(u ∈ U(2)$_{a.s.}$, one or two $u_i$ are possibly zero). For each perturbation direction, we have

$$[\hat{\lambda}_0] = \begin{pmatrix} \langle\langle \Lambda_x | u^\dagger \sigma_i u \otimes (\delta X^i - p_{x,u}^i \|\delta \vec{x}\|) | \Lambda_x \rangle\rangle & \dots \\ \vdots & \ddots \end{pmatrix} \quad (A.10)$$

in the basis $(u|\Lambda\rangle, uu_1|\Lambda\rangle, uu_2|\Lambda\rangle, uu_3|\Lambda\rangle)$. The perturbation theory needs to diagonalise $[\hat{\lambda}_0]$, but in this case the eigenbasis $(\tilde{u}_0|\Lambda\rangle, \tilde{u}_1|\Lambda\rangle, \tilde{u}_2|\Lambda\rangle, \tilde{u}_3|\Lambda\rangle)$ is not consistent with the direction $\vec{p}_{x,u}$. We need to find $u_*$ such that $[\hat{\lambda}_0]$ is diagonal (or at least is block-diagonal of shape $(1 \times 1) \oplus (3 \times 3)$), fixing a single consistent perturbation direction $\vec{p}_{x,u_*}$. We have then $\delta\vec{x} = \frac{\langle\langle \Lambda(x) | u_*^\dagger \sigma_i u_* \otimes \delta X^i | \Lambda(x) \rangle\rangle}{\vec{p}_{x,u_*} \cdot \vec{n}_{x,u_*}} \vec{p}_{x,u_*}$ with $\vec{n}_{x,u_*} = \langle\langle \Lambda(x) | u_*^\dagger \vec{\sigma} u_* | \Lambda(x) \rangle\rangle$.

## Appendix B. Paralinkable fuzzy spaces and entangled quantum paths

### Appendix B.1. Generalised displacement operators

For the sake of the simplicity, in the main text we have treated only the case of totally linkable fuzzy spaces, i.e. fuzzy spaces for which two quasicoherent states can be linked by a transformation $u_{yx} \otimes \mathbf{D}(y, x) \in SU(2)_{a.s.} \otimes e^{i\vec{x}}$. This assumption can be considered as too restrictive, but it is pertinent since it implies that the magnitude of the quantum entanglement be constant on $M_\Lambda$ (and so corresponds to fuzzy space homogeneous from the point of view of the quantum information properties). More precisely, if we consider the eigen density matrices: $\rho_\Lambda(y) = u_{yx}\rho_\Lambda(x)u_{yx}^\dagger$, the von Neumann entropy $S_\Lambda = -\text{tr}(\rho_\Lambda \ln \rho_\Lambda)$ is constant (this one can be considered as a measure of the entanglement since $S_\Lambda = 0$ for a pure state $\iff$ a separable quasicoherent state, and $S_\Lambda = \ln 2 = \sup S$ for the microcanonical state $\frac{1}{2}\text{id} \iff$ a maximally entangled quasicoherent state).

In this appendix we want to generalise the discussion by choosing weaker assumptions. For a fuzzy space $\mathfrak{M}$ supposed weakly non-degenerate, we said that two points $x, y \in M_\Lambda$ are paralinkable if it exists an orthonormal basis of $\mathbb{C}^2$, $B_x = (|O_x\rangle, I_x\rangle)$, two operators $\mathbf{D}^\alpha(y, x) \in e^{i\text{Env}(x)}$ (with $\alpha \in \{0, 1\}$) and $u_{yx} \in U(2)$ such that

$$|\Lambda(y)\rangle\rangle = u_{yx} P_\alpha(x) \otimes \mathbf{D}^\alpha(y, x) |\Lambda(x)\rangle\rangle \quad (B.1)$$

with $P_0(x) = |O_x\rangle\langle O_x|$ and $P_1(x) = |I_x\rangle\langle I_x|$. We can write

$$|\Lambda(x)\rangle\rangle = |O_x\rangle\otimes|\Lambda^0(x)\rangle\rangle + |I_x\rangle\otimes|\Lambda^1(x)\rangle\rangle \quad (B.2)$$

$$|\Lambda(y)\rangle\rangle = |O_y\rangle\otimes|\Lambda^0(y)\rangle\rangle + |I_y\rangle\otimes|\Lambda^1(y)\rangle\rangle \quad (B.3)$$

with $u_{yx}$ the matrix of basis change from $B_x$ to $B_y$. $p_\alpha(x) = \langle\Lambda^\alpha(x)|\Lambda^\alpha(x)\rangle$ is the probability to the spin state be $P_\alpha(x)$ if the quasicoherent state is $|\Lambda(x)\rangle\rangle$ ($p_1+p_0=1$). By construction $|\Lambda^\alpha(y)\rangle\rangle = \mathbf{D}^\alpha(y, x)|\Lambda^\alpha(x)\rangle\rangle$ (no summation onto the indices $\alpha$ which are at the same level). The eigen density matrix in $B_x$ is $\rho_\Lambda(x) = \begin{pmatrix} p_0(x) & c(x) \\ c(x) & p_1(x) \end{pmatrix}$ with the coherence defined by $c(x) = \langle\Lambda^1(x)|\Lambda^0(x)\rangle$. Two properties are relaxed with respect to the definition of linkable points, firstly the generalised displacement operator $P_\alpha(x) \otimes \mathbf{D}^\alpha(y, x)$ is not separable, and secondly $\mathbf{D}^\alpha \in e^{i\text{Env}(x)} \supsetneq e^{i\vec{x}}$. Since elements
of $\mathrm{Env}(\mathcal{X})$ are not self-adjoint, $\mathbf{D}^\alpha$ are not necessary unitary, but the normalisation condition implies that

$$ \sum_{\alpha} \langle \Lambda^{\alpha}(x) | \mathbf{D}^{\alpha}(y, x)^{\dagger} \mathbf{D}^{\alpha}(y, x) | \Lambda^{\alpha}(x) \rangle = 1 \qquad (B.4) $$

but $p_\alpha(y) = \langle \Lambda^\alpha(x) | \mathbf{D}^\alpha(y, x)^\dagger \mathbf{D}^\alpha(y, x) | \Lambda^\alpha(x) \rangle \neq \langle \Lambda^\alpha(x) | \Lambda^\alpha(x) \rangle = p_\alpha(x)$. The relaxation of the two assumptions permits to have a change of the occupation probabilities ($p_\alpha$) (and a change of coherence $c$) between $x$ and $y$, and so a change of the entanglement magnitude: $S_\Lambda(y) \neq S_\Lambda(x)$. But the entanglement class cannot be changed, we cannot link a separable state to an entangled state (as for example with $|\Lambda^1(x)\rangle = 0$ and $|\Lambda^1(y)\rangle \neq 0$). So a fuzzy space $\mathfrak{M}$ totally paralinkable keeps a certain degree of homogeneity since all its quasicoherent states are in the same entanglement class (but with various entropy values).

Let $\varrho_\Lambda^\alpha = |\Lambda^\alpha\rangle\langle\Lambda^\alpha|$ and $P_\Lambda^\alpha = \frac{\varrho_\Lambda^\alpha}{\mathrm{tr}\,\varrho_\Lambda^\alpha}$ (i.e. $\varrho_\Lambda^\alpha = p^\alpha P_\Lambda^\alpha$, and $\mathrm{tr}(P^1P^0) = |c|^2 \neq 0$). We have still $P_\Lambda = |\Lambda\rangle\langle\Lambda|$. We can introduce the following pure states of $\mathrm{Env}(\mathcal{X})$: $\omega_x = \mathrm{tr}(P_\Lambda(x)\bullet)$ and $\omega_x^\alpha = \mathrm{tr}(P_\Lambda^\alpha(x)\bullet)$, and after some algebra using the trace cyclicality we can show that $\omega_x$ is a convex combination of $(\omega_x^\alpha)$:

$$ \omega_x = p_\alpha(x)\omega_x^\alpha \qquad (B.5) $$

and by construction, we have

$$ \omega_y^\alpha = \frac{p_\alpha(y)}{p_\alpha(x)} \omega_x^\alpha \circ \mathrm{Ad}_{\mathbf{D}^\alpha(y,x)} \qquad (B.6) $$

(without summation of the indices $\alpha$).

The generalised displacement operator $P_\alpha \otimes \mathbf{D}^\alpha$ can be seen as a quantum entanglement between quantum paths on $\mathfrak{M}$ and local orientation (spin) of $\mathfrak{M}$. From $\omega_x$ to $\omega_y$, the quantum path $\mathbf{D}^0(y, x)$ is followed if the spin is initially in the pure state $P^0(x)$ whereas the quantum path $\mathbf{D}^1(y, x)$ is followed if the spin is initially in the pure state $P^1(x)$. So with respect to the discussion in main sections of this paper, in addition to an entanglement between quantum location and quantum spin (“local” entanglement of the state $|\Lambda(x)\rangle$) we can have also entanglement between quantum paths and quantum spin (“nonlocal” entanglement of displacement operator $P_\alpha(x) \otimes \mathbf{D}^\alpha(y, x)$, nonlocal since it is defined onto two points $x$ and $y$ in the $\mathbb{R}^3$ the classical space of the observer). From $\omega_x$ to $\omega_y$ the system follows “in parallel” the two quantum paths $\mathbf{D}^\alpha(y, x)$ with probabilities $p_\alpha(x)$ (this is the reason for which we said that the two points are “paralinkable”).

