arXiv subjects

Erasmo Recami

Publications and source records attributed to Erasmo Recami.

At least 37 records · Page 2Linked to original sources

arXiv2009

Localized Waves: A not-so-short Review

In the FIRST PART we present simple introductions to gaussian and Bessel waves, and to the Localized Waves (LW), pulses or beams, showing the important properties of the latter, and their applications whenever a role is played by a wave-equation (electromagnetism, optics, acoustics, seismology, geophysics, gravitation, elementary particle physics,...). The First Part ends with a historical APPENDIX, recalling how the geometrical methods of Special Relativity (SR) had predicted the most interesting LWs, i.e., the X-shaped pulses; and presenting a bird's-eye view of the experiments performed with evanescent waves (and/or tunnelling photons), and with the "localized Superluminal solutions". In the SECOND PART, after some more theoretical introduction, we develop a Generalized "Bidirectional Decomposition", and obtain several luminal and Superluminal non-diffracting solutions; we get a space-time focusing of X-Shaped pulses; and deal with chirped optical X-shaped pulses in material media. Finally, in the THIRD PART we investigate also the subluminal LWs, which, among the others, allow to emphasize the role of SR, in its extended, or rather non-restricted, formulation. We study in particular the topic of zero-speed waves, endowed with a static envelope: Namely, we show how localized wavefields can be constructed with high transverse localization, and with a longitudinal intensity pattern that assumes any desired shape within a chosen interval of the propagation axis. Such "Frozen Waves" promise to have even more applications. In between, we do not forget to briefly treat the case of not axially-symmetric solutions, in terms of higher order Bessel beams.

physics.optics↗

arXiv2008

Cherenkov radiation has nothing to do with X-shaped Localized Waves (Comments on "Cherenkov-Vavilov Formulation of X-Waves")

The Localized Waves are nondiffracting ("soliton-like") and self-reconstructing solutions to the wave equations, and are known to exist with subluminal, luminal and superluminal peak-velocities. The most studied ones are the "X-shaped" superluminal waves; which are associated with a cone, so that some authors [e.g., Walker and Kuperman, PRL 99 (Dec.2007) 244802] have been tempted to link them with Cherenkov radiation. However, the "X-waves" belong to a different realm, and exist even in the vacuum, independently of any media, as verified in a number of papers [listed e.g. in "Localized Waves" (J.Wiley; Jan.2008)]. We want to clarify the whole question on the basis of rigorous formalism and clear physical considerations. In particular, by explicit calculations based on Maxwell equations only, we show that: (i) the X-waves exist also inside both the front and the rear part of their double cone (that has nothing to do with Cherenkov's); (ii) they are to be found not via ad hoc assumptions, but by use of strict mathematical (or experimental) procedures; (iii) the ideal X-waves possess infinite energy, but finite-energy X-waves can be easily constructed (even without space-time truncations): And we do construct exact finite-energy solutions (totally free from backward-traveling waves); (iv) an actual attempt at comparing Cherenkov radiation with X-waves would lead one to consider the very different situation of the (X-shaped, too) field generated by a superluminal point-charge, a question exploited in previous papers [Recami et al., PRE 69 (2004) 027602]: We show explicitly, here, that the point-charge would not lose energy in the vacuum, and its field would not need to be continuously feeded by incoming side-waves (as it is the case, indeed, for an ideal, ordinary X-wave).

physics.optics↗

arXiv2008

A homage to E.C.G.Sudarshan: Superluminal objects and waves (An updated overview of the relevant experiments)

This writing has been prepared on the occasion of the 75th birthday of E.C.George Sudarshan, who (besides the originator of the V-A theory for weak-forces, of quantum optics --the quantum representation of coherent light--, of dynamical maps for open quantum systems, of the Zeno effect, etc.) was a pioneer, already in the sisties, also of the theory of the so-called "tachyons". This paper wishes to be a homage to E.C.G.Sudarshan, in connection with the last-mentioned pioneering work of his. After a brief theoretical introduction (based on the standard postulates of Special Relativity, and therefore extending it --or rather non-restricting it-- without any violations: for example, without any violations of the so-called Einstein causality), the main aim of this article is an updated presentation of the status-of-the-art of the "superluminal" experiments. In particular, we devote such a review to the phenomena met in tunneling through quantum (and classical) barriers [e.g., referring to the Genelalized Hartman Effect]; and in connection with the "Localized (nondiffracting) Solutions" of the wave equantions [e.g., of Maxwell equations]: especially of the "X-shaped" ones. The interested reader is provided with an extended Bibliography.

