Symplectic topology and geometric quantum mechanics

Maurice A. de Gosson, Franz Luef

A. E. F. Djemai

Dante Fabian

We extend the sphere-on-sphere geometric quantization mechanism of the Operator-Derived Dimension (ODD) framework to derive the full kinematical structure of quantum mechanics. Starting from the topological origin of Planck’s constant h as a winding invariant of the S1 intersection between the five-dimensional bulk and the four-dimensional causal horizon, we show that the symplectic form, canonical commutation relations, phase, spin, and the path-integral formulation arise uniquely and necessarily from the same geometry. Quantization is therefore not postulated but enforced by topology, while operator algebra emerges from the transverse slicing dynamics. This places the foundations of quantum mechanics on geometric and topological necessity rather than axiomatic assumption.

A. E. F. Djemai

Maurice A. de Gosson

A positive de…nite symmetric matrix quali…es as a quantum mechanical covariance matrix if and only if + 12 i~ 0 where 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 re…nement 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 Sera…ni.

Jamal Elfakir

This thesis, explores the quantum entanglement and evolution through both a geometric and dynamical perspective. The first part focuses on classical phase space and its central role in Hamiltonian mechanics, emphasizing the importance of symplectic structures in describing mechanical states. The study highlights the formal analogy between classical phase space and the Hilbert space used in quantum mechanics. The second part is devoted to the geometric description of quantum states through the projective structure of Hilbert space. Emphasis is placed on the geometric interpretation of quantum evolution, particularly via the Fubini-Study metric, associated symplectic structures, and the geometric phase acquired during unitary evolutions. The final two parts are dedicated to the study of spin systems (both two-body and many-body) under different interaction models (XXZ Heisenberg and all-range Ising). Both the dynamical aspects (evolution speed, entanglement, and the quantum brachistochrone problem) and the geometric and topological structures of the corresponding quantum states are analyzed.