The microscopic dynamics of quantum space as a group field theory

The microscopic dynamics of quantum space as a group field theory

Daniele Oriti Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
Am Mühlenberg 1 D-14476 Golm, Germany, EU

We provide a rather extended introduction to the group field theory approach to quantum gravity, and the main ideas behind it. We present in some detail the GFT quantization of 3d Riemannian gravity, and discuss briefly the current status of the 4-dimensional extensions of this construction. We also briefly report on some recent results obtained in this approach, concerning both the mathematical definition of GFT models as bona fide field theories, and possible avenues towards extracting interesting physics from them.

quantum gravity, loop quantum gravity, spin foam models, matrix models, non-commutative geometry, simplicial quantum gravity

I Introduction and basic ingredients

The field of non-perturbative and background independent quantum gravity has progressed considerably over the past few decades libro (). New research directions are being explored, new important developments are taking place in existing approaches, and some of these approaches are converging to one another. As a result, ideas and tools from one become relevant to another, and trigger further progress. The group field theory (GFT) formalism laurentgft (); iogft (); iogft2 (); VincentRenorm () nicely captures this convergence of approaches and ideas. It is a generalization of the much studied matrix models for 2d quantum gravity and string theory mm (). At the same time, it generalizes them, as we are going to explain, by incorporating the insights coming from canonical loop quantum gravity and its covariant spin foam formulation, and so it became an important part of this approach to the quantization of 4d gravity carlo (); review (); alex (); thesis (). Furthermore, it is a point of convergence of the same loop quantum gravity approach and of simplicial quantum gravity approaches, like quantum Regge calculus and dynamical triangulations iogft (), in that the covariant dynamics of the first takes the form, as we are going to see, of simplicial path integrals. More recently, tools and ideas from non-commutative geometry have been introduced as well in the formalism, and this has helped attempts to make tentative contact with effective models and quantum gravity phenomenology.

The goals of this paper are to explain the general idea behind the GFT formalism and its roots, discuss its relation with other current approaches to quantum gravity, detail to some extent the construction and features of GFT models in 3 and 4 dimensions, and finally report briefly on some recent results, concerning both an improved mathematical understanding of it and results with possible bearing on phenomenology. We do not intend to provide an up-to-date review or a status report of the subject, which has progressed enormously in the last few years, and we will refer instead to the literature for most of these recent developments. The models we will discuss in some more detail are in Euclidean signature, but the whole construction can be performed in the Lorentzian setting as well111Doing so does not require a modification of the general formalism, but only of some ingredients, and of course much additional care to the mathematical issues of dealing with non-compact groups..

i.1 A general definition

In very general terms, group field theories are an attempt to define quantum gravity in terms of combinatorially non-local quantum field theories on group manifolds or on the corresponding Lie algebras, related to the Lorentz or rotation group. The formalism itself is mostly characterized by the combinatorial non-locality that we are going to specify in the following, and the choice of group manifolds as domain of definition of the field, and in particular of the Lorentz/rotation groups, is dictated by the wish to model quantum gravity. Other choices could be devised easily, for different purposes (e.g. describing matter or gauge fields, capturing topological structures, etc). From this point of view, group field theories are a special case of tensor models tensor (), corresponding to a special choice of domain space for the fundamental tensors and, possibly, peculiar symmetries. We stick in fact to quantum gravity models in this contribution. Before introducing the formalism or specific models, let us motivate in some detail these basic ingredients: a quantum field theory framework, the use of group structures, the combinatorial non-locality.

i.2 A quantum field theory for quantum gravity?

It is actually easy to see why we may want to use a quantum field theory formalism (QFT) also for quantum gravity. Quantum field theory is the best formalism we have for describing physics at both microscopic and mesoscopic scales, both in high energy particle physics and in condensed matter physics, for both elementary systems and many-particle ones. And actually, even at large and very large scales, it is still field theory that we use, only we are most often able to neglect quantum aspects (e.g. General Relativity itself).

So, probably, a more relevant question is: can we still hope for a formulation of quantum gravity, for description of the microscopic structure of quantum space, in terms of a quantum field theory? The rationale behind such question is the following. We only know how to define quantum field theories on given background manifolds and we have under full control (including renormalization etc), only quantum field theories on flat spaces. Moreover, we have already tried to apply this formalism to gravity, formulating it as a quantum field theory of massless spin-2 particle propagating on a flat space, the gravitons, thought of as the carriers of the gravitational interaction, coupled universally to other matter and gauge fields. It has been the first approach to quantum gravity ever developed, by the great scientists of the past century carlohistory (). We know it does not work. The field theory defined by this approximation is not renormalizable. Quantum gravity, beyond the effective field theory level, is not such a quantum field theory.

This historically well founded objection, however, is not a no-go theorem, of course. In particular, the non-renormalizability result does not rule out the use of a quantum field theory formalism as such. What it rules out is the specific idea of a field theory of gravitons as a fundamental definition of the dynamics of quantum space. To the eyes of many, it rules out also the idea that the requirement of background independence with respect to spacetime is a dispensable one, supporting instead the belief that this should be the defining property of any sound quantum gravity formalism carlodiffeo (); leeback (), and possibly manifestly realized222In fact, even a theory formulated perturbatively around a given background can be background independent, if the physical observables computed from it turn out to be independent of the background used to compute them. However, this is not easy to realize and to verify explicitly, so the requirement of manifest background independence becomes more than just an aesthetic choice..

The more serious objection to the idea of using a quantum field theory formalism for quantum gravity, in fact, is that a good theory of quantum gravity should be background-independent, because it should explain origin and properties of spacetime itself, of its geometry and, maybe, its topology. But, as said, we know how to formulate quantum field theories only on given backgrounds.

Again, this does not rule out the use of the QFT formalism. It means however, that, if an (almost) ordinary QFT it should be, quantum gravity can only be a QFT on some auxiliary, internal or “higher-level”space333An alternative to this conclusion is represented by the Asymptotic Safety program asympsafety (). Although clearly based on a very different starting point and a different language, this program is not in contradiction with the GFT approach. We are however not going to discuss their possible relations, which would lead us astray from the focus of this paper..

