Editing Tachyon

{{Short description|Hypothetical faster-than-light particle}}
{{About|hypothetical faster-than-light particles|quantum fields with imaginary mass|Tachyonic field|other uses}}
{{Use dmy dates|date=June 2023}}
{{Infobox particle
| name = Tachyon
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| classification = [[Elementary particle]]
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| status = Hypothetical 
| theorized = 1967 
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}}

A '''tachyon''' ({{IPAc-en|ˈ|t|æ|k|i|ɒ|n}}) or '''tachyonic particle''' is a hypothetical [[particle]] that always travels [[Faster-than-light|faster than light]]. [[Physicist]]s posit that faster-than-light particles cannot exist because they are inconsistent with the known [[Scientific law#Laws of physics|laws of physics]].<ref name="Tipler" /><ref name="Randall 2005 p286">{{cite book |last=Randall |first=Lisa |date=2005 |title=Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions |isbn=9780060531089 |quote=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. |page=286 |publisher=Harper Collins }}</ref> If such particles did exist they perhaps could be used to send signals faster than light and into the past. According to the [[theory of relativity]] this would violate [[Causality (physics)|causality]], leading to [[logical paradox]]es such as the [[grandfather paradox]].<ref name="Tipler" /> Tachyons would exhibit the unusual property of increasing in speed as their energy decreases, and would require infinite energy to slow to the speed of light. No verifiable experimental evidence for the existence of such particles has been found.

In the 1967 paper that coined the term, [[Gerald Feinberg]] proposed that tachyonic particles could be made from excitations of a [[Quantum field theory|quantum field]] with [[imaginary mass]].<ref name="Feinberg 1967-1969" /> However, it was soon realized that Feinberg's model did not in fact allow for [[superluminal]] (faster than light) particles or signals and that tachyonic fields merely give rise to instabilities, not causality violations.<ref name="Aharonov-etal-1969" /> The term [[tachyonic field]] refers to imaginary mass fields rather than to faster-than-light particles.<ref name="Randall 2005 p286" /><ref name="Sen" />

== Etymology ==
The term ''tachyon'' comes from the {{langx|el|links=no|[[Wiktionary:ταχύς|ταχύς]]}}, ''tachus'', meaning ''swift''.<ref name="FoxKuperLipson">{{Cite journal |last1=Fox |first1=R. |last2=Kuper |first2=C. G. |last3=Lipson |first3=S. G. |date=1970 |title=Faster-Than-Light Group Velocities and Causality Violation |url=https://www.jstor.org/stable/77636 |journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences |volume=316 |issue=1527 |pages=515–524 |doi=10.1098/rspa.1970.0093 |jstor=77636 |bibcode=1970RSPSA.316..515F |issn=0080-4630|url-access=subscription }}</ref>{{rp|515|q= The name 'tachyon' was proposed by Feinberg (1967), from the Greek ταχύς ('swift').}} The complementary particle types are called [[Massless particle|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.

== History ==
Faster-than-light particles were discussed before the advent of relativity by such physicists as [[JJ Thomson]] and [[Arnold Sommerfeld]].<ref name="Sommerfeld" /><ref name="sudarshan62" />{{rp|718}} The possibility of existence of faster-than-light particles was also proposed by {{Ill|Lev Yakovlevich Shtrum|lt=|ru|Штрум, Лев Яковлевич}} in 1923.<ref>{{Cite journal |last1=Chashchina |first1=Olga |last2=Silagadze |first2=Zurab |date=13 April 2022 |title=Relativity 4-ever? |journal=Physics |volume=4 |issue=2 |pages=421–439 |arxiv=2107.10739 |doi=10.3390/physics4020028 |issn=2624-8174 |doi-access=free|bibcode=2022Physi...4..421C }}</ref>

In 1962<ref name="sudarshan62" /> and again in 1969<ref name="sudarshan69" /> Oleksa-Myron Bilanuik, Vijay Deshpande and [[E. C. George Sudarshan]] discussed the possibility of a class of faster-than-light particles consistent with special relativity. As part of their discussion they point out that light particles are never accelerated but rather are created with the full velocity of light. Similarly they argue that while accelerating normal matter beyond the speed of light is inconsistent with special relativity, this does not prevent creation of faster than light particles.<ref name="Feinberg 1967-1969" /> 

