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...

Many have written of the experience of mathematical beauty as being comparable to that derived from the greatest art. This makes it interesting to learn whether the experience of beauty derived from such a highly intellectual and abstract source as mathematics correlates with activity in the same part of the emotional brain as that derived from more sensory, perceptually based, sources. To determine this, we used functional magnetic resonance imaging (fMRI) to image the activity in the brains of 15 mathematicians when they viewed mathematical formulae which they had individually rated as beautiful, indifferent or ugly. Results showed that the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain, namely field A1 of the medial orbito-frontal cortex (mOFC), as the experience of beauty derived from other sources.

We study a non-local scalar quantum field theory in flat spacetime derived from the dynamics of a scalar field on a causal set. We show that this non-local QFT contains a continuum of massive modes in any dimension. In 2 dimensions the Hamiltonian is positive definite and therefore the quantum theory is well-defined. In 4-dimensions, we show that the unstable modes of the non-local d'Alembertian are propagated via the so called Wheeler propagator and hence do not appear in the asymptotic states. In the free case studied here the continuum of massive mode are shown to not propagate in the asymptotic states. However the Hamiltonian is not positive definite, therefore potential issues with the quantum theory remain. Finally, we conclude with hints toward what kind of phenomenology one might expect from such non-local QFTs.