The Fock space of loopy spin networks for quantum gravity

Abstract

In the context of the coarse-graining of loop quantum gravity, we introduce loopy and tagged spin networks, which generalize the standard spin network states to account explicitly for non-trivial curvature and torsion. Both structures relax the closure constraints imposed at the spin network vertices. While tagged spin networks merely carry an extra spin at every vertex encoding the overall closure defect, loopy spin networks allow for an arbitrary number of loops attached to each vertex. These little loops can be interpreted as local excitations of the quantum gravitational field and we discuss the statistics to endow them with. The resulting Fock space of loopy spin networks realizes new truncation of loop quantum gravity, allowing to formulate its graph-changing dynamics on a fixed background graph plus local degrees of freedom attached to the graph nodes. This provides a framework for re-introducing a non-trivial background quantum geometry around which we would study the effective dynamics of perturbations. We study how to implement the dynamics of topological BF theory in this framework. We realize the projection on flat connections through holonomy constraints and we pay special attention to their often overlooked non-trivial flat solutions defined by higher derivatives of the $\delta$-distribution.

Appendices

Appendix 1: Projective limits of loopy spin networks

The general framework of a projective family and the projective limit can be found in [34], where it is applied to define the kinematical Hilbert space of spin network states for loop quantum gravity. Here we apply these definitions to loopy spin networks, in order to define superposition states of potentially an infinite number of little loops. To this purpose, we focus on the flower graph, with a single vertex and an arbitrary number of loops attached to that central node.

In order to define precisely this idea of varying number of loops, we start with wave-functions over a finite number of loops and define a nesting, that is describe how to include a set of loops inside a larger one. We will identity the set of all potential loops with the set of integers. Finite sets of loops are defined as finite subsets of integers. Loops are labeled by the integers and are a priori distinguishable. For instance, a wave-function with the support on the loop number 2 and a wave-function on the loop number 277 are not the same though they both are one-loop states and depend on only one variable, as illustrated in Fig. 21.

Fig. 21

We distinguish the different potential loops and therefore consider the resulting wave-functions as different even when they excite the same number of loops

Mathematically, we consider the set of all finite subsets of integers $\mathcal {P}_{<\infty }(\mathbb {N})$. To each subset $E \in \mathcal {P}_{<\infty }(\mathbb {N})$, we associate the set $\mathrm {SU}(2)^{E}$ of colorings of the corresponding loops by $\mathrm {SU}(2)$ group elements. Then wave-functions on E are gauge-invariant functions over $\mathrm {SU}(2)^{E}$:

$$\begin{aligned} \{\Psi : \mathrm {SU}(2)^E \rightarrow \mathbb {C}~:~ \forall g \in \mathrm {SU}(2),~ \Psi (\{gh_{\ell }g^{-1}\}_{\ell \in E}) = \Psi (\{h_{\ell }\}_{\ell \in E})\}. \end{aligned}$$

(137)

Defining the scalar product using the Haar measure over $\mathrm {SU}(2)^{E}$, the Hilbert space ${\mathcal {H}}_{E}$ of quantum states on the loopy spin network defined by the subset E of loops is the $L^{2}(\mathrm {SU}(2)^{E}/\text {Ad}\mathrm {SU}(2))$.

The space of loops is equipped with a partial directed order given by the inclusion of subsets of integers. The partial directed order encodes how different subsets are nested within one another: a wave-function over the loop number 2 and a wave-function over the loop number 277 are different but they are both embedded in the larger class of wave-functions which depend on both loop number 2 and loop number 277 as illustrated in Fig. 22.

Fig. 22

Though different, two functions over two different loops can be embedded in a larger space of functions depending on several loops by identifying them with functions with trivial dependency on some loops

This partial ordering by inclusion of subsets induces a projective structure on the loop colorings by group elements. We define a projector $p_{EE'}$ defined for every pair of subsets $(E,E')$ such that $E \subseteq E'$ by:

This projector is simply the canonical restriction from the larger subset $E'$ to the smaller subset E. These projectors satisfy a key transitivity property:

$$\begin{aligned} \forall E,E',E'', \quad E\subset E'\subset E''\, \Longrightarrow p_{E'E''} \circ p_{EE'} = p_{EE''}, \end{aligned}$$

