DDiscrete fuzzy de Sitter cosmology
Maja Buri´c and Duˇsko Latas ∗ University of Belgrade, Faculty of Physics, P.O. Box 44SR-11001 Belgrade
Abstract
We analyze the spectrum of time observable in noncommutativecosmological model introduced in [5], defined by ( ρ, s = ) represen-tation of the de Sitter group. We find that time has peculiar property:it is not self-adjoint, but appropriate restrictions to the space of phys-ical states give self-adjoint extensions. Extensions have discrete spec-trum with logarithmic distribution of eigenvalues, t n ∼ (cid:96) log n +const,where (cid:96) characterizes noncommutativity and the usual assumption is (cid:96) = (cid:96) P lanck . When calculated on physical states, radius of the universeis bounded below by (cid:96) (cid:113) (cid:0) + ρ (cid:1) , which resolves the big bang sin-gularity. An immediate consequence of the model is a specific breakingof the original symmetry at the Planck scale. The expression ‘quantum space’ was introduced in the early days of quantummechanics by Heisenberg, along with ‘quantum derivative’ introduced by ∗ [email protected], [email protected] a r X i v : . [ h e p - t h ] J u l irac who observed that commutator is a derivation; ‘points’ of the quantumspace are ‘q-numbers’, operators. Today the idea that spacetime, as seenby quantum particles, is described by operators gives strong heuristic andphysical motivation for noncommutative geometry.There is a surprisingly simple covariantization of the usual flat space of quan-tum mechanics to curved noncommutative spaces. If we identify flat quantumspace with the Heisenberg algebra,[ ip i , x j ] = ∂ i x j = δ ji , (1.1)( (cid:126) = 1, p i , x j hermitian), curved quantum space can be defined by a movingframe e µα , [ ip α , x µ ] = e α x µ = e µα ( x ) (1.2)as in general relativity, [1]. Adding to the last relation property (which oneexpects in the quantum-gravity regime) that spacetime at the Planck scaleis discrete or has a minimal quantum of length, i.e. that coordinates may benon-commuting, [ x µ , x ν ] = i ¯ kJ µν ( x ) , (1.3)we have a general situation, a noncommutative algebra of coordinates andmomenta, A . In principle, A may not have a Schr¨odinger-type represen-tation of momenta through the partial derivatives; in fact, some represen-tations might be finite-dimensional. In this picture, position algebra (1.3)determines the structure of the ‘points’ of noncommutative space i.e. thealgebraic properties of coordinates, while (1.2) and the related commutatorsbetween momenta define the differential-geometric structure and enable tointroduce connection and curvature. Algebraic and geometric structures areintertwined by the assumption that one deals with operators i.e. by associa-tivity, [2].This is the general framework which we use. Its algebraic part is, in var-ious descriptions of noncommutative spaces, more or less invariant, whilethe differential-geometric part is specific in every approach: we use the non-commutative frame formalism of Madore. The frame formalism has proven2n many aspects successful, in particular in describing spaces of euclideansignature with finite-dimensional representations like the fuzzy sphere and anumber of other models in two and three dimensions, [1, 3, 4]. For furtherdevelopment of this concept it is crucial to provide realistic cosmological andastrophysical configurations in four dimensions: this is the main motivationfor our work.Noncommutative geometry is but one of the approaches to quantum grav-ity. Other approaches are perhaps, in the view of description in terms oflagrangian and quantization procedures, more fundamental. String theoryintroduces elementary substructure which after quantization, macroscopi-cally, gives spacetime geometry and classical gravity. In loop quantum grav-ity, vielbein and connection fields are basic variables which are quantizedin background-space independent way. In these approaches ‘quantum space’with its properties is a derived quantity or notion. But in most cases, be-ing effective or not, coordinates are operators in the Hilbert space of states:therefore models, algebras with physically plausible features are common tomany theories. We thus hope that properties of fuzzy de Sitter space and itsphysical interpretation discussed here will be of wider interest.The plan of the paper is as follows. In Section 2 we introduce fuzzy de Sitterspace as a unitary irreducible representation of the de Sitter group, i.e. iden-tify its coordinates and differential structure. In Section 3 we give Hilbertspace representation for a specific de Sitter space defined by ( ρ, ) represen-tation of the principal continuous series of SO (1 ,
