This article is divided in five parts, the first one having nothing to do with tachyons. In fact, to prepare the ground, in part I (sect.2) we shall merely show that special relativity - even without tachyons - can be given a form such to describe both particles and antiparticles. Part II is the largest one: initially, after some historical remarks and having revisited the postulates of special relativity, we presnt a review of the elegant ''model theory'' of tachyons in two dimensions; passing then to four dimensions, we review the main results of the classical theory of tachyons that do not depend on the existence of Superluminal reference frames (or that are at least independent of the explicit form of the Superluminal Lorentz ''transformations''). In particular, we discuss how tachyons would look like, i.e. their apparent ''shape''. Last but not least, all the common causality problems are thoroughly solved, on the basis of the previously reviewed tachyon kinematics. Part III deals with tachyons in general relativity; in particular the question of the apparent Superluminal expansions in astrophysics is reviewed. Part IV shows the interesting, possible role of tachyons in elementary-particle physcis and in quantum theory. In part V, the last one, we face the (still open) problem of the Superluminal Lorentz ''transformations'' in four dimensions, by introducing, for instance, an auxiliary six-dimensional space-time, and finally present the electromagnetic theory of tachyons: a theory that can be relevant also from the ''experimental'' point of view.

Classical tachyons and possible applications

The various experimental sectors of physics in which superluminal motions seem to appear are briefly mentioned. In particular, a bird's-eye 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 beams. The authors also present a series of new SLSs to the Maxwell equations, suitable for arbitrary frequencies and arbitrary bandwidths, some of them endowed with finite total energy. Among the others, the authors set forth an infinite family of generalizations of the classical 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.).

On the localized superluminal solutions to the Maxwell equations

In the First Part of this paper (which is mainly a review) we present simple, general and formal,
introductions to the ordinary gaussian waves and to the Bessel waves, by explicitly separating the case of beams from the case of pulses; and, afterwards, 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 important properties of the latter: 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.). The First Part of this review ends with an Appendix, wherein: (i) we recall how, in the seventies and eighties, the geometrical methods of Special Relativity (SR)
predicted --in the sense below specified-- the existence of the most interesting LWs, i.e., of the X-shaped pulses; and (ii) in connection with the circumstance that the X-shaped waves are
endowed with Superluminal group-velocities (as discussed in the first part of this paper), 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 tunnelling photons), and with the "localized Superluminal
solutions" to the wave equations.
In the Second Part of this work, we address in more detail various theoretical approaches leading to nondiffracting solutions of the linear wave equation in unbounded homogeneous media, as well as some interesting applications of
these waves. After some more introductory remarks (Sec.VI), we analyse in Section VII the general structure of the Localized Waves, develop the so called Generalized Bidirectional
Decomposition, and use it to obtain several luminal and Superluminal nondiffracting solutions of the wave equations. In Section VIII we present a method for getting a space-time focusing by a continuous superposition of X-Shaped pulses of different velocities. Section IX addresses the properties of chirped optical X-Shaped pulses propagating in material media without boundaries.
Finally, in the Third Part of this paper we "complete" our review by investigating also the not less interesting) case of subluminal
Localized Solutions to the wave equations, which, among the others, allow us to emphasize the remarkable role of SR, in its extended, or rather non-restricted, formulation. [For instance, the various Superluminal and subluminal LWs are expected to be transformed one into the other by suitable Lorentz transformations]. We start by studying --by means of various approaches-- the very peculiar topic of zero-speed waves: Namely, of the localized fields with a static envelope; consisting, for instance, in "light at rest". Actually,
in Section X we show how a suitable superposition of Bessel beams can be used to construct stationary localized wave fields with high transverse localization, and with a longitudinal intensity pattern that assumes any desired shape within a chosen interval 0< z<L of the propagation axis. We have called Frozen Waves such solutions: As we shall see, they can have a lot of noticeable applications. In between,
we do not forget to briefly treat the case of not axially-symmetric solutions, in terms of higher order Bessel beams.
In this review we have fixed
our attention especially on electromagnetism and optics: but results of the present kind are valid, let us repeat, whenever an essential role is played by a wave-equation.

Localized Waves: A not-so-short review

Publisher Summary The localized waves (LWs), also known as “non diffracting waves,” can indeed resist diffraction for a long distance in free space. The term “beam” refers to a monochromatic solution to the considered wave equation with a transverse localization of its field. Today, localized waves are well established theoretically and experimentally and have innovative applications not only in vacuum but also in material (linear or nonlinear) media, and are able to resist also dispersion. Their potential applications are being intensively explored, always with surprising results, in fields such as acoustics, microwaves, and optics, with great promise also in mechanics, geophysics, and even gravitational waves and elementary particle physics. Of note, the so-called “frozen waves” presented in this chapter are already being applied in high-resolution ultrasound scanning of moving organs in the human body.

Chapter 4 Localized Waves: A Review

In this paper we examine critically and in detail some existing definitions for the tunnelling times, namely: the phase-time; the centroid-based times; the Buttiker and Landauer times; the Larmor times; the complex (path-integral and Bohm) times; the dwell time, and finally the generalized (Olkhovsky and Recami) dwell time, by adding also some numerical evaluations. Then, we pass to examine the equivalence between quantum tunnelling and "photon tunnelling" (evanescent waves propagation), with particular attention to tunnelling with Superluminal group-velocities ("Hartman effect"). At last, in an Appendix, we add a bird-eye view of all the experimental sectors of physics in which Superluminal motions seem to appear.

Tunnelling times: An elementary introduction( ∗ )

From the Special Relativity Theory, the existence of Superluminal reference-frames and faster-than-light objects can be inferred, when its validity is not arbitrarily confined to subluminal velocities. The ‘Extended Theory of Relativity’ allows in particular extending the Doppler effect formulae to Superluminal sources. This point is here explored, due to its possible astrophysical interest. Some other questions are briefly considered, regarding the possibility of ‘radio contact’ between subluminal and Superluminal objects.

R. Mignani

Papers

The possibility of superluminal sources and their Doppler effect