A Discrete and Coherent Basis of Intertwiners
We construct a new discrete basis of -valent SU(2) intertwiners. This basis possesses both the advantage of being discrete, while at the same time representing accurately the classical degrees of freedom; hence it is coherent. The closed spin network amplitude obtained from these intertwiners depends on twenty spins and can be evaluated by a generalization of the Racah formula for an arbitrary graph. The asymptotic limit of these amplitudes is found. We give, for the first time, the asymptotics of 15j symbols in the real basis. Remarkably it gives a generalization of the Regge action to twisted geometries.
It has been observed long ago that the compositions of quantum states of angular momentum are related to geometrical objects Schwinger (); wigner (); Bargmann (). The simplest example is the Clebsch-Gordan coefficients which vanish unless the spins satisfy the so called triangle relations. A less trivial example is the Wigner 6j symbol which vanishes unless the spins can represent the edge lengths of a tetrahedron. This insight was one of the motivations which led Ponzano and Regge to use the 6j symbol as the building block for a theory of quantum gravity in three dimensions Regge:1961px (); PR (), together with the fact that the asymptotic limit of the 6j symbol is related to the discretized version of the Einstein-Hilbert action. In higher dimensions this line of thought led to the idea of spin foam models which are a generalization of Ponzano and Regge’s idea to a four dimensional model of quantum General Relativity. For a review see Perez:2012wv ().
Following a canonical approach, Loop Quantum Gravity came to the same conclusion: Geometrical quantities such area and volume are quantized Rovelli:1994ge (). In fact there are many remarkable similarities between LQG and spin foam models. For instance, the building blocks of both models are SU(2) intertwiners which represent quanta of space Barbieri:1997ks (); Baez:1999tk (); Rovelli:2006fw (). An intertwiner is simply an invariant tensor on the group. In other words if we denote by the dimensional vector space representing spin then an intertwiner is an element of the SU(2) invariant subspace of this tensor product which we will denote
The vectors in this Hilbert space will be referred to as -valent intertwiners since they are represented graphically by an -valent node. The legs of this node carry the spins which can be interpreted as the areas of the faces of a polyhedron OH (); LFetera1 (); LFetera2 (); Bianchi:2010gc () which is a consequence of the celebrated Guillemin-Sternberg theorem GS (). In this paper we will be focused on 4-valent intertwiners but many of the methods developed here can be extended to the -valent case.
Two types of basis for the space (1) are usually considered. The most common one is a discrete basis which diagonalizes a commuting set of operators invariant under adjoint action. Each of these operators is associated with a decomposition of an -valent vertex as a contraction of 3-valent ones and the basis elements are constructed by the contraction of 3-valent intertwiners along the corresponding channels. In the 4-valent case the different bases correspond to the channels and are labelled by one extra spin. Bases constructed in this way are orthonormal, but lack sufficient data to describe a classical geometry. For example a tetrahedron is uniquely determined by six quantities, such as the four areas and two of the dihedral angles between faces. Therefore the orthonormal basis of 4-valent intertwiners, having only five labels, is not suitable for the specification of a fixed tetrahedral geometry.
Another basis usually considered is the coherent state basis (introduced in Loop gravity by Livine Speziale coh1 ()) which overcomes this difficulty by introducing an overcomplete basis for which is labelled by a set of normal vectors. In the semiclassical limit these normal vectors satisfy the closure relation and can be identified with the vectors normal to the faces of a tetrahedron whose areas are proportional to the spins Conrady:2009px (); Freidel:2009nu (). These insights have since led to a reformulation of LQG in terms of twisted geometries twisted () obtained by matching the normal vectors corresponding to faces of classical polyhedra, but such that the shapes of the glued faces do not necessarily match. One of the drawbacks of this basis, however, is that it introduces a continuum of states to represent a simple finite dimensional Hilbert space. Moreover, in this representation the link with the real basis and its simplicity is lost.
What we want to investigate is whether there exists a basis which captures the essence of both constructions by being at the same time discrete and coherent. We provide here such a construction for the Hilbert spaces and focus our description to the 4-valent case, which exemplifies the main features of this basis. We will also show that it leads to a simple generalization of the Racah formula when we use these intertwiners in the spin-network evaluation of a general graph. This basis arises naturally in the expansion of generating functionals for spin networks which were provided in FH_exact (). Finally we present a new symbol corresponding to the coherent spin network amplitude on a 4-simplex which is a simple generalization of the symbol.
