To me, this question defines the fundamental tenet of special relativity. Once you define the speed of light as a constant in any inertial reference frame, you have to explain what happens to a wave of a given frequency propagating from one reference frame to another. For a continuous wave, that simply becomes a change in frequency as the oscillations pass "faster" or "slower" in the second reference frame.

While not my main area of expertise (BIG DISCLAIMER HERE), where the problem comes in is when you consider the particle nature of light. If we consider a photon as a propagating wave packet (e.g. one oscillation of the EM field) then the above explanation seems to break down since that one oscillation that is propagating away from the source at the speed of light in the source reference frame would necessarily appear to be propagating faster or slower than the speed of light in another reference frame moving relative to the first. THAT has to be fixed! I believe that's where your group velocity vs. phase velocity comes in. One corresponds to the speed of transfer of information (packet) while the other corresponds to the rate of change of the phase (frequency) of the wave. The point of SR is that in the second reference frame, the wave didn't get to the receiver any faster/slower than the speed of light in that reference frame, but it was just propagating at a higher or lower frequency the whole time. Of course the fact that the source may be much closer/further away by the time it arrives is immaterial. To the observer, the wave ALWAYS had the frequency of their reference frame.

Again, probably not the best way to explain it, but this is the way I visualize the problem. Of course every time I turn around, I question what that all really means, but at least I can see from this discussion how the whole theory came about.

r here is an independent variable and does not depent on time. Taking the time derivative produces E = const.

Dear Sofia,

In the case of the light in vacuum you don't need to separate velocity of group (derivative of angular frequency respect to k) from the phase velocity (angular frequency divided by k) because the dispersion law is

E=pc

And it is inmediate to see that both velocities are equal to c which is a constant.

Daniel,

in a wave-packet E cannot be fixed, i.e. ΔE = 0, unless the time duration of the wave-packet is infinite - see uncertainty principle. The linear momentum also cannot be fixed, otherwise the length of the wave-packet should be infinite.

About the phase velocity I was told that near an antenna it is greater than c. Is that true?

Dear Sofia,

The phase velocity can be higher than c but it doesn't depend of any antenna, what depends is only of the dispersion law that it follows in the physical medium. What never can happens is that the energy velocity is higher than c in every medium that you have.

The Fourier harmonics moves at different velocities as you said but all of them are forming a packet which moves at the group velocity (only defined assuming that is enough the first Taylor term approach in the dispersion law). One good place for seeing how the energy needs to follow lower velocities than light is in the Cherenkov effect.

https://en.wikipedia.org/wiki/Cherenkov_radiation

Dear Daniel,

I know from my teacher that near an antenna the phase velocity may exceed c. He was an electricity engineer. So am I but I didn't learn antennas. However, let's see if somebody is specialist in antennas and can tell us more precise things.

Now , please tell me, the dispersion function regards which physical quantity? Velocity, linear momentum, or energy? We have to pay attention that for particles possessing rest-mass, at low energies, the linear momentum is equal to mv, s.t. the velocity differs from the linear momentum only by a multiplicative constant, the rest mass. For photons there is no such thing.

Thanks for the article with Cherenkov radiation, I will look.

Feynman pretty much wrote the definitive work on why wave propagation in matter can appear to exceed the speed of light. However, that is the speed of light in the material!!! I strongly recommend taking a look at that. I don't recall exactly where he covers it, but I found the Feynman lectures published online by CalTech. http://www.feynmanlectures.caltech.edu/

Dear Sofia,

I don't know about antennas that they have electric charged in accelerated motion for producing radiation or vice verse. But I don't see how the group velocity could be higher than c (where? because it needs to finish having c?

The dispersion law is the frequency in function of wavelength, or the energy in function of the mementum for particles.

Dear Daniel,

An antenna works as a circuit with a capacitor and a coil, i.e. the current is variable and emits e.m. waves.

