A tachyon (//) or tachyonic particle is a hypothetical particle that always travels faster than light. Most physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such particles did exist, they could be used to build a tachyonic antitelephone and send signals faster than light, which (according to special relativity) would lead to violations of causality. No experimental evidence for the existence of such particles has been found.
The possibility of particles moving faster than light was first proposed by Robert Ehrilch and Arnold Sommerfeld, independently of each other. In the 1967 paper that coined the term, Gerald Feinberg proposed that tachyonic particles could be quanta of a quantum field with imaginary mass. However, it was soon realized that excitations of such imaginary mass fields do not under any circumstances propagate faster than light, and instead the imaginary mass gives rise to an instability known as tachyon condensation. Nevertheless, in modern physics the term "tachyon" often refers to imaginary mass fields rather than to faster-than-light particles. Such fields have come to play a significant role in modern physics.
The term comes from the Greek: ταχύ, tachy, meaning "rapid". The complementary particle types are called luxons (which always move at the speed of light) and bradyons (which always move slower than light); both of these particle types are known to exist.
Tachyons in relativity theoryEdit
In special relativity, a faster-than-light particle would have space-like four-momentum, in contrast to ordinary particles that have time-like four-momentum. Although in some theories the mass of tachyons is regarded as imaginary, in some modern formulations the mass is considered real, the formulas for the momentum and energy being redefined to this end. Moreover, since tachyons are constrained to the spacelike portion of the energy–momentum graph, they could not slow down to subluminal speeds.
In a Lorentz invariant theory, the same formulas that apply to ordinary slower-than-light particles (sometimes called "bradyons" in discussions of tachyons) must also apply to tachyons. In particular the energy–momentum relation:
This equation shows that the total energy of a particle (bradyon or tachyon) contains a contribution from its rest mass (the "rest mass–energy") and a contribution from its motion, the kinetic energy. When v is larger than c, the denominator in the equation for the energy is "imaginary", as the value under the radical is negative. Because the total energy must be real,[dubious ] the numerator must also be imaginary: i.e. the rest mass m must be imaginary, as a pure imaginary number divided by another pure imaginary number is a real number.
One curious effect is that, unlike ordinary particles, the speed of a tachyon increases as its energy decreases. In particular, approaches zero when approaches infinity. (For ordinary bradyonic matter, E increases with increasing speed, becoming arbitrarily large as v approaches c, the speed of light). Therefore, just as bradyons are forbidden to break the light-speed barrier, so too are tachyons forbidden from slowing down to below c, because infinite energy is required to reach the barrier from either above or below
In 1985, Chodos proposed that neutrinos can have a tachyonic nature. The possibility of standard model particles moving at superluminal speeds can be modeled using Lorentz invariance violating terms, for example in the Standard-Model Extension. In this framework, neutrinos experience Lorentz-violating oscillations and can travel faster than light at high energies. This proposal was strongly criticized.
A tachyon with an electric charge would lose energy as Cherenkov radiation—just as ordinary charged particles do when they exceed the local speed of light in a medium (other than a hard vacuum). A charged tachyon traveling in a vacuum, therefore, undergoes a constant proper time acceleration and, by necessity, its worldline forms a hyperbola in space-time. However reducing a tachyon's energy increases its speed, so that the single hyperbola formed is of two oppositely charged tachyons with opposite momenta (same magnitude, opposite sign) which annihilate each other when they simultaneously reach infinite speed at the same place in space. (At infinite speed, the two tachyons have no energy each and finite momentum of opposite direction, so no conservation laws are violated in their mutual annihilation. The time of annihilation is frame dependent.)
Even an electrically neutral tachyon would be expected to lose energy via gravitational Cherenkov radiation (unless gravitons are themselves tachyons), because it has a gravitational mass, and therefore increases in speed as it travels, as described above. If the tachyon interacts with any other particles, it can also radiate Cherenkov energy into those particles. Neutrinos interact with the other particles of the Standard Model, and Andrew Cohen and Sheldon Glashow used this to argue that the faster-than-light neutrino anomaly cannot be explained by making neutrinos propagate faster than light, and must instead be due to an error in the experiment. Further investigation of the experiment showed that the results were indeed erroneous.
Causality is a fundamental principle of physics. If tachyons can transmit information faster than light, then according to relativity they violate causality, leading to logical paradoxes of the "kill your own grandfather" type. This is often illustrated with thought experiments such as the "tachyon telephone paradox" or "logically pernicious self-inhibitor."
