1 Introduction↩︎

The discretization of continuous structures plays a pivotal role in both theoretical and applied mathematics, particularly in fields such as numerical analysis, computational physics, and data science. Traditional approaches to discretization often rely on heuristic or ad hoc methods, which can lead to inconsistencies and inefficiencies. Recent advances suggest that a more rigorous approach to discretization lies in preserving the essential features of the continuous structures while transitioning to a discrete framework. A key aspect of effective discretization is ensuring that the discrete model retains the underlying geometric and algebraic properties of the continuous one.
Structure-preserving discretization methods have gained significant attention due to their ability to retain key properties of continuous systems, such as symmetries, conservation laws, and geometric structures, when transitioning to discrete models. State-of-the-art approaches include finite element exterior calculus (FEEC) [1], [2], which ensures stability and convergence in numerical approximations of partial differential equations by preserving the de Rham complex. Geometric integration methods, such as symplectic integrators, maintain the symplectic structure in Hamiltonian systems, ensuring long-term accuracy in energy conservation [3], [4]. Discrete exterior calculus (DEC) extends these ideas to computational geometry by discretizing differential forms while preserving topological properties like divergence and curl [5], [6]. Recent advances have also focused on the relation between noncommutative geometries and discrete differential structures [7], [8]. This is in particular exemplified by the Berezin-Toeplitz quantization. This quantization discretizes certain types of Poisson algebras while maintaining algebraic structures in the transition from continuous to discrete systems [9]. Hence, quantization provides a robust framework for discrete geometry applications.
Although there are similarities among the different approaches of structure-preserving discretization, it is yet not clear how one can describe under a unified theory, eclectic methods ranging from finite exterior calculus to quantization; we identify this as a gap in the theory.
In this paper, we present an axiomatization of structure-preserving discretization by leveraging the language of commutative diagrams. Our primary objective is to formalize the process of discretization such that it maintains the integrity of underlying geometric and algebraic structures. We argue that by employing commutative diagrams, one can systematically capture and enforce the preservation of essential relationships between continuous and discrete models. A subsequent objective is to demonstrate that the structure-preserving discretization of differential structures, following the previous axioms, lead to noncommutative differential structure on operator algebras.
Commutative diagrams provide a visual and conceptual tool from category theory that allow us to represent the interrelations between various mathematical structures and their discretizations. By constructing and analyzing these diagrams, we can ensure that the discretization process respects the morphisms and relationships inherent in the continuous setting. This approach not only facilitates a clearer understanding of the discretization process but also helps in developing algorithms that are both theoretically sound and practically effective.
We demonstrate, by applying our axiomatization, that the discretization of a continuous differential structure is inherently given by a noncommutative geometry. This means that the resulting discrete structure is represented by operator algebras and the differentiation is realized by a commutator with a selfadjoint operator \(D\). The results presented in this work corroborate previous observations and constructions [10], [11]. The dictionary between continuous and discrete settings is identical to the dictionary between classical geometry and noncommutative one. This dictionary is summarized in the following table.

The Berezin-Toeplitz quantization is an archetypal example of deformation of a classical geometry given by a compact Kähler manifold \((M,g,\lbrace\cdot,\cdot\rbrace)\) to a noncommutative finite dimensional operator algebra \((M_n(\mathbb{C}),\left[ \cdot, \cdot \right] )\). Thus, it serves as a fundamental example of transition from a continuous to a discrete space. We use it in this work as a central application of our axiomatization in order to show that it satisfies the conditions of a structure preserving discretization.
The paper is organized as follows: Section 2 introduces the key concepts and axioms that form the foundation of our framework. Section 3 demonstrates how commutative diagrams can effectively formalize the preservation of structures and explains how noncommutative geometry emerges from the structure-preserving discretization of a differential algebra. Section 4 provides a comprehensive review of diverse examples of structure-preserving discretizations, showcasing the framework’s flexibility. Finally, Sections 5 and 6 apply these findings to the Berezin-Toeplitz quantization, highlighting its adherence to the proposed criteria; Section 6 being a walk through example on the sphere.
Through this work, we hope to contribute to a deeper understanding of discretization methods and to provide a foundation for future research in this area.

