# A simpler way of imposing simplicity constraints

###### Abstract

We investigate a way of imposing simplicity constraints in a holomorphic Spin Foam model that we recently introduced. Rather than imposing the constraints on the boundary spin network, as is usually done, one can impose the constraints directly on the Spin Foam propagator. We find that the two approaches have the same leading asymptotic behaviour, with differences appearing at higher order. This allows us to obtain a model that greatly simplifies calculations, but still has Regge Calculus as its semi-classical limit.

## I Introduction

One of the big open problems of modern theoretical physics is finding a non-perturbative definition of Quantum Gravity in 4 dimensions. Spin Foam models Rovelli:2011eq ; Perez:2012wv are an attempt at such a definition. The basic idea behind the framework is to start with a topological field theory, known as BF theory, which at the classical level can be constrained to a first-order formulation of General Relativity. More precisely, starting with the action

(1) |

and imposing the simplicity constraint reduces the theory to the Plebanski action for GR Plebanski:1977zz . At the quantum level, the idea is to start with the well-known partition function of BF theory and impose the simplicity constraints on expectation values of states. In the spin representation of Spin Foam models, like the EPRL-FK models Freidel:2007py ; Engle:2007uq ; Engle:2007qf ; Engle:2007wy , the boundary states are given by spin networks, so the simplicity constraint is implemented on the boundary spin network of a fundamnetal building block – a 4-simplex.

Spin Foam models have been recently rewritten in a holomorphic representation using spinors coh1 ; Livine:2007ya ; Freidel:2009nu ; Freidel:2009ck ; Freidel:2010tt ; Borja:2010rc ; Livine:2011gp ; Dupuis:2012vp ; Dupuis:2010iq ; Dupuis:2011dh ; Dupuis:2011wy . The standard framework for imposing the simplicity constraints was implemented in the holomorphic framework by Dupuis and Livine in Dupuis:2011fz , which we will refer to as the DL model. In the Riemannian case of the gauge group , the constraint imposes left and right spinors to be proportional to each other on the boundary spinor network of a 4-simplex.

In a recent article Banburski:2014cwa we have introduced a new Riemannian Spin Foam model in the holomorphic representation, which allowed the successful calculation of 4d Pachner moves. Rather than imposing the constraints on the boundary spinor networks, we impose them on the Spin Foam propagators. For the case of the 4-simplex, this effectively imposes the constraints not only on the boundary, but also in the bulk. This choice results in a reduction of two internal strands to one and allows to calculate the Spin Foam amplitudes much more efficiently.

Naively it is not obvious that imposing more constraints does not spoil the semi-classical limit of the Spin Foam model. In this article we calculate the asymptotic behaviour of the amplitude of a 4-simplex for both the DL model as well as the new model proposed in Banburski:2014cwa . We find first that the DL model’s asymptotics turn out to be the same as the EPRL-FK model Bianchi:2008ae ; Conrady:2008mk ; Barrett:2009gg ; Magliaro:2011dz ; Han:2011rf ; Han:2011re ; Han:2013gna ; Han:2013hna ; Han:2013ina ; Hellmann:2013gva , as was expected. Next, we study the asymptotics of the new model and find that due to non-trivial cancellations on-shell, it has the same 1st order behaviour, giving Regge Calculus Regge:1961px . The differences in asymptotics between the models reside in the Hessian, the overall normalization and the higher order terms as well as on the off-shell trajectories.

## Ii Holomorphic Spin Foam Models

In this section we will review the holomorphic Spin Foam models. We will start from a short review of the representation of SU(2) in terms of holomorphic functions of spinors, followed by a summary of the holomorphic simplicity constraints introduced by Dupuis and Livine in Dupuis:2011fz . We will finish this section by showing how the constraints can be imposed in two ways – on the boundary spinor network, as was done in the Dupuis-Livine model Dupuis:2011fz and on the Spin(4) projectors, as we have introduced in Banburski:2014cwa .

### ii.1 Holomorphic representation

Let us consider the Bargmann-Fock space Bargmann ; Schwinger of holomorphic functions on spinor space endowed with the Hermitian inner product

(2) |

where and is the Lebesgue measure on . We use the notation

and to denote the conjugate spinor . For the construction of Spin Foam amplitudes, we will be interested in the SU(2) invariant functions on spinors

(3) |

These invariant elements of form a space of -valent intertwiners. One way to construct an element of is to average a function of spinors over the group using the Haar measure. In this way we can construct a projector as

(4) |

where the projection kernel in the case is given by

(5) |

and is the normalized Haar measure over SU(2). In Freidel:2012ji it was shown that the group integration can be performed explicitly and in Banburski:2014cwa we have shown that the projector can rewritten as

(6) |

We will also refer to the kernel as a projector. From the fact that we can construct coherent intertwiner states Freidel:2013fia

(7) |

Spin Foam amplitudes are usually constructed as contractions of such coherent states. Before we construct these amplitudes, we have to discuss simplicity constraints.