(138)

so that the couple of sets $(\mathrm {SU}(2)^{E},p_{EE'})_{E,E' \in \mathcal {P}_{<\infty }(\mathbb {N})}$ form what is called a projective family. The projective limit $\overline{\mathrm {SU}(2)}$ is then defined by:

$$\begin{aligned} \overline{\mathrm {SU}(2)} = \Big \{ (g_{E})_{E \in \mathcal {P}} \in \times _{E\in \mathcal {P}} \mathrm {SU}(2)^{E} ~:~ E \subseteq E' \Rightarrow p_{EE'} g_{E'} = g_{E} \Big \} \end{aligned}$$

(139)

Intuitively, this corresponds to collections of colorings on all possible subsets of loops which are compatible with each other with respect to the inclusion. Therefore, these compatibility conditions between all finite samplings of the collection, as illustrated in Fig. 23, is the precise implementation of the notion of a coloring of an infinite number of loops.

Fig. 23

The projective limit is made of collections of colorings of finite subsets equipped with compatibility conditions: the coloring of a finite subset is the projection of the coloring of any larger subset (colour figure online)

We translate the projective structure to the space of wave-functions. The projectors $p_{EE'}$ for $E\subset E'$ turn into injections $I_{EE'}\equiv p^{*}_{EE'}$ defined by their pull-backs:

$$\begin{aligned} I_{EE'} : {\mathcal {H}}_{E}\rightarrow & {} {\mathcal {H}}_{E'} \nonumber \\ f\mapsto & {} p^{*}_{EE'}f : p^{*}_{EE'}f \big (\{g_{\ell }\}_{\ell \in E'}\big )=f \big (\{\{g_{\ell }\}_{\ell \in E}\}\big ), \end{aligned}$$

(140)

where $p^{*}_{EE'}f$ trivially depends on group elements living on loops of $E'$ which do not belong to the smaller set E. The compatibility conditions translates into an equivalence relation:

$$\begin{aligned} f_{E_1} \sim f_{E_2} \Leftrightarrow \forall E_3\in \mathcal {P},~E_1 \subseteq E_3,~E_2\subseteq E_3,~p^*_{E_1 E_3} f_{E_1} = p^*_{E_2E_3} f_{E_2} \end{aligned}$$

(141)

This allows to define wave-functions on the projective limit of the loop colorings $\overline{\mathrm {SU}(2)}$ and give a precise sense to functions over an infinite number of loops:

$$\begin{aligned} \mathcal {H}=\left( \bigcup _{E\in \mathcal {P}} \mathcal {H}_{E}\right) /\sim \end{aligned}$$

(142)

In order to make this projective limit less abstract and easier to handle, we use another representation. For every equivalence class of wave-functions in the projective limit ${\mathcal {H}}$, let us remove all the trivial dependency and pick its representant based on the smallest subset of loops. So, in practice, we define spaces of “proper states”, i.e. wave-functions that have no trivial dependency:

$$\begin{aligned} \mathcal {H}_{E}^0 = \left\{ f \in \mathcal {H}_{E} : \forall \ell \in E,\,\int _{\mathrm {SU}(2)}\mathrm {d}h_{\ell }\,f=0 \right\} . \end{aligned}$$

(143)

This is the space of functions really defined on the subset E, with an actual dependance on each loop and no constant term. The integral condition removes all the spin-0 components of the wave-functions. First, we show that an arbitrary wave-function over the subset E of loops can be fully decomposed into proper states with support on all the subsets of E:

Lemma 6.1

The following isomorphism holds as a pre-Hilbertian space isomorphism:

$$\begin{aligned} \forall E \in \mathcal {P}_{<\infty }(\mathbb {N}), \quad \mathcal {H}_{E} \simeq \bigoplus _{F \in \mathcal {P}(E)} \mathcal {H}_{F}^0, \end{aligned}$$

(144)

where the direct sum is over all subsets $F\subset E$. This isomorphism is realized through the projections $f_{F}=P_{E,F}f$ acting on wave-functions $f\in {\mathcal {H}}_{E}$:

$$\begin{aligned} f_{F}\big (\{h_{\ell }\}_{\ell \in F} \big ) \!=\! \sum _{\widetilde{F}\subset F} (-1)^{\#\widetilde{F}} \int \prod _{\ell \in E{\setminus } F}\mathrm {d}g_{\ell } \prod _{\ell \in \widetilde{F}}\mathrm {d}k_{\ell }\, f\big (\{h_{\ell }\}_{\ell \in F{\setminus } \widetilde{F}}, \{k_{\ell }\}_{\ell \in \widetilde{F}}, \{g_{\ell }\}_{\ell \in E{\setminus } F} \big ). \end{aligned}$$

