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Editorial Note

 Ignazio Licata

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MAJORANA IMPACT ON CONTEMPORARY PHYSICS

1

The Scientific Work Of Ettore Majorana: An Introduction

 Erasmo Recami

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A Brief bibliography of the scientific work of Ettore Majorana has been discussed.

2

 On the Hamiltonian Form of Generalized Dirac Equation for Fermions with Two Mass States

 Sergey. I. Kruglov

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Dynamical and non-dynamical components of the 20-component wave function are separated in the generalized Dirac equation of the first order, describing fermions with spin 1/2 and two mass states. After the exclusion of the non-dynamical components, we obtain the Hamiltonian Form of equations. Minimal and non-minimal electromagnetic interactions of particles are considered here.

3

Majorana Equation and exotics: Higher Derivative Models, Anyons and Noncommutative Geometry

 Mikhail S. Plyushchay

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In 1932 Ettore Majorana proposed an infinite-component relativistic wave equation for  particles of arbitrary integer and half-integer spin. In the late 80s and early 90s it was found that the higher-derivative geometric particle models underlie the Majorana equation, and that its (2+1)-dimensional analogue provides with a natural basis for the description of relativistic anyons. We review these aspects and discuss the relationship of the equation to the exotic planar Galilei symmetry and noncommutative geometry. We also point out the relation of some Abelian gauge field theories with Chern-Simons terms to the Landau problem in the noncommutative plane from the perspective of the Majorana equation.

4

Wave Equations, Renormalization and Meaning of the Planck's Mass: Some Qualitative Considerations

 Leonardo Chiatti

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The five-dimensional version of the quantum relativistic Klein-Gordon wave equation is assumed to be a more fundamental description for the dynamics of the single particle without spin. The meaning of the renormalization procedure in QFT and the Planck's mass one are briefly discussed from this point of view.

5

 Nonlinear Field Equations and Solitons as Particles

 Attilio Maccari

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Profound advances have recently interested nonlinear field theories and their exact or approximate solutions. We review the last results and point out some important unresolved questions. It is well known that quantum field theories are based upon Fourier series and the identification of plane waves with free particles. On the contrary, nonlinear field theories admit the existence of coherent solutions (dromions, solitons and so on). Moreover, one can construct lower dimensional chaotic patterns, periodic-chaotic patterns, chaotic soliton and dromion patterns. In a similar way, fractal dromion and lump patterns as well as stochastic fractal excitations can appear in the solution. We discuss in some detail a nonlinear Dirac field and a spontaneous symmetry breaking model that are reduced by means of the asymptotic perturbation method to a system of nonlinear evolution equations integrable via an appropriate change of variables. Their coherent, chaotic and fractal solutions are examined in some detail. Finally, we consider the possible identification of some types of coherent solutions with extended particles along the de Broglie-Bohm theory. However, the last findings suggest an inadequacy of the particle concept that appears only as a particular case of nonlinear field theories excitations.

6

 The Quantum Character of Physical Fields.

Foundations of Field Theories

 Ludmila. I. Petrova

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The existing field theories are based on the properties of closed exterior forms, which are invariant ones and correspond to conservation laws for physical fields. Hence, to understand the foundations of field theories and their unity, one has to know how such closed exterior forms are obtained. In the present paper it is shown that closed exterior forms corresponding to field theories are obtained from the equations modeling conservation (balance) laws for material media. It has been developed the evolutionary method that enables one to describe the process of obtaining closed exterior forms. The process of obtaining closed exterior forms discloses the mechanism of evolutionary processes in material media and shows that material media generate, discretely, the physical structures, from which the physical fields are formed. This justifies the quantum character of field theories. On the other hand, this process demonstrates the connection between field theories and the equations for material media and points to the fact that the foundations of field theories must be conditioned by the properties of material media. It is shown that the external and internal symmetries of field theories are conditioned by the degrees of freedom of material media. The classification parameter of physical fields and interactions, that is, the parameter of the unified field theory, is connected with the number of noncommutative balance conservation laws for material media.