### ii.2 Simplicity Constraints

In this section we will review the idea behind simplicity constraints and their holomorphic version Dupuis:2011dh . The basic idea behind Spin Foam models is to take a topological field theory – BF theory, and impose constraints to reduce it to General Relativity.

BF theory is defined on a principal bundle over a dimensional manifold , with a group and a connection . B is a form in the adjoint representation of . is a curvature 2 form. The action is defined as

(8) |

It is a topological field theory in the sense that all the solutions of equations of motion are locally gauge equivalent. One can prove that a bivector in or is a simple bivector if and only if there exists a vector such that . When this condition is satisfied, is constrained to be proportional to or . Using a parameter (the Barbero-Immirzi parameter) to distinguish these two sectors, we obtain the Holst action for gravity

(9) |

For the Riemannian 4d Spin Foam models, we use the gauge group , which is the double cover of . The holomorphic simplicity constraints are isomorphisms between the two representation spaces of : for any two edges which connect to the same node ,

(10) |

where is related to the Barbero-Immirzi parameter by

(11) |

The Eq.(10) can be only satisfied if there exists a unique group element for each node , such that

(12) |

It is interesting to notice that can be expressed purely in terms of left and right spinors as

(13) |

The holomorphic simplicity constraints imply the geometrical simplicity only when . This happens only when the closure constraints are satisfied, which in the holomorphic representation are imposed in the semi-classical limit, see Dupuis:2011dh .

Geometrically, each spinor defines a three vector through the equation,

(14) |

Thus around a node in a spin-network, each link (dual to a triangle in the simplicial manifold) is associated with two 3-vectors and given by the left and right spinors. These vectors correspond to the selfdual and anti-selfdual components of the field respectively :

(15) |

At the level of the vectors and the holomorphic simplicity constraints imply now

(16) |

which leads to the constraint that the norms of the selfdual and anti-selfdual components of the bivector have to be equal to each other:

(17) |

Thus the field is a simple bivector, and for the node there exists a common time norm to all the bivectors:

(18) |

This implies now that should be satisfied in the semi-classical limit.

### ii.3 Imposing constraints

We will now impose the holomorphic simplicity constraints on the Spin(4) BF theory in order to obtain a model of 4d Euclidean quantum gravity. In the EPRL model the simplicity constraints

(19) |

are imposed on the intertwiners as an operator equation, providing a map from Spin(4) to SU(2), which can be graphically denoted as

(20) |

In the FK model Freidel:2007py it was shown however, that instead of working with the intertwiners, one can work with coherent states whose norm is an area of the faces of a tetrahedron. The Eq.(19) is satisfied by working with states whose spin labels solve the constraints. Note however, that this approach to writing the coherent path integral doubles the number of variables, as one has to deal with independent vectors and . In Conrady:2009px Conrady and Freidel extended the construction into fully geomterical coherent states by using the work of Guillemin and Sternberg GS , which states that, for compact groups, “quantization commutes with reduction”.

Here we will discuss two natural ways of imposing the holomorphic simplicity constraints. The first one is to impose the contraints on the boundary spinor network defined by contraction of coherent states as done by Dupuis and Livine in Dupuis:2011fz , to which we will refer to as the DL model. This corresponds to the following gluing of 4-simplices

(21) |

with two copies of spinors and on the inside of a 4-simplex and one copy satisfying on its boundary. In this way, DL model constitutes a weakening of the simplicity constraints, making all the constraints imposed coherently. This imposition of constraints is dual to the FK one, as off-shell we have independent spins and , but only one copy of spinors . There are still however two independent copies of spinors and .

The other way of imposing the holomorphic simplicity constraints inspired by the Guillemin-Sternberg result is to impose the constraints on all the labels of the coherent states (7), or effectively on the Spin(4) projector. This approach satisfactorily reduces the two copies of spinors to a single copy and gives the model we have recently introduced in Banburski:2014cwa . Graphically this corresponds to

(22) |

with both for the boundary spinors and for the interior ones. In this way we obtain a model without the doubling of variables that the previous models exhibited, which in practice allows for much simpler calculations.