2 Axiomatization of the theory of discretizations↩︎

In the first section, we introduce our axiomatization of structure-preserving discretization. The axioms rely on Category theory tools that we introduce briefly.

2.1 Category theory↩︎

Let us recall that a category \(\mathcal{C}\) is defined by a class \(\mathrm{ob}(\mathcal{C})\) whose elements are called objects and a class \(\mathrm{hom}(\mathcal{C})\) whose elements are called morphisms or arrows. Each arrow \(f\) has a source object \(X\) and a target object \(Y\). A morphism \(f\) from \(X\) to \(Y\) is then denoted \(f:X\rightarrow Y\). The class of arrows \(\mathrm{hom}(\mathcal{C})\) is equipped with a binary operation called composition of arrows between two morphism \(f\) and \(g\): \[f:X\rightarrow Y, \quad g:Y\rightarrow Z \quad \text{define}\quad g\circ f:X\rightarrow Z\] such that \(f\circ g\) is an element of \(\mathrm{hom}(\mathcal{C})\). The composition is associative and there is exactly one identity morphism denoted \(1_X\) for every object \(X\) such that for every arrow \(f:X\rightarrow Y\): \[1_Y\circ f =f =f\circ f_X.\] One says that a morphism \(f:X\rightarrow Y\) is invertible and called an isomorphism (or more simply an invertible arrow) if there exists a morphism \(g:Y\rightarrow X\) such that \(f\circ g=1_Y\) and \(g\circ f=1_X\).

2.2 Axiomatization↩︎

In what will follow, we will always, unless stated otherwise, consider objects in the category of Banach spaces. Nevertheless, one should have in mind that many of the constructions can be extended to merely topological spaces.

We can also require stronger conservation properties between the continuous object \(C_1\) and its discretization \(C_1^n\).

2.3 Convergence↩︎

Consider a discretization \(\mathfrak{D}(f)\) of an arrow \(f\). Heuristically, the convergence of the discretization means that \((C_1^n,f_n)\) converges to \((C_1,f)\). Hence, to properly define a notion of convergence, one needs to compare elements between different levels of dicretization.
Let us recall that, the graph of an arrow \(f\), denoted \(\mathrm{Gr}(f)\), is defined as the subset of \(C_1\times C_2\) such that: \[\mathrm{Gr}(f)=\left\lbrace (x,y)\in C_1\times C_2 : y=f(x) \right\rbrace\] We denote by \(p_i:\mathrm{Gr}(f)\rightarrow C_i\) for \(i=1,2\) the obvious coordinate projections.

For a structure preserving discretization \(\mathfrak{D}(f)\), one can obtain an equivalent definition of convergence.

If we assume that the arrow \(f\) is also invertible, then one can deduce the following corollary relating convergence of \(\mathfrak{D}(f)\) and \(\mathfrak{D}(f^{-1})\).

3 Derivations and Noncommutative geometry↩︎

3.1 Derivations↩︎

The central focus of discretization theory is the study of differential equations derived from the differential structure of a manifold. In the following paragraph, we aim to demonstrate how the structure-preserving discretization of a smooth differential operator inevitably results in a noncommutative differential structure.
A differential structure means here the data of a unital algebra of smooth functions \(A\) and a derivation; we recall that a derivation \(d:A\rightarrow M\) is map such that:

• \(d\) is linear,

• \(d(1)=0\),

• \(d(ab)=d(a)b + ad(b)\).

where \(M\) is a \(A\)-bimodule. The data \((A,d)\) is called a differential algebra and constitute one of the fundamental structure that one may want to preserve under discretization. Consider a structure preserving discretization \((A_n,d_n)\) of a given differential algebra \((A,d)\) given by the following diagram:

We have established that one approximate an differential algebra \((A,d)\) of functions by a matrix algebra, then the discrete differential structure is inherently noncommutative. We now show a concrete case where such approximation exists.

