Quantum Cosmology from Group Field Theory Condensates: a Review

Quantum Cosmology from Group Field Theory
Condensates: a Review\star\star$\star$This paper is a contribution to the
Special Issue on Tensor Models, Formalism and Applications. The full
collection is available at
http://www.emis.de/journals/SIGMA/Tensor_Models.html

Steffen GIELEN † and Lorenzo SINDONI ‡

S. Gielen and L. Sindoni

† Theoretical Physics, Blackett Laboratory, Imperial College London, London
SW7 2AZ, UK<EMAIL_ADDRESS>
‡ Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, 14476 Golm,
Germany<EMAIL_ADDRESS>
Received February 29, 2016, in final form August 12, 2016; Published online
August 18, 2016

We give, in some detail, a critical overview over recent work towards deriving
a cosmological phenomenology from the fundamental quantum dynamics of group
field theory (GFT), based on the picture of a macroscopic universe as a
“condensate” of a large number of quanta of geometry which are given by
excitations of the GFT field over a “no-space” vacuum. We emphasise conceptual
foundations, relations to other research programmes in GFT and the wider
context of loop quantum gravity (LQG), and connections to the quantum physics
of real Bose–Einstein condensates. We show how to extract an effective
dynamics for GFT condensates from the microscopic GFT physics, and how to
compare it with predictions of more conventional quantum cosmology models, in
particular loop quantum cosmology (LQC). No detailed familiarity with the GFT
formalism is assumed.

group field theory; quantum cosmology; loop quantum gravity

83C45; 83C27; 83F05; 81T27

##  1 Introduction

The idea that spacetime and geometry should be emergent, i.e., an effective
description of the collective behaviour of different, “pre-geometric”
fundamental degrees of freedom, is now a common theme in most approaches to
quantum gravity [104]. In particular, in theories formulated fundamentally in
terms of discrete, combinatorial structures rather than continuum variables
(such as a spacetime metric or connection on a smooth manifold) in order to
avoid the problem of perturbative non-renormalisability of general relativity
[76], the emergent nature of a continuum description is a necessity, in order
to connect to other areas of physics and to our experience which take place in
the continuum. Showing how a continuum emerges is hence the challenge faced by
loop quantum gravity [8, 124, 131] and spin foam models [117, 125], causal
dynamical triangulations [4], causal set theory [82], as well as tensor models
[122] and group field theories [18, 91, 103], a challenge which has been
addressed with varying success. A separate notion of emergence of dynamical
spacetime arises in the AdS/CFT correspondence [95] for supersymmetric
theories, which posits that a continuum quantum spacetime geometry is in a
precise sense dual to a lower-dimensional conformal field theory on a fixed
geometry. These rather different notions of emergence all face the challenges
of (nonlocally [98]) mapping the fundamental degrees of freedom to the sought-
after emergent spacetime description.

In a group field theory (GFT), spacetime emerges from combinatorial
structures, labelled by additional geometric or pre-geometric data. In the
simplest representation of simple GFT models, this data consists of group
elements attached to the edges of a graph, typically of fixed valency. The
group elements themselves are interpreted as parallel transports of a
gravitational connection along these edges, and hence are analogous to the
variables of lattice gauge theory. Unlike in lattice gauge theory, the graph
or lattice does not carry a clear physical interpretation: graphs appear as
Feynman diagrams in the perturbative expansion around a GFT vacuum, and
quantities of interest such as the partition function contain a summation over
an infinite number of such graphs. In general, there is little reason to trust
perturbation theory, i.e., a truncation of this sum to a few simple graphs,
since there is no small expansion parameter and no interpretation of the
vacuum as a physically meaningful geometry. Indeed, viewing GFT as a second
quantisation formulation of loop quantum gravity (LQG) and hence adopting
operators for geometric observables such as volumes and areas from LQG [105],
the most natural GFT vacuum is a state in which expectation values for such
geometric observables vanish – it is a ‘no-space’ state, analogous to the
Ashtekar–Lewandowski vacuum [6] of LQG. An approximately smooth, macroscopic
geometry of physical interest can hardly be found as a small perturbation of
such a vacuum.

Since the early days of LQG, some effort has been invested into the
construction of states that can approximate continuum geometries, such as flat
space [10]. These necessarily excite a large number of degrees of freedom
compared to the Ashtekar–Lewandowski vacuum, and accordingly are described by
a large graph with many edges and vertices. One would hope to find states that
can play the role of a new vacuum of the theory around which a different
notion of perturbation theory can be defined, e.g., in order to compare LQG
with the perturbative quantisation programme of continuum general relativity.
Completing this task requires knowledge about the dynamics of many interacting
degrees of freedom, which has been difficult to extract from traditional LQG
and its spin foam formulation, partly due to the rather complicated
combinatorial structure of existing models. In the study of spin foam models
such as the currently favoured EPRL model [62], known results for transition
amplitudes are mainly restricted to asymptotic regimes, in particular in the
“large-$j$ limit”, see, e.g., [22, 81].

The situation calls for the development of systematic methods to treat many
degrees of freedom in LQG, beyond the truncation given by a fixed graph. One
possibility is to adopt methods from lattice gauge theory and use coarse
graining or renormalisation procedures in order to obtain solutions to the
dynamics and ultimately a theory with a well-defined continuum limit, where
diffeomorphism symmetry is restored [54]. The endpoint of such a
renormalisation group flow corresponds to a particular continuum phase,
describing the collective behaviour of many degrees of freedom (infinitely
many in the limit). In an interpretation of the setting as a theory of quantum
gravity, one would like to show that such a phase describes a smooth non-
degenerate spacetime geometry, such as Minkowski or de Sitter space.

In the GFT setting, one can also explore a different route. As a second
quantisation formulation of LQG, the main advantage of GFT is the availability
of quantum field theory methods when treating many degrees of freedom in the
language of quantum fields rather than $N$-particle states. Conceptually, the
situation is rather similar to the use of (non-relativistic) quantum field
theory in condensed matter physics, when one is interested in the collective
behaviour of a large number of interacting atoms or molecules, rather than
perturbative scattering around the Fock vacuum. One of the most relevant
examples of the successful application of quantum field theory in condensed
matter physics is the study of Bose–Einstein condensation and superfluidity
[118]. In this context, the condensation of many atoms into a common ground
state can be viewed as a transition from a perturbative phase around the Fock
vacuum (of zero atoms) into a condensed phase, with associated symmetry
breaking of the ${\rm U}(1)$ symmetry of the theory. Similarly, the language
of GFT allows a treatment of many degrees of freedom through quantum fields
rather than a fixed graph or “$N$-particle state”. The elementary building
block of a graph, a vertex of given valency, then plays the role of an atom of
geometry. In fact, this interpretation is quite literally the one in LQG: the
volume operator only has non-zero eigenvalues when acting on vertices [124,
131], so that such vertices do indeed form the “atoms of space”. Using not a
fixed number of such atoms but more general states that contain a
superposition of many (typically infinitely many) graphs is quite natural in
LQG where the (kinematical) Hilbert space is a direct sum of Hilbert spaces
associated to different graphs [124, 131]. The continuum limit of an infinite
number of such atoms is the usual thermodynamic limit of statistical field
theory. This view suggests that a very natural candidate for a configuration
of many degrees of freedom with high coherence and symmetry, and potentially a
candidate for a non-perturbative vacuum of LQG, should be given by a group
field theory condensate analogous to a Bose–Einstein condensate [101, 102].

Given that this idea of a macroscopic spacetime as a kind of condensate of
microscopic degrees of freedom is rather natural, it may not appear surprising
that it has already been suggested in other contexts. In LQG, it was suggested
by Koslowski [90] and shown in more detail by Sahlmann [127] that the
Ashtekar–Lewandowski construction, leading to a unique vacuum in LQG [93],
could be generalised to allow for a larger class of vacua, where expectation
values for geometric operators are shifted by a given classical field
configuration with respect to the Ashtekar–Lewandowski vacuum, so that one can
choose for example a vacuum corresponding to continuum flat space. One can
view the Koslowski–Sahlmann vacua as implementing a breaking of diffeomorphism
symmetry which would require these expectation values to vanish. There has not
yet been much work on understanding these vacua as a dynamically emerging
configuration; excitations over them are also still discrete structures as in
usual LQG.

There is also a more “bottom-up” motivation from quantum cosmology for
thinking of a macroscopic universe as a sort of condensate [35]. In quantum
cosmology, the traditional strategy since Misner [99] for obtaining a
manageable quantum model of the universe has been to perform a
“minisuperspace” reduction at the classical level, i.e., to assume that the
universe is spatially homogeneous, and to quantise only the remaining degrees
of freedom of geometry. The assumption of exact homogeneity means that one
focuses on the local evolution of a small patch around a single point in
space, as there are no spatial gradients by construction, and one imposes the
symmetry-reduced quantum Einstein equations as the dynamics of such a patch.
As explained in [35], this procedure can be thought of as a “single-patch”
quantum model in which one quantises a single “atom” or chunk of space, and
assumes that the dynamics of the atom is related to the effective dynamics at
large scales. This view would suggest passing to a “many-patch model” of many
interacting chunks of space, as a way to check the validity of this
assumption, and to extend the model to include inhomogeneities. Depending on
the length scales associated to these chunks, this idea can be related to the
separate universe picture in cosmology [133], where inhomogeneities in a
certain range of wavelengths can be included by modelling an inhomogeneous
universe as composed of many small, locally homogeneous patches. In such a
many-patch model, given that inhomogeneities are very small in the observed
Universe, one then considers states with weak interactions between different
patches. A simple ansatz for the many-patch wavefunction would be a product
state, so that there is a collective single-particle wavefunction describing
the state, again very similar to states used for (weakly interacting)
Bose–Einstein condensates. This idea was explored in [37] in a lattice model
starting from classical general relativity, where one finds that the existence
of interactions between patches leads to a nonlinear and nonlocal evolution
equation for the single-particle wavefunction, similar to the Gross–Pitaevskii
equation in condensed matter physics [118]. We will find very similar
qualitative features in the effective dynamics of GFT condensates.

The application of loop quantum gravity techniques to quantum cosmology led to
the development of loop quantum cosmology (LQC) [11, 32]. LQC combines “top-
down” and “bottom-up” elements: while the classical Friedmann equation is used
as a starting point for the dynamics, one also uses the kinematics of LQG, so
that the continuum Ashtekar–Barbero connection [5, 19] is replaced by finite
parallel transports along edges of a graph, and the Wheeler–DeWitt equation
becomes a difference rather than a differential equation. An open issue that
has attracted a lot of interest in recent years (see, e.g., [1, 30, 31, 61,
64] for various approaches) is how to embed the kinematics and dynamics of LQC
into full LQG, or rather derive LQC as an approximation or effective
description of the full theory of LQG when applied to almost exactly
homogeneous, macroscopic Universes like our own. In this review, we explore
another viewpoint in which one thinks of quantum cosmology as the effective
dynamics of a condensate of microscopic degrees of freedom of quantum
geometry. In short, “quantum cosmology is the hydrodynamics of quantum
gravity”. The condensate hypothesis provides methods for deriving an effective
description of the dynamics of realistic universes in GFT, which can then be
compared with conventional quantum cosmology models such as LQC, as well as
ultimately with observation.

###  1.1 Relation to analogue and emergent gravity

The philosophy we are following shares some common ground with the philosophy
behind the programmes of “analogue” and “emergent” gravity. Analogue models
for gravity [21] seek to emulate some physical properties of gravitational
systems, such as Hawking radiation [132], by non-gravitational systems usually
described by some form of hydrodynamics. Motivated by the derivation of the
Einstein equations from thermodynamics [86], and more generally the close link
between spacetime geometry and thermodynamics, following Boltzmann’s dictum
“if you can heat it, it must have microstructure” [115], emergent models go
one step further and aim to describe dynamical gravity as emerging from an
effective description of different fundamental degrees of freedom [128]. The
idea that spacetime is a condensate of fundamental “atoms of spacetime” has
been suggested in this context of emergent gravity [84], without an obvious
proposal for the nature of the “atoms”.

Group field theories provide a natural candidate for a theory of the atoms of
spacetime and their dynamics, and suggest that classical gravitational
dynamics can be found in an effective hydrodynamic description of the dynamics
of a GFT condensate, as suggested in the context of lower-dimensional models
in [110]. The results and methods covered in this review show the detailed
application of these ideas to cosmology.

###  1.2 Relation to studies of renormalisation and phase structure in LQG/GFT

The study of condensates in GFT can be seen as part of a larger effort aimed
at understanding the collective dynamics of many interacting degrees of
freedom of quantum geometry in loop quantum gravity and its generalisations,
including the more general setting of GFT. Given the interpretation of GFT
condensates as macroscopic continuum geometries, one would like to show that
such condensates correspond to a dynamical phase of a known GFT model, at
least in a certain range of the free parameters (coupling constants). One
would also like to understand the possible phase transitions, in particular
transitions between a “pre-geometric” and a “geometric” continuum phase such
as described by a condensate. Replacing the classical Big Bang singularity by
such a phase transition is the idea often dubbed geometrogenesis, and first
advocated in the context of “quantum graphity” [97]. Showing that
geometrogenesis can occur in the setting of GFT would mean a revolutionary
reinterpretation of the origin of space and time, as it would show the
emergence of a GFT phase with macroscopic geometric interpretation (in terms
of a four-dimensional, continuum spacetime, causal structure, a metric, etc.)
from a framework which is fundamentally formulated without any of these
structures, and instead just uses combinatorial and group-theoretic data.

The main tool for mapping out the phase diagram for GFT and related theories
is the renormalisation group, particularly its Wilsonian formulation based on
effective actions. Applying renormalisation tools to GFT has been an extremely
active field of research in recent years, see, e.g., [23, 27, 46, 47, 48]. A
major result is that many models studied so far (of the tensorial GFT type)
exhibit asymptotic freedom [23, 123], making them potential candidates for a
fundamental theory. The infrared limit of the renormalisation flow is then
important in order to understand the hypothesis that spacetime is a GFT
condensate (of many “small” quanta). Here, in some models evidence has
accumulated for phase transitions between a symmetric and a condensate-type
phase [25, 26], just of the type we are interested in, with an associated flow
to an interacting fixed point in the IR. So far, no such renormalisation
analysis has been performed for GFT models with direct relation to LQG, such
as the EPRL/FK GFT [24], but progress towards the study of more and more
sophisticated models has been rapid. Ultimately, one would aim to justify the
assumption of a GFT condensate in the models we are looking at from a more
rigorous analysis of this type.

