Abstract
It is shown that the geometry of quantum theory can be derived from geometrical structure that may be considered more fundamental. The basic elements of this reconstruction of quantum theory are the natural metric on the space of probabilities (information geometry), the description of dynamics using a Hamiltonian formalism (symplectic geometry), and requirements of consistency (Kähler geometry). The theory that results is standard quantum mechanics, but in a geometrical formulation that includes also a particular case of a family of nonlinear gauge transformations introduced by Doebner and Goldin. The analysis is carried out for the case of discrete quantum mechanics. The work presented here relies heavily on, and extends, previous work done in collaboration with M. J. W. Hall.
Highlights
It is known that quantum mechanics has a rich geometrical structure which allows for a geometric formulation of the theory
The basic elements of this geometrical reconstruction of quantum theory are the natural metric on the space of probabilities, the description of dynamics using a Hamiltonian formalism, and requirements of consistency (Kahler geometry)
The geometry of quantum theory can be derived from information geometry, the natural geometry on the space of probabilities, using only a few assumptions
Summary
It is known that quantum mechanics has a rich geometrical structure which allows for a geometric formulation of the theory. One successful strategy has been to start from a known formulation of quantum mechanics and to identify geometrical features that can be used for the reformulation of the theory. This paper inverts this procedure: the aim is to derive the geometry of quantum theory from geometrical structure that may be considered more fundamental, and to examine the assumptions that are needed to do this. The analysis is carried out for the case of discrete quantum mechanics; a similar approach has been carried out previously for continuous systems [2]
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