Therefore, one sees that if the domain \(\mathcal{H}\) of a continuous differential operator \(D\) admits a filtration family \(\lbrace\mathcal{H}_n\rbrace\), then we can construct the following diagram of discretization: \[\begin{tikzcd} \mathcal{H} \arrow[r,"D"] \arrow[d,"P_n"'] & \mathcal{H} \arrow[d,"P_n"] \\ \mathcal{H}_n \arrow[r,"D_n"] & \mathcal{H}_n \end{tikzcd} \label{diagram2}\tag{2}\] Thus, the lack of commutativity of the previous diagram i.e. the existence of a structure preserving discretization, in this case is measured by the degree \(\mathrm{deg}(D)\) of the operator \(D\). We can then restrict ourselves to subclass of operators with vanishing degrees.

Recall that two (possibly unbounded) self-adjoint operators \(A\) and \(B\) are said to commute if and only if all the projections in their associated projection-valued measures commute. Hence, a trivial example of increasing sequence of commuting projections is given by spectral projectors.
Similarly to the axioms of structure preserving discretization, the block diagonal condition might be too restrictive; one can then relax the commutativity condition.

We can now state a theorem on existence of structure preserving discretization.

3.2 \(C^*\)-algebras as a general setting for structure preserving discretizations↩︎

In fact, most cases of discretization and existence of projection maps \(\pi^n\) and section maps \(s^n\) fit it in the general framework of \(C^*\)-algebras. Let us briefly recall that a \(C^*\)-algebra is a Banach algebra equipped with an involution \(*\), compatible with the norm: \(\|x^*x\|=\|x\|^2\).

4 Zoology of structure preserving discretizations↩︎

In the following section, we review archetypical examples of structure preserving discretizations through the lens of our axiomatization.

4.1 Finite element exterior calculus↩︎

We start by recalling some of the basic definitions of finite element exterior calculus.

The framework of the finite element exterior calculus is given by the \(L^2\)-de Rham complex, represented by the following complex \[0\rightarrow \mathrm{H\Lambda}^0(\mathrm{\Omega})\xrightarrow[]{d}\mathrm{H\Lambda}^1(\mathrm{\Omega})\xrightarrow[]{d}\cdots \xrightarrow[]{d}\mathrm{H\Lambda}^n(\mathrm{\Omega})\rightarrow 0\] where \(\mathrm{\Omega}\subseteq \mathbb{R}^n\) is a Lipschitz domain. In order to give a numerical approximation of a PDE, the method builds a finite-dimensional subcomplex \[0\rightarrow \mathrm{\Lambda}_\hbar^1(\mathrm{\Omega})\xrightarrow[]{d}\cdots \xrightarrow[]{d}\mathrm{\Lambda}_\hbar^n(\mathrm{\Omega})\rightarrow 0\] of the de Rham complex. In order to construct this subcomplex, one shows that there exists morphism \(\pi^\hbar\) projecting the de Rham complex down to the appropriate subcomplex; so that each map \(\pi^\hbar_k:\mathrm{H\Lambda}^k\rightarrow \mathrm{\Lambda}^k_\hbar\) is a projection onto the subspace \(\mathrm{\Lambda}_\hbar^k\) and induces the following commuting diagram \[\begin{tikzcd} H\mathrm{\Lambda}^k \arrow[r,"d"] \arrow[d,"\pi^\hbar_k"'] & H\mathrm{\Lambda}^{k+1} \arrow[d,"\pi^\hbar_{k+1}"] \\ \mathrm{\Lambda}_\hbar^k \arrow[r,"d"] & \mathrm{\Lambda}_\hbar^{k+1} \end{tikzcd}\] In this case, the section maps \(s_k^{\hbar}\) are simply identity maps. Therefore, if we consider \(\lbrace H\mathrm{\Lambda}^k,d^k\rbrace\) as an object of the category of Hilbert complexes, then the data \((\lbrace \mathrm{\Lambda}^k_{\hbar},d\rbrace, \pi^\hbar_k)\) gives a strongly structure preserving discretization.