By the way, look what I found in Wikipedia on the page

https://en.wikipedia.org/wiki/Phase_velocity#Relation_to_group_velocity,_refractive_index_and_transmission_speed

"The phase velocity of electromagnetic radiation may – under certain circumstances (for example anomalous dispersion) – exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer. It was theoretically described by physicists such as Arnold Sommerfeld and Léon Brillouin."

I didn't find additional explanations.

Tunneling occurs faster than light, but tunneling refers to the passing though a potential barrier.

Anyway, why the group-velocity of any wave-packet of light (any photon) is the same? Because so say Maxwell's equations? It seems to me that the reason is deeper. I think that gluons, which also have rest-mass zero, also have group-velocity c (I am not sure, because gluons are confined particles s.t. I don't know if one can speak of their velocity).

Dear Sofia,

The reason is the one that I have told you at the beginning: electromagnetic waves in vacuum has a linear dispersive equation and therefore their group and phase velocities are the same. The Poynting vector gives them the momentum and they have a constant density of energy.

If you have a condenser and an induction in the circuit it means that you have a resonant frequency for selecting the kind of wave that you want, nothing more. The velocity needs a dispersive equation for distinguishing the things that you ask, but my knowledge of antennas is almost nothing, I only know how the radiation is created.

The group velocity is defined by a whole lot of separate waves, superimposing to create a wave packet, which goes along with

the particle. Therefore it is reasonable to assign c with the photon.

Dear all , there is just a caveat in using the concept " group velocity" if one applies the simple definition via d-omega/d-beta..This is only valid in regions of weak dispersion and of course the group velocity should not exceed the speed of light and in particular the group delay should bot become negative. However there are lots papers claiming superluminal propagation and usually the above mentioned caveat is not taken into account. One can easily produce zero and even negative group delays in the frame of that logic (just make a simple notch filter with 3 cables and measure it with the VNA in S21 and you jet negative group delays...)Of course phase velocities can easily exceed the value of c (in then old days there wre CRTs writing faster then the speed of light in realtime).or just visualize a strong laser pointer where the light spot can easily go faster than light over the moon.

Dear Fritz,

Can you be more explicit? You may introduce explanations in the same post.

For instance what you mean by beta? In literature β = v/c. Please check your formula, shuldn't there be dҟ instead of dβ?

"However there are lots papers claiming superluminal propagation . . ."

Of course, the penetration of a potentail barrier by tunneling is done at superluminal velocity. But a barrier means that the light does not travel in void, there is a field there.

". . . make a simple notch filter with 3 cables and measure it with the VNA in S21 and you jet negative group delays . . ."

What are VNA and S21? Not everybody belongs to your domain of competence, so, please be more explicit. (A lot of time ago when I was a student, I wanted to take a course about antennas, but the lecturer threw formulas without proving, and I abandonned.)

With kind regards,

Sofia

Hi Sofia

yes the beta I mentioned is v/c and k is often used like beta..

The well known claim that there there is superluminal propagation e.g at a waveguide below cutoff has led since more than 40 years to eternal discussions (similar in the near field of an antenna).VNA stands for vector network analyzer

And S21 is the forward transmission (one can also use S12 as backward transmission in this case)

Fritz,

Thanks for your explanations.

Now, the definition of the group velocity is not dω/d(v/c), where from did you take this formula? This derivative doesn't even have dimensions of velocity. The definition is dω/dҟ (look in Wikipedia at the page "group velocity"). The group velocity is defined as the velocity with which moves the wave-packet, more illustratively speaking, the peak of its envelope. (I see that you are specialized in antennas. Is dω/d(v/c) the formula with which you work? What is its physical justification?)

To your attention I am questioning the universal constant c, s.t. I speak of the group velocity in vacuum.