4) and solve the eigenvalueequation for the observable of cosmic time τ . In Section 4 we examine theobtained solutions and show how to redefine time to render it self-adjoint. Fi-nally, in the last section we discuss physical properties and some cosmologicalimplications of the given fuzzy geometry.3 Fuzzy de Sitter space
Our task is to study observable of time in cosmological model introducedin [5, 6]. In commutative geometry, four-dimensional de Sitter space can bedefined as an embedding in five-dimensional flat space [7], v − w − x − y − z = − L , ds = dv − dw − dx − dy − dz (2.4)where v ∈ ( −∞ , ∞ ) is the embedding time. Introducing tL = log v + w L , xL = xv + w , yL = yv + w , zL = zv + w (2.5)one obtains the line element in the FRW form, the ‘steady state universe’, ds = d t − e tL (cid:0) d x + d y + d z (cid:1) . (2.6)Time t ∈ ( −∞ , ∞ ) is defined only for v + w >
0: coordinates (2.5) coveronly half of the de Sitter space and the steady state space is incomplete,extendible.Fuzzy de Sitter space can be defined in an analogous manner. The generalidea, realized in all details for the fuzzy sphere [3], is to identify spacetimewith the algebra of a Lie group, realizing the embedding through the Casimirrelations: then in fact fuzzy spacetime is given by an irreducible representa-tion of a Lie group. We start with the group SO (1 ,
4) with generators M αβ ( α, β = 0 , , , , M αβ , M γδ ] = − i ( η αγ M βδ − η αδ M βγ − η βγ M αδ + η βδ M αγ ) , (2.7)the signature is η αβ = diag(1 , − , − , − , − v , w , x , y , z are embedding coordinates x α : they are proportional to the‘Pauli-Lubanski vector’ W α , W α = 18 (cid:15) αβγδη M βγ M δη , x α = (cid:96)W α . (2.8)Dimensional constant (cid:96) fixes the length scale of noncommutativity: depend-ing on physical interpretation, it can lie between the GUT scale and the4lanck length [6, 8]: usually one assumes (cid:96) ∼ (cid:96) P lanck . One of the twoCasimirs of SO (1 , η αβ W α W β = −W (2.9)defines the embedding equivalent to (2.4). We will for simplicity assume thatthe other Casimir operator Q = − M αβ M αβ (2.10)is also fixed, i.e. that fuzzy de Sitter space is given by a unitary irreduciblerepresentation (UIR) of the de Sitter group.All UIR’s of the SO (1 ,
4) are infinite-dimensional, labelled by two quantumnumbers: conformal weight ρ and spin s , [9], W = s ( s + 1) (cid:0)
14 + ρ (cid:1) , Q = − s ( s + 1) + 94 + ρ . (2.11)In the following we will use UIR’s of the principal continuous series, ρ ≥ s = 0 , / , , / , and the Hilbert space representations; in fact in thisconcrete calculation we use only the simplest nontrivial of them, ( ρ, s = ).Various choices of differential calculi on fuzzy de Sitter space were discussedin [5]. The simplest one which has the de Sitter metric as commutative(macroscopic) limit is the calculus generated by four momenta, translations ip i = M i + M i , i = 1 , , ip = M . When calculated,expression (1.2) for vielbein suggests to choose comoving coordinates propor-tional to W i , and cosmic time τ proportional to log( W + W ) [5], x i (cid:96) = W i , τ(cid:96) = log x + x (cid:96) = log ( W − W ) . (2.12)It is clear that correct identification of coordinates and momenta is very im-portant for understanding of various properties and limits of a given fuzzyspace, as well as for its physical interpretation. One way to see if noncom-mutativity improves the singularity structure of spacetime is to determinethe spectra of coordinates, in this case τ and x i , or (cid:80) ( x i ) . As found in [6],5pectra of x i are continuous in ( ρ, s ) representations; embedding time W /l has discrete spectrum. Here we wish to find eigenvalues of the cosmic time. ∗ Properties of the spectrum can be often inferred directly from the algebra.In this case we have relation[ iM , W − W ] = W − W , (2.13)which implies that the group action of dilatation M is given by e iαM ( W − W ) e − iαM = e α ( W − W ) . (2.14)The last formula means, apparently, that the spectrum of W − W is con-tinuous. Namely, it is easy to check formally that, if there is a nonzeroeigenvalue λ > W − W and the corresponding eigenvector ψ λ ,( W − W ) ψ λ = λ ψ λ , (2.15)then for every real α , e − iαM ψ λ is the eigenvector for the eigenvalue e α λ .This would mean that the spectrum consists of all real λ >