The term ''tachyon'' was coined by [[Gerald Feinberg]] in a 1967 paper titled "Possibility of faster-than-light particles".<ref name="Feinberg 1967-1969" /> He had been inspired by the science-fiction story "Beep" by [[James Blish]].<ref name="Benford-legends" /> Feinberg studied the kinematics of such particles according to [[special relativity]]. In his paper, he also introduced [[Tachyonic field|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; however, later updates from CERN on the [[OPERA experiment]] indicate that the [[2011 OPERA faster-than-light neutrino anomaly|faster-than-light readings]] were due to a faulty element of the experiment's fibre optic timing system.<ref name="CERN-2012-06-08" />

== Special relativity ==
In [[special relativity]], a faster-than-light particle would have [[spacelike]] [[four-momentum]],<ref name="Feinberg 1967-1969" /> unlike ordinary particles that have [[time-like]] [[four-momentum]]. While some theories suggest the mass of tachyons is [[imaginary number|imaginary]], modern formulations often consider their mass to be real,<ref name=":0">{{cite journal |last=Recami |first=E. |date=16 October 2007 |title=Classical tachyons and possible applications |journal=Rivista del Nuovo Cimento |volume=9 |issue=6 |pages=1–178 |doi=10.1007/BF02724327 |s2cid=120041976 |issn=1826-9850 |bibcode=1986NCimR...9e...1R}}</ref><ref name=":1">{{Cite journal |last=Vieira |first=R. S. |date=2011 |title=An introduction to the theory of tachyons |journal=Rev. Bras. Ens. Fis. |volume=34 |issue=3 |arxiv=1112.4187 |bibcode=2011arXiv1112.4187V}}</ref><ref name=":2">{{Cite journal |last1=Hill |first1=James M. |last2=Cox |first2=Barry J. |date=8 December 2012 |title=Einstein's special relativity beyond the speed of light |journal=Proceedings of the Royal Society A |volume=468 |issue=2148 |pages=4174–4192 |bibcode=2012RSPSA.468.4174H |doi=10.1098/rspa.2012.0340 |issn=1364-5021 |doi-access=free}}</ref> with redefined formulas for momentum and energy. Additionally, since tachyons are confined to the [[spacelike]] portion of the energy–momentum graph, they cannot slow down to subluminal (slower-than-light) speeds.<ref name="Feinberg 1967-1969" />

=== Mass ===
{{Main|Mass#Tachyonic particles and imaginary (complex) mass|Tachyonic field}}

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]]:
:<math>E^2 = (pc)^2+ (mc^2)^2 \;</math>
(where '''p''' is the relativistic [[momentum]] of the bradyon and '''m''' is its [[rest mass]]) should still apply, along with the formula for the total energy of a particle:
:<math>E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}.</math>
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 '''<math> v </math>''' (the particle's velocity) is larger than '''<math> c </math>''' (the speed of light), the [[Fraction|denominator]] in the equation for the energy is [[imaginary number|imaginary]], as the value under the square root is negative. Because the total [[energy]] of the particle must be [[real number|real]] (and not a [[Complex number|complex]] or imaginary number) in order to have any practical meaning as a measurement, 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).

In some modern formulations of the theory, the mass of tachyons is regarded as real.<ref name=":0" /><ref name=":1" /><ref name=":2" />

=== Speed ===
One curious effect is that, unlike ordinary particles, the speed of a tachyon ''increases'' as its energy decreases. In particular, <math> E </math> approaches zero when <math> v </math> approaches infinity. (For ordinary bradyonic matter, ''<math> E </math>'' increases with increasing speed, becoming arbitrarily large as ''<math> v </math>'' approaches '''''<math> c </math>''''', the [[speed of light]].) Therefore, just as bradyons are forbidden to break the light-speed barrier, so are tachyons forbidden from slowing down to below ''c'', because infinite energy is required to reach the barrier from either above or below.