We can use the quantum metric $\gamma$ with generalised displacement operators but only with its representation on $(\mathbf{d}X^i)$ and not with the one on $(\theta_i)$ (since the generators of $\mathrm{Ad}_{\mathbf{D}^\alpha}$ are not in $L_\mathcal{X}$ but in $L_{\mathrm{Env}(\mathcal{X})}$ – we recall that $(\theta_i)$ is the dual basis of $(L_{X^i})$ –). With $\mathbf{D}^\alpha(y, x) = e^{i\delta\Pi^\alpha}$ ($\delta\Pi^\alpha \in \mathrm{Env}(\mathcal{X})$), we have $\mathbf{d}X^i(L_{\delta\Pi^\alpha}) = -i[\delta\Pi^\alpha, X^i]$ by extension of noncommutative differential calculus from $\mathrm{Der}(\mathcal{X})$ to $\mathrm{Der}(\mathrm{Env}(\mathcal{X}))$. So the distance between $\omega_x$ and $\omega_y$ is

$$ \mathrm{dist}_{\gamma}(\omega_x, \omega_y) = \omega_x \left( \sqrt{ \begin{pmatrix} \gamma(L_{\delta\Pi^0}, L_{\delta\Pi^0}) & \gamma(L_{\delta\Pi^0}, L_{\delta\Pi^1}) \\ \gamma(L_{\delta\Pi^1}, L_{\delta\Pi^0}) & \gamma(L_{\delta\Pi^1}, L_{\delta\Pi^1}) \end{pmatrix} } \right) \qquad (B.7) $$

where the matrices are written in the basis $\mathcal{B}_x$. The distance between $\omega_x$ and $\omega_y$ is set to be the mean value (in the state $\omega_x$) of the average length of the two quantum
paths $\mathbf{D}^{\alpha}(y, x)$ ("average length" in the meaning of the averaging onto the quantum fluctuations associated with the quantum uncertainties $\Delta X^i$).

We can also define a distance without taking into account of the correlations between the two quantum paths:

$$ \operatorname{dist}_{\gamma}^{\text{off}}(\omega_x, \omega_y) = \omega_x \left( \left( \begin{array}{cc} \sqrt{\gamma(L_{\delta\Pi^0}, L_{\delta\Pi^0})} & 0 \\ 0 & \sqrt{\gamma(L_{\delta\Pi^1}, L_{\delta\Pi^1})} \end{array} \right) \right) \quad (B.8) $$

$$ = p_{\alpha}(x)\omega_{x}^{\alpha}(\sqrt{\gamma(L_{\delta\Pi^{\alpha}}, L_{\delta\Pi^{\alpha}})}) \quad (B.9) $$

We can also introduce spinors of linking vector observables:

$$ \delta \vec{\ell}_{yx} = \begin{pmatrix} \delta \vec{\ell}_{yx}^0 & \delta \vec{\ell}_{yx}^{01} \\ \delta \vec{\mathcal{C}}_{yx}^{10} & \delta \vec{\ell}_{yx}^1 \end{pmatrix}, \quad \delta \vec{\ell}_{yx}^{\text{off}} = \begin{pmatrix} \delta \vec{\ell}_{yx}^0 & \vec{0} \\ \vec{0} & \delta \vec{\ell}_{yx}^1 \end{pmatrix} \quad (B.10) $$

where the linking vector observables are $\delta \vec{\ell}_{yx}^{\alpha} = \mathbf{D}^{\alpha}(y, x)^{\dagger}[\vec{X}, \mathbf{D}^{\alpha}(y, x)] = -i[\delta\Pi^{\alpha}, \vec{X}] + \mathcal{O}(\|\delta\vec{x}\|^2)$ and the correlators are $\delta \vec{\mathcal{C}}_{yx}^{01} = \mathbf{D}^{0}(y, x)^{\dagger}[\vec{X}, \mathbf{D}^{1}(y, x)] = -i[\delta\Pi^{1}, \vec{X}] + \mathcal{O}(\|\delta\vec{x}\|^2)$. We can see that

$$ \delta \vec{\ell}_{yx}^{\text{off}} \cdot \delta \vec{\ell}_{yx} = \begin{pmatrix} \gamma(L_{\delta\Pi^0}, L_{\delta\Pi^0}) & \gamma(L_{\delta\Pi^0}, L_{\delta\Pi^1}) \\ \gamma(L_{\delta\Pi^1}, L_{\delta\Pi^0}) & \gamma(L_{\delta\Pi^1}, L_{\delta\Pi^1}) \end{pmatrix} + \mathcal{O}(\|\delta\vec{x}\|^2) \quad (B.11) $$

and then $\operatorname{dist}_{\gamma}(\omega_x, \omega_y) = \omega_x(\sqrt{\delta \vec{\ell}_{yx}^{\text{off}} \cdot \delta \vec{\ell}_{yx}})$ and $\operatorname{dist}_{\gamma}^{\text{off}}(\omega_x, \omega_y) = \omega_x(|\delta \vec{\ell}_{yx}^{\text{off}}|) = p_{\alpha}(x)\omega_x^{\alpha}(|\delta \vec{\ell}_{yx}^{\alpha}|)$.

### Example 2: Fuzzy hyperboloid

We can rewrite the quasicoherent state of the fuzzy hyperboloid as:

$$ |\Lambda(\alpha)\rangle\rangle = \left(1 - \sqrt{2}\sigma_1 \otimes (a - \alpha)\hat{N}_{\alpha}^{-1}X^3\right) \mathcal{D}(\alpha) \begin{pmatrix} |0\rangle \\ 0 \end{pmatrix} + \mathcal{O}(\epsilon^2) \quad (B.12) $$

where $\hat{N}_{\alpha} = (a^{\dagger} - \bar{\alpha})(a - \alpha) = \sum_{n=0}^{+\infty} n |n\rangle_{\alpha\alpha} \langle n|$, $\hat{N}_{\alpha}^{-1} = \sum_{n=1}^{+\infty} \frac{1}{n} |n\rangle_{\alpha\alpha} \langle n|$ being its pseudo-inverse, and where $\mathcal{D}(\alpha)$ is the displacement operator of the CCR algebra coherent states [11]. It follows that

$$ |\Lambda(\beta)\rangle\rangle = \left(1 - \sqrt{2}\sigma_1 \otimes (a - \beta)\hat{N}_{\beta}^{-1}X^3\right) \mathcal{D}(\beta) \times \mathcal{D}(\alpha)^{-1} \left(1 + \sqrt{2}\sigma_1 \otimes (a - \alpha)\hat{N}_{\alpha}^{-1}X^3\right) |\Lambda(\alpha)\rangle\rangle + \mathcal{O}(\epsilon) \quad (B.13) $$

and then after some algebra

$$ |\Lambda(\beta)\rangle\rangle = e^{i\operatorname{Im}(\alpha\beta)} (1 + \sqrt{2}\sigma_1 \otimes (a - \beta)\hat{N}_{\beta}^{-1} (\mathcal{D}(\beta - \alpha)X^3\mathcal{D}(\beta - \alpha)^{-1} - X^3)) \times \mathcal{D}(\beta - \alpha)|\Lambda(\alpha)\rangle\rangle + \mathcal{O}(\epsilon^2) \quad (B.14) $$

Let $W_{\beta\alpha} \equiv \sqrt{2}(a - \beta)\hat{N}_{\beta}^{-1}(\mathcal{D}(\beta - \alpha)X^3\mathcal{D}(\beta - \alpha)^{-1} - X^3)$, in the basis $|0\rangle, |1\rangle$ we have then

$$ |\Lambda(\beta)\rangle\rangle = e^{i\operatorname{Im}(\alpha\beta)} \begin{pmatrix} \mathcal{D}(\beta - \alpha) & W_{\beta\alpha}\mathcal{D}(\beta - \alpha) \\ W_{\beta\alpha}\mathcal{D}(\beta - \alpha) & \mathcal{D}(\beta - \alpha) \end{pmatrix} |\Lambda(\alpha)\rangle\rangle + \mathcal{O}(\epsilon^2) \quad (B.15) $$

in the new basis defined by $|\pm\rangle = \frac{|0\rangle \pm |1\rangle}{\sqrt{2}}$ we have

$$ |\Lambda(\beta)\rangle\rangle = e^{i\operatorname{Im}(\alpha\beta)} \begin{pmatrix} (1 + W_{\beta\alpha})\mathcal{D}(\beta - \alpha) & 0 \\ 0 & (1 - W_{\beta\alpha})\mathcal{D}(\beta - \alpha) \end{pmatrix} |\Lambda(\alpha)\rangle\rangle + \mathcal{O}(\epsilon^2) \quad (B.16) $$

$$ = e^{i\operatorname{Im}(\alpha\beta)} \begin{pmatrix} e^{W_{\beta\alpha}}\mathcal{D}(\beta - \alpha) & 0 \\ 0 & e^{-W_{\beta\alpha}}\mathcal{D}(\beta - \alpha) \end{pmatrix} |\Lambda(\alpha)\rangle\rangle + \mathcal{O}(\epsilon^2) \quad (B.17) $$

$$ = P_{\alpha} \otimes \mathbf{D}^{\alpha}(\beta, \alpha) |\Lambda(\alpha)\rangle\rangle + \mathcal{O}(\epsilon^2) \quad (B.18) $$
with $P_{\pm} = |\pm\rangle\langle\pm|$, and $D^{\pm}(\beta, \alpha) = e^{i\operatorname{Im}(\alpha\bar{\beta})}e^{\pm W_{\beta\alpha}}\mathcal{D}(\beta - \alpha)$. The fuzzy hyperboloid is then paralinkable (at the first order of the perturbation theory) except at $\alpha = 0$. $\delta\Pi_{\alpha}^{\pm} = -i(\delta\alpha a^{+} - \delta\bar{\alpha}a) \mp i\delta W_{\alpha} + \mathcal{O}(\max(\epsilon, |\delta\alpha|)^2)$, with $\delta W_{\alpha} = \sqrt{2}(a - \alpha)\tilde{N}_{\alpha}^{-1}[\delta\alpha a^{+} - \delta\bar{\alpha}a, X^3]$. The fuzzy hyperboloid model corresponds to a two-sheet hyperboloid, for which we have considered only the upper sheet. The second sheet is obtained by $\epsilon \to -\epsilon$ ($X^3 \to -X^3$). The quasicoherent states of the two sheets are related by a simple internal global gauge change:

$$|\Lambda_{-\epsilon}(\alpha)\rangle\rangle = \sigma_3 |\Lambda_{+\epsilon}(\alpha)\rangle\rangle \qquad (B.19)$$

