Updated 2015 by John Baez.
Original by Scott Chase.
There was a young lady named Bright,
Whose speed was far faster than light.
She went out one day,
In a relative way,
And returned the previous night!
— Reginald Buller
It is a well known fact that nothing can travel faster than the speed of light. At best, a massless particle travels at the speed of light. But is this really true? In 1962, Bilaniuk, Deshpande, and Sudarshan, Am. J. Phys. 30, 718 (1962), said "no". A very readable paper is Bilaniuk and Sudarshan, Phys. Today 22, 43 (1969). Here is a brief overview.
Draw a graph, with momentum (p) on the x-axis, and energy (E) on the y-axis. Then draw the "light cone", two lines with the equations E = ±p. This divides our 1+1 dimensional space-time into two regions. Above and below are the "timelike" quadrants, and to the left and right are the "spacelike" quadrants.
Now the fundamental fact of relativity is that
where E is an object's energy, p is its momentum, and m is its rest mass, which we'll just call 'mass'. In case you're wondering, we are working in units where c=1. For any non-zero value of m, this is a hyperbola with branches in the timelike regions. It passes through the point (p,E) = (0,m), where the particle is at rest. Any particle with mass m is constrained to move on the upper branch of this hyperbola. (Otherwise, it is "off shell", a term you hear in association with virtual particles — but that's another topic.) For massless particles, E² = p², and the particle moves on the light-cone.
These two cases are given the names tardyon (or bradyon in more modern usage) and luxon, for "slow particle" and "light particle". Tachyon is the name given to the supposed "fast particle" which would move with v > c. Tachyons were first introduced into physics by Gerald Feinberg, in his seminal paper "On the possibility of faster-than-light particles" [Phys. Rev. 159, 1089–1105 (1967)].
Now another familiar relativistic equation is
Tachyons (if they exist) have v > c. This means that E is imaginary! Well, what if we take the rest mass m, and take it to be imaginary? Then E is negative real, and E² − p² = m² < 0. Or, p² − E² = M², where M is real. This is a hyperbola with branches in the spacelike region of spacetime. The energy and momentum of a tachyon must satisfy this relation.
You can now deduce many interesting properties of tachyons. For example, they accelerate (p goes up) if they lose energy (E goes down). Furthermore, a zero-energy tachyon is "transcendent", or moves infinitely fast. This has profound consequences. For example, let's say that there were electrically charged tachyons. Since they would move faster than the speed of light in the vacuum, they should produce Cherenkov radiation. This would lower their energy, causing them to accelerate more! In other words, charged tachyons would probably lead to a runaway reaction releasing an arbitrarily large amount of energy. This suggests that coming up with a sensible theory of anything except free (noninteracting) tachyons is likely to be difficult. Heuristically, the problem is that we can get spontaneous creation of tachyon-antitachyon pairs, then do a runaway reaction, making the vacuum unstable. To treat this precisely requires quantum field theory, which gets complicated. It is not easy to summarize results here. But one reasonably modern reference is Tachyons, Monopoles, and Related Topics, E. Recami, ed. (North-Holland, Amsterdam, 1978).
But tachyons are not entirely invisible. You can imagine that you might produce them in some exotic nuclear reaction. If they are charged, you could "see" them by detecting the Cherenkov light they produce as they speed away faster and faster. Such experiments have been done but, so far, no tachyons have been found. Even neutral tachyons can scatter off normal matter with experimentally observable consequences. Again, no such tachyons have been found.
How about using tachyons to transmit information faster than the speed of light, in violation of Special Relativity? It's worth noting that when one considers the relativistic quantum mechanics of tachyons, the question of whether they "really" go faster than the speed of light becomes much more touchy! In this framework, tachyons are waves that satisfy a wave equation. Let's treat free tachyons of spin zero, for simplicity. We'll set c = 1 to keep things less messy. The wavefunction of a single such tachyon can be expected to satisfy the usual equation for spin-zero particles, the Klein-Gordon equation:
where □ is the D'Alembertian, which in 3+1 dimensions is just
The difference with tachyons is that m² is negative, and so m is imaginary.
To simplify the math a bit, let's work in 1+1 dimensions with co-ordinates x and t, so that
Everything we'll say generalizes to the real-world 3+1-dimensional case. Now, regardless of m, any solution is a linear combination, or superposition, of solutions of the form
where E² − p² = m². When m² is negative there are two essentially different cases. Either | p | ≥ | E |, in which case E is real and we get solutions that look like waves whose crests move along at the rate | p/E | ≥ 1, i.e., no slower than the speed of light. Or | p | < | E |, in which case E is imaginary and we get solutions that look like waves that amplify exponentially as time passes!
We can decide as we please whether or not we want to consider the second type of solution. They seem weird, but then the whole business is weird, after all.
(1) If we do permit the second type of solution, we can solve the Klein-Gordon equation with any reasonable initial data — that is, any reasonable values of φ and its first time derivative at t = 0. (For the precise definition of "reasonable", consult your local mathematician.) This is typical of wave equations. And, also typical of wave equations, we can prove the following thing: if the solution φ and its time derivative are zero outside the interval [−L, L] when t = 0, they will be zero outside the interval [−L− | t |, L + | t |] at any time t. In other words, localized disturbances do not spread with speed faster than the speed of light! This seems to go against our notion that tachyons move faster than the speed of light, but it's a mathematical fact, known as "unit propagation velocity".
(2) If we don't permit the second sort of solution, we can't solve the Klein-Gordon equation for all reasonable initial data, but only for initial data whose Fourier transforms vanish in the interval [−| m |, | m |]. By the Paley-Wiener theorem this has an odd consequence: it becomes impossible to solve the equation for initial data that vanish outside some interval [−L, L]! In other words, we can no longer "localize" our tachyon in any bounded region in the first place, so it becomes impossible to decide whether or not there is "unit propagation velocity" in the precise sense of part (1). Of course, the crests of the waves exp(−iEt + ipx) move faster than the speed of light, but these waves were never localized in the first place!
The bottom line is that you can't use tachyons to send information faster than the speed of light from one place to another. Doing so would require creating a message encoded some way in a localized tachyon field, and sending it off at superluminal speed toward the intended receiver. But as we have seen you can't have it both ways: localized tachyon disturbances are subluminal and superluminal disturbances are nonlocal.
[1] A. Bers, R. Fox, C. G. Kuper and S. G. Lipson, The impossibility of free tachyons, in Relativity and Gravitation, eds. C. G. Kuper and Asher Peres, New York, Gordon and Breach Science Publishers, 1971, pp. 41–46. This paper analyses the Klein-Gordon equation with imaginary mass, and shows that localized disturbances spread with at most the speed of light, but grow exponentially. Conclusion: "Hence free tachyons have to be rejected not on causality grounds but on stability grounds."
See also the relativity FAQ Faster than light.