We derive the two-valuedness and the angular momentum of a spin-1/2 from a rotation of 3-dimensional surface of a sphere existing in extra 4-dimensional space other than normal 3-dimensional space, in this paper. We will derive the two-valuedness of the spin as follows. We introduce 3-dimensional surface of a sphere S3 existing in extra 4-dimensional space (W, X, Y, Z) other than normal 3-dimensional space (x, y, z). We interpret the angle of rotation of the 3-sphere S3 as the phase of a wave function. We interpret the 3-sphere S3 as the absolute value of a wave function. We can express 3-sphere as the manifold with a constant sum of squares of the radius of two circles. When one circle's radius becomes the maximum, the other circle's radius becomes zero. Therefore, we can turn the circle inside out naturally. If we combine the circle turned inside out with the original circle, the manifold becomes a torus with a node. If we rotate the node of the torus by 360 degrees, we can turn the torus inside out. If we rotate the node of the torus 720 degrees, we can return the torus to the original state. This property is consistent with the property of the spin. We derive the angular momentum of the spin as follows. We make 3-dimensional solid sphere by removing one point from 3-sphere S3. On the other hand, we can make boundary like a 3-sphere S3 by removing one point from normal 3-dimensional space (x, y, z). We combine the boundaries of them. By repeating this, we can construct 3-dimensional helical space. The angle of rotation of the 3-sphere S3 is the angle of rotation of 3-dimensional helical space. On the other hand, we can interpret the angle of the rotation in the helical space as the coordinates (x, y, z) of the normal 3-dimensional space. Therefore, we can interpret the angular momentum of the 3-sphere S3 as the angular momentum of normal space.
We derive the two-valuedness and the angular momentum of a spin-1/2 from a rotation of 3-dimensional surface of a sphere existing in extra 4-dimensional space other than normal 3-dimensional space, in this paper. We will derive the two-valuedness of the spin as follows. We introduce 3-dimensional surface of a sphere S3 existing in extra 4-dimensional space (W, X, Y, Z) other than normal 3-dimensional space (x, y, z). We interpret the angle of rotation of the 3-sphere S3 as the phase of a wave function. We interpret the 3-sphere S3 as the absolute value of a wave function. We can express 3-sphere as the manifold with a constant sum of squares of the radius of two circles. When one circle's radius becomes the maximum, the other circle's radius becomes zero. Therefore, we can turn the circle inside out naturally. If we combine the circle turned inside out with the original circle, the manifold becomes a torus with a node. If we rotate the node of the torus by 360 degrees, we can turn the torus inside out. If we rotate the node of the torus 720 degrees, we can return the torus to the original state. This property is consistent with the property of the spin. We derive the angular momentum of the spin as follows. We make 3-dimensional solid sphere by removing one point from 3-sphere S3. On the other hand, we can make boundary like a 3-sphere S3 by removing one point from normal 3-dimensional space (x, y, z). We combine the boundaries of them. By repeating this, we can construct 3-dimensional helical space. The angle of rotation of the 3-sphere S3 is the angle of rotation of 3-dimensional helical space. On the other hand, we can interpret the angle of the rotation in the helical space as the coordinates (x, y, z) of the normal 3-dimensional space. Therefore, we can interpret the angular momentum of the 3-sphere S3 as the angular momentum of normal space.
The bipolar coordinates of a point may be loosely defined as the distances of the point from each of two reference points. Consistent definitions are developed for the bipolar coordinates of a point on a sphere with the reference points at opposite ends of a diameter, and it is shown that the bipolar coordinates so defined transform like Pauli spin functions under rotations. The electron is visualized as a point mass with charge e moving on the sphere with its motion expressed in terms of its bipolar coordinates. It is assumed observable only at two diametrically opposite points of its orbit. This picture allows an exact physical interpretation of the spin variable and the Pauli spin operators. The magnetic moment associated with the motion is $\frac{e\ensuremath{\hbar}}{2mc}$; the radius of the electron is taken as the radius of the sphere which is $\frac{\ensuremath{\hbar}}{2mc}$; the wave-length associated with the spinning motion is mch. Two related functions must be introduced to take care of reflections and these together with the two original functions transform like Dirac spin functions under rotations, reflections, and Lorentz transformations.