Best regards

Sofia, your calculation of the group velocity is incorrect. To obtain it, you should take dE/dp (or the gradient of E with respect to p) and not the derivative of φp at constant value. The relationship between E and p for elementary particles in vacuum is E2 = p2 c2+ m2 c4. From that you can derive |∇pE| E/p =vg vφ= c2 , which means that for elementary particles the product of group and phase velocity is c2 in vacuum. For photons, both velocities are equal to c (m=0). Note that for pure plane waves, the phase velocity is the only one appearing so your derivative actually produces the phase velocity and not the group velocity.

In materials, the relationship between E (or ω) and p (or k) is more complicated for photons, so both their group and phase velocities will differ from c. Note that the speed appearing in the Maxwell equations is the phase velocity. This holds for the standard vacuum Maxwell equations and it holds for the Fourier transformed Maxwell equations in materials.

Sofia: Sorry, I see that you did not actually want to calculate the group velocity. As far as the phase velocity goes, your calculation is correct, but for the rest, there are inwarranted conclusions.

In a material, both group and phase velocity will depend on p or k, in vacuum, both don't, so there is no dispersion. So the answer to your question is simply this: the universal constant c arising from the Maxwell equations is universal only in vacuum, where it is both the group and the phase velocity. In materials, there is a single speed only in the frequency-space Maxwell equations (in the time domain, they become more complicated due to absorption and dispersion) and that is the phase velocity, not the group velocity. Also, it is usually not equal to the vacuum speed of light.

Dear all

there had been a few years ago a lot of "noise"about superluminal claims (60 ns difference) for neutrino transmission (OPERA experiment) from CERN (Geneva) to Grand Sasso (not far from Rome)

In the end, a faulty optical fiber connector was identified as cause of the problem and a cross-check with the IKARUS experiment (also CERN -Gd Sasso neutrino beam but different detector there) confirmed this.This nothing really superluminal from there. However there are certain theories (also related to space curvature)that for cosmological dimensions something could happen. And of course for homogeneous plane waves in vacuo its like K. Kassner said above...

However be very careful in the nearfield of an antenna there one can also describe the field as a superposition of plane waves but there you can nicely "show" group velocities based on the simple definition, which are bigger than c.(but there one is in a region of strong dispersion[from the nearfield].

Klaus,

Thanks for noticing what in fact described my formula. Now, I don't care of what happens in a material or inside some potential barrier. The universal constant c does not refer to these situations. One of my teachers in Technion told me that near an antenna, the phase velocity may exceed c. But he gave me no additional explanations. As I said, I disliked the course on antennas, and didn't take it, s.t. until today I don't understand how the phase velocity may exceed c.

What I know a case when the phase velocity is not exactly PHYSICAL. Indeed, if we send a spot of light to a far screen, and move the toss-light, the spot may move on the screen at superluminal velocity. This it's easily understandable.

There is another nice way to visualize high phase velocities..assume you are standing on a beach and the (water) waves are coming with some shallow angle to the beach..then the speed of propagation of those water waves (which for water waves depends on the depth of the water) can be seen as a kind of group velocity but the speed with which the wave breaking front is running up the beach is a phase velocity.The phase velocity is something very clearly physical and can easily be measured (e.g via standing wave pattern in a waveguide) but also in acoustics. And what you teacher told you that the PHASE velocity near an antenna may exceed the speed of light is absolutely correct similar to the situation at a waveguide below cutoff.BUT there are many claims that also the group velocity in such cases exceeds v=c which can easily be "demonstrated" but one should be very careful before interpreting this as signal velocity (V>c) since it usually violates the underlaying approximation or condition to use that simple definition only for weak dispersion.

Fritz,

Your last post is really a tellegram. Can't you go more slowly? From your last sentence I understood nothing.

"there are many claims that also the group velocity in such cases exceeds v=c which can easily be "demonstrated" but one should be very careful before interpreting this as signal velocity (V>c) since it usually violates the underlaying approximation or condition to use that simple definition only for weak dispersion."

With the waves of water I understood you. But not with the group-velocity. Can't you give a simple example? The group-velocity is usually understood INTUITIVELY as the speed of energy transfer. It's a law of our universe that energy cannot travel in vacuum faster than c.