7

Relativistic Causality and

 Quasi -Orthomodular Algebras

 Renato.Nobili

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The concept of fractionability or decomposability in parts of a physical system has its mathematical counterpart in the lattice--theoretic concept of orthomodularity. Systems with a finite number of degrees of freedom can be decomposed in different ways, corresponding to different groupings of the degrees of freedom. The orthomodular structure of these simple systems is trivially manifest. The problem then arises as to whether the same property is shared by physical systems with an infinite number of degrees of freedom, in particular by the quantum relativistic ones. The latter case was approached several years ago by Haag and Schroer (1962; Haag, 1992) who started from noting that the causally complete sets of Minkowski spacetime form an orthomodular lattice and posed the question of whether the subalgebras of local observables, with topological supports on such subsets, form themselves a corresponding orthomodular lattice. Were it so, the way would be paved to interpreting spacetime as an intrinsic property of a local quantum field algebra. Surprisingly enough, however, the hoped property does not hold for local algebras of free fields with superselection rules. The possibility seems to be instead open if the local currents that govern the superselection rules are driven by gauge fields. Thus, in the framework of local quantum physics, the request for algebraic orthomodularity seems to imply physical interactions! Despite its charm, however, such a request appears plagued by ambiguities and criticities that make of it an ill--posed problem. The proposers themselves, indeed, concluded that the orthomodular correspondence hypothesis is too strong for having a chance of being practicable. Thus, neither the idea was taken seriously by the proposers nor further investigated by others up to a reasonable degree of clarification. This paper is an attempt to re--formulate and well--pose the problem. It will be shown that the idea is viable provided that the algebra of local observables: (1) is considered all over the whole range of its irreducible representations; (2) is widened with the addition of the elements of a suitable intertwining group of automorphisms; (3) the orthomodular correspondence requirement is modified to an extent sufficient to impart a natural topological structure to the intertwined algebra of observables so obtained. A novel scenario then emerges in which local quantum physics appears to provide a general framework for non--perturbative quantum field dynamics.

8

 Lorentz Invariant Majorana Formulation of Electrodynamics  in the Clifford Algebra Formalism

 Tomislav Ivezic

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 In this paper we present a new geometric formulation (Clifford algebra formalism) of the field equations, which is independent of the reference frame and of the chosen system of coordinates in it. This formulation deals with the complex 1-vector \Psi =E-icB (i is the unit imaginary), which is four-dimensional (4D) geometric generalization of Majorana's complex 3D quantity \Psi }=E-icB. When the sources are absent the field equations with the complex  \Psi  become Dirac-like relativistic wave equations for the free photon. In the frame of ``fiducial'' observers (the observers who measure fields are at rest) and in the standard basis the component form of the field equations with 4D \Psi  reproduces the component form of Majorana-Maxwell equations with 3D field \Psi . The important differences between the approach with the 4D \Psi  and that one with the 3D \Psi are discussed.

9

" Anticoherent " Spin States via the Majorana Representation

 Jason Zimba

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In this article we define and exhibit '' anticoherent" spin states, which are in a sense '' the opposite" of  the familiar coherent spin states.  Since the familiar coherent states are as "classical" as spin states can be, the anticoherent states may turn out to be better candidates for applications involving non-classical behaviors such as quantum entanglement. Thanks to the Majorana representation of spinors as 2s-tuples of points on the Riemann sphere, classes of anticoherent states are easy to find; the development of such examples also leads us into some curious geometry involving the perfect solids.

10

Stretching the Electron as Far as it Will Go

 G. W. Semenoff and P. Sodano

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 Effects associated with the existence of isolated zero modes of Majorana fermions are discussed. It is argued that the quantization of this system necessarily contains highly extended quantum states and that populating and depopulating such states by interacting with the quantum system leads to long-ranged teleportation-like processes. Also leads to spontaneous violation of fermion parity symmetry. A quasi-realistic model consisting of a quantum wire embedded in a p-wave superconductor is discussed as an explicit example of a physical system with an isolated Majorana zero mode.

11

Why do Majorana Neutrinos Run Faster than Dirac Neutrinos?

 Zhi-zhong Xing and He Zhang

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The \tau-lepton dominance in the one-loop renormalization-group equations (RGEs) of neutrinos sets a cute criterion to parametrize the 3x3 lepton flavor mixing matrix U: its elements U_{3i} (for i=1,2,3) should be as simple as possible. Such a novel parametrization is different from the ``standard" one used in the literature and can lead to greatly simplified RGEs for three mixing angles and the physical CP-violating phase(s), no matter whether neutrinos are Dirac or Majorana particles. We show that the RGEs of Dirac neutrinos are not identical with those of Majorana neutrinos even if two Majorana CP-violating phases vanish. As the latter can keep vanishing from the electroweak scale to the typical seesaw scale, it makes sense to explore the similarities and differences between the RGE running effects of Dirac and Majorana neutrinos. We conclude that Majorana neutrinos are in general expected to run faster (i.e., more significantly) than Dirac neutrinos.

12

Universe Without Singularities

A Group Approach to De Sitter Cosmology

 Ignazio Licata

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In the last years the traditional scenario of ``Big Bang'' has been deeply modified by the study of the quantum features of the Universe evolution, proposing again the problem of using ``local'' physical laws on cosmic scale, with particular regard to the cosmological constant role. The ``group extention'' method shows that the De Sitter group univocally generalizes the Poincaré group, formally justifies the cosmological constant use and suggests a new interpretation for Hartle-Hawking boundary conditions in Quantum Cosmology.

