Spin Foam Models of n-dimensional Quantum Gravity, and Non-Archimedan and Non-Commutative Formulations J.Manuel Garc´ıa-Islas 4 0 0 2 Centro de Investigacion en Matem´aticas n A.P. 402, 36000 a Guanajuato, Gto, M´exico J 3 email: [email protected] 1 1 v 8 Abstract. This paper is twofold. First of all a complete unified picture 5 of n-dimensional quantum gravity is proposed in the following sense: In spin 0 1 foam models of quantum gravity the evaluation of spin networks play a very 0 important role. These evaluations correspond to amplitudes which contribute 4 0 in a state sum model of quantum gravity. In [6], the evaluation of spin networks / as integrals over internal spaces was described. This evaluation was restricted c q to evaluations of spin networks in n-dimensional Euclidean quantum gravity. - r Here we propose that a similar method can be considered to include Lorentzian g : quantum gravity. We therefore describe the the evaluation of spin networks in v the Lorentzian framework of spin foam models. We also include a limit of the i X Euclidean and Lorentzian spin foam models which we call Newtonian. This r a Newtonian limit was also considered in [10]. Secondly, we propose an alternative formulation of spin foam models of quantum gravity with its corresponding evaluation of spin networks. This al- ternative formulation is a non-archimedean or p-adic spin foam model. The interest on this description is that it is based on a discrete space-time, which is the expected situation we might have at the Planck length; this description might lead us to an alternative regularisation of quantum gravity. Moreover a non-commutative formulation follows from the non-archimedean one. 1 1 Introduction In [10], a definition for the evaluation of spin networks in 3-dimensional quan- tum gravity was considered. This was done in a unified spirit of three cases of quantum gravity which are the Lorentzian, the Euclidean and the Newto- nian cases. 1 Particular attention was paid to the evaluation and asymptotics of the tetrahedron(3-simplex) network. Then it was suggested that the same work could be generalised by considering evaluations of spin networks in the n-dimensional case. The evaluation of spin networks in the n-dimensional case was considered and restricted to the Euclidean case in [6]. This evaluation was described as Feynman graphs. These are integrals over internal spaces. In this paper we generalise the work done in [10] and at the same time we complete in part the story of [6] by considering the evaluation of the n- dimensional Lorentzian and Newtonian n-simplex. We restrict ourselves to the case of the principal unitary irreducible representations of SO(n 1,1) and − ISO(n 1). − In section 2 we summarise the results on n-dimensional Euclidean quantum gravity studied in [6] and give the recipe of the evaluations of spin networks as simple graphs. 2 In section 3 we describe the case of n-dimensional Lorentzian quantum grav- ity and consider the case of spacelike spin networks, that is, graphs whose edges are all spacelike. In section 4 we describe the limitng case of Newtonian spin networks, a limit of both, Euclidean and Lorentzian quantum gravity. This case should be interpreted as a mathematical idea which physically is given when the speed of light is so large. However, we doubt its physical interpretation of a Newtonian quantum gravity as this may not have sense at all as it will be addressed in section 4. Anyway, it is still interesting to have a physical interpretation of the Newtonian spin networks and to explore its impotance for spin foam models, if any. The second part of the paper starts in section 5 where we propose a study of a p-adic and quantum deformation formulation of quantum gravity inspired on the spin foam models. There is a possible relation to a non-archimedean and non-commutative geometry respectively. This last section may be of great interest although future work is required to give it a precise formulation and to develop its possible importance. There is a thought that an adelic description of quantum gravity in terms of spin foams may be needed. 1This latter case was given its name first in [10], and should be interpreted as a mathe- matical limit. This is because physically, there may not be sense for quantizing Newtonian gravity. This is better expained in section 4. 2Aformulationoftheexistenceofhigherdimensionalgravitytheoriesappearsin[7],there- fore it is worth to study the evaluation of spin networks in any dimension. 