A baby Majorana quantum formalism

The aim of the present paper is to introduce and to discuss the most basic fundamental concepts of quantum physics by means of a simple and pedagogical example. An appreciable part of its content presents original results. We start with the Euclidean plane which is certainly a paradigmatic example of a Hilbert space. The pure states form the unit circle (actually a half of it), the mixed states form the unit disk (actually a half of it), and rotations in the plane rule time evolution through Majorana-like equations involving only real quantities for closed and open systems. The set of pure states or a set of mixed states solve the identity and they are used for understanding the concept of integral quantization of functions on the unit circle and to give a semi-classical portrait of quantum observables. Interesting probabilistic aspects are developed. Since the tensor product of two planes, their direct sum, their cartesian product, are isomorphic (2 is the unique solution to x^x= x\times x = x+x), and they are also isomorphic to C^2, and to the quaternion field H (as a vector space), we describe an interesting relation between entanglement of real states, one-half spin cat states, and unit-norm quaternions which form the group SU$(2)$. We explain the most general form of the Hamiltonian in the real plane by considering the integral quantization of a magnetic-like interaction potential viewed as a classical observable on the unit $2$-sphere. Finally, we present an example of quantum measurement with pointer states lying also in the Euclidean plane.