4.2 Transfer Operator and diffeomorphism group↩︎

In the following paragraph, we would like to illustrate how our axiomatization allows us to describe structure-preseving discretizations of diffeomorphism groups; we prove on practical examples how infinite group of symmetries are related to matrix groups.
Consider a smooth compact manifold \(X\) and its diffeomorphism group \(\mathrm{Diff}(X)\). In this section, we identify structure preserving discretization of diffeomorphism groups which play a central role in PDE theory.
The group \(\mathrm{Diff}(X)\) admits a natural action on the Hilbert space \(L^2(X)\), the space of square-summable half-densities on \(X\); by choosing a smooth measure \(m\) on \(X\) one can identify \(L^2(X)\) with the usual space of functions \(f\) on \(X\) which are square-summable with respect to \(m\). Hence, \(\mathrm{Diff}(X)\) admits an obvious unitary representation on \(H\) induced by its action on \(X\): for any \(\psi \in \mathrm{Diff}(X)\), define \[\widehat{\psi}:H\rightarrow H \qquad \widehat{\psi}(f)(x):= J_\psi(x)^{\frac{1}{2}}f(\psi^{-1}x)\] where \(J_\psi(x)=dm(\psi^{-1}x)/dm(x)\). We can restrict this representation to an operator \(\widehat{\psi}:C^\infty(X)\rightarrow C^\infty(X)\) and study the possible structure preserving discretization of \(\mathrm{GL}(C^\infty(X))\).
Fix a diffeomorphism \(\psi\), in order to present a structure preserving discretization of the pair \((C^\infty(X), \widehat{\psi})\), we need to specify in which category the object \((C^\infty(X), \widehat{\psi})\) belongs to: we will consider it as an object of the category of linear spaces. Therefore, following the definition, a structure preserving discretization is given by a commuting diagram \[\begin{tikzcd} C^\infty(X) \arrow[r,"\widehat{\psi}"] \arrow[d,"\pi^n"'] & C^\infty(X) \arrow[d,"\pi^n"] \\ A_N\arrow[r,"\widehat{\psi}_N"] & A_N \end{tikzcd} \label{diagram3}\tag{3}\] such that \(\lim_{n\rightarrow \infty}\|\pi^n(f)\|_\infty = \|f\|_\infty\) and \((A_N,\widehat{\psi}_N)\) is a pair of a vector space \(A_N\) and a linear map \(\widehat{\psi}_N\).

Therefore, diagram 3 becomes \[\begin{tikzcd} C^\infty(X) \arrow[r,"\widehat{\psi}"] \arrow[d,"\pi^n"'] & C^\infty(X) \arrow[d,"\pi^n"] \\ \mathbb{C}^N\arrow[r,"\widehat{\psi}_n"] & \mathbb{C}^N \end{tikzcd}\] and the existence of \(\widehat{\psi}_n\) is given by the Second Existence Theorem. We can then state the following theorem.

One could also be interested in subgroups of the diffeomorphism group. For instance, if the manifold \(X\) is equipped with a volume form \(d\omega\), we can define the scalar product between two function: \[\left\langle f,g \right\rangle =\int_X f\overline{g} \;d\omega\] Then, one can define the subgroup of diffeomorphisms preserving this inner product and denoted it by \(\mathrm{Diff}^+_0(X)\).
One can also define the subgroup of markovian diffeomorphism, i.e. the group of diffeomorphism \(\psi\) satisfying \(\psi(1)=1\). We denote this subgroup by \(\mathfrak{D}(\mathrm{Diff}_{\mathrm{m}}(X))\).
Let us denote by \(\mathrm{GL}^{\mathrm{st}}_N(\mathbb{C})\), the group of invertible stochastic matrices.