[64 Foundational] The 0-Sphere Model treats the electron not as a structureless point but as a small internal geometric region whose phase trembles in time. This paper asks where one of the most stubborn features of that electron comes from: the fact that its spin has exactly two values, and that the particle must be turned through two full revolutions — not one — before it returns to itself. The proposed answer is that it comes from a single, very ordinary mathematical operation: taking a square root. — The idea in one breath — Picture the electron’s internal world as the surface of a four-dimensional ball — a closed, finite space with no edges. On such a closed space there is a natural “averaging” operation, and its allowed values come out evenly spaced and labelled by whole numbers (0, 1, 2, …). These whole numbers describe the orbital, direction-like part of the internal motion. Nothing surprising yet. The surprise appears when you take the square root of that averaging operation. A square root is always defined only up to a sign — √4 is both +2 and −2 — so its allowed values arrive in matched positive–negative pairs. Those two signs are exactly the two spin states: up and down. The same square root, applied historically by Dirac in 1928 to the equation of a relativistic particle, is what gave physics the electron’s spin in the first place. This paper’s claim is that one and the same operation shows up at two stages of the model and reduces the two-valuedness of spin to plain geometry. — Why this matters — Two structural facts fall out for free, where before they had to be put in by hand: Why the values are discrete. Because the internal space is closed and finite (compact), the allowed values can only be separated, whole-number-labelled steps — never a continuous smear. Why there are exactly two of them. Because the square-root step inevitably splits each value into a + and a − partner. Two signs, two spin sheets. The two sheets join smoothly into the double cover of the rotation group — the precise mathematical reason a spin one-half object needs two full turns to come home. In plain terms: walk once around the rotations and you land on the other sheet; walk around twice and you finally return to where you started. — What is new, and what is not — The square-root operation itself is classical and is claimed as no one’s discovery — it has been textbook material since 1928. What is new is the recognition that the very same operation appears at two stages of the model, and that together they reduce the two-valuedness of spin to the geometry of a closed surface. This replaces a picture of two states being secretly present at once with a cleaner, time-sequential one: the electron visits up, then down, in turn, rather than holding both simultaneously. — Key Contributions — Spin’s two values as a square root. The two spin sheets are the matched plus/minus values that any square root must take — here, the square root of the natural averaging operator on the closed internal space. Closedness gives discreteness. The fact that the allowed values are separated, whole-number steps follows from the internal space being compact (finite and edgeless). One operation across two stages. The same square root acts both on the geometric internal stage (giving a clean, discrete set of states) and on the historical relativistic stage (the one Dirac used in 1928), the two being linked by a standard mathematical bridge. The double cover as a branch structure. The two sheets are the two faces of the familiar “two-storey” surface that the square-root function naturally lives on; their joining is the rotation group’s double cover — the two-valuedness of the square root made visible. A firmer footing for earlier results. The half-angle and g = 2 results obtained earlier from the fiber-bundle picture are placed on an operator-theoretic footing. A correction that keeps the good part. The earliest spin paper’s reading of four simultaneous states is replaced by a time-sequential one, while its elegant two-storey (Riemann-surface) picture is kept — now on firmer ground. — Series Position — This paper continues the half-angle internal-energy programme and gives an operator-theoretic foundation to the geometric g = 2 result of Paper #48, consolidates the threads of Paper #50 and Paper #57, and corrects the four-state reading of Paper #3 while retaining its two-sheeted Riemann surface. Through a standard mathematical bridge it points toward the metric-emergence programme of Paper #51 and Paper #63 as the destination it does not itself develop. It belongs to the series founded by Paper #1. — References (Hanamura papers) — # Title (abbreviated) DOI 1 A Model of an Electron Including Two Perfect Black Bodies 10.5281/zenodo.16759284 3 Distinguishing Electron Spin via Riemann Surface Guidance 10.5281/zenodo.17759634 10 Redefining Electron Spin and the Anomalous Magnetic Moment 10.5281/zenodo.17764997 48 Geometric Origin of g=2: U(1) Fiber Bundle 10.5281/zenodo.19227518 50 Rotation from Scalar Oscillation: Photon-Sphere Angular Momentum 10.5281/zenodo.19482145 51 Helical Trajectory, Line Integrals, Emergence of the Metric (v2) 10.5281/zenodo.20388056 57 Spherical Harmonic Imprinting and Gravitational Coupling 10.5281/zenodo.19932176 59 Proton-Trapped Electron and Spin-2 Gravitational Radiation 10.5281/zenodo.19932907 60 The Berry–Synge–Bonnet–Myers Triangle 10.5281/zenodo.19932922 63 From Line Integral to Covariant Derivative: A Reader's Map 10.5281/zenodo.20767589 — Series Context — The 0-Sphere Model is an ongoing research programme (2018–present) that derives spin, anomalous magnetic moment, Zitterbewegung, and emergent spacetime from the geometry and thermodynamics of a two-kernel electron model. All papers in the series are archived on Zenodo: Zenodo search: Hanamura, Satoshi
In four-dimensional or higher-dimensional space, spin angular momentum is not known well. For instance, we are not sure how many components it has in four-dimensional space. In this paper, the spin properties are theoretically derived and investigated in general-dimensional space. Before studying spin, we review the orbital angular momentum properties first, because they are theoretically well known even in higher-dimensional space. Then, for the orbital angular momentum, we make a special but natural assumption, and see that it leads a quantity similar to spin or Pauli matrices. It is remarkable that all the spin properties are derived from the one and only one assumption. Consistency of our consideration is checked by calculating the four-dimensional version of spin spherical harmonics and its eigenvalues. This theory may enable us to understand the double degeneracy of spin-1/2 particles without bringing in something called spin.
