En esta respuesta que, básicamente, va a repetir Leandro M. la buena respuesta de la utilización de fórmulas.
Supongamos por simplicidad de uso de las unidades donde $c=1=\hbar$. Considere la posibilidad de un spinless relativista complejo campo escalar
$$\tag{1} \left(\partial_t^2-\partial_x^2+m_0^2\right)\phi(x,t)~=~0 $$
en 1+1 dimensiones de espacio-tiempo. Las componentes real e imaginaria,
${\rm Re}(\phi)$ ${\rm Im}(\phi)$ , son independientes de los campos, desde la eq. de movimiento (1) es lineal.
La densidad Lagrangiana para un spinless relativista complejo campo escalar (1) es
$$\etiqueta{2} {\cal L}~=~|\partial_t\phi|^2 - {\cal V}, \qquad
{\cal V}~=~|\partial_x\phi|^2+ m_0^2 |\phi|^2. $$
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
$$\tag{3} \tilde{\phi}(p,t)~=~\int_{\mathbb{R}} \!dx~e^{-ipx} \phi(x,t),\qquad \phi(x,t)~=~\int_{\mathbb{R}} \!\frac{dp}{2\pi}~e^{ipx}\tilde{\phi}(p,t).$$
Then the wave equation (1) becomes a second-order linear ODE
$$\tag{4} \left(\partial_t^2+E_p^2\right)\tilde{\phi}(p,t)~=~0, $$
where
$$\tag{5} E_p~:=~\sqrt{p^2+m_0^2}.$$
The complete solution to the second-order linear ODE (4) is$^1$
$$ \la etiqueta{6} \tilde{\phi}(p,t)~=~\sum_{\pm} C_{\pm}(p)e^{\pm iE_pt}
~ = ~ (P)\cos(E_pt)+B(p)t~{\rm sinc}(E_pt),$$
where
$$ \tag{7} C_{\pm}(p)~=~\frac{1}{2}\left(A(p)\mp i \frac{B(p)}{E_p}\right)$$
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
$$\etiqueta{8} v_p ~:=~\frac{E_p}{|p|}
~\left\{ \begin{array}{c} > \cr = \cr <\end{array}\right\}~
1\quad\text{para}\quad m_0^2
~\left\{ \begin{array}{c} > \cr = \cr <\end{array}\right\}~0. $$
The group velocity is
$$\la etiqueta{9} v_g ~:=~\frac{dE_p}{d|p|} ~\stackrel{(5)}{=}~\frac{|p|}{E_p}~=~\frac{1}{v_p}
~\left\{ \begin{array}{c} < \cr = \cr >\end{array}\right\}~
1\quad\text{para}\quad m_0^2
~\left\{ \begin{array}{c} > \cr = \cr <\end{array}\right\}~0. $$
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
$$ {\rm supp}(\phi(\cdot,t))~:=~\overline{\{ x\in \mathbb{R}|\phi(x,t) \neq 0\}}~=~[a_-(t),a_+(t)]~\subset ~\mathbb{R}, $$
$$\la etiqueta{10}
a_+(t)~:=~\sup {\rm supp}(\phi(\cdot,t))~<~\infty, \qquad
a_-(t)~:=~\inf {\rm supp}(\phi(\cdot,t))~>~-\infty , \qquad $$
of the form of an interval with endpoints $-\infty<a_-(t)<a_+(t)<\infty$.
Vamos a definir para su posterior comodidad, el punto medio y la mitad de la longitud
$$\tag{11} c(t)~:=~\frac{a_+(t)+a_-(t)}{2}, \qquad b(t)~:=~\frac{a_+(t)-a_-(t)}{2}~\geq~0, $$
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
$$\tag{12} E_p~\sim~\pm p\quad\text{for}\quad |p|\to\infty.$$
If the compactly supported function $\phi(\cdot,t)\en {\cal L}^1(\mathbb{R})$ 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
$$\tag{13} \tilde{\phi}(p,t)~\sim~e^{-ia_{\pm}(t)p}\quad\text{for}\quad {\rm Im}(p)\to \pm \infty. $$
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,
$$\tag{14} \frac{da_{\pm}(t)}{dt}~=~\pm 1, $$
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 $m_0^2<0$.