In addition, we notice that a structure preserving diffeomorphism discretization of a diffeomorphism group induces a structure preserving discretization of the associated Lie algebra. Let us recall that the pullback representation gives the map \[\mathrm{Diff}(X)\rightarrow \mathrm{Lin}(C^\infty(X)) \quad \psi\mapsto (f\mapsto \widehat{\psi}(f))\] Differentiating it at the identity \(\psi=\mathrm{id}\) gives a representation of the algebra of vector fields \(\mathrm{Vect}(X)\) over the space \(X\) into the space of derivations: \[\mathrm{Vect}(X)\rightarrow \mathrm{Der}(C^\infty(X)) \quad \Psi\mapsto \mathcal{L}_\Psi\] Here the operator \(\mathcal{L}_\psi:C^\infty(X)\rightarrow C^\infty(X)\) is given by the derivative of a function \(f\) in the direction of the vector field \(X\), i.e. \[\mathcal{L}_\Psi f:=\left. \frac{d}{dt}\right|_{t=0}\widehat{\psi}_t(f),\] where \(\psi^t\) denotes the flow of \(\Psi\). Moreover, one can show that the map is a Lie-algebra anti-homomorphism: \[\mathcal{L}_{[\Psi,\Phi]}=\mathcal{L}_\Phi\mathcal{L}_\Psi-\mathcal{L}_\Psi\mathcal{L}_\Phi\] for any vector fields \(\Psi,\Phi\in \mathrm{Vect}(X)\).
Hence, using the structure preserving condition 4 and differentiating Diagram 3 at the identity we obtain the new diagram: \[\begin{tikzcd} C^\infty(X) \arrow[r,"\mathcal{L}_\Psi"] \arrow[d,"\pi^n"'] & C^\infty(X) \arrow[d,"\pi^n"] \\ A_N\arrow[r,"\mathcal{L}_{\Psi_N}"] & A_N \end{tikzcd}\] where \(\mathcal{L}_\psi\) and \(\mathcal{L}_{\psi_N}\) are derivations.

