TL;DR
This paper explores a new discretization scheme for 2+1D gravity in loop quantum gravity, revealing a family of phase space polarizations, including a dual polarization related to teleparallel gravity.
Contribution
It introduces a family of phase space polarizations in 2+1D loop quantum gravity, including a novel dual polarization linked to teleparallel gravity.
Findings
Identified a dual polarization of the discrete phase space.
Analyzed properties and symmetries of the dual polarization.
Suggested a relation between dual polarization and teleparallel gravity.
Abstract
In a recent paper, we introduced a new discretization scheme for gravity in 2+1 dimensions. Starting from the continuum theory, this new scheme allowed us to rigorously obtain the discrete phase space of loop gravity, coupled to particle-like "edge mode" degrees of freedom. In this work, we expand on that result by considering the most general choice of integration during the discretization process. We obtain a family of polarizations of the discrete phase space. In particular, one member of this family corresponds to the usual loop gravity phase space, while another corresponds to a new polarization, dual to the usual one in several ways. We study its properties, including the relevant constraints and the symmetries they generate. Furthermore, we motivate a relation between the dual polarization and teleparallel gravity.
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Dual 2+1D Loop Quantum Gravity on the Edge
**Barak Shoshany
- Perimeter Institute for Theoretical Physics,*
31 Caroline Street North, Waterloo, Ontario, Canada, N2L 2Y5** [email protected]
Abstract
In a recent paper, we introduced a new discretization scheme for gravity in 2+1 dimensions. Starting from the continuum theory, this new scheme allowed us to rigorously obtain the discrete phase space of loop gravity, coupled to particle-like “edge mode” degrees of freedom. In this work, we expand on that result by considering the most general choice of integration during the discretization process. We obtain a family of polarizations of the discrete phase space. In particular, one member of this family corresponds to the usual loop gravity phase space, while another corresponds to a new polarization, dual to the usual one in several ways. We study its properties, including the relevant constraints and the symmetries they generate. Furthermore, we motivate a relation between the dual polarization and teleparallel gravity.
Contents
1 Introduction
The theory of general relativity famously describes gravity as a result of the curvature of spacetime itself. Furthermore, the geometry of spacetime is assumed to be torsionless by employing the Levi-Civita connection, which is torsionless by definition. While this is the most popular formulation, there exists an alternative but mathematically equivalent formulation called teleparallel gravity[1, 2, 3], differing from general relativity only by a boundary term. In this formulation, one instead uses the Weitzenböck connection, which is flat by definition. The gravitational degrees of freedom are then encoded in the torsion of the spacetime geometry.
Loop quantum gravity[4]* *is a popular approach towards the formulation of a consistent and physically relevant theory of quantum gravity. In the canonical version of the theory[5], one starts by rewriting general relativity in the Hamiltonian formulation and quantizing using the familiar Dirac procedure[6]. One finds a fully constrained system, that is, the Hamiltonian is simply a sum of constraints.
In 2+1 spacetime dimensions, where gravity is topological[7], there are two such constraints:
- •
The Gauss (or torsion) constraint, which imposes zero torsion everywhere,
- •
The curvature (or flatness) constraint, which imposes zero curvature everywhere.
In the classical theory, it does not matter which constraint is imposed first. However, in the quantum theory, it does matter, since the Hilbert space is defined in terms of representations of the symmetries generated by the constraints.. The first constraint that we impose is used to define the kinematics of the theory, while the second constraint will encode the dynamics. Thus, it seems natural to identify general relativity with the quantization in which the Gauss constraint is imposed first, and teleparallel gravity with that in which the curvature constraint is imposed first.
Indeed, in loop quantum gravity, which is a quantization of general relativity, the Gauss constraint is imposed first. This is done by selecting, as the basis for the kinematical Hilbert space, the spin network basis[8] of rotation-invariant states. Then, the curvature constraint is imposed at the dynamical level in order to obtain the Hilbert space of physical states.
In [9], an alternative choice was suggested where the order of constraints is reversed. The curvature constraint is imposed first by employing the *group network *basis of translation-invariant states, and the Gauss constraint is the one which encodes the dynamics. This dual loop quantum gravity quantization is the quantum counterpart of teleparallel gravity, and could be used to study the dual vacua proposed in [10, 11].
In this paper, we will only deal with the classical theory. We will explore a family of discretizations which includes, in particular, three cases of interest:
- •
The loop gravity phase space, which is the classical version of the spin network basis[12]. This case was studied in detail in our paper [13] and is related to 2+1D general relativity. We will provide a more rigorous derivation of some results, in particular the discrete curvature constraint, and additional subtle details which were missing in our initial treatment. The phase space obtained in this case contains the phase space of spin networks, plus curvature and torsion excitations corresponding to edge modes which do not cancel.
- •
Dual loop gravity, which is the classical version of the group network basis. This case was first studied in [14] in the simple case where there are no curvature or torsion excitations. It is intuitively related to teleparallel gravity. Here, we will study this case carefully, incorporating the edge modes as was done in [13] for the loop gravity case. We will rigorously derive the discrete constraints and the symmetry transformations they generate. The resulting phase space will contain the phase space of group networks, plus the same curvature and torsion obtained in the previous case.
- •
A mixed phase space, containing both loop gravity and its dual, which is intuitively related to Chern-Simons theory[15], as we will motivate below. In this case our formalism should be related to existing results[16, 17, 18, 19, 20, 21, 22].
Crucial to our formalism is the separation of discretization into two steps. This procedure was first utilized, in the 3+1-dimensional case, in [23, 24], but without considering any curvature and torsion. The steps are as follows:
Subdivision, or decomposition into subsystems. More precisely, we define a cellular decomposition111The cells in this decomposition can take any shape.* *on our 2-dimensional spatial manifold. This structure has a dual structure, which as we will see, will be the spin network graph. 2. 2.
Truncation, or coarse-graining of the subsystems. In this step, we assume that there is arbitrary curvature and torsion inside each loop of the spin network. We then “compress” the information about the geometry into a single point, or vertex, inside the loop. Since the only way to probe the geometry is by looking at the holonomies and fluxes on the loops of the spin network, the observables before and after this truncation are the same.
The edge modes, mentioned earlier, are the final piece of our formalism. When discretizing gauge theories, and gravity in particular, a major problem is preserving gauge invariance despite the discreteness of the resulting theory. The presence of boundaries can be shown to introduce new degrees of freedom, called edge modes[25, 26, 27]222See also [28] for a more intuitive discussion and [29, 30, 31] for the case of 3+1-dimensional gravity., which may be used to *dress *observables and make them gauge-invariant. These edge modes are associated to new boundary symmetries, which transform them and control the gluing map between subsystems.
As we will see below, the edge modes at the boundaries of the cells in our cellular decomposition will mostly cancel with the edge modes on the boundaries of the adjacent cells. However, there will also be edge modes at the vertices of the cells, which will not have anything to cancel with. These degrees of freedom will survive the discretization process, and introduce a particle-like phase space[32, 33] for the curvature and torsion, which we then interpret as mass and spin respectively.