(145)

Its inverse is the re-summation of the projections:

$$\begin{aligned} f=\sum _{F\subset E} f_{F}. \end{aligned}$$

(146)

Proof

We proceed in two steps. First we check that each projection $f_{F}$ is a proper state,

$$\begin{aligned} \forall \ell \in F,\quad \int \mathrm {d}h_{\ell }\,f_{F}=0, \end{aligned}$$

and that re-summing these projections $\sum _{F\subset E} f_{F}$ yields f. Second, we check that the integral condition, ensuring that there is no spin-0 mode, also implies that the subspaces $\mathcal {H}_{F}^0$ are pairwise orthogonal, which concludes the proof. $\square$

This decomposition generalizes to the projective limit:

Proposition 6.2

The following isomorphism holds as a pre-Hilbertian space isomorphism:

$$\begin{aligned} \mathcal {H} \simeq \bigoplus _{E \in \mathcal {P}_{<\infty }(\mathbb {N})} \mathcal {H}_{E}^0. \end{aligned}$$

(147)

Proof

If $(f_E)_{E \in \mathcal {P}_{<\infty }(\mathbb {N})}$ is in $\bigoplus _{E \in \mathcal {P}_{<\infty }(\mathbb {N})} \mathcal {H}_{E}^0$, we define the set of subsets on which the state f does not vanish:

$$\begin{aligned} C_f = \{ E \in \mathcal {P}_{<\infty }(\mathbb {N}) : f_E \ne 0 \}. \end{aligned}$$

(148)

By definition of the direct sum, $C_f$ is finite, so we can define the finite subset $F = \cup _{E \in C_f} E$ and the re-summation map:

$$\begin{aligned} \phi : \bigoplus _{E \in \mathcal {P}_{<\infty }(\mathbb {N})} \mathcal {H}_{E}^0\rightarrow & {} \mathcal {H} \nonumber \\ (f_E)_{E \in \mathcal {P}_{<\infty }(\mathbb {N})}\mapsto & {} \left[ \sum _{E \in C_f} f_E\right] \end{aligned}$$

(149)

where the brackets refer to the equivalence class of the function. This map is obviously linear and we now look for a definition of its inverse. So let us consider a state in $\mathcal {H}$, that is an equivalence class s. We define the set of subsets of loops on which it has support:

$$\begin{aligned} D_s = \{ E \in \mathcal {P}_{<\infty }(\mathbb {N}) : \exists f \in s,~f \in \mathcal {H}_{E} \} \end{aligned}$$

(150)

Then we consider the smallest set in $D_s$, which can be defined¹⁷ as the intersection $F_s = \bigcap _{E \in D_s} E$. In a sense, this is the minimal support of the state s. We choose a representative $f^{s}$ of the equivalence class s in $F_{s}$. It is actually unique by definition of the equivalence relation. Then we consider the decomposition in proper states of $f^{s}$ over all subsets F of $F_{s}$ and define:

$$\begin{aligned} \psi : \mathcal {H}\rightarrow & {} \bigoplus _{E \in \mathcal {P}_{<\infty }(\mathbb {N})} \mathcal {H}_{E}^0 \nonumber \\ s\mapsto & {} \sum _{F\subset F_{s}}P_{F_{s},F} f^{s} \end{aligned}$$

(151)

It is direct to check that it is indeed the inverse of $\phi$. $\square$

This decomposition into proper states is very useful to visualize the space: each wave-function can be decomposed into a sum of wave-functions over a finite number of loops but with no trivial dependency. This gives a precise meaning to superpositions of numbers of loops.

