aa r X i v : . [ h e p - t h ] F e b On the area of the sphere in a Snyder geometry
P. Valtancoli
Dipartimento di Fisica, Polo Scientifico Universit´a di Firenzeand INFN, Sezione di Firenze (Italy)Via G. Sansone 1, 50019 Sesto Fiorentino, Italy
Abstract
We compute the area of a generic d -sphere in a Snyder geometry. Introduction
The quantization of gravity is expected to involve the discretization of some geometric quan-tities. From heuristic arguments ( based on Ehrenfest principle) Bekenstein [1]-[2] proposedthat the area of the event horizon must have a discrete spectrum: A n = 4 πr ∼ l p n n = 1 , , ... (1.1)where l p is the Planck length.Among the possible candidates for quantum gravity we can mention the theories withminimal length. These are the subject of several studies, and among these Snyder alge-bra stands out. This necessarily leads to a non-trivial deformation of quantum mechanics.Evidence of this is the fact that the symmetry group of the path integral ( the canonicaltransformations ) is modified by the presence of the minimal length [3].In this article we want to highlight another equally important aspect of the Snyder space,which connects to the Bekenstein conjecture of the horizon area of the black hole. In [4] itis considered a space-time described by a classical time and a non-commutative space R d defined by Snyder algebra and the area of the disc and sphere is shown to be quantized. Theirproof is based on manipulating Snyder algebra so that the eigenvalue problem relates to thatof the angular momentum in d = 2 and d = 3. In this work we use an explicit representationof Snyder algebra, which can be easily extended to the generic case of the sphere S d , andwe are able to understand the structure of the eigenvalues and of the eigenfunctions in anexhaustive way.This work then relates to our previous results on the quantization of the harmonic os-cillator in d dimensions in the presence of Snyder algebra, from which we deduce that thequantization of the area of the sphere S d is obtainable from the spectrum of the harmonicoscillator in a particular limit µ → µ parameter is introduced in [5] - [6] ). The area of the sphere is in general a function of the radial coordinate. In non-commutativegeometry the radial coordinate is replaced by a linear operator acting on an auxiliary Hilbertspace. The possible measures of the area of the sphere are identifiable with the eigenvaluesof this linear operator. In [4] the eigenvalues of the following operator have been computedin d = 2 and d = 3 1 R = d X i =1 ( X i ) (2.1)in the presence of Snyder algebra, an important example of non-commutative geometry.In d = 3 this definition coincides precisely with the area of the sphere.In this article we generalize the computation of the eigenvalues and eigenvectors of the b R operator to the case of a generic sphere S d using a particular representation of the Snyderalgebra in the compact variable ρ : X i = i ~ p − βρ ∂∂ρ i p i = ρ i p − β ρ < ρ < β (2.2)The results that we make explicit in this article can be obtained from the case of theharmonic oscillator in d dimensions in the limit of the parameter µ → S sphere, and then generalize to the case of thegeneric S d sphere.Let’s first discuss the spectrum of the operator b R in d = 2: b R ψ = − ~ (cid:20) (1 − β ( ρ x + ρ y )) (cid:18) ∂ ∂ρ x + ∂ ∂ρ y − β (cid:18) ρ x ∂∂ρ x + ρ y ∂∂ρ y (cid:19) (cid:19) (cid:21) ψ = − ~ (cid:20) ( 1 − βρ ) (cid:18) ∂ ∂ρ + 1 ρ ∂∂ρ − l ρ (cid:19) − β ρ ∂∂ρ (cid:21) ψ = R ψ (2.3)where l is the eigenvalue of the angular momentum in d = 2. Let z = √ β ρ , the eigenvalueequation becomes: (cid:20) (1 − z ) (cid:18) ∂ ∂z + 1 z ∂∂z − l z (cid:19) − z ∂∂z (cid:21) ψ = − R ~ β ψ (2.4)We need to isolate the contribution of the angular part ψ ( z ) = z l P ( z ) (2.5)where P ( z ) is a polynomial in the variable z (cid:20) (1 − z ) ∂ ∂z + (cid:18) lz − l ) z (cid:19) ∂∂z + (cid:18) R ~ β − l (cid:19)(cid:21) P ( z ) = 0 (2.6)2inally, we pass from the variable z to the variable x = z obtaining (cid:20) x (1 − x ) ∂ ∂x + (cid:18) ( l + 1) − (cid:18) l + 32 (cid:19) x (cid:19) ∂∂x + 14 (cid:18) R ~ β − l (cid:19) (cid:21) P ( x ) = 0 (2.7)a hypergeometric equation with coefficients ( a, b, c ): a = 12 l + 12 + s l + R ~ β + 14 ! b = 12 l + 12 − s l + R ~ β + 14 ! c = l + 1 (2.8)The general solution is therefore ψ ( z ) = z l F " l + 12 + s l + R ~ β + 14 ! , l + 12 − s l + R ~ β + 14 ! , l + 1 ; z (2.9)Polynomial solutions are obtained when the coefficient b is a negative integer: b = 12 l + 12 − s l + R ~ β + 14 ! = − n (2.10)from which we obtain as possible values of the area of the sphere in d = 2: R = ~ β (cid:20) n + 4 n (cid:18) l + 12 (cid:19) + l (cid:21) (2.11)To compare with the results of the article [4] we must introduce a new quantum number N = 2 n + l obtaining full agreement R = ~ β [ N ( N + 1) − l ] (2.12). Now let’s analyze the eigenfunctions in detail. When the coefficient b = − n , the coefficient a = n + l + hence the eigenfunction corresponding to the quantum numbers ( n, l ) is3 n,l ( z ) = z l F (cid:18) n + l + 12 , − n, l + 1; z (cid:19) (2.13)This particular hypergeometric function is nothing more than a Jacobi polynomial with α = l and β = − : F (cid:18) n + l + 12 , − n, l + 1; z (cid:19) = n ! Γ( l + 1)Γ( n + l + 1) P ( l, − ) n (1 − z ) (2.14)We can calculate the normalization of the eigenfunction ψ n,l ( z ) starting from the knownnormalization of the Jacobi polynomials: Z − (1 − w ) α (1+ w ) β P α,βn ( w ) P α,βm ( w ) dw = 2 α + β +1 n + α + β + 1 Γ( n + α + 1) Γ( n + β + 1)Γ( n + α + β + 1) n ! δ n,m (2.15)With simple steps we get Z dz z (1 − z ) − ψ n,l ( z ) ψ m,l ( z ) = 14 n + 2 l + 1 n ! Γ( n + ) ( Γ( l + 1) ) Γ( n + l + 1) Γ( n + l + ) δ n,m (2.16)Note the factor (1 − z ) − in the normalization due to the choice of the representation(2.2) of the Snyder algebra in the compact variable ρ . For d = 2 the linear operator to consider is the following: b R = − ~ (cid:20) (1 − βρ ) ∇ ρ − βρ ∂∂ρ (cid:21) = R ψ (3.1)where the operator ∇ ρ is generally defined by ∇ ρ = 1 ρ d − ∂∂ρ ρ d − ∂∂ρ − l ( l + d − ρ (3.2)Let’s set z = √ βρ again, the eigenvalue equation to be solved is the following:4 (1 − z ) (cid:18) ∂ ∂z + d − z ∂∂z − l ( l + d − z (cid:19) − z ∂∂z + R ~ β (cid:21) ψ = 0 (3.3)We isolate the angular factor as before: ψ ( z ) = z l P ( z ) (3.4)where P ( z ) is a polynomial in the variable z , from which we obtain: (cid:20) (1 − z ) ∂ ∂z + (cid:18) d − lz − ( d + 2 l ) z (cid:19) ∂∂z + (cid:18) R ~ β − l (cid:19) (cid:21) P ( z ) = 0 (3.5)We set x = z and we get a hypergeometric equation again (cid:20) x (1 − x ) ∂ ∂x + (cid:18) l + d − (cid:18) l + d + 12 (cid:19) x (cid:19) ∂∂x + 14 (cid:18) R ~ β − l (cid:19) (cid:21) P ( x ) = 0(3.6)with coefficients a = 12 l + d −
12 + s l + l ( d −
2) + ( d − R ~ β ! b = 12 l + d − − s l + l ( d −
2) + ( d − R ~ β ! c = l + d b is a negative integer: R = ~ β (cid:20) n + 4 n (cid:18) l + d − (cid:19) + l (cid:21) (3.8)We introduce again the total quantum number N = 2 n + l and obtain R = ~ β [ N ( N + d − − l ( l + d −
2) ] (3.9)which generalizes in a simple way the result contained in [4] for d = 2.5or l = N we obtain the Bekenstein quantization condition of the area of the eventhorizon: R = ~ β N (3.10)We compute the eigenfunctions associated with the spectrum (3.9) ψ n,l ( z ) = z l F (cid:18) n + l + d − , − n, l + d z (cid:19) = z l n ! Γ( l + d )Γ( n + l + d ) P ( l + d − , − ) n (1 − z ) (3.11)Also in this case the normalization of the eigenfunction ψ n,l ( z ) can be calculated withsimple steps from that of the Jacobi polynomials: Z dz z d − (1 − z ) − ψ n,l ( z ) ψ m,l ( z ) = 14 n + 2 l + d − n ! Γ( n + ) (Γ( l + d )) Γ( n + l + d ) Γ( n + l + d − ) δ n,m (3.12) In this article the calculation of the area of the sphere S d in the presence of the Snyderalgebra has been solved exactly and it coincides in every dimension with the Bekensteinconjecture for the area of the horizon of a black hole. This eigenvalue problem can be linkedto the case of the harmonic oscillator in the presence of Snyder algebra by means of anappropriate limit.Note that the eigenfunctions can be connected to tabulated functions, the Jacobi polyno-mials. It is a remarkable fact that the normalization of eigenfunctions faithfully derives fromthat of Jacobi polynomials. In particular, to satisfy the orthogonality condition, a non-trivialmeasure of the integral is required, which can be deduced from the known normalization ofJacobi polynomials with a simple coordinate transformation.These results encourage us to think that understanding the Snyder space is a necessarystep towards the quantization of gravity in (3 + 1) dimensions.6 Data Availability
The data supporting the findings of this study are available within the article [ and itssupplementary material].
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