One might expect that the geometry will be encoded in the constraints alone, by imposing that a loop of holonomies sees the curvature inside it and a loop of fluxes sees the torsion inside it. As we will see, while the constraints do indeed encode the geometry, the presence of the edge modes enforces the inclusion of the curvature and torsion themselves as additional phase space variables.
1.1 Basic Definitions and Notation
Consider a group , which is333The notation signifies the *cotangent bundle *of . a generalization of the Euclidean or Poincaré group. One possible option is
[TABLE]
but we will keep it general. The algebra for this group is given by
[TABLE]
where are the structure constants444They satisfy anti-symmetry and the Jacobi identity . For we have where is the Levi-Civita symbol.. The algebra indices go from 1 to , which is e.g. 3 for . The generators are the *rotation *generators, and they correspond to a non-Abelian group , while the generators are the *translation *generators, and they correspond to an Abelian normal subgroup .
Notation-wise, all Lie algebra elements and Lie-algebra-valued forms will be written in bold font to distinguish them from Lie group element or Lie-group-valued forms. Furthermore, we will use calligraphic font for or -valued forms (which will rarely be of interest) and Roman font for , or -valued forms.
Given any two Lie-algebra-valued forms of degrees and respectively, we define the graded commutator:
[TABLE]
We also define a dot (inner) product, also known as the Killing form, on the generators as follows:
[TABLE]
Given two Lie-algebra-valued forms, the dot product is defined to include a wedge product. Thus, if is a pure rotation and is a pure translation, which will usually be the case555In the general case, which will only be relevant for our discussion of Chern-Simons theory in the next subsection, for -valued forms and we have
(5)
, we have
[TABLE]
Finally, in addition to the exterior derivative and the interior product on spacetime, we introduce a variational exterior derivative and a variational interior product on field space. These operators act analogously to and , and in particular they are nilpotent, e.g. , and satisfy the graded Leibniz rule. Degrees of differential forms are counted with respect to spacetime and field space separately; for example, if is a 0-form then is a 1-form on spacetime, due to , and independently also a 1-form on field space, due to . The dot product defined above also includes an implicit wedge product with respect to field-space forms, such that e.g. if and are 0-forms.
1.2 The Chern-Simons Action and 2+1D Gravity
Let be a 2+1-dimensional spacetime manifold and let be a 2-dimensional spatial manifold such that where represents time. Let us also define the Chern-Simons connection 1-form , valued in :
[TABLE]
where is the -valued connection 1-form and is the -valued frame field 1-form. The -valued curvature 2-form is then defined as:
[TABLE]
and it may be split into
[TABLE]
where is the -valued curvature 2-form and is the -valued torsion 2-form, and they are defined in terms of and as
[TABLE]
where is the covariant exterior derivative.
In our notation, the Chern-Simons action is given by
[TABLE]
and its variation is
[TABLE]
From this we can read the equation of motion
[TABLE]
and, from the boundary term, the symplectic potential
[TABLE]
which gives us the symplectic form
[TABLE]
Furthermore, we can write the action666Here we use the following identities, derived from the properties of the dot product (4)and the graded commutator:
(16)
in terms of and :
[TABLE]
This is the action for 2+1D gravity, with an additional boundary term (which is usually disregarded by assuming has no boundary). Using the identity , we find the variation of the action is
[TABLE]
and thus we see that the equations of motion are
[TABLE]
and the symplectic potential is
[TABLE]
Of course, (19) and (20) may be easily derived from (13) and (14).
1.3 Phase Space Polarizations and Teleparallel Gravity
The symplectic potential (20) results in the symplectic form
[TABLE]
In fact, one may obtain the same symplectic form using a *family of potentials *of the form
[TABLE]
where the parameter determines the *polarization *of the phase space. This potential may be obtained from a family of actions of the form
[TABLE]
where the difference lies only in the boundary term and thus does not affect the physics. Hence the choice of polarization does not matter in the continuum, but it will be very important in the discrete theory, as we will see below.
The equations of motion for any action of the form (23) (or constraints, in the Hamiltonian formulation) are, as we have seen:
- •
The torsion (or Gauss) constraint ,
- •
The curvature constraint .
Now, recall that general relativity is formulated using the Levi-Civita connection, which is torsionless by definition. Thus, the torsion constraint can really be seen as *defining *the connection to be torsionless, and thus selecting the theory to be general relativity. In this case, is the true equation of motion, describing the dynamics of the theory.
In the teleparallel formulation of gravity we instead use the Weitzenböck connection, which is defined to be flat but not necessarily torsionless. In this formulation, we interpret the curvature constraint as defining the connection to be flat, while is the true equation of motion.
There are three cases of interest when considering the choice of the parameter . The case is the one most suitable for 2+1D general relativity:
[TABLE]
since it indeed produces the familiar action for 2+1D gravity. The case is one most suitable for 2+1D Chern-Simons theory:
[TABLE]
since it corresponds to the Chern-Simons action (17). Finally, the case is one most suitable for 2+1D teleparallel gravity:
[TABLE]
as explained in [34].
Further details about the different polarizations may be found in [14]. However, the discretization procedure in that paper did not take into account possible curvature and torsion degrees of freedom. In the rest of this paper, we will include these degrees of freedom in the discussion by generalizing our results in [13] to include all possible polarizations of the phase space.
2 The Discrete Geometry
2.1 The Cellular Decomposition
and Its Dual
We embed a cellular decomposition and a dual cellular decomposition in our 2-dimensional spatial manifold . These structures consist of the following elements, where each element of is uniquely dual to an element of :
[TABLE]
The 1-skeleton graph is the set of all vertices and edges of . Its dual is the spin network graph , the set of all nodes and links of . Both graphs are oriented, and we write to indicate that the edge starts at the vertex and ends at , and to indicate that the link starts at the node and ends at . Furthermore, since edges are where two cells intersect, we write to denote that the edge is the intersection of the boundaries and of the cells and respectively. If the link is dual to the edge , then we have that and ; therefore the notation is consistent. This construction is illustrated in Fig. 1 (taken from [13]).
For the purpose of doing calculations, it will prove useful to introduce disks around each vertex . The disks have a radius , small enough that the entire disk is inside the face for every . We also define punctured disks , which are obtained from the full disks by removing the vertex , which is at the center, and a *cut *, connecting to an arbitrary point on the boundary . Thus
[TABLE]
The punctured disks are equipped with a cylindrical coordinate system such that and ; note that is scaled by , so it has a period of 1, for notational brevity. The boundary of the punctured disk is such that
[TABLE]
where is the *inner boundary *at , is the cut at , and is the *outer boundary *at , and the point where the cut meets the outer boundary is . Note that . The punctured disk is illustrated in Fig. 2 (taken from [13]).