4.3 Berezin-Toeplitz Quantization↩︎

The Berezin-Toeplitz quantization bridges the gap between classical and quantum mechanics by transitioning from the Poisson algebra of smooth functions on a Kähler manifold to a finite-dimensional, noncommutative matrix algebra. This quantization method maintains essential geometric and algebraic properties, making it an ideal framework for discretizing continuous structures while ensuring structural integrity. In this section, the detailed process of Berezin-Toeplitz quantization is presented, and we demonstrate its role in the discretization of compact manifolds. We prove from first principle that it is a structure preserving method. It also provides a significant example to understand how continuous spaces are represented discretely in noncommutative settings.
The natural setting of the Berezin-Toeplitz quantization is a compact Kähler manifold \((M,\omega)\). Whereas the basic objects can be defined on an arbitrary Kähler manifold, some proofs rely on compactness to avoid technicalities. For this reason, we restrict ourselves to the compact setting.
A Kähler manifold \((M,\omega)\) is called quantizable if there exists an associated quantum line bundle \((L,h,\nabla)\), where \(L\) is a holomorphic line bundle \(L\) over \(M\) (i.e. locally a one-dimensional vector space), \(h\) a Hermitian metric on \(L\), and \(\nabla\) a connection compatible with the metric \(h\) and the complex structure.
We consider the space of holomorphic sections \(\mathrm{\Gamma}_{hol}(M,L)\) with value in the line bundle \(L\). The bundle \(L\) will be the stage for a finite dimensional quantization. For this, one needs to define the appropriate Hilbert space from \(\mathrm{\Gamma}_{hol}(M,L)\). the scalar product on the space of sections and the norm are given by \[\left\langle \varphi,\psi \right\rangle :=\int_Mh(\varphi,\psi)\mathrm{\Omega}, \quad \|\varphi\|:=\sqrt{\left\langle \varphi,\varphi \right\rangle },\] where \(\mathrm{\Omega}=\frac{1}{n!}\omega^{\wedge n}\) is the Liouville form, used as a volume form on \(M\). Let \(\mathrm{L}^2(M,L)\) be the \(\mathrm{L}^2\)-completion of \(\mathrm{\Gamma}_\infty(M,L)\), and \(\mathrm{\Gamma}_{hol}(M,L)\) be its finite dimensional (due to compactness) subspace of holomorphic sections. Let \(\mathrm{\Pi}^{(1)}:\mathrm{L}^2(M,L)\rightarrow \mathrm{\Gamma}_{hol}(M,L)\) be the projection. For \(f\in C^\infty(M)\), the Toeplitz operator \(T^{(1)}_f\) is defined to be \[T^{(1)}_f:=\mathrm{\Pi}^{(1)}(f\cdot): \quad \mathrm{\Gamma}_{hol}(M,L) \rightarrow \mathrm{\Gamma}_{hol}(M,L)\] The linear map \[T:C^\infty(M)\rightarrow \mathrm{End}\left(\mathrm{\Gamma}_{hol}(M,L) \right), \qquad f,\mapsto T_f\] is the Berezin-Toeplitz quantization map. In general, because \[T_fT_g\neq T_{fg}\] the quantization map is neither a Lie algebra homomorphism nor an associative algebra homomorphism.
Moreover, the Berezin-Toeplitz quantization map from the commutative algebra of functions to a noncommutative finite-dimensional matrix algebra. The finite dimensionality implies the loss of a lot of classical information. In order to recover this information, one should consider not just the bundle \((L,\nabla, h)\) alone but all its tensor powers \((L^{\otimes m},\nabla^{(m)},h^{(m)})\) and apply the construction for every \(m\in \mathbb{N}_0\). In this way, the direct sum of the spaces of holomorphic sections of \((L^{\otimes m},\nabla^{(m)},h^{(m)})\) gets identified with a Hilbert subspace \(\mathcal{H}\) called generalized Hardy space.
Thus, we have the following decomposition: \[\mathcal{H}=\bigoplus_m(\mathcal{H}_m,\left\langle \cdot,\cdot \right\rangle_m),\qquad \mathcal{H}_m=\mathrm{\Gamma}_{hol}(M,L^m).\] where we have defined the modified measure \[\left\langle \varphi,\psi \right\rangle_m :=\int_Mh(\varphi,\psi)\mathrm{\Omega}^{(m)}_\epsilon,\qquad \text{where}\quad\mathrm{\Omega}^{(m)}_\epsilon:=\epsilon^{(m)}(x)\mathrm{\Omega}(x)\] with \(\epsilon^{(m)}\) the Rawnsley’s epsilon function.

This infinite family should in some sense approximate the algebra \(C^\infty(M)\); one obtains a family of matrix algebras and maps \[T^{(m)}:C^\infty(M) \rightarrow \mathrm{End}(\mathrm{\Gamma}_{hol}(M,L^m)), \qquad f\mapsto T_f^{(m)}=\mathrm{\Pi}^{(m)}(f\cdot), m\in \mathbb{N}_0\]

In other words, for \(f\in C^\infty(M)\) we have the decomposition \[T_f=\prod_{m=0}^\infty T_f^{(m)}\] where \(T_f^{(m)}\) are exactly the restriction of \(T_f\) to \(\mathcal{H}^{(m)}\).

We introduce on \(\mathrm{End}(\mathrm{\Gamma}_{hol}(M,L^{(m)}))\) the Hilbert-Schmidt norm \[<A,C>_{HS} =\mathrm{Tr}(A^*C)\]

