Scalar Curvature of a Causal Set

Abstract

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,11.30.Cp The coexistence of Lorentz symmetry and fundamental, Planck scale spacetime discreteness has its price: one must give up locality. Since, if our spacetime is granular at the Planck scale, the "atoms of spacetime" that are nearest neighbours to a given atom will be of order 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 spatial volume on which Lorentz transformations act transitively. 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 embodies 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 fundamentally discrete approach to the problem of quantum gravity must face. These are to explain (1) how the fundamental 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 observation 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 discrete entities the underlying theory is based on.) Whenever a background principle or structure in a physical theory is abandoned in order to seek a dynamical explanation for that structure, the state we actually observe becomes 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 space of discrete manifolds in which the vast majority have no continuum approximation. T...

Generalized causal set d’Alembertians

“…However, one can also cite reasons why one might need to have ρ −4 , leading to a more long-range nonlocality. 3 Although these reasons are not conclusive, let us accept them 2 This process is known as Poisson sprinkling: given a spacetime M , let the discrete subset of points, C, be one particular realization of a Poisson process in M , and let the elements of C retain the causal relations they have when regarded as points of M . In order that the resulting precedence relation on C approximately encode the metric of M , one must exclude spacetimes with closed causal curves, for example by requiring M to be globally hyperbolic.…”

“…In this paper, we make a start on answering this question by analysing the "spectral properties" (Fourier transform) of a family of continuum operators (D) ρ . In section 2, we discuss the continuum operators corresponding to the original 2D [2] and 4D [3] causet d'Alembertians, and in section 3 we generalize the discussion to an infinite family of operators parametrized by a set of coefficients, {a, b n }, for which we derive explicit equations that ensure the usual flat space d'Alembertian is recovered in the infrared limit. Based on the UV behaviour of these operators (which we determine for all dimensions and coefficients {a, b n }), we propose a genuinely Lorentzian perturbative regulator for quantum field theory (QFT).…”

“…In order that the resulting precedence relation on C approximately encode the metric of M , one must exclude spacetimes with closed causal curves, for example by requiring M to be globally hyperbolic. 3 The issue here concerns the behavior of B (D) ρ for one particular sprinkling versus its behavior after averaging over all sprinklings. The latter converges to as ρ → ∞ but the former incurs fluctuations which grow larger as ρ → ∞ and which therefore will be sizable if ρ is the sprinkling density, −4 .…”

“…for causets derived by sprinkling M 2 ). The expression introduced in [2] was generalized to D = 4 dimensions in [3] and recently to arbitrary D in [4,5].…”

Suppression of non-manifold-like sets in the causal set path integral

“…The action we shall use, the Benincasa-Dowker action, was introduced in [10]. For a causal set C with n elements, it takes the general form [11,12] 1…”