The outer boundary of each disk is composed of arcs such that
[TABLE]
where is the number of cells around and the cells are enumerated . Similarly, the boundary of the cell is composed of edges and arcs such that
[TABLE]
where is the number of cells adjacent to or, equivalently, the number of vertices around . We will use these decompositions during the discretization process.
2.2 Truncating the Geometry to the Vertices
2.2.1 Motivation
Before the equations of motion (i.e. the curvature and torsion constraints ) are applied, the geometry on can have arbitrary curvature and torsion. We would like to capture the “essence” of the curvature and torsion and encode them on codimension 2 defects.
For this purpose, we can imagine looking at every possible loop on the spin network graph and taking a holonomy in around it. This holonomy will have a part valued in , which will encode the curvature, and a part valued in , which will encode the torsion.
A loop of the spin network is the boundary of a face . Since the face is dual to a vertex , the natural place to encode the geometry would be at the vertex. Thus, we will place the defects at the vertices, and give them the appropriate values in obtained by the holonomies.
The disks defined above are in a 1-to-1 correspondence with the faces . In fact, we can imagine deforming the disks such that they cover the faces, and their boundaries are exactly the loops . Thus, we may perform calculations on the disks instead on the faces.
This intuitive and qualitative motivation will be made precise in the following subsections.
2.2.2 The Chern-Simons Connection on the Disks
We define the Chern-Simons777Recall that we use calligraphic font to denote forms valued in the double , and bold calligraphic font for forms valued in its Lie algebra . connection on the punctured disk as follows:
[TABLE]
where:
- •
is a non-periodic -valued 0-form defined as ,
- •
is a periodic888By “periodic” we mean that, under , the non-periodic variable gets an additional factor of due to the term , while the periodic variable is invariant. (Recall that we are scaling by , so the period is and not .) -valued 0-form,
- •
is a constant element of the Cartan subalgebra of .
Note that this connection is related by a gauge transformation of the form to a connection defined as follows:
[TABLE]
The connection satisfies , so its curvature is . This curvature vanishes everywhere on the punctured disk (which excludes the point ), since . However, at the origin of our coordinate system, i.e. the vertex , is not well-defined, so we cannot guarantee that vanishes at itself.
In fact, we can show that it must not vanish there. If we integrate the curvature on the full disk using Stokes’ theorem, we get:
[TABLE]
where since we are using coordinates scaled by , and we used the fact that is constant. We conclude that, since vanishes everywhere on , and yet it integrates to a finite value at , the curvature must take the form of a Dirac delta function centered at :
[TABLE]
where is a distributional 2-form such that for any 0-form ,
[TABLE]
The final step is to gauge-transform back from to the initial connection defined in (31). The curvature transforms in the usual way, , so we get
[TABLE]
where we defined
[TABLE]
Note again that, while \boldsymbol{\mathcal{F}}\bigl{|}_{D_{v}} (on the full disk) does not vanish, \boldsymbol{\mathcal{F}}\bigl{|}_{v^{*}} (on the punctured disk) does vanish.
2.2.3 The Connection and Frame Field on the Disks
Now that we have defined the Chern-Simons connection 1-form and found its curvature on the disks, we split into a -valued connection 1-form a -valued frame field 1-form as defined in (7). Similarly, we split into a -valued curvature 2-form and a -valued torsion 2-form as defined in (9).
From (7) we get:
[TABLE]
where:
- •
is a non-periodic -valued 0-form and is a non-periodic -valued 0-form such that999This notation differs from the one we used in [13]. For the periodic variables, we used and in [13]. Here, we still use , but instead of we use due to this variable’s relation to the flux , as shown below. For the non-periodic variables, we used and in [13]. Here we use and in order to avoid introducing additional letters, which might be confusing. The circle above the letter conveys that it involves the angular variable and is thus non-periodic.
[TABLE]
- •
is a periodic -valued 0-form,
- •
is a periodic -valued 0-form,
- •
is a constant element of the Cartan subalgebra of , such that where is the Cartan generator,
- •
is a constant element of the Cartan subalgebra of , such that where is the Cartan generator,
- •
By construction .
The full expressions for and on in terms of and are as follows:
[TABLE]
Furthermore, from (9) we get:
[TABLE]
where represent the *momentum *and *angular momentum *respectively:
[TABLE]
In terms of and , we may write and on the disk as follows:
[TABLE]
It is clear that the first term in each definition is flat and torsionless, while the second term (involving and respectively) is the one which contributes to the curvature and torsion at . Since the punctured disk does not include itself, the curvature and torsion vanish everywhere on it:
[TABLE]
As before, while and do not vanish on the full disk , they do vanish on . We call this type of geometry a piecewise flat and torsionless geometry101010The question of whether the geometry we have defined here has a notion of a “continuum limit”, e.g. by shrinking the loops to points such that the discrete defects at the vertices become continuous curvature and torsion, is left for future work.. Given a particular spin network , and assuming that information about the curvature and torsion may only be obtained by taking holonomies along the loops of this spin network, the piecewise flat and torsionless geometry carries, at least intuitively, the exact same information as the arbitrary geometry we had before.
2.2.4 The Connection and Frame Field on the Cells
Now that we have defined and on the punctured disks , defining them on the cells is a piece of cake. The geometry inside the cells is flat and torsionless everywhere, not distributional. Thus, the expressions for and on are analogous to the first term in each of the expressions in (40), which is the flat and torsionless term:
[TABLE]
where is a -valued 0-form and is a -valued 0-form. Of course, by construction, the curvature and torsion associated to this connection and frame field vanish everywhere on the cell:
[TABLE]
2.3 Dressed Holonomies and Edge Modes
Consider the definition \mathbf{A}\bigl{|}_{c}=h_{c}^{-1}\mathrm{d}h_{c} for in terms of . Note that is invariant under the left action transformation for some constant . Thus, inverting the definition \mathbf{A}\bigl{|}_{c}=h_{c}^{-1}\mathrm{d}h_{c} to find in terms of , we get
[TABLE]
where is a path-ordered exponential, and is a new degree of freedom which does not exist in . The notation suggests that it is the holonomy “from to itself”, but it is in general not the identity! The notation is just a placeholder for the *edge mode *which “dresses” the holonomy.
For the “undressed” holonomy – which is simply the path-ordered exponential from the node to some point – we thus have
[TABLE]
Similarly, the definition \mathbf{A}\bigl{|}_{v^{*}}=h_{v}^{-1}\mathrm{d}h_{v}+h_{v}^{-1}\mathbf{M}_{v}h_{v}\thinspace\mathrm{d}\phi_{v} is invariant under , but only if is in , the Cartan subgroup of , since it must commute with . Inverting the relation \mathbf{A}\bigl{|}_{v^{*}}=\mathring{h}_{v}^{-1}\mathrm{d}\mathring{h}_{v}, we get
[TABLE]
where again the edge mode is a new degree of freedom. The undressed holonomy is then
[TABLE]
From (48) and (50), we may construct general path-ordered exponentials from some point to another point by breaking the path from to such that it passes through an intermediate point. If that point is the node , then we get
[TABLE]
and if it’s the vertex , we similarly get
[TABLE]
Furthermore, we may use the continuity relations (101) and (102) (to be discussed later) to obtain a relation between the path-ordered integrals and the holonomies and . If then we can write
[TABLE]
and if then we can write
[TABLE]
Note that, in particular,
[TABLE]
A similar discussion applies to the translational holonomies and , and one finds two new degrees of freedom, and .
