ V l , l ∈{ , , ..., Λ } , g ∈ SO ( ) . Moreover on H Λ the following resolution of identity holds:I = Λ ∑ l = C l (cid:90) SO ( ) d µ ( g ) P l , g , C l = l + π , P l , g = φφφ l , g (cid:104) φφφ l , g , ·(cid:105) . (5.13)14 nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore
We can parametrize g ∈ SO ( ) , the invariant measure and the integral over SO ( ) through theEuler angles ϕ , θ , ψ : g = e ϕ I e θ I e ψ I where I : = − , I : = −
10 0 01 0 0 ⇒ (5.14) πππ Λ ( g ) = e i ϕ L e i θ L e i ψ L , (cid:90) SO ( ) d µ ( g ) = π (cid:90) d ϕ π (cid:90) d θ sin θ π (cid:90) d ψ = π . (5.15)In (5.13) integration over ψ can be actually eliminated rescaling d µ by 2 π , i.e. one can integratejust over S , because the ψ ll are eigenvectors of L . The theorem holds also for Λ = ∞ , i.e. on L ( S ) , because on the latter the commutation relations [ L i , L j ] = i ε i jk L k are the same: the UR(5.12) is saturated by the spin coherent states φφφ l , g : = πππ Λ ( g ) Y ll ∈ V l , and (5.13) holds provided l runover N and we replace ψ ll by Y ll , πππ Λ by the (reducible) representation of SO ( ) on L ( S ) [4].Again, ∆ x , ∆ x , ∆ x may vanish separately, not simultaneously, because ( ∆ xxx ) ≥ ( ∆ xxx ) min ∼ Λ (5.16)Fixed a generic normalized vector ωωω ≡ Λ ∑ l = l ∑ h = − l ω hl ψ hl , for g ∈ SO ( ) let ωωω g : = πππ Λ ( g ) ωωω , P g : = ωωω g (cid:104) ωωω g , ·(cid:105) . (5.17)As the unitary representation πππ Λ of SO ( ) on H Λ is reducible , more precisely the direct sum of theirreducible representations ( V l , π l ) , l = , ..., Λ , completeness and resolution of the identity for thesystem S ω ≡ { ωωω g } g ∈ SO ( ) are not automatic. S ω is complete if for all l there exists at least one h such that ω hl (cid:54) = Theorem 4.2 in [4]. S ω ≡ { ωωω g } g ∈ SO ( ) is a strong SCS if l ∑ h = − l | ω hl | = l + ( Λ + ) ∀ l; it is alsofully O ( ) -covariant if ω hl = ω − hl . The following resolution of the identity on H Λ holds:id = ( Λ + ) π (cid:90) SO ( ) d µ ( g ) P g , P g : = ωωω g (cid:104) ωωω g , ·(cid:105) . (5.18)We can make the isotropy subgroup H ⊂ SO ( ) nontrivial choosing e.g. ωωω an eigenvectorof L ; correspondingly H = { e i ψ L | ψ ∈ R / π Z } (cid:39) SO ( ) . In particular φφφ β = ∑ Λ l = ψ l e i β l √ l + ( Λ + ) (with β ∈ ( R / π Z ) Λ + ) has zero eigenvalue. Setting φφφ β g = πππ Λ ( g ) φφφ β , we find that different raysare parametrized by g = e ϕ I e i θ I ∈ SO ( ) / SO ( ) . Hence (5.18) holds also with the (normalized )integration over just the coset space SO ( ) / SO ( ) (cid:39) S . Based on eqs. (58-59) of [4] we thus find Corollary 5.1. S β = { φφφ β g } g ∈ S is a family of fully O ( ) -covariant, strong SCSs, andid = ( Λ + ) π (cid:90) π d ϕ (cid:90) π d θ sin θ P β g , P β g = φφφ β g (cid:104) φφφ β g , ·(cid:105) (5.19)15 nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore for all β ∈ ( R / π Z ) Λ + . On it ( ∆ LLL ) is independent of β , while ( ∆ xxx ) is smallest on the φφφ g , with ( ∆ LLL ) = Λ ( Λ + ) , ( ∆ xxx ) (cid:12)(cid:12) φφφ g < Λ + . (5.20)Within the class of strong SCS, the φφφ g are closest to classical states(=points) of S , and inone-to-one correspondence with them: S ↔ S ≡ { φφφ g } g ∈ SO ( ) / SO ( ) (cid:39) S . O ( ) -invariant weak SCS on S Λ minimizing ( ∆ xxx ) Since ( ∆ xxx ) is O ( ) -invariant, so is the set W of states minimizing it; W is a weak SCS. Wecan recover the whole set from any element χχχ through rotations, W = (cid:110) χχχ g ≡ πππ Λ ( g ) χχχ (cid:111) g ∈ SO ( ) .Choosing χχχ so that (cid:104) xxx (cid:105) = |(cid:104) xxx (cid:105)| eee [whence (cid:104) x (cid:105) = |(cid:104) xxx (cid:105)| , ( ∆ xxx ) = (cid:104) xxx (cid:105) − (cid:104) x (cid:105) ], we find (cid:104) xxx (cid:105) χχχ g = (cid:12)(cid:12)(cid:12) (cid:104) xxx (cid:105) χχχ (cid:12)(cid:12)(cid:12) uuu g , where uuu g = guuu . We have shown that L χχχ =