So we can then look at General Relativity (GR) itself and try to identify some background (non-dynamical) structures that are already present in it and could provide or characterize such space. After all, GR is a classical field theory, and it is background independent with respect to the geometry of spacetime, which is the minimum of what we may want to have as dynamical quantum degrees of freedom in a quantum gravity theory.

i.3 Background structures in classical and quantum GR

The first background structure that comes to mind is the spacetime dimension. We have of course overwhelming experimental evidence for a 4-dimensional spacetime. But it is also true that we do not have a clear enough understanding of why this dimensionality should hold true at high energy, microscopic length scales or when all quantum effects of space dynamics are taken into account. So, it makes sense to look for alternatives, i.e. for a formalism in which the spacetime dimension is dynamical. Group field theories (as loop quantum gravity or simplicial quantum gravity and tensor models) fix the kinematical dimension of space at the very beginning, at least in the present formulation. However, on the one hand these approaches are not subjected to any obstruction to dimensional generalization; on the other hand this kinematical choice does not ensure that the dynamic dimension of spacetime in some effective continuum and classical description will match the kinematical one. The example of dynamical triangulations DT (), in fact, show that this matching is far from trivial, and that its achievement can be considered in fact an important result.

Another background structure is the internal, local symmetry group of the theory, i.e. the Lorentz group, which provides the local invariance under change of reference (tetrad) frame, and that is at the heart of the equivalence principle. It is a sort of “background internal space ”of the theory. This gives the primary conceptual motivation for using the Lorentz group (and related) in GFT. At the same time, as we are going to discuss in the following, this choice allows to incorporate in the GFT formalism what we have learned from canonical loop quantum gravity, as well as many of its results carlo (); thomas (). In fact, the role of the Lorentz group (and of its rotation subgroup) is brought to the forefront in the LQG formalism (see chapter by H. Sahlmann), which is based on the classical reformulation of GR as a background independent (and diffeomorphism invariant) theory of a Lorenz connection. Upon quantization, it gives a space of states which is an space of generalized connections, obtained as the projective limit of the space of cylindrical functions of finite numbers of group elements, representing parallel transports of the same connection along elementary paths in space. Thus spacetime geometry is encoded, ultimately, in these group-theoretic structures.

A background structure of GR is, in fact, also its configuration space, seen from the Hamiltonian perspective, and regardless of the specific variables used to parametrize it: the space of (spatial) geometries on a given (spatial) topology, coined “superspace”by Wheeler. In the ADM variables, this is a metric space in its own right giulini () and could be considered indeed a sort of “background meta-space: a space of spaces”. Let us sketch briefly how this background structure enters the quantization of the classical theory, at least at the formal level, in both canonical and covariant approaches. Loop quantum gravity, spin foam models and simplicial quantum gravity reformulate and make more precise and successful, to different degrees and in different ways, these ’historic’ approaches.

The canonical approach starts with a globally hyperbolic spacetime, with topology . For simplicity, is usually chosen to be compact and simply connected, with the topology of the 3-sphere . The wave function of canonical geometrodynamics are functionals on the space of 3-metrics on the 3-sphere, , and kinematical observables are functionals of the phase space variables, themselves built from the conjugate 3-metric and extrinsic curvature , turned into (differential) operators acting on the wave functions. Gravity being (classically) a totally constrained system, the dynamics is imposed by identifying the space of states (and associated inner product), i.e. the space of functionals on the space of 3-geometries (metrics up to spatial diffeomorphisms), which satisfy also the Hamiltonian constraint, and thus the Wheeler-deWitt equation . This, together with the identification of physical observables, defines the theory from a canonical perspective. A covariant definition of the dynamics can instead be looked for in sum-over-histories framework. Given the same (trivial) spacetime topology, and the same choice of spatial topology, consider all the possible geometries (spacetime metrics up to diffeomorphisms) that are compatible with it. Transition amplitudes (defining either the physical inner product of the canonical theory or ‘causal’ transition amplitudes, and thus Green functions for the canonical Hamiltonian constraint operator teitelboim ()), for given boundary configurations of the field (i.e. possible 3-geometries on the 3-sphere): and , would be given by a sum over spacetime geometries like:


i.e. by summing over all 4-geometries inducing the given 3-geometries on the boundary, with the amplitude modified by boundary terms if needed. The expression above is obviously purely formal, for a variety of well-known reasons. In any case, it looks like a prototype of a background independent quantization of spacetime geometry, for given spatial and spacetime topology, and given space of possible 3-geometries. Also, the physical interpretation of the above quantities presents several challenges, given that the formalism seems to be bound to a cosmological setting, where our usual interpretation of quantum mechanics is rather dubious. A good point about group field theory, and about LQG, spin foams and simplicial gravity, is that they seem to provide a more rigorous definition of the above formula, which is also ılocal in a sense to be clarified below.

i.4 Making topology dynamical: the idea of 3rd quantization

Making sense of a path integral quantization of gravity on a fixed spacetime is difficult and ambitious enough. However, one may wish to treat also topology as a dynamical variable in the theory, and try to implement a sort of “sum over topologies”alongside a sum over geometries, thus extending the latter to run over all possible spacetime geometries and not only over those which can live on a given topology. This has consequences on the type of geometries one can consider, in the Lorentzian case, given that a non-trivial spacetime topology implies spatial topology change fay () and this in turn forces the metric to allow either for closed timelike loops or for geometries which are degenerate (at least) at isolated points. This argument was made stronger by Horowitz horo () concluding that if degenerate metrics are included in the (quantum) theory, then topology change is not only possible but unavoidable and non-trivial topologies therefore must be included in the quantum theory. There are several other results that suggest the need for topology change in quantum gravity, including work on topological geons fayrafael (), in string theory greene (), and on wormholes as a possible explanatory mechanism for the small value of the cosmological constant banks (). Moreover, the possibility has been raised that all constants of nature can be seen as computable dynamical vacuum expectation values in a theory in which topology change is allowed coleman ().

