Do tachyons move faster than light?

A tachyon is a particle with an imaginary rest mass. This however does not mean it "travels" faster than light, nor that there's any conflict between their existence and the special theory of relativity.

The main idea here is that the typical intuition we have about particles -- them being billiard ball-like objects -- utterly fails in the quantum world. It turns out that the correct classical limit for quantum fields in many situations is classical fields rather than point particles, and so you must solve the field equations for a field with imaginary mass and see what happens rather than just naively assume the velocity will turn out to be faster than light.

The mathematical details are a bit technical so I'll just refer to Baez's excellent page if you're interested ( http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html ), but the conclusion can be stated very simply. There's two types of "disturbances" you can make in a tachyon field:

1) Nonlocal disturbances which can be poetically termed "faster than light" but which do not really represent faster than light propagation since they are nonlocal in the first place. In other words, you can't make a nonlocal disturbance in a finite sized laboratory, send it off to your friend in the andromeda galaxy and have them read the message in less time than it would take for light to get there. No, you could at best make a nonlocal disturbance that is as big as your laboratory, and to set that up you need to send a bunch of slower than light signals first. It's akin to telling all your friends all over the solar system to jump at exactly 12:00 am tomorrow: you'll see a nonlocal "disturbance" which cannot be used to send any information because you had to set it up beforehand.

2) Localized disturbances which travel slower than light. These are the only types of disturbances that could be used to send a message using the tachyon field, and they respect special relativity.

In particle physics the term "tachyon" is used to talk about unstable vacuum states. If you find a tachyon in the spectrum of your theory it means you're not sitting on the true vacuum, and that the theory is trying to "roll off" to a state of lower energy. This actual physical process is termed tachyon condensation and likely happened in the early universe when the electroweak theory was trying to find its ground state before the Higgs field acquired its present day value.

A good way to think about tachyons is to imagine hanging several pendulums on a clothesline, one after the other. If you disturb one of them, some amount of force will be transmitted from one pendulum to the next and you'll see a traveling disturbance on the clothesline. You'll be able to identify a "speed of light" for this system (which will really be the speed of sound in the string). Now you can make a "tachyon" in this system by flipping all the pendulums upside down: they'll be in a very unstable position, but that's precisely what a tachyon represents. Nevertheless, there's absolutely no way that you could send a signal down the clothesline faster than the "speed of light" in the system, even with this instability.

tl;dr: Careful consideration of tachyons makes them considerably different from science fiction expectations.

EDIT: As per jdlugosz's suggestion, I've included the link to Lenny Susskind's explanation.

http://youtu.be/gCyImLu0HSI?t=58m51s

In this answer we'll basically repeat Leandro M.'s good answer for a tachyonic field using formulas. (In contrast, note that the current version of the Wikipedia page mostly discusses the hypothetical notion of a tachyonic point particle, which by the very definition moves faster than the speed of light, which is widely believed to be irrelevant for modern physics, and which we will not discuss further here.)

Let us for simplicity use units where $c=1=\hbar$. Consider a spinless relativistic complex scalar field

$$ \left(\partial_t^2-\partial_x^2+m_0^2\right)\phi(x,t)~=~0 \tag{1}$$

in 1+1 spacetime dimensions. The real and imaginary components, ${\rm Re}(\phi)$ and ${\rm Im}(\phi)$, are independent fields, since the eq. of motion (1) is linear.

The Lagrangian density for a spinless relativistic complex scalar field (1) is

$$ \begin{align} {\cal L}~=~&|\partial_t\phi|^2 - {\cal V}, \cr {\cal V}~=~&|\partial_x\phi|^2+ m_0^2 |\phi|^2.\end{align}\tag{2} $$

A tachyonic mass $ m_0^2<0$ corresponds to a potential density ${\cal V}$ that is unbounded from below, which leads to an instability $|\phi|\to \infty$.

Let us perform a spatial Fourier transformation

$$\begin{align} \tilde{\phi}(p,t)~=~&\int_{\mathbb{R}} \!dx~e^{-ipx} \phi(x,t),\cr \phi(x,t)~=~&\int_{\mathbb{R}} \!\frac{dp}{2\pi}~e^{ipx}\tilde{\phi}(p,t).\end{align}\tag{3}$$

Then the wave equation (1) becomes a second-order linear ODE

$$ \left(\partial_t^2+E_p^2\right)\tilde{\phi}(p,t)~=~0, \tag{4}$$

where

$$ E_p~:=~\sqrt{p^2+m_0^2}.\tag{5}$$

The complete solution to the second-order linear ODE (4) is$^1$

$$ \begin{align} \tilde{\phi}(p,t)~=~&\sum_{\pm} C_{\pm}(p)e^{\pm iE_pt}\cr ~=~& A(p)\cos(E_pt)+B(p)t~{\rm sinc}(E_pt),\end{align}\tag{6}$$

where

$$ C_{\pm}(p)~=~\frac{1}{2}\left(A(p)\mp i \frac{B(p)}{E_p}\right)\tag{7}$$

are two integration constants. We next analyze various cases.