physics.class-ph↗

arXiv2007

Einstein, Sciascia, Majorana, Amaldi, e il Rapporto tra Intellettuali e Potere (Einstein, Sciascia, Majorana, Amaldi, and the Relationship of Intellectuals with the Power)

In Europe (e.g., in Italy), and in the States, the opinion is widely spreading that the negative consequences of modern progress are the fault of "Science". A lively debate on this topic took place among the famous writer Leonardo Sciascia and Italian physicists like Edoardo Amaldi and Emilio Segre', when Sciascia wrote his known book about the disappearance of Ettore Majorana, a book that presented Majorana (a very reserved theoretical physicist) as an example of the scientist that keeps his discoveries secret when afraid of possible evil applications. We wish to contribute now to such a question, since many special meetings did recently return to it, while celebrating the centenary of Majorana's birth (2006), or 30 years (2005) from Sciascia's book publication. It appeared natural to us to start with the figure of Einstein, who, even if extremely peaceable, supported the atomic bomb construction by his letter to Roosvelt. We discuss first the significance of part of Einsteins's scientific work (which flourished in particular one century ago): We seize this opportunity for presenting a derivation of the "twins paradox", so simple that it has been taught to Senior High School last-year students or University undergraduate students; all the present work, more in general, is addressed mainly to them. In the second part of this paper, we analyse the general meaning of Einstein's contributions to philosophy and pedagogy. The third part is entirely devoted to our main subject, i.e., to discussing the "Responsibility of the Intellectuals". The reader interested chiefly in this last part, can find it re-expounded in a self-contained way in the Appendix. More information in the Contents of this article.

physics.gen-ph↗

arXiv2007

L'articolo di Ettore Majorana su "Il valore delle Leggi Statistiche nella Fisica e nelle Scienze Sociali" (Ettore Majorana's article on "The value of Statistical Laws in Physics and in Social Sciences")

The mentioned article was written by Ettore Majorana, in a partially educational way, for a journal of Sociology; but he gave up publishing it (and threw it away). It appeared posthumous, thanks to Giovanni Gentile Jr. (the inventor of "parastatistics") in "Scientia" 36 (1942) 58-66. It has not been re-published, in Italian, till the beginning of 2006, when we made known some abridgements of it by Italian newspapers and by the journal "Fisica in Medicina". We don't know when was it written: perhaps in 1930. However, its central theme was still alive in Majorana's mind in 1934: in fact, on July 27, 1934, he will write to G.Gentile Jr. to expect that < >. Here, in Part I, we present a suitable reduction, edited by us, of Majorana's article; while in Part II we add a complete transcription of it. [Since the paper which appeared in "Scientia" contains some errors in the interpretation of Majorana's handwriting, the present versions have been very slightly "corrected" by us]. For the translations into English of Majorana's paper, see Refs.[5,6] below. A more extended Summary (in English, besides in Italian) can be found at the beginning of the present e-print. The interested reader can found all the known biographical documents --apart from the ones discovered during the last two years-- in the book by E.Recami, "Il Caso Majorana: Epistolario, Testimonianze, Documenti" (Mondadori, Milan, 1987 and 1991; Di Renzo Editore, Rome, 2000 and 2002); and in the e-prints arXiv:physics/9810023v4 [physics.hist-ph]; arXiv:0708.2855v1 [physics.hist-ph]; and arXiv:0709.1183 [physics.hist-ph].

physics.hist-ph↗

arXiv2007

Einstein e il Rinnovamento delle Scienze (Einstein and the Renewal of Science)

As it is well-known, the year 2005 has been the centenary of the "annus mirabilis" (1905) during which Albert Einstein published four fundamental papers of his. But already in 1979, for the centenary of Einstein's birth, the world celebrated his monumantal work. In Italy too, there appeared scientific books, and many semi-popularization (or popularization) articles. The present paper represents a talk delivered in Italian, at the invitation of the Nobel Foundation (Sanremo, IM; Italy), in time for its publication in 1979. This article has been however reprinted, much more recently, in 2002, by the "Centro DIEA", Faculty of Engineering, University of Bologna, Bologna, Italy [and its source-file, in html, has been prepared with DIEA's collaboration]. We present here a description of the human, philosophycal and scientific background, starting from which Einstein produced his amazing results: Indeed, we try to show why Einstein's writings today are still so important not only for pure physics (and technology!), but also for our epistemological understanding of the procedures followed by science, and by our own mind, in their development, as well as for our philosophical views about the world we live in; without forgetting the teachings that come (or should come) from Einstein's life and claims for modern pedagogy.