In this section we will give more details on these two alternatives before going on to show that indeed the two methods result in the same semi-classical limit.

#### ii.3.1 DL model

In Dupuis:2011fz ; Dupuis:2011dh the simplicity constraints in the DL model are imposed on the boundary spinor network state, as is usually done in EPRL-FK models written in terms of coherent states. The amplitude for a single 4-simplex is given by a product of contraction of coherent states for left and right sectors, with the simplicity constraints imposed on the boundary spinors as follows

(23) |

where is the set of tetrahedra labeled by . To make the comparison of this imposition of constraints to the one we will introduce in the next section, we notice that the DL model can be rewritten as a contraction of a product of projectors ,

(24) |

with and being two independent copies of spinors in the bulk, with no simplicity constraints imposed on them. This is exactly the expression we hinted to in (21) and it describes a mapping from the Spin(4) representation given by the spinors and into the SU(2) representation given by the spinor . The gluing of 4-simplex amplitudes in this language requires a product of two such projectors, giving a map , which can be graphically represented as

(25) |

where the tetrahedra 1 and 2 belong to different 4-simplices. Using this graphical notation, the amplitude for a single 4-simplex then contains a double strands in the bulk corresponding to the two copies of spinors and , and a single copy on the boundary corresponding to the spinors . This can be seen in Fig. 1.

#### ii.3.2 Constrained propagator

Since spin foam amplitudes for BF theory are constructed from contractions of projectors (5) into graphs corresponding to 4d quantum geometries, it is natural to impose the simplicity constraints directly onto the projectors themselves and hence on all the labels of the coherent states. Let us consider the Spin(4) projector obtained by taking a product of two SU(2) projectors

(26) |

where we use a prime to distinguish the left and right SU(2) sectors. We now impose the holomorphic simplicity constraints on both incoming and outgoing strands

This makes the two products of spinors proportional to each other, with the proportionality constant being , so we get that the constrained projector is given by

(27) |

where we have defined , . Note that by imposing the constraints, no longer satisfies the projection property , hence we will refer to it as the propagator. This is the object we alluded to graphically in Eq. (22). We can simplify this expression into a single sum by letting to arrive at the most compact form of the constrained propagator

(28) |

where we have recognized the power series expansion of the hypergeometric function

(29) |

We can now notice that the constrained Spin(4) propagator is just an SU(2) projector with non-trivial weights for each term that depend on the Barbero-Immirzi parameter. Note however that the power of in Eq.(27) tracks the homogeneity of the left and right SU(2) sectors and as such preserves the Spin(4) invariance.

The imposition of simplicity constraints on all of the spinors has the additional effect of modifying the measure of integration on

(30) |

The partition function can now be constructed from these constrained propagators. In Banburski:2014cwa we proposed the amplitude

(31) |

where is a face weight, the set is the set of integers satisfying and contraction of spinors according to the 2-complex on different edges is implied. The is a constrained propagator at fixed spins and it is given by

(32) |

An example of an amplitude of two 4-simplices glued along one tetrahedron is shown in Fig. 2. The main thing to note is that in this imposition of simplicity constraints, there is only one copy of strands, corresponding to spinors , to be contracted in the bulk of the 4-simplex amplitude. The boundary data is the same as in the DL model and is given by the single copy of spinors .

We thus see that the main difference between the two models is on the inside contraction in the 4-simplex. In the constrained propagator model, the simplicity constraints are imposed on both the boundary and the interior of the 4-simplex. In contrast, the standard approach is to have simplicity imposed only on the boundary, with the interior of the 4-simplex contractions identical to that of Spin(4) BF theory. The obvious worry one could have is that the constrained propagator model is over-constrained and does not lead to General Relativity in the semi-classical limit. We show in the next section however, that at least to the leading order, both models have the same asymptotic behaviour.

## Iii Asymptotics

In this section we will calculate the asymptotics of the two models with different imposition of simplicity constraints. First we show that the Dupuis-Livine model indeed has the same asymptotic behaviour as the EPRL-FK models. We then show that there are non-trivial cancellations in the asymptotic expansion of the constrained propagator model that lead to the same semi-classical limit as the DL model.

#### iii.0.1 The dihedral angle

Before we calculate the asymptotic expansion of the Spin Foam amplitudes, we have to understand how to reconstruct from our data the angle appearing in the classical area-angle Regge action Dittrich:2008va :

(33) |

where is the area of face shared by tetrahedra and , which share a common face with each other, and is the 4-d dihedral angle, which is the angle between the two 4-vectors normal to the two tetrahedra .