2 2 Euclidean spin networks A way to evaluate simple spin networks for the group SO(n) was introduced in [6]. In this section we give a description of this idea which follows similarly for the Lorentz group SO(n 1,1) and for the inhomogeneous group ISO(n 1) − − which will be discussed in the following sections. Simple spin networks of the group SO(n) are those constructed from special representations of SO(n). A special class of representations are the spherical harmonics [11] that appear in the decomposition in the decomposition of the space of fuctions L2(Sn−1) into irreducible components. These representations are labelled by integers ℓ and the spin networks, Σ, have edges labelled by ρ’s. The evaluation of these spin networks is given by the following rules which closelyresembletherulesofFeynmangraphsthatapearinquantumfieldtheory: -With every edge of the graph Σ, associate a propagator KE(x,y). ρ -Take the product of all these data and integrate over one copy of the ho- mogeneous space SO(n)/SO(n 1) = Sn−1 for each vertex. − Thus the evaluation formula is given by dx KE(x,y) (1) ZSn−1 Yv vYe ρe where ρ denotes the representation labelling the edge e. e The propagators KE(x,y) are given by the Gegenbauer polynomials Cp as ρ n described in [6]. In [10] we discussed that the propagators in n-dimensional quantum gravity for any case(Euclidean, Lorentzian, Newtonian) are given by zonal spherical functions of the respective group. In this Euclidean case these propagators are Legendre polynomials and are directly related to the Gegenbauer polynomials [19]. These propagators are then expressed in an integral form as Γ(n−1) π KE(r) = 2 (cosθ +icosψsinθ)σsinn−3ψdψ (2) ρ √πΓ(n−2) Z 2 0 where σ = p + iℓ. p is related to the dimension of the space-time as p = − (n 2)/2 and ℓ labels the irreducible representations os SO(n). − As a special case we have the n-simplex Kn+1, which is the complete graph with n+1 vertices.3 3A complete graph Kn+1 has n+1 vertices and an edge for each pair of vertices, so that it has n(n+1)/2 edges. 3 1 2 (n+1) 3 4 Its evaluation is given by the integral Kn+1 = dx ...dx KN(r(x,y)) (3) ZSn−1 1 n+1Ye ρe anditsasymptoticswhichwasstudiedin[6]andspecialisedtothe3-dimensional case in [10] is given by π cos (V θ +k ) (4) ij ij 4 (cid:16)Xi<j (cid:17) where V are volumes of (n 2)-simpleces and k is an integer expressed in terms ij − of the dimension n, [6]. In the 3-dimensional case we have that V are lengths ij of edges. The formula was obtained in [10] and in that case, k is the Hessian of the action. All these method can be generalised to include the Lorentzian case and then to the Newtonian limit. This is done in the following sections. 3 Lorentzian spin networks For the n-dimensional Lorentzian case we have n-dimensional Minkowski space given by Rn with bilinear form [x,y] = x y x y ... x y 0 0 1 1 n−1 n−1 − − − The unimodular group which leaves this bilinear form invariant is denoted by SO(n 1,1). The light cone, also known by null cone, C is the set of points − x Rn which satisfy [x,x] = 0. ∈ The subgroup of SO(n 1,1) of transformations preserving both sheets of − the cone is the connected component denoted by S (n 1,1). The language 0 − of representation theory is important in the spin foam models formulations of quantum gravity. For our group S (n 1,1), its representations are mainly 0 − divided in a set of discrete representations labelled by integers and in a set of continuous representations labelled by real numbers. In this paper we restrict ourselves to the continuos representations known as the principal unitary series. These representations are thought as labelling 4 spin networks with spacelike edges. A state sum model of quantum gravity is constructed by considering a triangulation of an n-dimensional manifold △ M and considering its dual complex , we construct a spin foam model by △ J labelling each face of by a principal unitary irreducible representation of the △ J group SO (n 1,1). The state sum model is the given by 0 − ∞ (M) = dρ A(f) A(e) A(v) (5) f Z Z 0 Yf Ye Yv where the integration is carried over the labels of all internal faces of the dual complex and A(f),A(e),A(v) are the amplitudes given to the faces,edges and vertives of the dual complex . These amplitudes are given by the evaluation △ J of certain labelled spin networks such as A(f) = ρ . .. A(e) = 1/ .. . n edges 1 2 (n+1) 3 A(v) = 4 Recall that the egdes of these graphs are labelled by principal unitary repre- sentations of SO (n 1,1). The vertex amplitude is given by the evaluation of 0 − the n-simplex graph. This n-simplex can be seen as a collection of n+1 points which are all joined together by edges. The evaluation of these graphs is analogous to the Euclidean case, so the recipe is: -To each edge of the spin network we associate a propagator. The propaga- tors are given by zonal spherical funtions for the continious representations of SO (n 1,1). 