guy duhaut
Born in 1948 into a farming family in a small village in eastern France, Guy moved to Nancy to attend middle and high school. He went on to study mathematics at the University of Nancy II. He married Marie-Laure Lamouroux, a fellow mathematics student, and the couple spent two years in Tunisia, where they taught at the Faculty of Mathematics. After returning to Nancy, Guy became a professor at Nancy II University. He taught a variety of courses and self-published several articles and books on mathematical physics. His two children, Sébastien and Noëmie— to whom some of these works are dedicated— scanned and published them after Guy’s death in 2024.
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Assuming an isotropic distribution of magnetic moment orientations and a Maxwellian velocity distribution, the resulting density profile on the detection screen is obtained through integration over all possible velocities. The derived distribution exhibits two symmetric maxima, corresponding to the two observed impact bands in the original 1922 experiment. This result provides a semi-classical explanation of the beam splitting, attributing it to the statistical distribution of magnetic moment orientations rather than discrete spin states.
The second section develops the theory of spherical harmonics as eigenfunctions of the Laplacian on the unit sphere, establishing their completeness and orthonormality. These functions are shown to generalize Fourier series to the sphere and are used to expand arbitrary functions in terms of angular modes.
The third section introduces Bessel functions, arising from the radial part of the Helmholtz equation in cylindrical and spherical coordinates. Their series representations, recurrence relations, and integral forms are discussed, along with their role in wave propagation and scattering problems.
Finally, the article explores the confluent hypergeometric function, its differential equation, series expansion, integral representations, and special cases such as Laguerre polynomials. These functions are essential in quantum mechanics and other areas involving radial equations.
Overall, the paper provides a unified framework for understanding classical special functions and their role in solving partial differential equations in physics.
L are vector spherical harmonics, constructed from classical scalar spherical harmonics.
The paper further develops a basis of vector functions on the sphere using these harmonics and applies the formalism to quantum wave equations. Specifically, it incorporates electromagnetic corrections from Pauli and Dirac into the Klein-Gordon equation using the van der Waerden–Kramers method, with a return to two-component wave functions as suggested by Weyl. The wave functions are valued in the quaternionic space and are represented via 2×2 complex matrices.
Finally, the study explores the fine structure of the quantum system by solving the second-order wave equation under the influence of electromagnetic potentials. The spectral properties of the associated operators are analyzed, leading to a refined understanding of the energy levels and their dependence on angular momentum and spin interactions.
Spontaneous Emission: The first section revisits classical electromagnetic radiation theory and adapts it to quantum currents derived from Dirac spinors. It derives expressions for radiated energy and transition intensities, including dipole approximations and spherical harmonics integrals. Applications to Lyman and Balmer transitions are discussed, with both relativistic and non-relativistic radial approximations.
Time-Dependent Electromagnetic Perturbations: The second part explores quantum transitions induced by time-varying electromagnetic fields, such as monochromatic waves and rapidly decaying pulses. Using perturbation theory and secular approximation, it analyzes transition probabilities, energy absorption, and emission. The Einstein A and B coefficients are derived, leading to the Planck radiation law through statistical equilibrium arguments.
Constant Magnetic Perturbation (Zeeman Effect): The final section examines the effect of a weak constant magnetic field on atomic energy levels using the Dirac equation. It includes the splitting of degenerate levels, secular determinant analysis, and the resulting spectral line shifts and intensities. Detailed examples, such as the 1S₁/₂ → 2P₁/₂ and 1S₁/₂ → 2P₃/₂ transitions, are provided with polarization and observational implications.
The document integrates advanced mathematical tools such as Clifford algebras, spherical spinors, and perturbation theory, offering a rigorous yet accessible framework for understanding radiation phenomena in quantum mechanics.
- Neutrinos and antineutrinos are described using helicized neutral wavefunctions, solutions to the Dirac equation.
- In beta decay (β-decay), the emitted neutral particle accompanying the electron is shown to be a right-handed neutrino.
- This finding challenges the “algebraic conservation of fermions” principle proposed by Reines and Cowan in 1956.
- The article analyzes the leptonic emission density and shows how it differs depending on whether a neutrino or antineutrino is involved.
- Experimental results, particularly from the Wu-Ambler-Hayward-Hoppes-Hudson experiment, support the conclusion that:
- The neutrino has right-handed helicity.
- The antineutrino has left-handed helicity.
Key contributions include:
- **Clifford-Based PCT Symmetries**: The P, C, and T transformations are constructed using Clifford conjugation and automorphisms, providing a unified and intrinsic algebraic description of discrete symmetries.
- **Lorentz Group and Spinor Representations**: The article details the spinorial action of the Lorentz group via the spin group and how it acts on wavefunctions and hermitian forms.
- **Charge Neutrality Conditions**: The framework distinguishes between wavefunctions of positive and negative charges and defines conditions for neutrality, showing how these are preserved under PCT transformations.
- **Weak Interaction Hamiltonian**: Using matrix-valued wavefunctions, the Hamiltonian for beta decay is constructed, incorporating both vector and axial-vector currents.
- **Leptonic Emission Densities**: The article derives expressions for lepton emission densities in beta decay, including cases with polarized and unpolarized nuclei, and connects them to experimental observables like angular asymmetries and correlation coefficients.
