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The Boolean Rotation Theorem, Riemann Sphere, and Wavefunction: Twistor Geometry of Polarization and Superposition Spinors

Yingrui Yang

Published Feb 2, 2026• 0 citations
Publisher

Abstract

Being either true or false, 1 or 0, the standard logic and the Boolean algebra traditionality never rotate. Thus, it can only account for polarization states but not superposition states. This paper proves the Boolean rotation theorem through complexifications. This result allows us to formulate polarization spinors as well as superpositions spinors. It provides a new understanding of the Riemann sphere of two-state systems. It also provides an alternative solution to the measurement of wavefunctions, which accounts for both the U-process and the R-process. The work reported in this paper formulates the Penrose twistor geometry of the polarization spinors and the superposition spinors.

DOI
10.20944/preprints202601.2414.v1

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Being either true or false, 1 or 0, the standard logic and the Boolean algebra traditionality never rotate. Thus, it can only account for polarization states but not superposition states. This paper proves the Boolean rotation theorem through complexifications. This result allows us to formulate polarization spinors as well as superpositions spinors. It provides a new understanding of the Riemann sphere of two-state systems. It also provides an alternative solution to the measurement of wavefunctions, which accounts for both the U-process and the R-process. The work reported in this paper formulates the Penrose twistor geometry of the polarization spinors and the superposition spinors.
In "Part I: Vector Analysis of Spinors", the author studied the geometry of two component spinors as points on the Riemann sphere in the geometric algebra of three dimensional Euclidean space. Here, these ideas are generalized to apply to four component Dirac spinors on the complex Riemann sphere in the complexified geometric algebra of spacetime, which includes Lorentz transformations. The development of generalized Pauli matrices eliminate the need for the traditional Dirac gamma matrices. We give the discrete probability distribution of measuring a spin 1/2 particle in an arbitrary spin state, assuming that it was prepared in a given state immediately prior to the measurement, independent of the inertial system in which measurements are made. The Fierz identities between the physical observables of a Dirac spinor are discussed.
This paper introduces the quaternionic wavefunction as a compact way to organize the standard spinor geometry of a spin one half particle. It shows how a two component complex state can be regrouped into quaternionic form in a way that makes the geometric relationship among spinors, the three sphere, and the transformation group SU(2) easier to see. The paper emphasizes that the mathematics is standard and that its contribution is expository. Its goal is to clarify how spin one half structure, unit quaternions, and rotational geometry fit together, while carefully distinguishing the full normalized state geometry from the projected Bloch sphere description. It is intended as a conceptual prelude to the broader RQM paper series, not as a claim of a new physical theory.
The present paper does not intend to provide the solution of the Riemann hypothesis. Rather, we propose an alternative approach to engage with Riemann hypothesis. We assume as our working hypothesis that the Riemann hypothesis is not soluble within the current framework of the analytic number theory. Rather, we treat it as an interesting platform of empirical investigations. Four possible paths to engage Riemann hypothesis are discussed. We introduce the notion of Riemann monad Antipodal and construct a Riemann sphere model through Möbius Transformation. It follows an introduction of the Riemann sphere of two-state systems by Penrose. We regard Riemann platform as a Gauge Structure, in which the ζ function is treated as a wavefunction that satisfies gauge symmetry by performing gauge transformations. Some empirical issues are discussed. These issues are: Statistical significance; the measurement problem and the Yes/No measurement; the Dirac -function; sample space and stochastic sampling.The present paper does not intend to provide the solution of the Riemann hypothesis. Rather, we propose an alternative approach to engage with Riemann hypothesis. We assume as our working hypothesis that the Riemann hypothesis is not soluble within the current framework of the analytic number theory. Rather, we treat it as an interesting platform of empirical investigations. Four possible paths to engage Riemann hypothesis are discussed. We introduce the notion of Riemann monad Antipodal and construct a Riemann sphere model through Möbius Transformation. It follows an introduction of the Riemann sphere of two-state systems by Penrose. We regard Riemann platform as a Gauge Structure, in which the ζ function is treated as a wavefunction that satisfies gauge symmetry by performing gauge transformations. Some empirical issues are discussed. These issues are: Statistical significance; the measurement problem and the Yes/No measurement; the Dirac Delta-function; sample space and stochastic sampling.
The concept of phase-space amplitudes for systems with continuous degrees of freedom is generalized to finite-dimensional spin systems. Complex amplitudes are obtained on both a sphere and a finite lattice, in each case enabling a more fundamental description of pure spin states than that previously given by Wigner functions. In each case the Wigner function can be expressed as the star product of the amplitude and its conjugate, so providing a generalized Born interpretation of amplitudes that emphasizes their more fundamental status. The ordinary product of the amplitude and its conjugate produces a (generalized) spin Husimi function. The case of spin-12\frac{1}{2}21​ is treated in detail, and it is shown that phase-space amplitudes on the sphere transform correctly as spinors under rotations, despite their expression in terms of spherical harmonics. Spin amplitudes on a lattice are also found to transform as spinors. Applications are given to the phase space description of state superposition, and to the evolution in phase space of the state of a spin-12\frac{1}{2}21​ magnetic dipole in a time-dependent magnetic field.
Classical spinor mechanics reproduces the algebraic framework of quantum computation-SU (2) rotations, tensor products, and exchange couplings-while remaining confined to separable correlations. By following the shared geometry of oscillators, polarization optics, and two-level systems, one can trace a continuous correspondence between classical and quantum dynamics and identify the operational boundary where quantum advantage appears. That boundary lies not in the equations of motion but in how correlations are accessed: classical spinor systems explore Hilbert space sequentially, whereas quantum systems do so coherently and in parallel. The distinction concerns realization rather than formalism-quantum evolution admits entangled superpositions and interference within a single trajectory, a geometric extension that underlies the exponential efficiency of Shor's factoring algorithm.