There has also been a lot of activity in the renormalisation analysis of
simplified and generalised spin foam models by coarse graining methods, aimed
at finding a continuum limit that restores diffeomorphism invariance [53, 56,
57]. Part of this effort has culminated in the development of a new
representation for LQG [13, 55], with a new vacuum that is in some sense dual
to the Ashtekar–Lewandowski vacuum of traditional LQG: it is a vacuum of
exactly flat connections but completely undetermined conjugate metric
variables. These results already show that phases of LQG and spin foam models
beyond the degenerate phase around the Ashtekar–Lewandowski vacuum exist,
giving further motivation to the study of GFT condensates, although the
detailed connection between these research programmes still needs to be
understood better.

###  1.3 Structure of this review

After giving some perspective in this introductory section, our aim is to give
a self-contained overview over the research of roughly the last three years on
condensates in GFT and their relation to quantum cosmology. To this end, in
Section 2 we introduce the basics of the GFT formalism, set up notation for
this review, and define the class of models we will be looking at. While some
of this material will be known to readers familiar with the GFT literature,
they might want to consult it for clarity on which GFT models we are studying
and on notational conventions. In Section 3, we outline the (kinematical)
definition of condensate states in GFT, mainly focusing on the simplest case
that is in direct analogy with real Bose–Einstein condensates. Many conceptual
and technical points regarding the interpretation of these condensates in
terms of spacetime geometry will be explained. We also touch upon a more
general class of condensate states introduced more recently. In Section 4, we
show how to extract information about the dynamics of such condensates from
the fundamental quantum GFT. We give different methods for extracting
effective dynamical equations, explain the relation to quantum cosmology and
the approximations and truncations involved. We also go into more detail as to
how such effective dynamics can be compared with models such as LQC. In
Section 5, we finally summarise the main results, open questions and future
directions of this approach.

##  2 Group field theory (GFT)

In this section we will present the main ideas and basic technical tools of
GFT that will be needed for the application to cosmology. The presentation
will be self-contained; for further background on the general setting of GFT
and recent progress, we refer to the reviews [18, 91, 103].

Group field theories are quantum (or statistical) field theories defined over
group manifolds, most commonly restricted to classes of theories which can be
designed to provide a possible definition of a path integral for quantum
gravity. In a nutshell, we are interested in models for which the perturbative
expansion of its partition function or field equation, and the perturbative
expansion of the correlation functions, generates a sum over graphs that can
be seen as dual to discretisations of manifolds.

In addition to the peculiar combinatorial structure, similar to that found in
matrix and tensor models after which GFTs are modelled, the graphs of GFT are
decorated by data related to the chosen group (group elements, representation
labels or Lie algebra elements). The role of these additional data is to store
geometric information (metric or connection data) beyond the mere
combinatorial/topological structure. In fact, this group-theoretic data
corresponds precisely to the basic variables of LQG, holonomies of a
connection and fluxes of a triad field. In this way, GFT attempts to define a
sum over discretised geometries which can be used, once a continuum limit is
identified, to obtain a path integral for quantum gravity.

The definitions given so far cover a fairly rich class of models. In this
review, we focus on simple models that have so far mainly been used in the
application to cosmology, but the constructions of GFT condensate states have
straightforward extensions to more complicated models. In particular, the
desired property of the sum over graphs – that it represents a sum over
combinatorial manifolds, or at least pseudomanifolds – is generally achieved
by introducing coloured models with more than one fundamental GFT field [78,
80]. We will not be discussing coloured models in any detail, but it should be
clear in the following that the main ideas are generally applicable to a wide
class of GFT models.

###  2.1 Definition of the basic models

There are two prototypes of GFT that are useful for applications to quantum
gravity, which both define a quantisation of topological BF theories over
random manifolds. The first one is the Boulatov model [41], whose partition
function can be naturally rewritten in terms of the Ponzano–Regge model [120]
for three-dimensional (Riemannian) quantum gravity.

The second model, defined by Ooguri [100], can be seen as the four-dimensional
counterpart of the Boulatov model. It generates a Feynman expansion which can
be related to the amplitudes of a topological BF quantum field theory, the
backbone of the spin foam approach to quantum gravity in four dimensions
[117]. For the application to cosmology, we focus on four dimensions, so that
we will be interested in models built on extending the Ooguri model from a
topological theory to a theory of quantum gravity. We will comment later on
various possible generalisations. We will also focus on the pure gravity case,
as most models so far aim to define pure quantum gravity. The extension to
matter coupling is of course crucial for a more realistic cosmology, and will
be discussed in Section 4.

The starting point is the choice of a Lie group $G$, which will select the
type of data appearing in the Feynman diagrams, and will be interpreted as the
local gauge group of gravity. Typical choices are $G=\mathrm{SU}(2)$,
$\mathrm{SO}(4)$, $\mathrm{Spin}(4)$ or $\mathrm{SL}(2,\mathbb{C})$. The
second main choice is the combinatorial structure of the elementary building
blocks used to build spacetime geometry. For four-dimensional spacetime and
three-dimensional space, the simplest possibility is to use simplicial
building blocks: the elementary building block of three-dimensional space is
then a (quantum) tetrahedron [20]. In the dual picture familiar from LQG and
spin foams, it corresponds to a central vertex with four open outgoing links,
with group-theoretic data associated to these links. The field theory is
accordingly defined on four copies of $G$ (see Fig. 1). With these choices,
the elementary quanta of the theory can be assembled into three-dimensional
simplicial geometries that represent the boundary states of the theory. This
interpretation in terms of boundary configurations of four-dimensional
geometries is then completed by the examination of the perturbative expansion
of the correlation functions that they encode, once the dynamics with the
correct combinatorial structure is introduced. It is the interplay between the
combinatorics of the equations of motion and the number of group arguments of
the fields, together with the gauge group used, that suggests the
interpretation of the quanta and their interactions in geometrical terms.
Indeed, with different dynamics one could also construct a field theory based
on fields with four arguments which has a $2+1$-dimensional interpretation,
with basic quanta interpreted in terms of quadrilaterals. Generalisations of
GFT in which the restriction to simplicial geometry is dropped and general
three-dimensional polyhedra appear as elementary quanta have been constructed
[109]. These generalisations are motivated primarily by the appearance in
canonical loop quantum gravity of graphs with vertices of arbitrary valency,
corresponding to general polyhedra [28], where a restriction to tetrahedra
seems arbitrary. Restricting to simplicial building blocks means one can keep
track of the topology of the graphs transparently, and is certainly the
simplest choice, on which we focus in the following.

Superficially, these choices seem to imply that we are assuming spacetime
remains four-dimensional up to the very smallest scales, but this is not true:
any useful notion of dimensionality would apply to the collective properties
of a number of elementary quanta, rather than a single one. Experience from
approaches to quantum gravity that fundamentally involve discrete structures
shows that one should then expect a dimensional flow when going from
macroscopic to microscopic scales, where the nature of this flow (and in
particular, the question of whether four dimensions are reached at large
scales) depends highly non-trivially on the details of the model, and not
simply on the “dimensionality” used for the fundamental building blocks of the
theory. For some results relevant in the GFT context, see [45].

Figure 1: GFT quanta as open spin network vertices (left) or quantum
tetrahedra (right).

After these basic choices, the field theory is specified by a choice of a type
of field and by the corresponding action. The most common choices are real or
complex scalar fields. While topological models can often be based on a real
field, in order to set up the Fock space structure required for the discussion
of GFT condensates, a complex field will be required. In either case, a simple
general form of the action, for compact $G$, is

$\displaystyle
S[\varphi]=\int\mathrm{d}g\,\mathrm{d}g^{\prime}\,\bar{\varphi}(g_{I})K(g,g^{\prime})\varphi(g_{I}^{\prime})+\frac{1}{5!}\int(\mathrm{d}g)^{5}V\big{(}g^{1},\ldots,g^{5}\big{)}\prod_{j=1}^{5}\varphi\big{(}g_{I}^{j}\big{)}+{\rm
c.c.},$ (2.1)

where here and in the following we are using a condensed notation. The
arguments of the field are $g^{i}\equiv
g^{i}_{I}=(g^{i}_{1},g^{i}_{2},g^{i}_{3},g^{i}_{4})$, and we are often
suppressing the lower index running over the four group elements. The symbol
$\int\mathrm{d}g$ stands for an integration over $G^{4}$, where $\mathrm{d}g$
is the Haar measure on $G^{4}$, normalised to unit total volume. Accordingly,
$\int(\mathrm{d}g)^{5}$ means integration over $G^{4\cdot 5}=G^{20}$. This
notation will be useful, as one is almost always interested in $G^{4}$ or
products of several $G^{4}$ domain spaces. In the few cases where a different
integration domain is needed, it will be specified explicitly (see, e.g.,
(2.5)), so that no confusion should arise.

One also needs to specify the function space that is assumed to be integrated
over in the GFT path integral. In the perturbative approach, we can assume
that this space is the space of square-integrable functions on $G^{4}$. Since
the interaction terms involve higher powers of the fields in convolution with
variously constructed kernels, it is necessary to check that these terms are
well-defined in the given space of functions and, if needed, to introduce a
regularisation, as we will discuss in one example below (2.2).

The kernels $K(g,g^{\prime})\equiv K(g_{I},g^{\prime}_{I})$ and
$V(g^{1},\ldots,g^{5})\equiv V(g^{1}_{I},\ldots,g^{5}_{I})$ determine the
details of the resulting Feynman amplitudes. The choice of an interaction term
containing five copies of the field in (2.1) is suggested by the
interpretation of these amplitudes in terms of four-dimensional flat simplices
glued along common boundaries, i.e., sharing tetrahedra on their boundaries.

This action can be generalised in many different ways. One obvious possibility
is to introduce more terms in the interaction potential, admitting an
interpretation of simplicial polytopes (i.e., polytopes whose boundaries are
made of simplices). For now, we focus on the well-understood simplicial case.
More importantly, in the construction of models relevant for quantum gravity
rather than topological field theory, one needs to include the so-called
simplicity constraints into the interaction $V$; we will detail this in
Section 2.5.

Apart from the motivation coming from the interpretation as simplicial
geometry, one can also consider different possible interaction terms in (2.1)
from a quantum field theory perspective, in particular by studying their
renormalisability. Indeed, the construction of renormalisable and
asymptotically free GFT models has been the focus of recent work, see Section
1.2.

An important property of the GFT field is a type of discrete gauge invariance,
similar to lattice gauge theory: as our group elements will be interpreted in
terms of parallel transports of a connection, from a vertex to the endpoint of
an edge, we should make sure that only gauge-invariant information is used. A
simple choice is to restrict the field configurations to those which are
diagonal under the right diagonal action of $G$ on $G^{4}$:

$\displaystyle\varphi(g_{1}h,g_{2}h,g_{3}h,g_{4}h)=\varphi(g_{1},g_{2},g_{3},g_{4}),\qquad\forall\,h\in
G.$ (2.2)

While this is the simplest option, there are other possibilities: for
instance, one can use general fields on $G^{4}$ but include into the potential
term a suitable projector that maps functions on $G^{4}$ onto functions on
$G^{4}/G$. In the following we will keep using $G^{4}$ as the domain of the
fields, even though we will impose gauge invariance directly on the field.
While this means that the parametrisation is redundant, it greatly simplifies
the construction of the models.

The symmetry (2.2) leads to some technical complications if one wants to
extend the construction to non-compact $G$, in particular
$G=\mathrm{SL}(2,\mathbb{C})$ as the Lorentz group in four dimensions. Indeed,
as one can see easily, for $G=\mathrm{SL}(2,\mathbb{C})$ the action (2.1) is
ill-defined; because of the symmetry property (2.2) of the field, there is a
redundant integration over $\mathrm{SL}(2,\mathbb{C})$ with infinite volume.
This issue can be fixed rather easily, either by working with a gauge-fixed
field on
$\mathrm{SL}(2,\mathbb{C})^{4}/\mathrm{SL}(2,\mathbb{C})\simeq\mathrm{SL}(2,\mathbb{C})^{3}$,
or by simply removing redundant integrations in the kinetic and interaction
terms, keeping arguments of the fields fixed. We refer to Appendix A of [72]
for a detailed discussion of these subtleties (and their resolution) for the
Ooguri model, but for the purposes of this review just stress that they pose
no substantial difficulty.

In order to finally be more concrete, choosing a real field
($\varphi=\bar{\varphi}$), the simple choices

$\displaystyle
K(g,g^{\prime})=\prod_{I}\delta\big{(}g^{\prime}_{I}(g_{I})^{-1}\big{)}$

for the kinetic term and

$\displaystyle
V\big{(}g^{1},\ldots,g^{5}\big{)}=\lambda\prod_{e}\delta\big{(}g^{s(e)}_{l(s(e))}\big{(}g^{t(e)}_{l(t(e))}\big{)}^{-1}\big{)}$
(2.3)

for the interaction term lead to the Ooguri model. Here we are denoting by $e$
the ten edges of a complete graph of five vertices (see Fig. 2), by $s(e)$ the
copy of the field associated to the source of the edge, by $t(e)$ the target,
and by $l(s(e))$ (and $l(t(e))$) the group element involved. This choice of
combinatorics encodes the pattern of gluings in order to form a 4-simplex out
of five tetrahedra. As we shall see later, by modifying the kernels $K$ and
$V$ it is possible to generate models that are related in a precise way to
spin foam models [117].

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$ Figure 2: Complete graph of five
vertices.

The partition function obtained from these ingredients,

$\displaystyle Z=\int\mathcal{D}\varphi\exp(-S[\varphi]),$

when treated as a perturbative expression in the GFT coupling constant
$\lambda$, generates a sum over graphs that can be seen as dual to simplicial
complexes. The amplitudes corresponding to the Ooguri model can be shown to be
the amplitudes of a discretised BF theory [100],

$\displaystyle
Z=\sum_{\Gamma}\lambda^{\\#V(\Gamma)}\mathcal{A}(\Gamma)=\sum_{\Delta(\Gamma)}\lambda^{\\#V(\Gamma)}\mathcal{A}_{BF}(\Delta(\Gamma)),$

with a sum over the graphs $\Gamma$ being reinterpreted as a sum over discrete
simplicial complexes $\Delta(\Gamma)$. It is precisely the possibility of
generating a sum over (random) simplicial complexes that makes GFT relevant
for the definition of a discrete path integral for the gravitational field.
The choice of the Euclidean partition function assumes implicitly that, after
its proper definition, it will be analytically continued, once a suitable
parameter has been introduced, to the quantum-mechanical partition function.
Since we will be working directly in the quantum operatorial version, this
choice plays no role in the rest of the presentation.

