Differential calculus on manifolds with boundary applications

Abstract

This paper contains a set of lecture notes on manifolds with boundary and corners, with particular attention to the space of quantum states. A geometrically inspired way of dealing with these kind of manifolds is presented, and explicit examples are given in order to clearly illustrate the main ideas.

Differential Geometry of Quantum States, Observables and Evolution

“…For a system with n levels (dim H = n) we would get n different orbits. The one of maximal dimension would be the bulk, while the boundary of the closed convex body S of quantum states would be the union of orbits of dimensions less than n. The geometry of S as developed in [6,7,16] will be exposed in Sect. 2.1.…”

“…We will briefly recall here the results of [6,7] concerning the geometry of the space of all states, pure or mixed for the qubit. Every 2 by 2 Hermitean matrix A may be written in the form:…”

“…It is now possible to construct a Lie-Jordan algebra (see for instance [6,7,14]) with commutative Jordan product • and Lie product {·, ·} on the space of observables (affine functions) out of the tensors R and Λ. Such algebra is defined by:…”

Covariant Jacobi brackets for test particles

“…[9] By associating a "symbol" to this differential operator by means of the functions e ±ipµx µ , see Ref. [10], we would find a dispersion relation for the momentum four-vector of the relativistic particle…”

Stratified manifold of quantum states, actions of the complex special linear group

“…Our understanding of the geometry of the space of quantum states is in constant evolution and there are different fields of application in which it is possible to use the knowledge we gain. For instance, geometrical ideas have been successfully exploited when addressing the foundations of quantum mechanics [4,7,8,10,13,20,22,23,25,31,35,40], quantum information theory [5,15,19,27,36,39,43,45,54], quantum dynamics [9,12,14,16,17,18,24], entanglement theory [3,6,11,29,30,34,48,49].…”