($\sigma_3 = |0\rangle\langle 0| - |1\rangle\langle 1| = |-\rangle\langle +| + |+\rangle\langle -| = \sigma_1^B$). $|\Lambda_{\pm\epsilon}(\alpha)\rangle\rangle$ are then linkable (with a trivial displacement operator, their distance by $\gamma$ is then zero). By definition the two sheets $M_{\Lambda, \pm\epsilon}$ are each connected but they are not connected each other. In contrast, the two sheets are each not linkable (they are only paralinkable except at $\alpha = 0$) but they are linked each other (more precisely, each point is linkable with its symmetric point by the reflection $x^3 \to -x^3$).

In the basis $\mathcal{B} = (|\pm\rangle)$ the quasicoherent state is

$$|\Lambda(\alpha)\rangle\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} |0\rangle_{\alpha} \\ |0\rangle_{\alpha} \end{pmatrix} - \sum_{n=1}^{+\infty} \frac{\alpha \langle n|X^3|0\rangle_{\alpha}}{\sqrt{2n}} \begin{pmatrix} |n-1\rangle_{\alpha} \\ |-n-1\rangle_{\alpha} \end{pmatrix} + \mathcal{O}(\epsilon^2) \quad (B.20)$$

It follows that the probabilities to follow the quantum paths $D^{\pm}$ are

$$p_{\pm}(\alpha) = \frac{1}{2} \mp \operatorname{Re} (\alpha \langle 1 | X^3 | 0 \rangle_{\alpha}) + \mathcal{O}(\epsilon^2) \qquad (B.21)$$

$$= \frac{1}{2} \mp \epsilon e^{-|\alpha|^2} \sum_{n=0}^{+\infty} \frac{\sqrt{r^2 + 3/2 + n} - \sqrt{r^2 + 1/2 + n}}{n!} |\alpha|^{2n} \operatorname{Re}(\alpha) \\ + \mathcal{O}(\epsilon^2) \qquad (B.22)$$

and the coherence is

$$c(\alpha) = \frac{1}{2} - \operatorname{Im} (\alpha \langle 1 | X^3 | 0 \rangle_{\alpha}) + \mathcal{O}(\epsilon^2) \qquad (B.23)$$

$$= \frac{1}{2} - \epsilon e^{-|\alpha|^2} \sum_{n=0}^{+\infty} \frac{\sqrt{r^2 + 3/2 + n} - \sqrt{r^2 + 1/2 + n}}{n!} |\alpha|^{2n} \operatorname{Im}(\alpha) \\ + \mathcal{O}(\epsilon^2) \qquad (B.24)$$

The linear entropy of the quasicoherent state is then

$$S_{\Lambda}^{\mathrm{lin}}(\alpha) = 1 - \operatorname{tr}(\rho_{\Lambda}^{2}) \qquad (B.25)$$

$$= 2\operatorname{Im}(\alpha \langle 1 | X^3 | 0 \rangle_{\alpha}) + \mathcal{O}(\epsilon^2) \qquad (B.26)$$

$$= 2\epsilon e^{-|\alpha|^2} \sum_{n=0}^{+\infty} \frac{\sqrt{r^2 + 3/2 + n} - \sqrt{r^2 + 1/2 + n}}{n!} |\alpha|^{2n} \operatorname{Im}(\alpha) \\ + \mathcal{O}(\epsilon^2) \qquad (B.27)$$

(the linear entropy is the first order approximation of the von Neumann entropy).

## Appendix B.2. Noncommutative gauge potentials

We can introduce a new noncommutative “nonabelian” gauge potential:

$$\mathfrak{A}_{\mathrm{D}} = i \begin{pmatrix} \mathbf{D}^{0\dagger} \mathrm{d}\mathbf{D}^{0} & \mathbf{D}^{0\dagger} \mathrm{d}\mathbf{D}^{1} \\ \mathbf{D}^{1\dagger} \mathrm{d}\mathbf{D}^{0} & \mathbf{D}^{1\dagger} \mathrm{d}\mathbf{D}^{1} \end{pmatrix} \in \Omega_{\mathrm{Der}}^{1}(\mathrm{Env}(\mathcal{X}), \mathrm{gl}(2, \mathbb{C})) \qquad (B.28)$$
for which we have $\mathfrak{A}_{\mathcal{D}(y,x)}(L_{\vec{x}}) = \delta \vec{\ell}_{yx}$. We have then

$$ \mathrm{tr} \left( \rho_{\Lambda}(x) \mathfrak{A}_{\mathcal{D}(y,x)}^{\mathrm{off}}(L_{X^i}) \right) = y^i - x^i \qquad (B.29) $$

with $\mathfrak{A}_{\mathcal{D}(y,x)}^{\mathrm{off}}$ the block diagonal matrix of blocks ($i\mathbf{D}^{\alpha\dagger}\mathbf{d}\mathbf{D}^{\alpha}$). The noncommutative “nonabelian” curvature $\mathfrak{F}_{\mathcal{D}} = d\mathfrak{A}_{\mathcal{D}} - i[\mathfrak{A}_{\mathcal{D}}, \mathfrak{A}_{\mathcal{D}}] \in \Omega_{\mathrm{Der}}^2(\mathrm{Env}(\mathfrak{X}), \mathrm{gl}(2, \mathbb{C}))$ is not zero.

$\mathfrak{A}_{\mathcal{D}}$ is a noncommutative equivalent of $\mathfrak{A}$ which has $A = \mathrm{tr}(\rho_{\Lambda}\mathfrak{A})$ as “abelianisation”. In a same way, we can define a noncommutative “abelianisation” of $\mathfrak{A}_{\mathcal{D}}$:

$$ \mathbf{A}_{\mathcal{D}} = \mathrm{tr}_{\mathbb{C}^2} (\rho_{\Lambda} \mathfrak{A}_{\mathcal{D}}) \qquad (B.30) $$

$$ = p_{\alpha} \mathbf{A}_{\mathcal{D}}^{\alpha} + \mathbf{C}_{\mathcal{D}} \in \Omega_{\mathrm{Der}}^{1}(\mathrm{Env}(\mathfrak{X})) \qquad (B.31) $$

with $\mathbf{A}_{\mathcal{D}}^{\alpha} = i\mathbf{D}^{\alpha\dagger}\mathbf{d}\mathbf{D}^{\alpha}$ and $\mathbf{C}_{\mathcal{D}} = i\mathbf{c}\mathbf{D}^{1\dagger}\mathbf{d}\mathbf{D}^{0} + i\bar{c}\mathbf{D}^{0\dagger}\mathbf{d}\mathbf{D}^{1}$.

The noncommutative “abelian” curvature is also nonzero in this case (if $p_{\alpha} \neq 0$):

$$ \mathbf{F}_{\mathcal{D}}[L_X, L_Y] = d\mathbf{A}_{\mathcal{D}}(L_X, L_Y) - i[\mathbf{A}_{\mathcal{D}}(L_X), \mathbf{A}_{\mathcal{D}}(L_Y)] \qquad (B.32) $$

$$ = i p_0 p_1 \zeta_{\alpha\beta} [\mathbf{A}_{\mathcal{D}}^{\alpha}(L_X), \mathbf{A}_{\mathcal{D}}^{\beta}(L_Y)] \\ + d\mathbf{C}_{\mathcal{D}}(L_X, L_Y) - i[\mathbf{C}_{\mathcal{D}}(L_X), \mathbf{C}_{\mathcal{D}}(L_Y)] \\ - i p_{\alpha} [\mathbf{A}_{\mathcal{D}}^{\alpha}(L_X), \mathbf{C}_{\mathcal{D}}(L_Y)] \qquad (B.33) $$

where $\zeta_{\alpha\beta} \equiv (-1)^{\alpha-\beta}$. We have also $\mathbf{A}_{\mathcal{D}}^{\mathrm{off}} = \mathrm{tr}_{\mathbb{C}^2} (\rho_{\Lambda}\mathfrak{A}_{\mathcal{D}}^{\mathrm{off}}) = p_{\alpha}\mathbf{A}_{\mathcal{D}}^{\alpha}$ and $\mathbf{F}_{\mathcal{D}}^{\mathrm{off}}(L_X, L_Y) = i p_0 p_1 \zeta_{\alpha\beta} [\mathbf{A}_{\mathcal{D}}^{\alpha}(L_X), \mathbf{A}_{\mathcal{D}}^{\beta}(L_Y)]$.

