## Quantum Gravity And Spacetime Ontology

As with the hole argument, shift argument, and other permutation symmetry arguments, there have been many grand proclamations about of the impact of quantum gravity on the issue of spacetime ontology and the debate between substantivalists and relationalists. I think it is fair to say that the received view amongst physicists working in the field of canonical quantum gravity is that the theory supports some form of relationalism. The most explicit defender of this view has been Rovelli (most explicit in: [1992] and [1997])—Smolin [2000; 2001], Baez [2001; 2006], and Crane [1993; 1995] paint similar philosophical stances. This has been largely backed up by philosophers who have taken an interest in the subject. Belot and Earman line up gauge invariant and non-gauge invariant interpretations with relationalism and substantivalism respectively; and, as we have seen, Belot sees reduced phase space and extended phase space quantizations respectively as similarly aligned. My key point is that the methods for dealing with gauge freedom (or not, as the case may be) do not bear any relation to spacetime ontology (as charted in the substantivalism vs relationalism debate), and either side of the debate can help themselves to any of the methods. Since these methods are central to the conclusions drawn in the quantum gravity context, we see that quantum gravity does not have the bearing on spacetime ontology that is often thought to hold. My conclusion is that the methods, although indeed central to our understanding of the structure of space and time, cannot in fact allow us to draw deeper metaphysical morals about the nature of this structure. I apply my structuralist stance in this context and argue that recent results in loop quantum gravity can be easily accommodated by my view.

Towards the end of their review of the problem of time, Belot and Earman make the following rather metaphysically weighty claims:

It would require considerable ingenuity to construct an (intrinsic) gauge invariant substantivalist interpretation of general relativity. And if one were to accomplish this, one's reward would be to occupy a conceptual space already occupied by relationalism. Meanwhile, one would forgo the most exciting aspect of substantivalism: it's link to approaches to quantum gravity, such as the internal time approach. To the extent that such links depend upon the traditional substantivalists' commitment to the existence of physically real quantities which do not commute with the constraints, such approaches are clearly unavailable to relationalists. ([Belot and Earman, 2001], pp. 248-9)

Their argument is based on the following line of reasoning: if spacetime points were real, then quantities like 'the curvature at point x' would be real too; but such quantities do not commute with the constraints, so spacetime points cannot be real after all. Substantivalists are then seen as being committed to the view that there are physically real quantities that do not commute with the constraints, and relationalists are committed to the denial of this. They have Karel Kuchar occupying the first position and Rovelli occupying the latter. In the next chapter I argue against the first alignment on the grounds that Kuchar is committed to the view that all physical quantities commute with the diffeomorphism constraint. It is true that Rovelli sees himself as occupying a relationalist position, and he sees this as following from complete gauge invariance. However, there are a number of reasons why this is problematic—recall for starters, from the previous chapter, that Rovelli and Kuchar can be 'permuted' over relationalist and substantivalist positions according to their taxonomy!

As I already mentioned, Belot connects the substantivalist/relationalist debate to the treatment of symmetries in Hamiltonian systems and their retention or removal respectively ([2001], p. 571).243

Let me first detach substantivalism from the internal time approaches. Belot ([1996], p. 241) claims that "substantivalism is ...a necessary condition for loyalty to the sort of approach to quantum gravity that Kuchar advocates"; namely an approach according to which observables commute with the diffeomorphism but not the Hamiltonian constraint. But although Kuchar might claim that his position is substantivalist (see Belot, ibid., p. 238), it is quite clear that a relationalist could just as well adopt it. Indeed, given that the diffeomorphism constraint is solved, Kuchar's position will come out as relationalist according to the received view— a view that Belot elsewhere endorses (see, e.g., Belot [2001]). According to Kuchar the lesson of the hole argument is that it is the geometry of a spatial manifold that has physical content: the diffeomorphism constraint should be solved for.

Next, let me disentangle the view that relationalists cannot adopt the view that there are some observables that do not commute with the constraints. I grant Belot and Earman's point that the reductive relationalist will be barred from those strategies that outlaw commutation with all of the constraints. However, as I hinted at above, the relationalist (even the reductive one) can help himself to Kuchar's position. The phase space there is a partially reduced one, with the gauge freedom generated by the diffeomorphisms of space modded out. This is a reasonable object for the relationalist even by Belot and Earman's lights. The fact that the observables are not to commute with the Hamiltonian constraint is no problem: the relationalist too might want to deny that the geometries relate by the Hamiltonian constraint are to be identified for exactly the reasons outlined by Kuchar. Thus, it is perfectly possible for a relationalist to deny Belot & Earman's condition.

Belot and Earman are agreed that the best (easiest) way to avoid the inde-terminism that arises in the hole argument, and gauge from gauge freedom in general, is to adopt a gauge invariant interpretation. However, they make the mistake of assuming that the way to achieve this is by giving a direct interpretation of the reduced phase space. They take such interpretations as showing, in the context of general relativity, there could not "be two possible worlds with the same geometry which differ only in virtue of the way this geometry is shared out over the existent spacetime points" ([2001], p. 228). This, they say, leads to relational-ism (in the absence of "an attractive form of sophisticated substantivalism"). They list several problems facing the reduced space accounts: the singular points corresponding to symmetric models; non-differentiability; and the unavailability of a set of coordinates able to separate the space's points. For these reasons they conclude that "a dark cloud hangs over the programme of providing gauge invariant interpretations of general relativity ...the present state of ignorance concerning

243 Likewise for other philosophical stances towards the symmetry arguments considered in this book. The idea is that 'substantivalism' and 'relationalism' are linked to a certain treatment of the symmetries in any theory formulated in a phase space description. Thus, one could be substantivalist or relationalist about vector potentials, for example; and this would simply correspond to endorsing an extended (direct) or reduced phase space formulation respectively. Of course, I argued earlier that these links can be severed for any theory.

the structure of the reduced phase space ... —and the lingering worry that this structure may be monstrous—should give pause to advocates of gauge invariant interpretations of the theory" (ibid., pp. 228-9). Perhaps this is a fair comment as far as the reduced space methods go; but such methods are not necessary for gauge invariant interpretations.

Thus, what I am denying here is that the various strategies used in responding to the problem of time and the hole argument (the analogous problem for space or spacetime) are related to interpretive stances regarding the nature of spacetime in general relativity. The strategies do not definitively support any such stance, nor do any such stances definitively support the strategies. Thus, what we have is an underdetermination of the various strategies and stances by each other. Whatever it is that pushes one towards a particular stance as regards the nature of spacetime, it cannot be the hole argument or the problem of time. The best these arguments can do is to tell us about the structure or spacetime, not its nature. However, as I argued in the previous section, for a structuralist, this is all one needs: nature just is structure!

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