All this, together with the analogy with string perturbation theory and the aim to solve some problems of the canonical formulation of quantum gravity, prompted the proposal of a “third quantization” formalism for quantum gravity kuchar (); giddingsstrominger (); guigan (). The general idea is to define a (scalar) field theory on superspace for a given choice of basic spatial manifold topology, e.g. the 3-sphere. This means turning the wave function of the canonical theory into an operator (acting on a new Hilbert space) , whose dynamics is defined by an action of the type:


with being the Wheeler-DeWitt differential operator of canonical gravity here defining the kinetic term (free propagation) of the theory, while is a generic, e.g. cubic, and generically non-local (in superspace) interaction term for the field, governing the topology changing processes. Notice that because of the choice of basic spatial topology needed to define the 3rd quantized field, the topology changing processes described here are those turning copies of the 3-sphere into copies of the same.

The quantum theory is “defined” by the partition function , in its perturbative expansion in “Feynman diagrams”:

Figure 1: The perturbative expansion of the 3rd quantized field theory in interaction processes for universes

The Feynman amplitudes will be given by the quantum gravity path integral (sum over geometries) for each spacetime topology (identified with a particular interaction process of universes). The one for trivial topology will represent a sort of one particle propagator, thus a Green function for the Wheeler-DeWitt equation444Note that this is in analogy with what happens in ordinary quantum field theory of point particles. Here the Feynman diagrams represent possible interaction processes of a certain number of (virtual) particles, and the Feynman amplitudes can be written in the form of sum over histories for the particles involved in these processes..

Other features of this (very) formal setting are: 1) the full classical equations of motions for the scalar field on superspace will be a non-linear extension of the Wheeler-DeWitt equation of canonical gravity, due to the interaction term in the action, i.e. of the inclusion of topology change; 2) the perturbative 3rd quantized vacuum of the theory will be the “no spacetime” state, and not any state with a semiclassical geometric interpretation in terms of a smooth geometry, say a Minkowski state.

We will see that the group field theory approach shares these general features, on top of proposing a new context to realize them 3rd ().

Notice that in this formalism for spacetime topology change, the spatial topology of each single-universe sector remains fixed, and the superspace itself remains a background structure for the “3rd quantized”field theory.

Notice also that, if one was to attempt to reproduce this type of setting in terms of the variables used in LQG, two background structures of classical GR, i.e. the Lorentz group and the superspace would be somehow unified, as superspace would have to be identified with the space of (generalized) Lorentz (or ) connections on some given spatial topology.

Something of this sort happens in group field theory, which can be seen as a sort of discrete non-local field theory on the space of geometries for building blocks of space. in turn given by group or Lie algebra variables. Before getting to the details of the GFT formalism, however, we want to motivate further the use of discrete structures and the associated non-locality.

i.5 A finitary substitute for continuum spacetime?

However good the idea of a path integral for gravity and its extension to a third quantized formalism may be, there has been no definite success in realizing them rigorously (beyond the minisuperspace-reduced contexts or semi-classical approximation). One is tempted to identify the main reason for the difficulties encountered to be the use of a continuum for describing spacetime, both at the topological and at the geometrical level. One can advocate the use of discrete structures as a way to regularize and make computable the above expressions, or to provide a more fundamental definition of the theory, with the continuum description emerging only in a continuum approximation of the corresponding discrete quantities, like hydrodynamics for large ensembles of many particles. This was in fact among the motivations for discrete approaches to quantum gravity as matrix models, or dynamical triangulations or quantum Regge calculus. And various arguments have been put forward to support the point of view that discrete structures provide a more fundamental description of spacetime. One possibility, suggested by various approaches to quantum gravity such as string theory or loop quantum gravity, is the existence of a fundamental length scale that sets a lower bound for distances and thus makes the notion of a continuum loose its physical meaning. Also, one can argue on both philosophical and mathematical grounds isham () that the very notion of “point”can correspond at most to an idealization of the nature of spacetime (see. e.g. non-commutative models of quantum gravity majid (); QGPhen ()). Spacetime points are indeed to be replaced, from this point of view, by small but finite regions and a more fundamental model of spacetime should take these local regions as basic building blocks. Also, the results of black hole thermodynamics seem to suggest that there should be a finite number of fundamental spacetime degrees of freedom associated to any region of spacetime, the apparent continuum being the result of the microscopic (Planckian) nature of them sorkinBH (). In other words, a finitary topological space sorkin () would constitute a better model of spacetime than a smooth manifold. These arguments also favor a simplicial description of spacetime, with the simplices being a finitary substitute of points. And these same arguments are reinforced by the results of LQG whose kinematical states are labelled by graphs thomas ().

Here is where GFTs provide a discrete or finitary implementation of the 3rd quantization idea, which is also a more ‘local’one, in the following sense. In GFT the spatial manifold is to be thought of as a collection of (glued) building blocks, akin to a many-particle state, and the field theory should be defined on the space of possible geometries of each such building block. Spatial topology is also allowed to change arbitrarily, if the building blocks are allowed to combine arbitrarily by the theory. These building blocks can be depicted as either fundamental simplices or (pieces of) finite graphs, as we will see, and the space of geometries of such discrete structures is then necessarily finite dimensional. So the basic dynamical objects whose interactions produce all possible geometries and topologies of space are fundamental constituents of a universe, as opposed to the ‘global’framework of formal 3rd quantization, where the system interacting is the whole universe itself.

So the GFT formalism incorporates insights from other approaches (loop quantum gravity and simplicial gravity) also in answering a second natural question that comes to mind when suggesting a quantum field theory formalism for the microstructure of space: a QFT of which fundamental quanta? Again, we know that these cannot be gravitons. They have to be quanta of space itself, excitations around a vacuum that corresponds to the absence of space .