## Appendix C. The Lorentz connection

### Appendix C.1. Frame changes on $\Omega$

**Property 15** Under a local external gauge transformation $J \in SO(3)_{\mathbb{R}^3}$, $\Omega$ becomes:

$$ \hat{\Omega}_{m}^{jk} = J_{j'j}^{j} J_{k'k}^{k} \Omega_{m'}^{j'k'} (J_{m}^{m'} + \partial_{m} J_{p}^{m'} x^{p}) + \frac{1}{2} J^{jl} \partial_{m} J^{k}_{l} \qquad (C.1) $$

$$ \hat{\Omega}_{m}^{i0} = J_{i'j}^{i} \Omega_{m'}^{i'0} (J_{m}^{m'} + \partial_{m} J_{p}^{m'} x^{p}) \qquad (C.2) $$

with $J_{j'j} = [J^t]^j_{j'} = [J^{-1}]^j_{j'}$ and $\Omega^{\mu\nu} = \Omega_m^{\mu\nu} \frac{\partial x^m}{\partial s^a} ds^a$ (($s^1, s^2$) being a local curvilinear coordinate system on $M_{\Lambda}$).

**Proof:**

$$ \mathrm{tr}(\sigma^i \hat{\mathfrak{A}}) = \mathrm{tr}(\sigma^i u_J^{-1} J^* \mathfrak{A} u_J) + i \mathrm{tr}(\sigma^i u_J^{-1} du_J) \qquad (C.3) $$

$$ = \mathrm{tr}(\hat{\sigma}^i J^* \mathfrak{A}) + i \mathrm{tr}(\hat{\sigma}^i du_J u_J^{-1}) \qquad (C.4) $$

$$ = J_{i'}^i \mathrm{tr}(\sigma^i J^* \mathfrak{A}) + i J_{i'}^i \mathrm{tr}(\sigma^i du_J u_J^{-1}) \qquad (C.5) $$

with $\hat{\sigma}^i = u_J \sigma^i u_J^{-1} = J_{i'j}^i \sigma^i$.

$$ J^* \mathfrak{A} = \mathfrak{A}_{m'}|_{J^{-1}x} (J_m^{m'} dx^m + \partial_m J_p^{m'} x^p dx^p) \qquad (C.6) $$

Since $J \in SO(3)$, we know that

$$ J_{i'}^i J_{j'j}^{j} \delta_{i'j'} = \delta_{ij} \qquad (C.7) $$

$$ \varepsilon_{i'j',k'} J_{i}^{i} J_{j'}^{k} = \varepsilon_{ij}^{k} J_{k}^{j'} \qquad (C.8) $$
(J preserves the Euclidean inner product and the vector cross product). We have then

$$ \varepsilon^{jk} {}_i J_{i'}{}^i \mathrm{Re} \operatorname{tr}(\sigma^{i'} \mathfrak{A}_{m'}) = \varepsilon^{j'k'} {}_{i'} J_{j'}{}^j J_{k'}{}^k \mathrm{Re} \operatorname{tr}(\sigma^{i'} \mathfrak{A}_{m'}) \quad (C.9) $$

$$ = - J_{j'}{}^j J_{k'}{}^k \Omega_{m'}^{j'k'} \quad (C.10) $$

Moreover

$$ J^i_j \sigma_i = u_j \sigma_j u_j^{-1} \quad (C.11) $$

$$ \Rightarrow dJ^i_j \sigma_i = du_j \sigma_j u_j^{-1} - u_j \sigma_j u_j^{-1} du_j u_j^{-1} \quad (C.12) $$

$$ \Rightarrow dJ^i_j \sigma_i = [du_j u_j^{-1}, J^i_j \sigma_i] \quad (C.13) $$

Then

$$ [J^{-1}]^j_i dJ^k_j \sigma_k = [du_j u_j^{-1}, \sigma_i] \quad (C.14) $$

$$ = \frac{1}{2} \operatorname{tr}(\sigma^l du_j u_j^{-1}) [\sigma_l, \sigma_i] \quad (C.15) $$

$$ = \iota \operatorname{tr}(\sigma^l du_j u_j^{-1}) \varepsilon_{li}{}^k \sigma_k \quad (C.16) $$

So $\operatorname{tr}(\sigma^l du_j u_j^{-1}) = \frac{1}{2} \varepsilon^{li}{}_k [J^{-1}]^j_i dJ^k_j$ and then

$$ \varepsilon^{jk} {}_i \iota \operatorname{tr}(\sigma^i du_j u_j^{-1}) = - \frac{1}{2} \varepsilon^{jk} {}_i \varepsilon^{il} {}_m [J^{-1}]^p{}_l dJ^m{}_p \quad (C.17) $$

$$ = - \frac{1}{2} (\delta^{jl} \delta_m^k - \delta_m^j \delta^{kl}) [J^{-1}]^p{}_l dJ^m{}_p \quad (C.18) $$

$$ = - \frac{1}{2} [J^{-1}]^{pj} dJ^k{}_p + \frac{1}{2} [J^{-1}]^{pk} dJ^j{}_p \quad (C.19) $$

$$ = \frac{1}{2} (J^{kp} dJ^j{}_p - J^{jp} dJ^k{}_p) \quad (C.20) $$

Finally we have

$$ \hat{\Omega}_m^{jk} = - \varepsilon^{jk} {}_i \mathrm{Re} \operatorname{tr}(\sigma^i \mathfrak{A}_m) \quad (C.21) $$

$$ = J_{j'}{}^j J_{k'}{}^k \Omega_{m'}^{j'k'} J_m{}^{m'} - \frac{1}{2} (J^{kp} \partial_m J^j{}_p - J^{jp} \partial_m J^k{}_p) \quad (C.22) $$

and

$$ \hat{\Omega}_m^{i0} = \mathrm{Im} \operatorname{tr}(\sigma^i \mathfrak{A}_m) \quad (C.23) $$

$$ = J_{i'}{}^i \Omega_{m'}^{i'0} J_m{}^{m'} \quad (C.24) $$

□

## Appendix C.2. Covariant derivatives on $M_\Lambda$

**Definition 8 (Covariant derivatives)** On $\mathbb{C}^2 \otimes \mathcal{H}_{M_\Lambda}$ and on $\mathcal{L}(\mathbb{C}^2)_{M_\Lambda}$ we define the covariant derivatives associated with the gauge potential $\mathfrak{A}$ by

$$ \nabla_a |\Phi\rangle\rangle = \left( \frac{\partial}{\partial s^a} - i \mathfrak{A}_a \right) |\Phi\rangle\rangle \quad (C.25) $$

$$ \nabla_a O = \frac{\partial O}{\partial s^a} - i[\mathfrak{A}_a, O]_\dagger \quad (C.26) $$

$$ \forall |\Phi\rangle\rangle \in \mathbb{C}^2 \otimes \mathcal{H}_{M_\Lambda} \text{ and } \forall O \in \mathcal{L}(\mathbb{C}^2)_{M_\Lambda}, \text{ with } [\mathfrak{A}_a, O]_\dagger = \mathfrak{A}_a O - O \mathfrak{A}_a^\dagger. $$

These definitions are consistent with the weak adiabatic approximation:

$$ \begin{aligned} |\Psi(t)\rangle\rangle &= \mathbb{T} e^{-i \int_0^t \mathfrak{A}_a s^a dt} |\Lambda(x(t))\rangle\rangle \\ &\Rightarrow \frac{d}{dt} |\Psi(t)\rangle\rangle = \mathbb{T} e^{-i \int_0^t \mathfrak{A}_a s^a dt} \nabla_a |\Lambda(x(t))\rangle\rangle|_{x=x(t)} \dot{s}^a \end{aligned} \quad (C.27) $$
and let $\rho(t) = \text{tr}_{\mathcal{H}} |\Psi(t)\rangle\langle\Psi(t)|$ be the spin density matrix, we have

$$ \begin{aligned} \rho(t)^{\dagger} &= \left( \mathbb{T}e^{-i \int_{0}^{t} \mathfrak{A}_{a} \dot{s}^{a} dt} \right)^{\dagger} \rho_{\Lambda}(x(t)) \mathbb{T}e^{-i \int_{0}^{t} \mathfrak{A}_{a} \dot{s}^{a} dt} \\ &\Rightarrow \text{tr} (\rho(t)^{\dagger} O(x(t))) \\ &= \text{tr} \left( \rho_{\Lambda}(x(t)) \mathbb{T}e^{-i \int_{0}^{t} \mathfrak{A}_{a} \dot{s}^{a} dt} O(x(t)) \left( \mathbb{T}e^{-i \int_{0}^{t} \mathfrak{A}_{a} \dot{s}^{a} dt} \right)^{\dagger} \right) \end{aligned} \quad (C.28) $$

where $\rho_{\Lambda} = \text{tr}_{\mathcal{H}} |\Lambda\rangle\langle\Lambda|$ is the eigen density matrix, and so

$$ \frac{d}{dt} \tilde{O}(t) = \mathbb{T}e^{-i \int_{0}^{t} \mathfrak{A}_{a} \dot{s}^{a} dt} \nabla_{a} O(x)|_{x=x(t)} \dot{s}^{a} \left( \mathbb{T}e^{-i \int_{0}^{t} \mathfrak{A}_{a} \dot{s}^{a} dt} \right)^{\dagger} \quad (C.29) $$

with $\tilde{O}(t) = \mathbb{T}e^{-i \int_{0}^{t} \mathfrak{A}_{a} \dot{s}^{a} dt} O(x(t)) \left( \mathbb{T}e^{-i \int_{0}^{t} \mathfrak{A}_{a} \dot{s}^{a} dt} \right)^{\dagger}$

