Jon Bain here…. and Mattingly begins!

James promises brevity and chattiness…

Q: your  framework assumes the notion of apparatus external to the system, eg a screen: can you address description of the whole cosmos? can you define when events occur–in the Bellian sense?
A: (not caught)Q: surely issues are well controlled by assuming that tau is a monotonic smooth function of time t? A: yes, but the framework is intended to be general.Q: Dirac observables are not all that one can observe, as my talk will show! Pashby looks forward to the questioner’s talk

He announces 4 main Sections: Evolving constants; Toy models; Event POVMs; Extended quantum mechanics

The overall claim will be that :

The two strategies—Eliminate time (Gambini Oporto Pullin) and Promote time (Brunetti & Fredenhagen and HOge)—are less enemies than they seem: they lead to a quantum mechanics using an extended phase space.

SEction 1 Evolving constants:

For classical mechanics of eg 3 point particles, he expounds the basic ideas of partial and complete observables with complete observables commuting with the constraints.

He then describes Dirac constrained quantization of this, invoking the algebraic quantization scheme of Ashtekar et al 1995.

Then he discusses how to treat the basic question

What is the value of the partial observables q or p, given that the time is t = tau?

The proposal of the first strategy (Gambini Oporto Pullin) is to integrate out the parameter tau, in effect by introducing an explcit partial observable, say Qb, for it.

Section 2 Toy models

This is illustrated by a toy model, providing a time-independent conditional probability.

He then pursues the question: what exactly are we conditioning on? This discussion leads in to the idea of Event POVMs, and so to

Section 3: Event POVMs

This is developed in terms of a specific example, and then the general approach of the second strategy is adumbrated. In particular: the operator normalization idea of Brunetti & Fredenhagen and Hoge.

Pashby describes how this construction is committed (through its use of operator normalization) to the event in question happening precisely once; and mentions how this leads to the further question how to model the times of an event that could happen more than once.

Section 4: extended phase space

Any operator in a conventional Hilbert space say L^2(Q) \equiv L^2(R^3) defines and operator in the extended Hilbert space L^2(Q \times R); where of course, R represents time.

He briefly relates the ideas of Brunetti & Fredenhagen and Hoge (2010) to:

a)    the Dirac- constrained quantization condition that the Hamiltonian constraint, as an operator, should kill the physical state;

b)   Dirac’s own ideas in 1927 for a quantum mechanics on an extended phase space.

He ends by expounding the idea of weights, and quantum partial observables.

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!!

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’

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.

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”.

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.

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.