To introduce how their dynamics is identified, and in the process motivate the peculiar combinatorial non-locality of GFT interactions (which is the price to pay for trying to maintain both ‘physical locality’and topology change), we take a short detour and discuss first briefly their lower-dimensional predecessors: matrix models.

i.6 A combinatorial non-locality: from point particles to extended combinatorial structures

Consider a point particle in 0+1 dimensions, with action . This action defines a trivial dynamics (for a trivial system), of course. What interests us here, however, is the combinatorial structure of its “Feynman diagrams”, i.e. the graphs that can be used as a convenient book-keeping tool in computing the corresponding partition function perturbatively in . These are simple 3-valent (because of the order of the “interaction”) graphs.

Figure 2: A Feynman graph for a point particle and the corresponding field theory

The fact that the Feynman diagrams of the theory are simple graphs like the above follows from 1) the point-like nature of the particle, and 2) the locality of the corresponding interaction, encoded in the identification of variables in the interaction part of the action. A less trivial system would be given by a relativistic particle, for example, or, better, a system of such interacting particles555Notice also that the relativistic particle is often taken as a sort of General Relativity in dimensions, and while this analogy has strong limitations (like all analogies), it is indeed very useful to grasp an intuitive understanding of several issues that show up in the (quantum) gravity context teitelboim ().. The combinatorial structure of the Feynman diagrams, now weighted by non-trivial amplitudes (convolutions of Feynman propagators for each interacting particle with identified initial/end points), will be the same as long as we do not allow for non-local interactions.

The same structure of diagrams is maintained, because the local nature of the interaction and the point-like nature of the corresponding quanta are maintained, also when moving to a field theory setting. Going from the above particle dynamics to the corresponding field theory (still dynamically rather trivial) means allowing for the creation and annihilation of particles, and then infinite number of degrees of freedom, because of the arbitrary number of particles that can be involved in any interaction process. Assume for simplicity that the dynamics is governed by the (trivial) action: . Now we have integrations over the position variables labeling the vertices, or, in momentum space, lines are labeled by momenta that sum to zero at vertices, and that are integrated over, reflecting a (potential) infinity of degrees of freedom in the theory. Still, the combinatorics of the diagrams is the same as in the corresponding particle case.

i.6.1 Matrix models and discrete and continuum 2d gravity

Now we move up in combinatorial dimension. Instead of point particles, let us consider 1-dimensional objects, that could be represented graphically by a line, with two end points. We label these two end points with two indices , and we represent these fundamental objects of our theory by matrices (with ) mm (), i.e. arrays of real or complex numbers replacing the “point”variables mm (). For simplicity, assume these matrices to be hermitian.

We want to move up in dimension also in the corresponding Feynman diagrams, i.e. we want to have diagrams that correspond to 2-dimensional structures, instead of 1-dimensional graphs. In order to do so, we have to drop the assumption of locality (that is, from a formal point of view, the simultaneous identification of all the arguments of the basic variables/fields appearing in the interaction). We define a simple action for , given by


We have also identified in the above formula the propagator and vertex term that will give the building blocks of the corresponding Feynman amplitudes. These building block can be represented graphically as follows:

The composition of these building blocks is performed by taking the trace over the indices in the kinetic and vertex term, and it represents the identification of the points labeled by the same indices, in the corresponding graphical representation. This identification of arguments is the higher-dimensional analog of the ‘locality’of usual field theory. Diagrams are then made of: (double) lines of propagation (made of two strands), non-local “vertices”of interaction (providing a re-routing of strands), faces (closed loops of strands) obtained after index contractions. The same combinatorics of indices (and thus of matrices) can be given a dual simplicial representation as well:

Therefore the Feynman diagrams used in evaluating the partition function correspond to 2-dimensional simplicial complexes of arbitrary topology, since they are obtained by arbitrary gluing of triangles (the interaction vertices) along common edges (as dictated by the propagator). Thus a discrete 2d spacetime emerges as a virtual construction, encoding the possible interaction process of fundamentally discrete quanta of space (the edges/matrices).

Figure 3: A (piece of) Feynman diagram for a matrix model, of which we give both direct and dual (simplicial) representation; the two parallel lines of propagation correspond to the two indices of the matrix; the extra line on the bottom indicates identification of the two edges of the triangles.

We can easily compute the Feynman amplitudes for the model:

where is the number of vertices of the Feynman diagram, the number of faces of the latter, and , again, the dimension of the matrices. We can then use the identity: , where are the numbers of vertices, edges and triangles of the simplicial complex dual to the Feynman diagram, is its Euler characteristics and its genus. Thus,

The same result can be obtained easily using the rescaling: , giving . Now the Feynman expansion gives a factor for each closed loop of strands (vertex of the simplicial complex), for each propagator (edge), for each vertex (triangle), to give the result above using .

We ask ourselves now what is the relation with gravity of this expansion and of the Feynman amplitudes of this theory. As long as we remain at the discrete level, we can of course only expect a relation with simplicial gravity. The general idea is that each Feynman amplitude will be associated to simplicial path integrals for gravity, discretized on the associated simplicial complex :

Now, consider continuum (Riemannian) 2d GR with cosmological constant , on a 2d manifold .

Its action is , where is the area of the surface. Consider then a simple discretization of the same. Chop the surface into equilateral triangles of area . The action will then be given by Using this discretization, and defining and , from our matrix model we get:

In other words, we obtain a (trivial) sum over histories of discrete GR on the given 2d complex, whose triviality is due to the fact that the only geometric variable associated to each surface is its area, the rest being only a function of topology. In addition to this sum over geometries, from our matrix model we obtain a sum over all possible 2d complexes of all topologies. In other words, the matrix model defines a discrete 3rd quantization of GR in 2d!

Having been assured that this theory has at least some relation with gravity at the discrete level, we can take it seriously and try to see if we can control it and find out if it has nice continuum properties, i.e. if it defines a nice continuum theory from its sum over discrete surfaces. This can be articulated in three basic questions: 1) can we control in any way the sum over triangulations defined by the perturbative expansion of the matrix model? 2) does the model have a critical behaviour corresponding to a continuum limit for trivial topology? 3) can we go further and define a continuum sum over topologies?