The main problem that we are left with, then, is to make sense of formal
expressions as this partition function. In this review, we mainly focus on the
attempts at the extraction of their physical content which follow the ideas of
condensed matter physics for many-body problems.

###  2.2 Second quantisation, states associated to closed graphs, and
operators

Before discussing the modification of the topological kernel (2.3) necessary
to formulate quantum gravity models and not just topological field theories,
we briefly present GFT from the canonical viewpoint of field operators. While
the operatorial version of the models does not add anything substantially new
to the general idea of obtaining spacetime from Feynman diagrams, it is the
key ingredient that will allow us to make significant progress in the
calculation of physical predictions. Indeed, the language of Fock spaces and
field operators acting on them will allow us to attack the problem of
understanding the physical implications of GFT using a number of powerful
techniques adapted from condensed matter physics. Thanks to this
reformulation, we will be able to design quantum states with clearer geometric
interpretation, which can include some educated guesses about possible
nontrivial nonperturbative aspects of the theory in a simple and controlled
manner. Additionally, the connections (and differences) to canonical LQG can
be stated more precisely [105], without going through the spin foam formalism.

The canonical formalism has clear advantages. Besides the construction of
interesting quantum states, in order to answer specific physical questions we
need to understand how to compute the expectation values of physical
quantities. The second quantised formulation that we are going to use will
allow us to translate and generalise geometric operators introduced in LQG.

The construction follows the standard construction of nonrelativistic quantum
field theories. It is important to stress, however, that we are not viewing it
as originating from the canonical quantisation of a field theory: there is no
canonical choice of “time variable” on the domain space of GFT that would
allow a definition of generalised coordinates and conjugate momenta, in order
to justify a particular symplectic structure and corresponding operator
algebra. Although it would be possible to define such an extension of GFT,
here we simply postulate the basic commutation relations and resulting Fock
space, motivated by the origins of GFT in LQG. Alternative constructions
leading to a different Fock quantisation are certainly possible.

The definition of the field operators involves a choice of statistics for
indistinguishable particles. We choose bosonic statistics, compatible with the
picture of spacetime as a condensate. The field operators acting on the Fock
space then obey the commutation relations

$\displaystyle\big{[}\hat{\varphi}(g),\hat{\varphi}^{\dagger}(g^{\prime})\big{]}=\mathbb{I}_{R}(g,g^{\prime}),\qquad\big{[}\hat{\varphi}(g),\hat{\varphi}(g^{\prime})\big{]}=\big{[}\hat{\varphi}^{\dagger}(g),\hat{\varphi}^{\dagger}(g^{\prime})\big{]}=0,$
(2.4)

where

$\displaystyle\mathbb{I}_{R}(g,g^{\prime})=\int_{G}\mathrm{d}k\prod_{I=1}^{4}\delta\big{(}g_{I}k(g^{\prime}_{I})^{-1}\big{)}$
(2.5)

represents a Dirac delta distribution on the homogeneous space $G^{4}/G$,
reflecting the choice of imposing gauge invariance at the level of the field
(2.2). Again, for non-compact groups, (2.5) needs to be replaced by a
regularised expression, corresponding to a Dirac delta on $G^{4}/G$.

From the commutation relations (2.4), one can now construct a Fock space in
the standard way, starting from a Fock vacuum $\left|0\right\rangle$. This
vacuum is interpreted as a “no-space” state with no quanta, analogous to the
Ashtekar–Lewandowski vacuum of LQG. The action of the ladder operators on this
state can be naturally interpreted as the creation of a four-valent open spin
network vertex; we are interpreting the state

$\displaystyle\left|g_{I}\right\rangle=\hat{\varphi}^{\dagger}(g_{I})\left|0\right\rangle$

as a single spin network vertex decorated with the four group elements
$g_{1},\ldots,g_{4}$.

States characterised by nontrivial distributions of the group-theoretic data
can be obtained straightforwardly from suitable integrations with
wavefunctions,

$\displaystyle\left|f\right\rangle=\int\mathrm{d}g\,f(g_{I})\hat{\varphi}^{\dagger}(g_{I})\left|0\right\rangle,\qquad
f(g_{I})=\langle g_{I}|f\rangle=\left\langle
0\right|\hat{\varphi}(g_{I})\left|f\right\rangle.$

This extends directly to multiparticle states. For these, we can add an
additional structure to our interpretation of the states, namely a graph.
Consider an $N$-particle state

$\displaystyle\left|f_{N}\right\rangle=\frac{1}{\sqrt{N!}}\int(\mathrm{d}g)^{N}\,f\big{(}g^{1},\ldots,g^{N}\big{)}\prod_{j=1}^{N}\hat{\varphi}^{\dagger}(g^{j})\left|0\right\rangle.$

The wavefunction $f(g^{1},\ldots,g^{N})=\langle
g^{1}_{I},\ldots,g^{N}_{I}|f_{N}\rangle$ will, in general, depend on all $4N$
group elements, modulo the vertex gauge invariances from (2.2). Notice that,
by bosonic symmetrisation, the wavefunction is completely symmetric under the
permutation of the tuples of group elements $g_{I}$ associated to each vertex.
Such generic $N$-particle states are interpreted graphically as states of $N$
open spin network vertices.

A special class of states, of particular importance in the LQG interpretation,
corresponds to the case in which the wavefunction is constructed out of a
generic multiparticle wavefunction with the application of a projection which
reduces the dependence of the wavefunction to products of the type
$(g_{I}^{i})^{-1}g_{J}^{j}$. This is achieved by $2N$ convolutions

$\displaystyle
f\big{(}\ldots,g_{I}^{i},\ldots,g_{J}^{j},\ldots\big{)}=\int_{G^{2N}}\mathrm{d}h_{IJ}^{ij}\,f\big{(}\ldots,h_{IJ}^{ij}g_{I}^{i},\ldots,h_{IJ}^{ij}g_{J}^{j},\ldots\big{)}.$
(2.6)

Since we are thinking of $g_{I}^{i}$ and $g_{J}^{j}$ as parallel transports of
a gauge connection along edges emanating from the vertex to the endpoint, it
is natural to interpret such a dependence as the gluing of the edge $I$
incident on the vertex $i$ with the edge $J$ incident on the vertex $j$, with
$(g_{I}^{i})^{-1}g_{J}^{j}$ being the parallel transport from the vertex $j$
to the vertex $i$ along a connecting edge.

In this way, we can attach to our states a graph label, based on the pattern
of convolutions of the arguments, and identify GFT Fock states with a direct
relation with the closed spin network wavefunctions in LQG. The precise
connection with the (kinematical) LQG states is a delicate issue, and we refer
to [105] for a more complete discussion. First, the inner product that we are
using here is the Fock space inner product, so that the graph label does not
automatically identify orthogonal subspaces of the Fock space: states
associated to different graphs, when they have the same number of quanta, are
not necessarily orthogonal. Second, wavefunctions are completely symmetrised
with respect to the permutations of vertices, while in the case of LQG the
wavefunctions do not necessarily have to have special symmetrisation
properties. Finally, GFT Fock states are not organised in terms of embedded
graphs. The graphs we can associate to them are purely combinatorial objects,
and make no reference to any background manifold structure, except to the
group manifolds from which the data decorating the graphs are picked. Thus,
they are closer to the setting of algebraic quantum gravity [73, 74], an
approach for defining canonical LQG without using embeddings or a
differentiable manifold structure.

Keeping in mind these subtleties, the Fock space language allows us to import
many of the concepts of LQG into the GFT setting [105]. In particular, one can
immediately translate the operators that are normally defined for spin
networks into the second quantised language that we have introduced. In order
to do so, we now consider the case $G=\mathrm{SU}(2)$, the gauge group of the
Ashtekar–Barbero formulation of classical general relativity [5, 19] that is
the basis of LQG.

The translation requires identifying the product of ladder operators that
corresponds to the first quantised operators of interest. Hermitian operators
are of particular significance, as they will store information about
observables. In order to be Hermitian, these operators have to be built with
(linear combinations of) strings of ladder operators possessing an equal
number of creation and annihilation operators. According to the number of
creation operators, operators will be distinguished as one-body operators,
two-body operators, and so on. The definition of these operators is necessary
to extract geometric information from states containing arbitrary
superpositions of states associated to different graphs. The translation of
intrinsically many-body operators (as, for instance, the holonomy computed on
a loop incident on several vertices) is not immediate, as the nature of the
GFT quanta as identical particles makes the identification of portions of the
graphs a tricky issue, in absence of further degrees of freedom that can be
used to distinguish between different kinds of vertices. Such degrees of
freedom could, for example, correspond to matter fields, or to an additional
“colour” label of multiple GFT fields (see Section 3.5 for some more
discussion on colouring in GFT). For our purposes, one-body operators are
enough to extract the relevant geometric information.

Generic one-body Hermitian operators have the form

$\displaystyle\widehat{X}=\int\mathrm{d}g\,\mathrm{d}g^{\prime}\,X(g,g^{\prime})\hat{\varphi}^{\dagger}(g)\hat{\varphi}(g^{\prime}),\qquad
X(g,g^{\prime})=\overline{X(g^{\prime},g)}.$

The simplest one-body operator is the number operator $\widehat{N}$, with
$X=1$,

$\displaystyle\widehat{N}=\int\mathrm{d}g\,\hat{\varphi}^{\dagger}(g)\hat{\varphi}(g).$

This operator extracts a specific and unambiguous graph property, the number
of vertices, or the average number of vertices in the case of a superposition
of graphs.

Other common one-body operators that are relevant in the GFT context are the
vertex volume operator $\widehat{V}$ and the plaquette area $\widehat{A}_{I}$.
They can be obtained by simply setting as $X(g,g^{\prime})$ the LQG matrix
elements of the desired operators, e.g.,

$\displaystyle\widehat{V}=\int\mathrm{d}g\,\mathrm{d}g^{\prime}\,V^{\rm
LQG}(g,g^{\prime})\hat{\varphi}^{\dagger}(g)\hat{\varphi}(g^{\prime}),$

where $V^{\rm LQG}(g,g^{\prime})\equiv\langle g_{I}|V^{\rm
LQG}|g^{\prime}_{I}\rangle$ is the matrix element of the volume operator
between single-vertex spin networks in LQG, and (replacing the matrix elements
by a differential operator)

$\displaystyle\widehat{A}_{I}=\int\mathrm{d}g\,\hat{\varphi}^{\dagger}(g)\left(\sum_{m,n=1}^{3}\delta_{mn}\mathrm{E}^{m}_{I}\mathrm{E}^{n}_{I}\right)^{1/2}\rhd\hat{\varphi}(g).$
(2.7)

Here $\mathrm{E}^{n}_{I}$ are the LQG flux operators, proportional to the Lie
derivatives on $\mathrm{SU}(2)$,

$\displaystyle\mathrm{E}^{n}_{I}f(g_{J})=\lim_{\epsilon\rightarrow
0}\mathrm{i}\kappa\,\frac{d}{d\epsilon}f\big{(}e^{-\mathrm{i}\sigma^{n}\epsilon/2}g_{I}\big{)},$

where $\sigma^{n}$ is a Pauli matrix and the operator only acts on the $I$-th
argument of the field, and $\kappa$ is a parameter with dimensions of area,
the “Planck area” of the theory, which in canonical LQG is usually set to
$\kappa=8\pi\gamma\hbar G_{\rm N}$ where $\gamma$ is the Barbero–Immirzi
parameter and $G_{\rm N}$ is the low-energy Newton’s constant111Introducing
$G_{\rm N}$ in this way, at the kinematical level in the microscopic theory,
raises the question of whether $G_{\rm N}$ can undergo renormalisation in LQG.
The results in the later sections of this review will show how, in GFT
condensate cosmology, the strength of the gravitational coupling to matter
emerges from the coupling constants of the GFT action, so that $G_{\rm N}$ has
a rather different origin and is subject to renormalisation.. Noting that
contracting the Lie derivatives with the Killing form leads to the Casimir
operator on $\mathrm{SU}(2)$, (2.7) can be simplified to

$\displaystyle\widehat{A}_{I}=\kappa\int\mathrm{d}g\,\hat{\varphi}^{\dagger}(g)\sqrt{-\Delta_{g_{I}}}\hat{\varphi}(g),$

where $\Delta_{g_{I}}$ is the Laplace–Beltrami operator on $\mathrm{SU}(2)$
acting on the $I$-th argument of $\hat{\varphi}$. This shows explicitly how
the (kinematical) area spectrum in LQG is related to the discrete spectrum of
the Laplacian on $\mathrm{SU}(2)$, multiplied by the parameter $\kappa$.

A basic property of one-body operators is that they are extensive: as it is
immediate to realise using Wick’s theorem, they generate a contribution from
each of the quanta of the state on which they act. This aspect will play a
role in our discussion of the condensate states and their geometrical content;
by only using one-body operators in extracting geometric data from the states,
we focus on quantities that are local at the level of each vertex or dual
tetrahedron.

###  2.3 Noncommutative dual formulation of GFT

Group field theories can be formulated in different equivalent
representations, using suitably defined Fourier transforms. The simplest one
is the spin representation (using the terminology for $\mathrm{SU}(2)$), which
is based on a Peter–Weyl decomposition. A slightly more sophisticated
representation of the theory is the one that makes uses of noncommutative
fluxes (see, e.g., [15, 77]). The basic idea is to use a noncommutative
Fourier transform that unitarily maps square-summable functions on the group
to square-summable functions on the corresponding Lie algebra $\mathfrak{g}$,

$\displaystyle\mathcal{F}(f)(B)\equiv\int_{G}\mathrm{d}g\,\mathrm{e}_{g}(B)f(g),$

where $\mathrm{e}_{g}(B)$ are plane waves associated to the particular
quantisation map chosen [77].

This allows a definition of Fourier transformed field operators acting on the
same Fock space,

$\displaystyle\hat{\tilde{\varphi}}(B_{I})=\int\mathrm{d}g\prod_{I=1}^{4}\mathrm{e}_{g_{I}}(B_{I})\,\hat{\varphi}(g_{I}),$

where, corresponding to the four group elements $g_{I}$,
$B_{I}=(B_{1},\ldots,B_{4})$ with $B_{I}\in\mathfrak{g}$.