As noted by [[Albert Einstein]], [[Richard C. Tolman]], and others, [[special relativity]] implies that faster-than-light particles, if they existed, [[Tachyonic antitelephone|could be used to communicate backwards in time]].<ref name="Benford" />

=== Neutrinos ===
In 1985, Chodos proposed that [[neutrino]]s can have a tachyonic nature.<ref name="Chodos" /> The possibility of standard model particles moving at faster-than-light speeds can be modeled using [[Lorentz invariance]] violating terms, for example in the [[Standard-Model Extension]].<ref name="Colladay" /><ref>
{{cite journal |last1=Colladay |first1=D. |last2=Kostelecky |first2=V. A. |date=1998 |title=Lorentz-Violating Extension of the Standard Model |journal=[[Physical Review D]] |volume=58 |issue=11 |pages=116002 |arxiv=hep-ph/9809521 |bibcode=1998PhRvD..58k6002C |doi=10.1103/PhysRevD.58.116002 |s2cid=4013391}}
</ref><ref>
{{cite journal |last=Kostelecky |first=V. A. |date=2004 |title=Gravity, Lorentz Violation, and the Standard Model |journal=[[Physical Review D]] |volume=69 |issue=10 |pages=105009 |arxiv=hep-th/0312310 |bibcode=2004PhRvD..69j5009K |doi=10.1103/PhysRevD.69.105009 |s2cid=55185765}}
</ref> In this framework, neutrinos experience [[Lorentz-violating neutrino oscillations|Lorentz-violating oscillations]] and can travel faster than light at high energies. This proposal was strongly criticized.<ref>
{{cite journal |last1=Hughes |first1=Richard J. |last2=Stephenson |first2=G. J. |date=1990 |title=Against Tachyonic Neutrinos |url=https://zenodo.org/record/1258487 |journal=Physics Letters B |volume=244 |issue=1 |pages=95–100 |bibcode=1990PhLB..244...95H |doi=10.1016/0370-2693(90)90275-B}}
</ref>

=== Superluminal information ===
[[File:Faster than light implies time travel diagram.svg|thumb|upright=1|[[Spacetime diagram]] showing that moving faster than light implies time travel in the context of special relativity. A spaceship departs from Earth from A to C slower than light. At B, Earth emits a tachyon, which travels faster than light but forward in time in Earth's reference frame. It reaches the spaceship at C. The spaceship then sends another tachyon back to Earth from C to D. This tachyon also travels forward in time in the spaceship's reference frame. This effectively allows Earth to send a signal from B to D, back in time.]]

If tachyons can transmit information faster than light, then, according to relativity, they violate causality, leading to logical paradoxes of the [[Grandfather paradox|"kill your own grandfather"]] type. This is often illustrated with thought experiments such as the "tachyon telephone paradox"<ref name="Benford" /> or "logically pernicious self-inhibitor."<ref name="Fitzgerald" />

The problem can be understood in terms of the [[relativity of simultaneity]] in special relativity, which says that different [[Inertial frame of reference|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#Spacetime intervals|spacetime interval]] between the events is 'space-like', meaning that neither event lies in the future [[light cone]] of the other.)<ref name="Jarrell" />

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.<ref name="Jarrell" /> 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.<ref name="Gron" /> This is known as the [[tachyonic antitelephone]].

==== Reinterpretation principle ====
<!-- "Reinterpretation principle" redirects here. -->

The '''reinterpretation principle'''<ref name="Feinberg 1967-1969" /><ref name="sudarshan62" /><ref name="Gron" /> 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.<ref name="Benford" /><ref name="Gron" /><ref name="Recami" /> 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<ref name="Benford" /> or a "logically pernicious self-inhibitor".<ref name="Fitzgerald" /> All forms of energy are positted to interact at least gravitationally, and many authors state that superluminal propagation in Lorentz invariant theories always leads to causal paradoxes.<ref name="Barcelo" /><ref name="Arkani" />

== Fundamental models ==
In modern physics, all fundamental particles are regarded as excitations of [[Quantum field theory|quantum fields]]. There are several distinct ways in which tachyonic particles could be embedded into a field theory.