In this section, we are going to draw a bridge between the classical problem of structure preserving discretization and the Berezin-Toeplitz quantization. We are going to show that the Berezin-Toeplitz quantization is in fact a structure preserving discretization of the Poisson algebra \((C^\infty(M), \left\lbrace \cdot, \cdot \right\rbrace )\), when \(M\) is a compact Kähler manifold.
Now that we have presented the background material on the Berezin-Toeplitz quantization, we can be described it by the following diagram. \[\begin{tikzcd} C^\infty(M)\otimes C^\infty(M) \arrow[r,"{\left\lbrace \cdot, \cdot \right\rbrace} "] \arrow[d,"T^{(m)}"] & C^\infty(M) \arrow[d,"T^{(m)}"] \\ M_m(\mathbb{C})\otimes M_m(\mathbb{C}) \arrow[r,"{\left[ \cdot, \cdot\right] } "] & M_m(\mathbb{C}) \end{tikzcd}\] Indeed, we prove that the Berezin-Toeplitz quantization satisfies the axiom of a structure preserving discretization of the pair \((C^\infty(M), \left\lbrace \cdot, \cdot \right\rbrace )\).

5 Noncommutative Laplacian as a discrete Laplacian↩︎

The Laplacian, which traditionally arises in the context of smooth Riemannian manifolds, is critical for understanding various physical phenomena, such as heat diffusion and wave propagation. However, when transitioning to discrete spaces, particularly in curved spaces, the formulation of the Laplacian requires a novel approach.
This section focuses on how the Berezin-Toeplitz quantization framework can be used to discretize the classical Laplacian, thereby creating a discrete version that operates on matrix algebras rather than smooth functions. The key idea is to show that the noncommutative Laplacian emerges naturally, following the axioms, as a discrete counterpart of the classical Laplace operator on compact Kähler manifolds. By leveraging the structure-preserving properties of the Berezin-Toeplitz quantization, the resulting discrete Laplacian retains important geometric and analytic features, offering a powerful tool for studying differential equations in noncommutative settings.
Consider again a compact Kähler manifold \((M,\omega)\). The Nash embedding theorem states that any Riemannian manifold can be isometrically embedded in the Euclidean space \(\mathbb{R}^d\) for sufficiently large \(d\). Thus, for a closed Poisson manifold \((M,\pi )\) with a metric \(g\), there exists an isometry embedding \[X:M\rightarrow \mathbb{R}^d\] for sufficiently large \(d\). We denote the embedding coordinate functions as \(\left\lbrace X^k \right\rbrace_{k=1,\dots, d}\). The word isometric means that the induced metric of the embedding is equal to the intrinsic metric \(g\) on \(M\): \[g_{ij}(x)=g_x(\partial_i,\partial_j)=\left\langle d_xX(\partial_i),d_xX(\partial_j) \right\rangle = \sum_{k=1}^d(\partial_iX^k)(\partial_jX^k) \label{coord}\tag{5}\] Using the smooth bivector \(\pi\in \mathfrak{X}(M)\), we can associate to any function \(f\) a derivation through the map defined by: \[\xi_f:C^\infty(M)\rightarrow \mathrm{Der}(M), \quad \xi_f(g)=\pi(df\wedge dg)\] We can now define the following family of self-adjoint differential operators \[\partial_k:C^\infty(M)\rightarrow C^\infty(M), \quad g\mapsto \partial_k(g):=\xi_{X^k}(g) =\left\lbrace X^k,g\right\rbrace\] for every \(k=1,\dots ,d\).

We also define the Bochner-Laplace operator for the metric \(g\) on \(M\) defined by \[\mathrm{\Delta}\varphi := -g^{ij}\nabla_i\nabla_j\varphi.\] using the covariant derivatives. There is a relation between the operator \(D\) and the Laplacian given by \[D^2=\mathrm{\Delta} +\text{lower order terms}\]

Therefore, we will derive structure preserving discretization of the differential algebra \((C^\infty(M),\mathcal{D})\) which will induce a discretization of the Laplacian.

An important question that remains to address is the convergence of the discrete Bochner-Laplacian to the continuous counterpart. In order to do so, we need to show, following Definition [conv], the limit \[\lim_{m\rightarrow \infty}\|\mathrm{\Delta}\varphi-\sigma^{(m)}\circ \mathrm{\Delta}_m\circ T^{(m)}\|=0\] We will need the following definition to tackle the convergence of unbounded linear operators.