The first question is whether we can control the sum over triangulations and over topologies at all. The answer is in the affirmative mm (): the reason is that the sum is manifestly governed by topological parameters and can be organized accordingly:

It is then apparent that, in the limit , only spherical (trivial topology, i.e. planar, i.e. of genus 0) contribute significantly to the sum.

The second question is whether, in this limit of trivial topology, one can also define a continuum limit and match the results of the continuum 2d gravity path integral. In order to study this continuum limit we expand in powers of , and it can be shown that:

so that, in the limit of large number of triangles, and for the coupling constant approaching the critical value (with critical exponent ), the free energy (logarithm of the partition function) can be shown to diverge. This is a signal of a phase transition. In order to identify this phase transition as a continuum limit we compute the expectation value for the area of a surface, assuming as above that each triangle contributes a constant area : , for large . We see that it also diverges when .

Thus we can send the area of each triangle to zero by sending the edge lengths to zero, , and simultaneously the number of triangles to infinity: (continuum limit), while tuning at the same time the coupling constant to its critical value , to get finite continuum macroscopic areas.

This defines a continuum limit of the matrix model. One can then show mm () that the results obtained in this limit match those obtained with a continuum 2d gravity path integral (when this can be computed explicitly), in the context of Liouville gravity mm ().

Let us now ask whether the 3rd quantized framework we have (in this 2d case) can also allow to understand and compute the contribution from non-trivial topologies in a continuum limit. The key technique is the so-called double-scaling limit mm (). One can first of all show that the contribution of each given topology of genus to the partition function is

where the last approximation holds in the limit of many triangles (necessary for any continuum limit, and corresponding to a thermodynamic limit), and where is a constant that can be computed.

Define now , so that we get:

We can then take the combined limits and , while holding fixed. The results of this double limit is a continuum theory to which all topologies contribute, and again match results with continuum Liouville gravity mm ().

The area of matrix models is vast and rich of results, not only in the 2d quantum gravity context, but ranging from condensed matter physics to hot topics in mathematical physics, from string theory to mathematical biology. For all of this, we can only refer to the literature mm ().

i.6.2 Tensor models

Now we generalize further in combinatorial dimension: from 2d to 3d. This is achieved by going from 1d objects, represented graphically as edges, and mathematically by matrices, to 2d objects, represented graphically as triangles, and mathematically by tensors tensor (). Obviously, every 2d discrete structure (squares, polygons, etc) could be a possible choice, but any other 2d structure can be built out of triangles, so we stick to what looks like the most fundamental choice. This also means that, in the Feynman expansion of the corresponding theory, we expect to generalize from 2d simplicial complexes to 3d ones. We then define tensors , with and an action for them given by

where the choice of combinatorics of tensors in the action, and of indices to be traced out is made so to represent, in the interaction term, four triangles (tensors) glued pairwise along common edges (common indices) to form a closed tetrahedron (3-simplex). Once more, the kinetic term dictates the gluing of two such tetrahedra along common triangles, by identification of the edges. From the action above we read out the kinetic and vertex term:

which can be represented graphically as

We can use them to expand perturbatively the partition function:

Feynman diagrams are again obtained by contraction of vertices with propagators over internal indices

By construction, Feynman diagrams are again formed by vertices, lines and faces, but now they can form also “bubbles”(3-cells), and are dual to 3d simplicial complexes.

Figure 4: A (piece of) Feynman diagram for a tensor model, of which we give both direct and dual (simplicial) representation; the three parallel lines of propagation (dual to the three edges in the triangles of the simplicial complex) correspond to the three indices of the tensor.

As a result is defined as a sum over all 3d simplicial complexes including manifolds as well as more singular complexes (i.e. singular complexes such that the neighbourood of some points is not homeomorphic to a 3-Ball), because we impose a priori no restriction on the gluing procedure of vertices by means of propagators.

Do these models provide a good definition of 3d quantum gravity? The answer, unfortunately, is no, at least in this simple formulation of them. We will discuss how more refined versions of the same models improve the situation (see also the detailed and up-to-date review tensorReview ()). None of the nice features of matrix models export to these tensor models. First of all there is no strong relation between the Feynman amplitudes of the above tensor model with 3d simplicial (classical or quantum) gravity. Even though 3d gravity is a topological theory, with no local propagating degrees of freedom and only locally flat solutions (in absence of a cosmological constant), it is still a highly non-trivial theory. The amplitudes of tensors models are too simple to capture either the flatness of geometry or the topological character of the quantum gravity partition function (as do Chern-Simons theory or discrete formulations like the Ponzano-Regge spin foam model). They do not have enough data in the amplitudes associated to each simplicial complex, or in boundary states. Second, there is no way to separate the contribution of manifolds from that of pseudo-manifolds, i.e. to suppress singular configurations or even to identify them clearly. Third, the expansion in sum over simplicial complexes cannot be organized in terms of topological invariants, and so there is no control over the topology of the diagrams summed over. Also the last two issues can be thought to be due to the lack of data and structure in the Feynman amplitudes of the theory. For example, one can consider (before going to full GFTs) slightly revised version of the same tensor models with fundamental variables , where the indices refer to the edges of a triangle and the indices refer to its vertices tensor ().

Clearly, the process of combinatorial generalization can be continued to higher tensor models whose Feynman diagrams will be higher simplicial complexes. It is clear, however, that the difficulties encountered with 3d tensor models are not going to be solved magically if we do not render the structure of the corresponding quantum amplitudes richer and more interesting. In particular, just as we do in the 1d case, i.e. in the case of particles, we could generalize matrix and tensor models in the direction of adding degrees of freedom, i.e. defining corresponding field theories. In the process, the indices of the tensor models will be replaced by variables living in appropriate domain spaces, and sum over indices by appropriate sums or integrals over these domain spaces, while maintaining their combinatorial pairing in the action. This pairing will make the resulting field theories combinatorially non-local, as we anticipated group field theories are. In fact, once more, this is in many ways the defining properties of group field theories.