=== Fields with imaginary mass ===
{{Main|Tachyonic field}}

In the paper that coined the term "tachyon", Gerald Feinberg studied Lorentz invariant quantum fields with [[imaginary mass]].<ref name="Feinberg 1967-1969" /> Because the [[group velocity]] for such a field is [[Superluminal motion|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.<ref name="susskind" /> Despite having no faster-than-light propagation, such fields are referred to simply as "tachyons" in many sources.<ref name="Sen" /><ref name="Greene" /><ref name="Kutasov" /><ref name="Gibbons" /><ref name="Randall 2005 p286" />

Tachyonic fields play an important role in modern physics. Perhaps the most famous is the [[Higgs boson]] of the [[Standard model of particle physics|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 theory|Ginzburg–Landau]] and [[BCS theory|BCS]] theories of superconductivity. Another example of a tachyonic field is the tachyon of [[bosonic string theory]].<ref name="Greene" /><ref name="Polchinski 1998" />

Tachyons are predicted by bosonic string theory and also the [[Super Virasoro algebra|Neveu-Schwarz]] (NS) and [[NS-NS sector|NS-NS sectors]], which are respectively the open bosonic sector and closed bosonic sector, of [[Superstring theory|RNS superstring theory]] prior to the [[GSO projection]]. However such tachyons are not possible due to the [[Tachyon condensation#Tachyon condensation in string theory|Sen conjecture]], also known as [[tachyon condensation]]. This resulted in the necessity for the [[GSO projection]].

=== Lorentz-violating theories ===
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.<ref name="Barcelo" /> A class of field theories of that type is the so-called [[Standard-Model Extension|Standard Model extensions]]. However, the experimental evidence for Lorentz invariance is extremely good, so such theories are very tightly constrained.<ref name="Glashow2">{{cite arXiv |eprint=hep-ph/0407087 |first=Sheldon Lee |last=Glashow |title=Atmospheric Neutrino Constraints on Lorentz Violation |date=2004 }}</ref><ref name="Coleman" />

=== Fields with non-canonical kinetic term ===
By modifying the [[kinetic term]] of the field, it is possible to produce Lorentz invariant field theories with excitations that propagate superluminally.<ref name="susskind" /><ref name="Arkani" /> 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.

== In fiction ==
{{Main|Tachyons in fiction}}

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 ''[[Tachyons in fiction|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]]'').<ref>{{Cite web |last=Wagstaff |first=Keith |title=The Science Behind ''Star Trek'' Technobabble |url=https://mashable.com/article/star-trek-science-technobabble/ |access-date=12 February 2021 |website=Mashable |date=15 July 2018 }}</ref>

== See also ==
* [[Lorentz-violating neutrino oscillations]]
* [[Massive particle]] – bradyon, aka tardyon
* [[Massless particle]] – luxon
* [[Retrocausality]]
* [[Tachyonic antitelephone]]
* [[Virtual particle]]
* [[Wheeler–Feynman absorber theory]]

== References ==
{{Reflist|refs=

<ref name="Aharonov-etal-1969">
{{cite journal
 |last1=Aharonov |first1=Y.
 |last2=Komar    |first2=A.
 |last3=Susskind |first3=L.
 |date=25 June 1969
 |title=Superluminal behavior, causality, and instability
 |journal=Physical Review
 |volume=182 |issue=5 |pages=1400–1403
 |doi=10.1103/PhysRev.182.1400
 |bibcode=1969PhRv..182.1400A
 |url=https://link.aps.org/doi/10.1103/PhysRev.182.1400
|url-access=subscription
 }}
</ref>