0 − -We multiply all these propagators and integrate over a copy of an internal space for each vertex. The internal spaces are homogeneous spaces in which the propagators are evaluated. For the group SO (n 1,1) the zonal spherical functions for the continuos 0 − represenations are given by the Legendre functions 5 Γ(n−1) π KL(r) = 2 (coshr cosθsinhr)σsinn−3θdθ (6) ρ √πΓ(n−2) Z − 2 0 where σ = p + iρ. p is related to the dimension of the space-time as p = − (n 2)/2, and ρ labels a continuous representation of SO (n 1,1). 0 − − Moreover, the homogeneous space is given by the (n 1)-dimensional hy- − perbolic space Hn−1 = SO (n 1,1)/SO(n 1). 4 ∞ 0 − − 3.1 The n-simplex The non-degenerate n-simplex is represented as the complete graph of (n+1) vertices denoted by Kn+1. Following the recipe for evaluation of spin networks, the evaluation of the n-simplex spin network graph would be given by the fol- lowing integral Kn = dx ...dx KL(r(x,y)) (7) Z 1 n−1 ρe H∞ Ye where we have a multiple integral over n vertices of the n-simplex. One of the integrals was dropped for regularisation in analogy to the 3-dimensional case [10], and the 4-dimensional case [3]. A problem that appears now is whether our integral (7) converges, that is, whether it is well defined. This problem also arised in the 3-dimensional case [10], and in the 4-dimensional case [3]. Here we give much evidence for its convergence in any dimension. This evidence is very close to a proof. 3.1.1 The convergence Before showing the evidence for the convergence of the n-simplex in any di- mension we first consider some other integral evaluations and show their con- vergence. Although this discussion follows closely the same ideas of the 4- dimensional case [1], it gives a generalisation to the n-dimensional case. We have that for ρ = 0 our kernel KL(r) is well defined for small r and for 6 ρ large values of r is asymptotic to [15] n 1 KρL(r) ∼ Aρ2(n−3)/2Γ( −2 )e(1−n2)r (8) where n is the dimension of the space-time and A is a factor which depends on ρ the representation. 4Letusdenotethe(n 1)-dimensionalhyperbolicspacesimplybyH∞. Thisnotationwill − be clear when we study the non-archimedeanformulation. 6 The theta symbol and more general graphs with loops: We prove first that the theta symbol converges in any dimension, that is the following evaluation converges ρ1 ρ 2 = dx K (x,y)K (x,y)K (x,y) (9) Z ρ1 ρ2 ρ3 ρ H∞ 3 This follows easily from the asymptotic behaviour of our kernel where our inte- gral can be written in the form n 1 3 A A A 23(n−3)/2 Γ( − ) dx e(1−n/2)re(1−n/2)re(1−n/2)r (10) ∼ ρ1 ρ2 ρ3 (cid:16) 2 (cid:17) ZH∞ The volume form gives ∼ Aρ1Aρ2Aρ323(n−3)/2(cid:16)Γ(n−2 1)(cid:17)3Z ∞dr e(2−2n)r (11) which for n 3 is finite. This implies that the theta symbol converges in any ≥ dimension greater than or equal to 3. Similarly, if we consider the evaluation of the edge amplitude A−1, or any e such graph of two vertices joined by k 3 edges, we have that its evaluation is ≥ given by ..... A A ...A 2k(n−3)/2 Γ(n−1) k ∞dr ek(1−n/2)re(n−2)r . ∼ ρ1 ρ2 ρk (cid:16) 2 (cid:17) Z ∼ Aρ1Aρ2...Aρk2k(n−3)/2(cid:16)Γ(n−2 1)(cid:17)kZ ∞dr e(k−2)2(2−n)r(12) which is finite. More general integrals: Now we consider the following integral I = dx K (x,x )K (x,x )...K (x,x ) (13) Z ρ1 1 ρ2 2 ρk k H∞ for fixed x ,...,x H . We prove that it converges. Using the fact that 1 k ∞ our kernel K (x,y)∈is asymptotic to A 2(n−3)/2Γ(n−1)e(1−n)r we have that our ρ ρ 2 2 integral I is asymptotic up to some factors, to dx e(1−n2)(r1+...rk) (14) ZH∞ 7 where r = d(x,x ). In [1] it was proved that we can find a barycentre b H+ i i ∈ for the points x , such that 5 i 1 r := d(x,b) (r +...+r ) (15) 1 k ≤ k Then in spherical coordinates around b we see that our integral I is bounded by ∞ ek(1−n2)rsinh(n−2)r dr (16) Z which converges for all n 3 and all k 3. ≥ ≥ Consider again the integral (14). We have that ∞ I sinh(n−2)re(1−n2)(r1+...rk) dr (17) ∼ Z Now we proceed as in [1] by breaking the integral over r into two parts by considering: 1 r +...+r kr , M = min (r +...