This week I’ve been studying GFT condensates, Bose-Einstein condensates and the Gross-Pitaevskii Equation and the emergence of spacetime. I have posted on spacetime as a GFT condensate and numerical work with python on Bose-Einstein condensates. This post is based on D. Oriti’s paper, ‘Disappearance and emergence of space and time in quantum gravity’
In this paper he looks at the disappearance of continuum space and time at a microscopic scale. These include arguments for the discrete spacetimes and non-locality in a quantum theory of gravity. He also discusses how these ideas are realized in specific quantum gravity approaches. He then considers the emergence of continuum space and time from the collective behaviour of discrete, pre-geometric atoms of quantum space such as quantum tetrahedra, and for understanding spacetime as a kind of condensate and presents the case for this emergence process being the result of a phase transition, called ‘geometrogenesis’. Oriti then discusses some conceptual issues of this scenario and of the idea of emergent spacetime in general. A concrete example is given in the form of the GFT framework for quantum gravity, and he illustrates a procedure for the
emergence of spacetime in this framework.
An example of emergent spacetime in the context of the GFT framework is given in this paper. It aims at extracting cosmological dynamics directly from microscopic GFT models, using the idea of continuum spacetime as a condensate, possibly emerging from a big bang phase transition.
GFTs 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 in the case of standard Bose condensates. The relevant ground states for different values of the parameters will correspond to the different macroscopic, continuum phases of the theory, with the dynamical transitions
from one to the other being phase transitions of the physical system called spacetime.
The fact that the relevant ground state for a proper continuum geometric phase would probably not be the GFT Fock vacuum can be argued also on the basis of the pregeomet ”meaning of it: it is a quantum state in which no pregeometric excitations at all are present, no simplices, no spin networks. It is a no space state, the absolute void. It can be the full non-perturbative, diffeo-invariant quantum state around which one defines the theory – in fact, it is analogous to the diffeoinvariant vacuum state of loop quantum gravity, but it is not where to look for effective continuum physics. Hence the need to change vacuum and study the effective geometry and dynamics of a different
As described in my last post it is possible to define an approximation procedure that associates 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 chosen is SO(3, 1) in the Lorentzian setting or SO(4) in the Riemannian setting and additional simplicity conditions are imposed, in the model, to reduce generic group and Lie algebra elements to discrete counterparts of a discrete tetrad and a discrete gravity connection.
A generic GFT state with a fixed number N of GFT quanta will be associated to a set of 4N Lie algebra elements: BI(m) , with m = 1, …,N running over the set of tetrahedra/vertices, I = 1, …, 4 indicating the four triangles of each tetrahedron. In turn, the geometricity conditions imply that only three elements are independent for each tetrehadron.
If the tetrahedra are embedded in a spatial 3-manifold M with a transitive
group action. The embedding is defined by specifying a location of the tetrahedra, i.e. associating for example one of their vertices with a point on the manifold, and three tangent vectors defining a local frame and specifying the directions of the three edges incident at that vertex. The vectors eA canbe interpreted as continuum tetrad vectors integrated over paths in M corresponding to the edges of the tetrahedron. Then, the variables gij(m) can be used to define the coefficients of continuum metric at a finite number N of points, as: gij(m) = g(xm)(vi(m), vj(m)), invariant under the action of the group SO(4).
The reconstruction of metric coefficients gij(m), at a finite number of points, from the variables associated to a state of N GFT quanta – such as quantum tetrahedra, depends only on the topology of the assumed symmetric manifold M and on the choice of group action H. The approximate metric will be homogenous if it has the same coefficients gij(m) at any m. This captures the notion of the metric being the same at every point. It also implies that the same metric would be also isotropic if H = R3 or H = SU(2).
The quantum GFT states obtained using the above procedure can be interpreted as continuum homogeneous quantum geometries. In such a second quantized setting, the definition of states involving varying and even infinite numbers of discrete degrees of freedom is straightforward, and field theory formalism is well adapted to dealing with their dynamics.
The crucial point, from the point of view of emergent spacetime and of the idea of spacetime as a condensate of quantum pregeometric and not spatio-temporal building blocks is that quantum states corresponding to homogeneous continuum geometries are exactly GFT condensate states. The hypothesis of spacetime as a condensate, as a quantum fluid, is realized in a literal way. The simplest state of this type -one-particle GFT condensate, for which we assume a bosonic quantum statistics, is
This describes a coherent superposition of quantum states of arbitrary number of GFT quanta, all of them described by the same distribution ϕo of pregeometric variables. The function ϕo is a collective variable characterizing such continuum geometry, and it depends only on invariant homogeneous geometric data. It is a second quantized state characterized by the fact that the mean value of the fundamental quantum operator φ is non-zero:
<ϕo|φ(gi)|ϕo> = ϕo(gi),
contrary to what happens in the Fock vacuum.
The effective dynamical equations for the condensate can be extracted directly from the fundamental GFT quantum dynamics.The generic form of the dynamics for the condensate ϕo is, schematically:
where Keff and Veff are modified versions of the kinetic and interaction kernels entering the fundamental GFT dynamics, reflecting the approximations needed to interpret ϕo as a cosmological condensate.This is a non-linear and non-local, Gross-Pitaevskii-like equation for the spacetime condensate function ϕo. In the simple case in which:
and we assume that the function ϕo depends on four such SU(2) variables. The final equation one gets for non-degenerate geometries) is:
that is the Friedmann equation for a homogeneous universe with constant curvature k.
- Constructing spacetime from the quatum tetrahedron: Spacetime as a Bose-Einstein Condensate (quantumtetrahedron.wordpress.com)
- Numerical work using python 1: Bose-Einsten condensation (quantumtetrahedron.wordpress.com)
- Physicists Eye Quantum-Gravity Interface (simonsfoundation.org)