Tachyon

Quantum field theory
History of...

A tachyon ( /ˈtæki.ɒn/) is a hypothetical subatomic particle that always moves faster than light. In the language of special relativity, a tachyon would be a particle with space-like four-momentum and imaginary proper time. A tachyon would be constrained to the space-like portion of the energy-momentum graph. Therefore, it cannot slow down to subluminal speeds.

The first hypothesis about tachyons is attributed to German physicist Arnold Sommerfeld. However, it was George Sudarshan,[1] Olexa-Myron Bilaniuk,[2] Vijay Deshpande,[2] and Gerald Feinberg[3] (who originally coined the term in the 1960s) who advanced a theoretical framework for their study. The name comes from the Greek: ταχύς (tachys, “swift, quick, fast, rapid”).

If tachyons are conventional, localizable particles that can be used to send signals faster than light, this would lead to violations of causality in special relativity. However, in the framework of quantum field theory, tachyons are understood as signifying an instability of the system and dismissed through tachyon condensation, rather than being treated as real faster-than-light particles, and such instabilities are described by tachyonic fields. Tachyonic fields have appeared theoretically in a variety of contexts, such as the bosonic string theory. According to the contemporary and widely accepted understanding of the concept of a particle, tachyon particles are too unstable to be treated as existent.[4] According to that theory, faster-than-light information transmission and causality violation with tachyons are impossible. Even if tachyons were conventional, localisable particles, they would still, by the Feinberg reinterpretation principle, preserve the basic tenets of causality in special relativity and not allow transmission of information faster than light,[3] contrary to what has been written in many works of science fiction.

Conventional massive particles that travel slower than the speed of light are sometimes termed "bradyons" or "tardyons" in contrast, although these terms are only used in the context of discussions about tachyons.

Despite the theoretical arguments against the existence of tachyon particles, experimental searches have been conducted to test the assumption against their existence; however, no experimental evidence[5] for the existence of tachyon particles has been found. Recently, time of flight measurements from the CERN-OPERA collaboration have indicated that muon neutrinos may exceed the speed of light. OPERA scientists have refrained from drawing conclusions from the faster-than-light neutrino anomaly, requesting that other groups first verify their result.[6]

Contents

Tachyons in relativistic theory

From a special relativity dynamics perspective a tachyon would be a particle with space-like four-momentum. The existence of such particles would pose intriguing problems in modern physics since they are faster than light.

Mass

There are two equivalent approaches to handling their kinematics:

E^2 = p^2c^2 %2B m^2c^4 \;
where p is the relativistic momentum of the bradyon and m is its rest mass still holds, along with the formula for the total energy of a particle:
E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}.
which is interpreted to mean that the total energy of a particle (bradyon or tachyon) contains a contribution from the rest mass (the "rest mass-energy") and a contribution from the body's motion, the kinetic energy.
However the energy equation has, when v is larger than c, an "imaginary" denominator, as the value inside the square root is negative. Because the total energy must be real then 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 quantum field theory imaginary mass would induce tachyon condensation; essentially a mathematical treatment of the 'particle' which yields a real result. One good way of illustrating this is by thinking of simple harmonic motion and the use of Euler's Formula to dissociate the 'imaginary' part of the motion which is of no meaning in the physical universe, although this is only a loose analogy.
E^2 %2B z^2c^4 = p^2c^2 \;
With this approach the energy equation becomes:
E = \frac{zc^2}{\sqrt{\frac{v^2}{c^2} -1}}.
And we avoid any necessity for imaginary masses, sidestepping the problem of interpreting exactly what a complex-valued mass may physically mean. Except, of course, when converting z back to m for interactions with non-tachyon particles. Both approaches are equivalent mathematically and have the same physical consequences.

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 Gregory Benford, among others, special relativity implies that tachyons, if they existed, could be used to communicate backwards in time[7] (see Tachyonic antitelephone article).