<ref name="Arkani">
{{cite journal
 |first1 = Allan    |last1 = Adams    |author1-link=Allan Adams
 |first2 = Nima     |last2 = Arkani-Hamed
 |first3 = Sergei   |last3 = Dubovsky    
 |first4 = Alberto  |last4 = Nicolis     
 |first5 = Riccardo |last5 = Rattazzi    
 |date= 2006
 |title = Causality, analyticity and an IR obstruction to UV completion
 |journal = Journal of High Energy Physics
 |volume = 2006  |issue = 10  |pages = 014
 |arxiv = hep-th/0602178  |doi = 10.1088/1126-6708/2006/10/014
 |bibcode = 2006JHEP...10..014A  |s2cid = 2956810
}}
</ref>

<ref name="Barcelo">
{{cite arXiv
 |last1=Barceló  |first1=Carlos
 |last2=Finazzi  |first2=Stefano
 |last3=Liberati |first3=Stefano
 |date=2010
 |title=On the impossibility of superluminal travel: The warp drive lesson
 |class=gr-qc
 |eprint=1001.4960
 |quote=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&nbsp;...
}}
</ref>

<ref name="Benford">
{{cite journal
 |last1=Benford |first1=G.   |author1-link=Gregory Benford
 |last2=Book    |first2=D.
 |last3=Newcomb |first3=W.
 |date=1970
 |title=The Tachyonic Antitelephone
 |journal=[[Physical Review D]]
 |volume=2  |issue=2  |pages=263–265
 |bibcode=1970PhRvD...2..263B
 |doi=10.1103/PhysRevD.2.263
}}
</ref>

<ref name="Benford-legends">
{{cite book
 |first=Gregory |last=Benford  |author-link=Gregory Benford
 |title=Old Legends
 |date=6 July 2013 |page=276
 |url=https://www.gwern.net/docs/radiance/2002-scholz-radiance#old-legends
 |quote=He told me years later that he had begun thinking about tachyons because he was inspired by James Blish's [1954] 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.
}}
</ref>

<ref name="CERN-2012-06-08">
{{cite press release
 |title       = Neutrinos Sent from CERN to Gran Sasso Respect the Cosmic Speed Limit
 |date        = 8 June 2012
|publisher   = [[CERN]]
 |url         = http://press-archived.web.cern.ch/press-archived/PressReleases/Releases2011/PR19.11E.html
 |access-date = 8 June 2012 |url-status = dead
 |archive-url = https://web.archive.org/web/20140222165941/http://press-archived.web.cern.ch/press-archived/PressReleases/Releases2011/PR19.11E.html
 |archive-date= 22 February 2014
}}
</ref>

<ref name="Chodos">
{{cite journal
 |last=Chodos |first=A.
 |date=1985
 |title=The neutrino as a tachyon
 |journal=[[Physics Letters B]]
 |volume=150  |issue=6  |pages=431–435
 |bibcode=1985PhLB..150..431C
 |doi= 10.1016/0370-2693(85)90460-5
|hdl=2022/20737
 |hdl-access=free
 }}
</ref>

<ref name="Coleman">
{{cite journal
 |last1 = Coleman |first1=Sidney R.
 |last2 = Glashow |first2=Sheldon L.
 |name-list-style = amp
 |date=1999
 |title=High-energy tests of Lorentz invariance
 |journal=[[Physical Review D]]
 |volume=59  |issue=11  |page=116008
 |arxiv=hep-ph/9812418  |bibcode=1999PhRvD..59k6008C 
 |s2cid=1273409  |doi=10.1103/PhysRevD.59.116008
}}
</ref>

<ref name="Colladay">
{{cite journal
 |last1=Colladay   |first1=D.
 |last2=Kostelecky |first2=V.A.
 |date=1997
 |title=CPT Violation and the Standard Model
 |journal=[[Physical Review D]]
 |volume=55 |issue=11 |pages=6760–6774
 |arxiv=hep-ph/9703464 |bibcode=1997PhRvD..55.6760C
 |doi=10.1103/PhysRevD.55.6760  |s2cid=7651433
}}
</ref>

<ref name="Feinberg 1967-1969">
{{cite journal
 |last=Feinberg |first=G.
 |date=1967
 |title=Possibility of faster-than-light particles
 |journal=[[Physical Review]]
 |volume=159  |issue=5  |pages=1089–1105
 |bibcode=1967PhRv..159.1089F
 |doi=10.1103/PhysRev.159.1089
}}<br/>
{{cite journal
 |last=Feinberg |first=G.
 |date=1978
 |title=[no title cited]
 |journal=[[Physical Review D]]
 |volume=17  |page=1651
 |doi=10.1103/physrevd.17.1651
}}
</ref>