0. We will showin the following that eigenvalues of W − W , calculated in the Hilbert spacerepresentation ( ρ, ), are in fact discrete. Namely, differential equation (3.9)corresponding to (2.15) has solutions of finite norm for all positive λ ∈ R ,which, due to appropriate functional-analysis theorems, means that W − W is not self-adjoint. The operator is only ‘formally symmetric’ because thedomains of W − W and ( W − W ) † are not equal. There are, however, self-adjoint extensions which we construct: each reduces the initial space of statesto the ‘subspace of physical states’, implying in consequence discreteness oftime. ∗ An important observation is that components W α are the Casimir operators of sub-groups of SO (1 , W of the SO (4) and W i of the SO (1 , W α could be in principle determined group-theoretically: by reduction of a given UIRof the SO (1 ,
4) to the sum of UIR’s of the corresponding subgroup. Similar strategy ispossible for τ which is one of two Casimir operators of the E (3) subgroup, generated by M i + M i and M jk : we have not succeeded to find the appropriate reduction formula inthe literature. Hilbert space representation
We work in the Hilbert space representation of the principal continuous series( ρ, s ), [10]. It is constructed in the familiar Bargmann-Wigner representationspace of the Poincar´e group with mass m > s , [11]. Generatorsof the Lorentz rotations are given by M µν = L µν + S µν , µ, ν = 0 , , , , (3.1)where S µν are spin generators, L ik = i (cid:16) p i ∂∂p k − p k ∂∂p i (cid:17) , L k = ip ∂∂p k , i, k = 1 , , p = (cid:112) m + ( p i ) . Generators of the Poincar´e translations,multiplication operators p µ , are used to define the remaining M µ by M µ = ρm p µ − m ( p ρ M ρµ + M ρµ p ρ ) . (3.2)This representation was used in [6]: we will introduce it here very briefly inorder to fix the notation and stress a couple of technical details and simpli-fications.Bargmann-Wigner space H for s = is the space of bispinors in mo-mentum representation, ψ ( (cid:126)p ), which are square-integrable solutions to theDirac equation. Using Dirac representation of γ -matrices, γ = (cid:32) I − I (cid:33) , γ i = (cid:32) σ i − σ i (cid:33) , ψ ( (cid:126)p ) can be written as ψ ( (cid:126)p ) = Φ( (cid:126)p ) − (cid:126)p · (cid:126)σp + m Φ( (cid:126)p ) (3.3)where Φ( (cid:126)p ) is an unconstrained spinor. Scalar product is given by( ψ, ψ (cid:48) ) = (cid:90) d pp ψ † γ ψ (cid:48) = (cid:90) d pp mp + m Φ † Φ (cid:48) . (3.4)Written in blocks of 2 × M αβ and W α have the form M =7 A BB A (cid:33) . Matrix elements of such operators are † ( ψ, M ψ (cid:48) ) = (cid:90) d p ψ † γ p M ψ (cid:48) = (cid:90) d pp Φ † (cid:18) A − p k σ k p + m A p i σ i p + m + [ B, p k σ k p + m ] (cid:19) Φ (cid:48) . Eigenvalue problem
M ψ = λψ can be written as a set of two spinor equa-tions: (cid:18) A − p k σ k p + m A p i σ i p + m + [ B, p k σ k p + m ] (cid:19) Φ = λ mp + m Φ (3.5) (cid:18) [ A, p k σ k p + m ] + B − p k σ k p + m B p i σ i p + m (cid:19) Φ = 0 . (3.6)One can easily check that the second equation is fulfilled for all solutions ofthe first, so essentially one has to solve (3.5).In our problem M = W − W , the blocks A and B are A = − m (cid:18) ρ − i (cid:19) p i σ i − i m p ( p + m ) ∂∂p i σ i , (3.7) B = − m (cid:15) ijk ( p + m ) p i ∂∂p j σ k − i m ( p + m ) . (3.8)Eigenvalue equation for W − W becomes (cid:18) − ρ m p i σ i − i p + m ) ∂∂p i σ i + i m p i ∂∂p i p j σ j (cid:19) Φ = λ Φ . (3.9)As W − W commutes with 3-rotations M ij , we can choose the eigenfunctionsin the form Φ λjm ( (cid:126)p ) = f ( p ) p φ jm ( θ, ϕ ) + h ( p ) p χ jm ( θ, ϕ ) , (3.10)where p is the radial momentum, p = − p i p i = p − m , and φ jm and χ jm are the eigenfunctions of the angular momentum. Using (3.10) we obtainradial equations for f and h :( p + 1) dfdp + iρf − j + p − f = 2 iλ hp , (3.11) † At this point we fix the relative positions of γ , 1 /p and M : this ordering is notessential and can be changed, but implies appropriate changes in relations which follow. p + 1) dhdp + iρh + j + p − h = 2 iλ fp . (3.12)Solutions to these equations are derived in Appendix 1. They are expressedin terms of the Bessel functions using variable z = (cid:113) p − mp + m ; this variablevaries in a finite interval, z ∈ (0 ,
