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Theoretical Investigation of Subluminal Particles Endowed with Imaginary Mass

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Theoretical Investigation of Subluminal Particles Endowed with Imaginary Mass Article Theoretical Investigation of Subluminal Particles Endowed with Imaginary Mass Luca Nanni Faculty of Natural Science, University of Ferrara, 44122 Ferrara, Italy; [email protected] Abstract: In this article, the general solution of the tachyonic Klein–Gordon equation is obtained as a Fourier integral performed on a suitable path in the complex w-plane. In particular, it is proved that this solution does not contain any superluminal components under the given boundary conditions. On the basis of this result, we infer that all possible spacelike wave equations describe the dynamics of subluminal particles endowed with imaginary mass. This result is validated for the Chodos equation, used to describe the hypothetical superluminal behaviour of the neutrino. In this specific framework, it is proved that the wave packet propagates in spacetime with subluminal group velocities and that it behaves as a localized wave for sufficiently small energies. Keywords: tachyons; Klein–Gordon equation; tachyon-like Dirac equation; relativistic wave packet PACS: 03.65.Pm; 03.65.Ge; 03.65.Db 1. Introduction The physics of tachyons is an intriguing and fascinating subject that has attracted the attention of physicists in both the pre-relativistic [1–3] and post-relativistic epochs [4–10]. Citation: Nanni, L. Theoretical However, the majority of the scientific community still considers it a speculative theory, and Investigation of Subluminal Particles Endowed with Imaginary Mass. there are only a few efforts aimed at finding the experimental evidence that could make it a Particles 2021, 4, 325–332. https:// candidate for extending the current Standard Model [11]. A comprehensive overview of the doi.org/10.3390/particles4020027 most relevant experiments dealing with the superluminal behaviour of waves and particles is detailed in [9,12–15]. If considered in the framework of quantum mechanics, tachyons Academic Editor: Kazuharu Bamba exhibit completely unexpected behaviours [16–18]. In particular, half-integer free fermions endowed with imaginary mass behave like wave trains propagating with subluminal group Received: 29 May 2021 velocities. The aim of this study is to verify whether this result can be generalized for any Accepted: 15 June 2021 particle regardless of its spin. The basic idea is that the individual components of spinor Published: 18 June 2021 wave functions, which are solutions of the tachyonic wave equations, must satisfy the tachyonic Klein–Gordon (TKG) equation as well [19]. Finding a general solution of the Publisher’s Note: MDPI stays neutral TKG equation without spacelike components means proving that any solution of the wave with regard to jurisdictional claims in equations of particles with negative mass squared represents a wave packet that propagates published maps and institutional affil- with subluminal group velocity. As proposed by Salesi, these particles are called pseudo- iations. tachyons (PTs), namely particles travelling with slower-than-light velocity but fulfilling the tachyonic energy-momentum relation E2 = p2c2 + m2c4, where m = im denotes the imaginary mass. The obtained result will be validated for the Chodos equation [20] which 1 describes the behaviour of a 2 -spin tachyon using a more different analytical approach Copyright: © 2021 by the author. than that discussed in [18]. Licensee MDPI, Basel, Switzerland. This article is an open access article 2. Preliminary Notions on the Fourier Transforms in the Complex Plane distributed under the terms and To facilitate the reading of Section3 of this article, a brief review on the theory of the conditions of the Creative Commons Fourier transform in the complex plane is presented [21]. Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Particles 2021, 4, 325–332. https://doi.org/10.3390/particles4020027 https://www.mdpi.com/journal/particles Particles 2021, 4 326 Let f (x) be an integrable function; its Fourier transform is defined as: ¥ Z F[ f (x)] = fe(w) = f (x)eiwxdx, (1) −¥ while the inverse Fourier transform is: ¥ h i 1 Z F −1 fe(w) = fe(w)e−iwxdw. (2) 2p −¥ Definitions (1) and (2) can be generalized by allowing w to take complex values. To this end, let us suppose that f (x) grows at most exponentially at infinity, i.e., f (x) = O ecjxj as x ! ±¥ and c > 0. Moreover, we split up f (x) as follows: f (x) = f+(x) + f−(x) : f+(x) = 0 8x< 0 and f−(x) = 0 8x >0 (3) The Fourier transform of the component f+(x) is: ¥ Z iRe(w)x −Im(w)x fe+(w) = f+(x)e e dx. (4) 0 It is evident that: ¥ Z −Im(w)x fe+(w) ≤ j f+(x)je dx, (5) 0 and the integral converges provided Im(w) > c. Therefore, fe+(w) exists and is holomor- phic for Im(w) > c. Next, we need to extend the Fourier inversion theorem to recover f+(x) from fe+(w). −ax h i To this end, let F+(x) = f+(x)e , where a > c, so that Fe+(w) = fe+(w + ia) exists and is holomorphic for Im(w) > c − a, in particular for w 2 R, since a > c. Thus, we can apply the Fourier transform inversion theorem obtaining: ¥ ¥ 1 R −iwx −ax 1 R −iwx F+(x) = 2p Fe+(w)e dw ) f+(x)e = 2p fe+(w + ia)e dw −¥ −¥ ¥ (6) 1 R −i(w+ia)x ) f+(x) = 2p fe+(w + ia)e dw. −¥ The final integral corresponds to an integration along a horizontal contour in the complex w-plane: ¥+ia Z 1 −iwx f+(x) = fe+(w)e dw. (7) 2p −¥+ia Suppose fe+(w) can be continued below Im(w) = c, so that it is holomorphic in some region W+ ⊃ fw : Im(w) > cg except for singularities at w = a1, a2, ··· . By the deformation theory, the inversion contour G+ = fx + ia : −¥ < x < ¥g may be deformed into W+ provided it passes above the singularities of fe+(w). Since the singularities of fe+(w) are below the inversion contour, for x < 0, we can close the contour at +i¥. This gives the expected result that f+(x) = 0 for x < 0. For x > 0, we would need to close the contour in Im(w) < 0, picking up the contributions from the singularities in fe+(w) and giving a nonzero value of f+(x). The same procedure works for f−(x) with everything upside down, namely fe−(w) can be continued above Im(w) = −c, so that it is holomorphic in some region W− ⊃ Particles 2021, 4 327 fw : Im(w) < −cg except for singularities at w = b1, b2, ··· . If there is a non-empty overlap region W = W+ \ W−nfaig [ bj , where faig are the singularities of fe+(w) and bj are the singularities of fe−(w), then the Fourier transform of f (x) is defined as: fe(w) = fe+(w) + fe−(w), (8) for w 2 W. Moreover, if G+ and G− can be deformed into the same contour G ⊂ W, with G above the singularities of fe+(w) and below the singularities of fe−(w), then: 1 Z f (x) = fe(w)e−iwxdw. (9) 2p G What is discussed in this section represents the method that will be implemented below to solve the TKG equation. 3. Dynamics of Wave Packets in the TKG Equation The TKG equation can be obtained by quantizing the tachyonic energy-momentum relation using the operators E ! i}¶/¶t and p ! −i}cr : ¶2 }2 − }2c2r2 − m2c4 y(r, t) = 0. (10) ¶t2 For free scalar particles, Equation (10) can be solved by Fourier transform in time, which allows decomposing the wave function y(r, t) in its monochromatic temporal com- ponents, using the initial conditions given for y(r, t)jr=0 and ¶y(r, t)/¶rjr=0. To simplify the following discussion, we suppose that the motion of the particle takes place along the z-axis. The solution is a wave packet represented by the following inverse Fourier transform: 1 Z h i y(z, t) = A(w)eiK(w)z + B(w)e−iK(w)z e−iwtdw, (11) 2p G where G is a path in the w-plane. The quantity w is defined as w = ±E/}, and, for a tachyon, it is always a real quantity except when E = ±mc2 (this last statement will be clarified in Section4 where we calculate the group velocity of the tachyonic wave packet). In fact: −1/2 −1/2 u2 u2 E = ±gmc2 = ± 1 − imc2 = ± − 1 mc2, (12) c2 c2 where g is the tachyonic Lorentz factor and u > c is the classical tachyonic velocity. Returning to Equation (11), the term K(w) is the dispersion relation whose characteristic equation is: w2 − w2 K2(w) = 0 where w2 = −m2c4/}2. (13) c2 0 Equation (13) is obtained from Equation (10) upon factorisation y(z, t) = j(z)e−iwt. The aim is to prove that Equation (1) admits a general solution in which no spacelike components arise. In other words, we want to prove that this solution is completely analogous to the one that solves the ordinary KG equation, where w is always real, and tends to the solution of the d’Alembert wave equation as the mass energy tends to zero. We point out that the spacelike components are those associated with complex values of w, and therefore of K(w). The coefficients A(w) and B(w) are calculated once the following boundary conditions have been set [21]: 0 f (t) = y(0, t) and f (t) = ¶y(z, t)/¶zjz=0. (14) Particles 2021, 4 328 As we will see, these conditions determine the choice of the path G of integration. Using the boundary conditions in Equation (14), we obtain [21]: 8 h 0 i ( ) = 1 R ¥ ( ) − f (t) iwt < A w 2 −¥ f t i K(w) e dt h 0 i (15) ( ) = 1 R ¥ ( ) + f (t) iwt : B w 2 −¥ f t i K(w) e dt The computation of the integrals in Equation (15) can be simplified if we consider f (t) as a semi-infinite wave train with a sharp edge.
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