Approaches to Quantum Gravity

The causal set approach to Quantum Gravity 397is that such a correspondence is necessary for any quantum theory, and so at leastsome of the histories in our Quantum Gravity SOH must be well approximatedby Lorentzian manifolds. A full justification of this point is beyond the scope ofthis chapter, but support may be taken from other quantum theories, both standardand speculative, and from some of the seminal writings on quantum mechanics[27; 28]. Some explanation can be found in [29].Here, the question is: when can a Lorentzian manifold (M, g) be said to be anapproximation to a causet C? Roughly, the order corresponds to the causal order ofspacetime, while the volume of a region corresponds to the number of elements representingit. 3 It is interesting to note that the manifold and the metric on it have beenunified into one structure, with counting replacing the volume measure; this is arealisation of Riemann’s ideas on “discrete manifolds” [34] (see also the translatedpassages in [5]). But a more exact definition of the approximation is needed.A causal set C whose elements are points in a spacetime (M, g), and whoseorder is the one induced on those points by the causal order of that spacetime, issaid to be an embedding of C into (M, g). 4 Not all causal sets can be embeddedinto all manifolds. For example, the causal set in figure 21.2 cannot be embeddedinto 1+1D Minkowski space, but it can be embedded into 2+1D Minkowski space.There are analogues to this causal set for all higher dimensions [35], and surprisinglythere are some causal sets that will not embed into Minkowski of any finitedimension. Thus, given a causal set, we gain some information about the manifoldsinto which it could be embedded. However, a manifold cannot be an approximationto any causal set that embeds into it; we could recover no volume information inbFig. 21.2. A Hasse diagram of the “crown” causet. This causet cannot be embeddedin 1+1D Minkowski space: if the above Hasse diagram is imagined asembedded into a 2D Minkowski spacetime diagram, the points at which elementsa and b are embedded are not correctly related. In no such embedding can theembedded elements have the causal relations of the crown causet induced onthem by the causal order of 1+1D Minkowski space. The causal set can howeverbe embedded into 2+1D Minkowski space, where it resembles a three-pointedcrown, hence its name.a3 While this is the stance taken in what might be called the “causal set Quantum Gravity program”, the causalset structure has also been useful elsewhere, although with different, or undefined, attitudes as to how itcorresponds to the continuum. See for example [30; 31; 32; 33].4 Really an embedding of the isomorphism class of that causet (the “abstract causet”). The distinction betweenisomorphism classes and particular instances of causal sets is not crucial for the purposes of this chapter, andwill be ignored.