Approaches to Quantum Gravity

The causal set approach to Quantum Gravity 401What does Lorentz invariance mean in this context? The answer should beguided by what is tested in the observations mentioned above. Let us begin withthe statement for theories on a background Minkowski spacetime. Here, Lorentzinvariance of a theory means that the dynamics should not distinguish a preferredLorentz frame. Next, we want to say that the Minkowski space is only anapproximation to some underlying discrete structure. In view of the statement ofLorentz invariance, we want to make sure that any dynamics on this approximatingMinkowski is not forced to pick a preferred Lorentz frame because of the discreteness.This leads to the following: if the underlying structure, in and of itself, servesto pick out a preferred direction in the Minkowski space, then Lorentz invariancehas been violated. This is the situation for lattice-like structures, and is arguably themost relevant statement for the current observational tests of Lorentz invariance.In contrast, by this criterion, the causal set provides a locally Lorentz invariantdiscrete structure – the only one considered in any approach to Quantum Gravity.This property is achieved thanks to the random nature of the discrete/continuumcorrespondence principle given above.As an analogy, consider a crystal, and a gas, as discrete systems of atoms whosebehaviour can be given an approximate continuum treatment. The crystal has aregular underlying structure that breaks rotational symmetry, and this symmetrybreaking can be observed macroscopically, by the existence of fracture planes andso on. The gas on the other hand has a random underlying structure, and the probabilitydistribution of the molecules’ positions at any time is rotationally invariant.There is no preferred direction in a gas that affects its behaviour in the effectivecontinuum treatment. We could “cook up” a direction from the positions of themolecules – in any region containing two molecules we can of course draw a vectorfrom one to the other. The point is that such “preferred directions” identifiableon microscopic scales have no effect on the bulk, continuum physics of the gas.Thus it is common to say that the behaviour of a gas is rotationally invariant.The Lorentz invariance of the causal set is similar. As previously noted, in thePoisson process, the probability for sprinkling n elements into a region dependson no property of that region other than the volume. In Minkowski spacetime,to establish Lorentz invariance of the Poisson process rigorously we need onlynote the theorems proving the existence and uniqueness of the process with thedistribution (21.1) for all measurable subsets of R d and its invariance under allvolume preserving linear maps (see e.g. [51]), which of course includes Lorentztransformations. In a curved spacetime, Lorentz invariance is to be understood tohold in the same sense that it holds in General Relativity: the equivalence of localLorentz frames.In some sense the situation is better than that for gases. In the case of a sprinklingof R 3 , a direction can be associated with a point in the sprinkling, in a way that