We find that this new basis of intertwiners is overcomplete, but is labelled by a set of discrete labels. Further, we show that the discrete basis generates the different orthonormal bases by simply summing over the extra labels. In this way many of the different variations of recoupling coefficients in the orthonormal bases can be generated from these more fundamental amplitudes. For example the different versions of the symbol can be obtained by summing over five spins of the symbol.
Finally we show that the semiclassical limit of the discrete basis corresponds to a classical framed tetrahedron in the same way as the coherent intertwiners. A framed tetrahedron is a tetrahedron with a unit vector in each of the faces representing a 2d frame. This frame vector is determined by the phase of the spinor and encodes the extrinsic geometry of the triangulation. The asymptotics of the symbol imply a classical action which, under certain geometricity constraints, is found to agree with the Regge action, which agrees with the analysis in Dittrich:2008ar () and Dittrich:2010ey (). When these constraints are not imposed the geometry is twisted in the sense of twisted ().
The paper is organized as follows. First we define the new basis of -valent intertwiners in the holomorphic representation and we describe the 3-valent case. Next we investigate the 4-valent case and construct the resolution of identity. We then compute the action of the invariant operators and show how the new basis relates to the orthonormal bases. Using generating functional techniques we compute the scalar product in the new basis, and use the relations with the orthogonal basis to generate the various other possible scalar products. We then discuss the utility of this basis in representing and computing spin network amplitudes and introduce the symbol in the 4-valent case. Finally we study the semi-classical behaviour of the new states and we show that they correspond uniquely to classical framed tetrahedra. Moreover, the asymptotics of the symbol is shown to admit an interpretation as a generalization of the Regge action to twisted geometry.
Ii The New Basis
One particularly useful representation of SU(2) is the so called Bargmann-Fock or holomorphic representation Bargmann (); Schwinger (). This space consists of holomorphic functions on spinor space endowed with the Hermitian inner product
where and is the Lebesgue measure on and we use the bra-ket notation for the scalar product of the two states defined by . The group SU(2) acts irreducibly on representations of spin given by the dimensional subspaces of holomorphic functions homogeneous of degree . The standard orthonormal basis with respect to this inner product is given by
which are simply the holomorphic represention of the SU(2) basis elements which diagonalize the operator . Here represents the components of the spinor .
In the following we will heavily use the fact that there are two SU(2) invariant products on spinor space, only one of which is holomorphic:111In fact because is holomoprhic it is automatically invariant. The other SU(2) invariant is :
where we use the notation
We will also use the notation to denote the conjugate spinor .
In the spin representation we define coherent states to be the holomorphic functionals
These states possess the characteristic property that their scalar product with any spin state reproduces the functional , that is
This follows from a direct computation which shows that This property implies that we can identify the label of with the state when evaluated on a spin functional. In the following we will use interchangeably the notation for the labels.
ii.1 Intertwiner Bases
In this holomorphic representation there are two natural and straightforward bases of -valent intertwiners, i.e. functions of the spinors which are SU(2) invariant and homogeneous of degree in . The first one is the Livine-Speziale coherent intertwiner basis coh1 (), and the second one is the discrete basis which is the new basis we want to study here.
The Livine-Speziale coherent intertwiners coh1 () are defined by group averaging as the following holomorphic functionals:
Note that the normalization of these states is different from coh1 () to better suit the Bargmann scalar product (2). These states are coherent in the sense that their scalar product reproduces the holomorphic functional, they are labelled by the continuous set of data and they resolve the identity:
This is shown by using the identity , which itself is proven by summing over and performing the Gaussian integration.
We will now show how to construct a new basis which is also coherent, resolves the identity, but is labelled by a discrete set. Since the product is holomorphic and SU(2) invariant it can be used to construct a complete basis of the intertwiner space by
This basis is labelled by non-negative integers with . Note that we are free to choose a phase convention and for simplicity we will choose it to be unity for now.222Later we will see that the asymptotic limit of the intertwiners will imply a canonical phase.
For a basis representing the subspace with fixed spins , we have homogeneity conditions which require the integers to satisfy
The sum of spins at the vertex is defined by and is required to be a positive integer. From the relation we see that these states satisfy the reality condition
Furthermore, from the coherency property (8) we can easily compute the overlap of these states with the coherent intertwiners:
where the last equality follows from the reality condition (11) and the fact that .
ii.2 3-valent Intertwiners
In the case there is only one intertwiner. Indeed, given the homogeneity restriction requires which can be easily solved by
In this case the fact that homogeneous functions of different degree are orthogonal implies that form an orthogonal basis 333One can also arrive at this basis by considering the respresentation space of symmetrized spinors. For details see appendix A of Rovelli:2004tv (). The two approaches are essentially the same, however in the holomorphic representation we have the advantage of tools like generating functionals and Gaussian integration. of (1).
