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The Physics of Ettore Majorana

Book Description

Through just a handful of papers, Ettore Majorana left an indelible mark in the fields of physics, mathematics, computer science and even economics before his mysterious disappearance in 1938. It is only now that the importance of Majorana's work is being realised: Majorana fermions are intensely studied today, and his work on neutrino physics has provided possible explanations for the existence of dark matter. In this unique volume, Salvatore Esposito explores not only Majorana's known papers but, even more interestingly, unveils his unpublished works as well. These include powerful methods and results, ranging from the atomic two-centre problem, the ThomasÐFermi model and ferromagnetism to quasi-stationary states, n-component relativistic wave equations and quantum scalar electrodynamics. Featuring biographical notes and contributions from leading experts Evgeny Akhmedov and Frank Wilczek, this fascinating book will captivate graduate students and researchers interested in frontier science as well as in the history of science.

Table of Contents

  1. Coverpage
  2. Halftitle page
  3. Title page
  4. Copyright page
  5. Contents
  6. Acknowledgments
  7. Part I Introducing the character
    1. 1 Life and myth
      1. 1.1 Fortunes and misfortunes of a genius
      2. 1.2 Family and university training
      3. 1.3 Lone physicist in the Fermi group
      4. 1.4 Leipzig–Rome–Naples: the later years
    2. 2 The visible side
      1. 2.1 Ten papers depicting the future
      2. 2.2 Introducing the Dirac equation into atomic spectroscopy
      3. 2.3 Spontaneous ionization
        1. 2.3.1 Anomalous terms in helium
        2. 2.3.2 Incomplete P′ triplets
        3. 2.3.3 Majorana–Fano–Feshbach resonances
      4. 2.4 Chemical bonding
        1. 2.4.1 Helium molecular ion
        2. 2.4.2 Majorana structures
      5. 2.5 Non-adiabatic spin-flip
        1. 2.5.1 Majorana sphere and a general theorem
        2. 2.5.2 Landau–Zener probability formula
        3. 2.5.3 Majorana’s holes
        4. 2.5.4 Majorana–Brossel effect
      6. 2.6 Nuclear forces
        1. 2.6.1 The Heisenberg model of nuclear interactions
        2. 2.6.2 Majorana’s exchange mechanism
        3. 2.6.3 Thomas–Fermi formalism and Yukawa potential
      7. 2.7 Infinite-component equation
        1. 2.7.1 A successful relativistic wave equation
        2. 2.7.2 Majorana equation
        3. 2.7.3 Infinite-dimensional representations of the Lorentz group
        4. 2.7.4 A difficult problem for Pauli and Fierz
        5. 2.7.5 Further elaborations
      8. 2.8 Majorana neutrino theory
        1. 2.8.1 “Symmetric” Dirac equation
        2. 2.8.2 Neutrino–antineutrino identity
        3. 2.8.3 Racah and the neutrinoless double β-decay
        4. 2.8.4 Pontecorvo and the neutrino oscillations
        5. 2.8.5 Majorana fermions
      9. 2.9 Complex systems in physics and economics
        1. 2.9.1 Genesis of paper N.10
        2. 2.9.2 Statistical laws in social sciences
        3. 2.9.3 A sensational success in econophysics
  8. Part II Atomic physics
    1. 3 Two-electron problem
      1. 3.1 A long-lasting success for quantum mechanics
      2. 3.2 Known solutions to the helium atom problem
        1. 3.2.1 Perturbative calculations
        2. 3.2.2 Variational method I
        3. 3.2.3 Self-consistent field method
        4. 3.2.4 Slater’s refinement
        5. 3.2.5 Variational method II: Hylleraas variables
        6. 3.2.6 Helium-like ions
      3. 3.3 Majorana empirical relations
      4. 3.4 Helium wavefunctions and broad range estimates
      5. 3.5 Accurate numbers and a general theory
        1. 3.5.1 A simpler alternative to Hylleraas’s method
        2. 3.5.2 Majorana’s variant of the variational method