The peculiarity of the Lie algebra representation is that, instead of the
ordinary product, a noncommutative star-product has to be used in order to
encode the noncommutative group multiplication. This product is defined in
terms of its action on plane waves,

$\displaystyle\mathrm{e}_{g}(B)\star_{B}\mathrm{e}_{h}(B)=\mathrm{e}_{gh}(B).$

While the Lie algebra representation presents a number of technical
difficulties due to the nontrivial aspects of the star-product, it offers a
dual, and perhaps clearer geometric interpretation of the GFT quanta. Let us
specialise again the discussion to $G=\mathrm{SU}(2)$, where the Lie algebra
variables $B_{I}$ can be naturally identified with the densitised triads
integrated over the plaquettes dual to the edges attached to a GFT vertex, see
(3.7)222The construction of a noncommutative Fourier transform, used for the
specific case of $\mathrm{SU}(2)$ (and hence $\mathrm{Spin}(4)$), can be
extended, at least formally, to the case of weakly exponential Lie groups
[77], including $\mathrm{SL}(2,\mathbb{C})$. What is so far missing is the
explicit construction of the plane waves and the detailed analysis of their
properties and ambiguities (related to the choice of a quantisation map) in
their definition [77]. Hence, so far there is no explicit construction of
Lorentzian models based on flux variables. However, there is also no obvious
reason why such a construction should not be possible..

This interpretation is corroborated by the following observation. Using the
(noncommutative) Dirac delta distribution in the Lie algebra representation,
given by

$\displaystyle\delta_{\star}(B)=\int_{G}\mathrm{d}g\,\mathrm{e}_{g}(B),$

we can easily see that the right gauge invariance of the group representation
translates into a closure condition of the fluxes, as

$\displaystyle\hat{\tilde{\varphi}}(B_{I})$
$\displaystyle=\int\mathrm{d}g\int_{G}\mathrm{d}h\prod_{I=1}^{4}\mathrm{e}_{g_{I}}(B_{I})\hat{\varphi}\big{(}g_{I}h^{-1}\big{)}=\int\mathrm{d}g\int_{G}\mathrm{d}h\prod_{I=1}^{4}\mathrm{e}_{g_{I}h}(B_{I})\hat{\varphi}(g_{I})$
$\displaystyle=\hat{\tilde{\varphi}}(B_{I})\star_{B_{I}}\delta_{\star}\left(\sum_{I=1}^{4}B_{I}\right).$
(2.8)

This closure condition $\sum_{I}B_{I}=0$ suggests that the state

$\displaystyle\left|B_{I}\right\rangle=\hat{\tilde{\varphi}}^{\dagger}(B_{I})\left|0\right\rangle$

can be interpreted as a closed tetrahedron, with $B_{I}$ being the bivector of
the face $I$.

The noncommutative counterpart of the convolution of group elements (2.6) that
leads to edges connecting different vertices is the insertion of a
noncommutative Dirac delta, $\delta_{\star}(B_{I}^{i}+B_{J}^{j})$, as can be
seen by Fourier transforming (2.6). The insertion of a star-Delta function now
can be interpreted as imposing the condition that the tetrahedron $i$ and the
tetrahedron $j$ are glued along the faces $I$ and $J$, respectively. The fact
that they are identified requires that the corresponding geometric data are
properly matched, thus the condition $B_{I}^{i}=-B_{J}^{j}$.

In this way, states built with the noncommutative dual representation are
naturally associated to three-dimensional simplicial complexes decorated with
data that are directly related to the intrinsic Riemannian geometry.
Therefore, they can be seen as triangulations of the spatial slices used in
the computation of quantum gravity amplitudes. It should be kept in mind,
however, that what we are gluing together are quantum tetrahedra. This will be
evident whenever we will try to compute the action of geometric operators on
these states. Despite the subtleties related to the noncommutative nature of
the Lie algebra variables, the flux representation allows a more transparent
construction of states having specific continuum metric properties, for
example corresponding to a particular spatially homogeneous metric as relevant
in cosmology.

###  2.4 Equations of motion in second quantisation

The tools that we have introduced can be used to design interesting states,
which can be interpreted in terms of spatial three-dimensional geometries. The
last ingredient that we need is the one that will generate the sum over four-
dimensional discrete geometries. This is achieved again in terms of the
perturbative expansion of a suitable dynamical field equation. As a simple
option, we can use the field equation from a classical action used in the
partition function,

$\displaystyle\frac{\delta
S[\hat{\varphi},\hat{\varphi}^{\dagger}]}{\delta\hat{\varphi}(g)}\left|\Phi\right\rangle=0,$
(2.9)

but other choices are possible. What is important is that the field equation
selecting the physical state $\left|\Phi\right\rangle$ can be split into a
free part, which admits for instance the Fock vacuum as a solution, and an
interaction term with the correct combinatorics.

The perturbative expansion of the partition function is now translated into
the expansion of $\left|\Phi\right\rangle$ in powers of a coupling constant
controlling the interaction term. Following the procedures of standard time-
independent perturbation theory in quantum mechanics, one then recovers a
diagrammatic expansion of the various correlation functions in the field
theory. To be more precise, in the previous sections we have seen how
convolutions of field operators can be interpreted in terms of discrete
geometric quanta, glued together to form specific geometrical structures.
Consider then two sets of strings of creation operators, each associated to a
graph, $\Gamma_{1}$, $\Gamma_{2}$, with data encoded in wavefunctions $f_{1}$,
$f_{2}$. Let us denote these operators as $\widehat{\Psi}(\Gamma_{i},f_{i})$,

$\displaystyle\widehat{\Psi}(\Gamma_{i},f_{i})=\int(\mathrm{d}g)^{\\#V(\Gamma_{i})}f_{i}(g^{i},\Gamma_{i})\prod_{j=1}^{\\#V(\Gamma_{i})}\hat{\varphi}^{\dagger}(g^{j}).$

It is easy to see that a correlation function such as

$\displaystyle
G(\Gamma_{1},f_{1};\Gamma_{2},f_{2})=\frac{\left\langle\Phi\right|\widehat{\Psi}^{\dagger}(\Gamma_{1},f_{1})\widehat{\Psi}(\Gamma_{2},f_{2})\left|\Phi\right\rangle}{\langle\Phi|\Phi\rangle},$
(2.10)

once the physical vacuum is treated perturbatively
$\left|\Phi\right\rangle=\left|0\right\rangle+\sum\limits_{i=1}^{\infty}\lambda^{i}\left|\Phi^{(i)}\right\rangle$,
can be rewritten in terms of a diagrammatic expansion, that can be interpreted
in terms of four-dimensional discrete geometries having as boundaries the
three-dimensional geometries dual to the graphs $\Gamma_{1}$, $\Gamma_{2}$,
with the proper inclusion of bulk and boundary geometric data, as depicted
schematically in Fig. 3 for the case of a lower-dimensional theory.

Figure 3: Given two boundary states, the perturbative expansion of the
correlation function (2.10) will generate all the simplicial complexes
compatible with them, and the corresponding amplitude.

The choice of the ingredients appearing in (2.9) (ladder operators and
kernels) will then specify which graphs are included in the perturbative
expansions of the correlation functions such as (2.10), together with the
corresponding amplitudes. Gravitational models can be designed so that (2.9)
leads to graphs and amplitudes that match the ones of the spin foam expansion.

The correlation functions in (2.10) should not directly be interpreted in
terms of transition amplitudes, as there is no notion of time evolution in
this timeless framework. They should be viewed instead as the analogue of the
matrix elements of the rigging map of LQG, enforcing the constraints that map
kinematical states into physical states.

As in the functional formulation, a proper definition of
$\left|\Phi\right\rangle$ must pass through the examination of the
perturbative series and its renormalisability, with all the related problems.
The true advantage of an operatorial formulation is the possibility of
developing a whole new approximation scheme for $\left|\Phi\right\rangle$ that
is not based on a perturbation expansion, and this is the reason why we prefer
this formulation of the theory.

If we view the theory defined in (2.9) as a peculiar many-body problem of the
kind encountered in condensed matter physics, we can try to abandon the
perturbative expansion of the physical state and introduce a completely
different approximation, based on a nonperturbative trial state
$\left|\Phi_{\mathrm{trial}}\right\rangle$ that will include our guesses about
the nature of the physical state of the theory, which will be presumably very
different from the Fock vacuum that is used for the perturbative expansion.
This is the point of view adopted when we are dealing with the GFT
condensates, as it will be explained. It will be also manifest how this
approach will make the investigation of the regime in which a large number of
quanta are involved much more productive.

###  2.5 Relation of GFT to tensor models and spin foam models

We have already mentioned that GFT is closely related to other approaches to
the statistical mechanics of random geometries.

The case of matrix models for random surfaces [51] is perhaps the case that is
understood with the greatest level of detail. The partition function, for a
finite size matrix model, is in fact an ordinary integral, and there are many
different techniques that can be used to determine its behaviour. The critical
properties of the model, describing the continuum limit of these models for
discrete random surfaces, can be extracted from a properly defined scaling
limit, in which the coupling constants are tuned to critical values while the
matrix size is sent to infinity.

Tensor models (see [80, 122] for reviews) are higher-dimensional
generalisations of matrix models. They involve tensorial objects with higher
rank, whose combinatorics is rich enough to permit the construction of Feynman
diagrams that can be put in correspondence with simplicial complexes. As in
the case of matrix models, tensor models are dealing only with the
combinatorial structure of the Feynman diagrams. In recent years, a variety of
results have been gathered about the combinatorial aspects of the partition
function of tensor models, leading to a first understanding of their critical
behaviour [40, 79].

There is a strong relationship between tensor models and GFT. In a certain
sense, one could view GFTs as tensor models with indices running on a
continuous set, a Lie group, instead of a discrete set. Conversely, a tensor
model might be pictured as a GFT based on a discrete group as
$\mathbb{Z}_{N}$, for instance. More generally, for compact groups, one can
rewrite, via the Peter–Weyl decomposition, a GFT as a generalised tensor model
involving various tensorial degrees of freedom, labelled by the irreducible
unitary representations of the given group (with a cutoff to obtain a finite
index range). This reduces to a conventional tensor model for $G={\rm U}(1)$.

GFT makes use of the same principles of the combinatorics of tensor models. In
addition, however, the additional group-theoretic data give rise to a much
richer variety of models, at fixed combinatorics, corresponding to the
possible ways to use this data to build the amplitudes. Most significantly,
the additional data of GFT allow us to construct models which can be rewritten
as simplicial path integrals for first-order gravitational theories [16, 17],
in particular Plebanski-type reformulations of general relativity that are
adapted to the Ashtekar variables for LQG.

The analysis of the GFT partition function then has to go beyond the
combinatorial aspects, and presents additional difficulties with respect to
the partition function for matrix and tensor models. Indeed, the GFT formalism
provides additional data in the amplitudes of the graphs, which are no longer
characterised by the combinatorial structure alone. Therefore, the
considerations leading to the identification of the classes of graphs
responsible for the dominant contributions to the partition functions need to
be revised. The hope is that the number of results obtained in the case of
tensor models, particularly on the problem of renormalisability, phase
transitions and the continuum limit, might be ultimately adapted and extended
to GFT. The renormalisation of GFT is an ongoing research programme (see
Section 1.2).

While GFT actions can obviously be of very general shapes, the special class
built on generalisations of the Ooguri model plays an important role for the
construction of candidates for quantum gravity models. The reason for this is
the precise relationship between GFT and spin foam models, a proposal for the
definition of a discretised sum-over-histories version of quantum gravity. In
turn, spin foams are closely related to loop quantum gravity, as spin foam
amplitudes are related to histories of spin networks. In four spacetime
dimensions, the question of whether spin foams and canonical LQG can be seen
as defining equivalent theories is a subtle issue and a focus of current
research [3]. The GFT formalism gives a different perspective on this issue:
the canonical and covariant descriptions of GFT simply correspond to the
standard dichotomy of defining a quantum field theory through a Fock
space/operator formalism or through the path integral. The two viewpoints on
GFT can then be respectively be related to canonical LQG and spin foam models,
without the need to directly map between canonical LQG and spin foams.

It is known that classical Einstein gravity without a cosmological constant
can be formulated in a variety of ways using different sets of degrees of
freedom and different variational principles (see, e.g., [116]). The Plebanski
formulation starts from a topological BF theory and supplements it with a term
imposing a certain set of constraints. These constraints make the theory
equivalent to a Palatini (i.e., a first order) formulation of vacuum general
relativity. Indeed, the role of these constraints is to break the gauge
symmetries of the topological theory, which would not possess the local
degrees of freedom required by general relativity333It is worth mentioning
that the three-dimensional case does not require the introduction of
constraints, as three-dimensional general relativity without a cosmological
constant can be written as a topological BF theory..

The spin foam approach to quantum gravity takes advantage of these results.
The basic idea is to start from the quantisation of BF theories, suitably
discretised on a cellular complex, and then to impose the necessary
constraints that would turn a topological theory into a gravitational theory
[12]. In the topological theory, the data associated to the complexes
(typically expressed in terms of representation labels) are allowed to take
values in the whole range allowed by the gauge group involved,
$\mathrm{Spin}(4)$ or $\mathrm{SL}(2,\mathbb{C})$. The imposition of the
simplicity constraints restricts the allowed data that can be attached to a
spin foam. In particular, when looking at the Lie algebra data, they are
equivalent to the requirement that these data can be rewritten in terms of
discretised tetrad fields. The various proposals in the literature try to
achieve this goal using a variety of related but different ideas and
techniques [117].

There is a precise correspondence between spin foam models and GFT. For any
given spin foam model, one can find a corresponding GFT action in a precise
way [121]: given the amplitude assigned to any given cellular complex by the
spin foam model, there is a GFT action whose Feynman graphs, dual to cellular
complexes, have matching amplitudes. Seen from the opposite direction, a GFT
action defines a spin foam model, once the partition function is rewritten in
terms of the Feynman expansion,

$\displaystyle
Z=\int\mathcal{D}\varphi\,\mathcal{D}\bar{\varphi}\,e^{-S[\varphi,\bar{\varphi}]}=\sum_{\sigma}\frac{\prod_{i}(\lambda_{i})^{N_{i}(\sigma)}}{{\rm
Aut}(\sigma)}\mathcal{A}_{\sigma}.$ (2.11)

Here we are labelling the Feynman graphs with the corresponding dual cellular
complexes $\sigma$, in order to make the correspondence with spin foam
approach more transparent. We are also including the possibility that several
different interactions terms are introduced, each with different kernels and
different combinatorial structure, and each controlled by a different coupling
constant $\lambda_{i}$. $N_{i}(\sigma)$ will be then the total number of
vertices of type $i$ appearing in the Feynman diagram. ${\rm Aut}(\sigma)$ is
the order of the automorphisms of $\sigma$. This is the combinatorial part of
the amplitude. It can be shown by direct calculations that the amplitude
$\mathcal{A}_{\sigma}$, the Feynman amplitude that the GFT model assigns to
$\sigma$, can be represented as a spin foam model.