Since there is only one holomorphic function it must be proportional to the Wigner 3j symbol
where the triangle coefficients can be found to be
Note that we could divide by to normalize this basis, but it will be simpler to instead work with these unnormalized states.
For there are more basis elements than the dimension of the intertwined space so the basis is no longer orthogonal. Indeed, since we have ’s satisfying relations (10) these intertwiners are labelled by integers. But this is clearly more that the dimension of the Hilbert space of -valent intertwiners, which is known to be labelled by integers, i.e. by contracting only 3-valent nodes. This means that the basis given above is overcomplete.
Another way to understand this counting is to recall that the algebra of gauge invariant operators acting on is given by for where denotes the angular momentum operator action in the direction. These operators satisfy the closure relation and the action of is given by multiplication by . These relations mean that we can express any instance of say, by a summation of operators depending on for . Thus a good basis of operator is for instance for and . There are such operators. They satisfy one relation that stems from the closure relation which is
This makes it clear that if we want to maximally represent these operators we need labels. These operators do not commute, therefore these labels represent an overcomplete basis. A maximal commuting subalgebra is of dimension .
For example, in the case the basis is labelled by integers while we need only one, and for it is labelled by integers while we need only two. Despite this overcompleteness we will be able to determine all of the necessary properties of these states and we will discover some interesting relations between the orthogonal bases on the one hand and coherent intertwiners on the other.
Iii The 4-valent case
We now focus on the case . A very convenient labelling of the basis is done in terms of three spins , , which refer to the three channels in which a 4-valent vertex can be split into two three valent ones. The relationship between these labels and the labels is given by
where , , and are such that the are non-negative integers. The constraints in (10) imply that , thus we also have
Summing over all shows that , , and are not independent but satisfy the relation
We can therefore label the -valent intertwiner basis by the four spins and two extra spins and we will henceforth denote by the corresponding integers in (19, 20). These integers cannot take arbitrary values, since are restricted by , this restriction444It is given by (22) (23) (24) is denoted by . In the case all spins are equal to this is simply , .
We will denote the corresponding basis by where
In the following we will omit the subscript and use the shorthand for notational simplicity when the context is clear and the external spins are fixed.
iii.1 Overcompletness and Identity Decomposition
As discussed above, the basis has one extra label and is thus overcomplete. We will now investigate the nature of the relations among these states which is summarized by the following theorem:
The states are not linearly independent; all the relations among them are generated by the fundamental relation
where stands for .
It turns out that the relation among the states is easily seen in the holomorphic representation. It is well known that the gauge invariant quantities are not independent, they satisfy the Plücker relation:
In order to write the effect of this relation on the states lets compute first the effect of multiplication by one monomial
where we used that , while , and . Performing similar computations for the different monomials we find that the multiplication by the Plücker relation can be implemented in terms of an operator whose image vanishes identically. It is defined by where is given by
here denotes . By shifting the parameters and and using that etc. we obtain the desired relation stated in the theorem. By taking powers of the operator we can generate many more relations which we will discuss in a later section. Despite the linear dependence of these states they admit a resolution of identity, consistent with a coherent state basis:
The resolution of identity on the space of 4-valent intertwiners has the simple form
We give a proof of this theorem in Appendix A which is specific to the 4-valent case. The resolution of identity in the -valent case has a similar form and follows from the relations (13). We will show that despite the fact that they are discrete, the basis shares many of the same properties as the coherent intertwiners such as the correspondence with classical tetrahedra in the semi-classical limit. In addition the states also possess a simple relation with the orthogonal basis as we will show in the next section .
iii.2 The Relation with the Orthogonal Basis
In the previous sections we introduced a new and overcomplete basis of the space of 4-valent intertwiners which provided a simple decomposition of the identity. On the other hand, the standard basis of 4-valent intertwiners is orthogonal, and is defined by the eigenstates of either of the invariant operators or or . We will denote these orthogonal bases by and and respectively. We would now like to investigate the action of the and channel operators and on as well as the relationship between the four bases: , , , .
