An example of geometrogenesis (cf Gielen, Oriti and Sindoni: 1303.3576)
We will have a change of vacuum, as we did in the condensed matter example; form a Fock vacuum to a coherent state.
That is: GFTs, like the quantum field theories describing the fundamental atoms in condensed matter systems, are defined usually in perturbative expansion around the Fock vacuum. In this approximation, they describe the interaction of quantized simplices and spin networks, in terms of spin foam models and simplicial gravity. The true ground state of the system, however, for non-zero couplings and for generic choices of the macroscopic parameters, will not be the Fock vacuum. The interacting system will organize itself around a new, non-trivial state, as we have seen in the case of standard Bose condensates. The relevant ground states for different values of the parameters (couplings, etc) will correspond to the different macroscopic, continuum phases of the theory, with the dynamical transitions from one to the other being indeed phase transition of the physical system we call spacetime.
There are three main results to report. [1] etc below
[1]: Oriti describes an approximation procedure that allows one to associate an approximate
continuum geometry to the set of data encoded in a generic GFT state. This applies to GFT models whose group and Lie algebra variables admit an interpretation in terms of discrete geometries, i.e. in which the group H chosen is SO(3, 1) in the Lorentzian setting or SO(4) in the Riemannian setting.
He begins with: A generic GFT state with a fixed number N of GFT quanta will be associated to a set of 4N Lie algebra elements: {BAmBm; I(m) } , with m = 1, …,N running over the set of tetrahedra/vertices, I = 1, …, 4 indicating the four triangles of each tetrahedron, (AB ) indicating Lie algebra components.
We imagine the tetrahedra being embedded in a spatial 3-manifoldM with a transitive group action of H . The embedding is defined by specifying a ‘location’ of the tetrahedra, i.e. associating for example one of their vertices with a point xm on the manifold, and three (tangent) vectors vi(m) , defining a local frame.
This gives a precise reconstruction procedure that gives an approximation to a continuum metric. But ONLY an approximation, in that it only specifies a continuum metric at a finite number of points N . The number of points can be understood as an approximation scale, as the number of observations in our sampling of a continuum metric. This in turn implies that a GFT state that aims at representing a continuum geometry, i.e. something that can be specified at an infinite number of points and that can be perturbed at each of them independently, should involve a number of quanta N that is on the one end allowed to vary and on the other hand allowed to go to infinity.
[2]: He identifies the quantum GFT states that, using the above procedure, can be interpreted as continuum homogeneous quantum geometries. Namely they are: GFT condensate states. So the hypothesis of spacetime as a condensate, as a quantum fluid, is therefore realized in quite a literal way.
(There is not just the one-particle GFT condensate, characterized by a single particle collective wave-function, analogous to the elementary coherent state mentioned above; there are states reflecting two-particle interactions, three-particle interactions etc.,)
[3]: One can derive effective dynamical equations for the condensate directly from the fundamental GFT quantum dynamics: on analogy with Section V’s deduction of the Gross-Pitaevski equation.
The convolutions in the interaction term of the action for the effective dynamics are as non-local as those in the fundamental GFT interaction, in their pairing of field arguments. We have thus a non-linear and non-local, Gross-Pitaevskii-like equation for the spacetime condensate function ϕ0 . Notice that it is an equation on superspace—that is, on the space of (homogeneous) geometries of a symmetric spacetime,; rather than an equation on spacetime itself. In this sense it is the same type of equation that are used in continuum quantum cosmology, with the basic variable being a “wave function of the universe”.
However, the differences with respect to quantum cosmology are crucial. For one, the equation is non-linear and thus it cannot be interpreted as an Hamiltonian constraint equation for a wave function. These non-linearities can be interpreted as taking into account indirectly, that is in the context of an equation for homogeneous quantum geometries, the effects of inhomogeneities, resulting from the fundamental interactions of the quantum building blocks of spacetime (again, this is in direct analogy with real atomic condensates). This type of non-linear and non-local equations have been in fact proposed independently and out of purely cosmological considerations.
Note that this equation arises as a “hydrodynamic equation” for the spacetime fluid, not as the quantization of a classical symmetry-reduced dynamics. In particular, it could be written (as can the Gross-Pitaevskii equation), in the tradition of Hamilton-Jacobi theory (cf. pilot-wave theory) using the polar decomposition: that is in terms of a probability density on the space of geometries and a quantum phase.
Beyond Spacetime 