In 1985 it was proposed by Chodos et al. that neutrinos can have a tachyonic nature.[8] Today, the possibility of having standard particles moving at superluminal speeds is a natural consequence of unconventional dispersion relations that appear in the Standard-Model Extension,[9][10][11] a realistic description of the possible violation of Lorentz invariance in field theory. In this framework, neutrinos experience Lorentz-violating oscillations and can travel faster than light at high energies. On the other hand, the above-mentioned proposal by Chodos et al. was strongly criticized by some researchers, [12] but this criticism was subsequently shown to be incorrect.[13] Since then, the idea has been studied by other researchers.[14]

Cherenkov radiation

Taking the formalisms of electromagnetic radiation and supposing a tachyon had an electric charge—as there is no reason to suppose a priori that tachyons must be either neutral or charged—then a charged tachyon must lose energy as Cherenkov radiation[15]—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, as we have seen, 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.

Quantum field theory

Quantizing tachyons shows that they must be spinless particles which obey Fermi-Dirac statistics; i.e., tachyons are scalar fermions, a combination which is not permitted for ordinary particles.[3] Like other fermions, tachyons must be created and annihilated in pairs.

Causality

Causality is a fundamental principle of theoretical particle physics (and thus a basis of arguments that various methods of potentially sending information faster than c or backwards in time are unworkable). If tachyons can be used to transmit information faster than light, then according to relativity, they can also be used to violate certain simple understandings of the causality principle using a scheme sometimes known as the "tachyon telephone paradox".[16] This 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, a seeming challenge to temporal causality in every frame. Nonetheless, the event cannot be said to lack causal structure as the events are still connected, albeit in this case in a backwards relationship. Mathematical details can be found in the tachyonic antitelephone article, and an illustration of such a scenario using spacetime diagrams can be found here.[18]

It has been argued that we can avoid the notion of tachyons traveling into the past using the Feinberg reinterpretation principle[3] which states that a negative-energy tachyon sent back in time in an attempt to challenge forward temporal causality can always be reinterpreted as a positive-energy tachyon traveling forward in time. This is because observers cannot distinguish between the emission and absorption of tachyons. For a tachyon, there is no distinction between the processes of emission and absorption, because there always exists a sub-light speed reference frame shift that alters the temporal direction of the tachyon's world-line, which is not true for bradyons or luxons. The attempt to detect a tachyon from the future (and challenge forward causality) can actually create the same tachyon and sends it forward in time (which is itself a causal event).

According to the Feinberg reinterpretation principle every tachyon detector will register tachyons in every possible detection mode; from the perspective of a frame where the registration by the "detector" preceded the activation of the "emitter", the "detector" in the past is actually spontaneously emitting tachyons, only some of which will be intercepted by the detector in the future. However, Feinberg's reinterpretation principle treats events in a way that the earlier event is defined as the "emission" and the later one the "detection", and one critic claims that it does not, in itself, solve the causality problems associated with tachyon information transmission.[16] This criticism however is ill-conceived as it completely ignores the causal structure of reality. For example, suppose experimenter A could selectively influence the rate at which tachyons from her emitter traveled to the detector of experimenter B, and B could measure changes in the rate tachyons arrived at his detector, so that A could transmit a message to B in binary code. Then in a frame where B's detector went off at an earlier time than A influenced her emitter, relabeling B's device as a "spontaneous emitter" and A's as a "detector" would be necessary to make the experiment intelligible as it wouldn't change the fact that B, by observing the changing rates of tachyons being spontaneously emitted from his device, could gain information about A's interactions with her own device in the future. In fact, this experiment would solidify the principle of causality even more since B can follow the causal link and make predictions about A's action in the future. Although remote, the possibility of backward causality is not a real challenge to the principle of causality, but rather a novel way of understanding an additional aspect of it.

