The Spinorial Origin of Quantum Mechanics (Preliminary Version)

Samuel Bernard-Bernardet, Benjamin Apffel

The spinorial ball is an interactive electronic device that we recently introduced to discuss the origin of spin-1/2 from the representation of the rotation group, without the framework quantum mechanics. Nevertheless, it can also be viewed as a macroscopic visualization of a quantum two-level system, and can thus be used to gain intuition on some generic features of qubits. In this article, we show how the ball can be used to fully visualize the evolution of a single qubit under arbitrary operators of SU(2). The Bloch sphere, the Hopf fibration, and the Berry phase can also be easily visualized and manipulated using this original device. We also discuss how the spinorial ball can be used to represent Hamiltonian evolution, and we describe an explicit mapping between the ball's motion and the evolution of a spin-1/2 system in arbitrary magnetic fields. An electronic implementation of projective measurement that matches the predictions of quantum mechanics is also proposed.

Jaume Giné

Tiberiu S. Tudor

In this paper we compare the rotation of a rigid body in the real three-dimensional Euclidean space E3 and its representation in the complex plane (Klein space), on one hand, with the transformation of polarization states of light (SOPs) by the phase-shifters figured in the complex plane and on the Poincaré sphere, on the other hand. Both the Klein space, in classical mechanics, and the Poincaré sphere, in polarization theory, are abstract spaces, whose construction is based on the classical stereographical projection between Riemann sphere and the simple complex plane C1. They are classical abstract spaces, even if they have been largely used for representing quantum spinorial physical realities too. At the interface of classical/quantum physics persist some misaperceptions about what is intrinsically of quantum origin and nature, and what is imported from the classical domain. In this context we examine some misunderstandings that take place in the field of these spaces. I shall focus on the double angle relationship between the rotation of representative points of the SOPs on the Poincaré sphere with respect to the corresponding rotations of the azimuthal and ellipticity angles of the “form of the SOPs”, at a transformation of state given by a phase shifter. This is a classical result, that is transferred on the sphere from the complex plane, on the basis of the stereographic bijective connection between the points on the sphere and those in the complex plane. In any textbook of quantum mechanics “the double angle/half angle problem” is presented as a pure quantum spinorial one, avoiding its classical face and origin. A quantum spinorial approach, obviously, recovers the classical results, together with the specific spinorial ones, but with regards to the double angle/half angle issue it is superfluous. We shall also briefly examine the classical and quantum spinorial content of what we know today under the global name of Pauli spin matrices. Often in papers or textbooks of physics the results are presented in a mélange in which it is difficult to establish from which point on one needs to appeal to spinorial or quantum aspects.

David Favrholdt

Mark S Swanson

Samuel Bernard-Bernardet, Emily Dumas, Benjamin Apffel

In quantum physics lectures, half-integer spins are generally introduced as “objects that do not come back to their original state after one full turn but that do after two.” As a consequence, students often consider this behavior to be purely quantum mechanical. However, spin-1/2 is above all a geometrical property of the rotations group and can, therefore, also have practical consequences at the macroscopic scale. To illustrate this, we introduce and describe in this work a new pedagogical tool named the spinorial ball. It allows students to concretely manipulate a macroscopic 1/2-spin, which helps them to build intuition as to how the latter behaves under rotations. This object can also be used to introduce several general concepts from the theory of Lie groups, such as group homomorphism and homotopy classes of loops through the example of the groups SU(2) and SO(3). The spinorial ball provides a macroscopic visualization of all these concepts, which are ubiquitous in quantum physics.