Generating Functions for Coherent Intertwiners

Generating Functions for Coherent Intertwiners

Abstract

We study generating functions for the scalar products of SU(2) coherent intertwiners, which can be interpreted as coherent spin network evaluations on a 2-vertex graph. We show that these generating functions are exactly summable for different choices of combinatorial weights. Moreover, we identify one choice of weight distinguished thanks to its geometric interpretation. As an example of dynamics, we consider the simple case of SU(2) flatness and describe the corresponding Hamiltonian constraint whose quantization on coherent intertwiners leads to partial differential equations that we solve. Furthermore, we generalize explicitly these Wheeler-DeWitt equations for SU(2) flatness on coherent spin networks for arbitrary graphs.

Introduction

Loop quantum gravity is an approach to quantizing general relativity where excitations are carried by embedded graphs so that the kinematical Hilbert space is spanned by (diffeomorphism equivalence classes of) such graphs. When restricted to a single graph, the kinematics is equivalent to lattice gauge theory and thus can be derived from a phase space with Wilson lines on links and their conjugate -valued electric fluxes. Moreover this phase space admits a natural interpretation in terms of discrete geometries known as twisted geometries twisted1 (); polyhedron ().

This is the frame where the dynamics has to be formulated and therefore the quantization heavily relies on representation theory. Typical objects from re-coupling theory are the Wigner 3nj-symbols, which depend on angular momenta (spins) built from sums of products of Clebsch-Gordan coefficients varshalovich (). They arise in (loop) quantum gravity as evaluations of the spin network states of geometry on the trivial connection and as the building blocks of the spinfoam transition amplitudes between those states.

While some basic properties have been known for several decades, a need for new results involving more and more spins have appeared and have led to some interesting progress. They come from various areas of physics, like quantum information spinnets-marzuoli (); 3nj-marzuoli (), semi-classical approximations for quantum angular momenta littlejohn (); Yu (), and quantum gravity 6jnlo (); pushing6j (); 6jmaite (); barrett-asym-summary (); recursion (); recursion6j_bis (); 3njsmall (); dowdall-handlebodies ().

While these spin network evaluations are very complicated, it has been noticed that they admit generating functions which are remarkably simple and can be written in a closed form. Schwinger calculated in his seminal paper schwinger:52 () some generating function for the Wigner 6j and 9j-symbols. Bargmann then derived them again through a different reasoning in bargmann:62 () using Gaussian integrals. Since then, generating functions for generic symbols have been evaluated, mostly on algebraic grounds wu-9j (); huang-wu-2j-15j (); labarthe (); schnetz (); garoufalidis (); costantino-generating ().

It has been recently understood that such generating functions are very useful for quantum gravity recursion_spinor (); jeff (). Indeed the discrete geometry of loop gravity states - twisted geometries - can be formulated classically with spinors twisted2 (), which are quantized as Schwinger’s bosonic operators. This way, loop quantum gravity wave-functions can be represented in a basis of coherent states spinor (); spinor_johannes (); un0 (); un1 (); un2 (); un3 (); un4 (); un4_conf (). In recursion_spinor (), it was shown that the wave-function for a flat geometry on the boundary of a tetrahedron (in the context of three-dimensional gravity) is just the Schwinger’s generating function for 6j-symbols when written in the appropriate coherent basis.

This result is exciting for the future. Indeed it means that working with some coherent state basis, one trades spin network evaluations to their generating functions, which are holomorphic functions of classical spinors. As often, we expect generating functions to be easier to handle than the symbols themselves. The usual difficulties can be translated into problems of complex analysis. For instance the asymptotic regime of Wigner symbols is hidden in the poles of the Schwinger’s generating functions.

Moreover, simple quantum gravity dynamics and aspects of more realistic dynamics have been formulated in terms of recursion equations on the amplitudes recursion (). For example, 3d gravity and the topological 4d BF model admit Wheeler-DeWitt equations which are difference equations solved by Wigner symbols recursion3d (); recursion4d (). When re-expressing those equations in a spinor coherent basis, they become partial differential equations recursion_spinor ().