Some arguments from quantum field theory suggest that even if tachyons exist, they cannot be used to transmit information faster than light, either because disturbances in the quantum field for a tachyon would not actually propagate faster than light, or because the tachyon is impossible to localize.[19] Without the possibility of faster-than-light information transmission, the problem of backwards-in-time information transmission can be avoided as well.

In the theory of general relativity, it is possible to construct spacetimes in which particles travel faster than the speed of light, relative to a distant observer. One example is the Alcubierre metric, another is of traversable wormholes. However, these are not tachyons in the above sense, as they do not exceed the speed of light locally.

Condensation

Main article: Tachyon condensation

In quantum field theory, a tachyon is a quantum of a field—usually a scalar field—whose squared mass is negative, and is used to describe spontaneous symmetry breaking: The existence of such a field implies the instability of the field vacuum; the field is at a local maximum rather than a local minimum of its potential energy, much like a ball at the top of a hill. A very small impulse (which will always happen due to quantum fluctuations) will lead the field to roll down with exponentially increasing amplitudes: it will induce tachyon condensation. It is important to realize that once the tachyonic field reaches the minimum of the potential, its quanta are not tachyons any more but rather have a positive mass-squared, such as the Higgs boson.

Technically, the squared mass is the second derivative of the effective potential, at a point where the first derivative is zero. So for a tachyonic field the second derivative is negative, meaning that the effective potential is at a local maximum rather than a local minimum. Therefore this situation is unstable and the field will roll down to another point, stopping only at a local minimum, where its quanta have a non-negative squared mass, so that it is not tachyonic any longer.[4]

Because a tachyon's squared mass is negative, it formally has an imaginary mass. This is a special case of the general rule, where unstable massive particles are formally described as having a complex mass, with the real part being their mass in usual sense, and the imaginary part being the decay rate in natural units.[4]

However, in quantum field theory, a particle (a "one-particle state") is roughly defined as a state which is constant over time; i.e., an eigenvalue of the Hamiltonian. An unstable particle is a state which is only approximately constant over time; However, it exists long enough to be measured. This means that if it is formally described as having a complex mass, then the real part of the mass must be greater than its imaginary part. If both parts are of the same magnitude, this is interpreted as a resonance appearing in a scattering process rather than particle, as it is considered not to exist long enough to be measured independently of the scattering process. In the case of a tachyon, the imaginary part of the mass is infinitely larger than the real part, and hence no concept of a particle can be attributed to it.

Even for tachyonic quantum fields, the field operators at spacelike separated points still commute (or anticommute), thus preserving the principle of causality. Therefore information never moves faster than light, according to quantum field theory.

Examples for tachyonic fields are all cases of spontaneous symmetry breaking. In condensed matter physics a notable example is ferromagnetism; in particle physics the best known example is the Higgs mechanism in the standard model.

Tachyons in theory of superfluid vacuum

Main article: Superfluid vacuum

In SVT the physical vacuum is viewed as the quantum superfluid which is essentially non-relativistic whereas the Lorentz symmetry is not an exact symmetry of Nature but rather the approximate description valid only for the small fluctuations of the superfluid background.[20] The proposed models have clear observational consequences related to the superluminal propagation and tachyons which are based on the fact that at very high velocities the behavior of the particle-like modes becomes distinct from the relativistic one — they can reach the speed of light limit at finite energy.[21] Among other predicted effects is the superluminal propagation and vacuum Cherenkov radiation.[22]

Tachyons in string theory

In string theory, tachyons have the same interpretation as in quantum field theory. However, string theory can, at least in principle, not only describe the physics of tachyonic fields, but also predict whether such fields appear.

Tachyonic fields indeed arise in many versions of string theory. In general, string theory states that what we see as "particles" —electrons, photons, gravitons and so forth—are actually different vibrational states of the same underlying string. The mass of the particle can be deduced from the vibrations which the string exhibits; roughly speaking, the mass depends upon the "note" which the string sounds. Tachyons frequently appear in the spectrum of permissible string states, in the sense that some states have negative mass-squared, and therefore imaginary mass. If the tachyon appears as a vibrational mode of an open string, this signals an instability of the underlying D-brane system to which the string is attached.[23] The system will then decay to a state of closed strings and/or stable D-branes. If the tachyon is a closed string vibrational mode, this indicates an instability in spacetime itself. Generally, it is not known (or theorized) what this system will decay to. However, if the closed string tachyon is localized around a spacetime singularity the endpoint of the decay process will often have the singularity resolved.