Referencias:
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La ayuda de los taquiones en El Original de Usenet Física de preguntas frecuentes.
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Notas a pie de página:
$^1$ La última forma de eq. (6) es manifiestamente libre de la raíz cuadrada de la ambigüedad (5) utilizando incluso las funciones, es decir, el coseno y la sinc función. La de Fourier transformada de onda $\tilde{\phi}(\cdot,t)$ es holomorphic iff los dos el coeficiente de funciones $A(\cdot)$$B(\cdot)$. Si la ola $\phi\in\mathbb{R}$ es real, entonces la transformada de Fourier transformada de onda satisfecho
$$\tag{15} \tilde{\phi}(p,t)^{\ast}~=~\tilde{\phi}(-p^{\ast},t),$$
iff
$$\tag{16} A(p)^{\ast}~=~A(-p^{\ast}), \qquad B(p)^{\ast}~=~B(-p^{\ast}). $$
See also the Schwarz reflection principle.
$^2$ Here is a rigorous proof of eq. (14). Assume that $\phi(\cdot,t\!=\!0)\in {\cal L}^2(\mathbb{R})$ is (i) square-integrable, and (ii) has compact support
$$\tag{17} -\infty~<~a_-(t\!=\!0) ~\leq ~a_+(t\!=\!0)~<~\infty.$$
[The square-integrability (i) is a technicality to get inside the realm of the Paley-Wiener (PW) theorem. Then by Cauchy-Schwarz's inequality, the function $\phi(\cdot,t\!=\!0)\in {\cal L}^1(\mathbb{R})$ is integrable.]
By shifting the $x$-axis if necessary, we may assume that the initial support midpoint $c(t\!=\!0)=0$ is zero, i.e.
$$\tag{18} \infty~>~a_0~:=~a_+(t\!=\!0)~=~-a_-(t\!=\!0)~\geq~0. $$
In this way we get an initial globally defined holomorphic Fourier transform of exponential type $a_0$
$$ \etiqueta{19} \forall p\in \mathbb{C}: ~~
|A(p)|~\stackrel{(6)}{=}~|\tilde{\phi}(p,t\!=\!0)|
~\stackrel{(3a)}{\leq}~ Ke^{a_0|p|}, $$
where
$$\tag{20} K~:=~\int_{\mathbb{R}}\!dx~ |\phi(x,t\!=\!0)|~=~\int_{[-a_0,a_0]}\!dx~ |\phi(x,t\!=\!0)|~<~\infty.$$
[Conversely, the ineq. (19) together with the Paley-Wiener (PW) theorem guarantees that the support $$\tag{21} {\rm supp}(\phi(\cdot,t\!=\!0))~\subseteq~[-a_0,a_0] $$ is inside the interval $[-a_0,a_0]$. The proof of eq. (21) is a straightforward exercise in closing an integration contour in the upper or lower half-plane of the complex $p$-plane.]
Assuming that the support ${\rm supp}(\phi(\cdot,t\!=\!t_0))$ remains compact for at least one other time slice $t_0\neq 0$, it is necessary that the coefficient function $B(\cdot)$ is an entire function of exponential type
$$ \etiqueta{22} \existe L,b_0>0~ \forall p\in \mathbb{C}: ~~
|B(p)|~\leq~ Le^{b_0|p|}. $$
It must be possible to choose $b_0\leq a_0$, because else the front velocity would be infinite, which is physically unacceptable.
Combining eqs. (19) and (22) with eq. (6), then for an arbitrary time slice $t$, we get a globally defined holomorphic Fourier transform of exponential type $a_0+|t|$,
$$\tag{23} \exists M>0~\forall p\in \mathbb{C}: ~~|\tilde{\phi}(p,t)|~\leq~ M e^{|m_0||t|}e^{(a_0+|t|)|p|} .$$
In eq. (23) we have used the triangle inequality
$$\tag{24} |E_p|~\stackrel{(5)}{\leq}~ \sqrt{|p|^2+|m_0|^2}~\leq~|p|+|m_0|. $$
Conversely, the ineq. (23) together with the PW theorem now guarantees that the support (10) is inside the interval
$$\tag{25} [-a_0-|t|,a_0+|t|]~\subset~\mathbb{R},$$
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 $|{\rm Im}(p)|\to\infty.$