It is well known that, up to normalization, the usual 4-valent intertwiner basis is obtained by the composition of two trivalent intertwiners. For now we will focus on the states, which in the holomorphic representation, are defined to be
as the quadratic combinations
The operator acts nontrivially only on a function of and its action amounts to replacing by , i-e
Using this we can now compute the action of on . First note that the action of on is given by and the action of is given by . Therefore the action of on is found to be
We are now in a position to discuss the physical interpretation of the spins and . From equation (34) we see that the operator is diagonal in the basis with eigenvalue . In Baez:1999tk () it is pointed out that if and are the classical area vectors of two faces of a tetrahedron then is equal to four times the area of the medial parallelogram between the two faces. The spins and would then be the areas of the other two medial parallelograms in the tetrahedron.
This interpretation, however, does not hold for the states as we will see by computing the action on . We will find the true correspondence with the classical variables when we study the semi-classical limit.
The action of on does not change the value of and it is given by
where stands for . Similarly the action of does not change the value of .
The action of on is given by while the action of on is
Now with this and the relation
we find the desired result. The action of can be deduced from a permutation exchanging and , under such a permutation and . Similarly under an exchange of and , and . ∎
While the and spins don’t share the interpretation of areas of parallelograms like in the orthogonal basis (since there are extra diagonal terms), it turns out that they are still closely related as we will now show. First of all, notice that the coefficient of the first term in (35) is the same as the eigenvalue in (34). Furthermore, if one sums over in (35) it can be seen that the last two terms cancel out because , … and so on. Therefore is proportional to . What we will now show in the following theorem is that the proportionality constant is exactly one.
The orthogonal basis is obtained from the basis by summing over the or channels
where and , , , . Now let , , , and as prescribed by (15). Then looking at the coefficient of
we get the conditions , , , and . These conditions are trivially satisfied if the are defined as in (19) and (20) which can be seen for instance by adding to the first condition. Notice, however that the LHS of (III.2), when expanded, is a sum over and whereas the RHS is a sum over , , and . Thus we get the identity
which implies .The other identities are obtained by permutation of indices. ∎
This last theorem shows that the and or bases are generated by the basis. This is particularly useful for instance when describing spin-network amplitudes containing 4-valent nodes since a choice of or basis must be made at every such node. The amplitude written in the basis however will generate all the different kinds of amplitudes by simply summing over the labellings. For example the symbol comes in five different kinds depending on the basis choice at the five nodes. Thus a new symbol labelled by 20 spins, i.e. ten , five ’s and five ’s, based on the basis would be a generator of these various symbols. Moreover this symbol would be the amplitude corresponding to the coherent 4-simplex. We will define this new symbol shortly.
Iv Scalar Products
In this section we will compute the scalar product in the basis and demonstrate the utility of Theorem III.4 by generating all the various other scalar products. Let us first make a general remark about the form of the scalar product that follows from the resolution of identity in (29). Let us split the scalar product into the naive product and the remainder:
The resolution of the identity implies that
This means that the reminder belongs to the algebra generated by the fundamental relation in (26). These relations can be derived by considering the product of the operator introduced previously: , where is the Plücker relation given in (27). Expanding using the multinomial theorem we find
where the summation coefficients are given by
and the sum is over , with and . From this result and the definition of the states , we can write this relation as
The coefficients in the sum vanish if any of the arguments in the factorials is negative. Note that for we recover the fundamental relation (26).
Now that we have determined the linear relations among the basis states we can deduce the exact form of the scalar product
The scalar product is given by
The proof of this formula is given in appendix B.
iv.1 Constraints Quantisation
We would like now to develop a deeper understanding of the construction just given of the scalar product. We have seen that the complexity of the scalar product comes from the imposition of the constraints . This suggest that we should be able to understand the previous construction in terms of constraint quantization. In order to do so, lets introduce the auxiliary Hilbert space with an orthogonal basis having and the scalar product
For this Hilbert space the decomposition of the identity takes the canonical form
We define the operator by
Its powers can be evaluated in terms of the coefficients introduced it the previous section, we find
The operator is not hermitian, however the operator
is an hermitian operator, being positive its kernel coincides with the kernel of . The intertwiner Hilbert space is defined as the quotient of this auxiliary Hilbert space by the relation . This means that , where with the projector onto the kernel of . This means that the intertwined states are related to the auxiliary states as
and the physical scalar is given by the matrix element of the projector
From the results of the previous section this projector can be explicitly constructed.
The projector onto the kernel of is explicitly given by
The proof is given in appendix B.