The prototype for a field theory of this non-local type and for a choice of domain space would be, for the 2d case:

with appropriate integrations over the domain space , and same identification of field arguments as in the indices of the matrix model, while for the field-theoretic generalization of the tensor model for we get:

The definition of good group field theory models for quantum gravity, of course, would require a careful choice of domain space and of classical action (kinetic and vertex functions). Here is where the input from other approaches is crucial, and where their characteristic structures are incorporated into the GFT formalism. In particular, the domain space of GFTs for quantum gravity in 3 and higher dimensions is chosen to be either a group manifold (from which the name of the formalism and the choice of notation in the examples of actions given above) or, more recently, the corresponding Lie algebra. This allows also a re-writing of the GFT action (and amplitudes) in terms of group representations, as we will see. We now motivate this choice for the domain space.

i.7 Ingredients from Loop Quantum Gravity and simplicial topological theories

Let us first look at Loop Quantum Gravity. This can be understood as an example of a theory initially defined in the continuum, but that ends up identifying, after quantization, discrete pre-geometric structures as a more fundamental set of building block for such continuum. These discrete, pre-geometric building blocks are then also used as the basic ingredients of GFTs. As we mentioned already, after passing to connection variables valued in the Lorentz algebra and suitable frame fixing of the tetrad variables to tetrads, GR becomes similar to a gauge theory for the gauge group , with classical phase space given by a connection 1-form and a conjugate electric field (triad 1-form). The difference in the Loop Quantum Gravity quantization is that one takes particular care of the diffeomorphism invariance that characterizes the theory thomas (). Due to the gauge fixing, therefore, what would be initially an gauge theory is reduced to an one, with a similar reduction of the conjugate variables to the connection. Let us see briefly (and simplifying considerably the construction) how the kinematical phase space of the theory is defined.

If we assume the –bundle to be trivial then every –connection can be seen as an –valued one–form on the three dimensional base manifold . Being a one–form can naturally (i.e. without referring to a background metric) be integrated along one–dimensional submanifolds of , namely along embedded edges :


The conjugate variable to the connection, the triad , being an –valued vector density it has a natural associated 2–form . This 2–form can be integrated along submanifolds of codimension one, namely analytic 2–surfaces (by means of appropriate parallel transports): Here is a smearing functions with values in , the topological dual of and are its components in a local basis.
To get quantities with a nicer behaviour under –transformations one introduces the holonomy


where denotes the path–ordering, which are of course group elements. For a graph with edges the holonomy assigns an element to every edge.

The rational for the above is that one is free to choose any parametrization of the classical phase space, provided any point in it can be identified by the coordinates chosen. In particular, one can specify the connection field at every point in the spatial manifold by providing its holonomies along all the paths embedded in the same manifold.

One then defines the space


of functions called cylindrical with respect to , i.e. that depend on only through the holonomies and is a continuous complex valued function.
The configuration space of the theory is defined to be the space (without reference to a specific graph ) as the space of functions that are cylindrical with respect to some graph. One can then show thomas () that the fluxes are vector fields on .

The classical Poisson algebra between cylindrical functions (including single holonomies) can be computed in full generality, i.e. for arbitrary graphs and surfaces . The basic feature is that holonomies Poisson commute, fluxes and holonomies have non-zero commutators depending on the intersection points between graphs to which holonomies are associated and surfaces on which the fluxes are smeared, while fluxes associated to any two surfaces (including coincident ones) do not commute. The non-commutativity of the fluxes even at the classical level is crucial for what follows. The algebra is in general rather complicated, depending on the specific surfaces chosen and their topological relations, to the point that the general commutator between fluxes is not known acz (); ioaristidebiancajohannes (). It simplifies considerably if one considers only “elementary”surfaces with single intersection points with the edges of the graph (this way, one has to label each state by both a graph and a set of dual surfaces to its edges). The fluxes are defined as:


where is the holonomy along the path from the starting point of the edge to the point on the surface .

In fact, considering a single link and a single elementary dual surface, also labelled by , the phase space of the theory reduces to the cotangent bundle of , , with fundamental variables being the holonomy along the link and the dual triad and fundamental Poisson brackets being


with the generators of the Lie algebra. The last bracket among fluxes is clearly the bracket.

Now, interestingly, this is also the kinematical phase space of discrete topological BF theory biancajimmy (); zapata (), with as gauge group, and thus it sets the basic kinematical stage for the simplicial path integral quantization of that theory. This fact is going to be crucial in the following, when we will discuss the GFT model for 1st order 3d gravity (which in 3d coincides with BF theory). Also simplicial BF theory, then, is based on a configuration space given by cylindrical functions for the gauge group; the difference with LQG (or with the covariant version of it) is therefore only in the constraints that one imposes on such functions to implement the dynamics.

i.8 Some mathematical tools

Given this phase space, and considering for now a single edge of any graph labelling the states, there is a natural Fourier transform that can be introduced and used to map between configuration and “momentum”space. This is the so-called non-commutative “group Fourier transform”, introduced first in PR3 (), and whose properties have been analysed in NCFourier (). Introduced first in the context of spin foam models, it has been recently used extensively in the GFT context ioaristide (); ioaristideflorian (), and then applied also in LQG ioaristidebiancajohannes (), to give a quantization of the theory in terms of metric (flux) variables. We introduce it briefly here, and then we will show its role in a GFT model for 3d gravity.

