LQG is then briefly compared to (ii) quantum Regge calculus and (causal) dynamical triangulations.

These are a more straightforward programme for discretizing GR, in that they triangulate spacetime (so all graphs are considered as embedded in spacetime) and assign each link its spatiotemporal length (in dynamical triangulations, all links being required to be of the same length). Then the dynamics is in terms of a sum over histories’ amplitudes. Most of the focus has been on proving phase transitions, and so with an eye on the continuum theory; and so with a less realistic construal of the discrete microscopic degrees of freedom.

This leads to a brief mention of (iii) matrix models; and (iv) causal set theory. As he proceeds, he briefly compares each program’s proposals with aspects of the preceding ones, especially as regards how realistically one tends to take (or should take!)  the postulated basic degrees of freedom.

Then he introduces the group field theory approach .The formalism is a field theory over a group manifold (or the corresponding Lie algebra) with the basic variable being a complex field φ (g1, g2, .., gd ) = φ12..d  : a function of d  group elements, for a d-dimensional spacetime (the most relevant case, therefore, being d  = 4). For example, we may have gi ∈ SO (4), for all i.

The field can be represented graphically as a (d-1)-simplex with field arguments associated to the d faces of it, or as a d-valent graph vertex, with field arguments associated to the links. (So for d=4, the field is represented as a 3-simplex ie as a tetrahedron.)

The dynamics is given by an action, which integrates (with the Haar measure) over the group manifold: (of course a multiple integral). The interaction term involves a non-standard (compared with the usual local QFT) convolution of fields defined in terms of their arguments (analogous to the tracing of indices in matrix models). Thus the specific pattern of convolution chosen depends on which requirement replaces the locality of standard QFTs.

For example, in d  = 4 and for gi ∈ SO (4), the interaction term might be a fivefold product with a specific identification of certain arguments of the field: say, φ1234 φ4567 φ7389 φ962 10 φ10 851 ; (so that we have a tenfold multiple integration with Haar measure). This  combinatorics of convoluted field arguments in the interaction matches the combinatorics of five tetrahedra glued along common triangles to form a 4-simplex. The propagator induces a simple gluing of 4-simplices across one shared tetrahedron, when used in perturbative expansion to form Feynman diagrams. Such Feynman diagrams are representable, by construction, as 4d simplicial complexes.

So in these models, one could say that the usual QFT locality is replaced by simpliciality , i.e. the requirement that interactions are associated to d-simplices.

He gives a survey of several programmes’ proposals for the quantum building blocks of spacetime. He begins with (i) Loop Quantum Gravity (LQG).

Turning to some details:

            (i) Loop Quantum Gravity (LQG):

This started off as a canonical quantization of continuum GR reformulated in terms

of first order variables (namely a Lorentz connection and a triad field). The quantization gives a direct sum of Hilbert spaces associated to graphs. For each given graph γ , a generic state is given by a wave function whose argument is an assignment of data to the elements of the graph. In one scheme, these are: group elements ge  representing holonomies of the gravitational connection along the links e  =1, …,E  of the graph (considered as embedded in the spatial manifold), so that the state is ψ(g1, …., gE ). In another scheme: one labels the links with representations of the Lorentz (or rotation) group.

Dynamics is given: (a) canonically, by a Hamiltonian constraint operator defined on graphs; (b) covariantly, by defining histories as graphs considered as embedded in spacetime (spin foams), and assigning amplitudes to histories, and then summing over amplitudes.

Recently the schemes have emphasized: (a) simplices, (b) thinking of the graph as dual to a (usually simplicial) cellular complex; (c) motivating the proposed assignment of amplitudes by considering lattice version of the gravitational path integral. This leads to:

Section II on ‘Hints about the disappearance of spacetime’. He begins by stressing such matters as spacetime singularties and black hole thermodynamics; and that the quantization of general relativity (or the quantization of some cousin theory that proceeds by quantizing its metric field) seems bound to lead to superpositions of geometrical and-or causal relations; and so to be difficult and-or radical.

There follows a survey of other such hints. This includes mention of (i) non-commutative geometry, (ii) doubly special relativity, (iii) the AdS/CFT duality, and in more detail: (iv) Jacobson’s representation of the Einstein field equations as equations of state, and the Liberati-Visser-et al. representation of sound waves in certain classical fluids by a massless scalar field propagating in a (3+1)-dimensional Lorentzian geometry. Example (iv) pre-figures the later discussion (in Section VIII, building on a condensed matter example in Section V) of schemes for obtaining a description of a classical spacetime and its dynamics, from the state of a BE condensate, or from some similar specific but collective state of a condensed matter system.