The problem can be understood in terms of the relativity of simultaneity in special relativity, which says that different inertial reference frames will disagree on whether two events at different locations happened "at the same time" or not, and they can also disagree on the order of the two events (technically, these disagreements occur when the spacetime interval between the events is 'space-like', meaning that neither event lies in the future light cone of the other).
If one of the two events represents the sending of a signal from one location and the second event represents the reception of the same signal at another location, then as long as the signal is moving at the speed of light or slower, the mathematics of simultaneity ensures that all reference frames agree that the transmission-event happened before the reception-event. However, in the case of a hypothetical signal moving faster than light, there would always be some frames in which the signal was received before it was sent so that the signal could be said to have moved backward in time. Because one of the two fundamental postulates of special relativity says that the laws of physics should work the same way in every inertial frame, if it is possible for signals to move backward in time in any one frame, it must be possible in all frames. This means that if observer A sends a signal to observer B which moves faster than light in A's frame but backwards in time in B's frame, and then B sends a reply which moves faster than light in B's frame but backwards in time in A's frame, it could work out that A receives the reply before sending the original signal, challenging causality in every frame and opening the door to severe logical paradoxes. Mathematical details can be found in the tachyonic antitelephone article, and an illustration of such a scenario using spacetime diagrams can be found in Baker, R. (2003)
The reinterpretation principle asserts that a tachyon sent back in time can always be reinterpreted as a tachyon traveling forward in time, because observers cannot distinguish between the emission and absorption of tachyons. The attempt to detect a tachyon from the future (and violate causality) would actually create the same tachyon and send it forward in time (which is causal).
However, this principle is not widely accepted as resolving the paradoxes. Instead, what would be required to avoid paradoxes is that unlike any known particle, tachyons do not interact in any way and can never be detected or observed, because otherwise a tachyon beam could be modulated and used to create an anti-telephone or a "logically pernicious self-inhibitor". All forms of energy are believed to interact at least gravitationally, and many authors state that superluminal propagation in Lorentz invariant theories always leads to causal paradoxes.
In modern physics, all fundamental particles are regarded as excitations of quantum fields. There are several distinct ways in which tachyonic particles could be embedded into a field theory.
Fields with imaginary massEdit
In the paper that coined the term "tachyon", Gerald Feinberg studied Lorentz invariant quantum fields with imaginary mass. Because the group velocity for such a field is superluminal, naively it appears that its excitations propagate faster than light. However, it was quickly understood that the superluminal group velocity does not correspond to the speed of propagation of any localized excitation (like a particle). Instead, the negative mass represents an instability to tachyon condensation, and all excitations of the field propagate subluminally and are consistent with causality. Despite having no faster-than-light propagation, such fields are referred to simply as "tachyons" in many sources.
Tachyonic fields play an important role in modern physics. Perhaps the most famous is the Higgs boson of the Standard Model of particle physics, which has an imaginary mass in its uncondensed phase. In general, the phenomenon of spontaneous symmetry breaking, which is closely related to tachyon condensation, plays an important role in many aspects of theoretical physics, including the Ginzburg–Landau and BCS theories of superconductivity. Another example of a tachyonic field is the tachyon of bosonic string theory.
Tachyons are predicted by bosonic string theory and also the Neveu-Schwarz (NS) and NS-NS sectors, which are respectively the open bosonic sector and closed bosonic sector, of RNS Superstring theory prior to the GSO projection. However such tachyons are not possible due to the Sen conjecture, also known as tachyon condensation. This resulted in the necessity for the GSO projection.
In theories that do not respect Lorentz invariance, the speed of light is not (necessarily) a barrier, and particles can travel faster than the speed of light without infinite energy or causal paradoxes. A class of field theories of that type is the so-called Standard Model extensions. However, the experimental evidence for Lorentz invariance is extremely good, so such theories are very tightly constrained.
Fields with non-canonical kinetic termEdit
By modifying the kinetic energy of the field, it is possible to produce Lorentz invariant field theories with excitations that propagate superluminally. However, such theories, in general, do not have a well-defined Cauchy problem (for reasons related to the issues of causality discussed above), and are probably inconsistent quantum mechanically.
The term "tachyon" was coined by Gerald Feinberg in a 1967 paper titled "Possibility of Faster-Than-Light Particles". He had been inspired by the science-fiction story "Beep" by James Blish. Feinberg studied the kinematics of such particles according to special relativity. In his paper he also introduced fields with imaginary mass (now also referred to as "tachyons") in an attempt to understand the microphysical origin such particles might have.
In September 2011, it was reported that a tau neutrino had traveled faster than the speed of light in a major release by CERN; however, later updates from CERN on the OPERA project indicate that the faster-than-light readings were due to "a faulty element of the experiment's fibre optic timing system".