Most students of science and engineering know that every elementary particle, such as electrons, neutrons, photons, neutrinos, etc., has a quantum mechanical property called “spin” which can be measured (perhaps not easily, but at least in principle) and has a quantized value, including zero. The vast majority of these students mentally visualize spin as the angular momentum associated with the elementary particle spinning or rotating about its own axis (like a top or a planetary object). This mental picture, although convenient and comforting, is actually somewhat crude and certainly incomplete. Landau and Lifshitz, in their classic textbook on quantum mechanics [1], wrote “[the spin] property of elementary particles is peculiar to quantum theory. [It] has no classical interpretation... It would be wholly meaningless to imagine the ‘intrinsic’ angular momentum of an elementary particle as being the result of its rotation about its own axis.” The simplistic notion of self-rotation about an axis, shown in Fig. 1.1, can- not explain many features of spin, such as why its magnitude cannot assume continuous values and why it is quantized to certain specific values. It also causes serious problems if taken too literally. As we will see later (Problem 1.2), if we think of an electron as a solid sphere of radius equal to the Lorentz radius e2/(4πǫ0m0c 2) (where e is the electron’s charge, m0 is the mass, c is the speed of light in vacuum, and ǫ0 is the dielectric constant of vacuum), then the velocity on the surface of a rotating electron would have to be many times the velocity of light in vacuum if such a rotation were to generate an angular momentum equal to the electron’s spin. Obviously that would not be permitted by the theory of relativity. Indeed, a deep understanding of quantum mechanics is required to understand how the spin property comes about. Its origin is in relativistic quantum mechanics and really was first appreciated by Paul Andrew Maurice Dirac when he derived the Dirac equation which is the cornerstone of relativistic quantum mechanics. Richard Feynman, noted for his distaste for mysticism in physics, wrote about the notion of “spin”: “It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is down deep in relativistic quantum mechanics.
In Section 3 we saw how to combine the intrinsic spin s and the orbital angular momentum L to form eigenstates of J 2 where J = L + s . The coefficients in ( 3.30 ) and ( 3.31 ) are special cases of a general class of transformation coefficients. The states on the r.h.s. of ( 3.30 ) and ( 3.31 ) are eigenstates of s 2 , s z , L 2 , L z . On the left hand side there are eigenstates of s 2 , L 2 , J 2 and J z . In general, we may start with any two sets ψ j 1 m 1 , ψ j 2 m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq354.tif"/> of two different systems. The spins j 1 and j 2 may be either orbital angular momenta or total spins of two particles, two intrinsic spins or one orbital angular momentum and one intrinsic spin. The latter case was considered in Section 3. In other cases j 1 may be the rank of an irreducible tensor operator and j 2 the angular momentum of a wave function or both j 1 and j 2 may be the ranks of two irreducible tensor operators. To simplify the notation we will use, for the transformation of the sets ψ j 1 m 1 , ψ j 2 m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq355.tif"/> under infinitesimal rotations, the expressions ( 6.20 ). This will not detract from the applicability of the following results to other tensorial sets. As pointed out several times above, the following mathematical derivations have general validity and are independent of the physical nature of ψ j 1 m 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq356.tif"/> and ψ j 2 m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq357.tif"/> . They are due only to the transformation properties of the two sets under rotations.
We study quantum theory as an example of entropic inference. Our goal is to remove conceptual difficulties that arise in quantum mechanics. Since probability is a common feature of quantum theory and of any inference problem, we briefly introduce probability theory and the entropic methods to update probabilities when new information becomes available. Nelson's stochastic mechanics and Caticha's derivation of quantum theory are discussed in the subsequent chapters. Our first goal is to understand momentum and angular momentum within an entropic dynamics framework and to derive the corresponding uncertainty relations. In this framework momentum is an epistemic concept – it is not an attribute of the particle but of the probability distributions. We also show that the Heisenberg's uncertainty relation is an osmotic effect. Next we explore the entropic analog of angular momentum. Just like linear momentum, angular momentum is also expressed in purely informational terms. We then extend entropic dynamics to curved spaces. An important new feature is that the displacement of a particle does not transform like a vector. It involves second order terms that account for the effects of curvature . This leads to a modified Schrödinger equation for curved spaces that also take into account the curvature effects. We also derive Schrodinger equation for a charged particle interacting with external electromagnetic field on general Riemannian manifolds. Finally we develop the entropic dynamics of a particle of spin 1/2. The particle is modeled as a rigid point rotator interacting with an external EM field. The configuration space of such a rotator is R 3 × S3 (S 3 is the 3-sphere). The model describes the regular representation of SU(2) which includes all the irreducible representations (spin 0, 1/2, 1, 3/2,...) including spin 1/2.