A Notations

Testing the imposition of the Spin Foam Simplicity Constraints

Abstract

We introduce a three-dimensional Plebanski action for the gauge group SO(4). In this model, the field satisfies quadratic simplicity constraints similar to that of the four-dimensional Plebanski theory, but with the difference that the field is now a one-form. We exhibit a natural notion of “simple one-form”, and identify a gravitational sector, a topological sector and a degenerate sector in the space of solutions to the simplicity constraints. Classically, in the gravitational sector, the action is shown to be equivalent to that of three-dimensional first order Riemannian gravity. This enables us to perform the complete spin foam quantization of the theory once the simplicity constraints are solved at the classical level, and to compare this result with the various models that have been proposed for the implementation of the constraints after quantization. In particular, we impose the simplicity constraints following the prescriptions of the so-called BC and EPRL models. We observe that the BC prescription cannot lead to the proper vertex amplitude. The EPRL prescription allows to recover the expected result when, in this three-dimensional model, it is supplemented with additional secondary second class constraints.

Introduction

Spin foam models (1); (2); (3); (4) constitute an exciting proposal for the definition of a background independent and non-perturbative quantization of general relativity. They were introduced originally (5) as a way to implement the dynamics of loop quantum gravity (6); (7), and can be thought of as representing a sum over histories of the gravitational field (8).

In three spacetime dimensions, pure gravity being a topological theory (in the sense that it has no local degrees of freedom), spin foam models can be obtained as an exact simplicial path integral for the first-order Palatini action, where the gauge group is taken to be in Riemannian signature (the whole symmetry group is larger than and depends on the sign of the cosmological constant ). This corresponds to the celebrated Ponzano-Regge (9) and Turaev-Viro (10) models when and respectively. Their vertex amplitudes are simply given by and (for a root of unity) coefficients. It has been shown (11) that the Ponzano-Regge amplitudes can be obtained from a canonical quantization of gravity in the spirit of loop quantum gravity, in which the kinematical states are given by spin networks, and the dynamics leads to a spin foam evolution. The Turaev-Viro amplitudes can also be obtained as the scalar product between physical states which are in turn given in terms of unitary representations of some quantum groups closely related to (12); (13). Therefore, in three dimensions, there is a clear understanding of the relationship between the so-called canonical and covariant approaches to quantum gravity (at least when the signature is Riemannian).

In four spacetime dimensions, the construction of such a correspondence between the canonical and spin foam quantizations is still an open problem (14); (15); (16). Since it is technically too involved to follow the construction of the three-dimensional theory and perform the straight simplicial path integral quantization of the four-dimensional Palatini action (see (17) for an attempt to follow this direction), spin foam models have to be derived using an alternative strategy. More precisely, the approach to four-dimensional spin foam models is based on the fact that gravity can be formulated as a constrained topological field theory, by virtue of the so-called Plebanski action (18). This latter is defined as the sum of a well-known topological action (19), plus a set of constraints on the field ensuring on-shell the condition that it comes from the exterior product of two tetrad one-form fields1. With this solution for the field, the Plebanski action reduces at the classical level to the usual Hilbert-Palatini action of first order general relativity. To derive a spin foam model, the following general strategy is adopted. First, the topological theory is discretized exactly on a two-complex, and the basic variables, i.e. the field and holonomies of the spacetime connection, are promoted to quantum operators. Then, the simplicity constraints are imposed at the quantum level as restrictions on group-theoretical data. The main challenge in this approach is essentially to impose consistently these second class simplicity constraints, and several proposals defining the different available spin foam models have been put forward in order to do so.

