arXiv:1001.2725v4 [gr-qc] 1 Nov 2011 The Scalar Curvature of a Causal Set Dionigi M. T. Benincasa and Fay Dowker Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom A one parameter family of retarded linear operators on scalar fields on causal sets is introduced. When the causal set is well approximated by 4 dimensional Minkowski spacetime, the operators are Lorentz invariant but nonlocal, are parametrised by the scale of the nonlocality and approximate the continuum scalar D’Alembertian □ when acting on fields that vary slowly on the nonlocality scale. The same operators can be applied to scalar fields on causal sets which are well approximated by curved spacetimes in which case they approximate □ − 1 $_{2}$R where R is the Ricci scalar curvature. This can used to define an approximately local action functional for causal sets. PACS numbers: 04.60.Nc,02.40.-k,11.30.Cp The coexistence of Lorentz symmetry and fundamen- tal, Planck scale spacetime discreteness has its price: one must give up locality. Since, if our spacetime is granu- lar at the Planck scale, the “atoms of spacetime” that are nearest neighbours to a given atom will be of or- der one Planck unit of proper time away from it. The locus of such points in the approximating continuum Minkowski spacetime is a hyperboloid of infinite spa- tial volume on which Lorentz transformations act tran- sitively. The nearest neighbours will, loosely, comprise this hyperboloid and so there will be an infinite number of them. Where curvature limits Lorentz symmetry, it may render the number of nearest neighbours finite but it will still be huge so long as the radius of curvature is large compared to the Planck length. Causal set theory is a discrete approach to quantum gravity which embod- ies Lorentz symmetry [1, 2] and exhibits nonlocality of exactly this form [3, 4]. Nonlocality looks to be simultaneously a blessing and a curse in tackling the twin challenges that any funda- mentally discrete approach to the problem of quantum gravity must face. These are to explain (1) how the fun- damental dynamics picks out a discrete structure that is well approximated by a Lorentzian manifold and (2) why, in that case, the geometry should be a solution of the Einstein equations. This is often referred to as the problem of the continuum limit but in the context of a fundamentally discrete theory in which the discreteness scale is fixed and is not taken to zero but rather the ob- servation scale is large, it is more accurately described as the problem of the continuum approximation. Consider first the problem of recovering a continuum from a quantum theory of discrete manifolds . (We adopt this term following Riemann [5] and use it to refer to causal sets, simplicial complexes, graphs, or whatever dis- crete entities the underlying theory is based on.) When- ever a background principle or structure in a physical the- ory is abandoned in order to seek a dynamical explana- tion for that structure, the state we actually observe be- comes a very special one amongst the myriad possibilities that then arise. The continuum is just such a background assumption. In giving it up, generally one introduces a However, nonlocality poses a danger when it comes to the second challenge of recovering Einstein’s equations. In the case of causal sets, the number of discrete man- ifolds of size N grows as e N $^{2}$/ 4 [14] and a local action would give causal set theory little chance of recovering the continuum. So the nonlocality of causal sets holds out hope that the theory has a continuum regime and indeed there exist physically motivated, classically stochastic dy- namical models for causal sets [15] in which the entropi- cally favoured configurations almost surely do not occur and those that do exhibit an intriguing hint of manifold- like-ness [16]. space of discrete manifolds in which the vast majority have no continuum approximation. There will therefore be a competition between the entropic pull of the huge number of noncontinuum configurations – choose one uni- formly at random and it will not look anything like our spacetime – and the dynamical law which must suppress the contributions of these nonphysical configurations to the path integral. The following general argument shows that a local dynamics for quantum gravity will struggle to provide the required suppression. Consider the parti- tion function as a sum over histories in which the weight of each discrete manifold is e − S where S is the real Wick rotated action. As we increase the observation scale, the sum will be over discrete manifolds with an increasing number, N , of atoms. If the action is local – which in a discrete setting translates to it being a sum over con- tributions from each atom – then it will grow no faster than N times some constant, α , and so each weight is no smaller than e − $^{αN}$. If the number of discrete mani- folds with N atoms grows faster than exponentially with N , and if the majority of these discrete manifolds are not continuumlike then they will overwhelm the parti- tion function and the typical configuration will not have a continuum approximation. Even when the number of discrete manifolds is believed to grow exponentially, en- tropy can still trump dynamics as was seen in the lack of a continuum limit in the Euclidean dynamical triangula- tions programme [6–9]. Causal dynamical triangulations do better, see, e.g. , [10–13], by restricting the class of triangulations allowed in the sum.