ICTP-East African Institute for Fundamental Research

KIST2 Building CST

Nyarugenge Campus

University of Rwanda

Kigali, Rwanda

# Orientations in the plane as quantum states

## Prof. Jean-Pierre Gazeau, Université Paris Diderot (Paris 7), France, will present an introduction to some of the most basic fundamental concepts of quantum physics using orientations or angles in the plane.

I will introduce and discuss some of the most basic fundamental concepts of quantum physics by using orientations or angles in the plane. Associating these quantum orientations with linear polarizations of light in the plane normal to its propagation constitutes the most appealing physical example of the presented formalism. The pure states form the unit circle (actually a half of it) and the mixed states form the unit disk (actually a half of it). Rotations in the plane rule time evolution through Majorana-like equations involving only real quantities for closed and open systems. Since the tensor product of two planes, their direct sum, and their cartesian product, are isomorphic (2 is the unique solution to x^x= x X x = x+x), and they are also isomorphic to C^2, and to the quaternion field H (as a vector space), I will describe an interesting relation between entanglement of real states, one-half spin cat states, and unit-norm quaternions which form the group SU(2). Finally, I will present an example of quantum measurement with pointer states lying also in the Euclidean plane.