  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.
