The primacy of dynamics

Theories of QG, in conclusion, start from some dynamical and discrete micro-structure and attempt to reconstruct the macroscopic continuous geometrical structure therefrom. The consequences of such a reconstruc- tion, if successful, amounts, according to their proponents, to a new con- ceptual framework in which space and time disappear, or become "non fundamental" in the microscopic level described by QG.

As an example, let us focus on Loop Quantum Gravity (LQG), whose basic building blocks are "loops", holonomies of the gravitational connec- tion. To make contact with experiment, LQG singles out a certain opera- tor as representing the geometrical magnitude of "area", defining it as the number of times the basic building block of the theory, the loop, crosses the "surface" ([Rov93], p. 810). It turns out that the spectrum of this op- erator is discrete, and so one of the predictions of LQG is that there is a minimal distance between any two spatial points ([Rov93], p. 811). This feature also allows LQG to remove the singularities that prevail in GTR and QFT.

The philosophical consequences of such a brilliant result are what we already know, as it can be read in one of the most important text on LQG:

In Newtonian and special relativistic physics, if we take away the dynamical entities – particles and fields – what remains is space and time. In general relativistic physics, if we take away the dynami- cal entities, nothing remains. The space and time of Newton and

Minkowski are reinterpreted as a configuration of one of the fields, the gravitational field. This implies that physical entities – particles and fields – are not all immersed in space, and moving in time. They do not live on spacetime. They live, so to say, on one another.

[. . .] One consequence is that the quanta of the field cannot live in spacetime: they must build "spacetime" themselves. This is precisely what the quanta of space do in loop quantum gravity.

[. . .] [I]n the quantum theory, where the field has quantized "gran- ular" properties and its dynamics is quantized and therefore only probabilistic, most of the "spatial" and "temporal" features of the gra- vitational field are lost. Therefore for understanding the quantum gravitational field we must abandon some of the emphasis on geom- etry. Geometry represents well the classical gravitational field, not quantum spacetime.

[Rov04], p. 20

Taken literally, the quote expresses two premises from which it con- cludes that geometry and spacetime "disappear" from LQG, only to emerge in the macroscopic limit. The first premise is relationalism, or the back- ground independence of LQG6; the second is the granular character of its basic building blocks. But, regardless of the premises, is this conclusion warranted? And, if so, what does it mean vis. a vis. the status of spacetime in QG?

Following two opponents of this literal approach, Hagar and Hemmo (see [HH13], p. 357-358), one can distinguish two possible readings of this so-called "disappearance" thesis. On the first, strong one, spacetime and geometry are not fundamental in the domain of applicability of LQG, but rather are the result of the kind of reconstruction offered by its proponents, hence reappear only in the low-energy limit of the theory. On the second, weak reading, spacetime and geometry are still present in LQG (and in its domain of applicability) and are as fundamental to the theory as are its dynamical building blocks, because "the dynamics only picks up the

6_{"Background independence [. . .] is a manifestation of relationalism [. . .]. Whether or}

not it is vindicated in classical GTR is still a matter of dispute among philosophers and physicists [. . .], but in the context of theories of quantum gravity, relationalism requires – on a minimal interpretation – that the defining equations of a theory will be independent of the actual shape of spacetime and the value of various fields within spacetime, and, in particular, will not refer to a specific coordinate system or a specific metric. Applying this strategy to quantum theories of gravity is not a trivial endeavor, given that conventional QFT is background dependent, and does require a fixed spacetime background" ([Hag14], p. 225).

structural properties of spacetime" ([HH13], p. 361; my italics). Thus, what presumably supervene on these building blocks, hence ’emerges’ in the low-energy limit, are specific classical features of spacetime and geometry, e.g., the Riemannian metric, or the local and global symmetry groups.

That one of the most important issues for QG theories is to reconstruct the precise geometrical features of relativistic spacetime from the algebraic structure of the dynamical theory is quite clear, considering the efforts at solving "mapping problems". In LQG, one of these is connected to the individuation of the area operator, which I mentioned above. Let’s see how two proponents of the "disappearance" thesis present this procedure.

What is believed to correspond to three-dimensional spatial struc- tures are so-called ’spin networks’. Spin networks can be thought of as networks of interwoven loops with ’spin’ representations sitting on both the network’s nodes and its edges. These spin representa- tions quantify the discretely valued quantum ’volume’ to which the node corresponds, and the discretely valued quantum ’area’ of the edge corresponding to the surface of adjacency of the connected ’vol- umes’. Although the dynamics of the theory is far from settled, the general scheme evolves these spin network states by having an ap- propriate Hamiltonian operator acting on them. [. . .] The resulting structure is taken to be the quantum analogue of a four-dimensional spacetime and is called ’spin foam’.

[. . .] The problem is that any natural notion of locality in LQG — one explicated in terms of the adjacency relations encoded in the funda- mental structure — is at odds with locality in the emerging space- time. In general, two fundamentally adjacent nodes will not map to the same neighbourhood of the emerging spacetime [. . .]. Hence the empirically relevant kind of locality cannot be had directly from the fundamental level.

[. . .] One influential idea based on so-called ’weave states’ proposes that the spacetime structure emerges from appropriately benign, i.e., semi-classical, spin networks which are in eigenstates of area and vol- ume operators (or perhaps other geometric operators), with eigenval- ues which approximate, for sufficiently large ’chunks’ of the spin net- work, the classical values of the standard area and volume functions of the corresponding approximated chunk of spacetime.

[HW13], p. 279

Whether the LQG theorists really want to reduce geometrical notions (areas and volumes) to dynamical entities (network’s edges and nodes) in

physics is doubtful, while it is clear that they – at least – feel the necessity to find algebraic structures that can be used to reconstruct the specific ge- ometrical framework that, as they say, would not be present at the level of the building blocks of the theory. Of course, our interest here is not to evaluate this attempts or the ambiguities of the philosophical literature on the possible readings of the dynamical approach. Instead, I wish to exam- ine, through the lenses of Hartmann’s critical ontology, the tenability of the strong reading.