Tachyon

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:

${\displaystyle E^{2}=(pc)^{2}+(mc^{2})^{2}\;}$

(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:

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

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 ${\displaystyle v}$ (the particle's velocity) is larger than ${\displaystyle c}$ (the speed of light), the denominator in the equation for the energy is imaginary, as the value under the square root is negative. Because the total energy of the particle must be real (and not a 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.[13][14][15]

One curious effect is that, unlike ordinary particles, the speed of a tachyon increases as its energy decreases. In particular, ${\displaystyle E}$ approaches zero when ${\displaystyle v}$ approaches infinity. (For ordinary bradyonic matter, ${\displaystyle E}$ increases with increasing speed, becoming arbitrarily large as ${\displaystyle v}$ approaches ${\displaystyle c}$, 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 Albert Einstein, Tolman, and others, special relativity implies that faster-than-light particles, if they existed, could be used to communicate backwards in time.[16]

In 1985, Chodos proposed that neutrinos can have a tachyonic nature.[17] 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.[18][19][20] In this framework, neutrinos experience Lorentz-violating oscillations and can travel faster than light at high energies. This proposal was strongly criticized.[21]

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 world line 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 2011 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.[22] Further investigation of the experiment showed that the results were indeed erroneous.

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.

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"[16] or "logically pernicious self-inhibitor."[23]

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).[24]

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.[24] 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.[25] This is known as the tachyonic antitelephone.

The reinterpretation principle[3][9][25] 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.[16][25][26] 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[16] or a "logically pernicious self-inhibitor".[23] 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.[27][28]

In the paper that coined the term "tachyon", Gerald Feinberg studied Lorentz invariant quantum fields with imaginary mass.[3] 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.[29] Despite having no faster-than-light propagation, such fields are referred to simply as "tachyons" in many sources.[5][30][31][32][2]

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.[30][33]

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.[27] 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.[34][35]

By modifying the kinetic energy of the field, it is possible to produce Lorentz invariant field theories with excitations that propagate superluminally.[29][28] 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.