An important ingredient that is explicitly added by the GFT approach is a
definite prescription for generating weights for amplitudes for different
cellular complexes in the sum over $\sigma$. These depend on the coupling
constants of the GFT action, as (2.11) shows explicitly.

We now present a concrete implementation of the ideas sketched so far,
restricting the analysis to the case of models and data relevant for four-
dimensional gravity. The starting point is the Ooguri model for
$\mathrm{Spin}(4)$ or $\mathrm{SL}(2,\mathbb{C})$, which will be modified to
include the restriction on the geometric data only. There are several similar
but inequivalent constructions, as the imposition of simplicity constraints is
not uniquely specified by the classical theory. Indeed, the translation of the
classical, continuum constraints into the discrete setting faces various
discretisation and ordering ambiguities related to the non-Abelian nature of
the variables. Furthermore, the simplicity constraints are characterised by a
free parameter, the Barbero–Immirzi parameter.

The idea of GFT and spin foam models is to work with graphs decorated with the
data of $\mathrm{SL}(2,\mathbb{C})$ or $\mathrm{Spin}(4)$ BF theories, and
then to restrict the sums to only those data that correspond to the geometric
degrees of freedom. For a finite value of the Barbero–Immirzi parameter, one
way to proceed is as follows. It is convenient to work with a field theory
based on $\mathrm{SU}(2)$, so that we can work directly with the geometric
data and use the tools that have been built to deal with LQG spin networks.
Following [60], we can define a map $S$ from the space of functions of
$\mathrm{SU}(2)$ into the space of functions of $\mathrm{Spin}(4)$ (or
$\mathrm{SL}(2,\mathbb{C})$)444Although, for the cases of interest, in the
$\mathrm{SL}(2,\mathbb{C})$ case they would be distributions.. Denoting with
$G_{I}$ the elements of the four-dimensional gauge group, the $\mathrm{SU}(2)$
field is mapped into

$\displaystyle(S\varphi)(G_{I})=\int\mathrm{d}g\,S(G_{I},g_{J})\varphi(g_{J}).$

This map $S$ takes care of restricting the $\mathrm{Spin}(4)$ data for the
Feynman diagrams to geometric data only, and it is the part of the
construction where the implementation of the simplicity constraints, encoded
in the so-called fusion coefficients [117], is located. The familiar
expressions for the various spin foam models can be recovered by expanding
these kernels in the appropriate basis functions (Wigner representation
matrices, coherent states, noncommutative plane waves555As discussed in [16,
17], it is possible to build GFT actions exploiting the relationship between
their flux representation, based on the noncommutative Fourier transform
discussed in Section 2.3, and simplicial path integrals that make direct use
of flux variables.), but the explicit form of this kernel is not very
illuminating.

The interaction term for the Ooguri model for $\mathrm{Spin}(4)$, built with
(2.3), can be used to define the interaction term for the $\mathrm{SU}(2)$
theory,

$\displaystyle\mathcal{V}_{\text{SF}}[\varphi,\overline{\varphi}]=\mathcal{V}_{\text{Ooguri}}[S\varphi,\overline{S\varphi}].$

The presence of the maps $S$ will then guarantee that, in the Feynman
diagrams, only the $\mathrm{Spin}(4)$ data solving the simplicity constraints
will be summed over. The Lorentzian case proceeds along similar lines,
although some additional care has to be used as the various expressions might
require regularisation, before they can be manipulated.

While this is not the only possible way to proceed, it is the simplest one
that allows us to work directly with geometric data and with the familiar
$\mathrm{SU}(2)$ spin networks. This choice will also make it easier to keep
track of the differences between different prescriptions for the simplicity
constraints. For the Barrett–Crane model, which corresponds to infinite
Immirzi parameter, the construction is slightly different as $\mathrm{SU}(2)$
has to be replaced with the homogeneous space
$\mathrm{SL}(2,\mathbb{C})/\mathrm{SU}(2)$ (or
$\mathrm{Spin}(4)/\mathrm{SU}(2)\simeq\mathrm{SO}(4)/{\rm
SO}(3)\simeq\mathrm{SU}(2)$ in the Riemannian case) [50].

##  3 GFT condensates: kinematics

After this discussion of the general formalism of GFT in terms of kinematics,
dynamics, and the relation to LQG and spin foams, in this section we review
the detailed construction and interpretation of condensate states in GFT.
There are close similarities to how one proceeds in the case of a non-
relativistic quantum field theory describing Bose–Einstein condensates [118],
but also important differences due to the fact that we are dealing with a
field theory of quantum geometry as a (background-independent) proposal for
quantum gravity. We largely follow the constructions and discussion given in
the original papers [71, 72], on which most of the work so far has been based.
In Section 3.5 we will show how these constructions can be generalised and
extended to states that might be closer to those of physical interest in
quantum gravity, and that have the advantage of also describing situations of
spherical symmetry.

###  3.1 Definitions and simplest condensates

A condensate phase of a quantum field theory can be characterised by an order
parameter, associated to a non-zero expectation value of a field operator in
this phase which would be zero in the standard Fock vacuum. In the simplest
case, this order parameter can be associated with the “condensate
wavefunction”, i.e., a complex-valued function on the configuration space of
the basic quantum field that fully determines the quantum state.

More concretely, we can focus on a particular subset of the space of general
$N$-particle states

$\displaystyle|\Psi\rangle:=\int(\mathrm{d}g)^{N}\Psi\big{(}g_{I}^{1},\ldots,g_{I}^{N}\big{)}\bigotimes_{i=1}^{N}|g_{I}^{i}\rangle$

in which the wavefunction $\Psi$ decomposes into a product of functions
depending on fewer arguments. Physically, this means neglecting correlations
between different particles either altogether or beyond a certain order (e.g.,
beyond 2-particle or 3-particle correlations). The two simplest cases are of
particular interest: in the first, $\Psi$ decomposes into one-particle
wavefunctions,

$\displaystyle\Psi\big{(}g_{I}^{1},\ldots,g_{I}^{N}\big{)}=\prod_{i=1}^{N}\psi\big{(}g_{I}^{i}\big{)},$
(3.1)

whereas in the second it involves two-particle correlations (this is known as
the Bogoliubov approximation in the context of Bose–Einstein condensates
[118]),

$\displaystyle\Psi\big{(}g_{I}^{1},\ldots,g_{I}^{N}\big{)}=\prod_{i=1}^{N/2}\psi\big{(}g_{I}^{2i-1},g_{I}^{2i}\big{)}+{\rm
permutations}$ (3.2)

where $N$ is even and the permutations are needed to ensure that the
wavefunction $\Psi$ is symmetric under the exchange of any two vertex indices,
compatible with the bosonic statistics for the GFT. This symmetry will be
automatically implemented once bosonic field operators are used.

States of the type (3.1) or (3.2) could be used as simple approximations to a
condensate, and capture much of the relevant physics of condensates. However,
they do not satisfy all of the desired properties; in particular, given that
they describe states of fixed particle number, the expectation value of the
field operators $\hat{\varphi}$ and $\hat{\varphi}^{\dagger}$ vanish. We hence
need to use states that are superpositions of states of the type (3.1) or
(3.2) for different $N$, with appropriate weights in the sum, and are led to
the definition of a single-particle condensate state

$\displaystyle|\sigma\rangle:=\mathcal{N}(\sigma)\exp\left(\int\mathrm{d}g\,\sigma(g_{I})\hat{\varphi}^{\dagger}(g_{I})\right)|0\rangle,$
(3.3)

where
$\mathcal{N}(\sigma)=\exp(-\frac{1}{2}\int\mathrm{d}g\,|\sigma(g_{I})|^{2})$
is a normalisation factor, and a dipole condensate state

$\displaystyle|\xi\rangle:=\mathcal{N}(\xi)\exp\left(\int\mathrm{d}g\,\mathrm{d}h\,\xi\big{(}g_{I}^{-1}h_{I}\big{)}\hat{\varphi}^{\dagger}(g_{I})\hat{\varphi}^{\dagger}(h_{I})\right)|0\rangle,$
(3.4)

with $\mathcal{N}(\xi)$ again a normalisation factor (given explicitly in
[72]). Note the definition of $\xi$ in (3.4), instead of an in principle
possible more general wavefunction $\psi(g_{I},h_{I})$. Because
$\hat{\varphi}^{\dagger}(g_{I})=\hat{\varphi}^{\dagger}(g_{I}h)$, the
functions $\sigma$ and $\xi$ satisfy $\sigma(g_{I})=\sigma(g_{I}h)$ and
$\xi(g_{I})=\xi(kg_{I}h)$ for all $h,k\in G$. For (3.3) to also represent a
state invariant under a “gauge transformation” acting simultaneously on the
open ends of the spin network represented by an open vertex, we also impose
$\sigma(g_{I})=\sigma(kg_{I})$ for all $k\in G$. (3.4) is built out of
elementary “dipoles” that correspond to gauge-invariant states in LQG, and are
characterised by a single wavefunction $\xi(g_{I})$ on $G\backslash G^{4}/G$,
as would be the possible states on a “dipole” in usual LQG [125].

The states (3.3) and (3.4) satisfy the property of wavefunction homogeneity
which requires the state to be determined by a single-particle wavefunction
($\sigma$ or $\xi$). They are superpositions of states of the type (3.1) and
(3.2) for different particle numbers, and hence rather different from the
states usually considered in LQG which are typically built on a fixed graph.

An important property of the state (3.3) is that it is an eigenstate of the
GFT field operator,

$\displaystyle\hat{\varphi}(g_{I})|\sigma\rangle=\sigma(g_{I})|\sigma\rangle.$

From this it immediately follows that
$\langle\hat{\varphi}(g_{I})\rangle=\sigma(g_{I})$, so that $\sigma(g_{I})$
represents the order parameter, and that more generally, any normal-ordered
$n$-point function involving $\hat{\varphi}(g_{I})$ and
$\hat{\varphi}^{\dagger}(g_{I})$ can be evaluated by replacing $\hat{\varphi}$
by $\sigma$ and $\hat{\varphi}^{\dagger}$ and $\bar{\sigma}$. In this very
simplest approximation, there are no correlations between the GFT quanta, and
in the mean-field approximation when fluctuations can be ignored,
$\sigma(g_{I})$ directly represents a classical GFT field configuration. This
property will be important in Section 3.3 when the geometric interpretation of
(3.3) will be discussed.

As is apparent from the appearance of an explicit normalisation factor in
(3.3) and (3.4), these states are normalisable for appropriate choices of the
functions $\sigma$ and $\xi$, and hence elements of the GFT Fock space defined
in Section 2. For (3.3), normalisability of the state is equivalent to a
finite expectation value of the number operator,

$\displaystyle
N:=\langle\sigma|\widehat{N}|\sigma\rangle=\langle\sigma|\int\mathrm{d}g\,\hat{\varphi}^{\dagger}(g_{I})\hat{\varphi}(g_{I})|\sigma\rangle=\int\mathrm{d}g\,|\sigma(g_{I})|^{2}<\infty.$
(3.5)

The thermodynamic limit $N\rightarrow\infty$ is described by states that are
no longer in the original GFT Fock space: this is standard in quantum field
theory, where in the limit corresponding to a phase transition one needs to
change representation to a different, unitarily inequivalent Hilbert space.
Here $N\rightarrow\infty$ corresponds to a continuum limit in which such a
transition is reached.

In practice, the difference between fixed particle number as in (3.1) and
fluctuating particle number as in (3.3) becomes unimportant for large $N$: an
elementary calculation shows that

$\displaystyle\left(\Delta
N\right)_{|\sigma\rangle}:=\sqrt{\langle\sigma|\widehat{N}^{2}|\sigma\rangle-(\langle\sigma|\widehat{N}|\sigma\rangle)^{2}}=\sqrt{\langle\sigma|\widehat{N}|\sigma\rangle}=\sqrt{N}$

and hence the relative uncertainty $(\Delta N)/N\sim 1/\sqrt{N}$ vanishes in
the thermodynamic limit; this is the well-known equivalence of the canonical
and grandcanonical ensembles in this limit.

The dipole condensate state (3.4) is no longer a coherent state for the GFT
field operator, but closer to squeezed states that appear in quantum optics
[96]. While a refinement of (3.3) that incorporates the usual notion of gauge
invariance in LQG and contains two-particle correlations, is already harder to
do computations with; no simple closed expression for general $n$-point
functions or quantities such as the average particle number is known.
Approximate expressions for correlation functions are discussed in [72], and
can be used for estimates of the error that one makes in using them.

The definition of the states (3.3) and (3.4) is mainly motivated by their nice
properties from the perspective of quantum field theory and by the analogy to
the well-studied case of Bose–Einstein condensates. They assume a truncation
to one- and two-particle correlations respectively, which can be seen as
corresponding to the situation of cosmological relevance, where we focus on
geometries that are spatially homogeneous or nearly homogeneous, so that
spatial gradients can be ignored in a first approximation. However, the
truncation also means that these states cannot be interpreted in the sense of
an extended graph in LQG: they would correspond to “graphs” consisting of a
large number of disconnected components formed by a single vertex or dipole,
respectively. Any topological information normally also contained in the graph
must come from elsewhere. One way to remedy this situation is to define
generalised condensates that are associated to a graph of fixed topology, as
we will outline in Section 3.5. Another is to make an assumption for the
topology of space and require self-consistency with the dynamics: e.g., if the
effective dynamics of isotropic and homogeneous geometries is given by the
Friedmann equation with positive spatial curvature, one infers that space
should have the topology of a 3-sphere. This viewpoint, advocated in [72] and
in some sense closer to the formalism of canonical LQG [124, 131], will be
outlined in Section 3.3.

###  3.2 “Condensate wavefunction” vs. “wavefunction of the universe”

The mean field $\sigma(g_{I})$ appearing in the definition (3.3) is often
referred to as the “condensate wavefunction”. Indeed, it is a wavefunction in
the sense that $|\sigma\rangle$ is a superposition of states with all possible
particle numbers, each of which is defined by a product wavefunction of the
form (3.1) where $\psi$ is replaced by $\sigma$: $\sigma$ is the wavefunction
of the “ground state” that the GFT quanta are supposed to be condensed into
(again, in this mean-field approximation where fluctuations over this
condensed phase are ignored).

