Why group velocity represents energy or information transmission? What relation between phase velocity and special relativity I have two question,


*

*why group  velocity represents energy or information transmission?

*what is the relation between phase velocity and Special relativity: why can it exceed C without violation of SR?
I didn't take any classes in special relativity or information theory
or quantum mechanics, my background only contains 


*

*general physics (basic knowledge about Special relativity)

*electromagnetism

*signals and systems (basic knowledge about signal transmission),
so please reply in a simple and easy way.
 A: The statement "group velocity represents energy or information transmission" is not entirely accurate. And that is a big can of worms, in fact. But let us start from the beginning. 
What is the dispersion relation?
Suppose that you have some quantity $u$ that depends on a coordinate $x$ and on time $t$.
Dispersion relation $\omega(k)$ is basically a statement, that if you have a coordinate dependence in a form of a wave with some wave vector $k$:
$$u(x)=Ae^{-ikx},$$
then you can instantly write the time dependence like
$$u(x,t)=Ae^{i\omega(k)t-ikx}.$$
What if you have more complex $u(x)$?
In that case you represent it as a sum (or an integral) of such waves. That is what they call "a Fourier transform":
$$u(x) = \int\limits_{-\infty}^\infty a(k)e^{-ikx}dk$$
Using the superposition principle, we can then write:
$$u(x,t) = \int\limits_{-\infty}^\infty a(k)e^{i\omega(k)t-ikx}dk$$
Now if you assume that $a(k)$ is peaked around some wave vector $k_0$, then 
you can derive that the resulting wave packet moves with a group velocity. 
You can look at the derivation here. 
But how is that the maximal speed of information transfer?
That is my point. Let us have a $\delta$-signal at $x=0$ -- its Fourier transform is uniform across all the frequencies:
$$u(x) = \delta(x) = \frac{1}{2\pi}\int\limits_{-\infty}^\infty e^{-ikx}dk$$
So the time evoultion would be:
$$u(x,t) = \frac{1}{2\pi}\int\limits_{-\infty}^\infty e^{i\omega(k)t-ikx}dk$$
And we are supposed to beleive that this expression gives $u(x,t)=0$ for $x>v_gt$ whatever $\omega(k)$ is.
And that is not true?
Yeah. The trick  is  that if you introduce dispersion and you want to preserve 
causality -- you must introduce some absorbtion as well.
The absorbtion is introduced as a complex component of $\omega(k)$ and the relations
between real and complex (disperion and absorbtion) components are called the Kramers Kronig relations.
Check this reference from the wiki for the formal description of the details of how the causality is related to those dispersion-absorbtion laws.  
A: If we have a wave of a well-defined direction and frequency, the dependence of the field $F$ (something that is waving) on position and time is 
$$ F = F_0 \cos (\omega t - k_x x - k_y y - k_z z ) $$
Adult physicists would use complex exponentials instead of the cosine but I decided to remove this potentially difficult piece of maths.
The argument of the cosine modulo $2\pi$ is the "phase". The group velocity is extracted from this phase. In the spacetime, one draws the hypersurfaces of constant phase, i.e.
$$ \omega t = \vec k \cdot \vec x $$
At a fixed $t$, this is an equation of a plane. If we watch what is happening with this plane as time goes by, it is moving in the transverse direction and the speed of the motion of this plane is 
$$v_{ph} = \frac{dx}{dt} = \frac{\omega }{ k}$$
That's called the phase velocity because we calculated it from the phase (by looking at which places the phase is constant, and how these places move in time).
In general, the phase velocity can exceed the speed of light because what is actually propagating by the phase velocity is "the plane on which the phase is constant" but this plane isn't a real physical object that carries information. It's just a fictitious place in the space defined by a mathematical property, "constant phase".
The fact that the phase velocity may exceed $c$ is analogous to the observation that if we sit at the center of a large hollow sphere with a lamp and we rotate the lamp, the illuminated trace of the lamp on the distant interior surface of the sphere may move faster than $c$ (it safely does if the radius of the sphere is large enough). But we are not transmitting anything from one place of the surface to another. Instead, the light goes from the center to the various points of the surface.
On the other hand, the group velocity does measure the actual propagation of material objects. We may create a "wave packet" by combining nearby frequencies. It is possible to derive that the center of such a wave packet will propagate by the group velocity
$$ v_g = \frac{d\omega}{dk} $$
and because this packet is a localized object that may carry information, relativity prohibits $v_g$ to exceed the speed of light in the vacuum $c$.