The group Fourier transform is based on the definition of plane waves


where in a basis of and the trace is taken in the fundamental representation. One can always identify with as a vector space and thus the can be interpreted as elements of . Denote the closure of the linear span of these elements as . We can then introduce a non–commutative product on the algebra of functions , starting from plane waves


and extending it to all of by linearity. endowed with this non–commutative product is refered to as .
Using these plane waves and the Haar measure on one defines the group Fourier transform as


As defined above, the Fourier transform is not invertible, but can be made so by suitable modification of the plane waves (which basically amounts to the multiplication by a polarization vector keeping track of the hemisphere of on which the group element belongs), as shown in NCFourier (). An alternative is to limit oneself to work with functions on and define plane waves with in place of in the above definition PR3 (); NCFourier (); we choose this second modification in the following. However, at times we do not indicate it explicitly in order to keep the notation simple and keep focusing on the main ideas. Therefore, using the star product (10), and these (modified) plane waves, we get a bijection


which maps functions on onto functions living on (equivalently, on the Lie algebra). The non–commutativity of (and of its Lie algebra) is taken into account via the star product (10). The extension of this group Fourier transform to functions of arbitrary numbers of group or Lie algebra elements ioaristide (); ioaristidebiancajohannes (), and to arbitrary groups, will be a crucial element of the GFT formalism, as it provides a duality of representations for the GFT field as a function of group elements or of Lie algebra elements. Clearly, it also plays an important role in dealing with simplicial BF theory, as we will see. In fact, the Feynman amplitudes of the GFT we will present, in the Lie algebra basis, will be given by the simplicial path integral for 3d BF theory (or 3d gravity). For an application of these tools to a simpler example, where the correctness of the result can however be explicitly checked, see iomatti ().

For arbitrary (square integrable) functions on groups, another type of generalization of the usual Fourier transform on is available. For compact groups (like the rotation group in any dimension), this is given by the Peter-Weyl decomposition of the function itself into irreducible representations. For it gives:

where (with repeated indices summed over) labels the irreducible representations of , are indices labelling a basis in the vector space of the representation (and its dual space), and are the Wigner representation matrices, here playing the role of plane waves. This maps functions of group elements to functions of representation labels, and can be inverted to give:

The extension of this decomposition to cylindrical functions is the basis of the spin network representation of Loop Quantum Gravity, in which basis states correspond to graphs , whose links are labelled by irreducible representations of , and vertices by intertwiners of the group, after imposition of gauge invariance. Similarly, the spin network basis provides an alternative representation for the GFT field and for the corresponding states. In this representation, as we will see, the field is represented not as a fundamental simplex but as a single spin network vertex, the building block for the construction of arbitrary spin networks.

i.9 The spin foam idea

A covariant path integral quantization of a theory based on spin networks will have as histories a higher-dimensional analogue of them: a spin foam review (); alex (); thesis (), i.e. a 2-complex (collection of faces bounded by links joining at vertices) with representations of the Lorentz (or ) group attached to its faces, in such a way that any slice or any boundary of it, corresponding to a spatial hypersurface, will be given by a spin network. Spin foam models review (); alex (); thesis () are intended to give a path integral quantization of gravity based on these purely algebraic and combinatorial structures.

In most of the current models the combinatorial structure of the spin foam is restricted to be topologically dual to a simplicial complex of appropriate dimension, so that to each spin foam 2-complex it corresponds a simplicial spacetime, with the representations attached to the 2-complex providing quantum geometric information to the simplicial complex. The models are then defined by an assignment of a quantum probability amplitude (here factorised in terms of face, edge, and vertex contributions) to each spin foam summed over, depending on the representations labeling it, also being summed over:

Figure 5: A spin network. Representations are here labelled by ’s and intertwiners by ’s.
Figure 6: A spin foam with two vertices representing a sequence (in a fictitious embedding spacetime) of two elementary transitions between spin networks.

One then has an implementation of a sum-over-histories for gravity in a purely combinatorial-algebraic context. We will show that this spin foam representation is characteristic of the GFT Feynman amplitudes, and that it is dual to the representation of the same in the form of a simplicial path integral, a duality stemming from the above duality of representations for functions on group manifolds. A multitude of results have been already obtained in the spin foam approach, for which we refer to review (); alex (); thesis ().

Let us summarize ingredients and aims of the GFT formalism, before entering the details of one specific GFT model. We want to define a quantum field theory of fundamental building blocks of quantum space, whose combinations can build up arbitrary spatial topological manifolds, and whose dynamics and interaction processes generate arbitrary spacetime topologies, thanks to a peculiar non-locality of field pairing in the interaction term of the GFT classical action. They are thus a sort of discrete or finitary, and local, 3rd quantization of gravity. In this, they represent a generalization of matrix models to arbitrary dimension. The arguments of the GFT field are given either as group elements or as Lie algebra elements or as group representations. In this, GFTs incorporate the kinematical description of geometry of Loop Quantum gravity and discrete topological BF theories. The Feynman amplitudes of the theory are, as we will see, given by simplicial path integrals or, equivalently, by spin foam models.

Ii Dynamics of 2d quantum space as a group field theory

We present here in some detail the construction and perturbative analysis of the GFT model for 3d Riemannian gravity, first introduced, in the group picture, by Boulatov boulatov (). The expansion in group representations of its Feynman amplitudes gives the Ponzano-Regge spin foam model PR1 (). It was recently reformulated in terms of non-commutative Lie algebra variables in ioaristide ().

ii.1 The kinematics of quantum 2d space in GFT: quantum simplices and spin networks

Consider a triangle in . We consider its (2nd quantized) kinematics to be encoded in the GFT field . We work here with real fields for simplicity only. The GFT field can be understood as living on the space of possible geometries for the triangle itself, or on the corresponding conjugate space. We parametrize the possible geometries for the triangle in terms of three Lie algebra elements attached to its three edges, and to be thought of as fundamental triad variables obtained by discretization of continuum triad fields along the edges of the same triangle, in line with both the LWG and discrete BF constructions.

The field is then a function

We do not assume any symmetry of the field under permutation of the arguments. Different choices are possible, as the field can be taken to be in any representation of the permutation group acting on its arguments. The choice of the representation made will influence the type of combinatorial complexes generated as Feynman diagrams of the theory iogft (). This will not concern us here. We will also see that a simple modification of the construction, defining ‘colored GFTs’tensorReview () can be used to make this choice somewhat irrelevant from the point of view of the same combinatorics.

