Linking covariant and canonical LQG II: Spin foam projector

Linking covariant and canonical LQG II: Spin foam projector

Antonia Zipfel antonia.zipfel@gravity.fau.de Institut für Quantengravitation, Department Physik Universität Erlangen
Staudtstrasse 7, D-91058 Erlangen, Germany
   Thomas Thiemann thomas.thiemann@gravity.fau.de Institut für Quantengravitation, Department Physik Universität Erlangen
Staudtstrasse 7, D-91058 Erlangen, Germany
Abstract

Abstract

In a seminal paper, Kaminski, Kisielowski an Lewandowski for the first time extended the definition of spin foam models to arbitrary boundary graphs. This is a prerequisite in order to make contact to the canonical formulation of Loop Quantum Gravity (LQG) whose Hilbert space contains all these graphs. This makes it finally possible to investigate the question whether any of the presently considered spin foam models yields a rigging map for any of the presently defined Hamiltonian constraint operators.

In the moment, a description of the KKL extension in terms of Group Field Theory (GFT) is out of technical reach because the interaction part of a GFT Lagrangian dictates the possible valence of a dual graph and so far is geared to duals of simplicial triangulations. To get rid of this restriction one would have to allow all possible interaction terms based on certain invariant polynomials of arbitrarily many gauge group elements what is currently out of technical control. Therefore one has to define the sum over spin foams with given boundary spin networks in an independent fashion using natural axioms, most importantly a gluing property for 2-complexes. These axioms are motivated by the requirement that spin foam amplitudes should define a rigging map (physical inner product) induced by the Hamiltonian constraint. This is achieved by constructing a spin foam operator based on abstract 2-complexes (rather than embedded ones) that acts on the gauge invariant kinematical Hilbert space of Loop Quantum Gravity by identifying the spin nets induced on the boundary graph of with states in .

In the analysis of the resulting object we are able to identify an elementary spin foam transfer matrix that allows to generate any finite foam as a finite power of the transfer matrix. It transpires that the sum over spin foams , as written, does not define a projector on the physical Hilbert space. This statement is independent of the concrete spin foam model and Hamiltonian constraint. However, the transfer matrix potentially contains the necessary ingredient in order to construct a proper rigging map in terms of a modified transfer matrix.

I Motivation

To quantize a Field Theory one can either choose a canonical approach, quantize the Hamiltonian and solve the Schrödinger equation, or a covariant one, which rests on the path integral description going back to Feynman’s famous PhD thesis Feynman:1948 (). In Loop Quantum Gravity (LQG), a background independent quantization of General Relativity, the canonical formulation lqgcan3 (); lqgcan2 () originates from a reformulation of the ADM action Arnowitt:1959ah () in terms of gauge connections by Ashtekar and Barbero Ashtekar:1986yd () while the covariant or spin foam model baez (); lqgcov (); Rovelli (), was initiated by Reisenberger’s and Rovelli’s ‘sum over histories’ ReisenbergerRovelli97 (). In both approaches many technical and structural difficulties arise from the constrained nature of GR deeply rooted in the diffeomorphism invariance of the theory. Particularly, the non-polynomial Hamiltonian constraint, although a quantization has been known for a while (see Thiemann96a ()), is challenging and up to today the physical Hilbert space cannot be determined satisfactorily. On the other hand, spin foam models suffer from second class constraint which cannot be implemented strongly. The covariant model has matured a lot but the correct treatment of the constraints is still under debate (see e.g. Alexandrov:2010un ()).

Even though both approaches differ significantly it was often emphasized in the past that they should converge to the same theory. Heuristically, the discrete time-evolution of a spin network on a spatial hypersurface, which defines a basis state in the gauge invariant kinematical Hilbert space of canonical LQG, leads to a colored 2-complex that is the main building block of spin foams. Therefore the partition functions defined by the latter can be either understood as propagator between two 3D geometries or as a rigging map, a generalized projector onto . This paper will especially focus on the latter train of thoughts.

