A duality in classical and quantum mechanics

J. M. Isidro

José M. Isidro

On classical phase spaces admitting just one complex-differentiable structure, there is no indeterminacy in the choice of the creation operators that create quanta out of a given vacuum. In these cases the notion of a quantum is universal, i.e., independent of the observer on classical phase space. Such is the case in all standard applications of quantum mechanics. However, recent developments suggest that the notion of a quantum may not be universal. Transformations between observers that do not agree on the notion of an elementary quantum are called dualities. Classical phase spaces admitting more than one complex-differentiable structure thus provide a natural framework to study dualities in quantum mechanics. As an example we quantise a classical mechanics whose phase space is a torus and prove explicitly that it exhibits dualities.

Mark P. Davidson

Philip Yu

J. M. Isidro

A recent notion in theoretical physics is that not all quantum theories arise from quantizing a classical system. Also, a given quantum model may possess more than just one classical limit. There is strong evidence for these facts in string duality and M-theory, and it has been suggested that they should also have a counterpart in quantum mechanics. In view of these developments we propose dequantization , a mechanism to render a quantum theory classical. Specifically, we present a geometric procedure to dequantize a given quantum mechanics (regardless of its classical origin, if any) to possibly different classical limits, whose quantization gives back the original quantum theory. The standard classical limit ℏ→0 arises as a particular case of our approach.

James G. Gilson