The three spin foam models for four-dimensional quantum gravity that have been studied the most are the Barrett-Crane (BC) model (20), the Engle-Pereira-Rovelli-Livine (EPRL) model (21); (22); (23), and the Freidel-Krasnov (FK) model (24); (25). They all rely on the general strategy outlined above, even though their construction requires to impose the simplicity constraints in drastically different ways. In the BC model, which was the first four-dimensional model to be introduced, the simplicity constraints are imposed as strong operator relations. This has the result of assigning simple representations of the gauge group to faces of the dual triangulation , and a specific unique intertwiner, known as the BC intertwiner (20); (26), to edges of . Due to difficulties in reproducing the correct semiclassical limit (27); (28); (29) and the structure of the graviton propagator (30), this model was partly discarded2, and the search for new models was motivated with the additional hope to relate the spin foam quantization to the canonical structure of loop quantum gravity. This search culminated with the introduction of the EPRL and FK models, which are both based on the introduction of a linear version of the (originally quadratic) simplicity constraints (24), and take as an important additional input the inclusion of the Barbero-Immirzi parameter. In these two models, the over-imposition of the simplicity constraints which is thought to be responsible for the issues with the BC vertex, is cured by using a weak imposition by means of coherent states for the FK model, or the Gupta-Bleuler scheme for the EPRL model. Our goal here is not to review extensively the details of these constructions, but rather to focus on the implementation of the simplicity constraints by introducing a model which allows for an immediate comparison between the alternative schemes. In fact, it has been argued on several occasions (33); (34); (35) that the current spin foam models miss the imposition of additional secondary second class constraints that are generated by the usual simplicity constraints. The usual point of view in the derivation of spin foam models is that it might be sufficient to impose the primary simplicity constraints consistently at all times, since the secondary constraints arise in fact as a consequence of this requirement. But even if this is true at the classical level, at the quantum level the secondary constraints are allowed to have non-vanishing fluctuations, and ought therefore be imposed as well. It is however very hard in the four-dimensional case to implement concretely this idea. In other words, we understand that there is a difficulty which needs to be resolved in the current four-dimensional spin foam models, but even if we have generic arguments indicating how to do so, a concrete realization has never been put forward.

Let us summarize the situation. At the classical level, it is clear that the Plebanski action is equivalent to the Palatini action once the field is forced to be simple. But since we do not know how to compute the simplicial path integral for the Palatini action, we try to compute that of the Plebanski action by first quantizing the unconstrained topological theory and then imposing the simplicity constraints at the quantum level. In fact, it is quite clear that we are restricted in our understanding of how to properly perform this last step, by the fact that we do no know what to “expect” from the spin foam quantization of the four-dimensional Plebanski action. For this reason, it would be nice to have a model in which we could work out explicitly the two alternative methods. By this, we mean that we would like to perform the spin foam quantization of the action in which the simplicity constraints have already been solved at the classical level, and then compare this result to the spin foam quantization in which the simplicity constraints are solved in the quantum theory. This would be a way to test the various proposals to deal with the simplicity constraints. It has already been argued in the literature (36) that dealing with second class constraints at the quantum level can lead to inconsistencies, and is not compatible with the quantization program à la Dirac. However, the argument alone is not fully convincing, since it was formulated on a finite dimensional model with no clear analogy with gravity, and since it does not give a clear explanation for how the various impositions of the constraints in the four-dimensional spin foam models should lead to a model with inconsistent physical predictions. It might very well be that despite the fact that we are dealing with second class constraints at the quantum level, there exists a preferred scheme in which the simplicity constraints can be imposed in a (yet-to-be defined) robust way to lead to an acceptable model for quantum gravity. In the end, the only way to discriminate between the various proposals is either to extract physical predictions to compare with experiments, or at least to test the strategies on toy models which bear a close analogies with gravity. Let us also point out that a model was introduced in (34) in order to illustrate the claim that additional secondary second class constraints should be taken into account in the spin foam models. This model is based on the idea of reducing a four-dimensional theory to an one by means of simplicity constraints, and demonstrates that this is only possible if certain secondary second class constraints are imposed. The weak point of this model is that the constraints that it imposes are not derived from the Hamiltonian analysis of a given Plebanski theory, but are instead put in by hand.

The aim of this paper is to formulate a robust model which allows to test the imposition of the spin foam simplicity constraints. Since pure gravity is always topological in three spacetime dimensions, it makes a priori no sense to have a Plebanski formulation in which the field is forced to be simple. In fact three-dimensional theory is already equivalent to gravity, when is a one-form field valued in the Lie algebra of . However, if one replaces the gauge group by (or its double cover ), the three-dimensional theory admits simplicity constraints similar to that of the four-dimensional theory. In the gravitational sector of solution to these simplicity constraint, the action becomes that of first order Riemannian gravity. In the topological sector, the theory is trivial. Therefore, at the classical level, the Plebanski theory that we introduce reduces to the Hilbert-Palatini action, whose spin foam quantization naturally leads to the Ponzano-Regge amplitudes. Now, it is also possible to perform the spin foam quantization of the three-dimensional theory, and to impose the simplicity constraints at the quantum level, mimicking in particular the prescription of the BC and EPRL3 models. This allows for a direct verification of whether any of these proposals leads to the proper vertex amplitude. Our construction can be summarized in the diagram of figure 1.