The subsequent analysis will be mainly based on Engle:2007uq () (EPRL-model) and Kaminski:2009fm () (KKL-model). Closely related to these is the FK-approach FK (). The boundary space of the EPRL/KKL-model can be formally identified with subspaces of which will be used here in order to define a spin foam operator for the canonical theory. Even if the operators are equipped with appropriate properties so that the sum has a chance to define a projector into , the object we obtain does not provide a rigging map. This conclusion is independent of the details of a spin foam model or of a Hamiltonian constraint. To prove that a method to split the operator into smaller building blocks is developed. This splitting procedure is also interesting from a purely technical point of view since it gives a better handle on the sum over all complexes in the EPRL/KKL-partition function. On the positive side, the splitting property just mentioned allows to extract a spin foam transfer matrix which, if proper regularized, defines a modified transfer matrix that potentially yields a proper rigging map.

The paper is organized as follows: The mathematical foundations for later manipulations of graphs and 2-complexes will be laid in section II. For the sake of self-containedness section III summarizes and compares a generalization of the EPRL-model with the KKL-model. Furthermore, we will review so-called projected spin networks Alexandrov:2002br (); Dupuis:2010jn () which provide a link between canonical and covariant LQG. In section IV.1 a general framework for merging both theories will be developed guided by the concept of rigging maps or group averaging methods for simpler constrained systems. On this basis, a list of properties that the operator should satisfy will be deduced. In section IV.3 a spin foam operator will be proposed that displays all the features worked out before. Section V contains the proof that each operator can be split into simple blocks based on 2-complexes which only contain a small number of internal vertices all connected to an initial spin net (see section V.1). This result can be used to show that the proposed projector is not of the required form (section V.2). However, may still contain the necessary information in order to construct a spin foam model using a modification of with the properties of a rigging map. We conclude by summarizing and discussing the results in section VI.

Ii Foams and graphs

The first part of this section gives a short review of the kinematical Hilbert space used in LQG focussing on spin network functions and will be followed by an introduction of piecewise linear complexes.

ii.1 Spin networks

The kinematical Hilbert space of canonical LQG is the space of complex valued, square-integrable functions of (generalized) connections on a spatial hypersurface embedded in space-time . A connection on a manifold can be reconstructed from the set of holonomies

(1)

along all (semianalytic) paths111A path is an equivalence class of curves under reparametrization and retracing. where denotes path ordering. Likewise, holonomies provide a map from the groupoid of paths into . Instead of evaluating a holonomy along a single path one can also use finite systems of path:

Definition 1.

A semianalytic graph embedded in is a finite set of oriented 222Taking the holonomy along a path always implies an orientation of the path.semianalytic paths (links ) which intersect at most in their endpoints (nodes ).

A graph is called closed if every node is the endpoint of at least two links and it is called connected if it cannot be written as the disjoint union of two graphs.

In the following, and will denote the set of all links and nodes in , respectively.

The Hilbert space is spanned by cylindrical functions

(2)

where is a function on and the scalar product is given by the Ashtekar-Lewandowski measure which reduces to the Haar measure of on every link (compare with Ashtekar:1994mh ()). More precisely, for a fixed graph with links is isomorphic to . Let be a labeling of the links by irreducible representations of dimension and and be magnetic indices associated to the target and source of . Since the matrix elements of the Wigner matrices , , define an orthogonal basis of the functions

(3)

build an orthonormal basis of . To restore gauge invariance one needs to assign an intertwiner to each node , that is a group homomorphism . At the node the space is formed by the tensor product of all irreducible representations assigned to the outgoing links at and equals the tensor product of all irreducible representations assigned to the ingoing links :

(4)

The space of all intertwiners, constitutes a Hilbert space when equipped with a scalar product

defined by the natural contraction of magnetic indices where denotes hermitian conjugation. Due to the compatibility conditions of recoupling theory (see appendix A) is finite dimensional. Equivalently we could define to be an invariant tensor

(5)

where is the contragredient representation. Therefore, we often will identify with the space of invariant tensors

(6)

equipped with the trace as inner product.

We are now ready to give an explicit definition of the gauge invariant kinematical Hilbert space , which will be mostly referred to as kinematical space since non-invariant elements will not be considered. The space is spanned by so called spin network functions

(7)

This function is truly gauge invariant if all magnetic indices are contracted or equivalently if the graph is closed.

An intertwiner depends in general on the ordering333Different orderings can be related by a change of basis in the intertwiner space. of the tensor product (6) which is why an orientation of the nodes has to be introduced indicating the order of the links.

Definition 2.

A spin network (short: spin net) consists of an oriented semianalytic graph, a labeling of links by irreps and an assignment of intertwiners to the nodes (see Fig. 1).

Figure 1: A link of a Spin-network with intertwiner associated to the source and associated to the target.