1. Waves localized in $p$-space (and hence non-local in $x$-space). Here we assume that the wave packet is almost monochromatic, so that the coefficient functions $p \mapsto C_{\pm}(p)$ are sharply peaked around a central momentum. Such a wave packet is hence non-local in $x$-space, cf. the Heisenberg uncertainty principle.

1a) Oscillatory case $p^2>-m_0^2$. The phase velocity is

$$ v_p ~:=~\frac{E_p}{|p|} ~\left\{ \begin{array}{c} > \cr = \cr <\end{array}\right\}~ 1\quad\text{for}\quad m_0^2 ~\left\{ \begin{array}{c} > \cr = \cr <\end{array}\right\}~0. \tag{8} $$

The group velocity is

$$ v_g ~:=~\frac{dE_p}{d|p|} ~\stackrel{(5)}{=}~\frac{|p|}{E_p}~=~\frac{1}{v_p} ~\left\{ \begin{array}{c} < \cr = \cr >\end{array}\right\}~ 1\quad\text{for}\quad m_0^2 ~\left\{ \begin{array}{c} > \cr = \cr <\end{array}\right\}~0. \tag{9} $$

The group velocity formula (9) is derived under the assumption that we may linearize the dispersion relation, i.e. the wave packet is assumed to be localized in $p$-space. In the tachyonic case $m_0^2<0$, the group velocity is faster than the speed of light.

1b) Exponentially growing/decaying case $p^2<-m_0^2$. Such non-travelling solutions (6) are only possible for tachyons $m_0^2<0$.

2. Waves localized in $x$-space. Assume that for each constant time slice $t$, the wave has compact support

$$ \begin{align} {\rm supp}(\phi(\cdot,t))~:=~&\overline{\{ x\in \mathbb{R}|\phi(x,t) \neq 0\}}\cr ~=~&[a_-(t),a_+(t)]~\subset ~\mathbb{R}, \cr a_+(t)~:=~&\sup {\rm supp}(\phi(\cdot,t))~<~\infty,\cr a_-(t)~:=~&\inf {\rm supp}(\phi(\cdot,t))~>~-\infty , \end{align}\tag{10} $$

of the form of an interval with endpoints $-\infty<a_-(t)<a_+(t)<\infty$. Let us define for later convenience the midpoint and the half-length

$$\begin{align} c(t)~:=&~\frac{a_+(t)+a_-(t)}{2},\cr b(t)~:=~&\frac{a_+(t)-a_-(t)}{2}~\geq~0, \end{align}\tag{11}$$

respectively.

   ^ phi    
   |            _____
   |           /     \_______________
   |          /    b           b     \
 --|---------|-----------|-----------|--------------> x
             a-          c           a+

$\uparrow$ Fig. 1. A wave $\phi(x)$ with compact support $[a_-,a_+]$ along the $x$-axis. Time $t$ is suppressed from the notation.

Up until now the Fourier variable $p$ has been real. However, the second-order linear ODE (4) and its solution (6) make sense for complex momentum $p\in\mathbb{C}$. We may hence take advantage of complex function theory. The square root (5) has an asymptotic behaviour

$$ E_p~\sim~\pm p\quad\text{for}\quad |p|\to\infty.\tag{12}$$

If the compactly supported function $\phi(\cdot,t)\in {\cal L}^1(\mathbb{R})$ is integrable, then the corresponding spatial Fourier transform $\tilde{\phi}(\cdot,t)$ is an entire function by Lebesgue's majorant theorem. Comparing eqs. (3a) and (10), the ultra-relativistic asymptotic behaviour is heuristically given as

$$ \tilde{\phi}(p,t)~\sim~e^{-ia_{\pm}(t)p}\quad\text{for}\quad {\rm Im}(p)\to \pm \infty. \tag{13}$$

A rigorous mathematical characterization$^2$ of this spatial Fourier transform is provided by the Paley-Wiener (PW) theorem.

Comparing eqs. (6), (12), and (13), we deduce that the front velocity is generically$^3$ the speed of light,

$$ \frac{da_{\pm}(t)}{dt}~=~\pm 1, \tag{14}$$

i.e. the endpoints $a_{\pm}(t)$ of the compact support move with the speed of light, independently of the mass square $m_0^2$. This is because the mass is not important in the ultra-relativistic limit (12). In particular, the support (10) of a position-localized wave packet does not expand faster than the speed of light, not even in the tachyonic case $m_0^2<0$.