Figure 1: The starting point of this investigation is the three-dimensional action . After imposition of the simplicity constraints , it reduces to the usual Hilbert-Palatini action, whose discretized partition function can be computed explicitly. The question we want to ask is whether it is possible to derive the same vertex amplitude by imposing the simplicity constraints at the quantum level on the partition function for the unconstrained action. In other words, does the imposition of the simplicity constraints à la BC or EPRL render this diagram commutative? If not, what is the additional input which is necessary in order to reconstruct the Ponzano-Regge model from the spin foam quantization of the three-dimensional Plebanski action?

We start in section I by introducing the classical framework for our analysis. In particular, we show how the three-dimensional action is reduced by the simplicity constraints to the usual action for three-dimensional gravity, and we perform the Hamiltonian analysis of the Plebanski theory to support this fact. Section II is devoted to the study of the quantum theory. We first recall the state sum model corresponding to the constrained classical action, and then perform the spin form quantization of the Plebanski theory. For this, we start by writing down the simplicial path integral for the theory, and then study the imposition of the simplicity constraints. In particular, we study the BC and EPRL prescriptions, in which the constraints are imposed strongly and weakly (in a sense that we will make more precise), respectively. We also study the so-called Warsaw modification (37); (38); (39) of the EPRL model in order to illustrate its signification on the three-dimensional model at hand. Our conclusion is that the BC prescription cannot lead to the proper vertex amplitude, whereas the EPRL prescription, when supplemented with the imposition of the additional secondary second class constraints, enables one to recover the expected result. This might be a strong indication of the fact that a weak imposition of the simplicity constraints in the quantum theory is an important ingredient in order to derive the vertex amplitude of four-dimensional quantum gravity. Nevertheless, we want to emphasize that in our three-dimensional model, none of above-mentioned proposals lead by themselves to the proper vertex amplitude, which is that of the Ponzano-Regge model. The reason for this is clear: the current spin foam models miss the imposition of the secondary second class constraints that are exhibited in the canonical analysis of the Plebanski theory. Indeed, we show how these constraints can be solved by imposing conditions on the spatial components of the connection, and how their imposition in the path integral with the EPRL prescription allows to recover the proper vertex amplitude.

I Classical theory – actions for three-dimensional gravity

To introduce our model, let us start with the familiar action, written in three dimensions and for the gauge group . In terms of an -valued one-form field (our notations are defined in appendix A) and the spacetime connection , it is given by

(1)

where denotes a trace in the Lie algebra , and is the curvature two-form associated to the connection. Clearly, this action is not that of Riemannian three-dimensional first order gravity, but this latter (or at least two copies of it) can be obtained if we use a decomposition of the fields into self-dual and anti self-dual components. Here we are not interested in doing this decomposition, because our goal is to mimic the constrained four-dimensional action which is the starting point for the spin foam quantization. To see that this is indeed possible, let us proceed in two steps. First, let us assume that the field is of the form

where is a zero-form in , and a Lie algebra-valued one-form. This is the three-dimensional analogue of the notion of simple two-form field that is used in four-dimensional Plebanski theory. Second, let us choose a gauge (which we will refer to as the time-gauge) in which . This gauge choice reduces the symmetry group to its subgroup. A key observation is that it is always possible to fix due to the presence of the “scaling” symmetry

(2)

for . With this choice of gauge, the action (1) becomes the usual action for gravity4

(3)

We have included the first equality to make explicit the fact that the three-dimensional Hilbert-Palatini action is simply a constrained action. This argument suggests that it is possible to obtain the action for gravity starting from (1) if we impose a simplicity-like condition on the field, and pick up a rotational subgroup of by making a gauge choice for (notice that this choice is not canonical).

We are going to see in this section that the simplicity of the field can be enforced with a Plebanski action whose Hamiltonian analysis will be detailed.

i.1 Plebanski action and simplicity constraints

As we mentioned above, the first things that we would like to do is to impose the simplicity of the field in the theory. Following what is done in four dimensions, we are going to introduce the Plebanski action as the sum of the action (1) plus a set of constraints on the field. Our aim is to reproduce simplicity-like conditions ensuring that we are indeed dealing with three-dimensional gravity. The action that we consider is5

(4)

where denotes the Hodge dual operator in , and is a symmetric Lagrange multiplier used to enforce the constraints

(5)

We want to understand the exact meaning of these constraints in the present context. The key difference with four-dimensional Plebanski theory is that we are dealing here with vectors, and not bivectors. Therefore, we need another notion of simplicity than the usual one used in spin foam models. As we have already mentioned earlier, we are going to say that a vector is simple if it can be written as

where is a zero-form in , and a one-form. The fact that the constraint defined by (5) can be used to obtain simple vectors is ensured by the following proposition:

Proposition 1.