However, those partial differential equations may be quite complicated. They actually depend on a choice of basis of coherent states, corresponding to a choice of combinatorial weights in the definition of the generating functions. The Schwinger’s choice which has been used so far in the literature is certainly a good choice to re-sum spin network evaluations for generic graphs as shown in jeff (), but some other choices may lead to simpler partial differential equations and saddle points with more straightforward geometric interpretation.

We investigate those ideas in the present paper using the special case of the graph with two vertices connected by links. It is obviously a good testing ground, already considered in 2vertex (), but it is also interesting in itself because the generating functions in this case generate scalar products of -valent intertwiners, which are central objects in quantum gravity.

In Sec. I we review the spinorial description of the LQG phase space and its quantization, presenting different bases of coherent intertwiners. In Sec. II we start to focus on the 2-vertex graph and show that the spin network evaluation in such coherent states basis is a generating function for the intertwiner scalar products, and can be written in terms of -invariant variables (cross-ratios). We also introduce several choices of combinatorial weights for the generating function, corresponding to different choices of coherent intertwiners.

In Sec. III we show that it is actually possible to calculate exactly these generating functions for different choices of combinatorial weights, at least in the case of the 2-vertex graph. We discuss the geometric content of their saddle point evaluations in Sec. IV, where it turns out that a natural geometric interpretation comes out when using a specific choice of weight which we will call the geometric choice.

The -flat dynamics (as in 3d gravity) is considered in Sec. V and it is found that the simplest Wheeler-DeWitt equation is obtained when considering the geometric generating function. Finally we give some preliminary calculations of generating functions for arbitrary graphs, and in particular obtain derive the Wheeler-DeWitt equations of flat dynamics.

The appendix A contains material on constrained Gaussian integrals, and B relates generating functions of scalar products of coherent intertwiners to generating functions of Wigner symbols.

I Coherent Intertwiners and Structure

We present a quick review of the spinorial framework for intertwiners as developed in un2 (); un3 (); un4 (); spinor (), following the previous identification of an action of the unitary group on the space of intertwiners un0 (); un1 (). In this setting, intertwiners appear as the quantization of classical polyhedra. We start by reviewing the spinor variables for polyhedra and their classical phase space. We then review their quantization into intertwiner states and the operators acting on the intertwiner space. This leads us to the definition of coherent intertwiners.

i.1 Spinors and Classical Setting for Intertwiners

In the following, we call a spinor a complex 2-vector , living in the fundamental representation of , and we define1 its dual spinor

(1)

We provide with the natural symplectic structure defined by the canonical Poisson bracket:

(2)

We further define the 3-vector obtained by projecting the spinor on the Pauli matrices:

(3)

where the Pauli matrices are normalized such that for and running from 1 to 3. Whenever there is no confusion, we will omit the argument for . This vector has norm and completely determines the spinor up a global phase. Moreover, the Poisson brackets of its components define a algebra:

(4)

Actually generates the action of on the spinors, for .

Now, the phase space for intertwiners with legs is defined by spinors living in satisfying the closure constraints:

(5)

These are first class constraints generating global transformations on all spinors. From a geometrical perspective, the constraint implies that the vectors define a unique convex polyhedron with faces: the vectors are the normal vectors to the faces. For more details on the reconstruction of this dual polyhedron, the interested reader is referred to polyhedron ().

Thus our intertwiner phase space is defined by the symplectic reduction by the closure constraints: is the space of spinors satisfying the closure constraints and up to global transformations. It describes the set of framed polyhedra with faces un1 (); un2 (). By “framed”, we mean that we have an extra phase attached to each face, which is the degree of freedom contained in compared to . These faces are mostly irrelevant when studying single intertwiners, but are needed when gluing those intertwiners into spin network states.