**Property 16** The covariant derivative onto $\mathcal{L}(\mathbb{C}^2)_{M_\Lambda}$ induces the following covariant derivative onto the 4-vectors $(v^0, \tilde{v}) \in \mathbb{R}^4_{M_\Lambda}$:

$$ \nabla_a \tilde{v} = \frac{\partial}{\partial s^a} \tilde{v} + \text{Re} \operatorname{tr}(\tilde{\sigma} \mathfrak{A}_a) \times \tilde{v} + \text{Im} \operatorname{tr}(\tilde{\sigma} \mathfrak{A}_a) v^0 + \text{Im} \operatorname{tr}(\mathfrak{A}_a) \tilde{v} \quad (C.30) $$

$$ \nabla_a v^0 = \frac{\partial}{\partial s^a} v^0 + \text{Im} \operatorname{tr}(\tilde{\sigma} \mathfrak{A}_a) \cdot \tilde{v} + \text{Im} \operatorname{tr}(\mathfrak{A}_a) v^0 \quad (C.31) $$

*Proof:* Let $O = v^0 \text{id} + v^i \sigma_i \in \mathcal{L}(\mathbb{C}^2)$ be the image of $(v^0, \tilde{v})$ by the isomorphism between $\mathbb{R}^4$ and $\mathcal{L}(\mathbb{C}^2)$ induced by the basis $(\text{id}, \sigma_1, \sigma_2, \sigma_3)$.

$$ \nabla_a O = \frac{\partial O}{\partial s^a} - i(\mathfrak{A}_a(v^0 \text{id} + v^i \sigma_i) - (v^0 \text{id} + v^i \sigma_i)\mathfrak{A}_a^\dagger) \quad (C.32) $$

$$ = \frac{\partial O}{\partial s^a} - \frac{i}{2} v^i (\text{tr}(\sigma^j \mathfrak{A}_a) \sigma_j \sigma_i - \text{tr}(\sigma^j \mathfrak{A}_a) \sigma_i \sigma_j) \quad (C.33) $$

$$ = \frac{\partial O}{\partial s^a} - \frac{i}{2} v^i \text{Re} \operatorname{tr}(\sigma^j \mathfrak{A}_a) [\sigma_j, \sigma_i] + \frac{1}{2} v^i \text{Im} \operatorname{tr}(\sigma^j \mathfrak{A}_a) \{\sigma_j, \sigma_i\} \quad (C.34) $$

$$ = \frac{\partial O}{\partial s^a} + \varepsilon_{ji}{}^k \text{Re} \operatorname{tr}(\sigma^j \mathfrak{A}_a) v^i \sigma_k + \delta_{ij} v^i \text{Im} \operatorname{tr}(\sigma^j \mathfrak{A}_a) \quad (C.35) $$

$$ = \left( \frac{\partial v^i}{\partial s^a} + \varepsilon^i_{jk} \text{Re} \operatorname{tr}(\sigma^j \mathfrak{A}_a) v^k + v^0 \text{Im} \operatorname{tr}(\sigma^i \mathfrak{A}_a) + \text{Im} \operatorname{tr}(\mathfrak{A}_a) v^i \right) \sigma_i \quad (C.36) $$

□

By restricting the covariant derivative to $\mathbb{R}^3_{M_\Lambda}$:

$$ \nabla_a \tilde{v} = \frac{\partial}{\partial s^a} \tilde{v} + \text{Re} \operatorname{tr}(\tilde{\sigma} \mathfrak{A}_a) \times \tilde{v} \quad (C.37) $$

we have

$$ \Gamma^i_{bc} = (\nabla_b e_c)^i \quad (C.38) $$

**Property 17** Let $g(t) = \mathbb{T}e^{-i \int_{0}^{t} \mathfrak{A}_{a} \dot{s}^{a} dt}$. If $t \mapsto x(s(t)) \in M_{\Lambda}$ is an autoparallel geodesic: $\ddot{s}^{a} + \Gamma^{a}_{bc} \dot{s}^{b} \dot{s}^{c} = 0$, then

$$ \frac{d}{dt} \left( g(t) \frac{d\tilde{P}_x}{dt} g(t)^{\dagger} \right) = \frac{i}{2} g(t) \left\{ (\mathfrak{A}_a - \mathfrak{A}_a^{\dagger}) \dot{s}^a, \frac{d\tilde{P}_x}{dt} \right\} g(t)^{\dagger} \quad (C.39) $$
Proof:

$$ \frac{d}{dt} \left( g \frac{d\mathcal{D}_x}{dt} g^\dagger \right) = g \left( \dot{s}^a \nabla_a \frac{d\mathcal{D}_x}{dt} \right) g^\dagger \quad (C.40) $$

$$ = g \left( \frac{\partial}{\partial s^a} \left( \frac{\partial \mathcal{D}_x}{\partial s^b} \dot{s}^b \right) \dot{s}^a - i \left[ \mathfrak{A}_a, \frac{\partial \mathcal{D}_x}{\partial s^b} \right]_\dagger \dot{s}^a \dot{s}^b \right) g^\dagger \quad (C.41) $$

$$ = g \left( \frac{\partial \mathcal{D}_x}{\partial s^b} \ddot{s}^b + \left( \frac{\partial^2 \mathcal{D}_x}{\partial s^a \partial s^b} - i \left[ \mathfrak{A}_a, \frac{\partial \mathcal{D}_x}{\partial s^b} \right]_\dagger \right) \dot{s}^a \dot{s}^b \right) g^\dagger \quad (C.42) $$

$$ = g \left( \frac{\partial \mathcal{D}_x}{\partial s^b} \ddot{s}^b + \left( \nabla_a \frac{\partial \mathcal{D}_x}{\partial s^b} \right) \dot{s}^a \dot{s}^b \right) g^\dagger \quad (C.43) $$

but $\frac{\partial \mathcal{D}_x}{\partial s^b} = -\sigma_i \frac{\partial x^i}{\partial s^b} = -\sigma_i e_b^i$, so

$$ \frac{d}{dt} \left( g \frac{d\mathcal{D}_x}{dt} g^\dagger \right) = -g \left( \sigma_i e_b^i \dot{s}^b + \nabla_a (\sigma_i e_b^i) \dot{s}^a s^b \right) g^\dagger \quad (C.44) $$

$$ = -g \left( e_b^i \ddot{s}^b + (\nabla_a e_b)^i \dot{s}^a s^b + \operatorname{Im} \operatorname{tr}(\mathfrak{A}_a) e_b^i \right) \sigma_i g^\dagger - \operatorname{Im} \operatorname{tr}(\sigma_i \mathfrak{A}_a) e_b^i g^\dagger \quad (C.45) $$

$$ \begin{aligned} & -g e_c^i (\ddot{s}^c + \Gamma^c_{ab} \dot{s}^a \dot{s}^b) g^\dagger \\ & \quad - \frac{i}{2} g \{ \mathfrak{A}_a - \mathfrak{A}_a^\dagger, \sigma_i e_b^i \} g^\dagger \dot{s}_a \dot{s}_b \end{aligned} \quad (C.46) $$

□

$\mathfrak{A}$ is almost surely self-adjoint, in the case where it is self-adjoint (implying $g^\dagger = g^{-1}$) we have then

$$ \frac{d}{dt} \left( g \frac{d\mathcal{D}_x}{dt} g^{-1} \right) = 0 \Rightarrow g(t) \frac{d\mathcal{D}_x}{dt} g^{-1}(t) = \left. \frac{d\mathcal{D}_x}{dt} \right|_{t=0} \quad (C.47) $$

and then

$$ \langle \langle \Lambda(x(t)) | \frac{d\mathcal{D}_x}{dt} | \Lambda(x(t)) \rangle \rangle = \langle \langle \Psi(t) | \frac{d\mathcal{D}_x}{dt} | \Psi(t) \rangle \rangle \quad (C.48) $$

$$ \Leftrightarrow \langle \langle \Psi(t) | \frac{d\mathcal{D}_x}{dt} | \Psi(t) \rangle \rangle = - \langle \langle \Lambda(x(t)) | \sigma_i | \Lambda(x(t)) \rangle \rangle \dot{x}^i = 0 \quad (C.49) $$

by using the adiabatic approximation $|\Psi(t)\rangle = g(t)|\Lambda(x(t))\rangle$ ($\langle\langle\Lambda|\sigma^i|\Lambda\rangle\rangle \partial_i$ is a normal vector of $M_\Lambda$ at $x$ whereas $\dot{x}^i \partial_i$ is a tangent vector of $M_\Lambda$ at $x$).