physics.hist-ph↗

arXiv2007

About Superluminal motions and Special Relativity: A Discussion of some recent Experiments, and the solution of the Causal Paradoxes

Some experiments, performed at Berkeley, Cologne, Florence, Vienna, Orsay, Rennes, etc., led to the claim that something seems to travel with a group velocity larger than the speed c of light in vacuum. Various other experimental results seem to point in the same direction: For instance, localized wavelet- type solutions to Maxwell equations have been found, both theoretically and experimentally, that travel with superluminal speed. [Even muonic and electronic neutrinos [it has been proposed] might be "tachyons", since their square mass appears to be negative]. With regard to the first-mentioned experiments, it was recently claimed by Guenter Nimtz that those results with evanescent waves (or tunneling photons) imply superluminal signal and impulse transmission, and therefore violate Einstein causality. In this note we want to stress that, on the contrary, all such results do not place relativistic causality in jeopardy, even if they referred to actual tachyonic motions: In fact, Special Relativity can cope even with superluminal objects and waves. For instance, it is possible (at least in microphysics) to solve also the known causal paradoxes, devised for faster than light motion, although this is not widely recognized yet. Here we show, in detail and rigorously, how to solve the oldest causal paradox, originally proposed by Tolman, which is the kernel of many further tachyon paradoxes (like J.Bell's, F.A.E.Pirani's, J.D.Edmonds' and others'). The key to the solution is a careful application of tachyon mechanics, as it unambiguously follows from special relativity. At Last, in one of the two Appendices, we propose how to evaluate the group-velocity in the case of evanescent waves. [PACS nos.: 03.30.+p; 03.50.De; 41.20.Jb; 73.40.Gk; 84.40.Az; 42.82.Et ]

physics.class-ph↗

arXiv2007

Sub-luminal wave bullets: Exact Localized subluminal Solutions to the Wave Equations

In this work it is shown how to obtain, in a simple way, localized (non- diffractive) subluminal pulses as exact analytic solutions to the wave equations. These new ideal subluminal solutions, which propagate without distortion in any homogeneous linear media, are herein obtained for arbitrarily chosen frequencies and bandwidths, avoiding in particular any recourse to the non-causal components so frequently plaguing the previously known localized waves. The new solutions are suitable superpositions of --zeroth-order, in general-- Bessel beams, which can be performed either by integrating with respect to (w.r.t.) the angular frequency, or by integrating w.r.t. the longitudinal wavenumber: Both methods are expounded in this paper. The first one appears to be powerful enough; we study the second method as well, however, since it allows dealing even with the limiting case of zero-speed solutions (and furnishes a new way, in terms of continuous spectra, for obtaining the so-called "Frozen Waves", so promising also from the point of view of applications). We briefly treat the case, moreover, of non-axially symmetric solutions, in terms of higher order Bessel beams. At last, particular attention is paid to the role of Special Relativity, and to the fact that the localized waves are expected to be transformed one into the other by suitable Lorentz Transformations. The analogous pulses with intrinsic finite energy, or merely truncated, will be constructed in another paper. In this work we fix our attention especially on electromagnetism and optics: but results of the present kind are valid whenever an essential role is played by a wave-equation (like in acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.)

physics.class-ph↗

arXiv2007

On the Localized superluminal Solutions to the Maxwell Equations

In the first part of this article the various experimental sectors of physics in which Superluminal motions seem to appear are briefly mentioned, after a sketchy theoretical introduction. In particular, a panoramic view is presented of the experiments with evanescent waves (and/or tunneling photons), and with the "Localized superluminal Solutions" (SLS) to the wave equation, like the so-called X-shaped waves. In the second part of this paper we present a series of new SLSs to the Maxwell equations, suitable for arbitrary frequencies and arbitrary bandwidths: some of them being endowed with finite total energy. Among the others, we set forth an infinite family of generalizations of the classic X-shaped wave; and show how to deal with the case of a dispersive medium. Results of this kind may find application in other fields in which an essential role is played by a wave-equation (like acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.). This e-print, in large part a review, was prepared for the special issue on "Nontraditional Forms of Light" of the IEEE JSTQE (2003); and a preliminary version of it appeared as Report NSF-ITP-02-93 (KITP, UCSB; 2002). Further material can be found in the recent e-prints arXiv:0708.1655v2 [physics.gen-ph] and arXiv:0708.1209v1 [physics.gen-ph]. The case of the very interesting (and more orthodox, in a sense) subluminal Localized Waves, solutions to the wave equations, will be dealt with in a coming paper. [Keywords: Wave equation; Wave propagation; Localized solutions to Maxwell equations; Superluminal waves; Bessel beams; Limited-dispersion beams; Electromagnetic wavelets; X-shaped waves; Finite-energy beams; Optics; Electromagnetism; Microwaves; Special relativity]

