# Theoretical Investigation of Subluminal Particles Endowed with Imaginary Mass

## Abstract

**:**

## 1. Introduction

## 2. Preliminary Notions on the Fourier Transforms in the Complex Plane

## 3. Dynamics of Wave Packets in the TKG Equation

## 4. Validation of the Obtained Result for Half-Integer Spin Tachyon

## 5. Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Sommerfeld, A. Simplified deduction of the field and the forces of an electron, moving in a given way. Proc. Amst. Acad.
**1904**, 8, 346–367. [Google Scholar] - Heaviside, O. Electrical Papers; Macmillian: London, UK, 1892; Volume II. [Google Scholar]
- Nahin, P.J. Oliver Heaviside: An accidental time traveller. Phil. Trans. R. Soc. A
**2018**, 376. [Google Scholar] [CrossRef] [Green Version] - Bilaniuk, O.M.P.; Dashpande, V.K.; Surdarshan, E.C.G. Meta Relativity. Am. J. Phys.
**1962**, 30, 718–723. [Google Scholar] [CrossRef] - Bilaniuk, O.M.P.; Surdarshan, E.C.G. Particles beyond the Light Barrier. Phys. Today
**1968**, 22, 43–51. [Google Scholar] [CrossRef] - Feinberg, G. Possibility of Faster-Than-Light Particles. Phys. Rev.
**1967**, 159, 1089–1105. [Google Scholar] [CrossRef] - Ecker, G. Quantum Field Theory with Space-Like Momentum Spectrum. Ann. Phys.
**1970**, 58, 303–321. [Google Scholar] [CrossRef] - Ehrlich, R. Faster-Than-Light Speeds, Tachyons, and the Possibility of Tachyonic Neutrinos. Am. J. Phys.
**2003**, 71, 1109–1114. [Google Scholar] [CrossRef] - Recami, E. Superluminal Waves and Objects: An Overview on the Relevant Experiments. J. Phys. Conf. Ser.
**2009**, 196, 012020. [Google Scholar] [CrossRef] - Recami, E. Tachyon Kinematics and Causality: A Systematic Thorough Analysis of the Tachyon Causal Paradoxes. Found. Phys.
**1987**, 17, 239–296. [Google Scholar] [CrossRef] - Bose, S.K. Aspects of Tachyon Theory. J. Phys. Conf. Ser.
**2009**, 196, 012022. [Google Scholar] [CrossRef] - Adamson, P.; Andreopoulos, C.; Arms, K.E.; Armstrong, R.; Auty, D.J.; Avvakumov, S.; Ayres, D.S.; Baller, B.; Barish, B.; Barnes, P.D., Jr.; et al. Measurement of Neutrino Velocity with the MINOS Detectors and NuMI Neutrino Beam. Phys. Rev. D
**2007**, 76, 072005. [Google Scholar] [CrossRef] [Green Version] - Mohseni Sadjadi, H.; Anari, V. Mass Varying Neutrinos, Symmetry Breaking, and Cosmic Acceleration. Phys. Rev. D
**2017**, 95, 123521. [Google Scholar] [CrossRef] [Green Version] - Backenstoss, G.; Daniel, H.; Koch, H.; Von der Malsburg, C.; Poelz, G.; Povel, H.P.; Schmitt, H.; Tauscher, L. π
^{−}mass and vacuum polarization. A re-evaluation. Phys. Lett. B**1973**, 43, 539–541. [Google Scholar] [CrossRef] - Lu, D.C.; Delker, L.; Dugan, G.; Wu, C.S.; Caffrey, A.J.; Cheng, Y.T.; Lee, Y.K. New High Accuracy Measurement of the Pionic Mass. Phys. Rev. Lett.
**1980**, 45, 1066–1069. [Google Scholar] [CrossRef] - Soucek, J. Quaternion Quantum Mechanics and the Description of Tachyons and Quarks. Czechoslov. J. Phys. B
**1979**, 29, 315–318. [Google Scholar] [CrossRef] - Schwartz, C. Toward a Quantum Theory of Tachyon Fields. Int. J. Mod. Phys. A
**2016**, 31, 1650041. [Google Scholar] [CrossRef] [Green Version] - Salesi, G. Slower-Than-Light Spin-1/2 Particles Endowed with Negative Mass Squared. Int. J. Mod. Phys. A
**1985**, 28, 5103. [Google Scholar] [CrossRef] [Green Version] - Jentschura, U.D.; Wundt, B.J. Pseudo-Hermitian Quantum Dynamics of Tachyonic Spin-1/2 Particles. J. Phys. A Math. Theor.
**2012**, 45, 444017. [Google Scholar] [CrossRef] [Green Version] - Chodos, A.; Hauser, A.I. The Neutrino as a Tachyon. Phys. Lett. B
**1985**, 150, 295–302. [Google Scholar] [CrossRef] - Partington, J.R. Fourier Transforms and Complex Analysis. In Harmonic Analysis and Rational Approximation; Fournier, J.D., Grimm, J., Leblond, J.R., Partington, J.R., Eds.; Lecture Notes in Control and Information Science; Springer: Berlin/Heidelberg, Germany, 2006; Volume 327. [Google Scholar]
- Kammler, D.W. A First Course in Fourier Analysis, 2nd ed.; Cambridge University Press: Cambridge, UK, 2008. [Google Scholar]
- Hecht, K.T. The Motion of Wave Packets: Fourier Analysis. In Quantum Mechanics; Graduate Texts in Contemporary Physics; Springer: New York, NY, USA, 2000. [Google Scholar]
- Jentschura, U.D. Tachyonic Field Theory and Neutrino Mass Running. Cent. Eur. J. Phys.
**2012**, 10, 749–762. [Google Scholar] [CrossRef] [Green Version] - Tanaka, S. Theory of Matter with Super Light Velocity. Prog. Theor. Phys.
**1960**, 24, 177–181. [Google Scholar] [CrossRef] [Green Version] - Park, S.T. Propagation of a Relativistic Electron Wave Packet in the Dirac Equation. Phys. Rev. A
**2012**, 86, 062105. [Google Scholar] [CrossRef] [Green Version] - Winful, H.G. Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox. Phys. Rep.
**2006**, 436, 1–69. [Google Scholar] [CrossRef] - Recami, E.; Mignani, R. Classical theory of tachyons (special relativity extended to superluminal frames and objects). Riv. Nuovo Cim.
**1974**, 4, 209–290. [Google Scholar] [CrossRef] - Dawe, R.L.; Hines, K.C. The Physics of Tachyons. Aust. J. Phys.
**1992**, 45, 591–620. [Google Scholar] [CrossRef] [Green Version] - Davies, P.C.W. Quantum Tunnelling Time. Am. J. Phys.
**2005**, 73, 23–27. [Google Scholar] [CrossRef] [Green Version] - Ramos, R.; Spierings, D.; Racicot, I. Steinberg, A.M. Measurement of the time spent by a tunnelling atom within the barrier region. Nature
**2020**, 583, 529–532. [Google Scholar] [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nanni, L.
Theoretical Investigation of Subluminal Particles Endowed with Imaginary Mass. *Particles* **2021**, *4*, 325-332.
https://doi.org/10.3390/particles4020027

**AMA Style**

Nanni L.
Theoretical Investigation of Subluminal Particles Endowed with Imaginary Mass. *Particles*. 2021; 4(2):325-332.
https://doi.org/10.3390/particles4020027

**Chicago/Turabian Style**

Nanni, Luca.
2021. "Theoretical Investigation of Subluminal Particles Endowed with Imaginary Mass" *Particles* 4, no. 2: 325-332.
https://doi.org/10.3390/particles4020027