Tachyons have appeared in many works of fiction. They have been used as a standby mechanism upon which many science fiction authors rely to establish faster-than-light communication, with or without reference to causality issues. The word tachyon has become widely recognized to such an extent that it can impart a science-fictional connotation even if the subject in question has no particular relation to superluminal travel (a form of technobabble, akin to positronic brain).
- Lisa Randall, Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions, p.286: "People initially thought of tachyons as particles traveling faster than the speed of light...But we now know that a tachyon indicates an instability in a theory that contains it. Regrettably, for science fiction fans, tachyons are not real physical particles that appear in nature."
- Tipler, Paul A.; Llewellyn, Ralph A. (2008). Modern Physics (5th ed.). New York: W.H. Freeman & Co. p. 54. ISBN 978-0-7167-7550-8.
... so existence of particles v > c ... Called tachyons ... would present relativity with serious ... problems of infinite creation energies and causality paradoxes.
- Feinberg, G. (1967). "Possibility of Faster-Than-Light Particles". Physical Review. 159 (5): 1089–1105. Bibcode:1967PhRv..159.1089F. doi:10.1103/PhysRev.159.1089. See also Feinberg's later paper: Phys. Rev. D 17, 1651 (1978)
- Aharonov, Y.; Komar, A.; Susskind, L. (1969). "Superluminal Behavior, Causality, and Instability". Phys. Rev. 182 (5): 1400–1403. Bibcode:1969PhRv..182.1400A. doi:10.1103/PhysRev.182.1400.
- A. Sen, "Rolling tachyon," JHEP 0204, 048 (2002). Cited 720 times as of 2/2012.
- Recami, E. (2007-10-16). "Classical tachyons and possible applications". Rivista del Nuovo Cimento. 9 (6): 1–178. Bibcode:1986NCimR...9e...1R. doi:10.1007/BF02724327. ISSN 1826-9850.
- Vieira, R. S. (2011). "An introduction to the theory of tachyons". Rev. Bras. Ens. Fis. 34 (3). arXiv:1112.4187. Bibcode:2011arXiv1112.4187V.
- Hill, James M.; Cox, Barry J. (2012-12-08). "Einstein's special relativity beyond the speed of light". Proc. R. Soc. A. 468 (2148): 4174–4192. Bibcode:2012RSPSA.468.4174H. doi:10.1098/rspa.2012.0340. ISSN 1364-5021.
- Benford, G.; Book, D.; Newcomb, W. (1970). "The Tachyonic Antitelephone". Physical Review D. 2 (2): 263–265. Bibcode:1970PhRvD...2..263B. doi:10.1103/PhysRevD.2.263.
- Chodos, A. (1985). "The Neutrino as a Tachyon". Physics Letters B. 150 (6): 431–435. Bibcode:1985PhLB..150..431C. doi:10.1016/0370-2693(85)90460-5.
- Colladay, D.; Kostelecky, V. A. (1997). "CPT Violation and the Standard Model". Physical Review D. 55 (11): 6760–6774. arXiv:hep-ph/9703464. Bibcode:1997PhRvD..55.6760C. doi:10.1103/PhysRevD.55.6760.
- Colladay, D.; Kostelecky, V. A. (1998). "Lorentz-Violating Extension of the Standard Model". Physical Review D. 58 (11): 116002. arXiv:hep-ph/9809521. Bibcode:1998PhRvD..58k6002C. doi:10.1103/PhysRevD.58.116002.
- Kostelecky, V. A. (2004). "Gravity, Lorentz Violation, and the Standard Model". Physical Review D. 69 (10): 105009. arXiv:hep-th/0312310. Bibcode:2004PhRvD..69j5009K. doi:10.1103/PhysRevD.69.105009.
- Hughes, Richard J; Stephenson, G.J (1990). "Against tachyonic neutrinos". Physics Letters B. 244 (1): 95–100. Bibcode:1990PhLB..244...95H. doi:10.1016/0370-2693(90)90275-B.
Bock, R. K. (9 April 1998). "Cherenkov Radiation". The Particle Detector BriefBook. CERN. Archived from the original on 18 December 2007. Retrieved 2011-09-23. Cite uses deprecated parameter
- Cohen, Andrew G. & Glashow, Sheldon L. (2011). "Pair Creation Constrains Superluminal Neutrino Propagation". Phys. Rev. Lett. 107 (18): 181803. arXiv:1109.6562. Bibcode:2011PhRvL.107r1803C. doi:10.1103/PhysRevLett.107.181803. PMID 22107624.
