Quantum Riemannian geometry of phase space and nonassociativity

Riemannian Geometry of Quantum Computation

An introduction is given to some recent developments in the differential geometry of quantum computation for which the quantum evolution is described by the special unitary unimodular group SU(2n). Using the Lie algebra su(2n), detailed derivations are given of a useful Riemannian geometry of SU(2n), including the connection, curvature, the geodesic equation for minimal complexity quantum compu...

Riemannian Geometry of Quantum Groups

Riemannian Geometry on Quantum Spaces

An algebraic formulation of Riemannian geometry on quantum spaces is presented, where Riemannian metric, distance, Laplacian, connection, and curvature have their counterparts. This description is also extended ∗email address: [email protected] to complex manifolds. Examples include the quantum sphere, the complex quantum projective spaces and the two-sheeted space.

Non–Commutative Geometry on Quantum Phase–Space

A non–commutative analogue of the classical differential forms is constructed on the phase–space of an arbitrary quantum system. The non– commutative forms are universal and are related to the quantum mechanical dynamics in the same way as the classical forms are related to classical dynamics. They are constructed by applying the Weyl–Wigner symbol map to the differential envelope of the linear...

Quantum and Braided Group Riemannian Geometry

We formulate quantum group Riemannian geometry as a gauge theory of quantum differential forms. We first develop (and slightly generalise) classical Riemannian geometry in a self-dual manner as a principal bundle frame resolution and a dual pair of canonical forms. The role of Levi-Civita connection is naturally generalised to connections with vanishing torsion and cotorsion, which we introduce...