# Tachyon

(Redirected from Tachyons)
Because a tachyon would always move faster than light, it would not be possible to see it approaching. After a tachyon has passed nearby, we would be able to see two images of it, appearing and departing in opposite directions. The black line is the shock wave of Cherenkov radiation, shown only in one moment of time. This double image effect is most prominent for an observer located directly in the path of a superluminal object (in this example a sphere, shown in grey). The right hand bluish shape is the image formed by the blue-doppler shifted light arriving at the observer—who is located at the apex of the black Cherenkov lines—from the sphere as it approaches. The left-hand reddish image is formed from red-shifted light that leaves the sphere after it passes the observer. Because the object arrives before the light, the observer sees nothing until the sphere starts to pass the observer, after which the image-as-seen-by-the-observer splits into two—one of the arriving sphere (to the right) and one of the departing sphere (to the left).

A tachyon or tachyonic particle is a hypothetical massive particle that always moves faster than light. The word comes from the Greek: ταχύς or tachys, meaning "swift, quick, fast, rapid", and was coined by Gerald Feinberg.[1] The complementary particle types are called luxon (always moving at the speed of light) and bradyon (always moving slower than light), which both exist.

Most physicists think that faster-than-light particles cannot exist because they are not consistent with the known laws of physics.[2][3] 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.[3] Potentially consistent theories that allow faster-than-light particles include those that break Lorentz invariance, the symmetry underlying special relativity, so that the speed of light is not a barrier.

In the 1967 paper that coined the term,[1] Feinberg proposed that tachyonic particles could be quanta of a quantum field with negative squared mass. However, it was soon realized that excitations of such imaginary mass fields do not in fact propagate faster than light,[4] and instead represent an instability known as tachyon condensation.[2] Nevertheless, negative squared mass fields are commonly referred to as "tachyons",[5] and in fact have come to play an important role in modern physics.

Despite theoretical arguments against the existence of faster-than-light particles, experiments have been conducted to search for them. No compelling evidence for their existence has been found. In September 2011, it was reported that a tau neutrino had travelled 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 resultant from "a faulty element of the experiment's fibre optic timing system".[6] It is important to note that many physicists do infer a possible link between the properties of neutrinos and theoretical tachyons.[7]

## Tachyons in relativistic theory

In special relativity, a faster-than-light particle would have space-like four-momentum,[1] in contrast to ordinary particles that have time-like four-momentum. It would also have imaginary mass. Being constrained to the spacelike portion of the energy–momentum graph, it could not slow down to subluminal speeds.[1]

### Mass

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:

$E^2 = p^2c^2 + m^2c^4 \;$

(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:

$E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}.$

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, 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.

### Speed

One curious effect is that, unlike ordinary particles, the speed of a tachyon increases as its energy decreases. In particular, $E$ approaches zero when $v$ 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.

As noted by Einstein, Tolman, and others, special relativity implies that faster-than-light particles, if they existed, could be used to communicate backwards in time.[8]

### Neutrinos

In 1985 Chodos et al. proposed that neutrinos can have a tachyonic nature.[9] 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.[10][11][12] In this framework, neutrinos experience Lorentz-violating oscillations and can travel faster than light at high energies. This proposal was strongly criticized.[13]

A tachyon with an electric charge would lose energy as Cherenkov radiation[14]—just as ordinary charged particles do when they exceed the local speed of light in a medium. 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, because it has a gravitational mass, and therefore increase 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 recently 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.[15]

### Causality

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"[8] or "logically pernicious self-inhibitor."[16]

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 spacetime interval between the events is 'space-like', meaning that neither event lies in the future light cone of the other).[17]

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.[17] 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 backwards 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 backwards 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.[18] 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)[19]

#### Reinterpretation principle

The reinterpretation principle[1][18][20] 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.[8][18][21] 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[8] or a "logically pernicious self-inhibitor".[16] 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.[22][23]

## Fundamental models

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 mass

In the paper that coined the term "tachyon", Gerald Feinberg studied Lorentz invariant quantum fields with imaginary mass.[1] 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.[4] Despite having no faster-than-light propagation, such fields are referred to simply as "tachyons" in many sources.[2][5][24][25][26][27]

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—in its uncondensed phase—has an imaginary mass. In general, the phenomenon of spontaneous symmetry breaking, which is closely related to tachyon condensation, plays a very 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.[24][26][28]

### 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.[22] A class of field theories of that type are the so-called Standard Model extensions. However, the experimental evidence for Lorentz invariance is extremely good, so such theories are very tightly constrained.[29][30]

### Fields with non-canonical kinetic term

By modifying the kinetic energy of the field, it is possible to produce Lorentz invariant field theories with excitations that propagate superluminally.[4][23] 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.