In most other respects, it can be quite misleading to think of $\sigma$ as a
usual quantum-mechanical wavefunction. First of all, the overall normalisation
of $\sigma$ is not arbitrary as in quantum mechanics, but corresponds to the
average particle number (see (3.5)). Indeed it is often useful to adopt a
parametrisation $\sigma(g_{I})=\sqrt{N}\sigma_{0}(g_{I})$ where $\sigma_{0}$
is normalised to one [118].

More importantly, there is no superposition principle and no direct
probability interpretation associated with $\sigma(g_{I})$, as the dynamics of
$\sigma$ is typically governed by nonlinear (and nonlocal) equations. At first
sight, this may appear confusing: are we proposing a radical nonlinear,
nonlocal modification of quantum mechanics for GFT? It should be clear, and
will become clearer in Section 4 below, that this is not the case, and all
dynamical equations are linear equations on the GFT Fock space. However, the
correspondence between a condensate state and its “wavefunction” is clearly
nonlinear,

$\displaystyle|\sigma\rangle+|\sigma^{\prime}\rangle\neq|\sigma+\sigma^{\prime}\rangle,$

and linear equations on the Hilbert space become nonlinear equations for
$\sigma$. Whatever the precise form of the effective dynamics for $\sigma$, it
will not make sense to directly interpret them as a Schrödinger-type or
Wheeler–DeWitt-type equation for a “wavefunction of the universe” as in
standard quantum cosmology [134], and common tools in quantum cosmology such
as the WKB approximation will have to be reinterpreted according to the rôle
of $\sigma$ as a mean-field configuration of the quantum field, rather than a
single-particle (“single-universe”) wavefunction.

To make this more precise, let us compare with the situation in condensed
matter physics. Here the “condensate wavefunction” describing the simplest
approximation to the Bose–Einstein condensate can be written as [118]

$\displaystyle\Psi(\vec{x})=\sqrt{\rho(\vec{x})}\exp(-{\rm
i}\theta(\vec{x})),$ (3.6)

where $\rho(\vec{x})$ defines the density and $\theta(\vec{x})$ the velocity
potential (for irrotational fluids, the velocity is
$\vec{v}\propto\vec{\nabla}\theta$) of the superfluid formed by the
condensate. $\Psi$ clearly has no probability interpretation; it already
describes a classical limit, the mean-field approximation, in which it
corresponds to a classical field configuration (given by two real functions
$\rho,\theta$ or one complex function $\Psi$).

A WKB approximation applied to (3.6) would correspond to a regime in which the
fluid density varies slowly over space while the velocity $\vec{v}$ of the
fluid is large, and such an approximation will not be valid in general, if we
remember that (3.6) corresponds to a ground-state configuration. For instance,
for the very simplest case of a non-interacting Bose gas, $\Psi(\vec{x})$ is
simply constant over space (over a box in which the gas is confined) but the
velocity $\vec{v}$ vanishes.

Phrased differently, a WKB approximation would correspond to a regime in which
the fundamental degrees of freedom, i.e., the individual atoms in the case of
the condensate or the degrees of freedom of quantum geometry in the case of
GFT, already behave semiclassically. While semiclassical properties of the
quantum state are an important consistency criterion in quantum cosmology
(see, e.g., [92]), if quantum cosmology arises from the hydrodynamics of
quantum gravity semiclassicality is not a restriction on the “wavefunction”
$\sigma$ but rather a statement about the validity of an ansatz of the form
(3.3), i.e., of the mean-field approximation. Early works on GFT condensates
advocated the use of a WKB approximation [71, 72], but it was understood in
[67, 69, 70] that this is not really justified. In this review, we will focus
on studies of the dynamics of GFT condensates that do not assume the validity
of a WKB regime. Nevertheless, interesting results such as [43] where the
dynamics of LQC and the effective dynamics of GFT condensates could be matched
in a WKB regime show that the role of WKB solutions in this setting deserves
to be better understood.

Because the “condensate wavefunction” is a mean-field configuration rather
than a quantum-mechanical wavefunction, there is no obvious map between the
formalism of GFT condensates and the Hilbert space formalism of Wheeler–DeWitt
quantum cosmology or LQC, but such a map is not required; what is needed is a
way to extract cosmological predictions from the effective dynamics of such
condensates. For this it has proven advantageous to focus on expectation
values of geometric operators that correspond to cosmological observables, as
we will detail in Section 4.4 and below. Viewed more practically, the absence
of a Hilbert space structure in this hydrodynamic setting might be seen as an
advantage rather than a problem in quantum cosmology, where a general
“wavefunction of the universe” has serious interpretational problems and one
usually uses a class of special, highly semiclassical states, so that the full
Hilbert space seems rather redundant when one is interested in the physical
predictions of quantum cosmology. In our setting, the semiclassical properties
expected from the universe are already encoded in the use of a restricted
class of (condensate) states.

###  3.3 Interpretation in terms of geometry

So far, the definition of GFT condensate states has mainly been motivated by
the analogy with Bose–Einstein condensation in condensed matter physics. In
order to relate this setting to quantum cosmology, we also need to show how
states such as (3.3) or (3.4) can be interpreted in terms of spacetime
geometry, i.e., in terms of (a probability distribution over) continuum
metrics and/or Ashtekar–Barbero connections. For this, a slight detour is
necessary in which we discuss in more detail the geometric interpretation of
the configuration space of the GFT field $\hat{\varphi}(g_{I})$ and,
accordingly, the wavefunctions $\sigma$ and $\xi$ appearing in (3.3) and
(3.4). Any interpretation proposed in this context is motivated by the close
relation of GFT to LQG and spin foam models, which do give a geometric
interpretation to the group-theoretic data associated to the states.
Eventually, it will have to be shown that such an interpretation is justified,
i.e., that the effective continuum geometry defined from it is governed by the
dynamics of a gravitational theory (Einstein’s equations, or a generalisation
of them), couples correctly to matter, and satisfies a notion of
diffeomorphism invariance. We will outline how this could be done for GFT
condensates in the discussion of dynamics in Section 4.

What we are looking for is an “inverse map”: given the parallel transports
$g_{I}$ of a connection along four edges meeting at a central vertex,

$\displaystyle g_{I}\sim\mathcal{P}\exp\left(\int_{e_{I}}A\right)$

or, in the dual non-commutative formulation introduced in Section 2.3, four
Lie algebra elements $B_{I}$ corresponding to bivectors associated to faces of
a dual tetrahedron (where we assume that simplicity constraints are either
trivial, for $G=\mathrm{SU}(2)$, or have been imposed properly),

$\displaystyle B^{AB}_{I}\sim\int_{\triangle_{I}}e^{A}\wedge e^{B},$ (3.7)

how do we reconstruct an (approximate) connection $A$ or a coframe field $e$
compatible with this data? This is a common problem in discrete approaches to
quantum gravity, and hard to address [38, 39]. Our task is somewhat easier
than in general because we are looking at specific states with high symmetry,
and expect the approximate continuum description to correspond to a
homogeneous or almost homogeneous geometry.

The following interpretation of the bivectors (3.7) in terms of a continuum
metric was given in [72]. First, the mean field $\sigma(g_{I})$666Here and in
the following, $\sigma(g_{I})$ stands for a general “condensate wavefunction”
with the required symmetries, and in particular can also stand for the
function $\xi(g_{I})$ used in (3.4). has two independent symmetries
$\sigma(g_{I})=\sigma(hg_{I}k)$ for all $h,k\in G$, which imply that its
noncommutative Fourier transform,

$\displaystyle\tilde{\sigma}(B_{I}):=\int\mathrm{d}g\prod_{I=1}^{4}e_{g_{I}}(B_{I})\,\sigma(g_{I}),$
(3.8)

satisfies (compare with the property (2.8) of the GFT field itself)

$\displaystyle\tilde{\sigma}(B_{I})=\delta_{\star}\left(\sum_{I}B_{I}\right)\star\tilde{\sigma}(B_{I}),\qquad\tilde{\sigma}(B_{I})=\tilde{\sigma}\big{(}kB_{I}k^{-1}\big{)}\qquad\forall\,k\in
G.$

These symmetries correspond to two symmetries in the continuum: the first is
the closure constraint implying that the four faces close to form a geometric
tetrahedron, and the second is the invariance under local frame rotations
which are classical discrete gauge transformations.

The next step is to assume that the continuum fields to be reconstructed vary
slowly over the scale of the tetrahedron, as must be the case for the
discretisation to approximately represent them. This means we can replace
$B^{AB}_{i}=\frac{l^{2}}{2}{\epsilon_{i}}^{jk}e^{A}_{j}e^{B}_{k}$ in terms of
three discrete triad vectors $e^{A}_{i}$, where $i=1,2,3$ and $l$ is a
coordinate length associated to the triangles that we leave free for now;
$B_{4}$ is replaced by $-\sum_{i}B_{i}$ using the closure constraint. The
gauge-invariant configuration space for each tetrahedron, invariant under the
transformations $B_{i}\mapsto kB_{i}k^{-1}$, is six-dimensional and can be
parametrised by the six “metric components”

$\displaystyle g_{ij}=e^{A}_{i}e^{B}_{j}\eta_{AB}=\frac{1}{4l^{2}{\rm
tr}(B_{1}B_{2}B_{3})}{\epsilon_{i}}^{kl}{\epsilon_{j}}^{mn}\tilde{B}_{km}\tilde{B}_{ln},\qquad\tilde{B}_{ij}:={\rm
tr}(B_{i}B_{j}),$ (3.9)

where $\eta_{AB}$ is the appropriate invariant inner product. The quantities
$g_{ij}$, constructed out of (3.7), should be scalars under diffeomorphisms.
They represent the 3-metric in a given frame that transforms covariantly under
diffeomorphisms, at a given fixed point in the tetrahedron (which can be
chosen for convenience to be one of the vertices). The continuum
interpretation of (3.9) is fixed after specifying this frame. In essence, the
freedom in choosing this frame corresponds to the freedom in rotating
tetrahedra in an embedding into a spatial manifold. This freedom is still
there because our construction is purely local on the level of each individual
tetrahedron and disregards any connectivity information in the states, as
seems appropriate when considering the simple condensates of Section 3.1.

In order to define a notion of spatially homogeneous geometries, we now
consider an embedding of a given configuration of tetrahedra into a continuum
manifold of topology $\mathfrak{G}/X$ where $\mathfrak{G}$ is a 3-dimensional
Lie group acting transitively on the manifold, and $X$ can be a discrete
subgroup [72]. $\mathfrak{G}$ defines the notion of homogeneity, and also
suggests a natural choice of frame, given by a basis of left-invariant vector
fields. The $g_{ij}$ in (3.9) are then interpreted as metric coefficients in
this frame, so that a spatially homogeneous geometry is precisely one in which
$g_{ij}$ are all constant in the continuum, or the same for a large number of
tetrahedra in the discrete. This then justifies the choice of state (3.1) or
(3.3) in order to describe continuum spatially homogeneous geometries, where
the continuum is strictly obtained in the limit $N\rightarrow\infty$.

Our use of an embedding in this context is very much analogous to how one
interprets spin network states in canonical LQG as encoding (distributional)
continuum geometric data on a given, continuum spatial hypersurface [124]. The
topological information that one may associate to a collection of $N$
tetrahedra (i.e., a disjoint union of $N$ three-balls) plays no role in this
continuum interpretation, and there would be no consistent way of using it if
we are looking at superpositions of different $N$, e.g., (3.3). It seems
plausible that, if cosmology and continuum geometry emerge from a hydrodynamic
approximation to quantum gravity, the topology of space should also be
emergent rather than determined by microscopic details, such a specific choice
of graph. Indeed, the fact that one might think of a superfluid fundamentally
as a collection of disconnected quantum atoms plays no role at the
hydrodynamic level. This viewpoint leaves open the question of how to choose
the embedding manifold, as we will discuss below. One might try to incorporate
a choice of topology into the definition of the state: in Section 3.5, we will
discuss a different class of condensates with definite topological
interpretation, which may be seen as an alternative proposal.

In (3.9), we have focused on the metric which is usually more important in
cosmology, but a similar construction of a continuum connection from the group
elements $g_{I}$, taking into account the invariance under $g_{I}\mapsto
hg_{I}k$, is possible. As discussed in some detail in [67] for
$G=\mathrm{SU}(2)$, if GFT condensates define spatially homogeneous continuum
geometries, the phase space of a tetrahedron can be mapped to the phase space
of spatially homogeneous continuum geometries, with the quotient space
$\mathrm{SU}(2)\backslash\mathrm{SU}(2)^{4}/\mathrm{SU}(2)\simeq\mathrm{SU}(2)^{3}/{\rm
Ad}_{\mathrm{SU}(2)}$ of gauge-invariant group variables corresponding to
continuum homogeneous connections and the quotient $\mathfrak{su}(2)^{\oplus
3}/{\rm Ad}_{\mathrm{SU}(2)}$ of gauge-invariant bivector variables
corresponding to continuum homogeneous metrics. This observation makes
explicit in which sense a tetrahedron can be seen as a “chunk of space” with
the same geometric data as a continuum geometry, and it provides an important
foundation for the idea of cosmology as hydrodynamics of GFT: it shows that
GFT condensate states, based on the idea of wavefunction homogeneity, capture
exactly the degrees of freedom needed for homogeneous cosmology, i.e.,
minisuperspace geometric degrees of freedom. The ambiguities in interpreting
these degrees of freedom are related to the ambiguity of defining an
isomorphism between GFT data and continuum cosmological variables.

Using (3.9) gives a direct interpretation of the gauge-invariant geometric
variables of a tetrahedron in terms of a continuum metric, which is relatively
simple and local at the level of each tetrahedron. There are some technical
difficulties in defining an operator version of (3.9), first from defining
inverse operators, and also from operator ordering ambiguities coming from
non-commutativity of the $\tilde{B}_{ij}$ [67]. Barring these, one could now
define an effective (spatially homogeneous) continuum metric out of
expectation values of such operators.

