### Tachyon

A **tachyon** or **tachyonic particle** is a hypothetical E. C. G. Sudarshan in 1962, although the term they used for it was "meta-particle".^{[2]}

Most physicists think that faster-than-light particles cannot exist because they are not consistent with the known laws of physics.^{[3]}^{[4]} 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.^{[4]} 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,^{[5]} and instead represent an instability known as tachyon condensation.^{[3]} Nevertheless, negative squared mass fields are commonly referred to as "tachyons",^{[6]} 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".^{[14]}

## 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 and proper time. 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]}^{[15]}^{[16]} In this framework, neutrinos experience Lorentz-violating oscillations and can travel faster than light at high energies. This proposal was strongly criticized.^{[17]}

### Cherenkov radiation

A tachyon with an electric charge would lose energy as Cherenkov radiation^{[21]}—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.^{[19]}

### 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."^{[20]}

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).^{[22]}

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.^{[22]} 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.^{[23]} 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)*^{[38]}

#### Reinterpretation principle

The **reinterpretation principle**^{[1]}^{[2]}^{[23]} 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]}^{[23]}^{[25]} 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".^{[20]} 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.^{[26]}^{[27]}

## 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.^{[5]} Despite having no faster-than-light propagation, such fields are referred to simply as "tachyons" in many sources.^{[3]}^{[6]}^{[28]}^{[29]}^{[30]}^{[31]}

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.^{[28]}^{[30]}^{[39]}

Tachyons are predicted by bosonic string theory and also the NS (which is the open bosonic sector) and NS-NS (which is the closed bosonic sector) sectors of RNS Superstring theory before GSO projection. However, due to the Sen conjecture—also known as tachyon condensation—this is not possible. 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.^{[26]} 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.^{[33]}^{[34]}

### 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.^{[5]}^{[27]} 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.^{[40]} 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,^{[36]} and more recent discussions happened in 1962^{[2]} and 1969.^{[37]}

## 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*).

## See also

- Massive particle (bradyon)
- Massless particle (luxon)
- Lorentz-violating neutrino oscillations
- Retrocausality
- Tachyonic antitelephone
- Wheeler–Feynman absorber theory