0. This implies that the isotropy subgroupis H = { e i ψ L | ψ ∈ R / π } (cid:39) SO ( ) whence W = { χχχ g ≡ πππ Λ ( g ) χχχ } g = e ϕ I e i θ I ∈ SO ( ) / SO ( ) (cid:39) S , uuu g =( sin θ cos ϕ , sin θ sin ϕ , cos θ ) . The (rays associated to) χχχ g are closest to classical states(=points)of S , and are in one-to-one correspodence with them: S ↔ S ≡ { φφφ g } g ∈ S . At order O ( / Λ ) χχχ coincides with the eigenvector (cid:98) χχχ of x with highest eigenvalue ( L (cid:98) χχχ = < ( ∆ xxx ) min = ( ∆ xxx ) χχχ g < ( Λ + ) . (5.21)
6. Outlook, comparison with the literature and final remarks
Imposing an energy cutoff E may: i) yield a simpler low-energy approximation T of a well-defined quantum theory T ; ii) make sense of T if T is well-defined while T is not (as in the caseof UV-divergent QFT); iii) help in figuring out from T a new theory valid also at energies E > E ,if E represents a threshold for new physics not accounted for by T .Denoting by H the Hilbert space of T , the cutoff is imposed projecting T on the Hilbertsubspace H characterized by energies E below E . The projected observables fulfill modified al-gebraic relations; in particular, space coordinates in general become noncommutative. Thus lowenergy effective theories with space(time) noncommutativity and lower bounds for space(time) lo-calization (as expected by any candidate theory of quantum gravity) may all naturally arise from theimposition of an energy cutoff. Mathematically, E can play the role of deformation parameter. If H remains finite-dimensional for all (finite) E , the latter may be replaced by a discrete parameterlike n = dim ( H ) , and T n ≡ T ( n ) make up a fuzzy approximation of T . If T lives on a manifold M , and in the Hamiltonian we include a suitable confining potential U n with a minimum on a sub-manifold N of M that becomes sharper and sharper as n → ∞ , we effectively induce a dimensionalreduction to a noncommutative quantum theory on N .In the present paper, after elaborating the arguments sketched in the previous two paragraphs,we have reviewed our application of the latter mechanism for the construction of a d -dimensional, O ( D ) -covariant fuzzy sphere ( d = , { S d Λ } Λ ∈ N ≡ { ( H Λ , A Λ ) } Λ ∈ N of finite-dimensional, O ( D ) -covariant ( D = d +
1) approximations of quantum mechanics (QM) of a spinlessparticle on the sphere S d ; xxx (cid:38)
1, and xxx essentially collapses to 1 as Λ → ∞ (see the Introduction).16 nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore
This result has been achieved imposing an energy-cutoff E = Λ ( Λ + d − ) on QM of a spinlessparticle in R D subject to a confining potential V ( r ; Λ ) that has a minimum on the sphere r = Λ → ∞ . A Λ is a fuzzy approximation of the whole algebraof observables of the particle on S d (phase space algebra), and converges to the latter in the limit Λ → ∞ . At least for D = ,
3, there is an O ( D ) -covariant ∗ -isomorphism A Λ (cid:39) π Λ [ U so ( D + )] ,where π Λ is a suitable irreducible representation of Uso ( D + ) on H Λ . The latter is a reducible representation of the subgroup O ( D ) (and of the Uso ( D ) ⊂ Uso ( D + ) subalgebra generated by the L i j ), more precisely the direct sum of all the irreducible representations fulfilling L ≤ Λ ( Λ + d − ) .A similar decomposition holds for the subspace C Λ ⊂ A Λ of completely symmetrized polynomialsin the x i acting as multiplication operators on H Λ . For instance, in the case d = H Λ (cid:39) Λ (cid:77) l = V l , C Λ (cid:39) Λ (cid:77) l = V l . (6.1)where ( V l , π l ) are the irreducible representations