Appendix 2: Holonomy constraint on $\mathrm {SU}(2)$

1.1 Distributions on $\mathrm {SU}(2)$ and conjugation-invariant solutions

We would like to impose the holonomy constraints for BF theory which read for a single group element:

$$\begin{aligned} \forall g \in \mathrm {SU}(2), \quad \widehat{\chi }\varphi \,(g)= \chi _\frac{1}{2}(g) \varphi (g) = 2 ,\varphi (g). \end{aligned}$$

(152)

If we stay in the strict framework of the Hilbert space $L^{2}(\mathrm {SU}(2))$, no square integrable function actually provides such an eigenvector for $\widehat{\chi }$ and we should solve this equation in the dual space. As is standard in quantum mechanics, the natural framework for solving the equation is a rigged-Hilbert space (or Gelfand triple), that is a triplet: $\mathcal {S} \subset \mathcal {H} \subset \mathcal {S}^*$. The space $\mathcal {H}$ is the Hilbert space. The smaller space $\mathcal {S}$ is provided with a stronger topology than the induced one and can thought of as the test function space, while its dual $\mathcal {S}^{*}$ is the space of continuous linear forms on ${\mathcal {S}}$ and defines the distribution space. The major property of $\mathcal {S}$ is to be small enough for the algebra of observables to be defined over it. Then the operator algebra can be naturally extended on $\mathcal {S}^{*}$ and thus on $\mathcal {H}$. For instance, an operator A defined on $\mathcal {S}$ acts on a (dual) state $\varphi$ be in $\mathcal {S}^*$ as:

$$\begin{aligned} \forall f \in \mathcal {S},~A\varphi (f) = \varphi (A^\dagger f). \end{aligned}$$

(153)

So let us be explicit for functions over $\mathrm {SU}(2)$. The Hilbert space $\mathcal {H}$ is the space of square-integrable functions. The space $\mathcal {S}$ is usually chosen to be the Schwarz space so that canonical position and momentum operator can be defined. Here, the rapid fall-off condition is not needed since we are dealing with a compact group, but we keep the smoothness requirement:

$$\begin{aligned} \mathcal {S} = \{\psi \in \mathcal {H} / \psi \in \mathcal {C}^\infty \} \end{aligned}$$

(154)

Regarding the topology, the space $\mathcal {S}$ is naturally endowed with the convergence on every $\mathcal {C}^k$ space. More precisely the space of $\mathcal {C}^k$ functions is equipped with the following norm:

$$\begin{aligned} \Vert f\Vert _{\mathcal {C}^k} = \sup _{0 \le i \le k} \Vert \partial _{a_1,\ldots ,a_i} f\Vert _\infty \end{aligned}$$

(155)

This norm has two nice properties. First, differentiation is continuous from $\mathcal {C}^k$ to $\mathcal {C}^{k-1}$. Second, the topology induced by the norms are finer as k goes to infinity. So the limit topology on $\mathcal {S} = \bigcap _{k \in \mathbb {N}} \mathcal {C}^k$goes as follows: a sequence of functions $f_{n\in {\mathbb {N}}}$ in $\mathcal {S}$ admits 0 as its limit if the sup-norm of all its derivatives $\Vert \partial _\alpha f_{n}\Vert _{\infty }$ go to 0 for arbitrary multi-index $\alpha$. This is topology is naturally finer than all the $\mathcal {C}^k$ topologies and the differentiation is still continuous. Provided with this topology, $\mathcal {S}$ is a Frechet space: it is complete and metrizable (though no norm is defined). Note that, although all the $\mathcal {C}^k$ are Banach spaces, their descending intersection $\bigcap _{k \in \mathbb {N}} \mathcal {C}^k$ is not.

Things are usually clearer and more explicit in the Fourier decomposition. Let us consider the Fourier decomposition of a function over $\mathrm {SU}(2)$ on the Wigner matrices:

$$\begin{aligned} f(g)=\sum _{j,a,b}f^{j}_{ab}D^{j}_{ab}. \end{aligned}$$

By the Fourier convergence theorem, smoothness actually translates into a rapid fall-off of the Fourier coefficients:

$$\begin{aligned} f\in {\mathcal {S}}\quad \Longleftrightarrow \quad \forall K\in {\mathbb {N}}, \quad \sum _{j} |f^{j}| d_j^K < +\infty , \end{aligned}$$

(156)

where $d_{j}=(2j+1)$ is the dimension of the spin-j representation and $|f^{j}|$ can equally be the sup-norm or the square-norm of the matrix $f^{j}_{ab}$. This also means that the Fourier coefficients of a distribution cannot diverge faster than polynomially:

$$\begin{aligned} \varphi \in {\mathcal {S}}^{*} \quad \Longleftrightarrow \quad \exists K, \quad \sum _{j} |\varphi ^{j}| d_j^{-K} < +\infty . \end{aligned}$$