## Appendix D. The categorical fibre bundle

Quantum mechanics deal with two evolution laws. A time continuous evolution laws (by the Schrödinger or the Dirac equations) associated with the smooth changes of the probabilities with respect to the time, and a discontinuous abrupt law (by the Born projection rule) associated with the state change after a measurement (and depending on the output of this one). This double aspect is also present with Fuzzy spaces. The Schrödinger-like equation is associated with the continuous evolution induced by the smooth moves of the probe $t \mapsto x(t)$. The length of the paths $\mathcal{C}$ in $M_\Lambda$ by $\gamma$ is a geometric measure of this evolution (by the energy uncertainty with probe misalignment). But, since the quasicoherent states are not separated $\langle\langle\Lambda(x)|\Lambda(y)\rangle\rangle \neq \delta(y-x)$, a Fuzzy space in the state $|\Lambda(x)\rangle$ can be projected in the
but the differentials are different (the one of $M_\Lambda$ for the first and the one of $\mathfrak{M}$ for the second). $\eta + A_D \in \Omega_{\text{Der}}^1(\mathfrak{X} \otimes \mathcal{C}^\infty(M_\Lambda))$ can be viewed as a noncommutative gauge potential onto a noncommutative manifold defined by $\mathfrak{X} \otimes \mathcal{C}^\infty(M_\Lambda) = \mathcal{C}^\infty(M_\Lambda, \mathfrak{X})$ ($\mathfrak{X}$-valued functions on $M_\Lambda$).

$B = d\eta \in \Omega^2(M_\Lambda, \mathbb{R})_{\text{Diff}M_\Lambda}$ is called curving [43].

**Example 2: Fuzzy surface plots**

For a diffeomorphism $\varphi \in \text{Diff}M_\Lambda$, the gauge potential-transformation is

$$ \eta_{\mathcal{D}(\varphi(\alpha), \alpha)} = -i \mathcal{D}(\varphi(\alpha), \alpha)^{\dagger} d\mathcal{D}(\varphi(\alpha), \alpha) \quad (D.7) $$

$$ \begin{aligned} = -i \left( \left( \frac{\partial \varphi}{\partial \alpha} - 1 \right) a^+ - \frac{\bar{\varphi}}{\partial \alpha} a \right) d\alpha \\ -i \left( \frac{\partial \varphi}{\partial \bar{\alpha}} a^+ - \left( \frac{\partial \bar{\varphi}}{\partial \bar{\alpha}} - 1 \right) a \right) d\bar{\alpha} \end{aligned} \quad (D.8) $$

$$ \begin{aligned} \omega_\alpha(\eta_{\mathcal{D}(\varphi(\alpha), \alpha)}) = -i \left( \left( \frac{\partial \varphi}{\partial \alpha} - 1 \right) \bar{\alpha} - \frac{\bar{\varphi}}{\partial \alpha} \alpha \right) d\alpha \\ -i \left( \frac{\partial \varphi}{\partial \bar{\alpha}} \bar{\alpha} - \left( \frac{\partial \bar{\varphi}}{\partial \bar{\alpha}} - 1 \right) \alpha \right) d\bar{\alpha} \end{aligned} \quad (D.9) $$

and so the adiabatic curving is

$$ B = d\omega_\alpha(\eta_{\mathcal{D}(\varphi(\alpha), \alpha)}) = -i \left( 2 - \frac{\partial \varphi}{\partial \alpha} - \frac{\partial \bar{\varphi}}{\partial \bar{\alpha}} \right) d\alpha \wedge d\bar{\alpha} \quad (D.10) $$

For $\varphi$ corresponding to an external gauge change $\varphi(x^1, x^2) = (\cos \vartheta x^1 + \sin \vartheta x^2, -\sin \vartheta x^1 + \cos \vartheta x^2) \iff \varphi(\alpha) = e^{-i\vartheta}\alpha$, we have $B = 2i(\cos \vartheta - 1)d\alpha \wedge d\bar{\alpha} = 4(\cos \vartheta - 1)dx^1 \wedge dx^2$.

**Appendix D.2. The fibre bundle on $M_\Lambda$**

Let $\{\mathcal{U}^\alpha\}$ be a good open cover of $M_\Lambda$. The choice of the quasicoherent state $|\Lambda(x)\rangle\rangle|\mathcal{U}^\alpha$ constitutes a trivialising local section of a line bundle over $M_\Lambda$. This one is the associated line bundle of a $U(1)$-principal bundle over $M_\Lambda$, $P_\Lambda$, of local trivialisation denoted by $\phi_P^\alpha : \mathcal{U}^\alpha \times U(1) \xrightarrow{\simeq} P_\Lambda|_{\mathcal{U}^\alpha}: \forall x \in \mathcal{U}^\alpha, h \in U(1), z \in \mathbb{C}$

$$ [\phi_P^\alpha(x, hh'), h'z]_{h' \in U(1)} = hz|\Lambda(x)\rangle\rangle|\mathcal{U}^\alpha \quad (D.11) $$

$P_\Lambda$ is endowed with a connection of local data the gauge potential $A$ and the curvature $F$. $P_\Lambda$ is the usual Berry-Simon adiabatic bundle [30, 31].

We suppose that $\forall x, y \in M_\Lambda, U_x \simeq U_y (U_x \subset SU(2)_{a.s.}$ such that $U_x \cdot |\Lambda(x)\rangle\rangle = \ker \mathcal{D}_x)$. We denote by $U_0$ the model group (possibly one of the groups $U_x$ at a certain point) and by $\phi_{U_x}^\alpha : U_0 \xrightarrow{\simeq} U_x$ the group homomorphism supposed continuous with respect to $x$ on $\mathcal{U}^\alpha$. Let $\bar{U}_0 = \bigcup_{u \in SU(2)} uU_0u^{-1}$ be the normal closure of $U_0$ and $\tilde{G} = SU(2) \times \bar{U}_0$ be the semi-direct product of groups defined by :

$$ \forall u, u' \in SU(2), \forall v, v' \in U_0 \quad (u', v')(u, v) = (u'u, u^{-1}v'uv) \quad (D.12) $$

here $SU(2)$ stands for the spin rotations induced by external gauge changes in $SO(3)$. $\tilde{G}$ acts on $|\Lambda\rangle\rangle$ as gauge changes via

$$ (u, v) \cdot |\Lambda(x)\rangle\rangle|\mathcal{U}^\alpha = u\phi_{Ux}^\alpha(v)|\Lambda(x)\rangle\rangle|\mathcal{U}^\alpha \quad (D.13) $$

the semi-direct product being chosen to be compatible with its action $(u', v')(u, v) \cdot |\Lambda(x)\rangle\rangle|\mathcal{U}^\alpha = u'\phi_{Ux}^\alpha(v')u\phi_{Ux}^\alpha(v)|\Lambda(x)\rangle\rangle|\mathcal{U}^\alpha$. These elements define a $\tilde{G}$-principal bundle
over $M_\Lambda$, $Q_\Lambda$, of local trivialisation $\phi_Q^\alpha : \mathcal{U}^\alpha \times \tilde{G} \xrightarrow{\simeq} Q_{\Lambda|\mathcal{U}^\alpha}$ defined by: $\forall x \in \mathcal{U}^\alpha, (u, v) \in \tilde{G}$

$$ \phi_Q^\alpha(x, u, v) = (x, u, \phi_{Ux}^\alpha(v)) \qquad (D.14) $$

If $U_x$ is trivial, $\tilde{G} = SU(2)$ and $Q_\Lambda$ is trivial. $\mathfrak{A}^{\text{off}} = \mathfrak{A} - \text{tr} \mathfrak{A}$ is the gauge potential of the connection endowing $Q_\Lambda$ which has $\mathfrak{F}^{\text{off}} = \mathfrak{F} - \text{tr} \mathfrak{F}$ as curvature. Since $i\Omega^{\mu\nu}\frac{1}{2}\{\sigma_\mu, \sigma_\nu\} = \mathfrak{A}^{\text{off}}$ (with $\sigma_0 = \text{id}$), $\mathfrak{A}^{\text{off}}$ and $\mathfrak{F}^{\text{off}}$ are just representations of the Lorentz connection and of the Riemann curvature. $Q$ is then the frame change principal bundle of $M_\Lambda$.

Let $G = U(1) \times \tilde{G} = U(2) \times \overline{U_0}$, the different gauge changes are associated with the following central extension of groups:

$$ 1 \to U(1) \to G \to G/U(1) = \tilde{G} \to 1 $$

It is known that a $G$-fibration compatible with $P_\Lambda$ and $Q_\Lambda$ is not a principal bundle but a categorical bundle (twisted bundle [44]), i.e. a bundle for which the total space is not a manifold but a category. Let $\mathcal{G}$ be the groupoid with $\text{Obj}(\mathcal{G}) = \tilde{G}$ and $\text{Morph}(\mathcal{G}) = (G \times G)/U(1) \equiv \{[g_2h, g_1h]_{h \in U(1)}\}$ with the following source, target and identity maps

$$ s[g_2, g_1] = g_1 U(1) \quad t[g_2, g_1] = g_2 U(1) \quad \text{id}_{gU(1)} = [g, g] \qquad (D.15) $$

where $gU(1) \in G/U(1) = \tilde{G}$ denotes a coset. The arrow composition (vertical composition of arrows) is defined by

$$ [g_3, g_2] \circ [g_2h, g_1] = [g_3h, g_1] \qquad (D.16) $$

and the groupoid low (horizontal composition of arrows) is defined by

$$ [g'_2, g'_1][g_2, g_1] = [g'_2 g_2, g'_1 g_1] \qquad (D.17) $$

The categorical bundle $\mathcal{P}_\Lambda$ over $\mathcal{M}_\Lambda$ is defined by

$$ \text{Obj}(\mathcal{P}_{\Lambda|\mathcal{U}^{\alpha}}) = \bigsqcup_{x \in \mathcal{U}^{\alpha}} G_x \subset \underline{U(2)}_{\mathcal{U}^{\alpha}} \qquad (D.18) $$

$$ \text{Morph}(\mathcal{P}_{\Lambda|\mathcal{U}^{\alpha}}) = \bigsqcup_{x \in \mathcal{U}^{\alpha}} \bigsqcup_{\varphi \in \text{Morph}\mathcal{U}^{\alpha}} \text{Hom}(G_x, G_x) \times G_x \\ \subset \underline{\text{Aut}U(2) \times U(2)}_{\mathcal{U}^{\alpha}} \qquad (D.19) $$

where $U(2)$ stands for the unitary operators of $\mathbb{C}^2$, $G_x \equiv \phi_P^\alpha(U(1)) \times SU(2) \times \overline{U_x}$; the projection functor $\mathcal{P}_\Lambda \to \mathcal{M}_\Lambda$ being canonically defined by the external direct product. The source, target and identity maps, and the composition of arrows are defined by $\forall g \in G_x, \forall f \in \text{Hom}(G_x, G_x), \forall f' \in \text{Hom}(G_{\varphi(x)}, G_{\varphi(x)})$:

$$ s(f, g) = g \qquad \text{id}_g = (\text{id}_{G_x}, g) \qquad (D.20) $$

$$ t(f, g) = f(g) \qquad (f', f(g)) \circ (f, g) = (f' \circ f, g) \qquad (D.21) $$