There are however several issues with (3.9), mainly related to its need to be
supplemented by a choice of frame in which the $g_{ij}$ define the 3-metric.
First of all, even though defining this frame by three left-invariant vector
fields is rather natural, the notion of left-invariance itself relies on the
choice of isometry group $\mathfrak{G}$, with the possible choices
corresponding to the homogeneous cosmological models according to the Bianchi
classification (see, e.g., [33]). The viewpoint taken in [72] is that this
choice should be fixed by self-consistency: one does not know initially which
of the Bianchi models is described by a particular condensate state, but its
effective dynamics will pick out at most one, as the Bianchi models all differ
by their “anisotropy potential” in the Hamiltonian constraint [33]. While this
indeed suggests an unambiguous association of condensate states with Bianchi
universes, a more fundamental understanding why a particular Bianchi geometry
would be selected is so far lacking. Perhaps more importantly, the definition
(3.9) uses only the bivectors $B_{1}$, $B_{2}$ and $B_{3}$, as $B_{4}$ has
been eliminated using the closure constraint $\sum_{I}B_{I}=0$. The four faces
of the tetrahedron are not treated “democratically”: an isotropic tetrahedron
would be one with three pairwise orthogonal edges of equal length, rather than
a fully equilateral one, see Fig. 4. One might prefer constructions in which
isotropy means equilateral tetrahedra [111]. The closure constraint also
breaks the ${\rm O}(3)$ symmetry of continuum geometry under a redefinition of
the basis of left-invariant vector fields; under such a redefinition,
$e_{i}\mapsto{O_{i}}^{j}e_{j}$ and $B_{i}\mapsto\pm{O_{i}}^{j}B_{j}$ which in
general does not preserve $\sum_{I}B_{I}=0$. This suggests that there should
be a preferred basis of left-invariant vector fields, and for Bianchi models
such a basis exists: it is the basis (defined up to the action of a
permutation group $\mathfrak{S}_{3}$) in which the metric is diagonal,
$g_{ij}=a^{2}_{i}\delta_{ij}$ with no summation over $i$. Hence, for
consistency, we are required to assume that off-diagonal components of (3.9)
vanish, at least as expectation values, which gives a restriction on possible
condensate states to define Bianchi models. Again, this is not a problem in
practice, but might suggest that the construction is not general enough777It
also appears that a similar issue would apply to connection variables, leading
to the conclusion that these would also have to be diagonalised. Then one
would be restricted to class A Bianchi models, for which a simultaneous
diagonalisation of the spatial metric and its conjugate momentum is possible
and a corresponding Hamiltonian formulation exists [33]..

Figure 4: An “isotropic tetrahedron” with three pairwise orthogonal edges.

So far, much of the work on GFT condensates has used geometric observables
that do not rely on (3.9) but are built out of more basic operators borrowed
from LQG (see Section 2.2). The perhaps most basic such operator is a volume
operator, which in the second quantised formalism simply corresponds to the
total volume contained in a state. The definition of the volume operator in
LQG is somewhat subtle, with different definitions arising from different
regularisation schemes [7, 126] and possible issues with a consistent
semiclassical limit [65]; these issues are less important for tetrahedra, for
which good agreement with Bohr–Sommerfeld quantisation has been found [29].
Given a definition of the volume operator, one can compute the expectation
value or higher moments of the volume of the universe. For a homogeneous,
isotropic geometry, this directly defines a scale factor $a$, proportional to
the third root of this total volume. For more general geometries, the total
volume is just the most coarse-grained geometric observable. Hence, for an
identification of the total volume with a quantity proportional to $a^{3}$ to
make sense, one needs to be confident that the state does define a homogeneous
and isotropic geometry. The embedding picture above gives an argument for why
condensates define spatially homogeneous geometries, but the notion of
isotropy must come from elsewhere if one does not want to use the definition
(3.9). For instance, one may define isotropy by equilateral tetrahedra, see
Section 4.5 below.

As discussed in Section 2.2, the GFT Fock space formalism also allows the
definition of “total area operators”, one-body operators that are simpler to
define than the total volume as they are directly built out of the basic flux
operators in LQG. They take the form

$\displaystyle\widehat{A}_{I}=\int\mathrm{d}g\,\hat{\varphi}^{\dagger}(g_{J})\,\sqrt{-\Delta_{g_{I}}}\hat{\varphi}(g_{J})$

and describe quantities such as “the total area of all $I$-th faces”. Their
geometric interpretation is not entirely clear in general, since one seems to
be summing over areas of disconnected surfaces, which is different from LQG
operators that correspond to the area of a 2-surface in space. For a
homogeneous and isotropic geometry, however, again the expectation value
$\langle\widehat{A}_{I}\rangle$ of such an operator has to be proportional to
the square of the scale factor. It turns out that
$\langle\widehat{A}_{I}\rangle$ divided by $N^{1/3}$ can be identified with a
cosmological scale factor, so that

$\displaystyle
a^{2}V_{0}^{2/3}:=\big{\langle}\widehat{A}_{I}\big{\rangle}N^{-1/3},$ (3.10)

where $V_{0}$ is a fiducial coordinate volume, akin to the volume of the
“fiducial cell” appearing in homogeneous models for quantum cosmology, such as
LQC [11, 32]. There are several arguments justifying the factor $N^{-1/3}$ in
(3.10). First, consistency requires that the Poisson brackets of the effective
cosmological quantities reduce to the ones of classical general relativity in
the limit of low curvature [68], which fixes this factor. One can also
motivate it heuristically by thinking of a large “fiducial cell” encompassing
part of the universe one is interested in, itself composed of $N$ elementary
cells. One would associate the cosmological area $a^{2}V_{0}^{2/3}$ to one of
the sides of the fiducial cell. The operator $\widehat{A}_{I}$, however, takes
into account all elementary cells, and hence overcounts by a factor $N^{1/3}$,
the number of “layers” within the fiducial cell. Similar ideas have been used
in LQC, where in order to connect lattice models and symmetry-reduced models
one defines averages of fluxes over a lattice, rather than using a total flux
(see, e.g., Section 3.6.1 in [34]). Also in that context, the number of layers
to average over would be $N^{1/3}$. The factor $N^{1/3}$ is also consistent
with a similar construction in [1], also rooted in LQC. All this suggests
(3.10) as the most natural and useful definition of a cosmological scale
factor $a$ from the “total area”.

The study of inhomogeneities in GFT condensates is still in its infancy, and
different ideas for their study have been put forward. One approach presumes
that the quantum notion of wavefunction homogeneity does not necessarily lead
to a universe that would be observed to be exactly homogeneous. Indeed, in any
scenario which is not a truncation to homogeneity but an embedding of
homogeneous geometries into a more general framework, one expects
inhomogeneities to be present as quantum fluctuations, as restricting
fluctuations over homogeneity to be exactly zero for both metric and
connection would seem to violate the Heisenberg uncertainty relations. A prime
example of this is inflation, where in the early universe quantum fluctuations
undergo a transition to classical inhomogeneities, visible as temperature
perturbations in the cosmic microwave background (CMB) today [89], in spite of
the initial state being homogeneous. Put differently, condensate states have
nonvanishing overlap with generic other states in the Fock space, in
particular states that one would associate with generic inhomogeneous
universes.

The mean-field approximation provides a direct way of implementing a
transition from quantum to classical inhomogeneities for quantum geometry as
represented by a GFT condensate [68]. Indeed, as discussed in Section 3.2, in
this mean-field approximation the “condensate wavefunction” represents a
classical field configuration – in the case of a real Bose–Einstein condensate
it defines the density profile for a fluid in the laboratory. In this sense,
in (3.3) the field $\sigma(g_{I})$ represents a classical field configuration
on the “connection minisuperspace” of group elements $g_{I}$, and its
noncommutative Fourier transform $\tilde{\sigma}(B_{I})$ (cf. (3.8)) a
classical field on the “metric minisuperspace” of bivectors $B_{I}$. For any
subset $C$ of such a minisuperspace, $\int_{C}|\sigma|^{2}$ counts the
expected number of “patches” with metric/connection data in $C$; in this
approximation, all “patches” have the same probability distribution of
geometric data but they need not all be in exactly the same configuration when
“measured” at a given moment. One distinguishes between wavefunction
homogeneity as introduced in Section 3.1 and statistical, classical
homogeneity, and is led to a picture of a “many-patch model” with a
statistical distribution of inhomogeneities, just as is observed in the CMB,
see Fig. 5. Some information about this statistical distribution can be
extracted from expectation values of more complicated operators such as

$\displaystyle\widehat{\alpha}_{I}=\kappa^{2}\int\mathrm{d}g\,\hat{\varphi}^{\dagger}(g_{J})(-\Delta_{g_{I}})\hat{\varphi}(g_{J}),$

by comparing them with the area $\langle\widehat{A}_{I}\rangle$. As argued in
[68], in the case of an isotropic geometry, and assuming a single scalar
perturbation $\psi(\vec{x})$, one can approximately identify

$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}x\,|\psi(\vec{x})|^{2}\approx\frac{V_{0}}{4}\left(\frac{\langle\widehat{\alpha}_{I}\rangle
N}{\langle\widehat{A}_{I}\rangle^{2}}-1\right),$ (3.11)

which reduces to zero in the exactly (quantum) homogeneous case where
$\langle\widehat{\alpha}_{I}\rangle N=\langle\widehat{A}_{I}\rangle^{2}$.

Hence, even without any information about the embedding of the patches into
space, limited information about perturbations can be extracted. The results
of [68] outline the potential for a systematic analysis of classical
inhomogeneities emerging from quantum fluctuations in the mean-field
approximation. For more complicated condensate states, the precise relation is
much less clear, and deserves further work.

Figure 5: One of the main goals of research on GFT condensates.

An essentially orthogonal point of view, though not incompatible with the
previous one, is that inhomogeneities should be included as fluctuations over
the exact condensate, i.e., as phonons [72], or more simply as generalisations
of the state (3.3) [69]. While looking at such perturbed condensates is no
problem in principle, one faces the problem of interpreting the background-
independent geometric data associated to the perturbation, given by group or
Lie algebra elements, in terms of continuum fields on a manifold, where the
manifold and the homogeneous background itself should be formed by the
unperturbed condensate. One expects some similarities with the
Born–Oppenheimer approximation in usual quantum cosmology (see, e.g., [87,
88]) in the separation of degrees of freedom into homogeneous and
inhomogeneous parts, where the “background” is formed by a semiclassical
wavefunction for the homogeneous mode. An interesting result of [69] was that
the ratio of the number of quanta in the homogeneous background and in the
fluctuation over it provides a natural small parameter not present in the
continuum. However, what is missing so far is a notion of separating
perturbation modes by “wavenumber”, i.e., a Laplacian or spectral
decomposition of modes with respect to the background geometry (presumably
unrelated to the Laplacian on the group $G$ used in studies of GFT
renormalisation [46, 47, 48]). The perturbations considered in [69] appear to
be merely perturbations in the homogeneous mode, not qualitatively different
from the unperturbed condensate.

###  3.4 Discreteness and continuum limit

The geometries defined by $N$-particle states in GFT or equivalently by a
fixed graph in LQG are clearly discrete. In order for them to admit even an
approximate continuum interpretation, $N$ must be very large, and the
curvature should be very small when compared to the length scale of a
tetrahedron. So far, there is no clear definition of a curvature operator in
the GFT setting, which would be needed, through an analysis of its expectation
values, to make this statement fully rigorous. In the more specific setting of
condensate states, it is possible to still make this idea more precise by only
referring to homogeneous degrees of freedom, as has been done in [72]; for
instance, a near-flatness condition can be imposed by demanding that the
average value of a function $\chi(g_{i}g_{4}^{-1}g_{j}g_{4}^{-1})$, where
$\chi$ is a character on the group, e.g., the trace in the $j=\frac{1}{2}$
representation for $G=\mathrm{SU}(2)$, is very close to $\chi(e)$ for all
$i,j$. For a more detailed analysis, in a specific example, of condensate
wavefunctions that correspond to such configurations of nearly flat
tetrahedra, and hence admit an approximate continuum picture, see [67].

As we have mentioned, $N\rightarrow\infty$ corresponds to a continuum limit,
which has to be understood as a phase transition, as it has been shown in
great detail in the case of matrix models for two-dimensional gravity [51].
For a phase transition to happen, the coupling constants of the GFT action
will have to be tuned to critical values (if they exist) at which the
expectation value of the number of quanta contained in the physical vacuum
diverges. Phase transitions are only strictly possible for an infinite number
of degrees of freedom, which means one is often mainly interested in the limit
$N\rightarrow\infty$. Simultaneously, this “infinite volume” limit has to be
accompanied by the divergence of correlation lengths of matter fields used to
probe spacetime geometry. It is only in the continuum limit that symmetries
such as diffeomorphism symmetry can be restored [54].

On the other hand, one of the central results of LQG has been the realisation
that, at least at the kinematical level, the spectra of area and volume
operators are discrete [126], although whether this implies discreteness for
physical states is far from clear [58]. If there is a physical discreteness in
the theory, we would be in a situation similar to causal sets [82] where there
is no continuum limit but only a “continuum approximation”. A GFT condensate
describing a universe similar to ours would then have an extremely large but
still finite number of degrees of freedom $N$, which for practical purposes
may still be approximated as infinite.

The formalism of GFT condensates for cosmology, in the way it has been set up
so far, essentially requires $N$ to be finite. First of all, this is because
the limit $N\rightarrow\infty$ corresponds to a change of representation to a
different Hilbert space, and while such a change may be highly desirable
eventually to describe a phase transition, we are performing all calculations
in the original Fock space around the perturbative GFT vacuum $\varphi=0$.
Furthermore, quantities of cosmological interest such as the scale factor $a$
in (3.10) are defined using $N$, and become ill-defined in the
$N\rightarrow\infty$ limit; we are defining approximate continuum quantities
from the discrete variables of GFT, before any double-scaling limit of the
usual type (discretisation scales go to zero, quantities of physical interest
remain finite) is performed. In order to extend these constructions to the
limiting case where the actual continuum limit is taken, a better
understanding of this limit in GFT and the change in Fock space representation
will be required. This said, it is also true that coherent states, even if
they have a finite expectation value of the number operator, have nonvanishing
projection over states with arbitrarily large number of quanta. Therefore,
these approximate states are taking into account the idea of a continuum
limit, even if in this limited sense. This is perhaps the most significant
difference with respect to states built with a given number of degrees of
freedom attached to a fixed graph.

Cosmologies based on a universe with a finite number of discrete degrees of
freedom can have interesting properties of immediate phenomenological
relevance. Perhaps the most striking is a proposal in the causal set approach
[130], in which the smallness of the cosmological constant $\Lambda$ is
conjectured to be a statistical effect arising from a very large but finite
number of constituents of the universe. One expects fluctuations in $\Lambda$
to be of the order $\Lambda\sim 1/\sqrt{N}$, and if $N$ is taken to be the
number of Planck volumes in a Hubble volume this turns out to be roughly the
observed value (this property is often considered part of the cosmological
constant problem, the “coincidence problem”). While more of a general argument
than a strict derivation, this idea shows how properties of the universe at
the largest scales might depend on a fundamental discreteness of spacetime.
More speculative ideas include attempts to derive deviations from scale
invariance in the CMB power spectrum from fundamental discreteness [59].

