Geometric quantum mechanics

What geometric quantum mechanics have taught us

It was generally believed that it was only classical mechanics that exhibited a natural Hamiltonian phase-space structure, to which one had to apply a suitable quantisation procedure to produce a very different kind of structure, namely, the complex Hilbert space of quantum mechanics together with a family of linear operators, corresponding to physical observables. However, with the development of geometric quantum mechanics it has become difficult to sustain this point of view, and quantum theory has come to be recognised more as a self-contained entity.

Indeed, according to the principles of geometric quantum mechanics, the physical characteristics of a given quantum system can be represented by geometrical features that are preferentially identified in the complex manifold of quantum states.

The approach to quantum mechanics achieved via its natural phase-space geometry offers insights into many of the more enigmatic aspects of the theory:

• Linear superposition of states
• Quantum entanglement
• Quantum probability
• Uncertainty relations
• Geometric phases
• The collapse of the wave function.

One of the goals of this research is to illustrate in geometrical terms the interplay between these aspects of quantum theory.