When the simplicity constraints are satisfied, there are three possible solutions for the field. They correspond respectively to:

  • A gravitational sector, in which there exists an -valued one form and a vector in such that

  • A topological sector, in which there exists an -valued one form and a vector in such that

  • A degenerate sector, in which and .

Proof.

Let us introduce the boost and rotational components of the field . They are given respectively by and . Each of these one-forms can be interpreted as a vector in , and the simplicity constraints simply mean that the vectors are orthogonal to the vectors , i.e.

(6)

Let us first assume that the three vectors form a basis of . Due to the simplicity constraints (6), the three remaining vectors are linked, and therefore they lie in the same plane, whose (non-necessarily unit) normal is denoted by . More precisely, there exist three vectors such that

(7)

Furthermore, since the vectors and are orthogonal, we can write to

(8)

where is a (non-vanishing) scalar, and is a real-valued one form. Due to the fact that the vectors form a basis, the three vectors form a basis as well. The solution given by (7) and (8) corresponds to the gravitational sector.

If we assume on the contrary that the three vectors form a basis, the same construction can be used to obtain the topological sector described in proposition 1.

When and , i.e. neither the nor the form a basis, we say that the field belongs to the degenerate sector.

It is clear from the expression of in the gravitational sector that in the time gauge, where we have and , the boost component is vanishing, while the rotational component reduces to the triad . This supports the heuristic argument that we have given at the beginning of this section in order to derive the action (3) from the theory. Note also that in the gravitational sector we have , while in the topological sector we have .

Let us conclude with a remark. There are 6 simplicity constraints acting on the 18 components of the field. Therefore we expect the simple field to be written in terms of 12 components only. Here, we have expressed in terms of the 16 components and . However, notice that there are 4 redundant components due to the presence of the following symmetries:

  • A rescaling symmetry, given by (2), which allows to remove one component (for instance, we can fix ).

  • Three translational symmetries acting in the non-degenerate sector. They are generated by a vector , and act like

    As a consequence, we can always choose , but here we do not make this choice in order not to break the symmetry of the theory.

i.2 Gravitational and topological sectors

Now that we have a good understanding of how the simplicity constraints can be used to define a sector of the Plebanski theory in which the vector is simple, let us see how it is possible to recover the action for three-dimensional gravity. For this, we are going to study separately the gravitational and topological sectors introduced in proposition (1).

Using the decomposition of into self-dual and anti self-dual generators (see appendix B), we can write the field as

where , and the action (1) therefore becomes

(9)

As usual, the equations of motion with respect to the connection lead to the torsion-free condition

and if , this relation can be inverted to find the torsion-free spin connection. This latter, when plugged back into the original action (9), leads to the sum of two second order Einstein-Hilbert actions,

(10)

each being defined with respect to a two-dimensional Urbantke-like metric (40) (in the sense that it is constructed with the field). In this expression, denotes the sign of (we refer the reader to appendix C for the calculation of ).

In the gravitational and topological sectors, the self-dual and anti self-dual components of the field are given by:

(11b)
  • gravitational sector: ,

  • topological sector: .

In each of these two sectors, we can now compute the two Urbantke metrics defined by the expressions above. In fact, from the simplicity constraints, we know already that . It is however less trivial to see that the metrics in the topological and gravitational sector are identical. A simple calculation shows that we have

(12)

where . In the time gauge, this metric reduces to .

Gathering these results on the Urbantke metric and the sign factors , we can conclude that the action (1) reduces in the gravitational sector to

while in the topological sector it becomes simply .

