Erasmo Recami

Recent developments in the time analysis of tunneling processes

Abstract In this paper we critically review and analyse the main theoretical definitions and calculations of the sub-barrier tunnelling and reflection times . Moreover, we propose a new, physically sensible definition of such durations, on the basis of a recent general formalism (already tested for other types of quantum collisions). Finally, we discuss some surprising results regarding the temporal evolution of the tunnelling processes.

UNIFIED TIME ANALYSIS OF PHOTON AND PARTICLE TUNNELLING

A unified approach to the time analysis of tunnelling of nonrelativistic particles is presented, in which Time is regarded as a quantum-mechanical observable, canonically conjugated to Energy. The validity of the Hartman effect (independence of the Tunnelling Time of the opaque barrier width, with Superluminal group velocities as a consequence) is verified for all the known expressions of the mean tunnelling time. Moreover, the analogy between particle and photon tunnelling is suitably exploited. On the basis of such an analogy, an explanation of some recent microwave and optics experimental results on tunnelling times is proposed. Attention is devoted to some aspects of the causality problem for particle and photon tunnelling. † Work partially supported by MURST, INFN, CNR and by I.N.P./PAN/Krakow ; e-mail addresses : recami@mi.infn.it ; olkhovsk@stenos.kiev.ua ; jakiel@alf.ifj.edu.pl

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.

Measurement of superluminal optical tunneling times in double-barrier photonic band gaps

Tunneling of optical pulses at 1.5 microm wavelength through double-barrier periodic fiber Bragg gratings is experimentally investigated in this paper. Tunneling time measurements as a function of the barrier distance show that, far from resonances of the structure, the transit time is paradoxically short--implying superluminal propagation--and almost independent of the barrier distance. This result is in agreement with theoretical predictions based on phase-time analysis and provides, in the optical context, an experimental evidence of the analogous phenomenon in quantum mechanics of nonresonant superluminal tunneling of particles across two successive potential barriers.

Superluminal motions? A bird-eye view of the experimental situation (†)

In this article, after a theoretical introduction and a sketch of some related long-standing predictions, a birds-eye view is presented—with the help of nine figures—of the various experimental sectors of physics in which Superluminal motions seem to appear (thus contributing support to those past predictions). In particular, a panorama is presented of the experiments with evanescent waves and/or tunnelling photons, and with the “localized Superluminal solutions” to the Maxwell equations (like the so-called X-shaped beams). The present review is brief, but is followed by a large enough bibliography to allow the interested reader deepening the preferred topic.

Propagation speed of evanescent modes

The group velocity of evanescent waves (in undersized waveguides, for instance) was theoretically predicted, and has been experimentally verified, to be superluminal (v(g)>c). By contrast, it is known that the precursor speed in vacuum cannot be larger than c. In this paper, by computer simulations based on Maxwell equations only, we show the existence of both phenomena. In other words, we verify the actual possibility of superluminal group velocities, without violating the so-called (naive) Einstein causality.

More about Lorentz transformations and tachyons: answer to the comments by Ramachandran, Tagare and Kolaskar

I n a no t e (1), RAMACHA~DI~AN et al. h a v e c o m m e n t e d a b o u t a r ecen t , b r ie f l e t t e r b y 0LKHOVSKY a n d one of us (2), r ega rd ing L o r e n t z t r a n s f o r m a t i o n s (LT) a n d fas ter t han l i gh t -pa r t i c l e s (3.~). E v e n if t h e i r c o m m e n t s d id n o t a p p e a r to us to be v e r y p e r t i n e n t , we are grafefu l to t h e m for th i s occasion, since i t allows us to c lar i fy our p r e l i m i n a r y l e t t e r (2), wh ich r e m a i n e d indeed a t t h e surface of t h e p rob lems . I n pa r t i cu la r , we shal l show t h a t ou r p rev ious cons ide ra t ions can be founded m u c h b e t t e r t h a n in ref. (2).

Tachyon kinematics and causality: a systematic thorough analysis of the tachyon causal paradoxes

The chronological order of the events along a spacelike path is not invariant under Lorentz transformations, as is well known. This led to an early conviction that tachyons would give rise to causal anomalies. A relativistic version of the Stückelberg-Feynman “switching procedure” (SWP) has been invoked as the suitable tool to eliminate those anomalies. The application of the SWP does eliminate the motions backwards in time, but interchanges the roles ofsource anddetector. This fact triggered the proposal of a host of causal “paradoxes.” Till now, however, it has not been recognized that such paradoxes can be sensibly discussed (and completely solved, at least “in microphysics”)only after the tachyon relativistic mechanics has been properly developed. We start by showing how to apply the SWP, both in the case ofordinary special relativity and in the case with tachyons. Then we carefully exploit the kinematics of the tachyon exchange between to (ordinary) bodies. Being finally able to tackle the tachyon causality problem, we successively solve the paradoxes of : (i) Tolman-Regge, (ii) Pirani, (iii) Edmonds, and (iv) Bell. Finally, we discuss a further, new paradox associated with the transmission of signals by modulated tachyon beams.