In order that labels a linearly independent set of states we require for all and exclude 2-valent nodes whose adjacent links have co-linear tangents444Not excluded are two-valent intertwiners whose tangents are not co-linear. The complex conjugate of a spin network can be obtained by reversing the orientation of all links of since .

The trace of a spin net is a map ,

(8)

defined by contracting the intertwiners.

ii.2 Complexes

Since piecewise linear cell complexes are fundamental for the construction of the covariant model, they will be briefly reviewed in the sequel to clarify the notation and set-up the ground for the later considerations. Good introductions to piecewise linear topology can be found for example in pl1 () and pl2 ().

Definition 3 (pl1 ()).

  • A compact -cell in , , is the convex hull of a finite set of affine independent points, called vertices, which span an n-dimensional affine subspace.

  • Let , be compact cells and be the hyperplane of dimension spanned by . If and , then is an m-face of . It is called proper if the dimension of is strictly lower than the dimension of . The set of all proper faces of is called the frontier of .

  • An n-complex is a finite union of compact -cells, , with at least one compact -cell such that the following two conditions hold:

    1. If then all faces of are in .

    2. If then either or is a common face of and .

  • The union of all cells of is called the underlying polyhedron .

A complex is a collection of all building blocks together with their gluing relations along common faces while the underlying polyhedron is the whole object glued together. If not necessary we will not make this explicit distinction to simplify the notation but it should be kept in mind that these are in principle different objects. For instance is a topological space while itself is just a set.

A compact n-cell is homeomorphic to an n-ball and the frontier homeomorphic to an -sphere (for a proof see e.g. pl1 (); pl2 ()). This can be understood by an easy example: Let be a 2-cell with a vertex in its interior and an edge joining and another vertex of (see Fig. 2). If would be in the frontier of then there would exist a straight line with . But such line would divide into two separate faces (figure on the right). Therefore the figure on the left of Fig. 2 is not a convex cell. On the other hand, it is also not a 2-complex since is not a face of the 2-cell . This is summarized by

Lemma 1.

Every vertex of a 2-cell is contained in exactly two 1-cells in the frontier of .

The reader might be concerned that convexity is to strong if 2-complexes shall describe the time evolution of a spin-network. Indeed, for a semianalytic link the ‘time-evolved’ face will certainly not define a 2-cell. Even if the link is approximated by p.l. 1-cells. Nevertheless, it is, of course, possible to approximate by a collection of convex faces which itself defines a 2-complex. Since such an approximation is somewhat arbitrary, the final model should be independent of this. Let us finally remark that the above lemma is still valid if we drop convexity as long as a face has no self-intersections, i.e. is homeomorphic to a 2-ball. The latter will be always assumed! Also all following assertions and theorems can be formulated and proven without using explicitly convexity. It is just convenient to keep it for the moment while it has to be relaxed later on555A more appropriate choice would be to define the model on ball rather than p.l.- complexes (see section IV.2)..

(a)

(b)
Figure 2: A face with a single internal vertex and edge is not a 2-complex. By contrast, the right picture is a 2-complex consisting of two faces glued together along their common 1-cell.

Let us now continue with the description of n-complexes. To efficiently characterize their local properties we introduce the following notations:

Notation 1.

  • A cell is called adjacent to a different cell if

  • The set of n-cells of is denoted by .

  • The interior of contains all points which are not contained in any proper face of

  • The vicinity of a cell is the set of all cells for which .666Note, is not a complex itself, since the faces of a cell are only contained in if they are adjacent to . Whereas the frontier of an -cell is an -complex.

  • The total number of cells in some set is denoted by

Definition 4.

  1. A complex is called connected if its underlying polyhedron is connected. Thus for any two sub-complexes , such that there exist at least one cell satisfying

  2. If and are two complexes then is called a subdivision of iff and every cell of is a subset of some cell of . A subdivision is called proper if .

For LQG only a special kind of 1- and 2-complexes, graphs and foams, are of interest. An abstract graph is a 1-complex without isolated vertices while a foam is a 2-complex whose boundary graph is closed (see below). For convenience 1-cells are called edges and 2-cells faces. Furthermore, vertices in a graph will be mostly called ‘nodes’ and labeled by while edges in a graph will be called mostly ‘links’ and labeled by to distinguish between graphs and 2-complexes.

A priori we also want to work with complexes without specifying an embedding. Thus, all attributes like orientation and coloring of a complex must be defined in a way independent of the embedding.