3 Discretizing the Symplectic Potential
3.1 The Choice of Polarization
Recall that there is a family of symplectic potential given by (22):
[TABLE]
We would like to replace and by their discretized expressions given by (45) and (38). Before we do this for each cell and disk individually, let us consider a toy model where we simply take and for some -valued 0-form and -valued 0-form over the entire manifold . We begin by calculating the variations of these expressions, obtaining
[TABLE]
[TABLE]
where we have defined the notation for the Maurer-Cartan form on field space. Thus, we have
[TABLE]
where we used the cyclicity of the dot product to cancel some group elements. Now, the first term is very simple; in fact, it is clearly an exact 2-form, and thus may be easily integrated. However, the second term is complicated, and it is unclear if it can be integrated. Nevertheless, we know that every choice of leads to the same symplectic form:
[TABLE]
Furthermore, we have seen from (23) that the difference between different polarizations amounts to the addition of a boundary term and is equivalent to an integration by parts. Thus, we employ the following trick. First we take in , so that it becomes the 2+1D gravity polarization:
[TABLE]
Then, in the discretization process, we obtain
[TABLE]
The integrand in an exact 2-form, and thus may be integrated in two equivalent ways:
[TABLE]
Note that the 1-forms and differ only by a boundary term of the form , and they may be obtained from each other with integration by parts, just as for the different polarizations. In fact, we may write:
[TABLE]
We claim that, even though technically both options are equivalent discretizations of the polarization in (56), there is in fact reason to believe that the choice of in (56) corresponds to the same choice of in (64)! We will motivate this by showing that the choice corresponds to the usual loop gravity polarization, which is associated with usual general relativity, while the choice corresponds to a dual polarization which, as we will see, is associated with teleparallel gravity.
3.2 Decomposing the Spatial Manifold
As we have seen, the spatial manifold is decomposed into cells and disks . The whole manifold may be recovered by taking the union of the cells with the *closures *of the disks (recall that the vertices are not in , they are on their boundaries):
[TABLE]
Here, we are assuming that the cells and punctured disks are disjoint; the disks “eat into” the cells. We can thus split into contributions from each cell and punctured disk :
[TABLE]
where
[TABLE]
Given the discretizations (45) and (38), we replace in (64) with or respectively, and then integrate using Stokes’ theorem to obtain:
[TABLE]
[TABLE]
In the next few subsections, we will manipulate these expressions so that they can be integrated once again to obtain truly discrete symplectic potentials.
3.3 The Vertex and Cut Contributions
The boundary splits into three contributions: one from the inner boundary (which is the vertex ), one from the cut , and one from the outer boundary . Thus we have
[TABLE]
where the minus sign comes from the fact that orientation of the outer boundary is opposite to that of the inner boundary. Here we will discuss the first two terms, while the contribution from the outer boundary will be calculated in Sec. 3.5.
Writing the terms in the integrand explicitly in terms of using (39), and making use of the identities
[TABLE]
[TABLE]
[TABLE]
we get
[TABLE]
[TABLE]
The integral on the inner boundary is easily calculated, since and obtain the constant values and on the inner boundary. Hence , and these expressions simplify to111111Here we used the identity to get and .
[TABLE]
[TABLE]
To evaluate the contribution from the inner boundary, we integrate from to . Then since
[TABLE]
we get:
[TABLE]
which may be simplified to
[TABLE]
Next, we have the cut . Since on the cut, we have a significant simplification:
[TABLE]
[TABLE]
In fact, the cut has two sides: one at and another at , with opposite orientation. Let us label them and respectively. Any term that does not depend explicitly on will vanish when we take the difference between both sides of the cut, since they only differ by the value of . Thus only the terms and survive. The relevant contribution from each side of the cut is therefore:
[TABLE]
where the point at is the vertex , and the point at and is labeled . Taking the difference between both sides of the cut, we thus get the total contribution:
[TABLE]
Adding up the contributions from the inner boundary and the cut, we obtain the vertex symplectic potential :
[TABLE]
3.4 The “Particle” Potential
Let be the component of parallel to :
[TABLE]
where and are the Cartan generator of rotations and translations respectively, and we remind the reader that the dot product is defined in (4) as and . Similarly, let be the component of parallel to :
[TABLE]
Let us now define a -valued 0-form , which is a 1-form on field space (i.e. a variation121212Despite the suggestive notation, in principle need not be of the form for some -valued 0-form . It can instead be of the form for some -valued 0-form . Its precise form is left implicit, and we merely assume that there is a solution for either or in terms of and .):
[TABLE]
and a -valued 0-form called the vertex flux:
[TABLE]
Then since we have
[TABLE]
and since we have
[TABLE]
Furthermore, since and we have
[TABLE]
Therefore (83) becomes
[TABLE]
This potential resembles that of a point particle with mass and spin . Note that the free parameter has been absorbed into and , so this potential is obtained independently of the value of and thus the choice of polarization!
3.5 The Edge and Arc Contributions
To summarize our progress so far, we now have
[TABLE]
where
[TABLE]
[TABLE]
and is given by (92). In order to simplify , we recall from Sec. 2.1 that the boundary of the cell is composed of edges and arcs such that
[TABLE]
while the outer boundary of the disk is composed of arcs such that
[TABLE]
where is the number of cells around . Importantly, in terms of orientation, and . We thus see that each edge is integrated over exactly twice, once from the integral over and once from the integral over with opposite orientation, and similarly each arc is integrated over twice, once from and once from with opposite orientation. Hence we may rearrange the sums and integrals as follows:
[TABLE]
where
[TABLE]
[TABLE]
Next, we note that the connection and frame field are defined using different variables on each cell and disk, but overall they must be continuous on the entire spatial manifold . This implies that the variables from each cell and disk, when evaluated on the edges and arcs, must be related via continuity relations, which are, for the edges ,
[TABLE]
and for the arcs
[TABLE]
where , , and are all constant and satisfy
[TABLE]
By plugging these relations into and and simplifying, using the identities
[TABLE]
where and , we find:
[TABLE]
[TABLE]
3.6 Holonomies and Fluxes
Let us label the source and target points of the edge as and respectively, and the source and target points of the arc as and respectively, where stands for “source” and for “target”:
[TABLE]
This labeling is illustrated in Fig. 3 (taken from [13]). We now define holonomies and fluxes on the edges and their dual links, and on the arcs and their dual line segments.