<ref name="Fitzgerald">
{{cite conference
 |first=P. |last=Fitzgerald
 |title=Tachyons, backwards causation, and freedom
 |series=PSA
 |book-title=Proceedings of the Biennial Meeting of the Philosophy of Science Association vol. 1970
 |conference=The Philosophy of Science Association, 1970 Biennial Meeting
 |volume=1970 |pages=425–426
 |date=1970
 |quote=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’&nbsp;...
}}
</ref>

<ref name="Gibbons">
{{cite journal
 |last1=Gibbons |first1=G.W.
 |date=13 June 2002
|title=Cosmological evolution of the rolling tachyon
 |journal=Physics Letters B
 |volume=537
 |issue=1–2
 |pages=1–4
 |arxiv=hep-th/0204008  |doi=10.1016/S0370-2693(02)01881-6
|bibcode=2002PhLB..537....1G
 |s2cid=119487619
 }}
</ref>

<ref name="Greene">
{{cite book
 |first=Brian |last=Greene  |author-link=Brian Greene
 |date=2000
 |title=The Elegant Universe
 |publisher=Vintage Books
}}
</ref>

<ref name="Gron">
{{cite book
 |last1=Grøn   |first1=Ø.
 |last2=Hervik |first2=S.
 |date=2007
 |title=Einstein's General Theory of Relativity: With Modern Applications in Cosmology
 |page=39
 |publisher=[[Springer (publisher)|Springer]]
 |isbn=978-0-387-69199-2
 |url=https://books.google.com/books?id=IyJhCHAryuUC&pg=PA39
 |quote=The tachyon telephone paradox cannot be resolved by means of the reinterpretation principle.
}}
</ref>

<ref name="Jarrell">
{{cite web
 |first=Mark |last=Jarrell
 |title=The Special Theory of Relativity
 |series=Electrodynamics course, chapter&nbsp;11
 |pages=7–11
 |publisher=[[University of Cincinnati]]
 |url=http://www.physics.uc.edu/~jarrell/COURSES/ELECTRODYNAMICS/Chap11/chap11.pdf
 |access-date=27 October 2006
 |archive-url=https://web.archive.org/web/20060913173236/http://www.physics.uc.edu/~jarrell/COURSES/ELECTRODYNAMICS/Chap11/chap11.pdf
 |archive-date=13 September 2006
}}</ref>

<ref name="Kutasov">
{{cite journal
 |last1 = Kutasov |first1 = David
 |last2 = Mariño  |first2 = Marcos
 |last3 = Moore   |first3 = Gregory
 |date=2000
 |title = Some exact results on tachyon condensation in string field theory
 |journal = Journal of High Energy Physics
 |volume = 2000 |issue = 10 |page = 045
 | doi=10.1088/1126-6708/2000/10/045 |arxiv = hep-th/0009148
 |bibcode = 2000JHEP...10..045K
 |s2cid = 15664546
}}
</ref>

<ref name="Polchinski 1998">
{{cite journal
 |last=Polchinski |first=J.
 |date=1998
 |title=String Theory
 |journal=Proceedings of the National Academy of Sciences of the United States of America
 |volume=95
 |issue=19
 |pages=11039–11040
 |publisher=[[Cambridge University Press]]
 |doi=10.1073/pnas.95.19.11039
 |pmid=9736684
 |pmc=33894
 |bibcode=1998PNAS...9511039G
 |place=Cambridge, UK
|doi-access=free
 }}
</ref>