(157)

The strong topology on ${\mathcal {S}}$ means that a sequence of smooth functions $f_{n}$ converges to 0 in ${\mathcal {S}}$ if and only if all the K-power sums go to 0:

$$\begin{aligned} \lim _{n\rightarrow \infty }f_{n}=0 \quad \Longleftrightarrow \quad \forall K\in {\mathbb {N}}, \quad \sum _{j} |f^{j}_{n}| d_j^K \quad \underset{n\rightarrow \infty }{\longrightarrow }0. \end{aligned}$$

(158)

This ensures that the evaluations of a distribution $\varphi$ will also converge $\varphi (f_{n})\rightarrow 0$.

Let us apply this to the holonomy constraints for functions invariant under conjugation on $\mathrm {SU}(2)$. In this case, all functions decompose on the characters,

$$\begin{aligned} \varphi (g)=\sum _{j\in \frac{{\mathbb {N}}}{2}}\varphi _{j}\chi _{j}(g), \end{aligned}$$

and the eigenvalue problem $\chi (g)\varphi (g)=2\varphi (g)$ translates into a recursion relation on the Fourier coefficients:

$$\begin{aligned} \rho \varphi _{0}=\varphi _{\frac{1}{2}},\quad \rho \varphi _{j\ge \frac{1}{2}}=\varphi _{j-\frac{1}{2}}+\varphi _{j+\frac{1}{2}}. \end{aligned}$$

(159)

Once we fix the initial condition $\varphi _{0}$, this recursive equation has a solution for every complex value $\rho \in {\mathbb {C}}$, but this does not systematically defines a solution state, in $L^{2}$ or a distribution. The solution to the recursion is given in terms of the two solutions $\mu _{\pm }$ of the quadratic equation $\mu ^{2}-\rho \mu +1=0$:

$$\begin{aligned} \forall j\in \frac{{\mathbb {N}}}{2}, \quad \varphi _{j}=\frac{\mu _{+}^{2j+1}-\mu _{-}^{2j+1}}{\mu _{+}-\mu _{-}}. \end{aligned}$$

(160)

For $\rho =2$, the discriminant $(\rho ^{2}-4)$ vanishes and this ansatz fails leads: instead of the power law, we get a linear growth $f_{j}=(2j+1)$, which leads back to the $\delta$-distribution peaked on the identity. For real values $|\rho |< 2$, the discriminant is negative and we get an oscillatory solution. Mapping $\rho$ to an angle $\theta$ defined as $\rho =2\cos \theta$, the two solutions are $\mu _{\pm }=\exp (\pm i \theta )$ and the Fourier coefficients are $\varphi _{j}=\sin (2j+1)\theta \,/\sin \theta \,=\chi _{j}(\theta )$, which gives a $\delta$-distribution fixing the class angle of the group element g to $\theta$. For $|\rho |>2$, the positive discriminant will leads to exponentially divergent coefficients $\varphi _{j}$ and do not define a proper distribution.

1.2 The holonomy constraint as a recursion relation

Let us write the holonomy constraint equation $\widehat{\chi }\,\varphi =2\,\varphi$ for distributions on $\mathrm {SU}(2)$ in the Fourier decomposition. We decompose the distribution $\varphi$ on the Wigner matrices:

$$\begin{aligned} \varphi (h)=\sum _{j,m,n} (2j+1)\varphi ^{j}_{nm}\,D^{j}_{mn}(h), \end{aligned}$$

with the magnetic moment indices $-j\le m,n\le +j$. We know from spin recoupling the action of a spin-$\frac{1}{2}$ Wigner matrix on an arbitrary spin (for example calculating the corresponding Clebsh–Gordan coefficients using the Schwinger representation of the ${\mathfrak {su}}(2)$ Lie algebra in terms of a pair of harmonic oscillators):

$$\begin{aligned} \forall A,B= & {} \pm , \quad D^{\frac{1}{2}}_{\frac{A}{2},\frac{B}{2}}(h)\,D^{j}_{m,n}(h) \nonumber \\= & {} \frac{1}{2j+1}\Bigg [AB\sqrt{(j+Am+1)(j+Bn+1)}\,D^{j+\frac{1}{2}}_{m+\frac{A}{2},n+\frac{B}{2}}(h) \nonumber \\&+\, \sqrt{(j-Am)(j-Bn)}\,D^{j-\frac{1}{2}}_{m+\frac{A}{2},n+\frac{B}{2}}(h) \Bigg ], \end{aligned}$$