In order to prove the theorem, we need the following lemma relating the Laplacian to the Poisson bracket.

We can also answer the question about the convergence of the spectrum of the Laplacian, which is crucial in many applications.

6 Example of the sphere \(\mathbb{S}^2\)↩︎

We show in the concrete example of the sphere \((\mathbb{S}^2,\lbrace\cdot, \cdot\rbrace)\), seen as a compact Poisson manifold, how a structure-preserving discretization of the differential structure of the space \(\mathbb{S}^2\) can be derived from first principles, including the discretization of the Laplacian operator. This is achieved following the set of axioms introduced in this work; the projection maps are provided by the Berezin-Toeplitz quantization. As a result, we recover the discrete Laplacian used in Zeitlin’s model for Euler’s equations on the sphere [9], [16]; hence proving that such model is structure-preserving in the sense of the present work. In a follow-up work [17], we extend this construction to compact surfaces with higher genus.
Let us recall that there exists an isometric embedding: \[\rho : \mathbb{S}^2\rightarrow \mathbb{R}^3, \qquad \rho(s):=(x,y,z)\] of the sphere onto \(\mathbb{R}^3\), such that \(x^2+y^2+z^2=1\). We consider the differential operator \(\mathcal{D}\) defined by \[\mathcal{D}(f)=\lbrace x,f\rbrace + \lbrace y,f\rbrace + \lbrace z,f\rbrace = - \left(\partial_xf + \partial_yf + \partial_zf \right)\] Finding a structure-preserving discretization of the differential algebra \((C^\infty(\mathbb{S}^2),\mathcal{D})\) is equivalent by our axiomatization to the existence of the following diagram \[\begin{tikzcd} C^\infty(\mathbb{S}^2) \arrow[r,"\mathcal{D}"] \arrow[d,"T^{(m)}"'] & C^\infty(\mathbb{S}^2) \arrow[d,"T^{(m)}"] \\ \mathrm{End}(\mathcal{H}_m)\arrow[r,"d_m"] & \mathrm{End}(\mathcal{H}_m) \end{tikzcd}\] such that \((\mathrm{End}(\mathcal{H}_m),d_m)\) is a differential algebra and \(\|T^{(m)}\circ\mathcal{D} - d_m\circ T^{(m)}\|\rightarrow 0\).
If we rewrite the spherical polynomial the spherical harmonics \(\left\lbrace Y_{\ell m}(\theta, \varphi)\right\rbrace\) as the space \(\mathcal{P}_{\mathrm{h}}[x_1,x_2,x_3]\) of harmonic homogeneous polynomial in \(3\) variables \(x_1,x_2,x_3\), then \(T^{(m)}\) can be taken as the projection onto the subspace \(\mathcal{P}^m_{\mathrm{h}}[x_1,x_2,x_3]\) of degree \(m\) homogeneous polynomials. In this case, \(T^{(m)}\) corresponds to the Berezin-Toeplitz quantization of \(\mathbb{S}^2\) and it satisfies the condition of a structure preserving discretization.
Following the procedure of the previous section, the continuous operators \((x,y,z)\) are replaced by \(m\)-dimensional matrices: \[T^{(m)}(x)=X_m, \quad T^{(m)}(y)=Y_m, \quad T^{(m)}(z)=Z_m,\] and the Lie algebra generated by \((X_m,Y_m,Z_m)\) is the Spin algebra \(\mathfrak{so}(m)\). Then, the differential operator \(\mathcal{D}\) is discretized using the Lie bracket of matrices and gives the derivation \(d_m\): \[d_m(A)=[X_m,A]+[Y_m,A]+[Z_m,A]\] Similarly, the Laplace operator on the sphere is replaced by \[\Delta_m(A)=[X_m,[X_m,A]]+[Y_m,[Y_m,A]]+[Z_m,[Z_m,A]],\] which is identified as the non-commutative Laplacian.

References↩︎