This shows that the gravitational sector to the solutions of the simplicity constraints corresponds indeed to three-dimensional gravity, whereas in the topological sector the action vanishes on-shell. This is exactly what happens in the four-dimensional Plebanski theory. Interestingly, the presence of a topological sector in this three-dimensional theory allows for the introduction of a Barbero-Immirzi parameter, but we shall come back to this point in section III.

i.3 Canonical analysis of the Plebanski action

In this subsection we perform the canonical analysis of the three-dimensional Plebanski theory. It is similar to the study of the four-dimensional Plebanski action (41). Using a decomposition of the spacetime manifold, (4) becomes

where the curvature and torsion two-forms are defined by6

(13)

We can see from this formula for the action that , and are non-dynamical variables, since the Lagrangian does not feature their time derivatives. The time component of the connection and are true Lagrange multipliers, enforcing respectively the torsion-free and simplicity constraints. However, since is involved quadratically in the simplicity constraint, it cannot be treated as a Lagrange multiplier. For this reason, we add to the Lagrangian the term

where and are new auxiliary -valued fields that do not affect the dynamics of the theory.

The basic variables of the theory are then the 12 spatial components of the connection, their 12 canonical momenta , and the 6 components with their momenta . The Poisson structure is given by

(14)

where . The total Hamiltonian is

(15)

We can now identify the primary constraints and compute their Poisson bracket with this Hamiltonian in order to study their evolution in time.

Primary and secondary constraints

We have the following 18 primary constraints:

The first set is obtained by varying the action with respect to , the second set by varying the action with respect to , and the third one by varying with respect to

Before going any further, let us introduce smeared variables in order to deal with the delta functions appearing in the Poisson brackets. To this end, we define the quantities

for any -valued smooth test function on . This enables us to compute

and

where stands for the covariant derivative with respect to the connection . Now we can compute the Poisson bracket of the primary constraints with the total Hamiltonian, to see if the requirement that they be preserved under the time evolution gives rise to secondary constraints.

a. Evolution of the constraints

First, it is easy to show that the conservation of does not lead to any secondary constraints. To see this, we can introduce the new constraint

(16)

and notice that it is in fact the generator of the infinitesimal gauge transformations on the phase space. Indeed, on the action of on the phase space variables (14) is given by

Since the Hamiltonian contains only terms with traces, it is left invariant under the action of this constraint, which therefore satisfies . The modification (16) of the constraint is permissible since it amounts to adding a term which is itself proportional to a constraint.

b. Evolution of the constraints

The requirement that the primary constraint be preserved in time leads to the following relation:

Among these 6 equations, 3 are in fact fixing the components (if we assume that the non-degeneracy condition holds), and the remaining 3 have to be imposed as secondary constraints. Projecting onto the vector , these secondary constraints can be written as

c. Evolution of the simplicity constraints

To study the evolution of the simplicity constraint, it is convenient to compute separately the Poisson bracket of its various components with the Hamiltonian. Requiring that the constraints and be preserved under time evolution leads to

(17a)
(17b)

These 3 equations determine 3 of the 6 components of the multiplier , and imply no secondary constraints on the dynamical variables of the phase space. Finally, a direct computation shows that the requirement that leads to the 3 secondary constraints

(18)

The Dirac algorithm stops here, and there are no tertiary constraints.

d. Summary of the analysis of the primary constraints

To summarize, we have a theory with 18 primary constraints consisting of 6 constraints , 6 constraints , and 6 constraints . They generate the 6 secondary constraints comprising the 3 constraints and the 3 constraints .

First and second class constraints

In the previous subsection, we have derived the primary and secondary constraints of the Plebanski theory. We are now going to split them between first class and second class constraints.

a. The first class constraints

First, it is easy to see that the constraints are first class. Indeed, they commute with all the other constraints since they generate the infinitesimal gauge symmetries.

The analysis of equation (17) suggests that amongst the 6 constraints , 3 are first class and 3 are second class. To see that this is indeed the case, let us decompose the set of constraints into

If the field does not belong to the degenerate sector, the previous set of constraints is equivalent to the requirement that . Furthermore, a direct computation (see the algebra of constraints in appendix D) shows that the constraints are first class. They are associated with the 3 components of that remain undetermined after taking (17) into account.