(161)

with the obvious action on the trivial spin $j=0$ mode. The action of the spin-$\frac{1}{2}$ character is obtained by adding the two operators for $A=B=\pm$:

$$\begin{aligned} \chi _{\frac{1}{2}}(h) = D^{\frac{1}{2}}_{+\frac{1}{2},+\frac{1}{2}}(h)+D^{\frac{1}{2}}_{-\frac{1}{2},-\frac{1}{2}}(h). \end{aligned}$$

We can then translate the functional equation $\chi _{\frac{1}{2}}\,\varphi =2\,\varphi$ into a recursion relation on the Fourier coefficients $\varphi ^{j}_{nm}$:

$$\begin{aligned} 2\varphi ^{0}= & {} \varphi ^{\frac{1}{2}}_{-\frac{1}{2},-\frac{1}{2}} + \varphi ^{\frac{1}{2}}_{+\frac{1}{2},+\frac{1}{2}}, \nonumber \\ 2(2j+1)\varphi ^{j}_{n,m}= & {} \varphi ^{j-\frac{1}{2}}_{n-\frac{1}{2},m-\frac{1}{2}}\sqrt{(j+m)(j+n)} + \varphi ^{j+\frac{1}{2}}_{n-\frac{1}{2},m-\frac{1}{2}}\sqrt{(j-m+1)(j-n+1)} \nonumber \\&+\, \varphi ^{j-\frac{1}{2}}_{n+\frac{1}{2},m+\frac{1}{2}}\sqrt{(j-m)(j-n)} \nonumber \\&+\, \varphi ^{j+\frac{1}{2}}_{n+\frac{1}{2},m+\frac{1}{2}}\sqrt{(j+m+1)(j+n+1)}. \end{aligned}$$

(162)

We easily check that $\varphi ^{j}_{nm}=\delta _{nm}=D^{j}_{nm}({\mathbb {I}})$ is a solution to these recursion relations, which corresponds to the $\delta$-distribution on the group $\varphi (h)=\delta (h)$. But straightforward computations also tell us that $\varphi ^{j}_{nm}=D^{j}_{nm}(J_{a})$ for $a=z,\pm$ are also solutions, or explicitly:

$$\begin{aligned} \varphi ^{j}_{nm}= \left| \begin{array}{l} D^{j}_{nm}(J_{z})=m\delta _{n,m}\\ D^{j}_{nm}(J_{+})=\sqrt{(j+m+1)(j-m)}\delta _{n,m+1}\\ D^{j}_{nm}(J_{-})=\sqrt{(j-m+1)(j+m)}\delta _{n,m-1} \end{array}\right. \end{aligned}$$

(163)

In general, we can show that for every vector $\vec {x}\in {\mathbb {R}}^{3}$, the coefficients $\varphi ^{j}_{nm}=D^{j}_{nm}(\vec {x}\cdot \vec {J})$ solve the recursion equation. Indeed, one differentiates $\chi _{\frac{1}{2}}(h)D^{j}_{nm}(h)$ and evaluates it at the identity,

$$\begin{aligned} \left. -i\partial _{x}\chi _{\frac{1}{2}}(h)D^{j}_{nm}(h)\right| _{{\mathbb {I}}} = \chi _{\frac{1}{2}}({\mathbb {I}})D^{j}_{nm}(\vec {x}\cdot \vec {J})+\chi _{\frac{1}{2}}(\vec {x}\cdot \vec {J})D^{j}_{nm}({\mathbb {I}}) =2D^{j}_{nm}(\vec {x}\cdot \vec {J}). \end{aligned}$$

(164)

Then one applies the recoupling relations given above to decompose $\chi _{\frac{1}{2}}D^{j}_{nm}$ onto the Wigner matrices for spins $j\pm \frac{1}{2}$ in order to prove that $\varphi ^{j}_{nm}=D^{j}_{nm}(\vec {x}\cdot \vec {J})$ do satisfy the required recursion relation. These coefficients actually correspond to the first derivative of the $\delta$-distribution, $\varphi (h)=\partial _{x}\delta (h)$.