3.6.1 Holonomies on the Links and Segments
The rotational131313Recall that we are dealing with a generalized Euclidean or Poincaré group where represents rotations and represents translations (or generalizations thereof). is valued in and is thus a rotational holonomy, while is valued in and is thus a translational holonomy. holonomy comes from the continuity relations (101). Its role is relating the variables on the cell to the variables on the cell . Now, in the relation , the holonomy on the left-hand side is from the node to a point on the edge . Therefore, the holonomy on the right-hand side should also take us from to . Since is the holonomy from to , we see that must take us from to . In other words, the holonomy is exactly the holonomy from to , along141414Since the geometry is flat, the actual path taken does not matter, only that it starts at and ends at . We may therefore assume without loss of generality that the path taken by is, in fact, along the link . the link .
Thus we define151515The change from lower-case to upper-case is only symbolic here, but it will become more meaningful when we define other holonomies and fluxes below. holonomies along the links :
[TABLE]
Similarly, the holonomy comes from the continuity relations (102), and it takes us from the vertex to the node . We define to be the line segment connecting to ; it is dual to the arc and its inverse is . We then define holonomies along the segments :
[TABLE]
The inverse holonomies follow immediately from the relations and :
[TABLE]
3.6.2 Fluxes on the Edges and Arcs
From the integral in the first term of (105), we are inspired to define *fluxes along the edges *:
[TABLE]
The tilde specifies that the flux is on the edge dual to the link ; the flux , to be defined below, is on the link, and similarly we will define to be the holonomy on the edge, while is the holonomy on the link.
The flux is a composition of two translational holonomies. The holonomy takes us from the point to the node , and then the holonomy takes us from to . Hence, the composition of these holonomies is a translational holonomy from to , that is, along161616Again, since the geometry is flat, the path passing through the node is equivalent to the path going along the edge . the edge , as claimed.
To find the inverse flux we use , and (101):
[TABLE]
Similarly, from the first integral in (106) we are inspired to define fluxes along the arcs :
[TABLE]
Note that this time, the two translational holonomies are composed at . As for the inverse, we define as follows and use (102) to find a relation with , taking into account the fact that and :
[TABLE]
In conclusion, we have the relations
[TABLE]
3.6.3 Holonomies on the Edges and Arcs
The holonomies and fluxes defined thus far will be used in the polarization. In the (dual) polarization, let us define holonomies along the edges and holonomies along the arcs :
[TABLE]
[TABLE]
As with , the holonomy starts from , goes to via , and then goes to via . Therefore it is indeed a holonomy along the edge . Similarly, the holonomy starts from , goes to via , and then goes to via . Therefore it is indeed a holonomy along the arc .
The difference compared to is that in we have rotational instead of translational holonomies, and the composition of holonomies is (non-Abelian) multiplication instead of addition. As before, the tilde specifies that the holonomy is on the edges or arcs and not the dual links or segments.
The variations of these holonomies are:
[TABLE]
[TABLE]
Thus, we see that they relate to the integrals in the second terms of (105) and (106).
Since , it is obvious that . Furthermore, by combining (117) with (102) we may obtain an expression for in terms of :
[TABLE]
If we now define
[TABLE]
then using the relations and , which come from the fact that , it is easy to see that . In conclusion, the inverses of these holonomies satisfy the relationships
[TABLE]
3.6.4 Fluxes on the Links and Segments
Just as we defined the holonomies on the links and segments from the variables and , which were used in the continuity relations (101) and (102), we can similarly define the fluxes on the links and segments from the variables and . These will, again, be used in the dual polarization.
Let us define fluxes along the links and segments :
[TABLE]
The factors of and are needed because they appear alongside the integrals in the variations (118) and (119). Thus, if we want the second terms in (105) and (106) to look like we want them to, we must include these extra factors in the definition of the fluxes. The fluxes are still translational holonomies between two cells (in the case of ) or a cell and a disk (in the case of ), but they contain an extra rotation at the starting point.
The inverse link flux follows from (101), (103) and , while the inverse segment flux follows from (102), (103) and :
[TABLE]
3.6.5 The Symplectic Potential in Terms of the Holonomies and Fluxes
With the holonomies and fluxes defined above, we find that we can write the symplectic potential on the edges and arcs, (105) and (106), as:
[TABLE]
[TABLE]
The full symplectic potential becomes:
[TABLE]
Notice how the holonomies and fluxes are always dual to each other: one with tilde (on the edges/arcs) and one without tilde (on the links/segments). For the polarization, the holonomies are on the links and segments and the fluxes are on their dual edges and arcs . This is the polarization considered in [13], and corresponds to the usual loop gravity picture. For the (dual) polarization, we have the opposite case: the fluxes are on the links and segments and the holonomies are on their dual edges and arcs . For any other choice of , we have a combination of both polarizations.
The phase space corresponding to for some flux and holonomy is called the holonomy-flux phase space, and it is the classical phase space of the spin networks which appear in loop quantum gravity.
4 The Gauss and Curvature Constraints
We have seen that, in the continuum, the constraints are . Let us see how they translate to constraints on the discrete phase space. There will be two types of constraints: the *curvature constraints *which corresponds to , and the *Gauss constraints *which correspond to . The constraints will be localized in three different types of places: on the cells, on the disks, and on the faces. After deriving all of the constraints and showing that they are identically satisfied in our construction, we will summarize and interpret them. The reader who is not interested in the details of the calculation may wish to skip to Sec. 4.4.
4.1 Derivation of the Constraints on the Cells
4.1.1 The Gauss Constraint on the Cells
The cell Gauss constraint will impose the torsionlessness condition inside the cells:
[TABLE]
As we have seen, is composed of edges and arcs such that
[TABLE]
Therefore we can split the integral as follows:
[TABLE]
where means “all cells adjacent to ” and means “all vertices adjacent to ”.
Using the fluxes defined in (111) and (114), we get 171717Note that in [13] we used a different convention for . This resulted in a relative minus sign between the two terms, which does not appear in this paper.
[TABLE]
This constraint is satisfied identically in our construction. Indeed, from (111) and (114) we have
[TABLE]
Since and (the end of an edge is the beginning of an arc and the end of an arc is the beginning of an edge), and (the end of the last arc is the beginning of the first edge), it is easy to see that the sum evaluates to zero.
4.1.2 The Curvature Constraint on the Cells
The cell curvature constraint will impose that inside the cells. An equivalent condition is that the holonomy around the cell evaluates to the identity:
[TABLE]
Since , we may decompose this as a product of path-ordered exponentials over edges and arcs:
[TABLE]
Furthermore, since the geometry is flat, we may deform the paths so that instead of going along the edges and arcs, it passes through the node . From (48) we have that
[TABLE]
so
[TABLE]
where we used the definition (116) of the holonomy on the edge. Note that the contribution from cancels. Similarly, we find
[TABLE]
where we used (121). Hence we obtain
[TABLE]
This is the curvature constraint on the cells. It is easy to show that it is satisfied identically in our construction. Indeed, using again the relations , and , we immediately see that
[TABLE]
as desired.