1) . Of two linearly independent solutionsfor fixed λ and j one is regular, f λj = C (cid:18) − z (cid:19) − iρ √ z J j (2 λz ) , h λj = iC (cid:18) − z (cid:19) − iρ √ z J j +1 (2 λz ) , (3.13)and therefore we conclude that the spectrum of W − W is positive realaxis, λ ∈ (0 , ∞ ). However, the given set of solutions is not orthonormal.The scalar product of two eigenfunctions is( ψ λjm , ψ λ (cid:48) j (cid:48) m (cid:48) ) = 2 δ jj (cid:48) δ mm (cid:48) C ∗ C (cid:48) (cid:90) zdz ( J j (2 λz ) J j (2 λ (cid:48) z ) + J j +1 (2 λz ) J j +1 (2 λ (cid:48) z )) . (3.14)As Bessel functions J j ( ζ ) are finite in any finite interval, integral (3.14) isbounded for λ = λ (cid:48) , i.e. all solutions are normalizable, which is in contradic-tion with the statement that they belong to continuous spectrum. Also theyare not orthogonal for λ (cid:54) = λ (cid:48) . Therefore, not all of formal solutions (3.13)can be the eigenfunctions of a self-adjoint operator, and self-adjointness is aproperty we would certainly like τ to have. The obtained result requires additional analysis. We started with a unitaryrepresentation of the SO (1 , M αβ . We defined W α by (2.8), as a sum of products of operatorswhich mutually commute. Therefore formally, W − W = τ /(cid:96) is hermitianand should have an orthonormal eigenbasis (discrete or continuous). Butin concrete representation we obtained a continuous set of eigenfunctions offinite norm which are not mutually orthogonal. Hence τ is not self-adjoint:9t can only be formally symmetric, with domain D ( τ ) unequal to the domainof its adjoint, D ( τ † ). To define self-adjoint extensions, if they exist, we needto resolve the issue of the domains.Problem is obviously in the radial equation. Separation of angular variablesgives a division of H into subspaces of fixed angular momentum j , in which τ reduces to operators T j : (cid:0) ψ jm , ( W − W ) ψ (cid:48) j (cid:48) m (cid:48) (cid:1) ≡ δ jj (cid:48) δ mm (cid:48) (cid:90) dz Φ † T j Φ (cid:48) (4.1)= 2 δ jj (cid:48) δ mm (cid:48) (cid:90) dz (cid:16) f ∗ h ∗ (cid:17) (cid:32) ρ z − z − i ( j + ) z − i ddz ρ z − z + i ( j + ) z − i ddz (cid:33) (cid:32) f (cid:48) h (cid:48) (cid:33) = δ jj (cid:48) δ mm (cid:48) (cid:90) dz (cid:16) F ∗ H ∗ (cid:17) (cid:32) − i ddz − i ddz (cid:33) (cid:32) F (cid:48) H (cid:48) (cid:33) . Functions F and H are defined by F = (cid:18) − z (cid:19) iρ z − j − f , H = (cid:18) − z (cid:19) iρ z j + h , (4.2)and they are introduced in Appendix 1 to solve the radial equation; theysimplify the matrix elements of T j as well as the scalar product, (cid:0) ψ jm , ψ (cid:48) j (cid:48) m (cid:48) (cid:1) = 2 δ jj (cid:48) δ mm (cid:48) (cid:90) dz (cid:16) F ∗ H ∗ (cid:17) (cid:32) z j +1 z − j − (cid:33) (cid:32) F (cid:48) H (cid:48) (cid:33) . Let us examine properties of T j . In order to find T † j we partially integrate(4.1), (cid:0) ψ jm , ( W − W ) ψ (cid:48) j (cid:48) m (cid:48) (cid:1) = − i δ jj (cid:48) δ mm (cid:48) (cid:90) dz (cid:18) F ∗ dH (cid:48) dz + H ∗ dF (cid:48) dz (cid:19) = i δ jj (cid:48) δ mm (cid:48) (cid:90) dz (cid:18) dF ∗ dz H (cid:48) + dH ∗ dz F (cid:48) (cid:19) − iδ jj (cid:48) δ mm (cid:48) (cid:0) F ∗ H (cid:48) + H ∗ F (cid:48) (cid:1) (cid:12)(cid:12) . (4.3)10e see that the ‘action’ of T j on functions, given by the first term in (4.3), isself-adjoint: but since the boundary term does not vanish, T j and T † j are notequal. This is in fact a definition of being ‘formally symmetric’, [12]. Theother signature of non-hermiticity are nonzero deficiency indices, i.e. theexistence of normalizable solutions to equations T j Φ = ± i Φ . We show inAppendix 2 that the deficiency indices of T j are ( n + , n − ) = (1 , H on which the boundary term vanishes: this subspace becomes the domain ofboth, redefined or ‘extended’ τ and τ † . A necessary condition for existenceof self-adjoint extensions is that deficiency indices n + and n − be equal. An-alyzing (4.3) in Appendix 2 we find that T j is self-adjoint if it is restrictedto subspace of functions (4.2) which satisfy F (0) = H (0) = 0 , H (1) = icF (1) , (4.4)