In order to classify all the solutions to these recursion relations, we need to determine the initial data required to implement the recursion. The problem is that the size of the matrices $\varphi ^{j}$ increases with the mode j. If we start with fixing $\varphi ^{0}$, then this determines only one component out of four of the next matrix $\varphi ^{\frac{1}{2}}$, thus leaving three new initial conditions to be freely chosen. The $4=(1+3)$ solutions corresponding to those required initial conditions are $\delta$ and the $\partial _{a}\delta$’s.

Moving up to the spin $j=1$ components, we have four recursion relations determining the matrix elements $\varphi ^{1}$ in terms of the $\varphi ^{\frac{1}{2}}$ matrix (indeed one relation per matrix elements of $\varphi ^{\frac{1}{2}}$). This leaves us with $5=(9-4)$ undetermined matrix elements, which we need to specify as initial conditions. Choosing vanishing $\varphi ^{0}$ and $\varphi ^{\frac{1}{2}}$ initial conditions, these five new initial conditions correspond to the five linearly-independent second-order-derivative solutions of the holonomy constraint: $\partial _{a}\partial _{b}$ and $\partial _{a}\partial _{a}-\partial _{b}\partial _{b}$ for $a\ne b$. So in total, we need to specify 9 initial conditions up to the spin 1 modes. And this will inexorably grow as we explore higher and higher spins, with $4j+1=[(2j+1)^{2}-(2j)^{2}]$ new initial conditions for the matrix $\varphi ^{j}$, thus reproducing the infinite dimensional space of solutions of the holonomy constraint, with the tower of higher and higher derivatives.

1.3 Laplacian constraint, double recursion and flatness equations

We introduce another constraint supplementing the holonomy constraint in order to truly impose flatness and get the $\delta$-distribution as unique solution: we impose the Laplacian constraint $(\widetilde{\Delta }-\Delta )=0$, where $\Delta =\partial ^{L}_{a}\partial ^{L}_{a}=\partial ^{R}_{a}\partial ^{R}_{a}$ is the usual Laplacian operator and $\widetilde{\Delta }=\partial ^{L}_{a}\partial ^{R}_{a}$ mixes the right and left derivations. These two operators do not change the spin j and act rather simply on the Wigner matrices:

$$\begin{aligned} \Delta D^{j}_{mn}(h)=-D^{j}_{mn}(hJ_{a}J_{a})=-j(j+1)D^{j}_{mn}(h) , \quad \Delta D^{j}_{mn}(h)=-D^{j}_{mn}(J_{a}hJ_{a}). \end{aligned}$$

(165)

Using the explicit action of the three ${\mathfrak {su}}(2)$ generators, and applying the Cauchy–Schwarz inequality to bound the sums, we can check that the operator $(\widetilde{\Delta }-\Delta )$ is positive. We can also translate the Laplacian constraint $\Delta \varphi =\widetilde{\Delta }\varphi$ into equations on the Fourier coefficient matrices $\varphi ^{j}$:

$$\begin{aligned} j(j+1)\varphi ^{j}_{nm}= & {} nm\,\varphi ^{j}_{nm} + \frac{1}{2}\varphi ^{j}_{n-1,m-1}\sqrt{(j+n)(j-n+1)(j+m)(j-m+1)}\nonumber \\&+\, \frac{1}{2}\varphi ^{j}_{n+1,m+1}\sqrt{(j-n)(j+n+1)(j-m)(j+m+1)}. \end{aligned}$$

(166)

This is a recursion at fixed spin j on the matrix elements of each $\varphi ^{j}$ independently. It works at fixed $(n-m)$, that is along the diagonals of the matrix, determining the matrix elements from, say, the highest weight components:

$$\begin{aligned} \begin{array}{ll} \varphi ^{j,j}&{}\rightarrow \,\varphi ^{j-1,j-1}\rightarrow \varphi ^{j-2,j-2}\rightarrow \dots \\ \varphi ^{j,j-1}&{}\rightarrow \, \varphi ^{j-1,j-2}\rightarrow \varphi ^{j-2,j-3}\rightarrow \dots \\ \varphi ^{j,j-2}&{}\rightarrow \,\varphi ^{j-1,j-3}\rightarrow \varphi ^{j-2,j-4}\rightarrow \dots \end{array} \end{aligned}$$

