2012/03/23 by Maurice A. de Gosson, de Gosson, Maurice A.
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Quantum chaos and dynamical systems #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1203.5310
openalex publication_date 2012/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A positive definite symmetric matrix σ qualifies as a quantum mechanical covariance matrix if and only if σ+(1/2)iℏΩ≥0 where Ω is the standard symplectic matrix. This well-known condition is a strong version of the uncertainty principle, which can be reinterpreted in terms of the topological notion of symplectic capacity, closely related to Gromov's non-squeezing theorem. We show that a recent refinement of the latter leads to a new class of geometric invariants. These are the volumes of the orthogonal projections of the covariance ellipsoid on symplectic subspaces of the phase space. We compare these geometric invariants to the algebraic "universal quantum invariants" of Dodonov and Serafini.