This document gives a brief description of tachyon theory, and explains why tachyons cannot be used for faster-than-light communication.

## Notes

This document is a qualitative, not quantitative, treatment of tachyon theory. To keep the mathematics simple and the diagrams comprehendible, the speed of light is assumed to be 1 space-unit per time-unit, and space is assumed to be one-dimensional.

## The momentum–energy graph

The graph shows momentum (p) on the x-axis and energy (E) on the y-axis. The two yellow lines show the ‘light cone’ (in four dimensional space-time this would be a ‘hypercone’) — that is, where E = ± p. The region above and below the two lines is known as ‘timelike’, and the region to the left and the right is known as ‘spacelike’.

## The standard particles

One of the famous equations of relativity states that E^{2} = m^{2}c^{4} + p^{2}c^{2}.
We are using units such that c = 1, so the equation becomes E^{2} = m^{2} + p^{2}.

The solutions of this are the blue hyperbolae shown on the graph.
Note that the hyperbolae cross the E-axis at E = m, consistent with E = m c^{2} when c = 1.

Particles on the upper branch are known as tardyons or bradyons (= “slow particles”) and they travel at subluminal speed. These include protons, neutrons and electrons. Particles with no mass move on the light cone, at the speed of light, and are known as luxons (= “light particles”). These include photons. Particles on the lower branch have negative mass, and these are the theorised ‘virtual particles’.

## Tachyons

Another famous relativistic equation is E = m(1 - (v/c)^{2})^{-½}.
As tachyons, the theorised faster-than-light particles, have v > c, the square root is of a negative number and therefore is imaginary.
But energy can’t be imaginary, so this tells us that the mass, m, is also imaginary and E is negative.

E^{2} = m^{2} + p^{2} becomes p^{2} - E^{2} = M^{2}, where M is real.
The solutions to this equation are the hyperbolæ in the spacelike region of the graph.
As E < 0 for a tachyon, the pink portion represents tachyons.

From the graph we can see an interesting property of tachyons - as they lose energy (E becomes more negative) their momentum *increases*, unlike ordinary particles.

## Electrically charged tachyons

If tachyons were electrically charged, as they travel faster than light they must produce Cherenkov radiation, lowering their energy and causing them to accelerate. Charged tachyons would therefore lead to a runaway reaction releasing an unlimited amount of energy. But as tachyon–antitachyon pairs would form spontaneously in a vacuum, the runaway reaction would lead to an unstable vacuum, so we must assume tachyons are uncharged.

## Tachyon communication

Let’s look at the relativistic quantum mechanics of tachyons, discussing only spin-free tachyons in two-dimensional space-time for simplicity.
These results still hold for other situations, but the mathematics is beyond the scope of this document.
The wave function for a spin-free tachyon would satisfy the Klein–Gordon equation ((d/dt)^{2} + (d/dx)^{2} + m^{2}) Φ = 0.

Any solution to this equation is a linear combination of solutions of the form Φ(t,x) = e^{-iEt + ipx}.
When m^{2} is negative (as with tachyons), we get two types of solution.
If |p| >= |E| the solutions look like waves propagating faster than light.
If |p| < |E| the solutions look like waves amplifying exponentially over time.
We can now decide whether to consider the second type of solution.

## If |p| < |E|

If we do consider the second type of solution, the Klein–Gordon equation can be solved for any reasonable initial data. We can prove that if Φ and dΦ/dt are zero outside some interval [-L,L] when t=0, they will be zero outside the interval [-L-|t|,L+|t|] at any time t. So localised disturbances do not propagate faster than light.

## If |p| >= |E|

If we do not consider the second type of solution, the Klein–Gordon equation can only be solved for initial data whose Fourier transformations vanish in [-|m|,|m|]. The Paley–Wiener theorem tells us we cannot solve the equation for initial data that vanish outside some interval [-L,L]. This means we cannot localise the wave.

## The final word on tachyon communication

To send a message faster than light using tachyons, we would have to encode the message on a localised tachyon field, and then send it off at superluminal speed. But this is impossible as local tachyon disturbances are subluminal, and superluminal tachyon disturbances cannot be localised.

## Further reading

Faster Than Light: Superluminal Loopholes in Physics, by Nick Herbert:

- Faster Than Light: Superluminal Loopholes in Physics at Amazon.com
- Faster Than Light: Superluminal Loopholes in Physics at Amazon.co.uk (for British readers)

Schrodinger’s Kittens and the Search for Reality, by John Gribbin:

- Schrodinger’s Kittens and the Search for Reality at Amazon.com
- Schrodinger’s Kittens and the Search for Reality at Amazon.co.uk (for British readers)