<ref name="Recami">
{{cite journal
 |first1=Erasmo  |last1=Recami
 |first2=Flavio  |last2=Fontana
 |first3=Roberto |last3=Garavaglia
 |date=2000
 |title=Special Relativity and Superluminal Motions: A Discussion of Some Recent Experiments
 |journal=International Journal of Modern Physics A
 |volume=15 |issue=18
 |pages=2793–2812
 |doi=10.1142/S0217751X00001403  | arxiv = 0709.2453
 |quote = it is possible ... to solve also the known causal paradoxes, devised for {{grey|[refuting]}} 'faster than light' motion, although this is not widely recognized yet.
}}
</ref>

<ref name="Sen">
{{cite journal
 |last=Sen |first=Ashoke
 |date=2002
 |title=Rolling tachyon
 |journal=Journal of High Energy Physics
 |volume=2002 |issue=4
 |page=048
 |doi=10.1088/1126-6708/2002/04/048
 |arxiv = hep-th/0203211
|bibcode=2002JHEP...04..048S
 |s2cid=12023565
 }}
</ref>

<ref name="Sommerfeld">
{{cite journal
 |last=Sommerfeld |first=A.
 |date=1904
 |title=Simplified Deduction of the Field and the Forces of an Electron Moving in Any Given Way
 |journal=KNKL. Acad. Wetensch
 |volume=7  |pages=345–367
}}
</ref>

<ref name="sudarshan62">
{{cite journal
 |last1=Bilaniuk  |first1=O.-M.P.
 |last2=Deshpande |first2=V.K.
 |last3=Sudarshan |first3=E.C.G.
 |date=1962
 |title='Meta' Relativity
 |journal=[[American Journal of Physics]]
 |volume=30 |issue=10 |pages=718
 |bibcode=1962AmJPh..30..718B
 |doi=10.1119/1.1941773
}}
</ref>

<ref name="sudarshan69">
{{cite journal
 |last1=Bilaniuk  |first1=O.-M.P.
 |last2=Sudarshan |first2=E.C.G.
 |date=1969
 |title=Particles beyond the Light Barrier
 |journal=[[Physics Today]]
 |volume=22 |issue=5 |pages=43–51
 |bibcode= 1969PhT....22e..43B
 |doi=10.1063/1.3035574
}}
</ref>

<ref name="susskind">
{{cite journal
 |last1 = Aharonov |first1 = Y.
 |last2 = Komar    |first2 = A.
 |last3 = Susskind |first3 = L.
 |title = Superluminal Behavior, Causality, and Instability
 |date=1969
 |journal = Phys. Rev.
 |volume = 182 |issue = 5  |pages = 1400–1403
 |doi = 10.1103/PhysRev.182.1400
 |bibcode = 1969PhRv..182.1400A
}}
</ref>

<ref name="Tipler">
{{cite book
 | last1 = Tipler    | first1 = Paul A.
 | last2 = Llewellyn | first2 = Ralph A.
 | title = Modern Physics
 | publisher = W.H. Freeman & Co.
 |date= 2008
 | edition = 5th | page = 54
 | location = New York, NY
 | isbn = 978-0-7167-7550-8
 |quote = ... so existence of particles {{math| ''v'' > ''c'' }} ... Called ''tachyons'' ... would present relativity with serious ... problems of infinite creation energies and causality paradoxes.
}}
</ref>

}} <!-- end of "refs=" -->

== External links ==
{{Wiktionary}}
{{Commons category}}
* {{cite web
 |title=The faster than light (FTL) FAQ
 |publisher=[[Iowa State University]]
 |url=http://www.public.iastate.edu/~physics/sci.physics/faq/FTL.html
 |archive-url=https://web.archive.org/web/20001121064300/http://www.public.iastate.edu/~physics/sci.physics/faq/FTL.html
 |archive-date=21 November 2000
}}
* {{ScienceWorld|title=Tachyon|urlname=physics/Tachyon}}
* {{cite web
 |title=Tachyons
 |series=Physics FAQ / Particle and Nuclear
 |department=Mathematics
 |publisher=[[University of California]]
 |place=Riverside, California
 |url=http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html 
}}

{{Particles}}
{{String theory topics |state=collapsed}}
{{Authority control}}

[[Category:Tachyons| ]]
[[Category:Hypothetical particles]]
[[Category:String theory]]
[[Category:Time travel]]