Putting this constraint with the holonomy constraint, we get a double recursion structure. The holonomy constraint realizes a recursion on the spin j, determining the matrix $\varphi ^{j}$ from the lower spin matrices, while the Laplacian constraint implements a recursion on the magnetic moment m within each matrix $\varphi ^{j}$:

$$\begin{aligned} \varphi ^{0} \rightarrow \left( \begin{array}{cc}\searrow &{} \searrow \\ \searrow &{} \searrow \end{array} \right) \rightarrow \left( \begin{array}{ccc}\searrow &{} \searrow &{} \searrow \\ \searrow &{} \searrow &{} \searrow \\ \searrow &{} \searrow &{} \searrow \end{array} \right) \rightarrow \left( \begin{array}{cccc}\searrow &{} \searrow &{} \searrow &{} \searrow \\ \searrow &{} \searrow &{} \searrow &{} \searrow \\ \searrow &{} \searrow &{} \searrow &{} \searrow \\ \searrow &{} \searrow &{} \searrow &{} \searrow \end{array} \right) \rightarrow \dots \end{aligned}$$

This allows to solve the problem of the infinite initial conditions needed for the holonomy constraint. We easily check that $\varphi _{nm}=\delta _{nm}$ is a solution:

$$\begin{aligned} j(j+1)=m^{2}+\frac{1}{2}(j+m)(j-m+1)+\frac{1}{2}(j-m)(j+m+1). \end{aligned}$$

But then, we completely solve the constraint and show that it implies that the function is invariant under conjugation:

Proposition 6.3

Let us consider the Laplacian constraint $(\Delta -\widetilde{\Delta })\,\varphi =0$ translated to the Fourier decomposition $\varphi (h)=\sum _{j,m,n}(2j+1){\mathrm {Tr}}\,\varphi ^{j}D^{j}(h)$. Then the each of the Fourier coefficient matrix $\varphi ^{j}$ at fixed spin j is proportional to the identity. This means that $\varphi$ is invariant under conjugation.

Proof

Let us fix $j\ge \frac{1}{2}$. The spin-0 component $\varphi ^{0}$ is unconstrained and left free. The recursion relation (166) allows to start with an element $\varphi _{j,j-M}$ with $0\le M \le 2j$ and to determine all of the following components along the corresponding diagonal, $\varphi _{j-N,j-M-N}$ for $0\le N\le (2j-M)$. One actually gets a relation in terms of combinatorial factors:

$$\begin{aligned} \varphi _{j-N,j-M-N}= \varphi _{j,j-M}\, \sqrt{\frac{(M+N)!}{M!N!}\,\frac{(2j)!(2j-M-N)!}{(2j-M)!(2j-N)!}}. \end{aligned}$$

(167)

In particular, one obtains for the other end of the diagonal:

$$\begin{aligned} \varphi _{-j+M,-j} = \varphi _{j,j-M}\,\frac{(2j)!}{M!(2j-M)!}. \end{aligned}$$

(168)

The trick is that the recursion relation (166) is symmetric under the exchange $(n,m)\leftrightarrow (-m,-n)$: we start now from the other end of the same diagonal and work our way back to the initial top element. Therefore the previous equality holds but in the opposite way:

$$\begin{aligned} \varphi _{j,j-M}= & {} \varphi _{-j+M,-j}\,\frac{(2j)!}{M!(2j-M)!} = \varphi _{j,j-M}\left( \frac{(2j)!}{M!(2j-M)!}\right) ^{2}, \nonumber \\&\text {thus either} \quad \frac{(2j)!}{M!(2j-M)!}=1 \quad \text {or}\quad \varphi _{j,j-M}=0. \end{aligned}$$

(169)

In the special case $M=0$, along the principal diagonal, the recursion relation simplifies and reads $\varphi ^{j,j}=\varphi ^{j-1,j-1}=\varphi ^{j-2,j-2}=\dots$. For all the other cases $M\ne 0$, the matrix elements must vanish. This concludes the proof that the matrix $\varphi ^{j}$ must be proportional to the identity. $\square$