+r ) 1 k x 1 k ≥ k therefore we have that M ∞ I e(1−n2)kM sinh(n−2)r dr + e(1−n2)krsinh(n−2)r dr ∼ Z Z M 1 M 1 ∞ e(1−n2)kM+(n−2)r dr + e(1−n2)kr+(n−2)r dr ∼ 2 Z 2 Z M Be(1−n2)kM+(n−2)M (18) ≤ where B is a constant factor. The triangle inequality implies that 1 r +...r r (19) 1 k ij ≥ k 1 − Xi<j for all x ,..,x where r = d(x ,x ). Therefore we have that 1 k ij i j 1 M r (20) ij ≥ k(k 1) − Xi<j which implies that n 1 I Bexp ((1 )k +(n 2)) r ) (21) ij ∼ − 2 − k(k 1) h − Xi<j i 5Although in [1] the case of the 3-dimensional hyperbolic space was considered, the proof for the existence of a barycentre works in any dimension 8 Now we can go back to the n-simplex and compute for instance the integral over x . This resembles an integral of the same kind as the integral I. Therefore n it is asymptotic to the equation (21) and we continue to integrate over the other variables. At some step we will have to change to spheroidal coordinates as was done in [1] for the 4-dimensional case. Following the procedure it can be expected that the integral that defines the evaluation of the n-simplex is bounded and therefore converges. 3.2 Asymptotics Once we have shown evidence for the convergence of the amplitude evaluation of the n-simplex, we calculate its asymptotics.6 First we notice that evaluating the integral of our kernel KL(r), we have that ρ 2(n−3)/2Γ((n 1)/2) KL(r) = − B(3−n)/2 (coshr) (22) ρ sinh(n−3)/2r σ+(n−3)/2 where B(3−n)/2 (coshr) are Legendre functions. We have that σ = p + iρ σ+(n−3)/2 − where p = (n 2)/2 so that − B(3−n)/2 (coshr) = B(3−n)/2 (coshr) (23) σ+(n−3)/2 −(1/2)+iρ For large value of ρ we have that our kernel can be approximated by the asymp- totic behaviour of B(3−n)/2 (coshr) which is given by −(1/2)+iρ 2 (3 n)π π ρ(2−n)/2cos ρr + − (24) rsinhr 4 − 4 (cid:16) (cid:17) We now have that our n-simplex evaluation can be written as Kn+1 = dx1...dxn−1 ei i<jǫijρijrij+(2−n)π/4 (25) Z P H∞ Yi<j The function given by S = ǫ ρ r +(2 n)π/4 is called the action. i<j ij ij ij − WerescaleallofourreprPesentations ρij byacommonfactorαρij andlookfor thebehaviourofourintegralformulawhenα . Weusethestationaryphase → ∞ method to evaluate the asymptotics of our integral formula (25). Moreover we restrict to the non-degenerate configurations where all r = 0. ij 6 Given our non-degenerate n-simplex, there are (n+1) timelike unit vectors n which are normals to the (n 1) simpleces. There is a notion of Lorentzian i − − angle, from which a Schl¨afli identity follows, [1]. This identity is given by 6The study of the asymptotics of amplitudes of spin networks in quantum gravity can be found in many other references such as [2], [4], [8], [10] 9 V dΘ = 0 (26) ij ij Xi≤j Some of our normal vectors are future pointing and some are past pointing. We know that our points x live on the hyperbolic hyperspace H which implies i ∞ that these vectors are all timelike and future pointing. Then x = a n where i i i a = 1 if n is future pointing, or a = 1 if n is past pointing. i i i i − By taking this into account we then vary the action and constrain such variation by a Lagrange multiplier which finally fixes the ǫ’s up to an overall sign. Wewillthenhaveanasymptoticbehaviourgivenbyanoscillatoryfunction analogous to the Euclidean one. 4 Newtonian spin networks This case of Newtonian spin networks has a mathematical sense, as it is a limit of the Euclidean and Lorentzian cases. However, its physical interpretation as a Newtonian quantum gravity theory is not clear to the author. First of all, as it is mentioned in [12], quantum gravity refers to the attempts to unify general relativity and quantum theory. If gravity was nothing but the Newtonian well known static force, the constuction of a corresponding quantum theory would be a simple and uninteresting affair. But even more, it may not have sense at all. This is because of the following reasoning: The non-degenerate scalar product which gives rise to the inhomo- geneous symmetry of space-time is interpreted physically as a limiting case of the Lorentzian one when the speed of light c . This is also correct form → ∞ the Newtonian theory point of view where the principle of the universal speed limit given by the velocity of ligth c is no longer true. In Newtonian theory a body can reach any speed and so light could travel so fast in a reference frame. Therefore c is not a constant any more and the Planck length Gh 1 2 ℓ = P c3 (cid:16) (cid:17) has no meaning at all as a constant. If light c , the Planck length L 0, P → ∞ → and so, space-time is classical. It is also possible that the name of Newtonian spin networks is not appro- priate. Any way, the inhomogeneous limit has sense and we have the right to study the evaluation of its spin networks. Mathematically the inhomogeneous group ISO(n 1) can be obtained by a − limit procedure from SO (n 1,1) [19]. 0 − The Newtonian spin networks for n-dimensional quantum gravity whose group is ISO(n 1), and whose bilinear form is degenerate should then be − evaluated by a zonal spherical function of this group. That is, 10