Note that to define globally $\mathcal{P}_\Lambda$ on the whole of $\mathcal{M}_\Lambda$ we need to consider for each open set $\mathcal{U}^\alpha$ the associated category $\mathcal{U}^\alpha$ with $\text{Obj}(\mathcal{U}^\alpha) = \mathcal{U}^\alpha$ and $\text{Morph}(\mathcal{U}^\alpha) = \text{Diff}\mathcal{U}^\alpha \times \mathcal{U}^\alpha$, within the functors $\cup$ and $\cap$ such that: $\text{Obj}(\mathcal{U}^\alpha \cup \mathcal{U}^\beta) = \mathcal{U}^\alpha \cup \mathcal{U}^\beta$, $\text{Obj}(\mathcal{U}^\alpha \cap \mathcal{U}^\beta) = \mathcal{U}^\alpha \cap \mathcal{U}^\beta$, $\text{Morph}(\mathcal{U}^\alpha \cup \mathcal{U}^\beta) = \text{Diff}(\mathcal{U}^\alpha \cup \mathcal{U}^\beta) \times \mathcal{U}^\alpha \cup \mathcal{U}^\beta \supset \text{Diff}\mathcal{U}^\alpha \cup \text{Diff}\mathcal{U}^\beta \times \mathcal{U}^\alpha \cup \mathcal{U}^\beta$, and $\text{Morph}(\mathcal{U}^\alpha \cap \mathcal{U}^\beta) = \text{Diff}(\mathcal{U}^\alpha \cap \mathcal{U}^\beta) \times \mathcal{U}^\alpha \cap \mathcal{U}^\beta \subset \text{Diff}\mathcal{U}^\alpha \cap \text{Diff}\mathcal{U}^\beta \times \mathcal{U}^\alpha \cap \mathcal{U}^\beta$ (where $\varphi \in \text{Diff}\mathcal{U}^\alpha$ is naturally extended in $\text{Diff}M_\Lambda$ by the identity on $M_\Lambda \setminus \mathcal{U}^\alpha$, $\text{Diff}\mathcal{U}^\alpha$ being
defined by the diffeomorphism of the open set $\mathcal{U}^{\alpha}$ asymptotically leaving invariant the
border of $\mathcal{U}^{\alpha}$).

The local trivialisation functor of $\mathcal{P}_{\Lambda}$ is $\phi_{\mathcal{P}}^{\alpha} : \mathcal{U}^{\alpha} \times \mathcal{G} \xrightarrow{\simeq} \mathcal{P}_{\Lambda|\mathcal{U}^{\alpha}}$ with: $\forall x \in \mathcal{U}^{\alpha}$, $\forall g \in G$, $\forall \varphi \in \text{Morph}\mathcal{U}^{\alpha}$, $\forall u_t, u_s \in U(2)$, $\forall v_t, v_s \in \overline{U}_0$, $\forall h \in U(1)$

$$
\phi_{\mathcal{P}}^{\alpha}(x, gU(1)) = \phi_{\mathcal{Q}}^{\alpha}(x, g) \qquad (\text{D.22})
$$

$$
\phi_{\mathcal{P}}^{\alpha}((\varphi, x), (u_t v_t h, u_s v_s)) = \\
\left( L \left( \phi_{\mathcal{P}}^{\alpha}(\varphi(x), h) \phi_{\mathcal{Q}}^{\alpha}(\varphi(x), u_t, v_t) \right) \circ \phi_{U\varphi(x)}^{\alpha} \circ \phi_{Ux}^{\alpha}^{-1}, \phi_{\mathcal{Q}}^{\alpha}(u_s, g_s) \right) \quad (\text{D.23})
$$

$L$ being the left action of $U(2)$ (as set of unitary operators of $\mathbb{C}^2$) on itself. This
definition is consistent with the source and target maps:

$$
s(\phi_{\mathcal{P}}^{\alpha}((\varphi, x), (u_t v_t h, u_s v_s))) = \phi_{\mathcal{Q}}^{\alpha}(x, u_s v_s) \quad (\text{D.24})
$$

$$
\begin{gather*}
t(\phi_{\mathcal{P}}^{\alpha}((\varphi, x), (u_t v_t h, u_s v_s))) = \\
\phi_{\mathcal{P}}^{\alpha}(\varphi(x), h) u_t u_s \phi_{U\varphi(x)}^{\alpha}(u_s^{-1} v_t u_s v_s) \tag{D.25}
\end{gather*}
$$

$\mathcal{G}$ acts on $|\Lambda\rangle$ as gauge changes via

$$
(u_s v_s) \cdot |\Lambda(x)\rangle|_{\mathcal{U}^\alpha} = u_s \phi_{Ux}^{\alpha}(v_s) |\Lambda(x)\rangle|_{\mathcal{U}^\alpha} \quad (\text{D.26})
$$

$$
[u_t v_t h, u_s v_s] \cdot_{\varphi} |\Lambda(x)\rangle|_{\mathcal{U}^{\alpha}} = \phi_{P}^{\alpha}(\varphi(x), h) u_t \phi_{U\varphi(x)}^{\alpha}(v_t) u_s \phi_{U\varphi(x)}^{\alpha}(v_s) \otimes \mathbf{D}(\varphi(x), x) |\Lambda(x)\rangle|_{\mathcal{U}^{\alpha}} \quad (\text{D.27})
$$

$\mathcal{P}_{\Lambda}$ is endowed with a 2-connection described by the gauge potentials $(\mathfrak{A}, \eta, H)$,
the fake curvature $\mathfrak{F}$ and the curving $B$ [8, 43]. Note that we have two gauge
potential-transformations, $\eta$ which is related to the arrows of the base category
$\mathcal{M}_{\Lambda}$ and $H$ which is related to the arrows of the structure groupoid $\mathcal{G}$ and defined
by $H\rho_{\Lambda} = -i\langle\Lambda(x)|\mathbf{D}(\varphi(x), x)^{-1}d\mathbf{D}(\varphi(x), x)|\Lambda(x)\rangle_*$.
$H$ is the non-abelian gauge
potential-transformation associated with $\mathfrak{A}$ (it plays the same role than $\eta$ for $A$).
$\text{tr}(\rho_{\Lambda}H) = \eta$, $H$ is then a gauge potential-transformation random variable which has
$\eta$ as mean value in the statistical distribution of states defined by $\rho_{\Lambda}$.

Let $U(\mathcal{X}) \subset \text{Env}(\mathcal{X})$ be the set of unitary operators in $\text{Env}(\mathcal{X})$. Let $\mathcal{E}$ be the category such that $\text{Obj}(\mathcal{E}) = \mathbb{C}^2 \otimes \mathcal{H}$, $\text{Morph}(\mathcal{E}) = U(\mathcal{X}) \times \mathbb{C}^2 \otimes \mathcal{H}$, with the following source, target and identity maps, and composition of arrows

$$
s(U, |\Psi\rangle) = |\Psi\rangle \qquad \text{id}_{\Psi} = (\text{id}, |\Psi\rangle) \tag{D.28}
$$

$$
t(U, |\Psi\rangle) = U|\Psi\rangle \quad (U', U|\Psi\rangle) \circ (U, |\Psi\rangle) = (U'U, |\Psi\rangle) \quad (\text{D.29})
$$