4.2 Derivation of the Constraints on the Disks
Since we have places the curvature and torsion excitations inside the disks, the constraints on the disks must involve these excitations – namely, and . We will now see that this is indeed the case.
4.2.1 The Gauss Constraint on the Disks
The disk Gauss constraint will impose the torsionlessness condition inside the *punctured181818As we have seen, we only have inside the *punctured disk ; at the vertex itself there is torsion, but is not part of . Instead, it is on its (inner) boundary. As can be seen from Fig. 2, the path we take here, as given by (140), does not enclose the vertex, and therefore the interior of the path is indeed torsionless. disks:
[TABLE]
The boundary is composed of the inner boundary , the outer boundary , and the cut :
[TABLE]
Hence
[TABLE]
where the minus signs represent the relative orientations of each piece. On the inner boundary , we use the fact that takes the constant value to obtain
[TABLE]
The outer boundary splits into arcs, and we use the definition (113) of the flux:
[TABLE]
On the cut , we have contributions from both sides, one at and another at with opposite orientation. Since on the cut, we have:
[TABLE]
and thus
[TABLE]
since has the value at and at on the cut.
Adding up the integrals, we find that the Gauss constraint on the disk is
[TABLE]
In fact, since this constraint is used as a generator of symmetries (as we will see below), it automatically comes dotted with a Cartan element , which commutes with . Therefore, the last term may be ignored, and the constraint simplifies to
[TABLE]
Thus it may also be written
[TABLE]
To see that this constraint is satisfied identically in our construction, let us combine (113) with (39) to obtain
[TABLE]
where we used a slight abuse of notation by using and to denote the corresponding angles, and . Let us now sum over the fluxes for each arc. Since (each arc ends where the next one starts) and (the last arc ends a full circle after the first arc began191919Recall that we are using scaled angles such that a full circle corresponds to 1 instead of !), we get
[TABLE]
Choosing without loss of generality the point to be at the beginning of the first edge, , and recalling that this point corresponds to the angle , we indeed obtain precisely the constraint (144).
4.2.2 The Curvature Constraint on the Disks
The disk curvature constraint will impose that inside the punctured disks202020Again, we only have inside the *punctured *disk ; at the vertex itself, there is curvature. However, the path of integration does not enclose the vertex, and therefore the interior of the path is indeed flat.. An equivalent condition is that the holonomy around the punctured disk evaluates to the identity:
[TABLE]
Let us describe the path of integration step by step, referring to Fig. 2:
- •
We start at , at the polar coordinates and .
- •
We take the path along the cut at from to .
- •
We go around the outer boundary of the disk at from to .
- •
We take the path along the cut at from to .
- •
Finally, we go around the inner boundary of the disk at from to , back to our starting point.
Let us evaluate each term individually. On and we have212121Note that the angle in the term in (50) refers to the difference in angles between the starting point and the final point, therefore it vanishes in this case since the path along the cut is purely radial. from (50):
[TABLE]
[TABLE]
On the inner boundary we have, again using (50),
[TABLE]
since is periodic. The minus sign comes from the fact that we are going from a larger angle to a smaller angle. Finally, on the outer boundary we have, splitting into arcs and then using (136) and ,
[TABLE]
In conclusion, the curvature constraint on the disks is
[TABLE]
In fact, we can multiply both sides by from the left and and obtain, after redefining ,
[TABLE]
This may be written more suggestively as
[TABLE]
Let us now show that this constraint is satisfied identically in our construction. From (117) we have
[TABLE]
and using the definition from (39) we get
[TABLE]
Now, consider the product
[TABLE]
This is a telescoping product; the term always cancels the term in the next factor in the product. After the cancellations take place, we are left only with , the product of exponents
[TABLE]
where we used the fact that the angles sum to 1, and . In conclusion:
[TABLE]
If we then choose, without loss of generality, the point (which defines the cut ) to be at (where is an arbitrarily chosen cell), we get
[TABLE]
and we see that the constraint is indeed identically satisfied.
4.3 Derivation of the Constraints on the Faces
We have seen that the Gauss constraints, as we have defined them, involve the fluxes on the edges and arcs. Since these fluxes are not part of the phase space for , these constraints cannot be imposed in that case. Similarly, the curvature constraints involve the holonomies on the edges and arcs and therefore will not work for the case . This is a result of formulating both constraints on the cells and disks, which then requires us to use the holonomies and fluxes on the edges and arcs which are on their boundaries.
Alternatively, instead of demanding that the torsion and curvature vanish on the cells and disks, we may demand that they vanish on the faces created by the spin network links. Since the (closures of the) faces cover the entire spatial manifold , this is entirely equivalent.
This alternative form is obtained by deforming (or expanding) the disks such that they coincide with the faces. The inner boundary is still the vertex . The outer boundary now consists of links , where and . The point on the outer boundary can now be identified, without loss of generality, with the node . Thus, the cut now extends from to .
Since the spatial manifold is now composed solely of the union of the closures of the faces, and not cells and disks, we only need one type of Gauss constraint and one type of curvature constraint. Let us derive them now.
4.3.1 The Gauss Constraint on the Faces
The face Gauss constraint will impose the torsionlessness condition inside the faces:
[TABLE]
The boundary is composed of the inner boundary , the outer boundary , and the cut :
[TABLE]
where the minus signs represent the relative orientations of each piece. On the inner boundary , we use the fact that takes the constant value to obtain as for above:
[TABLE]
On the cut , we have as before
[TABLE]
The outer boundary splits into links:
[TABLE]
Now, (102) can be inverted222222Note that (102) is only valid on the arc , which is the boundary between and . However, since we have expanded the disks, the arcs now coincide with the links, with every arc intersecting the two links connected to the node . Thus the equation is still valid at itself. to get
[TABLE]
Plugging into (167), we get
[TABLE]
In fact, we can get rid of the first two terms, since the sum is telescoping: each term of the from for is canceled232323Of course, and also cancel each other, but we choose to leave them. by a term of the form for . Thus we get
[TABLE]
Next, we note that from (101) we have
[TABLE]
and if we plug in (102) for , , and we get
[TABLE]
[TABLE]
From (172) we see that . Plugging this into (173), we get the simplified expression
[TABLE]
Therefore, we may rewrite (170) as:
[TABLE]
Finally, we recall from (123) the definition of the fluxes on the links:
[TABLE]
In the second equality we use the fact that, since we have deformed the disks, the source point of the edge lies on the spin network itself, and we can further deform the edge such that . Plugging into (175), we obtain
[TABLE]
Finally, from (102) we have , and we get
[TABLE]
Adding up the integrals in (164), we obtain the Gauss constraint on the faces:
[TABLE]
Just like the Gauss constraint on the disks, this can be simplified by noting that the constraint comes dotted with an element of the Cartan subalgebra, which commutes with :
[TABLE]
where we used the fact that and the part commutes with . Thus, Gauss constraint on the faces may be rewritten in a simplified way:
[TABLE]
Let us now show that this constraint is satisfied identically. We have from the definition of :
[TABLE]
The sum is telescoping, and every term cancels the previous one. However, in the term with , we have
[TABLE]
since , unlike , is not periodic. Therefore, the first and last terms do not cancel each other. If we furthermore choose , we get
[TABLE]
Then, using (178) we immediately obtain (179), as desired.