Finally, the last first class constraints are given by a new set , which is obtained by adding to a linear combination of constraints. In particular, is defined as being equal (up to a factor 2) to the Hamiltonian density (15):

where the values of the Lagrange multipliers , and , are those determined by the constraint analysis. When the Lagrange multipliers are not fixed by the Hamiltonian analysis, it is possible to fix them to the value zero in the expression for . Concerning the constraints , a direct calculation shows that they are given by

and that they generate as expected space diffeomorphisms on the phase space variables:

where is a vector field on , and denotes the smearing of the with .

b. The second class constraints

The remaining constraints, , and , are the second class constraints of the theory. In order to prove this, it is possible to compute the (square 12-dimensional) Dirac matrix , whose elements are given by Poisson brackets between the various (candidate) second class constraints, and show that its determinant is non-vanishing. This is indeed the case, since the Dirac matrix is given by

and it satisfies clearly . Here the elements of the matrix are given by

and we have , where denotes the three-dimensional volume. The matrix has elements determined by the Poisson brackets

and satisfies . The explicit form of the matrices and is not important in order to prove the invertibility of the Dirac matrix, and it is in fact easy to see that its determinant is given by7

Summary

To summarize, the first class constraints of the system are given by , and , and the constraints , and , are of second class. Therefore, we have 36 phase space variables and that are subject to 12 first class constraints (generating gauge symmetries) and 12 second class constraints, and the usual counting shows that there are no phase space degrees of freedom. This is of course to be expected in three-dimensional gravity.

As usual, the first class constraints of the theory are the infinitesimal generators of the gauge symmetries. The constraint is the generator of the internal symmetries. The 3 primary constraints appear because we have treated the lapse and the shift encoded in as dynamical variables. The constraints and therefore encode the vanishing of the momenta and conjugated to the lapse and the shift. The remaining constraints, and , are related to the scalar and vector constraints, i.e. to the spacetime diffeomorphisms.

Let us conclude this canonical analysis with an important remark concerning the second class constraint. The 3 constraints constrain 3 out of the 6 components of the variables , which are the conjugate momenta to . As a consequence, only 3 components of are left, and they correspond to the momenta and associated with the lapse and shift variables. Moreover, the lapse and the shift are the 3 independent components of that are left after imposing the 3 components of the simplicity constraints. The spatial part of the simplicity constraints ensures that the components can be expressed in terms of the zero-form and the one-form . Finally, the meaning of the 3 constraints is that 3 of the components of the connection can be expressed in terms of and . It is important to stress that these are secondary constraints which have been obtained from the requirement that be preserved in time. In other words, the usual simplicity constraints are of second class because there are in fact secondary constraints which do not commute with the primary ones. This point will be very important in the spin foam quantization that we perform in the next section. We are going to show the importance of imposing the secondary second class constraints in the spin foam models.

Ii Spin foam quantization

In this section, we study the quantization of the three-dimensional Plebanski theory. From the classical analysis of the previous section, it should be clear that the action (4) is equivalent in the gravitational sector to that of first order gravity, whose spin foam quantization is the Ponzano-Regge model. We can now try to reproduce the strategy of four-dimensional spin foam models, and see if the spin foam quantization of the full Plebanski theory leads to the Ponzano-Regge model as well. This is a well-posed question which will enable us to clarify the issue of imposing the simplicity constraints at the quantum level.

ii.1 The Ponzano-Regge model

We have seen in the previous section that when the simplicity constraints are imposed at the classical level, the gravitational sector of solutions to the Plebanski action (4) is equivalent to the Palatini action. It is therefore possible to compute explicitly the simplicial path integral of this theory, and we know that it leads to the Ponzano-Regge state sum model, whose vertex amplitude is given by an symbol. We recall some basic facts about the derivation of this result, and also introduce the notations that we will use in the next subsections.

The partition function of three-dimensional Riemannian gravity is formally given by

where denotes a trace in the Lie algebra . To compute the simplicial path integral, we introduce a simplicial decomposition of the spacetime manifold , along with its dual two-complex (42). This latter is consisting of vertices (dual to tetrahedra ), edges (dual to triangles ) and faces (dual to links ). Since the triad is a one-form, it is natural to integrate it along the one-cells (links) of to obtain -valued elements . The connection is discretized by computing its holonomy along the edges of . We can then discretize the curvature by taking the product of holonomies along the edges lying on the boundary of a face to form

With these variables, the discretized version of the action (3) can be written as

and the partition function becomes

(19)

where is the Lebesgue measure on , and the Haar measure on . It is now possible to perform the integral over to obtain

where the delta distribution over imposes the flatness of the connection. Using the Peter-Weyl decomposition, we can write

(20)

where the sum is taken over all the possible representations labeling the set of faces . For arbitrary cellular decompositions , let us call the number of faces meeting at every edge . In (20) we will have an integral over of products of representation matrices for the group element . We can therefore use the fact that