10 20 30 40 - Figure 1: Solutions to Eq. (4.5) for j = , c = 1.Let us check that eigenfunctions (3.13) can satisfy (4.4). First relation isclearly true, the second gives J j +1 (2 λ ) J j (2 λ ) = c = const , (4.5)11hat is, an equation for λ . This equation, as seen from Figure 1, has infinitelymany solutions for every real c ; the set of solutions is discrete. The otherway to see this is for large values of λ as, asymptotically, J j +1 (2 λ ) J j (2 λ ) ∼ − tan (cid:18) λ − (2 j + 1) π (cid:19) , λ → ∞ . (4.6)The eigenvalues can be labelled by a natural number n , and for large λ theybecome equidistant with period π/ c we can fix the valueof one of the λ ’s; the other eigenvalues are determined by (4 . c we obtain a different self-adjoint extension T ( c ) j , i.e. we havea one-parameter family: we can take for example c = 1 as a preferred choice.Let us check orthogonality. Using the recurrence relations between the Besselfunctions we find( ψ λjm , ψ λ (cid:48) j (cid:48) m (cid:48) ) = 2 C ∗ Cδ jj (cid:48) δ mm (cid:48) (cid:90) z dz ( J j (2 λz ) J j (2 λ (cid:48) z ) + J j +1 (2 λz ) J j +1 (2 λ (cid:48) z ))= δ jj (cid:48) δ mm (cid:48) | C | λ − λ (cid:48) (cid:16) λ (cid:48) J j (2 λ ) J (cid:48) j (2 λ (cid:48) ) + λ (cid:48) J j +1 (2 λ ) J (cid:48) j +1 (2 λ (cid:48) ) − λJ j (2 λ (cid:48) ) J (cid:48) j (2 λ ) − λJ j +1 (2 λ (cid:48) ) J (cid:48) j +1 (2 λ ) (cid:17) = − δ jj (cid:48) δ mm (cid:48) λ − λ (cid:48) | C | J j +1 (2 λ ) J j +1 (2 λ (cid:48) ) (cid:18) J j (2 λ ) J j +1 (2 λ ) − J j (2 λ (cid:48) ) J j +1 (2 λ (cid:48) ) (cid:19) , (4.7)where in the second line J (cid:48) a ( ζ ) denotes the derivative of J a ( ζ ). The lastexpression is zero for λ (cid:54) = λ (cid:48) for discrete set of eigenfunctions which satisfy(4.5), and we confirm that the given basis is orthogonal. Let us verify that fuzzy de Sitter space corresponds to an expanding cos-mology and discuss the absence of the big bang singularity. The (squared)radius of the universe is given by( x i ) = − (cid:96) W i W i (5.1)12nd its evolution can be traced by the expectation value (cid:104) ( x i ) (cid:105) in the eigen-states of time. Eigenvalue λ of W − W used in the previous calculation isrelated to the time eigenvalue t exponentially, t = (cid:104) τ (cid:105) = ( ψ λjm , τ ψ λjm ) = (cid:96) log λ . (5.2)Using Casimir relation (2.11), − W i W i = W + W − W , (5.3)and taking normalized eigenstates ψ λjm ( ψ λjm , ψ λjm ) = 2 C ∗ C (cid:90) dz z (cid:0) J j (2 λz ) + J j +1 (2 λz ) (cid:1) = 1 (5.4)we find (cid:104)− W i W i (cid:105) = W + (cid:104) ( W + W )( W − W ) (cid:105) = W + λ + 2 λ (cid:104) W (cid:105) . Expectation value (cid:104) W (cid:105) can be estimated. We have W = − (cid:32) p (cid:126)r · (cid:126)σ i(cid:126)L · (cid:126)σ + i i(cid:126)L · (cid:126)σ + i p (cid:126)r · (cid:126)σ (cid:33) ,therefore( ψ λjm , W ψ λjm ) = (cid:90) d pp Φ † λjm (cid:18) im (cid:126)p · (cid:126)σ p + m ) − m (cid:126)r · (cid:126)σp + m − im ( (cid:126)p · ∇ )( (cid:126)p · (cid:126)σ )( p + m ) (cid:19) Φ λjm = − i (cid:90) dz (1 − z ) (cid:18) F ∗ λj dH λj dz + H ∗ λj dF λj dz (cid:19) = λ C ∗ C (cid:90) dz z (1 − z ) (cid:0) J j (2 λz ) + J j +1 (2 λz ) (cid:1) . (5.5)Comparing the last integral with (5.4),0 ≤ (cid:90) dz z (1 − z ) (cid:0) J j (2 λz ) + J j +1 (2 λz ) (cid:1) ≤ (cid:90) dz z (cid:0) J j (2 λz ) + J j +1 (2 λz ) (cid:1) (5.6)we obtain that 0 ≤ ( ψ λjm , W ψ λjm ) ≤ λ , hence W + λ ≤ ( ψ λjm , − W i W i ψ λjm ) ≤ W + 2 λ . (5.7)13he expectation value of the radius of the universe is bounded below by (cid:96) √W : it does not vanish in physical states which lie in the domain of self-adjoint extensions τ ( c ) , i.e. can be expanded in the corresponding eigenbases.The radius, on the other hand, grows with time exponentially: for late timeswe have (cid:112) (cid:104)− W i W i (cid:105) ∼ λ = e t/(cid:96) .Another important point is