Using the non-commutative group Fourier transform PR3 (); NCFourier () introduced earlier, the same GFT field can be recast as a function of group elements.

Recapitulating and detailing a bit more its definition, this transform stems from the definition of plane waves as functions on , depending on a choice of coordinates on the group manifold, where , are times the Pauli matrices and ‘’ is the trace in the fundamental representation 666For more details see NCFourier (); ioaristide (); ioaristidebiancajohannes (). For and , we thus have

and the -product is the one defined in the previous section.

As said, Fourier transform and -product extend straightforwardly to functions of several variables like the GFT field (and generic cylindrical functions) so that

so that the GFT field can also be seen as a function of three group elements, thought of as parallel transports of the gravity connection along fundamental links dual to the edges of the triangle represented by , and intersecting them only at a single point.

In order to define a geometric triangle, the vectors (Lie algebra elements) associated to its edges cannot be independent. Indeed, they have to ‘close’ to form a triangle, i.e. they have to sum to zero. We thus impose a constraint on the field

by means of the projector , where is the element of the family of functions:

These play the role of Dirac distributions in the non-commutative setting, in the sense that777Although behaving like a proper delta distribution with respect to the star product, under integration, this is a regular function when seen as a function on . In fact, seen as a function of , is the regular function peaked on given by , where is the 1st Bessel function NCFourier ().

One can also show that .

In terms of the dual field , the same closure constraint implies invariance under the diagonal (left) action of the group on the three group arguments, imposed by projection :


Because of this gauge invariance, which is in fact imposed in the same way as the Gauss constraint is imposed on cylindrical functions in LQG ioaristidebiancajohannes (), the field can be best depicted graphically as a 3-valent vertex with three links, dual to the three edges of the closed triangle.

This object, both mathematically and graphically, will be the GFT building block of our quantum space.

One obtains another representation of the GFT field by means of Peter-Weyl decomposition into irreducible representations, in the same way as one obtains the spin network expansion of generic cylindrical functions in LQG.

The invariant field decomposes in representations as:


where is the Wigner invariant 3-tensor, the 3j-symbol.

This expansion in irreducible group representations can be understood as a representation of the GFT field in terms of the quantum numbers associated to the quantized geometry of the triangle it represents. The ’s label eigenvalues of the length operators corresponding to its edges, while the angular momentum indices encode directional degrees of freedom. This is confirmed by canonical analysis of the quantum geometry of the triangle, as well as by geometric quantization methods barbieri (); baezbarrett (); eteracarlo ().

Multiple fields can be convoluted (in the group or Lie algebra picture) or traced (in the representation picture) with respect to some common argument. This represents the gluing of multiple triangles along common edges, and thus the formation of more complex simplicial structures, or, dually, of more complicated graphs

The corresponding field configurations represent thus extended chunks of quantum space, or many-GFT-particle states. A generic polynomial GFT observables would be given by this type of construction, and thus be associated with a particular quantum space. This includes, of course, any open configuration, in which the arguments of the involved GFT fields are not all convoluted or contracted, representing a quantum space (not necessarily connected) with boundary.

We would like to point out that, as in the case of tensor models, combinatorial generalizations can be considered, since there is no a priori restriction on how many arguments a GFT field can have. Once closure constraint or gauge invariance has been imposed on such generalized field with arguments, it can be taken to represent a general n-polygon (dual to an n-valent vertex), and glued to others in order give a polygonized quantum space in the same way, as we have outlined for triangles.

ii.2 Classical (3rd quantized) dynamics of 2d space in GFT

We now define a classical dynamics for the GFT field we have introduced. The prescription for the interaction term, as in tensor models, is simple: four geometric triangles should be glued to one another, along common edges, to form a 3-dimensional geometric tetrahedron. The kinetic term should encode the gluing of two tetrahedra along common triangles, by identification of their edge variables. There is no other dynamical requirement at this stage.

Thus we take four fields and identify pairwise their edge Lie algebra elements (triad edge vectors), with the combinatorial pattern of the edges of a tetrahedron; this gives the potential term in the action, weighted by an arbitrary coupling constant. And we take two more fields and identify their arguments; this defines the kinetic term in the action. Naming , the combinatorial structure of the action is then

where it is understood that -products relate repeated indices as , with .

Notice once more that there is no difficulty (as there is none in matrix models) in defining or dealing with a combinatorial generalization of the action; given a building block , one can add other interaction terms corresponding to the gluing of triangles (or polygons) to form general polyhedra, or even more pathological configurations (e.g. with multiple identifications among triangles). The only restriction may come from the symmetries of the action, and for the wish to keep things manageable.

The structure of this action is best visualized in terms of diagrams, similar to those used in the discussion of tensor models. Kinetic and interaction terms identify a propagator and a vertex with combinatorial structure as


and an expression (obtained as usual in QFT by writing explicitly the kinetic and interaction parts of the action in the form of a convolution of unconstrained fields with a kinetic and an interaction kernel, respectively) given by:


with , where we have used ‘’ for triangle and ‘’ for tetrahedron. The group variables and arise from (14), and should be interpreted as parallel transports through the triangle for the former, and from the center of the tetrahedron to triangle for the latter.

The integrands in (17) factorize into a product of functions associated to strands (one for each field argument), with a clear geometrical meaning. The pair of variables associated to the same edge corresponds to the edge vectors as seen from the frames associated to the two triangles sharing it. The vertex functions state that the two variables are identified, up to parallel transport , and up to a sign labeling the two opposite edge orientations inherited from the triangles . The propagator encodes a similar gluing condition, allowing for a further mismatch between the reference frames associated to the same triangle in two different tetrahedra. In this non-commutative Lie algebra representation of the field theory, the geometric content of the action is indeed particularly transparent.

Using the group Fourier transform we can obtain a pure group representation of the theory (like in the original definition by Boulatov boulatov ()).