This work bridges abstract algebraic structures with physical observables in weak interactions, offering a mathematically elegant and physically insightful treatment of symmetry and charge in quantum field theory.
Key contributions include:
- **Charge-Invariant Formulation**: The interaction is expressed using a basis of the general linear algebra gl(2) instead of the usual su(2), allowing a more general treatment of charge symmetry.
- **Dirac Decomposition**: The Dirac operator is used to derive field equations for both vector and pseudo-scalar mesons, leading to analogues of Maxwell’s equations for strong interactions.
- **Meson Potentials and Fields**: The article derives explicit forms for meson potentials and their associated fields, distinguishing between vector and pseudo-scalar cases, and shows how these fields satisfy modified Dirac-type equations.
- **Hamiltonian Structure**: The Hamiltonians for free and interacting meson-nucleon systems are constructed, including terms that arise due to non-zero trace components in the Clifford algebra representation.
- **Charge Conservation and Gauge Conditions**: The work includes a detailed treatment of current conservation and gauge fixing (e.g., Lorenz gauge) in the context of strong interactions.
This approach provides a unified algebraic framework that parallels the treatment of electromagnetism but is adapted to the non-Abelian and short-range nature of the strong force.
Key contributions include:
- **Quaternionic Reformulation**: Vector calculus is expressed using quaternions, allowing a more unified treatment of rotations and spinorial actions.
- **Spherical Spinors**: These are introduced as quaternion-valued eigenfunctions of the angular momentum operator L. They generalize Pauli spinors and are used to expand functions on the sphere.
- **Dirac Equation**: The Dirac operator is constructed in a Clifford algebra framework. Plane wave solutions and their helicities are analyzed, and the equation is solved in the presence of a Coulomb potential using spherical spinor expansions.
- **Fine Structure and Zeeman Effect**: The fine structure of hydrogen-like atoms is derived without introducing imaginary units to make radial functions real. The Zeeman effect is studied in both weak and strong magnetic fields, recovering known results like the Paschen–Back and Lorentz triplets.
- **Transition Integrals**: The article provides detailed computations of radial and spherical integrals for electromagnetic transitions, including selection rules and approximations up to second order in perturbation theory.
The work is mathematically rigorous and bridges abstract algebraic structures with concrete physical applications, particularly in atomic and quantum field theory.
### **Chapter Summaries**
#### **I. Opérateurs de Dirac sur SO(3)**
- **Forme de Killing**: Defines the Killing form on so(3) and introduces the Clifford application to quaternions.
- **Action de SO(3)**: Establishes the action of SO(3) on quaternions and differentiable applications.
- **Invariant Operators**: Introduces invariant differential operators D and G, and their spectral properties.
#### **II. Spectral Analysis**
- **Spectral Properties**: Analyzes the spectra of operators D and G, showing their values and associated subspaces.
- **Irreducible Representations**: Discusses the irreducibility of representations within the defined subspaces.
### **Bibliography**
Includes references to foundational works in quantum mechanics, group theory, and representations, such as those by Hamilton, Darwin, Pauli, Dirac, Weyl, Casimir, Van der Waerden, Cartan, Chevalley, Gelfand, Naimark, Vilenkin, and Godement.
This document provides a detailed analysis of quaternionic representations and their applications in mathematical physics, offering insights into the structure and behavior of spinorial operators and their spectral properties.
Chapter Summaries
I. Spineurs
Quaternions et SU(2): Discusses the quaternion algebra and its relation to SU(2), including the homomorphism to SO(3).
Action de SO(3) sur les spineurs: Defines the action of SO(3) on spinors, establishing properties and invariance.
II. Opérateurs sur la sphère
Algèbres de Lie: Explores the Lie algebras su(2) and so(3), their bases, and associated subgroups.
Opérateur L: Introduces the operator L, its invariance, and its decomposition in spherical coordinates.
III. Polynômes harmoniques
Quaternisation: Examines the space of homogeneous harmonic polynomials valued in quaternions.
Décomposition: Analyzes the spectral decomposition of operator L and the associated eigenvalues and eigenfunctions.
IV. Representations
Rappels linéaires: Reviews linear representations and their properties.
Spinorielles: Focuses on spinorial representations, demonstrating their irreducibility.
Développement des spineurs sur la sphère: Discusses the development of spinors on the sphere and their orthogonal decomposition.
Bibliography
Includes references to foundational works in quantum mechanics, group theory, and representations, such as those by Heisenberg, Dirac, Pauli, Weyl, and others.
It is a mathematically rich and pedagogically structured treatise that explores the group SO(3) and its connections to geometry, algebra, and physics. Drawing from his university lectures in the early 1980s, Duhaut presents a unified framework that integrates Clifford algebras, Lie groups, differential operators, and representation theory. The work emphasizes the role of symmetry and invariance in mathematical physics, particularly in the context of spin, quantum mechanics, and integration on manifolds. Rather than offering a purely theoretical exposition, the text is designed to be accessible and applicable, with a focus on concrete operators (like Dirac and Laplacian), invariant measures (like Haar), and the structure of SO(3) as a model for physical systems with rotational symmetry.