You make a mistake initially:

"A photon is a wave-packet which can be Fourier expanded as"

A photon is not a wave packet. The wave packet represents the probability density of having the photon at this time-space position. In interactions with matter, the photon always has a well-defined frequency.

The wave associated with a propagating photon for which there is no information on its spatio-temporal position is a circular polarized plane wave.It is implicitly with this photon which is defined the light propagation velocity in the endless vacuum.

Phase velocity is the propagation velocity of the probability wave associated with the photon. In the infinite vacuum the probability wave is not deformed the shape of the wave packet remains invariant, it propagates at the velocity c. In the matter or in a waveguide this velocity may be greater than the speed of propagation in the infinite vacuum. An omega component of the wave packet can not be considered alone. It has a physical meaning only taken with the whole wave packet.

The reasoning error comes from the fact that photons are bosons: we can have any number of photons in @the same quantum state. For a single photon the wave packet represents a probability density. For a huge number of photons the wave packet represents the number of photons having a given frequency. To reason correctly it must be done for a single photon.The phase velocity affects the shape of the probability function but not the speed of each photon constituting the wave packet

Hi Jean,

I agree with your comment for a single photon..but the physics of all this can very nicely be visualized in a microwave experiment using a wave guide "measurement line"i.e. a waveguide (TE10) with has a longitudinal slit in the middle of the wide side (no wall current are intersected then) and a moveable e-field probe there.Of course in a microwave experiment one uses MANY photons..and such a setup can be operated in traveling wave and also standing wave mode with the output of the above mentioned E-field probe connected to port 1 of VNA(vector network analyzer) and port one from the VNA to the input of the (waveguide type) measurement line via some coax waveguide transition.And on the single frequency mode (CW=continuous wave) mode one can visisalize very nicely the phase velocity this way and then applying a modulation via synthetic pulse method using the time domain menu (and here is finally the relation the the Fourier transform mentioned earlier) one can also visualize the velocity of propagation of this (Gaussian shape) carrier modulated signa (RF burst with Gaussian envelope and +-sigma length of say 0.5 ns)

The group velocity is more important because it represents the velocity of the wave energy (wave packet) when the phase velocity represent the velocity of the wavefronts. in a non dispersive medium (vacuum), the phase velocity and the group velocity are equal.

This is my latest guess,

There are two different things, the photon, as monochromatic wave, and the bunch of waves or photons, whoes shape is constant in vacuum. The later acts as a particle, with just an average energy.

The group velocity of a wave is the velocity of the envelope of the wave packet. The procedure for derivation of the group velocity is as follows:

The wave packet can be considered to be a Gaussian wave packet with a carrier wave of wavenumber, $k_0$,

g(x)=e^{-\frac{1}{2} (\frac{x-x_0}{\sigma_x})^2}\ e^{j k_0 x}

where $k_0$ is the carrier wavenumber. The Fourier transform of this wave packet is:

\tilde g(k)=\frac{\sigma_x}{2\sqrt{\pi}} e^{-\frac{\sigma_x^2}{2}(k-k_0)^2}\ e^{j x_0 (k-k_0)}.

The time evolution of this equation is the wave packet at any time which is constructed by the superposition principle,

A(x,t)=\int_{-\infty}^\infty \tilde g(k) e^{j(k x-\omega(k) t)} dk

where $\tilde g(k)$ is the Fourier transform of the wave packet at time $t=0$. Linearizing the angular frequency which is accomplished by Taylor series approximation , $\omega (k) \approx \omega_o+(k-k_0) \frac{d\omega}{dk}$ and substituting into $A(x,t)$ one can obtain:

A(x,t)=e^{i(k_0 x -\omega_0)} \int_{-\infty}^\infty \tilde g(k) e^{j (k-k_0) (x-\frac{d\omega}{dk} t)} dk

The physical interpretation of this equation has become one of the most important concept in physics and engineering. The first term describes a wave propagating with the velocity of $\omega_0/k_0$ and the second term a wave packet which propagates with $\frac{d\omega}{dk}$.The previous is called phase velocity and is used for non-dispersive media and the later, known as group velocity, which represents the envelope of the wave packet traveling with this velocity while propagating through space. Our aim is to determine the group velocity in the case of a wave propagating inside an accelerating structure.

