# Discretization by quantization

My name is Erik Jansson and I am a PhD student in Professor Klas Modin’s group at the Department of Mathematical Sciences at Chalmers University of Technology and the University of Gothenburg. We work on discretizing certain types of equations using quantization theory. In a nutshell, the idea is to turn
the traditional perspective of physicists on its head: instead of seeing continuous equations as a large-scale limit of the quantum world, we view the quantized setting as a way to discretize spatially continuous
equations.

The model equation we first consider is the ideal incompressible Euler equation on the sphere S2, given by

ω̇ = ∇ψ ⋅ ∇ω,    Δψ = ω,

where ω is the vorticity and ψ is the stream function, related to the fluid velocity by the skew-gradient v = ∇ψ. It turns out that ψ ⋅ ∇ω is a Lie bracket on the space of smooth functions, turning C(S2) into an infinite-dimensional Lie algebra. In fact, this gives rise to rich geometric properties of the Euler equations. As an example, we have an infinite number of conserved quantities called Casimirs, given by

Ck(ω) = ∫S2ωk.

Further, the Euler equations have a remarkable interpretation as the geodesic equation on the infinite-dimensional group of volume-preserving diffeomorphisms Diffμ0(S2), equipped with the L2-metric [1].

The question is: Is there a discretization preserving this structure? This was answered already in 1991 by Vladimir Zeitlin, who approximated the Euler equations by using quantization theory, obtaining a matrix
flow on the Lie algebra 𝔰𝔲(N) analogous to the 2 − D Euler equations,

= [P,W],   ΔNP = W,

where W is the vorticity matrix and P is the stream matrix, and ΔN : 𝔰𝔲(N) → 𝔰𝔲(N) is a quantized Laplacian [3,6]. It turns out that the Zeitlin flow is isospectral, preserving the spectrum of W. This means that the quantized Casimirs, given by Ck(W) = Tr (Wk) are preserved.

Zeitlin’s approximation of the Euler equations, starting from random initial conditions.

Lately, there has been a resurgence of interest in Zeitlin’s model. In particular, our group, sometimes together with researchers in the Netherlands, have started to study Zeitlin’s model both as a tool to
understand Euler’s equation, but also as an interesting equation in its own right [2,4,5].

Further, as noted above, the Euler equations are geodesic equations on the group of volume-preserving diffeomorphisms. My own research has centered on extending Zeitlin’s model from the volume-preserving
diffeomorphism group to the full diffeomorphism group Diff (S2). This relies on three crucial steps:

1. Recognizing that any velocity field on S2 can be decomposed into an incompressible, divergence free, part, and a gradient, curl-free, part.

2. Proving that any vector field can be generated by a complex-valued function ψ = f + ig, where f generates the divergence free part and g the curl-free part.

3. Establishing that this in fact corresponds to complexifying the infinite-dimensional Lie algebra of smooth functions.

These steps allow us to apply Zeitlin’s model to the real part as well as to the imaginary part of ψ, opening the door for structure preserving quantization-based discretizations not only for incompressible fluid equations, but also to a wide range of equations such as the shallow water equations or KdV equations. Furthermore, the potential applications extend beyond fluid dynamics, encompassing fields such as optimal transport, gradient flows, and shape analysis.

References

[1] Vladimir I. Arnold. Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications ‘ a l’hydrodynamique des fluides parfaits. Ann. de l’Institut Fourier, 16(1):319–361, 1966.

[2] P. Cifani, M. Viviani, and K. Modin. An efficient geometric method for incompressible hydrodynamics on the sphere. J. Comput. Phys., 473:111772, 2023.

[3] Jens Hoppe and Shing-Tung Yau. Some properties of matrix harmonics on S2. Commun. Math. Phys. , 195(1):67–77, July 1998.

[4] K. Modin and M. Viviani. Canonical scale separation in 2d incompressible hydrodynamics. J. Fluid Mech. , 943:A36, 2022.

[5] Klas Modin and Milo Viviani. A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics. J. Fluid Mech. , 884, 2020.

[6] Vladimir Zeitlin. Finite-mode analogs of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure. Phys. D: Nonlinear Phenom. , 49(3), April 1991.