2,390 research outputs found
Is there a Jordan geometry underlying quantum physics?
There have been several propositions for a geometric and essentially
non-linear formulation of quantum mechanics. From a purely mathematical point
of view, the point of view of Jordan algebra theory might give new strength to
such approaches: there is a ``Jordan geometry'' belonging to the Jordan part of
the algebra of observables, in the same way as Lie groups belong to the Lie
part. Both the Lie geometry and the Jordan geometry are well-adapted to
describe certain features of quantum theory. We concentrate here on the
mathematical description of the Jordan geometry and raise some questions
concerning possible relations with foundational issues of quantum theory.Comment: 30 page
Homotopes and Conformal Deformations of Symmetric Spaces
Homotopy is an important feature of associative and Jordan algebraic
structures: such structures always come in families whose members need not be
isomorphic among other, but still share many important properties. One may
regard homotopy as a special kind of deformation of a given algebraic
structure. In this work, we investigate the global counterpart of this
phenomenon on the geometric level of the associated symmetric spaces -- on this
level, homotopy gives rise to conformal deformations of symmetric spaces. These
results are valid in arbitrary dimension and over general base fields and
-rings.Comment: 28 pages, 2nd corrected versio
Exponential families, Kahler geometry and quantum mechanics
Exponential families are a particular class of statistical manifolds which
are particularly important in statistical inference, and which appear very
frequently in statistics. For example, the set of normal distributions, with
mean {\mu} and deviation {\sigma}, form a 2-dimensional exponential family.
In this paper, we show that the tangent bundle of an exponential family is
naturally a Kahler manifold. This simple but crucial observation leads to the
formalism of quantum mechanics in its geometrical form, i.e. based on the
Kahler structure of the complex projective space, but generalizes also to more
general Kahler manifolds, providing a natural geometric framework for the
description of quantum systems. Many questions related to this "statistical
Kahler geometry" are discussed, and a close connection with representation
theory is observed. Examples of physical relevance are treated in details. For
example, it is shown that the spin of a particle can be entirely understood by
means of the usual binomial distribution. This paper centers on the
mathematical foundations of quantum mechanics, and on the question of its
potential generalization through its geometrical formulation
- …