For more information you can check out my recent paper:

Conference Paper NEW ANALYTICAL DERIVATION OF GROUP VELOCITY IN TW ACCELERATI...

The group velocity is vg = d omega / d k while the phase velocity is vf = omega / k. In dispersive media they are not the same. In anisotropic elastic materials they are also different. Photons are second quantization quasiparticles of the electromagnetic field they are pure transversal modes with spin equal to one. Phonons are second quantized quasiparticles of the elastic field. They are composed by longitudinal and transverse modes, they also have spin equal to one or to zero. Regards.

There are MANY papers claiming that one can calculate or measure in many different structures (including evanescent wave setups i.e. waveguide below cutoff or in the nearfield of some antenna) group velocities well above c. The caveat (Jackson statement) there however is the limit of validity of the simple definition of the group velocity as mentioned earlier in this discussion. This simple definition (via domega/dbeta) is ONLY valid in regions of small (neglegible) dispersion. Using a vector networkanalzyer and just 2 cables (a real fun experiment and very well know to anyone working with (periodic) notch filters) its very easy to build a simple notch filter in transmission..and when select the function "group delay"on the instrument you can easily demonstrate negative values for the group delay in the notch (i.e. the" signal arrive before it was sent") which normally terminates the discussion

1) In vacuum group velocity of light is equal to phase velocity.

2) In media speed of light is not universal.

3) Photon is not a wave packet.

4) In quantum mechanics group velocity corresponds to classical velocity of a particle.

5) Phase velocity in quantum mechanics is not definite because energy is determined up to arbitrary constant.

Dear Prof. Sofia D. Wechsler Nice answer on photons given by Dr Mikhail G. Ivanov Regards. Group and phase velocity in vacuum are the same since the photon spectra is linear in k. omega = +/- c k. The light cone in Landau-Lifshitz II Vol. Classical field theory.

By the way in the 60´s Academic Lev Landau himself suggested the 5th Vol on Quantum Field Theory, could be replaced by the QED book with Profs. Berestetsky and Akhiezer.

What would be the reason for that? Any suggestions Dear Prof. Christian Baumgarten ?

Phase is relative, not a single measurement observable. With wavefunction collapse the coherent phase is lost, and what is measured is an amplitude, a lump of incoherent heat - the remains of the 'group' of coherent coupled modes that comprised the wavefunction, each with its own relative phase.

Sofia - the view that Fourier analysis yields dispersion seems to ignore the role of vacuum wavefunction impedances excited by the photon, the impedance of the .511 MeV 'mass gap' of quantum field theory. Reflections of photon fields due to impedance mismatches provide confinement.

In discrete space / time, it is impossible to define the concept of limit transition and, accordingly, the differentiation operation.

As one of the methods of analytical study of such systems is quantum theory. It introduces the concept of a state function as an analytic function covering the corresponding discrete set. In this case, physical quantities turn into operators acting on a state function, and observables into matrix elements (averages) from these states.

From these positions, the phase and frequency are in the same ratio as the coordinate and momentum.

Thus, quantum theory, in fact, is a statistical theory, and classical optics is the first stage of quantization. From here it’s followed "superlight" speeds, as well as tunnel junctions.

It is necessary to distinguish between the speed of light propagation in a vacuum and the speed of propagation of the maximum (e.g. gravitational) interaction. The first, especially in the field of hard x-rays) can be noticeably smaller than the second.

The connection between the quantum approach and the decomposition of Karhunen – Loeve