We replace the usual notion of quantum cell from statistical mechanics by that of "quantum blob". A quantum blob is the transform, by a linaer symplectic transformation, of a phase space ball with radius equal to the square root of h-bar. The intersection of a quantum blob with any symplectic plane is an ellipse with area one half of h. This very special property, which is closed related to the principle of the symplectic camel, leads to a symplectic invariant statement of the uncertainty principle. We moreover prove that the average of a phase space Gaussian over a quantum blob is the Wigner transform of a minimum uncertainty Gaussian.
We invite the reader (presumably an upper level undergraduate student) to a journey leading from the continent of Classical Mechanics to the new territories of Quantum Mechanics. We'll be riding the symplectic camel and have William of Occam as travel companion, so no excess baggage is allowed. The first part of our trip takes us from the symplectic egg to Gromov's non-squeezing theorem and its dynamical interpretation. The second part leads us to a symplectic formulation of the quantum uncertainty principle, which opens the way to new discoveries.
We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke-Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho--Stark principle from the point of view of quantum polarity.