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.
Abstract Let H be either a complex inner product space of dimension at least two or a real inner product space of dimension at least three, and let us fix an
α∈(0,2π). The purpose of this paper is to characterise all bijective transformations on the projective space P(H) which preserve the quantum angle
α (or Fubini–Study distance
α) between lines in both directions. (Let us emphasise that we do not assume anything about the preservation of other quantum angles). For real inner product spaces and when
H=C2 we do this for every
α, and when H is a complex inner product space of dimension at least three we describe the structure of such transformations for
α≤4π. Our result immediately gives an Uhlhorn-type generalisation of Wigner’s theorem on quantum mechanical symmetry transformations, that is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability
cos2α in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when
H=C2 and
α=4π where an additional possibility occurs. (Note that the classical theorem of Uhlhorn is the solution for the
α=2π case). Usually in the literature, results which are connected to Wigner’s theorem are discussed under the assumption of completeness of H; however, here we shall remove this unnecessary hypothesis in our investigation. Our main tool is a characterisation of bijective maps on unit spheres of real inner product spaces which preserve one spherical angle in both directions.
T99 provides a geometric interpretation of the Born rule within a real symplectic-metric framework. Rather than treating quantum probabilities as fundamental postulates, the theorem derives them as normalized projection weights onto an observable subspace of a larger real state space. The construction begins with a real vector space
R equipped with a compatible metric-symplectic-complex structure
(R,g,ω,J), where \[J^2=-I,\qquadg(Ju,Jv)=g(u,v).\] The state space is assumed to be split into complementary components \[R=O\oplusO',\] where
O is the observable subspace and
O′=JO is its conjugate counterpart. The theorem introduces the observable projection operator
ΠO:R→O and an orthogonal decomposition of the observable sector into measurement directions
O=⨁iOi, with associated projectors
Πi. Every state
v∈R admits a unique decomposition \[v=o+h,\qquado\in O,\qquadh\in O'.\] The observable component is
o=ΠOv, while the conjugate component lies outside the directly observed sector. A central result is that the conjugate sector is metrically equivalent to the observable sector. Because
g(Jv,Jv)=g(v,v), The conjugate component possesses the same norm structure as the observable component. It is therefore not a lower-order correction or negligible remainder, but a geometrically equivalent part of the larger real state space. The theorem then derives the Born readout formula as a projection law. For each measurement direction
Oi,
Pi(v)=∥v∥2∥Πiv∥2. The probabilities satisfy
0≤Pi(v)≤1, and
∑iPi(v)=PO(v)=∥v∥2∥ΠOv∥2. Whenever the state is normalized entirely within the observable sector,
∥ΠOv∥=∥v∥, The probabilities satisfy
∑iPi(v)=1. The Born rule, therefore, emerges as the normalized squared projection of the full state onto observable measurement directions. Choosing the complex structure
J recovers the familiar complex formulation. The real metric and symplectic structures combine into the Hermitian inner product
g(⋅,⋅)+iω(⋅,⋅), and the projection formula becomes
Pi=∣⟨i∣ψ⟩∣2. The standard Born rule is thus recovered as the complex representation of the underlying projection geometry. Several consequences follow. Components lying entirely in the conjugate sector contribute no direct observable readout when measurements are restricted to the observable slice. The observable sector nevertheless remains a complete probability space, since normalized states within
O satisfy the usual probability-sum rule. Standard quantum mechanics, therefore, appears as the special case in which measurements access only the observable sector and no probability weight leaks from the conjugate component into observable readout. T99 provides a geometric foundation for the Born rule within the reduced Q5 framework. The theorem interprets quantum probabilities as observable-slice projections of a larger real symplectic structure and identifies the standard complex Born rule as the closed-observable-sector limit of that broader geometric description.