## History

As mentioned above, the term "tachyon" was coined by Gerald Feinberg in a 1967 paper titled "Possibility of Faster-Than-Light Particles".[1]"He had been inspired by the science-fiction story "Beep" by James Blish.[31] 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.

The first hypothesis regarding faster-than-light particles is sometimes attributed to German physicist Arnold Sommerfeld in 1904,[32] and more recent discussions happened in 1962[20] and 1969.[33]

## 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 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).

Also referenced in the movie, "K-PAX". Kevin Spacey's character claims to have traveled to Earth at Tachyon speeds.

## References

1. 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)
2. ^ a b c Lisa Randall, Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions, p.286: "People initially thought of tachyons as particles travelling 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."
3. ^ a b Tipler, Ralph A.; Llewellyn (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."
4. ^ a b c Aharonov, Y.; Komar, A.; Susskind, L. (1969). "Superluminal Behavior, Causality, and Instability". Phys. Rev. (American Physical Society) 182 ({5},): 1400–1403. Bibcode:1969PhRv..182.1400A. doi:10.1103/PhysRev.182.1400.
5. ^ a b A. Sen, "Rolling tachyon," JHEP 0204, 048 (2002). Cited 720 times as of 2/2012.
6. ^ "Neutrinos sent from CERN to Gran Sasso respect the cosmic speed limit" (Press release). CERN. 8 June 2012. Retrieved 2012-06-08.
7. ^ Feinberg, G. (1997). "Tachyon". Encyclopedia Americana 26. Grolier. p. 210.
8. ^ a b c d Benford, G.; Book, D.; Newcomb, W. (1970). "The Tachyonic Antitelephone". Physical Review D 2 (2): 263. Bibcode:1970PhRvD...2..263B. doi:10.1103/PhysRevD.2.263.
9. ^ Chodos, A. (1985). "The Neutrino as a Tachyon". Physics Letters B 150 (6): 431. Bibcode:1985PhLB..150..431C. doi:10.1016/0370-2693(85)90460-5.
10. ^ 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.
11. ^ 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.
12. ^ 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.
13. ^ R. J. Hughes and G. J. Stephenson, Jr., Against tachyonic neutrinos, Phys. Lett. B 244, 95–100 (1990).
14. ^ Bock, R. K. (9 April 1998). "Cherenkov Radiation". The Particle Detector BriefBook. CERN. Retrieved 2011-09-23.
15. ^ Cohen, Andrew G. and Glashow, Sheldon L. ("2011",). "Pair Creation Constrains Superluminal Neutrino Propagation". Phys.Rev.Lett. "107",: "181803",. doi:10.1103/PhysRevLett.107.181803.
16. ^ a b 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..."
17. ^ a b Mark, J. "The Special Theory of Relativity". University of Cincinnati. pp. 7–11. Archived from the original on 2006-09-13. Retrieved 2006-10-27.
18. ^ a b c 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."
19. ^ . Baker, R. (12 September 2003). "Relativity, FTL and causality". Sharp Blue. Retrieved 2011-09-23.
20. ^ a b 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.
21. ^ 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].
22. ^ a b 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..." [1]
23. ^ a b Allan Adams, Nima Arkani-Hamed, Sergei Dubovsky, Alberto Nicolis, Riccardo Rattazzi, "Causality, Analyticity and an IR Obstruction to UV Completion", JHEP 0610 (2006) 014 [2].
24. ^ a b Brian Greene, The Elegant Universe, Vintage Books (2000)
25. ^ Kutasov, David and Marino, Marcos and Moore, Gregory W. (2000). "Some exact results on tachyon condensation in string field theory". JHEP 0010: 045. arXiv EFI-2000-32, RUNHETC-2000-34.
26. ^ a b NOVA, "The Elegant Universe", PBS television special, http://www.pbs.org/wgbh/nova/elegant/
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31. ^ "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." pg276 of Gregory Benford's "Old Legends"
32. ^ 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.
33. ^ 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.