4.3.2 The Curvature Constraint on the Faces
The face curvature constraint will impose that inside the faces. As before, an equivalent condition is that the holonomy around the face evaluates to the identity:
[TABLE]
On and we have as before
[TABLE]
[TABLE]
On the inner boundary we have
[TABLE]
Finally, we decompose the outer boundary (which is now a loop on the spin network) into links:
[TABLE]
From (55) we know that
[TABLE]
and therefore
[TABLE]
where we used the choice and the fact that the product is telescoping, that is, each term cancels the term which follows it, except the first and last terms, which have nothing to cancel with.
Joining the integrals, we get
[TABLE]
From (102) we find that
[TABLE]
and thus
[TABLE]
For the last step, since we have the identity on the right-hand side, we may cycle the group elements and rewrite the constraint as follows:
[TABLE]
Switching to the notation of (108) and (109), we rewrite this as
[TABLE]
An even nicer form of this constraint is
[TABLE]
In other words, the loop of holonomies on the left-hand side would be the identity if there is no curvature, that is, .
To show that this constraint is satisfied identically, we use (52) with and :
[TABLE]
Comparing with (55), we see that
[TABLE]
and therefore
[TABLE]
We now use this to rewrite the left-hand side of (196) as follows:
[TABLE]
Again, we have a telescoping product, and after canceling terms we are left with
[TABLE]
which is exactly (196) after using (108) and (109).
4.4 Summary and Interpretation
In conclusion, we have obtained242424One might wonder about the appearance of in (204) and (206), since the true phase space variable is , defined implicitly in (87) as a function of and . It is possible that there is an expression for these two constraints in terms of instead of , but since we only have an **implicit **definition for in terms of its variation , it is unclear how to obtain it. For now, we simply assume that both and are phase space variables. See also footnote 12. Gauss constraints and curvature constraints for each cell , disk and face :
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
The Gauss constraint on the cell can also be written as
[TABLE]
It tells us that the sum of fluxes along the edges and arcs surrounding is zero, as expected given that the interior of is flat. Alternatively, we may say that the sum of fluxes along the edges is prevented from summing to zero by the presence of the fluxes on the arcs.
The Gauss constraint on the punctured disk can also be written as
[TABLE]
It tells us that the sum of fluxes on the arcs of the disk is prevented from summing to zero due to the torsion at the vertex , as encoded in the parameter . Note that if , that is, there is no torsion at , then the constraint becomes simply
Importantly, notice that the sum on the right-hand side of (208) is over all the fluxes on the arcs surrounding a particular cell , while the sum on the left-hand side of (209) is over all the fluxes on the arcs surrounding a particular disk . While the sums look alike at first sight, they are completely different and one cannot be exchanged for the other.
The Gauss constraint on the face can also be written as
[TABLE]
It tells us that the sum of fluxes on the link forming the boundary of the face is prevented from summing to zero due to the torsion at the vertex , as encoded in the parameter .
The curvature constraint on the cell is
[TABLE]
It is analogous to the cell Gauss constraint, and imposes that the product of holonomies along the boundary of the cell is the identity.
The curvature constraint on the punctured disk can also be written as
[TABLE]
On the left-hand side, we have a loop of holonomies around the vertex . If , that is, there is no curvature at , then the right-hand side becomes the identity, as we would expect. Otherwise, it is a quantity which depends on the curvature. The curvature constraint on the disks is thus analogous to the Gauss constraint on the disks, with torsion replaced by curvature.
Finally, the curvature constraint on the face can also be written as
[TABLE]
It has the same meaning as the one on the disks, except that the loop of holonomies around the vertex is now composed of links instead of arcs.
5 The Constraints as Generators of Symmetries
5.1 The Discrete Symplectic Form
The discrete symplectic potential we have found is
[TABLE]
In the second line, we can use (114), that is, , to write
[TABLE]
Thus, the labels and may be freely exchanged. Using the identity , we find that the corresponding symplectic form is
[TABLE]
We now look for transformations252525The transformations will be given by the action of the Lie derivative where is the variational interior product with respect to . In the literature the notation is often used instead, but we avoid it in order to prevent confusion with the variational exterior derivative . with parameters and such that:
[TABLE]
[TABLE]
We will see that the proportionality coefficients will be -dependent.
5.2 The Gauss Constraints as Generators of Rotations
5.2.1 The Gauss Constraint on the Cells
Let us consider the rotation transformation with parameter defined by
[TABLE]
such that any other variables (in particular, those unrelated to the particular of choice) are unaffected. Applying it to and using the identity , we get:
[TABLE]
However, the first and last triple products in each line cancel each other, and we are left with:
[TABLE]
Hence this transformation is generated by the cell Gauss constraint , given by (202), as long as .
5.2.2 The Gauss Constraint on the Disks
Next we consider the rotation transformation with parameter defined by
[TABLE]
such that any other variables (in particular, those unrelated to the particular of choice) are unaffected. Importantly, we choose the 0-form to be valued in the Cartan subalgebra, so it commutes with and . Applying the transformation to and using the identities and , we get:
[TABLE]
Isolating and using the fact that it commutes with and , we see that most terms cancel262626In this calculation, we make use of the Jacobi identity:
(219)
, and we get:
[TABLE]
Hence this transformation is generated by the disk Gauss constraint , given by (203), as long as .
5.2.3 The Gauss Constraint on the Faces
Lastly, we consider the rotation transformation with parameter defined by
[TABLE]
such that any other variables (in particular, those unrelated to the particular of choice) are unaffected, and such that
[TABLE]
where is valued in the Cartan subalgebra. Applying the transformation to , we get after a calculation analogous to the one we did for the disks,
[TABLE]
The variation of the Gauss constraint (204) is
[TABLE]
but since is in the Cartan we have , so this simplifies to
[TABLE]
Thus, in conclusion,
[TABLE]
and this transformation is generated by the face Gauss constraint , given by (204), as long as .
5.3 The Curvature Constraints as Generators of Translations
5.3.1 The Curvature Constraint on the Cells
For the curvature constraint on the cells, we would like to find a translation transformation with parameter such that
[TABLE]
First, we should calculate . Recall that
[TABLE]
To simplify the calculation, let us define such that we may write
[TABLE]
where we omit the subscript on for brevity. Then
[TABLE]
where . Hence
[TABLE]
where for . For conciseness, we may define such that and, for ,
[TABLE]
and write
[TABLE]
Plugging in back, and using the identity
[TABLE]
we get
[TABLE]
Now, if we transform only the dual fluxes and (for a particular ), then we get
[TABLE]
Comparing with (232), we see that if we take
[TABLE]
we will obtain
[TABLE]
as required. Hence this transformation is generated by the cell curvature constraint , given by (205), as long as .