Before moving to the quantization and to intertwiner states, let us further define -invariant observables on the constrained phase space and describe the natural action carried by the space of spinors, which will be essential later on. First, we identify the following -invariant observables given by the scalar products on the spinors with each other and their dual un0 (); un1 (); un2 (); un3 (); un4 ():

(6)

The satisfy , while the ’s are holomorphic, anti-symmetric and satisfy the Plücker relations

(7)

The standard scalar products between 3-vectors are easily expressed in terms of these observables:

(8)

The Poisson brackets of the ’s and form a closed Lie algebra (for more details see un2 (); un3 (); un4 (). In particular, the ’s form a -algebra:

(9)

Actually, as shown in un2 (); un3 (), the ’s generate the natural -action on the spinors: , for . The key point is that this action commutes with the closure constraints and is cyclic for fixed total area . Defining the completely squeezed configuration,

(10)

we can indeed get arbitrary spinors satisfying the closure constraints by acting with unitary matrices on this set of spinors (and appropriately rescaling by the total area):

(11)

The closure constraints come from the unitarity of the matrix .

i.2 Quantization and Coherent Intertwiners

Following the previous work un2 (); un3 (); un4 (); spinor (); spinor_johannes (), we quantize this classical phase space as a set of harmonic oscillators:

(12)

So now we have a couple of harmonic oscillators attached to each leg of the intertwiner. We then quantize the vectors and observables , and , using normal ordering when necessary,

(13)
(14)

The commutators between these operators reproduce exactly the algebra of the Poisson brackets. The operators are the generators of the algebra attached to the -th leg. Then the total energy on the -th leg, , commutes with these generators and give the spin of the -representation. More precisely, we have Schwinger’s representation for the algebra and we can easily go between the standard oscillator basis labeled by the number of quanta and the usual magnetic momentum basis for spin systems by diagonalizing the operators and :

(15)

So fixing the total energy of the two harmonic oscillators, we fix the spin of the -representation attached to the leg . Calling the Hilbert space of a single harmonic oscillator, this allows us to decompose the tensor product in -representations:

(16)

where we write for the -representation of spin .

We now consider copies of this representation of , and impose the closure constraints , which amount to require the invariance under the global -action. This means that we are looking at -invariant states in the tensor product of the -representations living on the legs around the vertex, i.e. intertwiners between the spins . This defines the Hilbert space of -valent intertwiners from our collection of harmonic oscillators,

(17)

The operators , and commutes with the generators of the global -transformations, , and thus act on the Hilbert space of intertwiners . As shown in un0 (); un1 (), the operators form a -algebra at the quantum level and generate a action on intertwiner states, similarly to the -action on the sets of classical spinors. These -transformations leave invariant the total area . This leads to the following decomposition of the space of -valent intertwiners:

(18)

Each subspace carries an irreducible representations of generated by the operators un1 (). Moreover this endows the Hilbert space with a Fock space structure, with the operators acting as annihilation operators going from to while the operators act as annihilation operators going from to un2 ().

We can then build coherent states for each of those Hilbert spaces, from the irreducible representations to the whole Hilbert space of -valent intertwiners . The coherent intertwiners on are obtained by group averaging over the harmonic oscillator coherent states. Coherent states on and on are obtained by projecting them at fixed total area or at fixed spins . Nevertheless, coherent intertwiners were slowly constructed in the reverse order, with a first definition of the Livine-Speziale coherent intertwiners ls (), then the definition of the coherent states un2 () and finally the introduction of the final coherent intertwiners un3 (); un4 (). We summarize their definitions and properties below.