13

Majorana and the Investigation of  Infrared Spectra of Ammonia

 Elisabetta. Di Grezia

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An account is given on the first studies on the physics of ammonia, focusing on the infrared spectra of that molecule. Relevant contributions from several authors, in the years until 1932, are pointed out, discussing also an unknown study by E.Majorana on this topic.

14

Exact Solution of Majorana Equation via Heaviside Operational Ansatz

 Valentino A. Simpao

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In context of a transformation between Majorana and Dirac wavefunctions, it suffices to solve the related interactive Dirac problem and then apply the transformation of variables on the Dirac wavefunction in order to obtain the Majorana wavefunction of the given Majorana equation. Clearly, this connection between solutions continues to hold if the free Majorana and Dirac equations are each coupled to an external gauge field [e.g., Electromagnetism] via the minimum coupling prescription. Applying the formal solution scheme Heaviside Operational Ansatz[heretofore, HOA] put forward in [ EJTP 1 (2004), 10-16], provides an exact quadrature solution for the massive minimum-coupled Majorana equation in terms of the solution of the corresponding massive minimum-coupled Dirac equation.

15

A Logical Analysis of Majorana’s Papers on Theoretical Physics

 A. Drago and S. Esposito

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 We study two celebrated Majorana's papers through a method of investigation which relies upon the recently recognized distinction between classical logic and several kinds of non-classical logics, i.e. the failure of the double negation law. This law fails when a double negated sentence is not equivalent to the corresponding positive sentence, owing to the lack of scientific evidence of the latter one. All recognized double negated sentences inside the text of each paper are listed; the mere sequence of such sentences giving the logical thread of Majorana's arguing. This one is recognized to be of the Lagrangian kind, which mixes logical arguing and mathematical calculation; i.e. the author puts a fundamental problem which is solved by anticipating the mathematical hypothesis able to solve it, and then by drawing from this hypothesis the mathematical consequences in order to reach to desired result.  Furthermore the rethoric of presentation used by Majorana results to be a juridical one, owing to his style of presenting the laws to which an ideal theoretical physicist has to conform his mind in order to solve the problem at issue.

16

Four Variations on Theoretical Physics by Ettore Majorana

 Salvatore. Esposito

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An account is given of some topical unpublished work by Ettore Majorana, revealing his very deep intuitions and skillfulness in Theoretical Physics. The relevance of the quite unknown results obtained by him is pointed out as well.

17

 The Majorana Oscillator

 Eliano Pessa

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At present the expression ‘Majorana oscillator’ does not appear to be in use in theoretical physics. However, the author of this paper heard it in the Seventies, during private conversations with the late Prof. B.Touschek. This little contribution tries to explore the possible meanings of this expression and introduces a new field equation, generalizing the one already introduced by Majorana himself, which could describe a hypothetical ‘Majorana oscillator’.

18

Scattering of an \alpha Particle by a Radioactive Nucleus

Unpublished 1928

 Ettore Majorana

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 In the following we reproduce, translated into English, a section of Volumetto II, a notebook written by Majorana in 1928 when he was still a Physics student at the University of Rome (see S. Esposito, E. Majorana jr, A. van der Merwe and E. Recami (eds.)  Ettore Majorana: Notes on Theoretical Physics, Kluwer, New York, 2003). This study was performed by the author when he was preparing his Thesis work on ``The Quantum Theory of Radioactive Nuclei'' (unpublished), whose supervisor was E. Fermi.

S. Esposito

19

Comments on a Paper by Majorana Concerning Elementary Particles

 David. M. Fradkin

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An early paper (1932) by Majorana, that has received but scant attention, is reexamined in light of later developments. This pioneering paper constructs a relativistically invariant theory of arbitrary spin particles, develops and utilizes infinite dimensional representations of the homogeneous Lorentz group, and provides a mass spectrum for elementary particles. The relevance of Majorana’s approach and results to later and current research is explained.

Reprinted with permission from the AMERICAN JOURNAL OF PHYSICS, Volume 34, Issue 4, pp. 314-318. Copyright 1966, American Association of Physics Teachers

We reproduce here the historical D. M. Fradkin 1966 paper whose role among the physicists of high energy was decisive; since then espressions like "Majorana mass", "Majorana spinors" and "Majorana neutrino" have become usual. The paper is based upon the work Teoria di Particelle con Momento Intrinseco Arbitrario, translated by Italiam from Edoardo Amaldi.

Ignazio Licata