Pointillisme à la Signac and construction of a quantum fiber bundle over convex bodies. (English) Zbl 1528.81180
Summary: We use the notion of polar duality from convex geometry and the theory of Lagrangian planes from symplectic geometry to construct a fiber bundle over ellipsoids that can be viewed as a quantum-mechanical substitute for the classical symplectic phase space. The total space of this fiber bundle consists of geometric quantum states, products of convex bodies carried by Lagrangian planes by their polar duals with respect to a second transversal Lagrangian plane. Using the theory of the John ellipsoid we relate these geometric quantum states to the notion of “quantum blobs” introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle. We show that the set of equivalence classes of unitarily related geometric quantum states is in a one-to-one correspondence with the set of all Gaussian wavepackets. We emphasize that the uncertainty principle appears in this paper as geometric property of the states we define, and is not expressed in terms of variances and covariances, the use of which was criticized by J. B. M. Uffink and J. Hilgevoord [Found. Phys. 15, No. 9, 925–944 (1985; doi:10.1007/BF00739034)].
MSC:
| 81S30 | Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics |
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
| 81P55 | Special bases (entangled, mutual unbiased, etc.) |
| 14J42 | Holomorphic symplectic varieties, hyper-Kähler varieties |
| 70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |
| 15A04 | Linear transformations, semilinear transformations |
| 55S40 | Sectioning fiber spaces and bundles in algebraic topology |
| 32M25 | Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions |
Keywords:
Lagrangian frame; symplectic group; polar duality; Gaussian wavepackets; Wigner transform; quantum fiber bundleReferences:
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