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Abstract: We discuss the hints for the disappearance of continuum space and time at microscopic scale. These include arguments for a discrete nature of them or for a fundamental non-locality, in a quantum theory of gravity. We discuss how these ideas are realized in specific quantum gravity approaches. Turning then the problem around, we consider the emergence of continuum space and time from the collective behaviour of discrete, pre-geometric atoms of quantum space, and for understanding spacetime as a kind of “condensate”, and we present the case for this emergence process being the result of a phase transition, dubbed “geometrogenesis”. We discuss some conceptual issues of this scenario and of the idea of emergent spacetime in general. As a concrete example, we outline the GFT framework for quantum gravity, and illustrate a tentative procedure for the emergence of spacetime in this framework. Last, we re-examine the conceptual issues raised by the emergent spacetime scenario in light of this concrete example.

**2013-09-27 15:36:39**: 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.

**2013-09-27 15:40:42**: 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.

**2013-09-27 15:43:18**: 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:

**2013-09-27 15:50:12**: 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.

**2013-09-27 15:53:01**: One main way of understanding GFTs is as a second quantized field theory of spin

network vertices, each corresponding to a quantum of the field φ and labelled by the d group or Lie algebra elements, constructed in such a way that its quantum states are generic superpositions of spin networks and its Feynman diagrams are spin foams. For any given spin foam model, there exists a choice of GFT action (thus, a specific GFT model), such that the corresponding Feynman amplitudes are the chosen spin foam amplitudes. It can be seen, therefore, as a possible (second quantized) incarnation of the LQG programme. And: Using the dual simplicial formulation of spin networks and spin foams, GFTs can also be understood as a second quantization of simplicial geometry, in which simplices can be created and annihilated in fundamental interaction processes, as well as change size and shape (with group or Lie algebra elements characterizing areas of triangles, discrete curvature, etc).

The main advantages compared to other formulations of LQG or spin foams are first of all that GFTs offer a complete definition of the quantum dynamics, e.g. a clear prescription for the weights to be used in the sum over spin foams and for how this should be generated; second, that one can take advantage of more or less standard QFT tools in studying the theory, despite the fully background independent context (from the point of view of physical spacetime). This in particular would be a key asset to study the physics of large numbers of LQG degrees of freedom (large spin networks and spin foams), e.g. using powerful tools like the renormalization group.

**2013-09-27 15:57:06**: Oriti endorses the compatibility of reduction and emergence; citing eg Nagel for the idea of reduction and Butterfield for the compatibility, its involving taking a limit of a parameter; and for the emergence occurring before the limit, since the limit is, typically, unphysical.

Turning to the emergence of spacetime:–

1) he joins Huggett and Wuthrich in rejecting Maudlin’s general objection against the idea of emergent spacetime, based on empirical coherence.

2) He emphasizes that there are two very different types of approximations/limits that have to be taken, in order to start from the discrete structures of the programmes he has surveyed so as to arrive at classical GR: a continuum limit and a classical limit. Apart from needing a precise definition of both of them, we must also be careful about the order in which they are taken. It is entirely possible that a continuum spacetime stems from the quantum properties of its fundamental building blocks, and would not be achieved if only their classical properties were considered.

**2013-09-27 16:04:46**: He turns to: a concrete non-spacetime example of emergence; as prototype for spacetime proposal

He sketches the way that starting from a coherent state in an non-relativistic quantum field theory of bosons, one can derive, for certain parameter values, the Gross-Pitaevski equation for the coherent state’s defining wave-function phi, where phi is now considered as a classical collective variable (and the Gross-Pitaevski equation is classical, albeit non-linear and containing hbar).

This illustrates both the points made above (i): The compatibility of reduction and emergence, since the hydrodynamic Gross-Pitaevskii equation is deduced from the microscopic theory, but exhibits novel behavior relative to that theory. (ii): Maudlin’s objection, i.e. his requirement that a theory postulate local beables, being too strong. Thus he writes: (quoting from the paper, Section V):

Now suppose that we did not have access to the atoms at all, experimentally, but still, because we were so smart, we had managed to guess the microscopic theory [i.e. the non-relativistic quantum field theory of bosons], and then even so smart as to guess the appropriate approximate ground state of the system [i.e. the coherent state] … Suppose that we insisted on the idea that the macroscopic fluid we lived in (yes, we are supposing we are unfortunate but smart fishes swimming in the cold superfluid) was only emergent from something totally different, and unaccessible, and almost unthinkable, the “atoms of the fluid”. Suppose that we had managed even to find an approximate derivation of the hydrodynamic equation from the microscopic theory, and thus to hypothetically explain some features of the fluid in terms of its [postulated] atoms. In such a situation, . . . [the microscopic theory], despite not allowing ‘fluid beables’ to describe the atoms, [would surely] still be “physically salient”.

**2013-09-27 16:08:37**: the idea of geometrogenesis

The idea of spacetime as a collective, emergent configuration of a large number of quantum gravity building blocks—in particular as a condensate—has been argued for by several authors and from a variety of standpoints. As we will see, this suits well the group field theory framework. So the idea is: continuum spacetime is a collective state of large numbers of GFT building blocks (and perhaps specific to one phase of the system)

In this framework, it turns out that:

We identify the process of quantum spacetime condensation with a known, even if not understood, physical process: namely the big bang singularity. [That is], we identify the coming of the universe, that is of space and time, into being with the physical condensation of the “spacetime atoms”.

And: From this perspective, cosmological singularities, i.e. divergences in curvature invariants in GR, could be a sign of the breaking down of the continuum description of spacetime in a more specific sense than generally understood: they would signal the breakdown of the hydrodynamic approximation of the system of spacetime atoms, and at the same time signal the onset of a phase transition.

Note that on this view (i.e. proposing both the spacetime condensate and the geometrogenesis), the emergence of spacetime continuum and geometry will be the result of the quantum properties of the atoms of spacetime. It will be a quantum phenomenon. Therefore, the order in which the two key limits/approximations needed to recover GR, the semi-classical and the continuum limit, have to be taken is clear: one has first to understand the continuum limit of the quantum system and only then one can take a classical limit and hope to recover a GR-like dynamics.

**2013-09-27 16:21:26**: The above example represents the first derivation of cosmological continuum spacetime and geometry, including its dynamical aspects, from a microscopic quantum gravity theory, as far as we are aware.

Final slide invites discussion of such matters as:

(I): crucial role of quantum properties in emergence; (ii) ontological status of non-spatiotemporal regimes and of the ‘atoms of space’

**2013-09-27 16:24:33**: Pressed on dynamics: He agrees that for GFT approach, notions of unitarity, as well as locality, are obscure…but only time for the one question!!