      6. 3.6 Conclusions
    2. 4 Thomas–Fermi model
      1. 4.1 Fermi universal potential
        1. 4.1.1 Thomas–Fermi equation
        2. 4.1.2 Numerical and approximate solutions
        3. 4.1.3 Mathematical properties
      2. 4.2 Majorana solution of the Thomas–Fermi equation
        1. 4.2.1 Transformation into an Abel equation
        2. 4.2.2 Analytic series solution
        3. 4.2.3 Numerical tables
      3. 4.3 Mathematical generalizations
        1. 4.3.1 Frobenius method
        2. 4.3.2 Scale-invariant differential equations
      4. 4.4 Physical applications
        1. 4.4.1 Modified Fermi potential for heavy atoms
        2. 4.4.2 Second approximation for the atomic potential
        3. 4.4.3 Atomic polarizability
        4. 4.4.4 Applications to molecules
      5. 4.5 Conclusions
  9. Part III Nuclear and statistical physics
    1. 5 Quasi-stationary nuclear states
      1. 5.1 Probing the atomic nucleus with α particles
      2. 5.2 Scattering of α particles on a radioactive nucleus
        1. 5.2.1 Quantum-mechanical theory
        2. 5.2.2 Thermodynamic approach
      3. 5.3 Transition probabilities of quasi-stationary states
        1. 5.3.1 Transitions from a discrete into a continuum state
        2. 5.3.2 Transitions into two continuous spectra
        3. 5.3.3 Transitions from a continuum state
      4. 5.4 Nuclear disintegration by α particles
        1. 5.4.1 Statement of the problem
        2. 5.4.2 The appropriate wavefunction
        3. 5.4.3 Cross section
      5. 5.5 Conclusions
    2. 6 Theory of ferromagnetism
      1. 6.1 Towards a statistical theory of ferromagnetism
        1. 6.1.1 Molecular fields
        2. 6.1.2 Heisenberg theory
        3. 6.1.3 Later refinements
      2. 6.2 Majorana statistical model
        1. 6.2.1 Distribution function
      3. 6.3 Solution of the model in the continuum limit
        1. 6.3.1 Partition function at finite temperature
        2. 6.3.2 Mean magnetization
      4. 6.4 Applications and further results
        1. 6.4.1 Particular ferromagnetic geometries
        2. 6.4.2 Critical temperature and dimensionality
      5. 6.5 Conclusions
  10. Part IV Relativistic fields and group theory
    1. 7 Groups and their applications to quantum mechanics
      1. 7.1 The “Gruppenpest” in quantum mechanics
      2. 7.2 Unitary transformations in two dimensions
        1. 7.2.1 D[sub(j)] representation and group generators
      3. 7.3 Three-dimensional rotations
        1. 7.3.1 Group generators
      4. 7.4 Application: the anomalous Zeeman effect
      5. 7.5 Lorentz group and its representations
        1. 7.5.1 n-dimensional Dirac matrices
        2. 7.5.2 Special case: maximum allowed p for fixed n
        3. 7.5.3 Non-Hermitian operators
        4. 7.5.4 Infinite-dimensional unitary representations
      6. 7.6 Conclusions
    2. 8 Dirac equations and some alternatives
      1. 8.1 Searching for an equation
        1. 8.1.1 Massive photons and the DKP algebra
        2. 8.1.2 Dirac–Fierz–Pauli formalism
        3. 8.1.3 General equations for arbitrary spin
      2. 8.2 Majorana n-component spinor equations
        1. 8.2.1 The 16-component equation for a two-particle system
        2. 8.2.2 Equation for a six-component spinor
        3. 8.2.3 Five-component equation
      3. 8.3 Parallel lives (and findings)
      4. 8.4 Conclusions
  11. Part V Quantum field theory
    1. 9 Scalar electrodynamics
      1. 9.1 Early quantum electrodynamics
        1. 9.1.1 Quantum field formalism
        2. 9.1.2 Particles and antiparticles
        3. 9.1.3 Pauli–Weisskopf theory
      2. 9.2 Majorana theory of scalar electrodynamics I
      3. 9.3 Majorana theory of scalar electrodynamics II
      4. 9.4 Application to the nuclear structure
      5. 9.5 Conclusions
    2. 10 Photons and electrons