discreteness of time that, as explained, also comesthrough the self-adjointness of τ . Though hermiticity is a usual condition inquantum mechanics, we rarely deal with operators that do not have uniqueself-adjoint extensions. This is related to the fact that quantum mechanicsis defined on the flat unbounded space: one can expect boundary effects incurved spaces, spaces which are bounded or singular (geodesically incom-plete, or with curvature singularities). In this context, formally symmetrichamiltonians with a one-parameter family of self-adjoint extensions appearin various physical situations (and mathematical setups) in general relativityand cosmology, [14, 15, 16, 17]. The interpretation of non-uniqueness of theextension varies: from understanding that it is a further quantization am-biguity [16], to that it renders a definition of spacetimes that are singularfor ‘quantum probes’ (as in some cases, classically singular spacetimes canappear completely regular for quantized particles), [15]. Wald relates thenecessity to choose one of the extensions with the fact that the initial-valueproblem is classically ill-defined at naked singularity, and regards the possi-bility of constructing a self-adjoint extension as a resolution to the singularityproblem, [14].The last point of view is in some sense close to our example, though weare extending time and not the hamiltonian. Discreteness of time becomesrelevant in the ‘deep quantum region’ λ → t → −∞ , near theclassical boundary through which the steady-state model can be extended tothe complete de Sitter space. For values away from the Planck scale timeis almost continuous: the difference between its consecutive eigenvalues ismacroscopically negligible, t n +1 − t n ≈ (cid:96) log (1 + 1 n ) . (5.8)14iscreteness obtained by requiring self-adjointness in known in other casesof quantum spaces. One example is the q -deformed Heisenberg algebra,[ p, x ] = − i + ( q − xp (5.9)and its unitary representations, [18, 19]. The analysis shows that coordinate x is not self-adjoint, but the self-adjoint extensions exist; both x and p havediscrete spectra. Another interesting case is the minimal-length Heisenbergalgebra, [ p, x ] = − i − iβp , (5.10)which is in [20] represented in the Schr¨odinger representation. Again it isfound that x has a one-parameter family of self-adjoint extensions which putsits spectrum on lattice.The q -deformed Heisenberg algebra (Manin plane) has, as symmetry, thequantum group SU q (2), so it is natural to ask whether in our model symmetrygets deformed as well. As shown in [5], our choice of frame in fact breaksthe SO (1 ,
4) invariance, and a priori symmetries of fuzzy de Sitter spaceare rotations and time translations, SO (3) × U (1). Here U (1) denotes thedilatation subgroup, U (1) = { e iαM | α ∈ R } , the dilatation generator playsthe role of the hamiltonian, H = M : it evolves the eigenstates of time,(2.14).If we keep the standard notion that symmetry is defined by group of transfor-mations, the choice of a self-adjoint extension τ ( c ) is spontaneous symmetrybreaking. This can be seen easily: the elements of U (1) do not preserve thespace of physical states defined by (4.4) for arbitrary values of parameter α . However, there is a subgroup of dilatations, U ( c ) (1), determined by theallowed values of α which preserve condition (4.4): it is represented non-linearly. For large eigenvalues, (4.4) becomes periodic and λ equally spaced:subgroup U ( c ) (1) becomes in this limit (in this region of physical parameters),the additive group of integers. In the continuum approximation (cid:96) → Acknowledgement
We thank Igor Salom and Ilija Buri´c for variousdiscussions on representation theory. The work was supported by the Ser-bian Ministry of Education, Science and Technological Development GrantON171031, and by the COST action MP 1405 “Quantum structure of space-time”.
Appendix 1
In this appendix we solve the radial equations (3.9). In the signature whichwe use (cid:126)p = ( p i ) , (cid:126)L = ( L i ) , (cid:126)σ = ( σ i ) , (cid:126)r = ( x i ) = i ∂∂p i ,(cid:126)p · (cid:126)σ = − p i σ i , σ i σ j = − η ij − (cid:15) ijk σ k , ( (cid:126)r · (cid:126)σ )( (cid:126)p · (cid:126)σ ) = i (cid:16) p ∂∂p + (cid:126)L · (cid:126)σ (cid:17) . The eigenvalue equation (3.9) is (cid:16) m ρ ( (cid:126)p · (cid:126)σ ) −