Definition 5.

  • The orientation of an edge determines source and target vertex of .

  • Suppose consist of edges then define a one-to-one map so that and is a vertex of for all and .

  • The face orientation is the equivalence class of under cyclic permutations.

Because constitutes a closed loop (lemma 1) there exist exactly two inequivalent orientations (cyclic/anticyclic) of a 2-cell . Furthermore, induces an edge orientation choosing and . This orientation is not unique if the edge is contained in the frontier of more than one face, i.e. the induced orientation of can be opposite to that of on the common edge . In this case the orientation of is antidromic to that of , otherwise it is dromic (see Fig. 3). Due to convexity and intersect at most in one edge so that this definition is consistent. Even in the more general case, when faces are allowed to intersect in more than one edge but the frontiers , are still homeomorphic to , the induced orientation on all common edges are either all opposed or all equal.

(a)
(b)
Figure 3: A 2-cell can be oriented by successively counting the edges in . This induces an orientation (black arrows) on with . When the face is subdivided by an edge (red) then the faces are oriented such that the induced orientation on is preserved. Thus and are oriented antidromic and here is ingoing to while is outgoing of the red edge.

Independently from the face orientation one can still assign an edge orientation. If the induced orientation of agrees with this independent orientation then is ingoing otherwise it is called outgoing with respect to the given edge.

Besides the above, the labeling by intertwiners (see below) requires an ordering:

Definition 6.


Let be an n-cell of the complex and the set of all cells in the vicinity of then the bijection

(8)

is called an ordering of . Two orderings are equivalent if they only differ by cyclic permutations.

In contrast to face orientations there exist more than just two inequivalent orderings, for instance a four valent internal edge has six inequivalent orderings.

ii.3 Foams

As mentioned above, not all 2-complexes can be used in LQG. For example, if one adds a single vertex, which is not contained in any edge or face, to a given 2-complex then this is still a well-defined 2-complex but does not give rise to a well-defined spin foam amplitude. To link canonical and covariant LQG we additionally need a method how to associate graphs and 2-complexes.

Definition 7.

  • The interior of a 2-complex is the set of all faces, all edges, which are contained in more than one face, and all vertices contained in more than one internal edge.

  • The boundary graph of a 2-complex is the set of all edges (links) contained in only one face and vertices (nodes) contained in only one internal edge .

  • A graph is said to border iff there exists a one-to-one (affine) map mapping each face and each edge of to a unique face and a unique internal edge in respectively.

  • A 2-complex whose boundary graph is the disjoint union of connected graphs bordering is called a foam.

We alert the reader that by definition a graph has no faces.

In the literature the boundary graph of a foam is often defined by either just the combinatorial definition (see e.g. Rovelli ()) or just by bordering graphs (see appendix of lqgcov ()). Neither of this is sufficient since for example is in general not a well-defined graph. Particularly, if the intersection point of two or more boundary links is contained in several internal edges then and consequently is not even a 1-complex. On the other hand, a graph bordering does not have to be closed.

Lemma 2.

Let be a foam then the boundary graph is the disjoint union of closed connected graphs. A face intersects a connected graph at most in one link .

Proof.

Suppose is not closed then there is at least one node adjacent to one and only one link in the boundary graph. Since is bordering , is also an endpoint of an internal edge . But is contained in only one face, namely the face generated by and consequently . .

Since whenever a connected graph borders there exists a one-to-one affine map , this implies that a face cannot intersect in more than one link. ∎

Note, lemma 2 does not exclude faces intersecting the boundary graph in several disconnected graphs , .

Lemma 3.

Let be an internal vertex of the foam then all edges are internal.

Proof.

Suppose is an element of but since then is not a graph. 

Lemma 2 and lemma 3 also show that every face has at least two internal edges.

Definition 8.


Subdivide all edges adjacent to an internal vertex by a vertex in the interior of and all faces by an edge with endpoints and whenever and . This yields a 1-complex called vertex boundary graph.

Since every is contained in at least two faces (lemma 3) and because of lemma 1, is the disjoint union of closed connected graphs!

Figure 4: The face orientation induces an orientation on the boundary links while the ordering on the internal edge induces the ordering on the boundary node (red) such that is the unique link contained in .
Definition 9.