(21)

projects onto the space of intertwiners between the representations coloring the faces meeting at the edge . Then, all the intertwiners meeting at a vertex can be contracted to define a vertex amplitude . Finally, the partition function can be written as a sum over spin foam amplitudes

To clarify the meaning of this formula, let us assume that the cellular decomposition is simplicial. In this case, vertices in are four-valent, while edges are three-valent. The vertex amplitude is therefore given by a contraction of four three-valent intertwiners, which is the so-called symbol. The final result of this calculation is

(22)

where the labels are associated to the six faces dual to the links a tetrahedron . Notice that the sum over intertwiners has now disappeared since there is only a unique (up to normalization) three-valent intertwiner. In other words, the intertwiner space

is one-dimensional when satisfy the triangular inequalities, and zero-dimensional otherwise.

It is convenient at this point to introduce the diagrammatic notations (43), in which lines are used to represent unitary irreducible representations, and boxes to represent integrals over the group defining a projector on the intertwiner space following (21). With this notation, the partition function defining the Ponzano-Regge model can be represented as follows:

Once the integrations are performed, we obtain a contraction a four three-valent intertwiners for each vertex dual to a tetrahedron, and the partition function becomes (22).

ii.2 Quantization of the sector

Spin foam models in four dimensions take as a starting point the four-dimensional Plebanski action, and are derived by first quantizing the topological part of the action, and then imposing the simplicity constraints at the quantum level.

Here we follow the same procedure, and start by quantizing the part of the Plebanski action (4). This can be done very easily along the lines of the standard construction introduced in the previous subsection, with the only difference that is now replaced by . Therefore, the field is discretized by assigning an element to each face dual to a one-cell in . Once we perform the integration over in the simplicial path integral, we are left with

where now we have a product of delta distributions on the group , which can again be evaluated with the Peter-Weyl decomposition. Using the fact that , it is possible to write an irreducible representation of as a tensor product of two irreducible representations of . The discretized partition function for the three-dimensional theory then takes the form

Now, we have to implement the simplicity constraints in this state sum in order to ensure that it describes quantum three dimensional gravity. Indeed, since at the classical level the gravitational sector of the Plebanski action corresponds to the action for gravity, whose spin foam quantization corresponds to (22), we expect that a proper imposition of the simplicity constraints will lead to the same result.

ii.3 Imposition of the simplicity constraints

In the simplicial path integral that we have written for the three-dimensional theory, the one-forms have been discretized by assigning an element to the faces of , which are dual to the links defining the boundary of the two-cells (triangles) . These vectors in fact determine the geometry of the triangles, and they satisfy the closure constraint

This is just the discrete analogue of the continuous Gauss law which ensures gauge invariance under the action of . Additionally, the discretized field is required to satisfy the discretized version of the simplicity constraints , which are given by8

diagonal simplicity: (23)
off-diagonal simplicity: (24)

These simplicity constraints do not distinguish between the topological and gravitational sectors, since they are left unchanged if we change for .

Just like in the four-dimensional theory (24); (35), the simplicity constraints can be linearized, and are equivalent to the requirement that

(25)

Indeed, in the gravitational sector where we have , it is clear that the linear simplicity constraint is satisfied. This linearized version has therefore the advantage of selecting the gravitational sector.

The quantization of the theory is based on the symplectic structure of the topological part of the action. In the action, the variable is canonically conjugated to the connection , and the quantization rule is therefore simply to identify the discretized field with the generators of the Lie algebra . The next step is to impose the simplicity constraints on these generators following the different spin foam models that have been introduced in the literature.

The imposition of the constraints is done before the integration over the connection components defining the intertwiners. Schematically, we can represent this imposition of the constraints as a white box acting on the group generators between two neighboring triangles. Our task is therefore to implement the simplicity constraints in such a way that the partition function reduces to

(26)

Here we have represented the implementation of the constraints via on one triangle only for the sake of clarity. More explicitly the meaning of this graphical notation is that we have an integral over the holonomies, together with a yet-to-be defined implementation of the simplicity constraints . In other words, we have

(27)

and the implementation of the simplicity constraints is done before the integration over the holonomies that defines the intertwiners leaving on the edges.

The BC prescription

We are going to start by solving the diagonal simplicity constraint. It is known that this constraint is equivalent to the requirement that the pseudo-scalar quadratic Casimir operator of defined by vanishes, which constrains the representations to be simple. Indeed, in terms of the self-dual and anti self-dual generators, (23) takes the form

which implies the restriction to simple representations of