of O ( ) characterized by LLL = l (+ ) . As Λ → ∞ these respectively become the decompositions of L ( S ) and of C ( S ) that acts on L ( S ) .Localization in configuration and angular momentum space can be measured through the O ( D ) -invariant square uncertainties ( ∆ xxx ) (see section 2.2) and ( ∆ LLL ) ; for d = , S d Λ . Section 2.4 is a coinciseintroduction to SCS. In sections 4.1, 5.1 we have studied the eigenvalue equation of a coordinate x i (slightly improving the results of [3]) and its relation with the minimization of ( ∆ xxx ) for d = , ( ∆ xxx ) make up a O ( D ) -invariant weak SCS W d (sections 4.3, 5.3). In sec-tions 4.2, 5.2 we have presented the class of O ( D ) -invariant, strong SCS, in particular the one S d minimizing ( ∆ xxx ) within the class.Let us compare S Λ with the seminal fuzzy sphere S n of Madore-Hoppe [17, 18]. The ∗ -algebra A n (cid:39) M n ( C ) of observables on S n is generated by hermitean coordinates x i ( i = , ,
3) fulfilling [ x i , x j ] = i (cid:112) l ( l + ) ε i jk x k , xxx : = x i x i = , l ∈ N / , n = l + . (6.2)In fact L i = x i (cid:112) l ( l + ) make up the standard basis of so ( ) in the irreducible representation ( π l , V l ) . Hence the spectrum of all x i is Σ x i = (cid:110) m / (cid:112) l ( l + ) | m = − l , − l , ..., l (cid:111) . We note that:i) Contrary to (5.6), eq. (6.2) are not covariant under the whole O ( ) , in particular under parity x i (cid:55)→ − x i , but only under SO ( ) .ii) Contrary to the Λ → ∞ limit of (6.1), in the l → ∞ limit H = V l remains irreducible anddoes not invade L ( S ) .iii) By Theorems 4.1, 5.1, the spectrum of any coordinate x i on either S Λ or S n fulfills the twoproperties listed in section 2.3. The former fulfills also one not shared by the latter: theeigenstate of x with maximal eigenvalue, which is very localized around the North poleof S , is a L = L , see fig. 2 right. As Λ → ∞ the latter becomes thegeneralized eigenstate (distribution) 2 δ ( θ ) / sin θ (cid:39) δ ( x ) δ ( x ) on S concentrated on the17 nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore
North pole (here θ is the colatitude); the classical counterpart of this property is that theclassical particle on S in the position xxx = ( , , ) has zero L ( z -component of the angularmomentum), because L = ( LLL ) = (cid:0) xxx × ppp (cid:1) = . On the contrary, on S n this property is lost; as the x i are obtained by rescaling the L i there is nolonger room for independent observables playing the role of angular momentum operators.iv) On our fuzzy sphere S Λ the states with minimal space uncertainty ( ∆ xxx ) make up a weak SCS W , and ( ∆ xxx ) W < ( Λ + ) ; the strong SCS S with minimal ( ∆ xxx ) has ( ∆ xxx ) S < Λ + ( ∆ xxx ) min = l + l = Λ ).Properties i)-iii) in particular show why in our opinion { C Λ } Λ ∈ N can be interpreted as the space offunctions on fuzzy configuration space S Λ , while { A n } n ∈ N of Madore-Hoppe should be interpretedonly as the space (actually, the algebra) of functions on a fuzzy spin phase space S n . As for iv), itwould be also interesting to compare distances between two maximally localized states on our S Λ (either in W or in S ) and on the Madore-Hoppe FS [43].Ref. [5] begins to apply in detail our approach to spheres S d with d ≥
