As part of a probabilistic reconstruction of quantum theory (QT), we show that spin is not a purely quantum mechanical phenomenon, as has long been assumed. Rather, this phenomenon occurs before the transition to QT takes place, namely in the area of the quasi-classical (here better quasi-quantum) theory. This borderland between classical physics and QT can be reached within the framework of our reconstruction by the replacement
p→M(q,t), where
p is the momentum variable of the particle and
M(q,t) is the momentum field in configuration space. The occurrence of spin, and its special value
1/2 , is a consequence of the fact that
M(q,t) must have exactly three independent components
Mk(q,t) for a single particle because of the three-dimensionality of space. In the Schrödinger equation for a "particle with spin zero", the momentum field is usually represented as a gradient of a single function
S. This implies dependencies between the components
Mk(q,t) for which no explanation exists. In reality,
M(q,t) needs to be represented by three functions, two of which are rotational degrees of freedom. The latter are responsible for the existence of spin. All massive structureless particles in nature must therefore be spin-one-half particles, simply because they have to be described by
4 real fields, one of which has the physical meaning of a probability density, while the other three are required to represent the momentum field in three-dimensional space. We derive the Pauli-Schrödinger equation, the correct value
g=2 of the gyromagnetic ratio, the classical limit of the Pauli-Schrödinger equation, and clarify some other open questions in the borderland between classical physics and QT.
This paper considers spin - j representation of the Lie group SU(2) and its applications to contemporary physics from Observer's Mathematics point of view (see www.mathrelativity.com). This mathematics was introduced by authors based on denial of infinity idea. We proved here the following. Theorem. In Observers Mathematics the probability of spin - j transformation is a homomorphism (representation) of SU(2) to the set of matrix transformations of a linear space of polynomial functions is less than 1. Theorem. The probability of elementary fermions have half-integer spin in Observer's Mathematics is less than 1. The probability of elementary bosons have integer spin in Observer's Mathematics is less than 1.
This note examines the hypothesis that the wave function is a combination of rotating and non-rotating wave modes, spin expressing the rotation of these components rather than the rotation of a particle-like object. For the photon the hypothesis is nearly literal: the transverse components of a circularly polarized wave rotate, and the associated angular momentum is mechanically measurable. A scalar counterexample shows that the rotation of components is not sufficient: spin is fixed by the way the components transform under rotations of physical space. The resulting hierarchy is: the relative phase is a measurable coordinate within a sector, the weight
m is an axial winding degree, the spin
j labels the irreducible representation of
SU(2), and univalence
(−1)2j separates superselection sectors as a rule on the algebra of observables. The formulation avoids the historical superluminal-velocity objection, which constrains rigid bodies rather than internal rotations of wave components. Any rotational ontology of spin must involve a relational, extended, or topologically attached wave configuration, never a point particle endowed with a mere internal axis. A soldering audit sharpens the programme-facing conclusion into two formal results: an exact Veronese-type obstruction excludes the first internal candidate for selecting the compact real form of the carrier square, and the currently derived data are shown to leave the spinorial soldering indeterminate — the missing data being a phase-sensitive real-form selector, an independent rotation action on the emergent space, and a Casimir-independent normalisation. Interpretive outlook (not a result): these findings leave open two structural readings. A future derived and sufficient soldering could make rotations effective symmetries of projected geometry, whereas an irreducible residual indeterminacy would be consistent with treating the spinorial carrier as prior to space.