Fiction

Main article: Tachyons in fiction

Tachyons appear in many works of fiction. It has 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 "sound" even if the subject in question has no particular relation to superluminal travel (a form of technobabble, akin to positronic brain).

See also

References

  1. ^ Bilaniuk, O.-M. P.; Sudarshan, E. C. G. (1969). "Particles beyond the Light Barrier". Physics Today 22 (5): 43–51. doi:10.1063/1.3035574. 
  2. ^ 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. 
  3. ^ a b c d 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. 
  4. ^ a b c Peskin, M. E.; Schroeder, D. V. (1995). An Introduction to Quantum Field Theory. Perseus Books. 
  5. ^ Feinberg, G. (1997). "Tachyon". Encyclopedia Americana. 26. Grolier. p. 210. 
  6. ^ "OPERA experiment reports anomaly in flight time of neutrinos from CERN to Gran Sasso" 2011-9-23
  7. ^ 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. 
  8. ^ 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. 
  9. ^ 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. 
  10. ^ 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. 
  11. ^ 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. 
  12. ^ R. J. Hughes and G. J. Stephenson Jr., Against tachyonic neutrinos, Phys. Lett. B 244, 95-100 (1990).
  13. ^ A. Chodos et al., Null experiments for neutrino masses, Mod. Phys. Lett. A7, 467 (1992).
  14. ^ List of publications on the tachyonic neutrino idea (may be incomplete). InSPIRE database.
  15. ^ Bock, R. K. (9 April 1998). "Cherenkov Radiation". The Particle Detector BriefBook. CERN. http://rd11.web.cern.ch/RD11/rkb/PH14pp/node26.html. Retrieved 2011-09-23. 
  16. ^ a b Grøn, Ø.; Hervik, S. (2007). Einstein's General Theory of Relativity: With Modern Applications in Cosmology. Springer. pp. 39. ISBN 978-0387691992. http://books.google.com/?id=IyJhCHAryuUC&lpg=PR1&pg=PA39#v=onepage&q. 
  17. ^ a b Mark, J.. "The Special Theory of Relativity". University of Cincinnati. pp. 7–11. Archived from the original on 2006-09-13. http://web.archive.org/web/20060913173236/http://www.physics.uc.edu/~jarrell/COURSES/ELECTRODYNAMICS/Chap11/chap11.pdf. Retrieved 2006-10-27. 
  18. ^ Baker, R. (12 September 2003). "Relativity, FTL and causality". Sharp Blue. http://www.theculture.org/rich/sharpblue/archives/000089.html. Retrieved 2011-09-23. 
  19. ^ Chase, S. I. (March 1993). "Tachyons". Usenet Physics FAQ. University of California, Riverside. http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html. Retrieved 2009-12-03. 
  20. ^ G. E. Volovik, The Universe in a helium droplet, Int. Ser. Monogr. Phys. 117 (2003) 1-507.
  21. ^ K. G. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences, Grav. Cosmol. 16 (2010) 288-297 ArXiv:0906.4282.
  22. ^ K. G. Zloshchastiev, Vacuum Cherenkov effect in logarithmic nonlinear quantum theory, Phys. Lett. A 375 (2011) 2305-2308 ArXiv:1003.0657.
  23. ^ Sen, A. (1998). "Tachyon condensation on the brane antibrane system". Journal of High Energy Physics 8 (8): 12. arXiv:hep-th/9805170. Bibcode 1998JHEP...08..012S. doi:10.1088/1126-6708/1998/08/012. 

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