An oriented foam is a foam whose edges and faces are oriented such that all faces touching the boundary graph are ingoing to . Furthermore, all internal edges carry an ordering which induces an ordering on the boundary nodes by where is the unique internal edge with and is the unique face containing the wedge spanned by and the boundary link (see Fig. 4) 777An ordering of internal vertices is not necessary..

Since borders , internal edges intersecting the boundary graph in a connected graph are either all in- or all outgoing of corresponding to the embedding respectively . If all internal edges are outgoing of it is called initial and otherwise final.

Below, subdivisions of oriented foams play a major role for example in order to construct vertex graphs or to analyze equivalence classes within the model. A subdivision of a foam should again yield a well defined foam, e.g. it is not allowed to split a boundary link without splitting the ingoing face as well. Moreover, the orientation of should be preserved: Suppose we split an edge by a vertex , then the new edges obey , and , if is internal then , inherit the order of . If then is adjacent to only two boundary links and the order is unique.

Let be two vertices such that linking and by an edge in yields two new faces . The new faces inherit the orientation of so that the induced orientation on all old edges is preserved while on the orientations of and are antidromic. Therefore, the direction of can be chosen freely (see Fig. 3).

For example, the induced as well as the free edge orientation on the half-edges connecting and vertices of a vertex graph (see definition 8) is preserved whereas in is oriented such that the wedge spanned by is outgoing (see Fig. 5).

(a)

(b)
Figure 5: The vertex boundary graph can be constructed by cutting out the vertex along the faces adjacent to . Here, the dotted lines indicate the splitting edges and the bold black points the vertices . The edges are oriented, such that the corresponding wedge is outgoing.

Another important example is gluing of (non-oriented) foams along common closed components of their boundary graphs: Suppose is isomorphic to then a new foam can be constructed by identifying defining a subdivision of where is removed. The same can be done for oriented complexes if their orientations match so that is an oriented subdivision of . Consequently, the orientations of faces glued together must be antidromic and if the internal edge is ingoing to then the corresponding edge must be outgoing of (see Fig. 6).888Since boundary links inherit the orientation of the faces intersecting this implies that the orientation of in is opposite to that of in and strictly speaking they are not isomorphic. But since in a subdivision the orientation of splitting edges is not determined the gluing is still well-defined when assuming that is not oriented.

(a)

(b)
Figure 6: Two foams can be glued together along a common closed component of their boundary graphs if the face orientation (green arrows) and internal edge orientations (black arrows) match.

ii.4 Spin foams

Similar to the coloring of graphs in section II.1, foams will be labeled by representation data of a gauge group . In LQG we are especially interested in the cases respectively . Since is a compact semisimple Lie group the representation theory is comparably easy and therefore we will focus on the latter.

A spin foam consists of an oriented foam and an assignment of a Hilbert space (irreducible representation space of ) to every face . This induces a Hilbert space999The total order of the tensor product in (9) is determined by the edge order. on every edge

(9)

and an invariant subspace spanned by intertwiners . To each internal edge we associate an operator in such a way that the domain of is associated to the source of and the image of is associated to the target of .

Let () be the total number of faces ingoing to (outgoing from) the edge and be a basis of of with then is a tensor of rank

(10)

The expansion of in a basis of reads

(11)

and, following the above, the dual is attached to the source and to the target of .

(a)
(b)

(c)
Figure 7: The intertwiner space associated to target/source of an internal edge is independent of the edge orientation and depends only on the face orientations since (compare the two figures on the left). The convention for assigning the Hilbert spaces and orientations of boundary and vertex graphs is chosen such that it agrees with the convention for spin nets. Compare the figures on the right with Fig. 5 and Fig. 1

The marking of the bulk induces a spin net structure on : A boundary link contained in the unique face is labeled by and a node is labeled by , if the internal edge adjacent to is ingoing, and by the dual intertwiner if is outgoing. By lemma 2 each boundary link in is adjacent to exactly two internal edges which are either both ingoing to or both outgoing of and therefore, if is ingoing to it is outgoing of . In both cases is associated to while the dual is associated to the source (see Fig. 7). In fact, whether the dual or the original Hilbert space is associated to a node only depends on the face orientation and the whole model can be formulated without specifying edge orientations (see Kisielowski:2011vu ()). However, in the subsequent discussion it is more convenient to keep all orientations as defined above.