      1. 10.1 Photon wave mechanics
        1. 10.1.1 Majorana–Oppenheimer formulation of electrodynamics
        2. 10.1.2 Lorentz-invariant wave theory
        3. 10.1.3 Two-component theory
        4. 10.1.4 Field quantization
      2. 10.2 Dynamical theory of electrons and holes
      3. 10.3 Conclusions
  12. Part VI Fundamental theories and other topics
    1. 11 A “path integral” approach to quantum mechanics
      1. 11.1 Dirac and Feynman’s mathematical approach
      2. 11.2 Majorana’s physical approach
      3. 11.3 Conclusions
    2. 12 Fundamental lengths and times
      1. 12.1 Introducing elementary space-time lengths
      2. 12.2 Quasi-Coulombian scattering
      3. 12.3 Intrinsic time delay and retarded electromagnetic fields
      4. 12.4 Conclusions
    3. 13 Majorana’s multifaceted life
      1. 13.1 Majorana as a student
        1. 13.1.1 Melting point shift due to a magnetic field
        2. 13.1.2 Determination of a function from its moments
        3. 13.1.3 WKB method for differential equations
      2. 13.2 Majorana as a phenomenologist: spontaneous and induced ionization of a hydrogen atom
        1. 13.2.1 Hydrogen atom placed in a high potential region
        2. 13.2.2 Ionization of a hydrogen-like atom in an electric field
      3. 13.3 Majorana as a theoretician: a unifying model for the fundamental constants
      4. 13.4 Majorana as a mathematician
        1. 13.4.1 Improper operators
        2. 13.4.2 Cubic symmetry
      5. 13.5 Majorana as a teacher
  13. Part VII Beyond Majorana
    1. 14 Majorana and condensed matter physics
      1. 14.1 Spin response and universal connection
      2. 14.2 Level crossing and generalized Laplace transform
      3. 14.3 Majorana fermions and Majorana mass: from neutrinos to electrons
        1. 14.3.1 Majorana’s equation
        2. 14.3.2 Analysis of Majorana neutrinos
        3. 14.3.3 Majorana mass
        4. 14.3.4 Majorana electrons
      4. 14.4 Majorinos
        1. 14.4.1 Kitaev chain
        2. 14.4.2 Junctions and the algebraic genesis of majorinos
        3. 14.4.3 Continuum majorinos
    2. 15 Majorana neutrinos and other Majorana particles: theory and experiment
      1. 15.1 Weyl, Dirac, and Majorana fermions
        1. 15.1.1 Particle−antiparticle conjugation
        2. 15.1.2 Dirac dynamics and the Majorana condition
        3. 15.1.3 Fermion mass terms and U(1) symmetries
        4. 15.1.4 Feynman rules for Majorana particles
      2. 15.2 C, P, CP, and CPT properties of Majorana fermions
      3. 15.3 Mixing and oscillations of Majorana neutrinos
        1. 15.3.1 Neutrinos with a Majorana mass term
        2. 15.3.2 General case of Dirac + Majorana mass term
        3. 15.3.3 Dirac and pseudo-Dirac neutrino limits in the D + M case
      4. 15.4 Seesaw mechanism of neutrino mass generation
      5. 15.5 Electromagnetic properties of Majorana neutrinos
      6. 15.6 Majorana particles in SUSY theories
      7. 15.7 Experimental searches for Majorana neutrinos and other Majorana particles
        1. 15.7.1 Neutrinoless double β-decay and related processes
        2. 15.7.2 Other lepton-number-violating processes
      8. 15.8 Baryogenesis through leptogenesis and Majorana neutrinos
      9. 15.9 Miscellaneous
      10. 15.10 Summary and conclusions
  14. Appendix Molecular bonding in quantum mechanics
    1. A.1 On the meaning of quantum state
    2. A.2 Symmetry properties of a system in classical and quantum mechanics
    3. A.3 Resonance forces between states that cannot be symmetrized for small perturbations and spectroscopic consequences. Theory of homopolar valence according to the method of bonding electrons. Properties of the symmetrized states that are not obtained from non-symmetrized ones with a weak perturbation
  15. References
  16. Author index
  17. Subject index