physics.class-ph↗

arXiv2007

On Localized "X-shaped" Superluminal Solutions to Maxwell Equations

In this paper we extend for the case of Maxwell equations the "X-shaped" solutions previously found in the case of scalar (e.g., acoustic) wave equations. Such solutions are localized in theory, i.e., diffraction-free and particle-like (wavelets), in that they maintain their shape as they propagate. In the electromagnetic case they are particularly interesting, since they are expected to be Superluminal. We address also the problem of their practical, approximate production by finite (dynamic) radiators. Finally, we discuss the appearance of the X-shaped solutions from the purely geometric point of view of the Special Relativity theory. [PACS nos.: 03.50.De; 1.20.Jb; 03.30.+p; 03.40.Kf; 14.80.-j. Keywords: X-shaped waves; localized solutions to Maxwell equations; Superluminal waves; Bessel beams; Limited-dispersion beams; electromagnetic wavelets; Special Relativity; Extended Relativity].

physics.optics↗

arXiv2007

Gli Strumenti Scientifici di Interesse Storico di tre antiche Scuole Medie Superiori di Bergamo ("Lussana", "Vittorio Emanuele", e "Quarenghi")

Wherever old Universities exist [in Europe, for example: Italy being not the last one], there can be found Scientific Instruments of Historical Interest, e.g., for Physics -which is the case we are concerned with here;- even if proper attention has been paid to such an exceptional heritage only during the last half a century. [We studied ourselves the material existing at the University of Catania, Catania, Italy, almost 600 years old]. However, even where ancient universities do not exist, a lot of Historical Scientific Instruments are often owned by the local Senior High Schools. By the present "small book" we want to make known the antique physics instruments that are in possession of three Senior High Schools of Bergamo, Italy: the "Istituto Tecnico Vittorio Emanuele", the "Liceo Scientifico Lussana", and the "Istituto per Geometri Quarenghi". Namely, we catalogue and describe: 61 instruments of the "Vittorio Emanuele" (52 of the second half of the XIX century, and 9 of the first half of the XX century); 100 instruments of the "Lussana" (belonging essentially to the first half of the XX century); and 26 instruments, mainly topographical in this case, of the "Quarenghi" (8 of the second half of the XIX century, and 16 of the first half of the XX century); without forgetting to add suitable explanations about their functioning. Our aim will be reached if we happen to stimulate other colleagues to perform similar work in their towns. The present small book is in Italian, but it contains fifty-one (51) self-explanatory photographs in color, and nineteen (19) black-and-white sketches. Actually, we can put in the Archives this small book, instead of others, also because -with the help of the typesetter- we eventually suceeded in reducing its whole size to less than 1 M-b.

physics.ed-ph↗

arXiv2007

Aspetti Moderni della Fisica Greca [Modern Aspects of Ancient Greek Physics]

Plutarchus, circa 100 AD, in his early book on "astrophysics" --in which he exposed, in a sense, a general theory of gravitation-- wrote the noticeable passage: < >. While Posidonius (circa 135-51 BC) had written: < >. These two examples show how "modern" was part of the ancient Greek physics, frequently overlooked since it is sometimes known through the praiseworthy translations of scholars often competent in humanities, more than in science. [A more extended English summary appears at the beginning of this article, which is in Italian.]

physics.ed-ph↗

arXiv2007

The Scientific Manuscripts left Unpublished by Ettore Majorana (with outlines of his life and work)