The formalism of LQC is based on fundamental discreteness in an essential way.
Indeed, its departures from the classical continuum Friedmann equation or
Wheeler–DeWitt quantum cosmology arise from using the input of a non-zero
minimal area from the kinematics of LQG [11, 32]. In the derivation of the
modifications to the continuum dynamics, the picture of a cosmological
universe as defined on a large but fixed graph is used rather explicitly,
e.g., in [9], without using a full LQG calculation. A crucial question in this
context is how the (locally finite) number of degrees of freedom evolves with
the expansion of the universe. The viewpoint emerging from [9] was that the
volume of (a comoving portion of) the universe and the number of fundamental
degrees of freedom should essentially be proportional, so that the volume per
elementary cell is a constant (given by its minimal, Planckian value). More
generally, LQC suggests a lattice refinement interpretation in which both
quantities could evolve as the graph is refined in the evolution of the
universe [36]. We will see later on (Section 4) that the picture of lattice
refinement and the resulting form of holonomy corrections in LQC can be
grounded in a natural way in the formalism of GFT condensates.

The picture of a graph to which new degrees of freedom are continuously added
as the universe evolves leads to many open questions. One needs to ensure that
any notion of homogeneity one wants to apply is preserved under such an
evolution, which itself is fundamentally irreversible. For some work on this
type of quantum dynamics, using Hilbert spaces of evolving dimension, see
[83]. The open issue of how lattice refinement, as apparently required in LQC,
is implemented dynamically in LQG provides additional motivation for the study
of GFT condensates and their effective dynamics, which may provide a link
between LQG and LQC by dynamically realising lattice refinement. Note that
lattice refinement requires a dynamically changing number of degrees of
freedom $N$, which indicates that working on a fixed graph in LQG may not be
the appropriate setting for cosmology.

###  3.5 Generalised condensates and generalisation to spherical symmetry

As discussed in Section 3.1, the simplest condensate states of the form (3.3)
or (3.4), while having attractive mathematical properties and a clear
motivation by analogy with Bose–Einstein condensates, have shortcomings in
their interpretation in terms of quantum geometry. In particular, we have seen
that the absence of a connected graph means that there is no information about
the topology of space, and that it is difficult to localise excitations, as
seems required if one wants to explicitly add perturbations. This suggests
defining generalised condensates on a nontrivial, connected graph. Here we
review the essential ideas and technical points of this definition; full
details can be found in [106].

Generalised condensates, just as the simple condensates of Section 3.1, are
based on the idea of wavefunction homogeneity, i.e., the property of a many-
particle state to be fully determined by a single-particle wavefunction.
Denote this wavefunction again by $\sigma$ and define operators

$\displaystyle\hat{\sigma}(h_{I})=\int\mathrm{d}g\,\sigma(h_{I}g_{I})\hat{\varphi}(g_{I}),\qquad\hat{\sigma}^{\dagger}(h_{I})=\int\mathrm{d}g\,\bar{\sigma}(h_{I}g_{I})\hat{\varphi}^{\dagger}(g_{I}).$

Now fix a graph $\Gamma$ which encodes the desired topology, in the sense that
it corresponds to the dual of a triangulation of a manifold of this topology,
and define the “seed state”

$\displaystyle|{\rm
seed}\rangle\propto\int(\mathrm{d}h)^{\\#V(\Gamma)}\prod_{e\in
E(\Gamma)}\delta\big{(}h^{v}_{I}(h^{w}_{J})^{-1}\big{)}\prod_{u\in
V(\Gamma)}\hat{\sigma}^{\dagger}(h^{u}_{I})|0\rangle,$

where $V(\Gamma)$ denotes the vertices in $\Gamma$ and $E(\Gamma)$ the edges
of $\Gamma$; again we are using a slight shorthand notation where each edge
$e$ is characterised as outgoing as the $I$-th link from the vertex $v$ and
incoming as the $J$-th link into the vertex $w$.

The “seed state” is of the type often used in LQG, a state on a fixed graph,
with a small number of degrees of freedom with respect to the
Ashtekar–Lewandowski vacuum. To define from it a type of condensate
representing some sort of continuum limit, and to incorporate a notion of
coarse graining related to ideas such as [56], one now defines refinement
operators which preserve the topology of the graph and the notion of
wavefunction homogeneity. Both properties can be implemented by basing the
refinement operators on Pachner moves [114], and by defining them using the
wavefunction $\sigma$ that was already used for the seed state. For instance,
in order to implement a $1\rightarrow 4$ Pachner move one should find an
operator $\widehat{\mathcal{M}}$ satisfying [106]

$\displaystyle\big{[}\widehat{\mathcal{M}},\hat{\sigma}^{\dagger}(h_{1},h_{2},h_{3},h_{4})\big{]}=\int_{G^{6}}\mathrm{d}g\,\left(\hat{\sigma}^{\dagger}(h_{4},g_{1},g_{2},g_{3})\,\hat{\sigma}^{\dagger}(g_{3},h_{3},g_{4},g_{5})\right.$
$\displaystyle\left.\hphantom{\big{[}\widehat{\mathcal{M}},\hat{\sigma}^{\dagger}(h_{1},h_{2},h_{3},h_{4})\big{]}=}{}\times\hat{\sigma}^{\dagger}(g_{5},g_{2},h_{2},g_{6})\hat{\sigma}^{\dagger}(g_{6},g_{4},g_{1},h_{1})\right),$

corresponding to the addition of three new vertices and six new edges to the
graph; in this move, a tetrahedron is replaced by a more refined triangulation
of a 3-ball, homeomorphic to the tetrahedron that was removed. Crucially, the
action of $\widehat{\mathcal{M}}$ does not create disconnected components but
only refines the given graph.

Once such an operator has been found, one can define generalised condensate
states

$\displaystyle|\Psi_{F}\rangle=F\big{(}\widehat{\mathcal{M}}\big{)}|{\rm
seed}\rangle$ (3.12)

for some function $F$. A natural choice motivated by technical simplicity and
the analogy to the simple condensates of Section 3.1 would be an exponential.
Even for this choice, expectation values for operators of geometric interest
are quite difficult to compute. An interesting general result shown in [106]
is that, for a certain class of refinement moves and initial seed states,
expectation values of one-body operators of the general form

$\displaystyle\widehat{\mathcal{A}}=\int\mathrm{d}g\,\mathrm{d}h\,A(g_{I},h_{I})\hat{\varphi}^{\dagger}(g_{I})\hat{\varphi}(h_{I})$

are extensive, i.e., reduce to a product of the average particle number
$N=\langle\widehat{N}\rangle$ and a single-particle expectation value given in
terms of the wavefunction $\sigma$. This result is important for cosmology,
where the scaling of observables with $N$ is a main property of interest.

The construction of generalised condensates leads in a rather straightforward
way to a notion of spherical symmetry. The seed state can be chosen to
correspond to a topology $S^{2}\times[0,1]$, interpreted as a spherical shell,
where one needs to define different types of vertices that keep track of the
inner boundary, outer boundary, and interior of the shell [106]. Gluing many
such shells together using the inner boundary of one and the outer boundary of
the next one, one obtains a more general configuration of spherical symmetry
with nontrivial dependence along the radial direction, given by different
wavefunctions $\sigma_{r}$ where $r$ labels the different shells.

For the construction of generalised condensates, it has proven advantageous to
make use of tools developed in the context of coloured tensor models (see,
e.g., [80, Section 4] for a concise presentation of the relevant procedures).
The simple GFT models based on a single field are replaced with coloured
models, possessing additional labels that facilitate the reconstruction of the
topological structure of the boundary state, given the pattern of convolutions
of the wavefunction. While the additional labels do not have an obvious
counterpart in LQG, they are useful to deal with the problem of isolating the
various contributions to quantities computed for families of different graphs.
As the particular colouring introduced requires bipartite graphs, Pachner
moves have to be replaced with _melonic refinement moves_. These refinement
moves allow a better handling of the construction of a refined foliation,
keeping fixed the topology. These small extensions are only required to make
the construction easier to be dealt with in concrete cases, and do not change
the substance of the general principles that we have discussed.

A first very interesting application of this formalism has been given in [107]
where it was shown how to define a horizon entanglement entropy for such
generalised condensates. There, thanks to the techniques developed in the
second quantised formulation of GFT and of tensor models, it has been possible
to get a first hint of the effect of superposing different graphs, a crucial
aspect of the condensate picture, on a key physical quantity for semiclassical
gravity, namely the entanglement entropy of subregions. It was shown that the
entanglement entropy associated to an isolated horizon [52], defined in terms
of GFT condensates, receives a (combinatorial) Boltzmann entropy contribution
from the structure of the state in terms of its graph components, as the graph
proliferation associated to the refinement moves involved has the correct
scaling with the number of tetrahedra in the state. This feature of the
generalised coherent states, in turn, gives rise to an area law with
subleading logarithmic corrections, as expected from semiclassical gravity.
This result already underscores the usefulness of GFT condensates and tensor
model techniques for calculations that address the physics of many GFT degrees
of freedom.

##  4 GFT condensates: dynamics

In Section 3, we described the construction of GFT condensate states at the
kinematical level, and given their possible interpretation in terms of
geometry and cosmology. The next step is to study the effective dynamics of
such states, as extracted from the quantum dynamics of a given GFT model or
class of GFT models. The restriction to a class of states such as (3.3) or
(3.12) can be sufficient for obtaining manageable equations, but often further
approximations are needed. Showing that there is a relation between the
resulting effective dynamics and the dynamics of cosmological spacetimes in
general relativity, such as the Friedmann equation for FRW backgrounds or
dynamical equations for linearised perturbations, would not only give further
support for the geometric interpretation of the abstract group-theoretic data
appearing in GFT that we have put forward, but more importantly also give
evidence that a certain GFT model, or class of models, can become serious
contenders for a theory of quantum gravity in four dimensions. In this vein,
one can hope to use GFT condensates to discriminate between different models,
by restricting only to those that lead to an acceptable cosmological
phenomenology. Ideally, a consistent picture of geometrogenesis would not only
lead to a resolution of the Big Bang singularity in a consistent quantum
theory, but also obviate the need for introducing inflationary models with
their known theoretical problems [85].

As we will show in this section, the basic strategy for deriving effective
dynamics for condensates from the fundamental GFT dynamics is rather
straightforward, and in many ways analogous to how one proceeds in the case of
Bose–Einstein condensates in condensed matter physics. We will give some
detail on the resulting equations, while trying to be as general as possible,
given the multitude of models in the GFT literature.

It has proven more difficult to identify interesting solutions to the
resulting equations and to map them to an effective classical or semiclassical
cosmological description. This is mainly because the resulting equations are
almost always nonlinear and nonlocal with a peculiar type of nonlocality, and
not many analytical solution techniques for such equations are known.
Numerical techniques, while certainly a possible way forward, have not been
employed much, and most of the work so far has focused on general and
foundational aspects of the formalism.

###  4.1 Gross–Pitaevskii equation as effective dynamics of a Bose condensate

In the interest of making this review self-contained for readers who are not
familiar with quantum field theory techniques in condensed matter physics, let
us briefly review how the Gross–Pitaevskii equation, a certain type of
“nonlinear Schrödinger equation”, emerges from the effective dynamics of a
quantum field theory for the atoms of a Bose–Einstein condensate.

In the simplest example, this quantum field theory can be defined by an action

$\displaystyle
S[\Psi,\bar{\Psi}]=\int_{\mathbb{R}^{3}\times\mathbb{R}}\\!\mathrm{d}x\,\mathrm{d}t\left\\{\frac{{\rm
i}\hbar}{2}\left[\bar{\Psi}\frac{\partial\Psi}{\partial
t}-\Psi\frac{\partial\bar{\Psi}}{\partial
t}\right]-\left(\frac{\hbar^{2}}{2m}|\vec{\nabla}\Psi|^{2}-\mu|\Psi|^{2}+V(\vec{x})|\Psi|^{2}+\frac{\lambda}{2}|\Psi|^{4}\right)\\!\right\\},$

where the scalar field $\Psi$ and its complex conjugate $\bar{\Psi}$,
functions on non-relativistic flat spacetime $\mathbb{R}^{3}\times\mathbb{R}$,
are conventionally treated as independent in the variation (this is basically
equivalent to splitting the complex field $\Psi$ into two real components).
Here $\mu$ represents a chemical potential and $\lambda$ is a coupling
representing the (weak) interactions between atoms. We assume that the atoms
have zero spin and that there is only a single species, but generalisations to
spin degrees of freedom and to several species are straightforward. More
importantly, it is assumed that the condensate is examined in a dilute limit,
in which the interatomic distance is larger than the range of the
interactions. This action should be seen as the first approximation of a more
complicated theory which will include more terms describing all the possible
interaction channels between the atoms. It is an effective field theory for
the order parameter controlling a phase transition, whose regime of validity
has to be checked a posteriori. However, it is also a generic prototype for a
variety of different situations in which a weak coupling approximation,
related to the low density of the fluid, can be applied.

Because the action is real, variations with respect to $\Psi$ and $\bar{\Psi}$
lead to only one independent complex equation, known as the Gross–Pitaevskii
equation,

$\displaystyle{\rm i}\hbar\frac{\partial\Psi}{\partial
t}=-\frac{\hbar^{2}}{2m}\Delta\Psi-\mu\Psi+V(\vec{x})\Psi+\lambda|\Psi|^{2}\Psi.$
(4.1)

This is the classical field equation for a complex field $\Psi$, nonlinear in
$\Psi$ due to the last term.

Proceeding with canonical quantisation, the first term in the action tells us
that $\Psi$ and $\bar{\Psi}$ are canonically conjugate, and they should
satisfy the Heisenberg algebra as operators, analogous to the GFT commutation
relations (2.4). A Fock space is constructed in which $\hat{\Psi}(\vec{x})$ is
a sum of annihilation operators and $\hat{\Psi}^{\dagger}(\vec{x})$ is a sum
of creation operators with respect to a Fock vacuum $|0\rangle$. The
excitations created by the ladder operators represent atoms.

The quantum dynamics given by the operator version of (4.1) is nonlinear and
explicit solutions are hard to obtain in general. To proceed, one uses, in the
simplest case, a mean-field approximation in which the operators are replaced
by classical fields, i.e.,

$\displaystyle\hat{\Psi}(\vec{x})=\psi(\vec{x}){\bf
1}+\hat{\chi}(\vec{x}),\qquad\hat{\Psi}^{\dagger}(\vec{x})=\bar{\psi}(\vec{x}){\bf