3; this allows a firstcomparison with the rest of the literature. The 4-dimensional fuzzy spheres introduced in [21], aswell as the ones of dimension d ≥ End ( V ) , where V carries a particular irreducible representation of both Spin ( D ) and Spin ( D + ) (and therefore ofboth U so ( D ) and U so ( D + ) ); as xxx is central, it can be set xxx = O ( D ) -covariant and Snyder-like. The fuzzy spherical harmonics are elements,but do do not close a subalgebra, of End ( V ) , i.e. the product Y · Y (cid:48) of two spherical harmonics isnot a combination of spherical harmonics. This is exactly as in our models, i.e. C Λ is a subspace,but not a subalgebra, of A Λ . (One can introduce a product in C Λ by projecting the result of Y · Y (cid:48) tothe vector space C Λ , but this will be non-associative; associativity is recovered in the Λ → ∞ limit).In [46, 24] the authors consider also the construction of a fuzzy 4-sphere S N through a re-ducible representation of U so ( ) on a Hilbert space V obtained decomposing an irreducible rep-resentation π of U so ( ) characterized by a triple of highest weights ( N , , n (cid:48) ) ; so End ( V ) (cid:39) π [ U so ( )] , in analogy with our results. The elements X i of a basis of the vector space so ( ) \ so ( ) play the role of noncommuting cartesian coordinates. Hence, the O ( ) -scalar xxx = X i X i is no longercentral, but its spectrum is still very close to 1 provided N (cid:29) n (cid:48) , because then V decomposes onlyin few irreducible SO ( ) -components, all with eigenvalues of xxx very close to 1; if n (cid:48) = xxx ≡ V carries an irreducible representation of O ( ) ), and one recovers the fuzzy 4-sphere of[21]. On the contrary, in our approach xxx ≡ x i x i (cid:39) x i = f ( L ) X i f ( L ) , with suitable functions f , f , rather than the X i .Many other aspects and applications of the general approach described in this paper and ofthese new fuzzy spheres deserve investigations. We hope that progresses can be reported soon.18 nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore
Figure 5:
The spectra of A n + (red eigenvalues) and A n (green eigenvalues) interlace.
7. Appendix
Consider a sequence { A n } n ∈ N of hermitean tridiagonal matrices with zero diagonal elements A n = a . . . a a . . . a a . . . . . . a n −
00 0 0 0 0 . . . a n − a n − . . . a n − . (7.1)For all n the matrix A n is nested into A n + , more precisely is the upper diagonal block of the latter.We arrange the (necessarily real) eigenvalues α nh of A n in decreasing order, α n ≥ α n ≥ ... ≥ α nn . Proposition 7.1.
The spectrum Σ A n = (cid:8) α nh (cid:9) nh = depends only on the | a h | , h = , ..., n − . Forall n ∈ N , a h (cid:54) = for all h = , ..., n implies that all Σ A h , Σ A h + interlace, i.e. between any twoconsecutive eigenvalues of A h + there is exactly one of A h (see fig. 5). As a particular consequence, all eigenvalues are simple, and the inequalities ≥ are strict. Proof.
The eigenvalue equation for A n reads p n ( α ) =
0, where the lhs is the polynomial of degree n defined by p n ( α ) = det ( α I n − A n ) (here I n is the unit matrix); p n ( α ) = ( α − α n ) ... ( α − α nn ) implies that p n ( α ) > α > α n . We easily find α = α = | a | , α = −| a | , so theclaim is true for n =
1. Applying Laplace rule with the last two rows of the determinant of α I n + − A n + = α I n − − A n − Tn − − a n − Tn − n − − a n − n − α − a n − a n α (here 0 k is the row with k zeroes, 0 Tk its transpose column) we find the recurrence relation p n + ( α ) = α p n ( α ) − | a n | p n − ( α ) . Now assume that the claim is true for all m ≤ n , with a generic n >
1; bythe previous relation also p n + ( α ) , and its roots, depend only on the | a h | , and p n ( α nh ) = p n + ( α nh ) = α nh p n ( α nh ) − | a n | p n − ( α nh ) = −| a n | ( α nh − α n − )( α nh − α n − ) ... ( α nh − α n − n − ) . (7.2)19 nergy cutoff, noncommutativity and fuzzyness: the O ( D ) -covariant fuzzy spheres Gaetano Fiore
By the induction hypothesis, α n > α n − > α n > α n − > ... > α nn − > α n − n − > α nn ; (7.3)choosing h = α n + > α n such that p n + ( α n + ) =
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