We present a panoramic view of the main scientific manuscripts left unpublished by the brightest Italian theoretical physicist of the XX century, Ettore Majorana. We deal in particular: (i) with his very original "study" notes (the so-called "Volumetti"), already published by us in English, in 2003, c/o Kluwer Acad.Press, Dordrecht & Boston, and in the original Italian language, in 2006, c/o Zanichelli pub., Bologna, Italy; and (ii) with a selection of his research notes (the so-called "Quaderni"), that we shall publish c/o Springer, Berlin. We seize the present opportunity for setting forth also some suitable -scarcely known- information about Majorana's life and work, on the basis of documents (letters, testimonies, different documents...) discovered or collected by ourselves during the last decades. [A finished, enlarged version of this paper will appear as the editors' Preface, at the beginning of the coming book "Ettore Majorana - Unpublished Research Notes on Theoretical Physics", edited by S.Esposito, E.Recami, A.van der Merwe and R.Battiston, to be printed by Springer verlag, Berlin].

physics.hist-ph↗

arXiv2007

Ettore Majorana's Inaugural Lecture on Theoretical Physics (Naples, Italy; Jan.13, 1938)

Ettore Majorana was a member of Enrico Fermi's research group in Rome, Italy. Fermi did regard Majorana as much brihter than himself as far as theoretical physics was concerned (more information can be found particularly in the arXives' e-print physics/9810023, in Italian, and refs therein, and also in the recent multilanguage arXiv:0708.2855v1 [physics.hist-ph]). In 1937 Majorana partecipated in the national Italian competition, for a chair in theoretical phyics, requested by Emilio Segre' at that time at Palermo University: Other competitors being GC.Wick, G.Racah, and G.Gentile jr. After a proposal of the judging Commette, chaired by E.Fermi, Majorana got a full-professorship at Naples University, for exceptional scientific merits, outside the competition normal procedures. In this e-print we make known the notes prepared by Majorana for his Inaugural Lecture (and discovered long ago, in 1973, by one of the present editors (ER)),together with some comments of ours: everything being both in English (first article) and in Italian (second article, with a short Bibliography at its end). The present articles have been prepared on the occasion of the Centenary (2006) of Majorana's birth. The preliminary notes for his Inaugural Lecture reveal Majorana's interest not only for scientific research, but also for the best didactical methods to be followed in order to teach classical and quantum physics in the most effective way (while his approach to Special Relativity is known to us from his lecture notes, published elesewhere). P.S.: Il Riassunto in Italiano appare all'inizio della versione italiana.

physics.hist-ph↗

arXiv2007

Ettore Majorana's Scientific (and Human) Personality

This article, in English, represents part of the invited talk delivered by the author at the "Internat.Conf.Majorana Legacy and Phys. XXI century" held at Dep. of Phys.,University of Catania, Italy, in Oct.2006: Conference organized for celebrating the 100th anniversary of the birth (in Catania) of the Italian theoretical physicist Ettore Majorana; probably the brightest Italian theoretician of the XX century (Enrico Fermi regarded him as the brightest in the world of his time), even if to some people Majorana is still known chiefly for his mysterious disappearance, in 1938, when he was 31. In this writing we outline the significance of his main publications, as well as his life: The biographical data being based on letters, documents, testimonies discovered or collected by the author during almost 30 years, and contained in the book by E.Recami, "Il Caso Majorana: Epistolario, Testimonianze, Documenti" (initially published, in 1987 and 1991, by Mondadori, Milan, and presenly published, in 2002, by Di Renzo Editore, Rome: http://www.direnzo.it). At last, some information and comments are added with regard to the scientific manuscripts left unpublished by Majorana. ----- Ettore Majorana: Un cenno alle sue Opere (e Vita). Il secondo articolo rappresenta la versione italiana di parte del seminario su invito tenuto dall'autore alla "Internat.Conf.Majorana's Legacy and Phys. of XXI Century" che ha avuto luogo nell'ottobre 2006 presso il Dip. di Fis. della Univ. di Catania: Conferenza organizzata per celebrare il centenario della nascita (a Catania) del fisico teorico Ettore Majorana; probabilmente il piu` geniale teorico italiano del XX secolo (Enrico Fermi lo considero` il piu` brillante del suo tempo nel mondo), anche se a qualcuno Majorana e` ancora noto soprattutto per la sua misteriosa scomparsa, avvenuta nel 1938, quando aveva 31 anni. In questo scritto descriviamo in sintesi tanto il significato delle sue piu` importanti pubblicazioni, quanto la sua vita: i dati biografici essendo basati su lettere, documenti, testimonianze scoperti o raccolti dall'autore durante quasi 30 anni, e contenuti nel libro di E.Recami: "Il Caso Majorana: Epistolario, Testimonianze, Documenti" (pubblicato inizialmente da Mondadori, Milano, nel 1987 e 1991, e al presente pubblicato, nel 2002, da Di Renzo Editore, Roma: http://www.direnzo.it). Infine, si aggiungono informazioni e commenti circa i manoscritti scientifici lasciti inediti dal Majorana.

