Do tachyons move faster than light?

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

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

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

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

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

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

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

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

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

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

The group velocity is

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

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

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

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 $pinmathbb{C}$. We may hence take advantage of complex function theory. The square root (5) has an asymptotic behaviour

$$ E_p~sim~pm pquadtext{for}quad |p|toinfty.tag{12}$$

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

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

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 $phiinmathbb{R}$ 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 $a_0$

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

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 $t_0neq 0$, it is then necessary that the coefficient function $B(cdot)$ is an entire function of exponential type

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

It must be possible to choose $b_0leq a_0$, 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 $a_0+|t|$,

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

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 $|{rm Im}(p)|toinfty.$

if tachyon is faster than speed of light it means that the theory that was given by the albert einstien is wrong .because he mentioned that the substance travelling at a speed of light should have infinite amount of energy

Here is another viewpoint- that tachyons are photons in their own frame of reference. The electrons that we know and love always seem to have the same characteristics, such as spin. Is this because only one spin value is allowed, or do the electrons simply not interact with photons put out by electrons with a different spin to themselves? Notice that when you are listening to one wavelength, you are excluding all the other possible wavelengths, even though they exist. If we could speed up the spin values of an atom, it might then seem to disappear from our known world! If we doubled the spin rate, it might be emitting photons that travel twice as fast as any photon we can intercept! But in its' own frame of reference, tempo would be twice as fast, so it would only record the photons as going at the speed of light. The only way to utilise such physics would be to plan for a trip to a star with antimatter rockets, just as we would in normal space, and add some tempo-boosting machinery. The rocket would seem to disappear to us on Earth, and take just a few years to reach the star, though the astronauts would have aged more than us. So Hyperspace is simply the same space with particles and photons that ignore us.