We can find the expression for the 4-d dihedral angle using Eq. (18) from the section on simplicity constraints:

(34) |

Using the expression of eq.(13), we can write the cosine of dihedral angle in terms of spinors,

(35) |

From the above two expressions, we can see that to decide the cosine of the dihedral angle , we need the data of two group elements associated with two nodes (tetrahedra), or the data of both left and right spinors of any one strand from each of the two tetrahedra. In summary,

Let us recall additionally, that the models we consider have Spin(4) symmetry, so we can rotate these results by a Spin(4) transformation .

#### iii.0.2 The asymptotics of DL model

An apparent difference between the holomorphic simplicity constraints and the ones in Euclidean EPRL/FK models is that they are constraints on spinors. However, they lead to the same constraint between spins,

(36) |

for the coherent intertwiners in the large limit Dupuis:2011fz . In this section, we briefly show that for the amplitude of a 4-simplex, the DL model has the same action at critical points as EPRL/FK models for Barbero-Immirzi parameter .

We can rewrite the amplitude (23) of a 4-simplex by expanding it in power series as

(37) |

Now that we have made the summation over spins explicit, we can re-exponentiate this expression to get the effective action of a 4-simplex amplitude with

(38) |

where the numerical factor is given by

(39) |

It is important to note that this action is complex-valued. To study the asymptotic behaviour of the amplitude, we have to separate the real and imaginary parts. The real part of the action is

(40) |

In the asymptotic analysis of complex functions the main contribution to the integral comes from critical points, which are stationary points of the action for which the real part is maximized. The critical point equations we get from variation of spinors are the closure constraints

(41) |

and the orientation condition requiring certain vectors to be anti-parallel, which we get from the maximization of the real part of the action:

(42) |

Using the relation (14) between vectors and spinors, we find that these conditions imply that the action of group elements on a spinor rotates it up to a phase into :

(43) |

This implies that the following identity holds

(44) |

The reconstruction theorem from Barrett:2009gg tells us now that given non-degenerate boundary data satisfying the closure constraint (41) and a set of group elements solving the orientation condition (42), we can reconstruct a geometric 4-simplex with the B field given by

(45) |

with the outward-pointing normal obtained by acting with the Spin(4) element on the vector .

At this point, it is clear that the critical action of DL model is exactly the same as the one calculated in the asypmptotic analysis of the EPRL model in Barrett:2009gg , and the imaginary part of the action reads

(46) |

where . To relate this to the area-angle Regge action, we have to relate the ’s to the dihedral angle. We cannot directly use our expression in Eq. (35) for the dihedral angle, since we no longer have the information about both the left and right spinors. We can however use the result of the reconstruction theorem from the Eq.(45) to construct the dihedral angle by the data as follows

(47) |

Notice however that we can obtain the same trace from the Eq. (44), which tells us that we can identify the cosine between the phase and the dihedral angle

(48) |

In Barrett:2009gg it has been shown explicitly that the phase difference and the dihedral angle can be identified up to a sign, which is due to the relative orientation of the bivector and 4-simplex. The angle can be shown to be proportional to Barrett:2009gg .

Hence the semi-classical limit of the Dupuis-Livine model is the same as the EPRL-FK models and is given by the action

(49) |

Since in Loop Quantum Gravity the spectrum of the area operator is given by , in the large spin limit we have obtained exactly the area-angle Regge action Regge:1961px ; Dittrich:2008va .

#### iii.0.3 The asymptotics of constrained propagator model

Let us now finally show that the constrained propagator model also leads to the same semi-classical limit as the EPRL-FK models. We first have to rewrite the amplitude in terms of group variables. Recall that we can write an SU(2) propagator as

(50) |

Thus taking two copies of such projectors and constraining them both in the and in the spinors, we get that the constrained propagator (28) can be written as

(51) |

The 4-simplex amplitude is now just a simple contraction of 5 such propagators. To compare it however to the amplitude in the DL model, we have to perform the integrate out the spinors in order to have the same number of variables. After the integration, the amplitude becomes

(52) |

We can see that there is a mixing between left and right sectors – while in the DL model the left and right group elements are multiplied separately as in Eq.(37), here the relevant group elements become a combination . Expanding this in a power series it would seem we would get four independent terms. However, since in the large limit the holomorphic simplicity constraints imply that we have , one can show that only three summations are independent, so the amplitude can be written as

(53) |

with the the spins satisfying