There is a canonical projection functor $\mathcal{E} \to \mathcal{P}\mathcal{X}$, defined by the $|\Psi\rangle \mapsto \text{tr}(|\Psi\rangle\langle\Psi|\cdot)$ and the projection induced by the quotient $U(\mathcal{X})/\mathcal{Z}(\text{Env}(\mathcal{X})) \simeq \text{InnAutEnv}(\mathcal{X})$. $\mathcal{E} \to \mathcal{P}\mathcal{X}$ can be viewed as a noncommutative bundle of flat connection defined by the gauge potential $\mathbf{A}_{\mathbf{D}} \in \Omega_{\text{Der}}^{1}(\mathcal{X})$. The functor $\omega: \mathcal{M}_{\Lambda} \to \mathcal{P}\mathcal{X}$ permits to consider the associated bundle $\mathcal{P}_{\Lambda} \times_{\mathcal{G}} \mathcal{E}_{|\omega(\mathcal{M}_{\Lambda})|}$ over $\mathcal{M}_{\Lambda}$, defined by

$$
\mathrm{Obj}(\mathcal{P}_{\Lambda} \times_{\mathcal{G}} \mathcal{E}_{|\omega(\mathcal{M}_{\Lambda})|}) = \{[p\tilde{g}^{-1}, \tilde{g}|\Lambda(x)\rangle]\}_{\tilde{g} \in \tilde{G}}, p \in \mathrm{Obj}(\mathcal{P}_{\Lambda})_x \} \quad (\text{D.30})
$$

$$
\begin{align*}
\text{Morph}(\mathcal{P}_{\Lambda} \times_{\mathcal{G}} \mathcal{E}_{|\omega(\mathcal{M}_{\Lambda})|}) &= \\
& \quad \{[(f, p)[g_t h, g_s]^{-1}, \\
& \quad (g_t g_s \mathbf{D}(\varphi(x), x) g_s^{-1} g_t^{-1}, h g_t g_s |\Lambda(x)\rangle)\}_{[g_t h, g_s] \in G \times G / U(1)} \\
& \quad , (f, p) \in \text{Morph}(\mathcal{P}_{\Lambda})(\varphi, x) \} \tag{D.31}
\end{align*}
$$

The strong adiabatic transport formulae: $e^{-i \int_{\mathcal{E}} A} |\Lambda(x)\rangle\rangle$ and $e^{-i \int_{\mathcal{E}} (A+\eta_{\mathcal{E}})} \mathbf{D}(\varphi(x), x) |\Lambda(x)\rangle\rangle$, corresponds to the action of the source and the target on the arrow $[e^{-i \int_{\mathcal{E}} \eta_{\varphi}}, e^{-i \int_{\mathcal{E}} A}]$,
and the weak adiabatic transport formulae: $\mathbb{P}e^{-i \int_{\mathfrak{e}} \mathfrak{A} |\Lambda(x)\rangle\rangle}$ and $\mathbb{P}e^{-i \int_{\mathfrak{e}} (\mathfrak{A}+H_{\varphi})} \otimes \mathbf{D}(\varphi(x), x)|\Lambda(x)\rangle\rangle$ corresponds to the action of the source and the target on the arrow $[\mathbb{P}e^{-i \int_{\mathfrak{e}} (\mathfrak{A}+H_{\varphi})}(\mathbb{P}e^{-i \int_{\mathfrak{e}} \mathfrak{A}})^{-1}, \mathbb{P}e^{-i \int_{\mathfrak{e}} \mathfrak{A}}]$. The adiabatic transport of $|\Lambda\rangle\rangle$ along the pseudo-surface $(\varphi, \mathcal{C}) \in \text{Diff } M_{\Lambda} \times \mathcal{P}M_{\Lambda}$ can be then viewed as the horizontal lift of the pseudo-surface $(\varphi, \mathcal{C})$ in $\mathcal{P}_{\Lambda}$.

## Appendix D.3. Noncommutative versus categorical geometries

<table><thead><tr><th rowspan="2">Noncommutative geometry</th><th colspan="2">Categorical geometry</th></tr><tr><th>Objects</th><th>Arrows</th></tr></thead><tbody><tr><td>InnAutEnv(Χ) × P(Μ)<br>automorphisms and pure states</td><td>M<sub>Λ</sub><br>eigenmanifold</td><td>Diff M<sub>Λ</sub> × M<sub>Λ</sub><br>eigenmanifold diffeomorphisms</td></tr><tr><td>Env(Χ)<br>quantum observables</td><td>C<sup>∞</sup>(M<sub>Λ</sub>)<br>classical observables</td><td>Diff M<sub>Λ</sub> × C<sup>∞</sup>(M<sub>Λ</sub>)<br>transformations of observables</td></tr><tr><td>Der(Χ)<br>derivatives (commutators)</td><td>TM<sub>Λ</sub><br>tangent vectors</td><td>TM<sub>Λ</sub>Diff M<sub>Λ</sub><br>diff. dependent tangent vectors</td></tr><tr><td>dl<sup>2</sup><sub>x</sub> / γ = δ<sub>ij</sub>dX<sup>i</sup> ⊗ dX<sup>j</sup><br>square length observable /<br>noncommutative inner product</td><td>γ<sub>ab</sub>ds<sup>a</sup>ds<sup>b</sup><br>square length<br>of the mean path</td><td>dist<sub>γ</sub>(ω<sub>x</sub>, ω<sub>ϕ(x)</sub>)<br>average length<br>of the quantum paths</td></tr><tr><td>A<sub>D</sub> ∈ Ω<sup>1</sup><sub>Der(Χ)</sub><br>linking vector observable</td><td>A ∈ Ω<sup>1</sup>(M<sub>Λ</sub>, R) /<br>Λ ∈ Ω<sup>1</sup>(M<sub>Λ</sub>, u(2)<sub>a.s.</sub>)<br>shift vector /<br>Lorentz connection</td><td>η ∈ Ω<sup>1</sup>(M<sub>Λ</sub>, R)<sub>Diff M<sub>Λ</sub></sub> /<br>H ∈ Ω<sup>1</sup>(M<sub>Λ</sub>, u(2)<sub>a.s.</sub>)<sub>Diff M<sub>Λ</sub></sub><br>gauge potential transformations</td></tr><tr><td>0<br>noncommutative flatness</td><td>F ∈ Ω<sup>2</sup>(M<sub>Λ</sub>, R) /<br>F̃ ∈ Ω<sup>2</sup>(M<sub>Λ</sub>, u(2))<sub>a.s.</sub><br>contorsion / Riemann curvature</td><td>B ∈ Ω<sup>2</sup>(M<sub>Λ</sub>, R)<sub>Diff M<sub>Λ</sub></sub> /<br>dH - iH ∧ H<br>curvings</td></tr></tbody></table>

The category having $\text{Diff } M_{\Lambda} \times C^{\infty}(M_{\Lambda})$ as arrow set is endowed with the following source, target and identity maps: $s(\varphi, f) = f$, $t(\varphi, f) = f \circ \varphi$, $\text{id}_f = (\text{id}_{M_{\Lambda}}, f)$; and with the following arrow composition: $(\varphi', f \circ \varphi) \circ (\varphi, f) = (\varphi \circ \varphi', f)$.

If we consider a paralinkable fuzzy space (Appendix B) we have a complete symmetry for the gauge fields between the noncommutative geometry and the categorical geometry, with $A_D \in \Omega^1_{\text{Der}(\mathbf{X})}$, $\mathfrak{A}_D \in \Omega^1(\text{Env}(\mathbf{X}), \mathfrak{gl}(2, \mathbb{C}))$, $F_D \in \Omega^2_{\text{Der}(\mathbf{X})}$ and $\mathfrak{F}_D \in \Omega^2(\text{Env}(\mathbf{X}), \mathfrak{gl}(2, \mathbb{C}))$.

The two geometries are related by applications associated with $\omega = \text{tr}(P_{\Lambda}\bullet) \in \mathcal{P}(\mathfrak{M})_{M_{\Lambda}}$:

$$ \begin{array}{ccc} \text{InnAutEnv}(\mathcal{X}) \times \mathcal{P}(\mathfrak{M}) & \xleftarrow{\omega} & \text{Diff } M_{\Lambda} \times M_{\Lambda} \\ \mathcal{P}(\mathfrak{M}) & \xleftarrow{\omega} & M_{\Lambda} \\ \text{Der}(\mathcal{X}) & \xrightarrow{\pi_x \omega_{x*}} & T_x M_{\Lambda} \\ \Omega^1_{\text{Der}(\mathcal{X})} & \xleftarrow{\omega_x^* \pi_x^*} & \Omega^1_x M_{\Lambda} \end{array} $$

In a same way, the different metrics are related by $\omega$:
<figure><img src="image_11.png" /></figure>

where $e^{L_{\varphi}} = \text{Ad}_{\mathcal{D}(\varphi(x),x)}$ (this relation holds only if $\varphi(x)$ and $x$ are linkable).

It is interesting to point out the relation between these two nonlocal geometries. Noncommutative geometry is nonlocal because of the quantum uncertainties $\Delta_x X^i$ and of the non-separability of quasicoherent states $\langle\langle\Lambda(y)|\Lambda(x)\rangle\rangle \neq \delta(y-x)$. The categorical geometry is nonlocal because of the arrows which model the possibility to jump “abruptly” from the source to the target (the arrows being associated with the displacement operator). For the metric properties, the two notions of nonlocality are intrinsically related : $d\ell^2 \to \gamma$ (square length of the mean path $\leftrightarrow$ distance between objects) $\gamma \to \text{dist}_{\gamma}$ (average length of the paths $\leftrightarrow$ length of arrows). The reason of this relation is obvious, Obj.$\mathcal{M}_{\Lambda}$ is the manifold of the mean values (in the quasicoherent states) of $\vec{X}$, whereas $\omega(\text{Morph.}\mathcal{M}_{\Lambda}) = \{(\text{Ad}_{\mathcal{D}(\varphi(x),x)}, \omega_x)\}_{x \in M_{\Lambda}, \varphi \in \text{Diff}_{M_{\Lambda}}}$ is associated with the displacement operator $\mathcal{D}(y, x)$ which models the transformation of the quasicoherent state for a measurement with the probe at $x$ followed by a measurement with probe at $y$, without any intermediate measurement. $\mathcal{D}(y, x)$ can be then assimilated to the quantum path between the two measurements (submitted to the quantum fluctuations).

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