- P. Fitzgerald, "Tachyons, Backward Casuation, and Freedom", PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1970 (1970), pp. 425–426: "A more powerful argument to show that retrocausal tachyons involve an intolerable conceptual difficulty is illustrated by the Case of the Logically Pernicious Self-Inhibitor..."
- Mark, J. "The Special Theory of Relativity" (PDF). University of Cincinnati. pp. 7–11. Archived from the original (PDF) on 2006-09-13. Retrieved 2006-10-27.
Grøn, Ø.; Hervik, S. (2007). Einstein's General Theory of Relativity: With Modern Applications in Cosmology. Springer. p. 39. ISBN 978-0-387-69199-2.
The tachyon telephone paradox cannot be resolved by means of the reinterpretation principle.
- . Baker, R. (12 September 2003). "Relativity, FTL and causality". Sharp Blue. Retrieved 2011-09-23.
- Bilaniuk, O.-M. P.; Deshpande, V. K.; Sudarshan, E. C. G. (1962). "'Meta' Relativity". American Journal of Physics. 30 (10): 718. Bibcode:1962AmJPh..30..718B. doi:10.1119/1.1941773.
- Erasmo Recami, Flavio Fontana, Roberto Garavaglia, "About Superluminal motions and Special Relativity: A Discussion of some recent Experiments, and the solution of the Causal Paradoxes", International Journal of Modern Physics A15 (2000) 2793–2812, abstract: "it is possible...to solve also the known causal paradoxes, devised for "faster than light" motion, although this is not widely recognized yet." [emphasis added].
- Carlos Barceló, Stefano Finazzi, Stefano Liberati, "On the impossibility of superluminal travel: the warp drive lesson", Second prize of the 2009 FQXi essay contest "What is Ultimately Possible in Physics?", p.8: "As a matter of fact, any mechanism for superluminal travel can be easily turned into a time machine and hence lead to the typical causality paradoxes..." 
- Allan Adams, Nima Arkani-Hamed, Sergei Dubovsky, Alberto Nicolis, Riccardo Rattazzi, "Causality, Analyticity and an IR Obstruction to UV Completion", JHEP 0610 (2006) 014 .
- Brian Greene, The Elegant Universe, Vintage Books (2000)
- Kutasov, David; Marino, Marcos & Moore, Gregory W. (2000). "Some exact results on tachyon condensation in string field theory". JHEP. 2000 (10): 045. arXiv:hep-th/0009148. Bibcode:2000JHEP...10..045K. doi:10.1088/1126-6708/2000/10/045. arXiv EFI-2000-32, RUNHETC-2000-34.
- NOVA, "The Elegant Universe", PBS television special, https://www.pbs.org/wgbh/nova/elegant/
- G. W. Gibbons, "Cosmological evolution of the rolling tachyon," Phys. Lett. B 537, 1 (2002)
- J. Polchinski, String Theory, Cambridge University Press, Cambridge, UK (1998)
- Glashow, Sheldon Lee (2004). "Atmospheric neutrino constraints on Lorentz violation". Cite journal requires
- Coleman, Sidney R. & Glashow, Sheldon L. (1999). "High-energy tests of Lorentz invariance". Phys. Rev. D59 (11): 116008. arXiv:hep-ph/9812418. Bibcode:1999PhRvD..59k6008C. doi:10.1103/PhysRevD.59.116008.
- "He told me years later that he had begun thinking about tachyons because he was inspired by James Blish's  short story, "Beep". In it, a faster-than-light communicator plays a crucial role in a future society but has an annoying final beep at the end of every message. The communicator necessarily allows sending of signals backward in time, even when that's not your intention. Eventually, the characters discover that all future messages are compressed into that beep, so the future is known, more or less by accident. Feinberg had set out to see if such a gadget was theoretically possible." pg276 of Gregory Benford's "Old Legends"
- Sommerfeld, A. (1904). "Simplified deduction of the field and the forces of an electron moving in any given way". KNKL. Acad. Wetensch. 7: 345–367.
- Bilaniuk, O.-M. P.; Sudarshan, E. C. G. (1969). "Particles beyond the Light Barrier". Physics Today. 22 (5): 43–51. Bibcode:1969PhT....22e..43B. doi:10.1063/1.3035574.
"Neutrinos sent from CERN to Gran Sasso respect the cosmic speed limit" (Press release). CERN. 8 June 2012. Archived from the original on 22 February 2014. Retrieved 2012-06-08. Cite uses deprecated parameter
 See, e.g. TV series Journeyman. https://en.wikipedia.org/wiki/Journeyman_(TV_series)