He begins (cf. Section I) with a gracious and collegial agreement with the organizers, Huggett and Wuthtrich, that, so to speak, quantum gravity is too important to be left to the physicists! That is: the conceptual problems are hard, and intriguing enough, that philosophers should participate now, and not wait to comment the final theory.

Question from Fletcher: what methodology applies when we identify a conflict between different concepts of time in our network of theories?  How do we decide how to modify our concepts?

K: In Barbour’s account, for example, treat his Leibnizian principles as unshakeable.  How can other concepts be modified to make them work?  But we need to appeal to idea of the “best option or fix” where a full theory of these desiderata is now missing.

Her conclusion: ACA seems to apply usefully to Rovelli’s concept of time.  A number of conceptual problems remain for his picture.

Now we’re considering Rovelli’s strategy for quantum gravity.

Spacetime is dynamical.  Like other dynamical fields, it must be quantum in nature.  Put together, these criteria lead us to demand a background-independent quantum field theory with a field that in some sense represents spacetime.

In particular, Rovelli wants to put time “on a par” with other physical variables.

He starts from relationist principles, according to which fundamentally there are only physical objects and space is nothing more than a relational property of them.

A question: doesn’t Rovelli think that spacetime is one of the dynamical objects as the theory, which doesn’t sound like a relational property of other objects?

Kon’s answer: this is Rovelli’s broader picture of spacetime, not his specifically quantum picture.

Rovelli’s “partial observable” for time is highlighted (a measurement of local succession of events).  Are there conceptual conflicts between this picture and Rovelli’s commitment to relationism?  Not necessarily, since we might understand succession in purely relational terms.

But since observations occur at an instant, this particular temporal concept must be privileged, so it’s not clear that time is “on an equal footing” with space and other physical variables.

A question: how is time different from space in this way?

Kon: being at a particular place is defined by being at a particular instant.  But she can see the point—perhaps spatial relations could provide a framework for defining temporal ones in a similar way.

A question from Mattingly: why should we care about whether conceptual analysis applies to other theories if we have QG, presumably the best theory?

K’s answer: the idea of “best scientific theory” is problematic.  We need an account that can make sense of the way different theories share concepts.

Her Alternative (ACA) to Jackson:

1. Identify theories being examined.  For folk theories, we consult our intuitions, for scientific theories we consult our best interpretation of them, making sure to treat multiple different theories as parts of a conceptual network.

2. Determine what role “time” plays in the whole network.  Do its roles in different theories clash, or exhibit redundancy or irrelevancy?  If so, we must either eliminate the concept or ‘engage in metaphysics.’ (to quote Jackson)

3. Integrate an analysis of *folk time* with our analysis of time in physics.

Applying Jackson’s CA to time.  An assumption for simplicity’s sake: the present is the only feature constitutive of folk time.  (We’re analyzing the folk concept of “the present.”)

This works great with Lorentzian relativity, Kon’s first example.  We can keep the concept of “the present.”  In relativity this concept gets eliminated.  Looks like the right result!

But the problem with Jackson’s JCA: we can’t just read metaphysics off a scientific theory the way Jackson assumes.

In an improved account of CA, we should look for three desiderata:

1. Don’t problematically read scientific concepts off a theory

2. Apply to more than one scientific theory

3. Include an analysis of folk time rather than just eliminating the concept

For Jackson, a concept is the meaning of a term.  For him, a concept’s structure and reference is fixed by its theoretical role (a descriptivist model rather than a “causal reference” model).  J uses conceptual analysis to solve the “location problem” — the problem of reconciling the phenomena with one’s fundamental ontology (locating the world in the fundamental stuff).

Two steps:

1. Find out the referents of upper-level terms (establish our shared “folk theory” via reflection on linguistic intuitions).  This tells us the theoretical role of the upper-level concepts (e.g. “water”).

2. Determine whether statements involving the concept to be analyzed are entailed by lower-level descriptions of reality.  (If not, we can eliminated the concept.)  Example: our folk concept of water as potable liquid can be reconciled with our theoretical entity H2O, so we have “located” water in H2O.

Kon notes that Jackson assumes the lower-level description should come from our best scientific theory.