  • Coherent States:

    They are defined by acting with the creation operators on the vacuum of the harmonic oscillators, to build the standard coherent states for the harmonic oscillators, and then by projecting to a fixed total energy in order to fix the spin . We denote them , with a spin label and a spinor ,

    (19)

    Their norm is easy to compute: . They are coherent states à la Perelomov, i.e. they transform covariantly under -transformations (e.g. un2 (); un3 ()),

    (20)

    where acts on the spinor as a matrix in the fundamental -representation2. Furthermore, these states are the tensor power of the states in the spin- representation, . Finally, these states are semi-classical. They are peaked with minimal uncertainty around the expectation values of the -generators :

    (21)
  • LS Coherent Intertwiners:

    Coherent intertwiners were first introduced in ls () from tensoring together coherent states and group-averaging in order to get -invariant states. This was re-cast in terms of spinors in un2 (); un3 (). Such a -valent coherent intertwiner is labeled by a list of spins and spinors attached to each leg and defined by

    (22)

    The norm and scalar product of these LS coherent intertwiners can be expressed as a finite sum of ratios of factorials un2 (). Such formulas are also directly deduced from the scalar product of the coherent states described below.

    An important point is that it is not required that the classical spinors labeling the states satisfy the closure constraint. One can show that the LS coherent intertwiners defined by closed sets of spinors are nevertheless dominant and that those labeled by spinors which do not satisfy the closure are exponentially suppressed ls (). This is done by computing asymptotically their norm in the large spin regime and showing that closed sets of spinors dominate the integral over coherent states in the decomposition of the identity on the Hilbert space . Such peakedness properties have been useful to define the EPRL-FK spinfoam models eprl (); fk (); ls2 ().

    As done in spinor (), it is possible to compute the action of the and operators on these states by commuting their action with the operators defining the coherent states3. This gives

    (23)
  • Coherent States:

    They are defined on for fixed total area , un2 (),

    (24)

    They are superpositions of LS coherent intertwiners un2 () as follows

    (25)

    From their definition above, one can prove that they are covariant under the -action un2 (), hence the name of coherent states,

    (26)

    where the arbitrary Hermitian matrix generates the unitary transformation. Their scalar products and norms are explicitly known un2 (),

    (27)
    (28)

    When the closure constraints are satisfied, i.e. when , the norm simplifies to where is the total area . The action of the operators and reads un3 (); spinor ():

    (29)

    Finally, all these properties allow to compute exactly the expectation values of the operators un2 ():

    (30)

    where we assumed that the spinors satisfy the closure condition. Let us emphasize that this expectation value is exact while the expectation values of the -observables on the LS coherent intertwiners are only known asymptotically in the large spin limit.

  • Coherent Intertwiners:

    The last notion of coherent intertwiners was introduced in un3 (). They truly represent coherent states on the spinorial phase space: they are simply labeled by a phase space point, i.e. spinors (up to global rotations). More explicitly, they are defined as the eigenstates of the annihilation operators (which is possible since the operators all commute with each other). Their expansions in the previous bases are

    (31)

    The last equality shows that these coherent intertwiners are the group averaging of the standard (unnormalized) coherent states for the harmonic oscillators. Using the above expansion onto the states and the action of the annihilation operators on them4, we easily show that

    (33)

    We can similarly compute the action of the other -invariant operators and ,

    (34)

    We further compute the norm and scalar product of these states un3 ():

    (35)
    (36)

    where the are the modified Bessel functions of the first kind. Finally, we also give the expectation values of the -operators:

    (37)

    The asymptotic behavior of the coherent states for large area is given by5

    (38)

    showing that these coherent intertwiners have the right semi-classical behavior.

i.3 From spinors to SU(2) invariant variables

In this article it will be convenient to work sometimes with invariant variables instead of spinors. One way to get them is is as follows. First we consider the variables formed from . But they are not independent variables, due to the Plücker relations (7). The latter exhaust the dependence relations between the , and can be solved to extract the invariant content. One uses the Plücker relations to express some of the in terms of others. Depending on which of them we eliminate, one gets different sets of variables. For instance, one can choose

(39)

where the variables , for are the cross-ratios,

(40)

We will show how that is done in practice in the section II.2.