12 ( p + m ) ( (cid:126)r · (cid:126)σ ) − m ( (cid:126)p · (cid:126)r )( (cid:126)p · (cid:126)σ ) (cid:17) Φ = λ Φ . (5.11)We use the Ansatz which separates angular and radial variables,Φ λjm ( (cid:126)p ) = f ( p ) p φ jm ( θ, ϕ ) + h ( p ) p χ jm ( θ, ϕ ) , (5.12)with p = − p i p i = p − m . The φ jm and χ jm are the spinor eigenfunctions16f M ij M ij and M ; they are orthonormal and satisfy φ jm = (cid:126)p · (cid:126)σp χ jm , ( (cid:126)L · (cid:126)σ ) φ jm = ( j − ) φ jm ,χ jm = (cid:126)p · (cid:126)σp φ jm , ( (cid:126)L · (cid:126)σ ) χ jm = − ( j + ) χ jm . (5.13)Introducing (5.12) we obtain radial equations( p + 1) dfdp + iρf − j + p − f = 2 iλ hp , (5.14)( p + 1) dhdp + iρh + j + p − h = 2 iλ fp . (5.15)In order to simplify them we rescale momentum to be dimensionless, p → mp , p → mp , p ∈ (0 , ∞ ), p ∈ (1 , ∞ ). Equations decouple when we introducenew functions F , H by f = ( p + 1) − iρ − j +14 ( p − j +14 F, h = ( p + 1) − iρ + j +14 ( p − − j +14 H. (5.16)We then obtain( p − d Fdp + 2( p + j ) dFdp + 4 λ ( p + 1) F = 0 , (5.17)( p − d Hdp + 2( p − j − dHdp + 4 λ ( p + 1) H = 0 , (5.18)and additional relations dFdp = 2 iλ ( p + 1) j − ( p − − j − H, dHdp = 2 iλ ( p + 1) − j − ( p − j F. (5.19)Equations (5.17-5.18) reduce to the Bessel equation ζ d Ydζ + ζ dYdζ + ( ζ − a ) Y = 0 (5.20)by compactification of the independent variable. Introducing z as z = (cid:114) p − p + 1 (5.21)17oth equations reduce to (5.20) for ζ = 2 λz ∈ (0 , λ ). In equation (5.17), a = j ; in (5.18), a = − j − J a ( ζ ), J − a ( ζ ) or J a ( ζ ), Y a ( ζ ) ( a is half-integer, so J − j − ( ζ ) = ( − j − Y j +1 ( ζ )).Therefore, F ∼ J j , J − j and H ∼ J j +1 , J − j − . Taking into account addi-tional relations (5.19) which are satisfied through recurrence relation1 ζ ddζ ζ a J a ( ζ )) = ζ a − J a − ( ζ ) , (5.22)we obtain two linearly independent solutions: F λj = Cz − j J j (2 λz ) , H λj = iCz j +1 J j +1 (2 λz ) , (5.23)˜ F λj = ˜ Cz − j J − j (2 λz ) , ˜ H λj = − i ˜ Cz j +1 J − j − (2 λz ) . (5.24)As the Bessel functions around ζ = 0 behave as J a ( ζ ) ∼ a + 1) (cid:18) ζ (cid:19) a (5.25)the second solution diverges, ˜ ψ λjm ∼ ζ − j − , so we have one regular solution, f λj = C (cid:18) − z (cid:19) − iρ √ z J j (2 λz ) , h λj = iC (cid:18) − z (cid:19) − iρ √ z J j +1 (2 λz ) . (5.26)It exists for every real λ . But J a ( − ζ ) = ( − a J a ( ζ ), so the spectrum can berestricted to the positive real axis, λ > ψ λjm , ψ λ (cid:48) j (cid:48) m (cid:48) ) = 2 δ jj (cid:48) δ mm (cid:48) (cid:90) dz ( f ∗ f (cid:48) + h ∗ h (cid:48) ) = 2 δ jj (cid:48) δ mm (cid:48) (cid:90) dz ( z j +1 F ∗ F (cid:48) + z − j − H ∗ H (cid:48) )= 2 δ jj (cid:48) δ mm (cid:48) C ∗ C (cid:48) (cid:90) zdz ( J j (2 λz ) J j (2 λ (cid:48) z ) + J j +1 (2 λz ) J j +1 (2 λ (cid:48) z )) . (5.27)It is nonzero for λ (cid:54) = λ (cid:48) , and finite for each λ , which as we discuss in the text,is a problem. Singular solutions do not have the right normalization to be18igenfunctions of the continuous spectrum: similarly to (5.27), we have( ˜ ψ λjm , ˜ ψ λ (cid:48) j (cid:48) m (cid:48) ) = 2 δ jj (cid:48) δ mm (cid:48) ˜ C ∗ ˜ C (cid:48) (cid:90) dz ( J − j (2 λz ) J − j (cid:48) (2 λ (cid:48) z ) + J − j − (2 λz ) J − j (cid:48) − (2 λ (cid:48) z )) . This integral is divergent in the lower limit, but the divergence depends on j and not on the difference λ − λ (cid:48) i.e. it does not have the required form δ ( λ − λ (cid:48) ) . Appendix 2
We start with the deficiency indices of T j . To determine them we need tosolve equations T j Φ = ± i Φ . (5.28)This is in fact not difficult: solutions to these equations are the same assolutions to (5.11) for λ = ± i : the Bessel functions of imaginary argumenti.e. the modified Bessel functions I a ( ζ ) and K a ( ζ ), I a ( ζ ) = i − a J a ( iζ ) , K a ( ζ ) = π i a +1 ( J a ( iζ ) + iY a ( iζ )) . (5.29)As before, a = ± j, ± ( j + 1). The modified Bessel functions have similarbehavior around zero as the Bessel functions: K a ( ζ ) is divergent and thecorresponding solution has infinite norm. This implies that equation T j Φ = i Φ has just one regular solution, F + = Cz − j I j (2 z ) , H + = − Cz j +1 I j +1 (2 z ) . (5.30)Similarly there is one regular solution ( F − , H − ) to equation T j Φ = − i Φ .This means that deficiency indices of T j are ( n + , n − ) = (1 , T j isnot a self-adjoint operator but can be extended to one.Next, let us briefly recall the procedure of constructing self-adjoint extensionsof formally symmetric operators. We use notation of [12], where also proof ofthe main technical result which we use can be found. We can write equation194.3) abstractly as(Φ , T j Φ (cid:48) ) = ( T j Φ , Φ (cid:48) ) + B (Φ , Φ (cid:48) ) = ( T † j Φ , Φ (cid:48) ) , (5.31)where the boundary term B (Φ , Φ (cid:48) ) is a bilinear form, which in our case reads B (Φ , Φ (cid:48) ) = ( F ∗ H (cid:48) + H ∗ F (cid:48) ) | . (5.32)Apparently, the domain of T j is given by all normalizable functions Φ , Φ (cid:48) that satisfy B (Φ , Φ (cid:48) ) = 0, or in our case F (0)= H (0)=0, F (1)= H (1)=0.Then D ( T † j ) = H and obviously the two domains are not equal, D ( T j ) ⊂ H .To achieve self-adjointness, one should relax the condition which determines D ( T j ) and restrict D ( T † j ). This is done effectively by finding n + linearlyindependent functions Φ k (more precisely, n + linearly independent vectorscorresponding to their boundary values), in our case one, Φ , that satisfy B (Φ k , Φ l ) = 0 . ∀ k, l . (5.33)The domain of a self-adjoint extension of T j is then defined as a set offunctions Φ, D ( T j ) = D ( T † j ) = { Φ | B (Φ , Φ k ) = 0 , ∀ k } . (5.34)In principle, boundary term (4.3) is a combination of values at both boundarypoints but often the constraints can be imposed separately. It is possible todo it in our case as wel: we can choose F (0) = H (0) = 0, in accordance withbehavior of the eigenfunctions of τ which constitute a basis. If, at the otherboundary, we denote the values of Φ as F (1) = σ e iβ , H (1) = iσ (cid:48) e iβ (cid:48) , (5.35)we find iβ = iβ (cid:48) + nπ . Constants β , σ and σ (cid:48) are real numbers, so the domainof the self-adjoint extension T ( c ) j is a set of functions that satisfy F (0) = H (0) = 0 , H (1) = ± i σσ (cid:48) F (1) = icF (1) , c ∈ R . (5.36)20 eferences [1] J. Madore, “An Introduction To Noncommutative Differential GeometryAnd Its Physical Applications,” Lond. Math. Soc. Lect. Note Ser. (2000).[2] M. Buric, T. Grammatikopoulos, J. Madore and G. Zoupanos, JHEP (2006) 054 [hep-th/0603044].[3] J. Madore, Class. Quant. Grav. (1992) 69, J. Hoppe, “Quantumtheory of a massless relativistic surface and a two-dimensional boundstate problem”, Ph.D. Thesis, MIT, 1982.[4] A. P. Balachandran, S. Kurkcuoglu and S. Vaidya, Singapore, Singapore:World Scientific (2007) 191 p. [hep-th/0511114].[5] M. Buric and J. Madore, Eur. Phys. J. C (2015) no.10, 502[arXiv:1508.06058 [hep-th]].[6] M. Buric, D. Latas and L. Nenadovic, Eur. Phys. J. C (2018) no.11,953 [arXiv:1709.05158 [hep-th]].[7] S. W. Hawking and G. F. R. Ellis, “Large Scale Structure of Space-Time”, Cambridge University Press (1975)[8] M. Buric, D. Latas, V. Radovanovic and J. Trampetic, Phys. Rev. D (2007) 097701 doi:10.1103/PhysRevD.75.097701 [hep-ph/0611299].[9] J. Dixmier, Bull. Soc. Math. France 89 (1961) 9.[10] P. Moylan, J. Math. Phys. (1983) 2706, P. Moylan, J. Math. Phys. (1985) 29.[11] V. Bargmann and E. P. Wigner, Proc. Nat. Acad. Sci. (1948) 211.[12] V. Hutson, J. Pym, M. J. Cloud, “Applications of Functional Analysisand Operator Theory”, Elsevier Science (2005)2113] N. Dunford and J. T. Schwartz, “Linear Operators”, J. Wiley & Sons(1957).[14] R. M. Wald, J. Math. Phys. (1980) 2802. doi:10.1063/1.524403[15] G. T. Horowitz and D. Marolf, Phys. Rev. D (1995) 5670doi:10.1103/PhysRevD.52.5670 [gr-qc/9504028].[16] A. A. Andrianov, C. Lan, O. O. Novikov and Y. F. Wang, Eur.Phys. J. C (2018) no.9, 786 doi:10.1140/epjc/s10052-018-6255-5[arXiv:1802.06720 [hep-th]].[17] S. Gryb and K. P. Y. Th´ebault, Class. Quant. Grav. (2019) no.3,035009 doi:10.1088/1361-6382/aaf823 [arXiv:1801.05789 [gr-qc]].[18] J. Schwenk and J. Wess, Phys. Lett. B (1992) 273.[19] A. Hebecker, S. Schreckenberg, J. Schwenk, W. Weich and J. Wess, Z.Phys. C (1994) 355.[20] A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D (1995) 1108[hep-th/9412167].[21] M. Fichtmuller, A. Lorek and J. Wess, Z. Phys. C71