physics.hist-ph↗

arXiv2007

Localized Waves: A scientific and historical introduction

In the first part of this paper (mainly a review) we present general and formal (simple) introductions to the ordinary gaussian waves and to the Bessel waves, by explicitly separating the cases of the beams from the cases of the pulses; and, finally, an analogous introduction is presented for the Localized Waves (LW), pulses or beams. Always we stress the very different characteristics of the gaussian with respect to the Bessel waves and to the LWs, showing the numerous and important properties of the latter w.r.t. the former ones: Properties that may find application in all fields in which an essential role is played by a wave-equation (like electromagnetism, optics, acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.). In the second part of this paper (namely, in its Appendix), we recall at first how, in the seventies and eighties, the geometrical methods of Special Relativity (SR) had predicted --in the sense below specified-- the existence of the most interesting LWs, i.e., of the X-shaped pulses. At last, in connection with the circumstance that the X-shaped waves are endowed with Superluminal group-velocities (as carefully discussed in the first part of this article), we briefly mention the various experimental sectors of physics in which Superluminal motions seem to appear: In particular, a bird's-eye view is presented of the experiments till now performed with evanescent waves (and/or tunneling photons), and with the "localized Superluminal solutions" to the wave equations.

physics.gen-ph↗

arXiv2007

Structure of the Nondiffracting (Localized) Waves, and some interesting applications

Since the early works[1-4] on the so-called nondiffracting waves (called also Localized Waves), a great deal of results has been published on this important subject, from both the theoretical and the experimental point of view. Initially, the theory was developed taking into account only free space; however, in recent years, it has been extended for more complex media exhibiting effects such as dispersion[5-7], nonlinearity[8], anisotropy[9] and losses[10]. Such extensions have been carried out along with the development of efficient methods for obtaining nondiffracting beams and pulses in the subluminal, luminal and superluminal regimes[11-18]. This paper (partly a review) addresses some theoretical methods related to nondiffracting solutions of the linear wave equation in unbounded homogeneous media, as well as to some interesting applications of such waves. In section II we analyze the general structure of the Localized Waves, develop the so called Generalized Bidirectional Decomposition, and use it to obtain several luminal and superluminal (especially X-shaped) nondiffracting solutions of the wave equation. In section III we develop a space-time focusing method by a continuous superposition of X-Shaped pulses of different velocities. Section IV addresses the properties of chirped optical X-Shaped pulses propagating in material media without boundaries. Finally, in Section V, we show how a suitable superposition of Bessel beams can be used to obtain stationary localized wave fields, with a static envelope and a high transverse localization, and whose longitudinal intensity pattern can assume any desired shape within a chosen interval of the propagation axis.

physics.gen-ph↗

arXiv2007

Introduction of a Quantum of Time ("chronon"), and its Consequences for Quantum Mechanics

In this review-article, we discuss the consequences of the introduction of a quantum of time tau_0 in the formalism of non-relativistic quantum mechanics (QM) by referring ourselves in particular to the theory of the "chronon" as proposed by P.Caldirola. Such an interesting "finite difference" theory, forwards --at the classical level-- a solution for the motion of a particle endowed with a non-negligible charge in an external electromagnetic field, overcoming all the known difficulties met by Abraham-Lorentz's and Dirac's approaches (and even allowing a clear answer to the question whether a free falling charged particle does or does not emit radiation), and --at the quantum level-- yields a remarkable mass spectrum for leptons. After having briefly reviewed Caldirola's approach, we compare one another the new Schroedinger, Heisenberg and density-operator (Liouville-von Neumann) pictures resulting from it. Moreover, for each representation, three (retarded, symmetric and advanced) formulations are possible, which refer either to times t and t-tau_0, or to times t-tau_0/2 and t+tau_0/2, or to times t and t+tau_0, respectively. It is interesting to notice that, e.g., the "retarded" QM does naturally appear to describe QM with friction, i.e., to describe dissipative quantum systems (like a particle moving in an absorbing medium). In this sense, discretized QM is much richer than the ordinary one. When the density matrix formalism is applied to the solution of the measurement problem in QM, very interesting results are met, so as a natural explication of "decoherence".

quant-ph↗