A new vacuum for loop quantum gravity

Loop quantum gravity [3] is a background independent approach to the quantization of gravity. Much of the structure and properties of the current theory are based on the mathematically rigorous quantum representation of the kinematical observable algebra, as constructed by Ashtekar et al [4]. A typical state in this Ashtekar–Lewandowski (AL) representation has totally degenerate spatial geometry almost everywhere—spatial geometry can only be excited along one-dimensional graph like structures. This makes the construction of states describing large scale physics difficult.

Different representations were sought after [5–7], however, these efforts led to the F-LOST uniqueness theorem [8], stating that the AL representation is the only one based on a (spatially) diffeomorphism invariant vacuum and satisfying a number of technical assumptions³ . We present here a construction of a new representation for loop quantum gravity that circumvents some of the technical assumptions of the uniqueness theorem. This new representation is unitarily inequivalent, and in a certain sense dual to the AL representation—typical states are now flat almost everywhere, spatial geometry is excited everywhere, and curvature can be excited on points (in (2+1)D) or along edges (in (3+1)D). Whereas the AL vacuum can be understood as a totally squeezed state in which the expectation values and fluctuations of the fluxes vanish, the new vacuum is a totally squeezed state in which the expectation values of curvature operators (encoded in the holonomy) and the associated fluctuations vanish.

We expect that this representation will very much facilitate the extraction of large scale physics from loop quantum gravity and that it gives a better starting point for the construction of physical states, satisfying the Hamiltonian constraints (Wheeler–deWitt equation) of the theory. Indeed this representation provides the physical states for (2+1) gravity. The underlying vacuum of this representation is the vacuum of BF theory which underlies the spin foam approach. We therefore hope that the new representation will much more directly connect loop quantum gravity to spin foams, in which curvature excitations are also distributional [11]; see [12] for related discussions involving the classical theory.

Both the AL representation and the new representation constructed here are based on an inductive limit construction for the Hilbert space of the theory, which is essential to achieve a representation of the continuum theory. This Hilbert space arises from a family of Hilbert spaces, labelled by discrete structures (graphs or triangulations) of different refinement degree, equipped with embedding maps that describe how coarser states embed into Hilbert spaces based on more refined discretizations. For this construction the choice of underlying vacuum is central: it also determines the embedding of states, the additional degrees of freedom on the refined discretization are put into the vacuum state. The AL representation and the new representation choose vacua that are dual to each other and have opposite peakedness properties.

The representation proposed in the present work is motivated by [1, 2], where a construction principle for the physical vacuum and Hilbert space is provided. In particular, [1] suggests the use of refining time evolution [13, 14] to define the embedding maps for the inductive limit Hilbert space. Indeed, here we will use the (refining) BF time evolution maps to this end, however, applied to loop quantum gravity states, not necessarily coinciding with BF physical states. The construction here therefore confirms the possibility to construct a physical Hilbert space with the methods of [1].

The setup in this work will be a simplicial version of LQG (see also [15]). To make the notion of a BF vacuum for LQG concrete, we have to (a) define the vacuum state, and (b) define a notion of refinement. We also have to specify a cylindrically consistent (with the notion (b) of refinement) observable algebra. The latter will be different from the standard holonomy–flux algebra of LQG, which allows us to evade the uniqueness theorems. We will then show that the Hilbert space of gauge–invariant functions can be generated from the new vacuum by (exponentiated) flux observables.

We will restrict ourselves to (2+1)-dimensional gravity, for which the BF state is indeed the physical vacuum. The (3+1)-dimensional case will be discussed in [16].

Here we sketch the setup of our construction. We consider a two-dimensional orientable smooth manifold, with an atlas of coordinate charts and an auxiliary metric. To this manifold, we associate a set of embedded triangulations ${{\Delta }^{*}}$, their dual complex Δ, and denote by Γ the one-skeleton of Δ. Each dual triangulation consists of (three-valent) vertices v, oriented edges e, and faces. By embedded, we mean that the vertices carry coordinates. We assume that the edges of the triangulation (denoted by e*) are geodesics in the auxiliary metric. We also assume that the triangulation is sufficiently fine in order to ensure that the geodesics are well-defined.

Concerning the group-theoretic data, we associate to each edge a space $\mathcal{F}(G)$ of functions over the group G. The topology of this space will be specified later on. The group G in question can be finite or a compact semi-simple Lie group.

The notion of refinement that we are going to use is based on Pachner moves. For triangulated two-dimensional surfaces these consist of the so-called 1–3, 3–1, and 2–2 moves, and two such triangulated surfaces are PL-homeomorphic if one can be transformed into the other by a finite number of these moves. Since we are interested in refining operations, we will consider only 1–3 and 2–2 Pachner moves. This is sufficient to ensure that topologically equivalent triangulations have common refinements. The notion of geometric triangulations includes embedding information for the vertices, and at least for the planar case that any two such triangulations have a common refinement [17].

Our task is to find observables that are cylindrically consistent, i.e. commuting with the refinement operations. We will first focus on closed holonomies and then on integrated simplicial fluxes.

The action of the Pachner moves can be deduced directly from the geometrical interpretation of BF theory as describing flat connections. Let ${{h}_{\gamma }}$ denote the holonomy along a closed path γ. Gluing a tetrahedron to the surface might change this path to $\gamma ^{\prime}$. However, since this amounts to adding only pieces of flat holonomies, the deformation of the path will not change the holonomy, and one can write that ${{h}_{\gamma }}={{h}_{{{\gamma }^{\prime }}}}$. This determines the action of the Pachner moves uniquely, and ensures that closed holonomy observables are cylindrically consistent.

1–3 Pachner moves. Consider a graph Γ consisting of a vertex v_A of Δ and its three edges⁴ e₁, e₂ and e₃. Gluing a tetrahedron to the triangulation changes the graph Γ to $\Gamma ^{\prime}$ as depicted in figure 1. There are now three vertices, ${{v}_{B}}$, ${{v}_{C}}$, and ${{v}_{D}}$, and three new edges.

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We now require that the holonomies stay the same, i.e. that ${{h}_{\gamma }}={{h}_{{{\gamma }^{\prime }}}}$, where γ is a path in Γ, and $\gamma ^{\prime}$ the path in $\Gamma ^{\prime}$ that goes along the new edges but stays in the same face. This automatically imposes cylindrical consistency of the holonomy operators. Thus we need the replacement rules:

$$\begin{eqnarray*}&&\begin{array}{ccccccccccccccc} g_{3}^{-1}{{g}_{1}}\to {{g}_{3^{\prime} }}g_{6^{\prime} }^{-1}g_{1^{\prime} }^{-1},\qquad g_{1}^{-1}g_{2}^{-1}\to {{g}_{1^{\prime} }}{{g}_{4^{\prime} }}{{g}_{2^{\prime} }}, \\ \quad {{g}_{2}}{{g}_{3}}\to g_{2^{\prime} }^{-1}{{g}_{5^{\prime} }}g_{3^{\prime} }^{-1}. \\ \end{array}\end{eqnarray*}$$

The holonomy around the new face is trivial, $g_{6^{\prime} }^{-1}{{g}_{4^{\prime} }}{{g}_{5^{\prime} }}=\mathbb{I}$. Implementing the conditions ${{h}_{\gamma }}={{h}_{{{\gamma }^{\prime }}}}$ with delta functions, we define the wave function $\psi ^{\prime}$ after the move as

$$\begin{eqnarray*}&&\begin{array}{ccccccccccccccc} {{\psi }^{\prime }}({{g}_{1^{\prime} }},...,{{g}_{6^{\prime} }},...) \\ \quad =\int \delta (g_{1}^{-1}{{g}_{3}}{{g}_{3^{\prime} }}g_{6^{\prime} }^{-1}g_{1^{\prime} }^{-1})\delta ({{g}_{2}}{{g}_{1}}{{g}_{1^{\prime} }}{{g}_{4^{\prime} }}{{g}_{2^{\prime} }}) \\ \quad \delta (g_{3}^{-1}g_{2}^{-1}g_{2^{\prime} }^{-1}{{g}_{5^{\prime} }}g_{3^{\prime} }^{-1})\psi ({{g}_{1}},{{g}_{2}},{{g}_{3}},...)\ {\bf d}{{g}_{1}}{\bf d}{{g}_{2}}{\bf d}{{g}_{3}} \\ \quad =\delta (g_{6^{\prime} }^{-1}{{g}_{4^{\prime} }}{{g}_{5^{\prime} }})\psi {{({{g}_{1}},{{g}_{2}},{{g}_{3}},...)}_{\left| \begin{array}{ccccccccccccccc} {{g}_{1}}=g_{1^{\prime} }^{-1} \\ {{g}_{2}}=g_{2^{\prime} }^{-1}g_{4^{\prime} }^{-1} \\ {{g}_{3}}={{g}_{6^{\prime} }}g_{3^{\prime} }^{-1} \\ \end{array} \right.}}, \\ \end{array}\end{eqnarray*}$$

where $\delta (\cdot )$ is the group delta function and ${\bf d}g$ is the Haar measure. Here we used the gauge invariance of the wave function ψ at v_A to gauge fix ${{g}_{1}}$–${{g}_{1}}=g_{1^{\prime} }^{-1}$.

The Pachner move has added three new edges, hence three new holonomy variables. These are restricted by a delta function, and there are in addition two gauge degrees of freedom. Hence no 'true' degree of freedom was added. By construction, the action of the 1–3 Pachner moves commutes with the action of (closed) holonomy operators.

2–2 Pachner moves. The study of the 2–2 Pachner move (represented in figure 2) is similar. We adopt the following replacement rule for a path $\gamma \to \gamma ^{\prime}$. Let γ be a path starting along the edge e_i, going possibly through e₃, and ending along e_j, with $i,j=1,2,4,5$. Then we define $\gamma ^{\prime}$ in an obvious way as passing through the corresponding replaced edges ${{e}_{{{i}^{\prime }}}}$.

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For a gauge-invariant wave function we can again use gauge fixing at ${{v}_{A}}$ and ${{v}_{B}}$, e.g. ${{g}_{2}}=g_{2^{\prime} }^{-1}$ and ${{g}_{3}}=\mathbb{I}$. The new wave function $\psi ^{\prime}$ after the 2–2 Pachner move becomes

$$\begin{eqnarray*}&&\begin{array}{ccccccccccccccc} {{\psi }^{\prime }}({{g}_{1^{\prime} }},...,{{g}_{5^{\prime} }},...) \\ =\psi {{({{g}_{1}},...,{{g}_{5}},...)}_{\left| \begin{array}{ccccccccccccccc} {{g}_{1}}=g_{1^{\prime} }^{-1}g_{3^{\prime} }^{-1} \\ {{g}_{2}}=g_{2^{\prime} }^{-1} \\ {{g}_{3}}=\mathbb{I} \\ \end{array} \right.\left| \begin{array}{ccccccccccccccc} {{g}_{4}}=g_{4^{\prime} }^{-1}g_{3^{\prime} }^{-1} \\ {{g}_{5}}=g_{5^{\prime} }^{-1} \\ \end{array} \right.}}. \\ \end{array}\end{eqnarray*}$$

Again, the action of the 2–2 Pachner moves does commute with closed holonomy operators.

In the spin representation the Pachner moves appear as the gluing of Ponzano–Regge [18] amplitudes corresponding to a tetrahedron to the hypersurface. The holonomy operators appear as tent moves in the spin representation [19], and the closed holonomies and Pachner moves commute due to the Biedenharn–Elliot identity [16].

Since we are considering a triangulation and its dual graph Γ, we will be interested in the so-called 'simplicial' or 'geometrical' fluxes [12, 15, 20, 21]⁵ , as opposed to the standard fluxes of LQG [3]. In the case $G={\rm SU}(2)$ these are defined by

$$\begin{eqnarray*}&&{{X}_{e}}={{\int }_{{{e}^{*}}}}{{h}_{{{e}^{*}}(t),e(0)}}{{E}_{a}}\left( {{e}^{*}}(t) \right){{({{\dot{e}}^{*}})}^{a}}(t)h_{{{e}^{*}}(t),e(0)}^{-1}{\bf d}t,\end{eqnarray*}$$

where $t\in [0,1]$ is a parametrization of the edge e* such that e and e* intersect at $t=1/2$, and ${{h}_{{{e}^{*}}(t),e(0)}}$ is the parallel transport from the point ${{e}^{*}}(t)$ to the source vertex $e(0)$ of e and E_a are Lie algebra valued co-triads.

The flux variables expresses the vector for the edge e* in the reference frame of the source of e. Since a 2–2 move replaces an edge of ${{\Delta }^{*}}$ by a transversal edge, we see that a flux variable in itself cannot be cylindrically consistent.

Thus we introduce 'integrated' flux observables ${{{\bf X}}_{{{\pi }^{*}}}}$ associated to paths ${{\pi }^{*}}$ in the one-skeleton ${{\Gamma }^{*}}$ of the triangulation ${{\Delta }^{*}}$ [22]. Essentially, these flux observables are defined as the sum of the individual fluxes X_e associated to the edges e dual to the elements e* of the path ${{\pi }^{*}}$ (possibly inverted to get the same orientation). Thus, whereas the e* add up a path along the edges of the triangulation ${{\Delta }^{*}}$, the edges e (dual to e*) may not be connected to each other.

To make this construction consistent and well-defined, the individual fluxes have to be transported to a common reference point, which we define as the source ${{e}_{1}}(0)\;=:\pi (0)$ of the edge dual to the first (triangulation) edge $e_{1}^{*}$ in the path ${{\pi }^{*}}$. Hence we define

$$\begin{eqnarray*}&&{{{\bf X}}_{{{\pi }^{*}}}}:=\mathop{\sum }\limits_{{{e}^{*}}\subset {{\pi }^{*}}}{{h}_{e(0),\pi (0)}}{{X}_{e}}h_{e(0),\pi (0)}^{-1},\end{eqnarray*}$$

where ${{h}_{e(0),\pi (0)}}$ is the holonomy going from the source of the edge e to $\pi (0)={{e}_{1}}(0)$.

This integrated flux observable corresponds geometrically to the displacement vector between the nodes ${{\pi }^{*}}(0)$ and ${{\pi }^{*}}(1)$ (in the triangulation). Under a gauge transformation, it transforms by the adjoint action of the group in the frame of the source vertex ${{e}_{1}}(0)$.

To define the composition of these flux observables suppose that ${{{\bf X}}_{\pi _{1}^{*}}}$ is a flux observable defined along $\pi _{1}^{*}$ and ${{{\bf X}}_{\pi _{2}^{*}}}$ is defined along $\pi _{2}^{*}$ with ${{\pi }_{1}}(1)={{\pi }_{2}}(0)$. Transporting the fluxes into the same frame, we have the composition

$$\begin{eqnarray}&&{{{\bf X}}_{\pi _{2}^{*}}}\;\circ \;{{{\bf X}}_{\pi _{1}^{*}}}={{{\bf X}}_{\pi _{1}^{*}}}+h_{{{\pi }_{1}}}^{-1}{{{\bf X}}_{\pi _{2}^{*}}}{{h}_{{{\pi }_{1}}}},\end{eqnarray} \tag{ 1 }$$

where ${{h}_{{{\pi }_{1}}}}$ is the holonomy from ${{\pi }_{1}}(0)$ to ${{\pi }_{2}}(0)$.

It is convenient to introduce the ribbon picture, which simultaneously represents the graph Γ and its dual ${{\Gamma }^{*}}$ as depicted in figure 3. A ribbon can be assigned holonomy and flux data with the composition rule $({{g}_{1}},0)\;\circ \;(\mathbb{I},{{X}_{1}})=({{g}_{1}},{{X}_{1}})$. Ribbons can then be composed following a path ${{\pi }^{*}}$. This defines the integrated fluxes through

$$\begin{eqnarray}&&({{g}_{2}},X_{2}^{-1})\;\circ \;({{g}_{1}},{{X}_{1}})=\left( {{g}_{2}}{{g}_{1}},{{{\bf X}}_{{{\pi }^{*}}}}={{X}_{1}}+g_{1}^{-1}X_{2}^{-1}{{g}_{1}} \right),\end{eqnarray} \tag{ 2 }$$

which is a semi-direct product structure.

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These fluxes are cylindrically consistent. Indeed, one can understand heuristically that the Pachner moves add only pieces of flat geometry, for which the Gauss constraints are satisfied. Thus the vector associated to the path ${{\pi }^{*}}$ does not change under a Pachner move.

In order to show this explicitly, we have once again to specify the replacement rules, that specify the integrated flux observables before and after the Pachner moves in terms of elementary fluxes and holonomies, and argue that the observables commute with the Pachner moves.

For the 1–3 Pachner move, the invariance of the integrated fluxes is immediate since the edges e* of the triangulation defining the path ${{\pi }^{*}}$ cannot be affected by the move. However, in order to show this explicitly, one has to keep track of the new holonomies that are introduced by the 1–3 move. Our replacement rule is such that the new path in the ribbon graph after the 1–3 move has to start from the same source $\pi (0)$, end at the same terminal point $\pi (1)$, go along the same flux variables (or their inverse), and transport these fluxes using the new holonomies.

Concerning the 2–2 move, if the path ${{\pi }^{*}}$ does not go along the internal edge of the triangulation that is being flipped by the move, the invariance of the integrated flux follows from the same reasoning as in the 1–3 case discussed above. If the path (or a portion of it) corresponds to the edge that is being flipped, one has to use the replacement rule depicted on figure 4. Cylindrical consistency then follows from the Gauss law, i.e. the fact that the three edges (fluxes) of a triangle close (add to zero). This replacement rule is unique as long as one follows the arrows of the ribbon representation.

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Gauge-invariant observables can be obtained by taking the trace over two integrated fluxes transported to a common reference frame, or by group averaging. Details on the Poisson algebra of these fluxes and holonomies will appear in [16].

To simplify the discussion, let us choose a gauge and consider a fixed triangulation. A gauge can be found via a maximal tree⁶ $\mathcal{T}$ in Γ with root r. Group elements associated to edges t of the tree are gauge-fixed to the identity. Leaves ℓ are edges not included in the tree and in one-to-one correspondence with the fundamental cycles of Γ. The cycle ${{c}_{\ell }}$ associated to the leave ℓ has the same starting vertex and orientation as ℓ. All other edges in the cycle are elements t of the tree. A BF vacuum state, peaked on a locally and globally flat connection, is given by ${{\eta }_{{\rm BF}}}={{\prod }_{\ell }}\delta ({{h}_{{{c}_{\ell }}}})\ \dot{=}{{\prod }_{\ell }}\delta ({{g}_{\ell }})$, where $\dot{=}$ denotes the gauge-fixed expression and ${{g}_{\ell }}$ is the group element associated to the leave ℓ.

Consider the exponentiated integrated flux observables associated to the leaves ℓ and transported to the root r. These act as right translations by group elements ${\rm A}{{{\rm d}}_{{{t}_{\ell }}}}({{h}_{\ell }})$. We will denote the exponentiated flux observables by ${{R}_{\{{\rm A}{{{\rm d}}_{{{t}_{\ell }}}}({{h}_{\ell }})\}}}$, where ${{t}_{\ell }}$ is the holonomy associated to the unique path going from the source of the leave ℓ to the root r along the tree, and ${\rm A}{{{\rm d}}_{g}}(h)=gh{{g}^{-1}}$. The action on the vacuum is

$$\begin{eqnarray*}&&{{R}_{\{{\rm A}{{{\rm d}}_{{{t}_{\ell }}}}({{h}_{\ell }})\}}}{{\eta }_{{\rm BF}}}\dot{=}\mathop{\prod }\limits_{\ell }\delta ({{g}_{\ell }}{{h}_{\ell }}).\end{eqnarray*}$$

For Abelian groups, ${{R}_{\{{{h}_{\ell }}\}}}{{\eta }_{{\rm BF}}}$ can be understood as the dual of the (gauge-invariant) spin network basis, labelled by group elements $\{{{h}_{\ell }}\}$ instead of representation labels.

The AL measure can be characterized by the evaluations of the positive linear functional ${{\mu }_{{\rm AL}}}$ on a spin network basis, which itself can be generated by the application of holonomy observables ${{\psi }_{\{j\}}}(\{g\})$ (evaluated in a representation j) to the AL vacuum state ${{\eta }_{{\rm AL}}}(\{{{g}_{e}}\})\equiv 1$. The holonomies ${{\psi }_{\{j\}}}(\{g\})$, with $\{j\}$ denoting the set of representations labeling the edges, are multiplication operators and lead to the spin network basis. The functional ${{\mu }_{{\rm AL}}}$ is defined as ${{\mu }_{{\rm AL}}}({{\psi }_{\{j\}}})={{\delta }_{\varnothing ,\{j\}}}$, which is non-vanishing iff all labels j are trivial.

To construct the new measure we will proceed similarly, but dualizing every ingredient. Instead of a constant function in holonomies, we consider a constant⁷ function in the fluxes. The spin network basis generated by holonomies is replaced by a dual basis, generated by exponentiated flux observables.

To define a measure in analogy with the AL one, we again have to dualize and therefore to switch to the 'spin representation' (or flux representation [20]). We denote the action of the shifts ${{R}_{\{{\rm A}{{{\rm d}}_{{{t}_{\ell }}}}({{h}_{\ell }})\}}}$ on the BF vacuum as states ${{\tilde{\chi }}_{\{{{h}_{\ell }}\}}}:={{R}_{\{{\rm A}{{{\rm d}}_{{{t}_{\ell }}}}({{h}_{\ell }})\}}}{{\eta }_{{\rm BF}}}$.

To make the analogy with the AL measure, we should understand the ${{\tilde{\chi }}_{\{{{h}_{\ell }}\}}}$ as function on the fluxes. For an Abelian group we can define ${{\tilde{\chi }}_{\{{{h}_{\ell }}\}}}(\{{{j}_{\ell }}\}):={{\prod }_{\ell }}{{\chi }_{{{j}_{\ell }}}}({{h}_{\ell }})$, where ${{\chi }_{j}}$ is the character of the representation j of the Abelian group. These functions then satisfy ${{\tilde{\chi }}_{\{{{h}_{\ell }}\}}}(\{{{j}_{\ell }}\}){{\tilde{\chi }}_{\{h_{\ell }^{\prime }\}}}(\{{{j}_{\ell }}\})={{\tilde{\chi }}_{\{{{h}_{\ell }}h_{\ell }^{\prime }\}}}(\{{{j}_{\ell }}\})$. To achieve a similar multiplication property for e.g. $SU(2)$ we can employ the flux representation [20], where the ${{\chi }_{j}}(\cdot )$ are replaced by so called plane wave functions on the group and the (discrete) representation label j by a continuous label $X\in {{\mathbb{R}}^{3}}$ denoting the flux variable. In the following we however do not need to work in this spin or flux representation, we only use this as argument that the spin j_e of the spin network basis in the AL representation is replaced by the group label ${{h}_{\ell }}$ for the states ${{\tilde{\chi }}_{\{{{h}_{\ell }}\}}}$ in the new representation.

In the generic case of non-Abelian groups, these states are gauge-variant at the root r, and the space of gauge-invariant functions is therefore parametrized by ${{G}^{|L|}}/{\rm A}{{{\rm d}}_{G}}$, where ${\rm A}{{{\rm d}}_{G}}$ denotes the action of the gauge group at the root. This can be cured by group averaging, which consists of considering

$$\begin{eqnarray*}&&\mathcal{G}({{\tilde{\chi }}_{\{{{h}_{\ell }}\}}}):=\int {{R}_{\{{\rm A}{{{\rm d}}_{{{t}_{\ell }}u}}({{h}_{\ell }})\}}}{{\eta }_{{\rm BF}}}{\bf d}u,\end{eqnarray*}$$

where $u\in G$ is the group averaging parameter.

We can now define a measure on this space of functions by

$$\begin{eqnarray}&&{{\mu }_{{\rm BF}}}\left( \mathcal{G}({{\tilde{\chi }}_{\{{{h}_{\ell }}\}}}) \right)=\mathop{\prod }\limits_{\ell }\tilde{\delta }({{h}_{\ell }}).\end{eqnarray} \tag{ 3 }$$

In the special case of finite groups, $\tilde{\delta }$ is the Kronecker symbol. For Lie groups, there are two possibilities:

• (a)  
  $\tilde{\delta }$ is the group delta function;
• (b)  
  $\tilde{\delta }(h)=1$ if $h=\mathbb{I}$, and is vanishing otherwise.

For example, the choice (b) leads for $G=U(1)$ to a Bohr compactification of the dual $\mathbb{Z}$ to $U(1)$ [23], which turns the vacuum into a normalizable state.

For (a), the resulting inner product coincides with the one of square integrable functions ${{L}_{2}}\left( {{G}^{|E|}},{{{\bf d}}^{|E|}}g \right)$ with the Haar measure on G. Indeed, one can verify that in this case

$$\begin{eqnarray} {{\langle \mathcal{G}({{\tilde{\chi }}_{\{{{h}_{\ell }}\}}}),\mathcal{G}({{\tilde{\chi }}_{\{h_{\ell }^{\prime }\}}})\rangle }_{{{L}_{2}}}} & = & \int \mathop{\prod }\limits_{\ell }\delta \left( h^{\prime -1}_{\,\ell }u{{h}_{\ell }}{{u}^{-1}} \right)\ {\bf d}u \\ {} & = & {{\mu }_{{\rm BF}}}\left( \overline{\mathcal{G}({{\tilde{\chi }}_{\{{{h}_{\ell }}\}}})}\mathcal{G}({{\tilde{\chi }}_{\{h_{\ell }^{\prime }\}}}) \right). \\ \end{eqnarray}$$

We leave the precise investigation of the measure (b) in the non-Abelian case for future explorations.

So far we have discussed the inner product on a fixed triangulation. To compare two states on two different triangulations, we need to consider a common refinement of these triangulations. The inner product has to be cylindrically consistent, so that it does not depend on the precise choice of common refinement.

However, if we choose in (3) the group delta functions on the right-hand side, we see that 1–3 moves will lead to additional factors of $\delta (\mathbb{I})$ for the measure and the inner product. To cure these divergencies for the inner product we can choose a (heat kernel) regularization for the delta functions, and divide the inner product of two states by the norm of a reference state (in this case the BF vacuum), leading to a modified inner product (see also [24])

$$\begin{eqnarray}&&\langle {{\psi }_{1}},{{\psi }_{2}}\rangle ^{\prime} =\mathop{{\rm lim} }\limits_{\varepsilon \to 0}\frac{{{\langle {{\psi }_{1}},{{\psi }_{2}}\rangle }_{\varepsilon }}}{{{\langle {{\eta }_{{\rm BF}}},{{\eta }_{{\rm BF}}}\rangle }_{\varepsilon }}},\end{eqnarray} \tag{ 4 }$$

where indicates the regulator.

Let us also mention a notion of spatial diffeomorphism invariance for the vacuum. Consider a triangle subdivided by a 1–3 move but with the inner vertex placed at two different positions leading to two states ${{\psi }_{1}}$ and ${{\psi }_{2}}$. One can find different ways to obtain common refinements, but these lead to the same state. Thus the inner product identifies the two states which differ by a vertex translation. This nicely reflects the notion of vertex translations as a diffeomorphism symmetry [25] (see also [26] for a similar mechanism for the physical inner product of (2+1) gravity).

Similarly to the AL projective limit construction [4] leading to a quantum extended configuration space $\bar{\mathcal{A}}$ of connections, we expect that the present construction will lead to a quantum extension $\bar{\mathcal{E}}$ of the space of fluxes. Previous attempts [20, 27] to define such a space used the standard LQG refinement maps and therefore failed. This is related to how holonomies and fluxes behave under the various refinement maps. The standard LQG refinement maps lead to a composition of holonomies, whereas fluxes stay constant [27]. The new BF embedding map composes fluxes as in (1), whereas holonomies are refined so that curvature is left invariant. Thus with the BF refinement maps fluxes are coarse grained. This could be useful for developing geometric coarse graining procedures [28].

We sketched the construction of an alternative representation for loop quantum gravity, in which typical states describe non-degenerate geometries with curvature defects. This is opposed to the AL representation supporting states with degenerate geometries almost everywhere. We therefore expect that the new representation will very much facilitate the extraction of large scale physics from loop quantum gravity, and further develop its applications to cosmology, in particular if the representation can be generalized to allow for states with a homogeneous instead of flat connection. Future work will make many notions more precise and provide the construction for the (3+1)-dimensional case. We will have the same cylindrically consistent observables as in (2+1), however, the integrated fluxes will describe the addition of normals belonging to pieces of a surface.

Our construction requires the modification of certain ingredients of loop quantum gravity, in particular the introduction of triangulations⁸ as opposed to just one-complexes (graphs). This necessity was also discussed in [29] in the context of a path integral quantization of BF theory.

The new representation presented here is unitarily inequivalent to the AL representation. One might wonder how this new representation evades the conclusion of the F-LOST uniqueness theorem [8], stating that the AL representation is the only representation of the holonomy–flux algebra, satisfying a number of conditions. As mentioned, one point is that the holonomy–flux algebra is changed—it is essential to use 'simplicial' fluxes that also incorporate the parallel transport. A precise definition of the observable algebra underlying the new representation will appear in [16]. Additionally, the new representation only allows exponentiated fluxes, which will not lead to weakly continuous families. In fact in the case of the $U(1)$ structure group we encounter the Bohr compactification of the dual $\mathbb{Z}$ of $U(1)$. This—as is known from loop quantum cosmology [31] in the case of operators encoding the holonomy (and allows us to evade the Stone–von Neumann uniqueness theorem for quantum mechanics)—requires the exponention of the fluxes. On a more heuristic level, the fact that the vacuum state is given by group delta functions forbids the action of fluxes as derivative operators and requires exponential fluxes that translate the arguments of the delta functions, as described in the main text.

Among possible future directions we mention the construction of a Hilbert space based on non-compact groups (which is required for Lorentzian signature in three-dimensions and for the self-dual Lorentzian theory in four-dimensions) [30], which might be allowed in the BF framework, but is not possible to construct with the AL construction. Furthermore, similar to [9], which shifts the expectation value of fluxes, we can attempt to obtain vacua peaked on homogeneous instead of flat connections, opening up applications to loop quantum cosmology [31]. This also relates to the question of whether one can derive a quantum group structure for loop quantum gravity [32].

Finally, let us comment on possible definitions of dynamics. The refinement by Pachner moves does suggest an implementation of the dynamics via Pachner moves, as discussed in [1, 14]. This would connect to the covariant spin foam picture [33]. An infinitesimal implementation of the dynamics via Hamiltonian constraints also seems to be possible. The construction of Thiemann [34] based on the AL vacuum leads to a finite result since the Hamiltonian acts only on (dual) nodes where volume is concentrated. A similar mechanism can be expected for the BF vacuum, where the Hamiltonian will only act on triangulation vertices (or edges in four-dimensions), where curvature is concentrated (see also [35]). This corresponds more directly to diffeomorphisms as vertex translations [36], as implemented without anomalies for various flat and homogeneous geometries [37, 38].

The question of the constraint algebra will however remain open for cases with propagating degrees of freedom [39]. Anomalies resulting from discretization artifacts can be avoided with an exact or perfect discretization, leading to a possibly non-local Hamiltonian [1, 40]. Alternatively, a renormalization procedure might lead to a restoration of diffeomorphism symmetry in the continuum limit [1, 41]. In this respect, the work presented here realizes for (2+1)-dimensional gravity the construction of a continuum limit of loop quantum gravity (and spin foam quantization) via dynamical cylindrical consistency, which has been outlined in [1, 2]. Thus, we can hope that such a construction leading to a physical vacuum is applicable for (3+1)-dimensional gravity as well.

We thank V Bonzom, L Freidel, W Kamiński and L Smolin for discussions. MG thanks the Perimeter Institute for hospitality. This research was supported by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. MG is supported by the NSF grant PHY-1205388 and the Eberly research funds of The Pennsylvania State University.