#020: Ensemble spaces as convex spaces

Category: Geometry - Tags: Synthetic geometry, Affine geometry, Projective geometry

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A recurrent idea in our project is that ensembles are the prime physical objects while pure states are the abstraction. The goal, then, is to find the minimum set of axioms that are always justified for statistical ensembles. Classical mechanics and quantum mechanics should be the specialization of such spaces. Current findings are summed up here

Mathematical problem. We found appropriate axioms for convex spaces, which an ensemble space can be shown to satisfy. The idea is that one can alwayre create convex combinations \(\sum_i p_i x_i\) of ensembles \(x_i\), as long as \(\sum_i p_i = 1\) and \(p_i >= 0\).

Classical discrete spaces correspond to symplexes. Classical continuum spaces correspond to the space of differentiable function defined over a symplectic manifold, which also integrate to one. Quantum spaces correspond to the space of positive semi-definite Hermitian operators with trace one.

We are still looking for the appriate axioms for entropy, for subspaces and the proper characterization of the different cases in terms of these objects.

Physical significance. The convex combinations represent the statistical mixtures. The focus on ensembles as basic objects is explained by the reproducibility of physical laws. A law, in fact, never applies to a single case, but estabilish a ``whenever X, we have Y’’ relationship. Therefore the objects must be repeatable procedure of measurements and preparation. This means that X and Y must be, conceptually, ensembles.