References:

1. Tachyons at The Original Usenet Physics FAQ.

--

Footnotes:

$^1$ The latter form of eq. (6) is manifestly free of the square root ambiguity (5) by using even functions, i.e. the cosine and sinc function. The Fourier-transformed wave $\tilde{\phi}(\cdot,t)$ is holomorphic iff the two coefficient functions $A(\cdot)$ and $B(\cdot)$ are. If the wave $\phi\in\mathbb{R}$ is real, then the Fourier-transformed wave satisfied

$$ \tilde{\phi}(p,t)^{\ast}~=~\tilde{\phi}(-p^{\ast},t),\tag{15}$$

iff

$$ A(p)^{\ast}~=~A(-p^{\ast}), \qquad B(p)^{\ast}~=~B(-p^{\ast}).\tag{16} $$

See also the Schwarz reflection principle.

$^2$ Here is a rigorous proof of eq. (14). Assume that $\phi(\cdot,t\!=\!0)\in {\cal L}^2(\mathbb{R})$ is (i) square-integrable, and (ii) has compact support

$$ -\infty~<~a_-(t\!=\!0) ~\leq ~a_+(t\!=\!0)~<~\infty.\tag{17}$$

[The square-integrability (i) is a technicality to get inside the realm of the Paley-Wiener (PW) theorem. Then by Cauchy-Schwarz's inequality, the function $\phi(\cdot,t\!=\!0)\in {\cal L}^1(\mathbb{R})$ is integrable.]

By shifting the $x$-axis if necessary, we may assume that the initial support midpoint $c(t\!=\!0)=0$ is zero, i.e.

$$ \infty~>~a_0~:=~a_+(t\!=\!0)~=~-a_-(t\!=\!0)~\geq~0. \tag{18}$$

In this way we get an initial globally defined holomorphic Fourier transform of exponential type $a_0$

$$ \forall p\in \mathbb{C}: ~~ |A(p)|~\stackrel{(6)}{=}~|\tilde{\phi}(p,t\!=\!0)| ~\stackrel{(3a)}{\leq}~ Ke^{a_0|p|}, \tag{19}$$

where

$$\begin{align} K~:=~&\int_{\mathbb{R}}\!dx~ |\phi(x,t\!=\!0)|\cr ~=~&\int_{[-a_0,a_0]}\!dx~ |\phi(x,t\!=\!0)|~<~\infty.\end{align}\tag{20}$$

[Conversely, the ineq. (19) together with the Paley-Wiener (PW) theorem guarantees that the support

$$ {\rm supp}(\phi(\cdot,t\!=\!0))~\subseteq~[-a_0,a_0] \tag{21}$$

is inside the interval $[-a_0,a_0]$. The proof of eq. (21) is a straightforward exercise in closing an integration contour in the upper or lower half-plane of the complex $p$-plane.]

Assuming that the support ${\rm supp}(\phi(\cdot,t\!=\!t_0))$ remains compact for at least one other time slice $t_0\neq 0$, it is then necessary that the coefficient function $B(\cdot)$ is an entire function of exponential type

$$ \exists L,b_0>0~ \forall p\in \mathbb{C}: ~~ |B(p)|~\leq~ Le^{b_0|p|}.\tag{22} $$

It must be possible to choose $b_0\leq a_0$, because else the front velocity would be infinite, which is physically unacceptable.

Combining eqs. (19) and (22) with eq. (6), then for an arbitrary time slice $t$, we get a globally defined holomorphic Fourier transform of exponential type $a_0+|t|$,

$$ \exists M>0~\forall p\in \mathbb{C}: ~~|\tilde{\phi}(p,t)|~\leq~ M e^{|m_0||t|}e^{(a_0+|t|)|p|} .\tag{23}$$

In eq. (23) we have used the triangle inequality

$$ |E_p|~\stackrel{(5)}{\leq}~ \sqrt{|p|^2+|m_0|^2}~\leq~|p|+|m_0|. \tag{24}$$

Conversely, the ineq. (23) together with the PW theorem now guarantees that the support (10) is inside the interval

$$ [-a_0-|t|,a_0+|t|]~\subset~\mathbb{R},\tag{25}$$

i.e. the front velocity is less or equal to the speed of light, as we wanted to show.

$^3$ We assume a generic situation, where the coefficient functions $C_{\pm}(p)$ do not vanish for $|{\rm Im}(p)|\to\infty.$