updated: 22-MAR-1993 by SIC;

original by Scott I. 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). I give here 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 E^{2} - p^{2} = m^{2}.
(Let's take c=1 for the rest of the discussion.) For any non-zero value of
m (mass), this is an 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,
E^{2} = p^{2}, 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. v.159, pp.1089--1105 (1967)]).

Now another familiar relativistic equation is E =
m*[1-(v/c)^{2}]^{-1/2}. 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 E^{2} - p^{2} = m^{2} < 0. Or, p^{2} -
E^{2} = M^{2}, 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 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. However,
one reasonably modern reference is *Tachyons, Monopoles, and Related
Topics*, E. Recami, ed. (North-Holland, Amsterdam, 1978).

However, 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. 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:

(BOX + m^{2})phi = 0

where BOX is the D'Alembertian, which in 3+1 dimensions is just

BOX = (d/dt)^{2}- (d/dx)^{2}- (d/dy)^{2}- (d/dz)^{2}.

The difference with tachyons is that m^{2} is *negative*, and m is
imaginary.

To simplify the math a bit, let's work in 1+1 dimensions, with co-ordinates x and t, so that

BOX = (d/dt)^{2}- (d/dx)^{2}

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

phi(t,x) = exp(-iEt + ipx)

where E^{2} - p^{2} = m^{2}. When m^{2} 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 waves
that amplify exponentially as time passes!

We can decide as we please whether or not we want to consider the second sort of solutions. They seem weird, but then the whole business is weird, after all.

1) If we *do* permit the second sort of solution, we can solve the
Klein-Gordon equation with any reasonable initial data - that is, any
reasonable values of phi 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 phi
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.