Quaternions, Spinors and the Hopf Fibration: Hidden Variables in Classical Mechanics

Rotations in 3 dimensional space are equally described by the SO(3) and SU(2) groups. These groups are shown to be isomorphic as they are both generated by the unit quaternion, thereby generating the same kinematics in 3D space. The Hopf Fibration describes the projection between the hypersphere S of the quaternion in 4D space, and the unit sphere S in 3D space. Great circles in S are mapped to points in S via the 6 Hopf maps and their respective stereographic projections. The higher and lower dimensional spaces are connected via the S fibre bundle which consists of the global, geometric and dynamics phases. The global phase is quantized in integer multiples of 2π and presents itself as a natural hidden variable of Classical Mechanics. Hidden variable theories of Quantum Mechanics have long been a topical theme due to the demonstrable non-locality and indeterminacy of the science. Quantum Mechanics was born at the turn of the 20th century when technology became sufficiently advanced so as to allow experiments on the smallest scales. The properties of light and atoms being measured were unable to be accounted for by the Classical Mechanics of that era, and so it became apparent early on that a more sophisticated theory of the fundamental processes is needed. In the absence of such a theory, many properties of light, atoms and electrons were catalogued while an underlying mathematical basis to justify their presence was unknown. In this vacuum of rational explanations for observed phenomena one axiom within Quantum Mechanics has risen above the fore to attain popular account for the unexplainable this axiom is the Born rule. Also known as the Copenhagen Interpretation of Quantum Mechanics, it states that the square modulus of the complex numbers describes the probable result of an experimental measure. Furthermore, that the fundamental particles exist in a superposition of probable states prior to measurement, and only following a measurement does the state, or wave-function collapse to provide the measured result. This article takes the view that the Born rule axiom is inherently incorrect, and is founded on an improper understanding of the underlying mathematics. In addition it is here demonstrated that hidden variables do exist, in the sense that phenomenological effects observed at the smallest scales are signatures measured from higher dimensional spaces. A proper and complete mathematical theory of these higher dimensional spaces is required. These statements are justified in this article in two parts. (i) In the opening section of this article (section 1) it is demonstrated unequivocally that the SU(2) and SO(3) groups are isomorphic. The dynamics of 3 dimensional ar X iv :1 60 1. 02 56 9v 13 [ qu an tph ] 1 N ov 2 02 1 Quaternions, Spinors and the Hopf Fibration: Hidden Variables in Classical Mechanics 2 space described by the SO(3) picture is equally represented in the SU(2) picture. This is in no doubt, as the quaternion is foundational structure of both pictures. Quaternion rotations in SU(2) are identical to quaternion rotations in SO(3). Quantum Mechanics has not only failed to recognise this, and failed to utilize the algebra correctly, but it has also introduced absurdities such as the Born rule for use in what can only be described as complete flights of fancy. In so doing it is easily demonstrated that there is a widespread misunderstanding of quaternion algebra. (ii) The quaternion is a 4 dimensional mathematical object which describes 3D rotations in full generality. As a consequence of the dimensional mismatch, information from the 4D space is both lost and retained. This is eloquently captured in the Hopf Fibration [1] [2], as it is demonstrated that 1) great circles in S correspond to points in S (information is lost), and 2) The signature of the 4 dimension (aka Hidden Variables) is encoded in the 3D kinematics through the global, geometric and dynamic phases of the S fibre bundle (information is retained). The phases of the S fibre bundle are calculated from the 3 dimensional kinematics, and are demonstrated in due course. These are the principle points of this article. To begin, it is first explicitly shown in section 1 that the SU(2) and SO(3) pictures are isomorphic. All rotations in SO(3) are equally described in SU(2). In section 2 the 6 maps of the Hopf Fibration are defined alongside their respective stereographic projections. Thereafter the phases of the S fibre bundle are presented in closed form. In section 3 it is shown how the Born rule is applied to the unit quaternion to describe particle spin in vain, representative of the widespread misunderstandings of quaternion algebra in Quantum Mechanics.