Quantum Monads in Phase Space and Related Toeplitz Operators
Maurice de Gosson
TL;DR
The work develops a phase-space quantum framework built on quantum blobs, the minimum-uncertainty phase-space cells, and their bijection with generalized Gaussian states. It constructs blob-based Toeplitz/anti-Wick operators by smoothing Weyl observables with Gaussian windows tied to a blob, yielding positive, physically interpretable density operators. By embedding this in the Weyl–Wigner–Moyal formalism and leveraging the Feichtinger algebra, frames, and metaplectic covariance, the authors establish a canonical blob quantization and a density-matrix interpretation for phase-space quantum mechanics. The approach provides a geometrically grounded, spectrally stable method for phase-space quantization with potential applications in quantum state estimation and phase-space analysis of quantum systems.
Abstract
In earlier work, we introduced quantum blobs as minimum-uncertainty symplectic ellipsoids in phase space. These objects may be viewed as geometric monads in the Leibnizian sense, representing the elementary units of phase-space structure consistent with the uncertainty principle. We establish a one-to-one correspondence between such monads and generalized coherent states, represented by arbitrary non-degenerate Gaussian wave functions in configuration space. To each of these states, we associate a classs of Toeplitz operators that extends the standard anti-Wick quantization scheme. The mathematical and physical properties of these operators are analyzed, allowing for a generalized definition of density matrices within the phase-space formulation of quantum mechanics.