Similarly, the coloring of induces a spin net on vertex boundary graphs , see definition 8. This yields a natural contraction of the intertwiners by

(12)

where is the half-edge of adjacent to and we assumed that all edges are incoming to . Note, that all intertwiners which are not assigned to boundary nodes can be contracted in this way defining the spin foam trace

(13)

To simplify the notation we did not display whether is a dual intertwiner or not and we will continue to do so if not explicitly necessary. When, in addition, group elements are attached to all boundary links then one obtains the spin foam partition function

(14)

Notice that no claim about convergence of (14) is made at this point for generic which therefore may only define a ‘distributional’ linear functional on the boundary space spanned by spin nets based on . To fix one’s intuition, consider the following easy but important example:

Definition 10.

The trivial evolution is an oriented foam which has no internal vertices and whose boundary graph is the disjoint union of two graphs and such that there exist a (non-oriented) isomorphism .

Since by definition the boundary links of inherit the orientation of the face in which they are contained and since for every face there are two links , and it follows that the orientation of is opposite to the one of . Moreover, each internal edge is adjacent to two nodes in the boundary graph, w.l.o.g. fix and , so that the spin net on is dual to the one induced on . Concluding,

(15)

The partition function (14) is invariant if one adds or removes faces labeled by the trivial representation. Later on we will also include additional face amplitudes such that is also invariant under colored subdivisions defined in the following

Definition 11.

A colored subdivision of a spin foam is an oriented subdivision of such that for the new colored foam holds

  1. if ; and

  2. if then and is a two-valent intertwiner

  3. if such that then .

Two spin foams and can be glued together along a common graph if the orientation matches and the induced spin network functions on are mutually conjugated. Then , where is contained in and respectively, and where .

ii.5 Triangulations and foams

Before we conclude the mathematical part and give a physical motivation for the above model we will briefly discuss triangulations of 4-manifolds and relations to foams as defined in definition 7. One of the main ingredients of covariant LQG is the truncations of degrees of freedom by introducing a triangulation of space-time. A triangulation of a smooth compact n-manifold is a triple where is a (simplicial) complex and a piecewise differential homeomorphism (see appendix B for details and an extension to non-compact manifolds). In 1940 Whitehead whitehead:1940 () proved101010Originally Whitehead proved the assertion in the category but already extended it to -triangulations. To ensure uniqueness up to p.l. homeomorphisms and ensure that is a p.l. manifold the embedding map must be sufficiently smooth, i.e. is not enough (for a counter example see cannon1979 ()). that any smooth manifold has an essentially unique triangulation up to p.l. homeomorphisms. Moreover, the underlying polyhedron is a p.l. manifold which means that any point in the interior of has a neighborhood which is p.l. homeomorphic to an -simplex. Thus, any -cell in the interior of is a proper face of two -cells and the set of all -cells contained in only one -cell induces a proper triangulation of the boundary of whereupon implies that any lower dimensional () cell must be contained in at least two higher dimensional cells.

Let be a (simplicial) n-complex triangulating where labels the dimension of the cell and is the number of cells. Let denote the barycenter of . The barycenters of n-cells define the dual vertices. The one-cell dual to is the union of the edge joining and and the edge joining and . Inductively, the dual cell of is defined to be the -dimensional subset of all points for which there exist , , such that where is a point in some dual to a cell in the vicinity of (see Fig. 8). The set of all dual cells is the dual complex of .

(a)
Figure 8: In two dimensions the complex dual to the black triangulation can be constructed by joining the barycenter (blue vertices) of the triangles where the blue face is dual to the red vertex.

In general and are not collinear and thus dual cells are not convex but compact polyhedra.

Lemma 4 (pl1 ()).

If is a p.l. n-manifold and an m-cell then is a p.l. -ball (or equivalently: p.l. homeomorphic to an -simplex). If then the cell dual to in the subcomplex is an -ball in the frontier of .

Obviously, is generically not a cell-complex in the strict sense of definition 3. Yet, it is a ball complex111111For a proof see pl1 (), that is a collection of m-balls, , which obey

  1. , if where is the interior of

  2. is a finite union of balls of lower dimension in and every dual m-ball, , lies in the frontier of at least one -ball.

From the third property and lemma 4 follows immediately that the subset , containing all -balls , which are adjacent to only one -ball, and all balls in their frontier , is dual to the subcomplex .

Definition 12.

Let be a triangulation of a compact -manifold then the dual 2-complex is the set obtained by removing all balls of dimension greater than two from and additionally all 2-balls from .