On the shape of tachyons

SummaryWe study some aspects of the experimental behaviour of tachyons, in particular by finding out their « apparent » shape. A Superluminal particle, which in its own rest frame is spherical or ellipsoidal (and with an infinite lifetime), would « appear » to a laboratory frame as occupying the whole region of space bound by a double cone and a twosheeted hyperboloid. Such a structure (the tachyon « shape ») rigidly travels with the speed of the tachyon. However, if the Superluminal particle has a finite lifetimein its rest frame, then in the laboratory frame it gets afinite space extension. As a by-product, we are able to interpret physically the imaginary units entering—as is well known—the transverse co-ordinates in the Superluminal Lorentz transformations. The various particular or limiting cases of the tachyon shape are thoroughly considered. Finally, some brief considerations concerning possible experiments to look for tachyons are added.RiassuntoSi studiano alcuni aspetti del comportamento dei taohioni, in particolare trovando quale « apparirebbe » la loro forma. Una particella Superluminale, che sia sferica o ellissoidale (e con vita di durata finita) nel proprio riferimento a riposo, a un osservatore nel laboratorio sembrerebbe occupare Tintera regione di spazio limitata da un doppio cono e da un iperboloide a due falde. Tale struttura (la « forma » del tachione) viaggerà rigidamente con la velocità del tachione. Si noti però che, se la particella Superluminale ha una vita finita (nel suo riferimento a riposo), allora nel laboratorio essa risulta avere un’estensione spazialefinita. Come conseguenza della precedente analisi, siamo in grado d’interpretare fisicamente le unità immaginarie che entrano — come noto — nelle coordinate trasversali per azione delle trasformazioni di Lorentz Superluminali. Si esaminano dettagliatamente i vari casi particolari o casi limite della forma dei tachioni. Infine, si aggiungono alcune considerazioni circa eventuali esperimenti atti alla ricerca effettiva dei tachioni.РезюмеМы исследуем некоторые аспекты экспериментального поведения тахионов, в частности, посредством нахождения их кажущейся формы. Суперлюминальная частица, которая в своей собственной системе соординат является сферической или эллипсоидальной (и с бесконечным временем жизни), в лабораторной системе координат представляется занимающей всю область пространства, ограниченную двойным конусом и гиперболоидом с двумя слоями. Такая структура (« форма » тахиона) движется со скоростью тахиона. Однако, ески суперлюминальная частица имеет конечное время жизни в своей собственной системе координат, то в лабораторной системе координат эта частица занимает конечное пространство. Как вспомогательный результат, мы можем физически интерпретировать мнимые единицы, входящие, как известно, в попереченые координаты в суперлюминальных преобразованиях Лоренца. Подробно исследуются различные частные и предельные случаи формы тахиона. В заключение, проводится обсуждение возможных экспериментов по наблюдению тахионов.

Generalized Lorentz transformations in four dimensions and superluminal objects

SummaryA new groupG of Lorentz transformations (LT) in four dimensions, generalized also for Superluminal frames, is introduced and particularly studied in its physical implications. With the help of a « principle of duality »—implied byG—between subluminal and Superluminal frames, the meanings of « inertial frame », « equivalence », « principle of relativity », « covariance » may be correspondingly extended. A biunivocal correspondence exists between bradyonic and tachyonic velocities, which we find to be a particular conformal mapping (inversion). Since the groupG consists of generic rotations in space-time, it includes,e.g., also the total-inversion operation (PT). Moreover (for a non « charge »-free universe), it is shown that our generalized special relativity requires covariance underCPT. A « tachyonization principle » is formulated, on the basis of which relativistic physical laws (of mechanics and electrodynamics, at least) can be easily extended to tachyons. Many simple applications are performed of the generalized LT’s (velocity composition law, comparison of the length and time units, Doppler effect, refraction index, …), either useful to clarify our problem or interesting in astrophysics.RiassuntoUn nuovo gruppoG di trasformazioni di Lorentz (LT) in quattro dimensioni, generalizzato anche per sistemi di riferimento Superluminali, è introdotto e studiato particolarmente nelle sue implicazioni fisiche. Con l’aiuto di un « principio di dualità » — implicato daG — tra sistemi subluminali e Superluminali, è possibile estendere il significato di « riferimento inerziale », « equivalenza », « principio di relatività », « covarianza ». Tra velocità bradioniche e tachioniche esiste una corrispondenza biunivoca, che risulta essere una particolare corrispondenza conforme (inversione). Poiché il gruppoG consiste di rotazioni generiche nello spazio-tempo, esso include per esempio anche l’operazione di inversione totale (PT). Inoltre (per un universo con « cariche »), si mostra che la nostra relatività ristretta generalizzata richiede la covarianza perCPT. Si formula un « principio di tachionizzazione », in base al quale le leggi fisiche relativistiche (quelle almeno della meccanica e dell’elettrodinamica) possono essere facilmente estese al caso dei tachioni. Si applicano le LT generalizzate ad alcuni semplici casi (legge di composizione delle velocità, confronto di unità di tempo e di lunghezza, effetto Doppler, indice di rifrazione, …) utili per chiarire il nostro problema o di interesse in astrofisica.РеэюмеВводится новая группаG преобраэований Лорентца в четырех иэмерениях, обобшенная также для сверхсветяшихся систем отсчета. Исследуются фиэические применения новой группыG. Испольэуя «принцип дуальности» между субсветяшимися и сверхсветяшимися системами отсчета, может быть расщирен фиэический смысл понятий « инерциальной системы отсчета», «зквивалентности», « принципа относительности » и « ковариантности ». Сушествует соответствие между скоростями брадионов и тахионов, которое получается как реэультат конкретного конформного отображения (инверсии). Так как группаG содержит врашения в пространстве и времени, то она включает, например, также операцию полной инверсии(РТ). В случае эарядовой инверсии наща обобшенная специальная теория относительности требует ковариантности относительноОРТ. Формулируется «принцип тахиониэации», на основе которого релятивистские фиэические эаконы (по крайней мере, механики и злектродинамики) могут быть легко обобшены для тахионов. Рассмотрено много простых применений обобшенных преобраэований Лорентца (эакон сложения скоростей, сравнение единиц длины и времени, зффект Допплера, козффициент преломления и т.д.), полеэных либо для прояснения нащей проблемы, либо интересных в астрофиэике.