5.3.2 The Curvature Constraint on the Disks
As in the cell case, we would like to find a translation transformation with parameter such that
[TABLE]
where
[TABLE]
First, we should calculate . Let us define, omitting the subscript on for brevity,
[TABLE]
[TABLE]
and
[TABLE]
Then we may calculate similarly to the previous subsection
[TABLE]
Note that for we have
[TABLE]
and since we are imposing , we get simply
[TABLE]
Furthermore, using the fact that
[TABLE]
we see that
[TABLE]
Therefore, we finally obtain the result
[TABLE]
Now, let us take
[TABLE]
where is a 0-form valued in the Cartan subalgebra, and calculate . We find that, since , the terms cancel out and we are left with
[TABLE]
We may now derive the appropriate transformation. If we transform only the segment flux and the vertex flux (for a particular ), then we get
[TABLE]
Comparing with (248), we see that if we take
[TABLE]
we will obtain, since ,
[TABLE]
as required. Hence this transformation is generated by the disk curvature constraint , given by (206), as long as .
5.3.3 The Curvature Constraint on the Faces
We would now like to find a translation transformation with parameter such that
[TABLE]
where
[TABLE]
As before, to calculate we define, omitting the subscript on for brevity,
[TABLE]
[TABLE]
[TABLE]
Then a similar calculation to the previous section gives
[TABLE]
and if we take
[TABLE]
where is a 0-form valued in the Cartan subalgebra, we get
[TABLE]
We may now derive the appropriate transformation. If we transform only the edge flux and the vertex flux (for a particular ), then we get
[TABLE]
Comparing with (259), we see that if we take
[TABLE]
we will obtain
[TABLE]
as required. Hence this transformation is generated by the face curvature constraint , given by (206), as long as .
5.4 Conclusions
We have found that the Gauss constraints and curvature constraints for each cell , disk and face , given by (202), (203), (204), (205), (206) and (207), generate transformations with rotation parameters and translations parameters as follows:
[TABLE]
[TABLE]
The Gauss constraint on the cell generates rotations of the holonomies on the links and segments connected to the node and the fluxes on the edges and arcs surrounding :
[TABLE]
where is a -valued 0-form.
The Gauss constraint on the disk generates rotations of the holonomies on the segments connected to the vertex and the fluxes on the arcs surrounding , as well as the holonomy and flux on the vertex itself:
[TABLE]
where is a 0-form valued in the Cartan subalgebra of .
The Gauss constraint on the face generates rotations of the fluxes on the links surrounding and the holonomies on their dual edges , as well as the holonomy and flux on the vertex itself:
[TABLE]
where is a 0-form valued in the Cartan subalgebra of and .
The curvature constraint on the cell generates translations272727Note that the curvature constraints do not transform any holonomies, since the holonomies are unaffected by translations. of the fluxes on the links and segments connected to the node :
[TABLE]
where
[TABLE]
and is a -valued 0-form.
The curvature constraint on the disk generates translations of the fluxes on the segments connected to the vertex , as well as the flux on the vertex itself:
[TABLE]
where
[TABLE]
is a 0-form valued in the Cartan subalgebra of , and .
The curvature constraint on the face generates translations of the fluxes on the edges dual to the links surrounding the face , as well as the flux on the vertex itself:
[TABLE]
where
[TABLE]
and is a 0-form valued in the Cartan subalgebra of .
Importantly, in the case , the usual loop gravity polarization, the curvature constraints on the cells and disks do not generate any transformations since . Similarly, for the case , the dual polarization, the Gauss constraints on the cells and disks do not generate any transformations since . Of course, the reason for this is that, as we noted earlier, these constraints are formulated in the first place in terms of holonomies and fluxes which only exist in a particular polarization. Thus for we must instead use the curvature constraint on the faces282828Which is indeed what we did in [13]., and for we must instead use the Gauss constraint on the faces.
In the hybrid polarization with , all of the discrete variables exist: there are holonomies and fluxes on both the links/edges and the arcs/segments. Therefore, in this polarization all 6 types of constraints may be consistently formulated using the available variables, and all of them generate transformations.
6 Summary and Outlook
In this paper, we generalized the work of [13] to include the most general possible discretization. We discovered a family of polarizations of the discrete phase space, given by different values of the parameter . Of these, the three cases of interest are , and .
In the case, which is the one we discussed in [13], the holonomies are on the links (and segments) and the fluxes are on their corresponding edges (and arcs), as in the familiar case of loop gravity. The Gauss constraints on the cells and disks generate rotations for all of the discrete variables, while the curvature constraints on the faces generate translations only for the fluxes on the edges and vertices.
In the case, the positions of the holonomies and fluxes are reversed. The holonomies are on the edges (and arcs) and the fluxes are on their corresponding links (and segments). The curvature constraints on the cells and disks generate translations for all of the fluxes, while the Gauss constraints on the faces generate rotations only for the fluxes on the links, holonomies on the edges, and fluxes and holonomies on the vertices.
Finally, in the case, we have the variables for both polarizations simultaneously. All 6 types of constraints exist, and each of them generates its associated transformations.
Intuitively, we may now conclude that the polarization corresponds to usual 2+1D general relativity, while (the dual polarization) corresponds to teleparallel gravity. This intuition is motivated by the fact that, as we have seen, in the polarization the holonomies and fluxes switch places, and thus the curvature and torsion (and their respective constraints) also switch places.
Since 2+1D general relativity has curvature but zero torsion, and teleparallel gravity has torsion but zero curvature, it makes sense to claim that these polarizations are related. Indeed, this is why we used the same parameter in both (22) and (56). Since the choice in (22) corresponds to Chern-Simons theory , we may further claim that the polarization in the discrete case is a discretization of Chern-Simons theory. Thus:
- •
The polarization corresponds to 2+1D general relativity,
- •
The polarization corresponds to Chern-Simons theory,
- •
The polarization corresponds to teleparallel gravity.
A discussion of quantization in different polarizations is provided in [9]. There, it is shown that in the case, the Gauss constraint is imposed at the kinematical level while the curvature constraint encodes the dynamics. In the , the roles of the constraints are reversed. This again motivates a relation between the case and teleparallel gravity. The relation of the case to Chern-Simons theory is motivated in [14]. We leave a more in-depth discussion and analysis of the relations between the case and teleparallel gravity, and between the case and Chern-Simons theory, to future work.
Following our exhaustive study of discretization of 2+1D gravity, it is our goal to adapt this discretization scheme to the physically relevant case of 3+1D gravity. While in the 2+1D case there is only one place where an integration may be performed in two different ways, in the 3+1D case there are two such integrations, since we have one more dimension. We expect to find both 3+1D general relativity and 3+1D teleparallel gravity as different polarizations of the discrete phase space. The discretization in 3+1 dimensions will be presented in an upcoming paper [35].
6.1 Acknowledgments
The author would like to thank Laurent Freidel and Florian Girelli for their mentorship, and the anonymous referee for helpful comments. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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