Duality, Quantum Mechanics and (Almost) Complex Manifolds

The classical mechanics of a finite number of degrees of freedom requires a symplectic structure on phase space C, but it is independent of any complex structure. On the contrary, the quantum theory is intimately linked with the choice of a complex structure on C. When the latter is a complex–analytic manifold admitting just one complex structure, there is a unique quantisation whose classical limit is C. Then the notion of coherence is the same for all observers. However, when C admits two or more nonbiholomorphic complex structures, there is one different quantisation per different complex structure on C. The lack of analyticity in transforming between nonbiholomorphic complex structures can be interpreted as the loss of quantum–mechanical coherence under the corresponding transformation. Observers using one complex structure perceive as coherent the states that other observers, using a different complex structure, do not perceive as such. This is the notion of a quantum–mechanical duality transformation: the relativity of the notion of a quantum.