That gives (complex) variables per intertwiner, which is the expected counting in agreement with the standard -valent tree unfolding ( spins plus internal spins) of intertwiners. The choice of which are eliminated corresponds to a choice of cross-ratios. We expect that choice to be equivalent to the choice of a tree to unfold the intertwiner, as already shown in holquantumtet () for .

Then, one can try to use those variables to build coherent intertwiners. The simplest way is to re-express the coherent intertwiners described above. It can be done as in holquantumtet () by studying the action of on spinors to extract the dependence on the . One gets for LS intertwiners

(41)

where is a state which only depends on the cross-ratios, and . Other equivalent choices of factorization can be obtained by acting with elements of the permutation group to exchange some links holquantumtet ().

A direct derivation of (41) via the scalar product and the Plücker identities will be given in the section II.2.

The scalar product between the states is known in the case , holquantumtet () but not in general6. In this paper we will show a generic formula for this scalar product. It has the following polynomial form which is different of that of holquantumtet () for .

Result 1.

The scalar product admits the form

(42)

We will show this result in the section II.2 and also build generating functions for these scalar products.

Ii Evaluations of Coherent Spin Networks as Generating Functions

ii.1 Coherent Spin Networks on the 2-Vertex Graph

Classical Phase Space on the 2-Vertex Graph

Figure 1: The 2-vertex graph: the two vertices and are linked by edges. We choose all the edges oriented in the same way with as their source vertex while is the target vertex. The spinors live around the vertex while the spinors live on the legs of the vertex . By courtesy of Iñaki Garay from spinor ().

Let us consider the 2-vertex graph, made of two vertices and connected by links, as pictured on the figure 1, and start by describing the classical phase space of spinors on that graph, as defined in un4 (); spinor (); spinor_johannes (); sfcosmo (). We have two sets of spinors, at the vertex and at the vertex , both satisfying the closure constraints, and , which translates into in terms of 3-vectors. Moreover, we impose matching conditions, or equivalently for all edges . All these constraints form a first class system. While the closure constraints generate transformations at each vertex, the matching conditions generate -phase multiplications on the spinors on each edge . The resulting phase space on the 2-vertex graph is then defined as the symplectic reduction . This constrained phase space has dimension and can be identified with the gauge invariant holonomy-flux phase space of loop quantum gravity on the 2-vertex graph, for which the configuration space is defined as the set of group elements up to global left and right translations, . Furthermore, the space of holomorphic functions on is shown to be isomorphic to the space of -functions on i.e. to the Hilbert space of spin network functions on the 2-vertex graph spinor (); spinor_johannes ().

This isomorphism is realized through the reconstruction of the holonomies from the spinor variables as first shown in twisted2 () and further investigated in spinor (); spinor_johannes ():

(43)

Indeed, when we assume the matching condition, i.e. that the spinors have equal norm, , this is the unique group element mapping the spinor on .

A more detailed analysis of the classical phase space associated to the 2-vertex graph, its various parameterizations, its geometrical interpretation and its relevance for defining cosmological settings in loop (quantum) gravity can be found in sfcosmo ().

Flat configurations are defined by up to gauge transformations at the two vertices, i.e. all equal to one fixed group element for all edges . This group element maps all the spinors on their counterpart . In particular, it implies that the -invariant observables and are equal for both sets of spinors and , i.e. and .

We can actually go further. Indeed, as shown by proposition 1.2 in un4 (), the are a complete set of -invariant observables. Assuming the closure constraints on both sets of spinors and , then assuming , i.e. , for all pairs of (different) edges is equivalent to the existence of a group element such that for all edges :

(44)

This fully characterizes the flat configurations on the 2-vertex graph.

Quantum States

Quantum states of geometry on this graph, as defined by loop quantum gravity, are gauge invariant functions of the holonomies along its edges, i.e. functions of group elements with a invariance at both vertices: