Inequivalent coherent state representations in group field theory

In this paper we will use the following notation and conventions. The base manifold of GFT is considered to be $G^{\times n}$ with $G=SU(2)$ and some fixed $n\in\mathbb{N}$; it will be denoted, $M\doteq G^{\times n}$. A generalization of statements from this paper to compact Lie groups other than $SU(2)$ is straightforward, but a treatment of non-compact base manifold requires more care. Throughout the whole paper the letter h is reserved as an element of G, and ${\rm d}h$ refers to the Haar measure on G, the Haar integral on G is denoted by $\int_{G}\, (\cdot)\, {\rm d}h$. The Haar measure is normalized to 1, $\int_{G}\, {\rm d}h=1$, and is invariant under left and right multiplication and inversion on G, that is for an integrable function f and $h_{1}, h_{2}\in G$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \int_{G}\,f\left(h_{1}h\,h_{2}\right){\rm d}h =\int_{G}\,f(h)\,{\rm d}h. \nonumber \end{align} \tag{ 1 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \int_{G}\,f\left(h^{-1}\right)\,{\rm d}h =\int_{G}\,f(h)\,{\rm d}h. \nonumber \end{align} \tag{ 2 }$$

It is a unique measure on G with this properties.

The letters x and y are reserved for elements of M, and ${\rm d}x$ refers to the Haar measure on M, the Haar integral on M is denoted by $\int_{M}\, (\cdot)\, {\rm d}x$. The Haar measure ${\rm d}x$ is, as above, normalized to 1 and invariant under left and right multiplication as well as inversion on M. Whenever necessary, we will use subscripts for the components of x and write $x=\left(x_{1}, x_{2}, \cdots, x_{n}\right)\in M$. We denote the Lie algebra of M by $\mathfrak{m}$ and by convention choose it to be isomorphic to the space of right invariant vector fields on M.

We denote the space of square integrable functions on M by $L^{2}\left(M, {\rm d}x\right)$ and define the bracket $(\cdot, \cdot)_{L^{2}}$, such that for any $f, g\in L^{2}\left(M, {\rm d}x\right)$,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle (\,f,g)_{L^{2}}=\int_{M}\,\overline{f}(x)g(x)\,{\rm d}x. \nonumber \end{align} \tag{ 3 }$$

The real and imaginary part of expressions are referred to as ${\rm Re}(\cdot)$ and ${\rm Im}(\cdot)$, respectively. The Dirac-delta distribution on M is denoted $\delta(\cdot)$ and satisfies

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f(y)=\int_{M}\,\delta\left(y\,x^{-1}\right)\,f(x)\,{\rm d}x, \nonumber \end{align} \tag{ 4 }$$

where $y\, x^{-1}$ denotes the group product between y and x⁻¹.

Throughout the paper we will use different norms on several different spaces. We will introduce them in the text whenever we use them, but here we summarize the notation for better overview:

$\|\cdot\|_{L^{2}}=\sqrt{(\cdot, \cdot)_{L^{2}}}$ is the L² norm, $\|\cdot\|_{k, \infty}$ is the family of semi norms with respect to which the space of smooth functions is complete, in particular $\|\cdot\|_{\infty}$ is the supremums norm for smooth functions, $\|f\|_{\infty}=\sup_{x\in M}\left|f(x)\right|$, $\|\cdot\|_{\star}$ refers to the $C^{\star}$-norm, $\|\cdot\|_{\mathcal{H}}$ refers to the Hilbert space norm for whatever Hilbert space is in question, and $\|\cdot\|_{\rm op}=\sup_{x\in\mathcal{H}}\|\cdot x\|_{\mathcal{H}}$ is the operator norm for bounded linear operators on the Hilbert space $\mathcal{H}$.

Group field theory is a field theoretical description of spin networks and simplicial geometry. It can be formulated in terms of functional integrals [1, 3, 47, 48] or in operator language [49]. In the latter, the natural starting point is a Fock space spanned by creation and annihilation operators $\varphi^{\dagger}(x), \varphi(x)$, acting on the Fock vacuum of zero quanta $|o)$, such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \varphi^{\dagger}(x)|o) =|x), \quad \varphi(x)|o) =0. \nonumber \end{align} \tag{ 5 }$$

In models with a simplicial or more general topological interpretation, like the ones related to loop quantum gravity and simplicial quantum gravity, one requires the operators to be invariant under the right multiplication by an arbitrary element of the group G, such that for all $h\in G$ we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \varphi\left(x_{1,}x_{2},\cdots,x_{n}\right)=\varphi\left(x_{1}h,x_{2}h,\cdots,x_{n}h\right).\label{eq:Closure constraint} \nonumber \end{align} \tag{ 6 }$$

This symmetry requirement is called the closure constraint [18–21, 50]. The functions that satisfy the closure constraint are called gauge invariant functions. The canonical commutation relation (CCR) between the fields without closure constraint is given by,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left[\varphi(x),\varphi^{\dagger}(y)\right]=\delta\left(x\,y^{-1}\right).\label{eq: CCR-2} \nonumber \end{align} \tag{ 7 }$$

And for gauge invariant fields the CCR read

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left[\varphi(x),\varphi^{\dagger}(y)\right]=\int_{G}\;\prod_{j=1}^{n}\,\delta\left(x_{j}hy_{j}^{-1}\right)\,{\rm d}h. \nonumber \end{align} \tag{ 8 }$$

The Fock space created by these operators can be understood as a kinematical Hilbert space $\mathcal{H}_{\mathrm{kin}}$, formed by generic quantum states on which no dynamics has yet been imposed. As in any background-independent formulation of quantum gravity, one expects the quantum dynamics to be encoded in a finite set of constraint operators $C_{i}:\mathcal{H}_{\mathrm{kin}}\to\mathcal{H}_{\mathrm{kin}}$ for ${\rm i}\in\left\{1, \cdots, N\right\}$. Following the idea of Dirac quantization the role of C_i is twofold: first to select the space of physical states formally as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{H}_{\mathrm{phys}}=\left\{|\psi)\in\mathcal{H}_{\mathrm{kin}}\,|\,C_{i}|\psi)=0\:\forall {\rm i}\in\left\{1,\cdots,N\right\} \right\} ,\label{eq:Physical Hilbert space} \nonumber \end{align} \tag{ 9 }$$

and second, select the relevant observables $\mathcal{O}$ by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left[C_{i},\mathcal{O}\right]=0\qquad \forall {\rm i}\in\left\{1,\cdots,N\right\} . \nonumber \end{align} \tag{ 10 }$$

Any concrete choice of such operators C_i defines a different GFT model. In analogy to this, the constraint operators in LQG would be the diffeomorphism constraints and the Hamiltonian constraint, but the GFT constraints cannot be directly interpreted as diffeomorphisms or Hamiltonian constraints, since the degrees of freedom of the GFT theory do not live on a continuous spacetime manifold where diffeomorphisms would be defined and act as symmetry transformations.

A treatment of constraint systems can be technically challenging [3, 51, 52]. In particular, if the zero eigenvalue lies in the continuous part of the spectrum of C_i the states $|\psi)$, that satisfy $C_{i}|\psi)=0$, are not contained in the kinematical Hilbert space, and one needs to generalize the construction using the notion of rigged Hilbert spaces [53]. This is already the case for finite-dimensional systems in the presence of gauge symmetries like reparametrization invariance, and it is an even more severe issue in continuum quantum gravity. There, it can be partially tackled by the method of refined algebraic quantization [3, 54], but experience with quantum field theories tells us that we need Hilbert spaces other than Fock to describe an infinite number of interacting degrees of freedom [22]. Hence, GFT combines both types of difficulties: a constrained system without explicit Hamiltonian evolution, and the need to study an infinite number of degrees of freedom.

To approach this problem and establish its rigorous operator formulation, we use the algebraic formalism for quantum statistical mechanics in GFT. In the following we will put the above formulation of GFT in algebraic terms and construct Hilbert spaces with infinite particle number as representations of the GFT algebra of observables. This will require the definition of a Weyl algebra for GFT.

The first step in the construction of an algebraic formulation is the construction of the algebra of observables. In GFT, by convenience, we choose this algebra to be the Weyl algebra. The later is a $C^{\star}$-algebra that is constructed over a symplectic space of the classical theory. For that reason we start our discussion of the algebraic construction with a definition of the suitable symplectic space in GFT.

1.2.1. Symplectic space of GFT.

We begin with the space of smooth, complex valued functions on M that we denote by $\mathcal{S}=\mathcal{C}^{\infty}(M)$. Let $L_{x}:M\to M$ denote the left and $R_{x}:M\to M$ the right multiplication on M by $x\in M$ and denote the pull-back of $f\in\mathcal{S}$ by L_x (respectively R_x) as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_{x}^{\star}f =f\circ L_{x} \quad \left({\rm respectively }~~R_{x}^{\star}f=f\circ R_{x}\right). \nonumber \end{align} \tag{ 11 }$$

Lemma 1. $\mathcal{S}$ is closed under translations; that is for any $y\in M$ and $f\in\mathcal{S}$ the functions $L_{y}^{\star}f$ and $R_{y}^{\star}f$ are again in $\mathcal{S}$. Moreover, $L_{y}^{\star}$ and $R_{y}^{\star}$ leave the L²-bracket, $(\cdot, \cdot)_{L^{2}}$, invariant.

Proof. The first statement follows from smoothness of the maps L_x and R_x. The second statement is a direct consequence of the left (respectively right) invariance of the Haar measure ${\rm d}x$. That is for $f, g\in\mathcal{S}$ and $y\in M$,

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left(L_{y}^{\star}f,L_{y}^{\star}g\right)_{L^{2}} & =\int_{M}\,\overline{f}(yx)g(yx)\,{\rm d}x\nonumber \\ & =\int_{M}\,\overline{f}(x)g(x)\,{\rm d}x\nonumber \\ & =(\,f,g)_{L^{2}}. \nonumber \end{align*}$$

And similar for $R_{y}^{\star}f$. □

Let $X_{i}\in\mathfrak{m}$ be a Lie algebra element of M, then X_i acts as a derivation on smooth functions such that for $f\in\mathcal{S}$ , $I\subset\mathbb{R}$ an interval containing zero and $t\in I$,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle X_{i}f(x)\doteq\partial_{t}\,f\left({\rm e}^{tX_{i}}\,x\right)\vert_{t=0},\label{eq:derivation on S} \nonumber \end{align} \tag{ 12 }$$

where ${\rm e}^{tX_{i}}$ denotes the exponential map on M [55].

Lemma 2. $\mathcal{S}$ equipped with topology induced by the family of semi-norms

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left\{\|f\|_{k,\infty}=\|X_{1}\cdots X_{k}f(g)\|_{\infty}:X_{1},\cdots,X_{k}\in\mathfrak{m};\forall k\in\mathbb{N}\right\} , \nonumber \end{align*}$$

is a complete, topological, locally convex, vector space.

Proof. See⁴ [55]. □

When the topology of $\mathcal{S}$ will be important in our discussion we will denote this topological space by $\mathcal{S}_{\infty}.$

Since M is compact, every smooth function on it is finite integrable and we can equip $\mathcal{S}$ with the norm-topology induced by the norm,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \|f\|_{L^{2}}^{2}=\int_{M}\,\bar{f}(x)\,f(x)\,{\rm d}x. \nonumber \end{align} \tag{ 13 }$$

Lemma 3. $\mathcal{S}$ equipped with the norm topology is not complete and its completion is the space of square integrable functions on M.

Proof. See [55]. □

To distinguish this topological space from the above, we will denote it $\mathcal{S}_{L^{2}}$ whenever this will be necessary.

Let $h\in G$ and $D:G\to M$ be a diagonal map such that $\newcommand{\e}{{\rm e}} D_{h}\equiv D(h)=(h, \cdots, h)$. We say f satisfies the closure constraint (or f is gauge invariant) if

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle R_{D_{h}}^{\star}f =f \ \qquad \forall h\in G. \nonumber \end{align} \tag{ 14 }$$

We denote the space of functions that satisfy the closure constraint by $\mathcal{S}_{G}$.

Proposition 4. $\mathcal{S}$ can be decomposed in complementary subspaces $\mathcal{S}_{G}$ and $\mathcal{S}_{NG}$ such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{S}_{\infty}=\mathcal{S}_{G}+\mathcal{S}_{NG}, \nonumber \end{align} \tag{ 15 }$$

and $\mathcal{S}_{G}\cap\mathcal{S}_{NG}=\left\{0\right\}$. Where $\mathcal{S}_{G}$ is a space of gauge invariant functions and $\mathcal{S}_{NG}$ is a space of functions that do not satisfy the closure constraint.

Proof. Let P define an operator on $\mathcal{S}$ and pointwise acting as

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle (Pf)(x)=\int_{G}\,\left(R_{D_{h}}^{\star}f\right)(x)\,{\rm d}h. \nonumber \end{align*}$$

P is linear since it is a composition of linear operators, $R_{D_{h}}^{\star}$ and $\int_{G}\, (\cdot)\, {\rm d}h$. We show that the image of P is in $\mathcal{S}_{\infty}$. By [55, lemma 2.1] it is enough to show that $\|Pf\|_{k, \infty}<\infty$ for any $k\in\mathbb{N}$. For an arbitrary fixed k we get

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|Pf\|_{k,\infty} & =\sup_{x\in M}\left|X_{1}\cdots X_{k}(Pf)(x)\right|\nonumber \\ & =\sup_{x\in M}\left|X_{1}\cdots X_{k}\int_{G}\,\left(R_{D_{h}}^{\star}f\right)(x)\,{\rm d}h\right|. \nonumber \end{align*}$$

By lemma 1 the integrand is a smooth function and can be upper bounded by $\sup_{x\in M}\left|\left(R_{D_{h}}^{\star}f\right)(x)\right|$. Hence, by dominant convergence theorem

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|Pf\|_{k,\infty} \leqslant\int_{G}\,\sup_{x\in M}\left|X_{1}\cdots X_{k}\,\left(R_{D_{h}}^{\star}f\right)(x)\right|\,{\rm d}h \nonumber \end{align*}$$

For any fixed $h\in G$ we have

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle X_{1}\cdots X_{k}\,\left(R_{D_{h}}^{\star}f\right)(x) =\partial_{t_{1}}\cdots\partial_{t_{k}}\,f\left({\rm e}^{t_{1}X_{1}}\cdots {\rm e}^{t_{k}X_{k}}\,x\,D_{h}\right), \nonumber \end{align*}$$

where all derivatives are taken at zero. Since $x\, D_{h}\in M$ it follows that

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \sup_{x\in M}\left|X_{1}\cdots X_{k}\,\left(R_{D_{h}}^{\star}f\right)(x)\right|=\sup_{x\in M}\left|X_{1}\cdots X_{k}\,f(x)\right| \nonumber \end{align*}$$

and we obtain

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|Pf\|_{k,\infty}\leqslant\|f\|_{k,\infty}. \nonumber \end{align*}$$

Therefore, $P:\mathcal{S_{\infty}}\to\mathcal{S}_{\infty}$, is a continuous linear operator on $\mathcal{S}.$

Further, by right invariance of the Haar measure it follows that P²f  =  Pf. By [56, theorem 1.1.8] it follows that $\mathcal{S}_{\infty}$ can be decomposed as

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{S}_{\infty} =\mathcal{S}_{G}+\mathcal{S}_{NG}, \nonumber \end{align*}$$

where $\mathcal{S}_{G}=P\mathcal{S}_{\infty}$ and $\mathcal{S}_{NG}=(1-P)\mathcal{S}_{\infty}$ and $\mathcal{S}_{G}\cap\mathcal{S}_{NG}=\left\{0\right\}$. □

Lemma 5. P is an orthogonal projector on $L^{2}\left(M, {\rm d}x\right)$.

Proof. P is bounded on $\mathcal{S}_{L^{2}}$ since for any $f\in\mathcal{S}$ we have by right invariance of the Haar measure

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|Pf\|_{L^{2}}^{2}=\int_{M}\,\int_{G}\,\left|f\left(x\,D_{h}\right)\right|\,{\rm d}h\,{\rm d}x=\int_{M}\,\left|f(x)\right|\,{\rm d}x=\|f\|_{L^{2}}^{2}. \nonumber \end{align*}$$

Let $f, g\in\mathcal{S}$. Then by Fubini and the invariance of the Haar measure under right multiplication and inversion, we have

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle (\,f,Pg)_{L^{2}} & =\int_{M}\,\overline{f}(x)\,\left(\int_{G}\,\left(R_{D_{h}}^{\star}g\right)(x)\,{\rm d}h\right)\,{\rm d}x\nonumber \\ & =\int_{M}\,\left(\int_{G}\,\overline{\left(R_{D_{h}}^{\star}f\right)}(x)\,{\rm d}h\right)\,g(x)\,{\rm d}x\nonumber \\ & =(Pf,g)_{L^{2}}. \nonumber \end{align*}$$

And for $h_{1}, h_{2}\in G$ we have

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle (PPf)(x) & =\int_{G}\int_{G}\left(R_{D\left(h_{1}\right)}^{\star}R_{D\left(h_{2}\right)}^{\star}f\right)(x)\,{\rm d}h_{1}\,{\rm d}h_{2}\nonumber \\ & =\int_{G}\int_{G}\left(R_{D\left(h_{1}\right)}^{\star}R_{D\left(h_{2}\right)}^{\star}f\right)(x)\,{\rm d}h_{1}\,{\rm d}h_{2}\nonumber \\ & =\int_{G}\int_{G}\left(\left(R_{D\left(h_{1}h_{2}\right)}\right)^{\star}f\right)(x)\,{\rm d}h_{1}\,{\rm d}h_{2}\nonumber \\ & =\int_{G}\left(\left(R_{D_{h}}\right)^{\star}f\right)(x)\,{\rm d}h=(Pf)(x). \nonumber \end{align*}$$

Therefore, P is an orthogonal projection on the dense domain of $L^{2}\left(M, {\rm d}x\right)$ and extends uniquely to the whole $L^{2}\left(M, {\rm d}x\right)$ by continuity. □

Theorem 6. The space $\mathcal{S}_{G}=P\mathcal{S}$ is dense in $PL^{2}\left(M, {\rm d}x\right)$—the image of the orthogonal projection P on $L^{2}\left(M, {\rm d}x\right)$.

Proof. Since $PL^{2}\left(M, {\rm d}x\right)$ is given by the projection P, it is a closed subspace of $L^{2}\left(M, {\rm d}x\right)$. By lemma 3 the set $PL^{2}\left(M, {\rm d}x\right)\cap\mathcal{S}$ is dense in $PL^{2}\left(M, {\rm d}x\right)$. Further, any $f\in PL^{2}\left(M, {\rm d}x\right)\cap\mathcal{S}$ is an almost-everywhere gauge invariant function that is smooth. Define g  =  f  −  Pf. Then g vanishes almost everywhere and is smooth. Hence g is zero everywhere, and we get $f\in\mathcal{S}_{G}$ and $PL^{2}\left(M, {\rm d}x\right)\cap\mathcal{S}\subseteq\mathcal{S}_{G}$. The opposite inclusion, $\mathcal{S}_{G}\subseteq PL^{2}\left(M, {\rm d}x\right)\cap\mathcal{S}$, is obvious since any $f\in\mathcal{S}_{G}$ is square integrable and $\mathcal{S}_{G}\subseteq\mathcal{S}$ by lemma 4. □

To proceed with the construction of the symplectic space we equip $\mathcal{S}_{\infty}$ with a symplectic form $\mathfrak{s}:\mathcal{S}\times\mathcal{S}\to\mathbb{R}$ defined for any $f, g\in\mathcal{S}$ by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathfrak{s}(\,f,g)={\rm Im}\left[(\,f,g)_{L^{2}}\right]. \nonumber \end{align} \tag{ 16 }$$

Restricting $\mathcal{S}$ to the subspace $\mathcal{S}_{G}$ we obtain the symplectic form on $\mathcal{S}_{G}$ that we denote by the same symbol $\mathfrak{s}$.

The above theorem ensures that after quantization, the one particle Hilbert space, that is given by the L² closure of the $\mathcal{S}_{G}$ will be that of a quantized polygon [50]. However, the symplectic structure of our space is different from the symplectic structure of a single polygon.

Remark 7. The space $\mathcal{S}_{G}$ is not closed under right multiplication, meaning that in general for $f\in\mathcal{S}_{G}$ and $y\in M$ not of the diagonal form (that is $y\neq D_{h}$ for any $h\in G$), the function $R_{y}^{\star}f$ will not be gauge invariant. To see this we observe

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left(R_{y}^{\star}f\right)(x) =f(xy), \nonumber \end{align*}$$

which is in general not equal to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f\left(xD_{h}y\right)=\left(R_{y}^{\star}f\right)\left(x\,D_{h}\right). \nonumber \end{align} \tag{ 17 }$$

For this reason we choose the definition of the Lie algebra to be given by right invariant vector fields on M (generated by left translation) to ensure that for any $f\in\mathcal{S}_{G}$, the function X_if stays in $\mathcal{S}_{G}$.

1.2.2. The Weyl algebra of GFT.

To define the Weyl algebra from the space $\mathcal{S}$ we follow the standard procedure presented for example in [51, 52] and that we recall below for convenience.

First we define a $\star$-algebra $\mathcal{A}\left(\mathcal{S}\right)$ such that:

• 1.  
  The elements of $\mathcal{A(S)}$ are complex valued functions on $\mathcal{S}$ with support consisting of a finite subset of $\mathcal{S}$. It follows that $\mathcal{A(S)}$ is a vector space.
• 2.  
  Then we define a $\newcommand{\e}{{\rm e}} \ell^{1}$ norm on $\mathcal{A(S)}$ by

  $$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|A\|_{1} =\sum_{f\in\mathcal{S}}|A(\,f)|. \nonumber \end{align*}$$

  The sum on the right hand side is well defined since each element in $\mathcal{A}\left(\mathcal{S}\right)$ is supported on a finite subset of $\mathcal{S}.$
• 3.  
  For $f\in\mathcal{S}$ we define functionals $W_{(\,f)}\in\mathcal{A}\left(\mathcal{S}\right)$ such that for any $g\in\mathcal{S}$

  $$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle W_{(\,f)}(g) & =\left\{\begin{array}{@{}ll@{}} 1 & {\rm if}\:f=g\,{\rm pointwise} \nonumber \\ 0 & {\rm otherwise} \end{array}\right.. \nonumber \end{align*}$$

  These functionals form a dense linear subspace in $\mathcal{A(S)}$.
• 4.  
  We then define the multiplication on that subspace by

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle W_{(\,f)}\cdot W_{(g)} ={\rm e}^{-\frac{\imath}{2}\mathfrak{s}(\,f,g)}\,W_{(\,f+g)}. \nonumber \end{align*}$$

  and extend it to the full $\mathcal{A(S)}$ by linearity.
• 5.  
  Finally, we define the involution $W_{(\,f)}^{\star}=W_{(-f)}$.

Closing $\mathcal{A(S)}$ in the $\newcommand{\e}{{\rm e}} \ell^{1}$ norm provides a Banach $^{\star}$-algebra $\mathbb{A}\left(\mathcal{S}\right)$. This algebra can be represented by bounded linear operators on some Hilbert space. Denoting the space of all non-degenerate, irreducible representations of $\mathbb{A}\left(\mathcal{S}\right)$ by Irreps, we define the Weyl algebra.

Definition 8. The Weyl algebra is a $C^{\star}$-algebra over $\mathcal{S}$ obtained by completion of $\mathbb{A}\left(\mathcal{S}\right)$ in the $C^{\star}$-norm

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \|W_{(\,f)}\|_{\star}:=\sup_{\pi\in {\rm Irreps}}\|\pi\left(W_{(\,f)}\right)\|_{\mathcal{H}}.\label{eq:c-star norm} \nonumber \end{align} \tag{ 18 }$$

We denote it $\mathfrak{A}\left(\mathcal{S}\right)$.

Lemma 9. For any $x\in M$ the maps $\alpha_{x}$ and $\beta_{x}$ from $\mathfrak{A}\left(\mathcal{S}\right)$ to $\mathfrak{A}\left(\mathcal{S}\right)$ defined such that for any $f\in\mathcal{S}$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \alpha_{x}\left(W_{(\,f)}\right) =W_{\left(L_{x}^{\star}f\right)}, \quad \beta_{x}\left(W_{(\,f)}\right) =W_{\left(R_{x}^{\star}f\right)}, \nonumber \end{align} \tag{ 19 }$$

and extended to the whole $\mathfrak{A}\left(\mathcal{S}\right)$ by linearity are $\star$-automorphisms.

Proof. By definition $\alpha_{x}$ and $\beta_{x}$ are linear. Further let $f, g\in\mathcal{S}$, then by lemma 1

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle \alpha_{x}\left(W_{(\,f)}W_{(g)}\right) & =\alpha_{x}\left(W_{(\,f+g)}{\rm e}^{-\frac{\imath}{s}{\rm Im}(\,f,g)_{L^{2}}}\right)\nonumber \\ & ={\rm e}^{-\frac{\imath}{2}{\rm Im}(\,f,g)_{L^{2}}}W_{\left(L_{x}^{\star}f+L_{x}^{\star}g\right)}\nonumber \\ & ={\rm e}^{-\frac{i}{2}{\rm Im}\left(L_{x}^{\star}f,L_{x}^{\star}g\right)_{L^{2}}}W_{\left(L_{x}^{\star}f+L_{x}^{\star}g\right)}\nonumber \\ & =W_{\left(L_{x}^{\star}f\right)}W_{\left(L_{x}^{\star}g\right)}\nonumber \\ & =\alpha_{x}\left(W_{(\,f)}\right)\alpha_{x}\left(W_{(g)}\right). \nonumber \end{align*}$$

Also

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \alpha_{x}\left(W_{(\,f)}^{\star}\right) & =\alpha_{x}\left(W_{(-f)}\right) =W_{\left(-L_{x}^{\star}f\right)} =\left[\alpha_{x}\left(W_{(\,f)}\right)\right]^{\star}. \nonumber \end{align*}$$

We can similarly address $\beta_{x}$. □

Restricting $\mathcal{S}$ to $\mathcal{S}_{G}$ we obtain a subset $\mathfrak{A}_{G}$ defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathfrak{A}_{G}=\overline{{\rm span}\left\{W_{(\,f)}\in\mathfrak{A}\left(\mathcal{S}\right)\vert\,f\in\mathcal{S}_{G}\right\} }^{\|.\|_{\mathfrak{A}\left(\mathcal{S}\right)}}, \nonumber \end{align} \tag{ 20 }$$

where $\overline{\circ}^{\|.\|_{\mathfrak{A}\left(\mathcal{S}\right)}}$ denotes the closure in the $\mathfrak{A}\left(\mathcal{S}\right)$-$C^{\star}$-algebra norm.

Theorem 10. $\mathfrak{A}_{G}$ is a maximal $C^{\star}$-sub-algebra of $\mathfrak{A}\left(\mathcal{S}\right)$ that satisfies $\forall A\in\mathfrak{A}_{G}$, $\beta_{D_{h}}(A)=A$ for any $h\in G$.

Proof. $\mathfrak{A}_{G}$ is spanned by Weyl elements of the form $W_{(\,f)}$ with $f\in\mathcal{S}_{G}\subset\mathcal{S}$, hence, $\mathfrak{A}_{G}\subset\mathfrak{A}\left(\mathcal{S}\right)$. Since $\mathcal{S}_{G}$ is closed under addition, and multiplication by real numbers, $\mathfrak{A}_{G}$ is closed under multiplication and involution,

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle W_{(\,f)}W_{(g)} & =W_{(\,f+g)}{\rm e}^{-\frac{\imath}{2}{\rm Im}(\,f,g)}\in\mathfrak{A}_{G}, W_{(\,f)}^{\star} =W_{(-f)}\in\mathfrak{A}_{G}. \nonumber \end{align*}$$

To show that $\mathfrak{A}_{G}$ is invariant under $\beta_{D_{h}}$ for any $h\in G$ let $\left(A_{n}\right)_{n\in\mathbb{N}}$ be a Cauchy sequence in $\mathfrak{A}_{G}$ such that

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle A_{n}=\sum_{i=0}^{n}c_{i}W_{\left(\,f_{i}\right)}\qquad {\rm with}\quad c_{i}\in\mathbb{C},\quad f_{i}\in\mathcal{S}_{G} \nonumber \end{align*}$$

and that converges to $A\in\mathfrak{A}_{G}$. Choose $h\in G$. Then by lemma 9 $\beta_{D_{h}}$ is a $\star$-automorphism on $\mathfrak{A}\left(\mathcal{S}\right)$ and the sequence $\left(\beta_{D_{h}}\left(A_{n}\right)\right)_{n\in\mathbb{N}}$ is a Cauchy sequence in $\mathfrak{A}\left(\mathcal{S}\right)$ that converges to $\beta_{D_{h}}(A)\in\mathfrak{A}\left(\mathcal{S}\right)$. However, if $f_{i}\in\mathcal{S}_{G}$ then $\beta_{D_{h}}\left(W_{\left(\,f_{i}\right)}\right)=W_{\left(R_{D_{h}}^{\star}f_{i}\right)}=W_{\left(\,f_{i}\right)}$ and the two sequences are identical in $\mathfrak{A}_{G}$. Thus, the limit points have to be equal and we get, $\beta_{D_{h}}(A)=A$. The fact that $\mathfrak{A}_{G}$ is maximal follows from proposition 4 and the fact that we can decompose, $\mathcal{S}=\mathcal{S}_{G}+\mathcal{S}_{NG}$ with $\mathcal{S}_{G}\cap\mathcal{S}_{NG}=\left\{0\right\}$. □

Corollary 11. The Weyl algebra over $\mathcal{S}_{G}$, denoted $\mathfrak{A}\left(\mathcal{S}_{G}\right)$, is a maximal $C^{\star}$-sub-algebra of $\mathfrak{A}\left(\mathcal{S}\right)$ whose elements are invariant under $\beta_{D_{h}}$ for any $h\in G$.

Proof. This follows from the fact that $\newcommand{\e}{{\rm e}} \eta:\mathfrak{A}\left(\mathcal{S}\right)\to\mathfrak{A}\left(\mathcal{S}_{G}\right)$ defined on Weyl elements by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \eta\left(W_{(\,f)}\right)=W_{(Pf)}, \nonumber \end{align} \tag{ 21 }$$

and extended to $\mathfrak{A}\left(\mathcal{S}\right)$ by linearity is an invertible $\star$-homomorphism from $\mathfrak{A}_{G}$ to $\mathfrak{A}\left(\mathcal{S}_{G}\right)$. The later is obvious since on $\mathfrak{A}_{G}$, η acts as an identity. □

This concludes our construction of the Weyl algebra for GFT. In the following we will not distinguish between the algebra $\mathfrak{A}\left(\mathcal{S}\right)$ and $\mathfrak{A}\left(\mathcal{S}_{G}\right)$ since all the following statements equally apply to both cases. For that reason we will use $\mathfrak{A}$ to refer to the Weyl algebra (gauge invariant or not) and use $\mathcal{S}$ for the space of smooth function (gauge invariant or not). $\mathcal{S}_{\infty}$ and $\mathcal{S}_{L^{2}}$ then refer to the corresponding topological spaces (gauge invariant or not). In section 2.4, however, we will use the gauge invariant algebra $\mathfrak{A}\left(\mathcal{S}_{G}\right)$ since it is more relevant for GFT's with simplicial interpretation.

In order to deal with states directly at the level of the algebra, we briefly introduce the concept of algebraic states. An algebraic state is a linear, positive, normalized functional on the algebra $\mathfrak{A}$,

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \omega:\mathfrak{A}\to\mathbb{C} \nonumber \end{align*}$$

such that for any $A\in\mathfrak{A}$ we get

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \omega\left(A^{\star}A\right) & \geqslant0 \omega\left(\mathds{1}\right) =1.\nonumber \end{align} \tag{ 22 }$$

The first inequality is the condition of positivity and the second is the normalization. Specifically for the Weyl algebra the positivity condition reads as follows:

Definition 12. The functional $\omega:\mathfrak{A}\to\mathbb{C}$ is positive if, for any finite $N\in\mathbb{N}$ and any set of complex coefficients $\left\{c_{n}\right\} _{n\in\left\{0, \cdots, N\right\} }$ and test functions $\left\{\,f_{n}\in\mathcal{S}\right\} _{n\in\left\{0, \cdots, N\right\} }$, the following holds

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle \sum_{n,m}^{N}c_{n}\bar{c}_{m}\,\omega\left(W_{\left(\,f_{n}-f_{m}\right)}\right){\rm e}^{-\imath\frac{{\rm Im}\left(\,f_{n},f_{m}\right)}{2}} \geqslant0. \nonumber \end{align*}$$

By the GNS construction, every algebraic state provides a triple $\left(\mathcal{H}_{\omega}, \pi_{\omega}, \vert\Omega)\right)$, where $\mathcal{H}_{\omega}$ is a Hilbert space, $\pi_{\omega}:\mathfrak{A}\to\mathcal{L}\left(\mathcal{H}_{\omega}\right)$ is a representation of $\mathfrak{A}$ in terms of bounded linear operators on $\mathcal{H}_{\omega}$, and the state vector $\vert\Omega)\in\mathcal{H}_{\omega}$, such that $\forall A\in\mathfrak{A}$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \omega(A)=\left(\Omega\vert\pi_{\omega}[A]\vert\Omega\right).\label{eq:Expectation value} \nonumber \end{align} \tag{ 23 }$$

This representation is unique, up to unitarily equivalence [57].

The algebra of observables $\pi_{\omega}\left(\mathcal{\mathfrak{A}}\left(\mathcal{S}\right)\right)$ on the GNS Hilbert space $\mathcal{H}_{\omega}$ is a sub-algebra of bounded linear operators on $\mathcal{H}_{\omega}$, that we denote $\mathfrak{M}$. The commutant of $\mathfrak{M}$ is a subset of bounded linear operators of $\mathcal{L\left(H_{\omega}\right)}$ on $\mathcal{H}_{\omega}$ such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathfrak{M}^{\prime}=\left\{A\in\mathcal{L}\left(\mathcal{H}_{\omega}\right)\vert\,\forall B\in\mathfrak{M}\:AB=BA\right\} . \nonumber \end{align} \tag{ 24 }$$

Usually, $\mathfrak{M}$ is not closed in the strong operator topology on $\mathcal{H}_{\omega}$. This is because the $C^{\star}$- norm (equation (18)) is stronger than the operator norm. The closure of $\mathfrak{M}$ in the strong (or equivalently, weak) operator topology is called the von Neumann algebra and is equal to the bicommutant of $\mathfrak{M}$ by the von Neumann theorem (see for example [58]). We denote the von Neumann algebra of the ω-GNS representation $\mathfrak{M}^{''}$.

The center of the von Neumann algebra is then defined as $Z=\mathfrak{M}^{\prime}\cap\mathfrak{M}^{''}$. A state is called factor if the center of its von Neumann algebra contains only multiples of identity.

A state ω is called pure if it can not be written as a convex combination of two or more states

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \omega=\lambda\omega_{1}+(1-\lambda)\omega_{2}\qquad 0<\lambda<1 \nonumber \end{align*}$$

where $\omega_{1}, \omega_{2}, \omega$ are pairwise distinct. Otherwise it is called mixed. The GNS representation of a state is irreducible if and only if the state is pure [58, theorem 2.3.19]. The GNS representation of a state is irreducible if the state is factor.

Most algebraic states are mere mathematical artifacts, and one needs a prescription for selecting interesting specific states that can be considered of physical relevance. One strategy is to rely on the quantum dynamics, encoded in a constraint operator. From the algebraic point of view the constraint operator is therefore related to the choice of the folium, or conversely, information about the constraint operator is partly encoded in the algebraic state.

We will not discuss the constraint operator explicitly, since little is known at present about the constraint operators underlying specific GFT models. Instead and reasonably, using the following criteria starting from the Fock representation of GFT, we consider state sequences that satisfy two conditions:

• 1.  
  All states in the sequence are coherent states.This is mainly motivated by the use of GFT coherent states in the extraction of an effective continuum dynamics in the series of works [40, 43, 44, 46]. Of course, coherent states are also key for the classical approximation of any QFT, and routinely used in particle physics, many-body systems and condensed matter theory, which provides further motivation.
• 2.  
  The particle number of the limit state diverges.As described above, it is reasonable to expect that quantum states that describe smooth geometries contain infinitely many particles. This is only possible if the particle number operator in the corresponding representation is formally divergent and by consequence, if the corresponding representation is non-Fock.

In the next section we provide simple explicit examples for GFT representations that satisfy these two properties.

The Weyl algebra $\mathfrak{A}$ admits the Fock representation, which is given by the GNS representation of the algebraic state

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \omega_{\rm F}\left(W_{(\,f)}\right)={\rm e}^{-\frac{\|f\|_{L^{2}}^{2}}{4}}.\label{eq:Fock state} \nonumber \end{align} \tag{ 25 }$$

Since the above state is regular, i.e. the function $\Omega(t):=\omega_{\rm F}\left(W_{(tf)}\right)$ for $t\in I\subset\mathbb{R}$ and any fixed $f\in\mathcal{S}$ is smooth, the generator of the Weyl operator exists by Stone's theorem [59]. Denoting the corresponding GNS triple by $\left(\mathcal{H}_{\rm F}, \pi_{\rm F}, \vert o)\right)$, we can write

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle (o|\pi_{\rm F}\left[W_{(\,f)}\right]|o)=(o|{\rm e}^{\imath\Phi_{\rm F}(\,f)}|o), \nonumber \end{align} \tag{ 26 }$$

where $\Phi_{\rm F}(\,f)$ is an essentially self-adjoint generator of $\pi_{\rm F}\left[W_{(\,f)}\right]$ in the Fock representation, defined on the dense domain

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle D\left(\Phi_{\rm F}\right) =\left\{\sum_{i}^{N}c_{i}\,\pi_{\rm F}\left[W_{\left(\,f_{i}\right)}\right]|o)|\,c_{i}\in\mathbb{C},f_{i}\in\mathcal{S},\,N\in\mathbb{N}\right\} . \nonumber \end{align*}$$

We can obtain the action of $\Phi_{\rm F}(\,f)$ on $D\left(\Phi_{\rm F}\right)$ by differentiation. For any $\vert\psi)\in D\left(\Phi_{\rm F}\right)$ and appropriate set of complex coefficients $\left\{c_{i}\right\} _{i\in\left\{0, \cdots, N\right\} }$ and test functions $\left\{\,f_{i}\right\} _{i\in\left\{0, \cdots, N\right\} }$ such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \vert\psi)=\sum_{i=0}^{N}c_{i}\,\pi_{\rm F}\left[W_{\left(\,f_{i}\right)}\right]\vert o), \nonumber \end{align} \tag{ 27 }$$

we get

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle (o\vert\Phi_{\rm F}(\,f)\vert\psi)=-\imath\partial_{t}\,\omega_{\rm F}\left(W_{(tf)}\sum_{i=0}^{N}c_{i}\,W_{\left(\,f_{i}\right)}\right)\vert_{t=0}.\label{eq:Phi generator} \nonumber \end{align} \tag{ 28 }$$

In particular we obtain for any $f\in\mathcal{S}$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle (o\vert\Phi_{\rm F}(\,f)\vert o)=0, \nonumber \end{align} \tag{ 29 }$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \|\Phi_{\rm F}(\,f)\vert o)\|_{\mathcal{H}}^{2}=(o\vert\Phi_{\rm F}(\,f)\Phi_{\rm F}(\,f)\vert o)=\frac{1}{2}\|f\|_{L^{2}}^{2}.\label{eq:square of Phi_F} \nonumber \end{align} \tag{ 30 }$$

By similar calculations it follows that the operators $\Phi_{\rm F}(\,f)$ satisfy the commutation relation, for any $f, g\in\mathcal{S}$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle \left[\Phi_{\rm F}(\,f),\Phi_{\rm F}(g)\right]=\imath{\rm Im}\left[(\,f,g)_{L^{2}}\right].\label{eq:CCR for field operators} \nonumber \end{align} \tag{ 31 }$$

We call $\Phi_{\rm F}(\,f)$ the field operator of GFT.

We can also define the creation and annihilation operators by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle \psi_{\rm F}(\,f) =\frac{1}{\sqrt{2}}\left[\Phi_{\rm F}(\,f)+\imath\Phi_{\rm F}(\imath f)\right]\label{eq:anihilation operator} \nonumber \end{align} \tag{ 32 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle \psi_{\rm F}^{\dagger}(\,f) =\frac{1}{\sqrt{2}}\left[\Phi_{\rm F}(\,f)-\imath\Phi_{\rm F}(\imath f)\right], \nonumber \end{align} \tag{ 33 }$$

with $\psi_{\rm F}(\,f){}^{\dagger}=\psi_{\rm F}^{\dagger}$$(\,f)$, such that $\psi_{\rm F}(\,f)$ is anti-linear in f, $\psi_{\rm F}^{\dagger}(\,f)$ is linear in f, both are closed on $D\left(\Phi_{\rm F}\right)$ and fulfill the canonical commutation relations

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left[\psi_{\rm F}(\,f),\psi_{\rm F}(g)\right]=\left[\psi_{\rm F}^{\dagger}(\,f),\psi_{\rm F}^{\dagger}(g)\right]=0 \nonumber \end{align} \tag{ 34 }$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left[\psi_{\rm F}(\,f),\psi_{\rm F}^{\dagger}(g)\right]=(\,f,g). \nonumber \end{align} \tag{ 35 }$$

From equations (32), (28) and (30) it follows that

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|\psi_{\rm F}(\,f)\vert o)\|_{\mathcal{H}}^{2} =(o\vert\psi_{\rm F}^{\dagger}(\,f)\psi_{\rm F}(\,f)\vert o)=0, \nonumber \end{align*}$$

and therefore

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \psi_{\rm F}(\,f)\vert o)=0, \nonumber \end{align} \tag{ 36 }$$

for all $f\in\mathcal{S}$. Hence, $\vert o)$ is the Fock vacuum with respect to the annihilation operator $\psi_{\rm F}(\,f)$ and the space $\mathcal{H}_{\rm F}$ is spanned by polynomials of creation operators $\psi_{\rm F}^{\dagger}(\,f)$ applied on $\vert o)$.

The Fock state is pure and hence the GNS representation of $\omega_{\rm F}$ is irreducible [60]. Also the Fock representation is the unique representation (up to unitary equivalence) in which the particle number operator N exists, formally given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle N =\sum_{i\in\mathbb{N}}\psi_{\rm F}^{\dagger}\left(\,f_{i}\right)\psi_{\rm F}\left(\,f_{i}\right)\label{eq:particle number operator} \nonumber \end{align} \tag{ 37 }$$

for some complete orthonormal basis $\left\{\,f_{i}\right\} _{i\in\mathbb{N}}$ of $L^{2}\left(M, {\rm d}x\right)$.

Usually coherent states are characterized as eigenstates of the annihilation operators in the Fock representation, and hence require a notion of the Hilbert space for their very definition. In the algebraic approach, this characterization is avoided by introducing a generalized notion of coherent states directly at the level of the algebra. This is described in [61, 62]. Below we briefly summarize some of the results of that work that will be important for our discussion.

Definition 13. Let $\Gamma:\mathcal{S}_{\infty}\to\mathbb{C}$ be a continuous linear form on $\mathcal{S}_{\infty}$. A state ω defined on the Weyl elements as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle \omega_{\Gamma}\left(W_{(\,f)}\right)=\omega_{\rm F}\left(W_{(\,f)}\right)\,{\rm e}^{\imath\,\sqrt{2}{\rm Re}\left[\Gamma(\,f)\right]},\label{eq:coherent state} \nonumber \end{align} \tag{ 38 }$$

and extended to $\mathfrak{A}\left(\mathcal{S}\right)$ by linearity, is called a coherent state. It is pure and regular [61].

With this definition the Fock state is the special case of the above family of coherent states for $\Gamma=0$.

Any linear functional Γ corresponds to a well defined state [61]. It should be noticed that there exist even more general definitions of coherent states, but this is the one that most closely reflects the condition of being an eigenfunction of the annihilation operator.

proposition The state ω of the above form is equivalent to the Fock state, if and only if Γ is continuous on $\mathcal{S}_{L^{2}}$.

The detailed proof of this proposition is presented in [62], but we provide an intuitive sketch.

Assume that Γ is a continuous functional on $\mathcal{S}_{L^{2}}$, and hence, it extends by continuity to $L^{2}\left(M, {\rm d}x\right)$. Then by the Riesz lemma there exists an $\gamma\in L^{2}\left(M, {\rm d}x\right)$ such that for any $f\in S_{L^{2}}$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Gamma(\,f)=\int_{M}\,f(x)\cdot\bar{\gamma}(x)\,{\rm d}x, \nonumber \end{align} \tag{ 39 }$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \|\Gamma\|_{\rm op}=\|\gamma\|_{L^{2}}. \nonumber \end{align} \tag{ 40 }$$

The state $\omega_{\Gamma}$ provides a GNS triple $\left(\mathcal{H}_{\Gamma}, \pi_{\Gamma}, \vert\Gamma)\right)$. It is not difficult to see that in this case the GNS Hilbert space is Fock and that $\overline{L(\,f)}$ is the eigenvalue of the state vector $|\Gamma)$ [61], i.e.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \psi_{\Gamma}(\,f)|\Gamma)=\overline{\Gamma(\,f)}|\Gamma)=(\,f,\gamma)_{L^{2}}|\Gamma).\label{eq:Eigenvalue equation} \nonumber \end{align} \tag{ 41 }$$

Since the representation is Fock, the particle number operator, equation (37), exists and its expectation value is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle (\Gamma|N|\Gamma)=\sum_{i\in\mathbb{N}}\left|\Gamma\left(\,f_{i}\right)\right|^{2}=\|\gamma\|=\|\Gamma\|_{\rm op}. \nonumber \end{align} \tag{ 42 }$$

That is, the particle number is given by the L² norm of γ or equivalently the operator norm of Γ. When Γ is discontinuous on $\mathcal{S}_{L^{2}}$ and, hence, unbounded on $L^{2}\left(M, {\rm d}x\right)$ the global particle number is ill-defined and the representation can not be Fock.

The non-Fock coherent states are hence classified by functionals Γ which are continuous on $\mathcal{S}_{\infty}$ but discontinuous on $\mathcal{S}_{L^{2}}$, sometimes called the space of tempered microfunctions.

By the Riesz–Markov theorem every functional Γ on $\mathcal{S}_{\infty}$ is of the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Gamma(\,f)=\int_{M}\:f(x)\,{\rm d}\nu(x),\label{eq:Riesz decomposition theorem} \nonumber \end{align} \tag{ 43 }$$

for some Baire measure ν.

From this we can easily state

Corollary 15. If  Γ is invariant under left multiplication i.e. for any $y\in M$, $\Gamma\left(L_{y}^{\star}f\right)=\Gamma(\,f)$ for any $f\in\mathcal{S}$, then the coherent state $\omega_{\Gamma}$ is Fock.

Proof. Let Γ be invariant under left translations. Then for any $f\in\mathcal{S}$ we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Gamma\left(L_{y}^{\star}f\right)=\int_{M}\:L_{y}^{\star}f(x)\,{\rm d}\nu(x)=\Gamma(\,f)=\int_{M}\:f(x)\,{\rm d}\nu(x), \nonumber \end{align} \tag{ 44 }$$

hence the measure ν is a left-invariant measure on M. By uniqueness of the Haar measure, ${\rm d}\nu=c\cdot{\rm d}x,$ for some $c\in\mathbb{R}$. Then by Hölder's inequality $\left|\Gamma(\,f)\right|\leqslant c\|f\|_{L^{2}}$, and hence Γ is continuous on $L^{2}\left(M, {\rm d}x\right)$. □

2.2.1. Remarks on the discontinuity of Γ.

Our procedure to construct inequivalent representations is fairly straightforward. By the above discussion, we simply need to construct a sequence of continuous functionals $\Gamma_{n}$ on $\mathcal{S}_{\infty}$ that converge pointwise to a functional $\Gamma_{\infty}$ unbounded on $L^{2}\left(M, {\rm d}x\right)$. Here we provide a very simple example in which the sequence of regular measures converges to a pure point measure. It should be clear, however, that any measure that satisfies the property of being unbounded on $L^{2}\left(M, {\rm d}x\right)$ leads to a new inequivalent representation.

Let us first define the Dirac measure $\nu_{D}$, such that for any open $U\subset M$ and $\mathds{1}\in M$ denoting the identity on M,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nu_{D}(U)=\left\{\begin{array}{@{}ll@{}} 1 & {\rm if }\quad \mathds{1}\in U \nonumber \\ 0 & {\rm otherwise} \end{array}\right.. \nonumber \end{align} \tag{ 45 }$$

It follows that on smooth functions $f\in\mathcal{S}$ we have,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nu_{D}(\,f)=f\left(\mathds{1}\right). \nonumber \end{align} \tag{ 46 }$$

Such a Riesz functional is continuous on $\mathcal{S}_{\infty}$, since

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left|\nu_{D}(\,f)\right|=\left|f\left(\mathds{1}\right)\right|\leqslant\|f\|_{\infty}. \nonumber \end{align} \tag{ 47 }$$

However, it is unbounded on $L^{2}\left(M, {\rm d}x\right)$ due to the possible singular behavior of functions at sets of Haar measure zero.

Assume further a contracting sequence of open sets $\left\{U_{n}\right\} _{n\in\mathbb{N}}$ around the identity $\mathds{1}\in M$, such that $U_{n+1}\subset U_{n}$ and $\cap_{n\in\mathbb{N}}U_{n}=\left\{\mathds{1}\right\}$, and consider a sequence of measures defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm d}\nu_{n}=\frac{\chi_{U_{n}}}{\left|U_{n}\right|}{\rm d}x,\label{eq:Sequence of measures} \nonumber \end{align} \tag{ 48 }$$

where $\chi_{U_{n}}$ is the characteristic function on U_n,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \chi_{U_{n}}(x)=\left\{\begin{array}{@{}ll@{}} 1 & {\rm if }~x\in U_{n} \nonumber \\ 0 & {\rm otherwise} \end{array}\right., \nonumber \end{align} \tag{ 49 }$$

and $\left|U_{n}\right|=\int_{U_{n}}\, {\rm d}x$.

Lemma 17. On $\mathcal{S}$ the sequence of functionals defined by (48) converges to the Dirac measure in the distributional sense. That is for any $f\in\mathcal{S}$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \lim_{n\to\infty}\nu_{n}(\,f)=\nu_{D}(\,f)=f\left(\mathds{1}\right). \nonumber \end{align} \tag{ 50 }$$

Proof. Since f is continuous, we can find for some $\newcommand{\e}{{\rm e}} \epsilon>0$ a neighborhood $\newcommand{\e}{{\rm e}} N_{\epsilon}\left(\mathds{1}\right)$ around $\mathds{1}$ on M such that $\newcommand{\e}{{\rm e}} \forall x\in N_{\epsilon}\left(\mathds{1}\right)$ $f(x)$ is in an -ball around $f\left(\mathds{1}\right)$ in $\mathbb{C}$. Since the sequence is contracting $\newcommand{\e}{{\rm e}} \exists N\in\mathbb{N}$ such that $\forall n>N$, $\newcommand{\e}{{\rm e}} U_{n}\subset N_{\epsilon}\left(\mathds{1}\right)$ then

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left|\nu_{n}(\,f)-\nu_{D}(\,f)\right| & =\left|\frac{1}{\left|U_{n}\right|}\int_{M}\,f(x)\,\chi_{U_{n}}(x){\rm \,d}x-f\left(\mathds{1}\right)\right|\nonumber \\ & =\frac{1}{\left|U_{n}\right|}\left|\int_{M}\,\chi_{U_{n}}(x)\left(\,f(x)-f\left(\mathds{1}\right)\right){\rm d}x\right|\nonumber \\ & \leqslant\frac{1}{\left|U_{n}\right|}\int_{M}\,\chi_{U_{n}}(x)\left|f(x)-f\left(\mathds{1}\right)\right|{\rm d}x\nonumber \\ & \leqslant\epsilon. \nonumber \end{align*}$$

□

At every finite n the measure $\nu_{n}$ is absolutely continuous with respect to the Haar measure and by the above proposition 14 every state

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle \omega_{n}\left(W_{(\,f)}\right):=\omega_{\rm F}\left(W_{(\,f)}\right)\cdot {\rm e}^{\imath\sqrt{2}{\rm Re}\left[\Gamma_{n}(\,f)\right]}, \nonumber \end{align} \tag{ 51 }$$

is equivalent to the Fock one. Where $\Gamma_{n}(\,f)\doteq\int_{M}\, f(x)\, {\rm d}\nu_{n}$. From the convergence of the measure, the convergence of the algebraic sequence is obvious.

Lemma 18. The sequence of states $\omega_{n}$ converges in the $w^{\star}$-topology to $\omega_{D}^{\mathds{1}}$, defined on Weyl elements such that for each $f\in\mathcal{S}$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle \omega_{D}^{\mathds{1}}\left(W_{(\,f)}\right)\doteq\omega_{\rm F}\left(W_{(\,f)}\right)\cdot {\rm e}^{\imath\sqrt{2}{\rm Re}\left(\Gamma[\,f]\left(\mathds{1}\right)\right)}, \nonumber \end{align} \tag{ 52 }$$

and extended by linearity to the whole $\mathfrak{A}$.

Proof. For any $W_{(\,f)}\in\mathfrak{A}$ we have

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle \left|\omega_{n}\left(W_{(\,f)}\right)-\omega_{D}^{\mathds{1}}\left(W_{(\,f)}\right)\right| & =\left|\omega_{F}\left(W_{(\,f)}\right)\right|\left|e^{\imath\sqrt{2}\Re\left[\int f\,\text{d}\nu_{n}\right]}-e^{\imath\sqrt{2}\Re\left[\int f\,\text{d}\nu_{D}\right]}\right|\\ & =\left|\omega_{F}\left(W_{(\, f)}\right)\right|\left|e^{\imath\sqrt{2}\Re\left[\int f\,\text{d}\nu_{n}-\int f\,\text{d}\nu_{D}\right]}-1\right|\\ & \to0. \end{align*}$$

By linearity of the state and the product property of the Weyl algebra, this extends to the whole algebra $\mathfrak{A}$. □

At finite n the representation is Fock, the particle number operator exists and the particle number of the nth member of the sequence is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \|\Gamma_{n}\|_{\rm op}=\frac{1}{\left|U_{n}\right|}. \nonumber \end{align} \tag{ 53 }$$

But with increasing n the particle number grows since the volume of U_n shrinks. At the limit point the total particle number diverges and the corresponding representation becomes inequivalent to the Fock one.

We can define states $\omega_{D}^{x}$ peaked at points $x\in M$ using the automorphisms $\alpha_{x^{-1}}$ introduced in the previous section such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \omega_{D}^{x}=\omega_{D}^{\mathds{1}}\circ\alpha_{x^{-1}}. \nonumber \end{align} \tag{ 54 }$$

We will show in the next section that each of the states $\omega_{D}^{x}$ leads to an inequivalent representation and breaks translation invariance.

In this section we will focus on the algebra $\mathfrak{A}\left(\mathcal{S}_{G}\right)$, since it is more relevant for GFT's with simplicial interpretation, however, all the constructions can be directly applied to $\mathfrak{A}\left(\mathcal{S}\right)$ leading to similar results.

We now construct an explicit representation that is generated by the above algebraic state following the construction in [64].

Take $L^{2}\left(M, {\rm d}\nu_{D}^{x}\right)$ to be the space of L² functions with respect to the Dirac measure concentrated at $x\in M$, i.e. for any $f\in\mathcal{S}_{G}$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nu_{D}^{x}(\,f)=f(x). \nonumber \end{align} \tag{ 55 }$$

The space $L^{2}\left(M, {\rm d}\nu_{D}^{x}\right)$ is one-dimensional. For any $f\in\mathcal{S}_{G}$ define commuting operators $A(\,f)$ and $B(\,f)$ on $L^{2}\left(M, {\rm d}\nu_{D}^{x}\right)$ such that for any $\varphi\in L^{2}\left(M, {\rm d}\nu_{D}^{x}\right)$ and $f\in\mathcal{S}_{G}$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left[A_{x}(\,f)\varphi\right](x) =f(x)\varphi(x) \qquad \left[B_{x}(\,f)\varphi\right](x) =\bar{f}(x)\varphi(x). \nonumber \end{align*}$$

We define the state vector

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle |\Omega_{D}^{x})\equiv |o)\otimes1 \nonumber \end{align} \tag{ 56 }$$

where 1 is the constant function on M and $|o)$ is the Fock vacuum. Further we define unitary operators

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle W_{(\,f)}^{x}={\rm e}^{\frac{\imath}{\sqrt{2}}\left[\psi_{\rm F}(\,f)+\psi_{\rm F}^{\dagger}(\,f)\right]}\otimes {\rm e}^{\frac{\imath}{\sqrt{2}}\left[A(\,f)+B(\,f)\right]}, \nonumber \end{align} \tag{ 57 }$$

where $\psi_{\rm F}(\,f), \psi_{\rm F}^{\dagger}(\,f)$ are the Fock operators. We denote the closure of the space generated by polynomials of operators $W_{(\,f)}^{x}$ acting on $\vert\Omega_{D}^{x})$ by $\mathcal{H}_{x}$. It follows that the operator algebra spanned by $W_{(\,f)}^{x}$ for $f\in\mathcal{S}_{G}$ is equivalent to $\mathfrak{M}$ (the $\omega_{D}^{x}$-GNS representation of the Weyl algebra $\mathfrak{A}\left(\mathcal{S}_{G}\right)$) since the expectation values coincide,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle (\Omega_{D}^{x}|W_{(\,f)}^{x}|\Omega_{D}^{x})={\rm e}^{-\frac{\|f\|_{L^{2}}^{2}}{4}}\cdot {\rm e}^{\imath\sqrt{2}{\rm Re}\left[\,f(x)\right]}.\label{eq:representation} \nonumber \end{align} \tag{ 58 }$$

Irreducibility and cyclicity of this representation are inherited from the Fock representation since $PL^{2}\left(M, \nu_{D}\right)$ is one-dimensional.

Let f_y be a real valued function on M defined such that for some fixed $a\in\mathbb{R}$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f_{y}(x)=\left\{\begin{array}{@{}ll@{}} a & {\rm if}\quad \exists h\in G\,~{\rm such ~that}~ {x=y\,D_{h}} \nonumber \\ 0 & {\rm else} \end{array} \right., \nonumber \end{align} \tag{ 59 }$$

clearly $f_{y}\in PL^{2}\left(M, {\rm d}x\right)$ and is zero almost everywhere with respect to the Haar measure.

Lemma 19. Let $\left\{\,f_{n}\vert\,f_{n}\in\mathcal{S}_{G}\right\} _{n\in\mathbb{N}}$ be a sequence that converges to f_y in the L²-norm. Then the limit $\lim_{n\to\infty}W_{\left(\,f_{n}\right)}^{x}$ exists in $\mathfrak{M}$. We call this element $W_{\left(\,f_{y}\right)}^{x}$. Moreover, $W_{\left(\,f_{y}\right)}^{x}$ is in the center and there exists a complex number $c\in\mathbb{C}$ such that $W_{\left(\,f_{y}\right)}^{x}=c\mathds{1}$.

Proof. Since $f_{y}\in PL^{2}\left(M, {\rm d}x\right)$ and $\mathcal{S}_{G}$ is dense in $PL^{2}\left(M, {\rm d}x\right)$ there exists a Cauchy sequence $\left\{\,f_{n}\vert\,f\in\mathcal{S}_{G}\right\}$ that converges to f_y. Then for n and m large enough and any $g\in\mathcal{S}_{G}$ we get by direct calculation

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle \|\left(W_{\left(\,f_{n}\right)}^{x}-W_{\left(\,f_{m}\right)}^{x}\right)W_{(g)}^{x}\vert\Omega_{D}^{x})\|_{\mathcal{H}} & =2(\Omega_{D}^{x}\vert\Omega_{D}^{x}) -2{\rm Re}\left[{\rm e}^{-\frac{\|f_{n}-f_{m}\|_{L^{2}}^{2}}{4}}{\rm e}^{\imath\sqrt{2}{\rm Re}\left[\,f_{n}(x)-f_{m}(x)\right]}\right]\nonumber \\ & \times{\rm Re}\left[{\rm e}^{-\frac{\imath}{2}{\rm Im}\left[\left(\,f_{n}-f_{m},g\right)_{L^{2}}+\left(-f_{m}-g,f_{n}+g\right)_{L^{2}}\right]}\right] \leqslant\epsilon. \nonumber \end{align*}$$

Since $\vert\Omega_{D}^{x})$ is cyclic we can reach every element of $\mathcal{H}_{x}$ acting on it by polynomials of Weyl operators. Hence, the sequence $\left\{W_{\left(\,f_{n}\right)}^{x}\vert\,f_{n}\in\mathcal{S}_{G}\right\}$ is a Cauchy sequence in the strong operator topology and therefore converges to an element in the von Neumann algebra $\mathfrak{M}^{''}$. We call this element $W_{\left(\,f_{y}\right)}^{x}$. For any $f\in\mathcal{S}_{G}$ we have

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle W_{(\,f)}^{x}W_{\left(\,f_{y}\right)}^{x} & =\lim_{n\to\infty}W_{\left(\,f+f_{n}\right)}^{x}{\rm e}^{-\frac{\imath}{2}{\rm Im}\left[\left(\,f,f_{n}\right)\right]} =W_{\left(\,f+f_{y}\right)}^{x}=W_{\left(\,f_{y}\right)}^{x}W_{(\,f)}^{x}, \nonumber \end{align*}$$

where the second equality follows from the fact that $W_{\left(\,f+f_{n}\right)}^{x}$ is a Cauchy sequence and f_y is zero almost everywhere. Hence, the element $W_{\left(\,f_{y}\right)}^{x}$ is in the center of the von Neumann algebra $\mathfrak{M}^{''}$. Since the state $\omega_{D}^{x}$ is pure the center contains only multiples of identity, thus there exists a $c\in\mathbb{C}$ such that $W_{\left(\,f_{y}\right)}=c\mathds{1}$. □

2.4.1. Breaking of translation symmetry.

We show that for any $x\in M$ the state $\omega_{D}^{x}$ breaks translation symmetry in the sense that for any non-trivial $y\in M$ the translation automorphism $\alpha_{y}$ can not be represented by a unitary operator on $\mathcal{H}_{x}$.

Corollary 20. Let $x, y\in M$ with $y\neq\mathds{1}$, then the states $\omega_{D}^{x}$ and $\omega_{D}^{yx}$ are inequivalent.

Proof. $\omega_{D}^{x}$ and $\omega_{D}^{yx}$ are pure states and therefore factor. By [58, proposition 2.4.27] factor states $\omega_{D}^{x}$ and $\omega_{D}^{yx}$ are (quasi)-equivalent if and only if the state $\omega=\frac{1}{2}\left(\omega_{D}^{x}+\omega_{D}^{yx}\right)$ is factor as well. Therefore it is enough to show that the center of the von Neumann algebra of ω is non-trivial. For some fixed $a\neq0\in\mathbb{R}$ define the function

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f_{yx}(z)=\left\{\begin{array}{@{}ll@{}} a & {\rm if }\quad \,\exists h\in G\quad M\ni z=yx\,D_{h} \nonumber \\ 0 & {\rm else} \end{array}\right.. \nonumber \end{align} \tag{ 60 }$$

The GNS triple of ω is given by

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left(\mathcal{H}=\mathcal{H}_{x}\oplus\mathcal{H}_{yx},\pi_{\omega}=\pi_{\omega_{D}^{x}}\oplus\pi_{\omega_{D}^{yx}},\vert\Omega_{D})=\vert\Omega_{D}^{x})\oplus\vert\Omega_{D}^{yx})\right), \nonumber \end{align*}$$

By lemma 19 there exists an element $W_{\left(\,f_{xy}\right)}$ in the von Neumann algebra of ω as a limit of an appropriate sequence of operators $W_{\left(\,f_{n}\right)}$, and $W_{\left(\,f_{yx}\right)}$ is in the center of the von Neumann algebra. However, a direct calculation shows that

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle (\Omega_{D}\vert W_{\left(\,f_{xy}\right)}\vert\Omega_{D})&=\frac{1}{2}\left[(\Omega_{D}^{x}\vert W_{\left(\,f_{yx}\right)}^{x}\vert\Omega_{D}^{x})+(\Omega_{D}^{yx}\vert W_{\left(\,f_{yx}\right)}^{yx}\vert\Omega_{D}^{yx})\right] =\frac{1}{2}\left(1+{\rm e}^{\imath\sqrt{2}a}\right). \nonumber \end{align*}$$

Hence, $W_{\left(\,f_{y}\right)}\neq\mathds{1}$ and the GNS representation of ω is not factor. By theorem [58, proposition 2.4.27] the states $\omega_{D}^{x}$ and $\omega_{D}^{yx}$ are inequivalent. □

Since $\omega_{D}^{x}$ and $\omega_{D}^{yx}$ are inequivalent, the translation automorphism $\alpha_{y}$ can not be implemented by an unitary operator for any non-trivial $y\in M$. Hence, the translation symmetry is broken and moreover for $x, z\in M$ not of the diagonal form the states $\omega^{x}$ and $\omega^{z}$, lead to inequivalent representations since they are related by translation $y=zx^{-1}\in M$.

Notice that the automorphism $\alpha_{x}$ implements the isometry of the base manifold and hence the above representations break the isometry transformation. This is rather different from ordinary field theory, in which Poincaré symmetry is not allowed to be broken [57, 60, 64, 65]. Again, this is possible because no spacetime interpretation is attached to the GFT base manifold.