Since property two and three listed above still hold every 1-ball in is contained in at least one 2-ball . A 1-ball is adjacent to exactly one 2-ball if and only if by lemma 4. As above we will call 1-balls contained in more than one -ball internal, otherwise it is called external. Again by lemma 4, every vertex of dual to a 4-cell must be the intersection of several internal edges. By the above construction every node in the boundary is the barycenter of a 3-cell and therefore the endpoint of exactly one internal 1-ball. Besides that, the dual 1-complex of is closed (every node of must be contained in at least two 1-balls), otherwise would not be empty, and bordering . This proves the first part of

Theorem 1.

If is the 2-complex dual to a triangulation of a compact 4-manifold then is combinatorially equivalent to a foam , i.e. there exists a bijection121212This map is defined on the complexes not on the underlying polyhedra! Furthermore, is a p.l. complex in the strict sense of definition 3. mapping each n-cell of to an n-cell of preserving the gluing relations (if is a common face of and then is a common face of and ). Moreover, is p.l. homeomorphic to .

Proof.

To prove that and are p.l. homeomorphic we construct the following subdivision and : Since dual cells are by construction the underlying polyhedra of cell-complexes p.l. homeomorphic to m-balls, we can fix a point in the interior of a dual face in such a way that the straight lines connecting and any barycenter or any vertex of lies in . When splitting every face in that way we obtain a simplicial complex which is a subdivision of . On the other hand, cells in are already convex so that one can choose any point in the interior of each face and each edge . By joining the points as above one can find a simplicial subdivision of which is combinatorially equivalent to . Define by , if is a vertex of and the corresponding vertex of , and extend it linearly. This gives the desired p.l. homeomorphism mapping -cells of to -cells of . ∎

Iii Covariant Quantum Gravity

iii.1 BF-theory and EPRL-model

The covariant quantization of GR is based on the observation that gravity is closely related to topological BF-theories. These theories are defined on the principal -bundle over a smooth -dimensional manifold with connection . The basic fields are the curvature and a (Lie) algebra -valued -form . Classically, the BF-action

(16)

for four dimensions with gauge group in euclidean models respectively in lorentzian ones is equivalent to the Holst action HolstAction () iff the -field can be expressed in terms of tetrads and the Hodge dual

(17)

The wedge product is taken with respect to the external indices, the trace in (16) contracts the internal indices and is the Barbero-Immirzi parameter. The variation of (16) with respect to the -field constrains the curvature to vanish and formally the path integral is given by

(18)

To obtain a covariant model of LQG we will first discretize,then quantize and finally implement the simplicity constraints (17).

iii.1.1 Discretized BF-theory

If is a simplicial triangulation of a closed manifold then the vector space of formal linear combinations of -cells in equipped with the scalar product is isometric to the space of -forms with scalar product . Furthermore, there is a one-to-one correspondence between the operations and operations in (see sen-2000-61 ()). For example the hodge dual acts on cells by mapping to dual cells.

Within this scheme the fields of BF-theory are smeared on -cells and on the dual faces such that

(19)

Remarkably, this step is independent of the chosen triangulation due to the topological nature of BF-theory. Only after the implementation of the simplicity constraint rendering the theory local the discretization yields a truncation of local degrees of freedom.

Recall that connections of a gauge theory are naturally regularized by holonomies along paths and therefore the ‘measure ’ in (18) can be replaced by . Similarly, the curvature is regularized along a loop enclosing a compact 2-d-surface since in second order approximation with . Thus the curvature integral in (19) can be replaced by

(20)

for . Here, are group elements attached to the edges bounding a face in the dual 2-complex131313For the following it is not important that is a ball-complex and the reader can safely assume that is a foam. equipped with an orientation. The order of the group elements in (20) is determined up to cyclic permutations by the orientation of the face and equals if is ingoing and if is outgoing of . Combining equation (20) and (19), (18) can be approximated by

(21)

The above procedure can be easily generalized to arbitrary 4-manifolds: If is non-compact one has to pass over to locally finite complexes (see appendix B). In order to keep everything finite we will not bother about this but always assume that is a compact region of space-time. In the case that has a non-empty boundary the action (16) must be supplemented by a boundary term in order to leave the equations of motions unaltered (see e.g. Oriti:2002hv ()). Without going into too much detail, can be constructed as in (21) just that the integral is only taken over bulk-variables.

Following Ding:2010fw (), we split each edge into half-edges and , where is adjacent to the source and to the target, and reorientat