Approaches to Quantum Gravity

36 R. D. Sorkin(as one might have expected) when K is held fixed as N increases. The means areaccurate by construction, in the sense that they exactly 7 reproduce the continuumexpression ¯B K φ (which in turn reproduces φ to an accuracy of around 1% forK > ∼200). (It should also be possible to estimate the fluctuations analytically, butIhavenottriedtodoso.)In any case, we can conclude that “discretized D’Alembertians” suitable forcausal sets do exist, a fairly simple one-parameter family of them being given by(3.7). The parameter ɛ in that expression determines the scale of the nonlocalityvia ɛ = Kl 2 , and it must be ≪ 1 if we want the fluctuations in Bφ to be small. Inother words, we need a significant separation between the two length-scales l andλ 0 = K −1/2 = l/ √ ɛ.3.2 Higher dimensionsSo far, we have been concerned primarily with two-dimensional causets (onesthat are well approximated by two-dimensional spacetimes). Moreover, the quotedresult, (3.3) cum(3.6), has been proved only under the additional assumption offlatness, although it seems likely that it could be extended to the curved case. Moreimportant, however, is finding D’Alembertian operators/matrices for four and otherdimensions. It turns out that one can do this systematically in a way that generalizeswhat we did in two dimensions.Let me illustrate the underlying ideas in the case of four dimensional Minkowskispace M 4 .InM 2 we began with the D’Alembertian matrix B xy , averaged over sprinklingsto get ¯B(x − y), and “discretized” a rescaled ¯B to get the matrix (B K ) xy .Itturns out that this same procedure works in four dimensions if we begin with thecoefficient pattern 1 −33−1 instead of 1 −2 1.To see why it all works, however, it is better to start with the integral kernel andnot the matrix (now that we know how to pass between them). In M 2 we found ¯Bin the form of a delta-function plus a term in p(ξ) exp(−ξ), where ξ = K v(x, y),and v(x, y) was the volume of the order-interval 〈y, x〉, or equivalently – in M 2 –Synge’s “world function”. In other dimensions this equivalence breaks down andwe can imagine using either the world function or the volume (one being a simplepower of the other, up to a multiplicative constant). Whichever one chooses, the realtask is to find the polynomial p(ξ) (together with the coefficient of the companiondelta-function term).To that end, notice that the combination p(ξ) exp(−ξ) can always be expressedas the result of a differential operator O in ∂/∂K actingonexp(−ξ). But then,7 Strictly speaking, this assumes that the number of sprinkled points is Poisson distributed, rather than fixed.