How do the group and phase velocities of pilot waves compare?

[Azurite]

How fast are the de Broglie pilot waves? Do they move at the particle group or phase velocity?

Is the phase velocity really faster than c as it weaves around the particle?

 

[Demystifier]

As for all other waves, the peak of the wave packet moves with the group velocity.

 

[Azurite]

As for all other waves, the peak of the wave packet moves with the group velocity.

Can the phase velocity (or whatever) ever be faster than c?

Is it not the velocity of the individual oscillations of the pilowaves is always greater than the velocity of light?

 

[muscaria]

Can the phase velocity (or whatever) ever be faster than c?

Is it not the velocity of the individual oscillations of the pilowaves is always greater than the velocity of light?

Why are you bringing the velocity of light into the picture?
The pilot wave is just the wavefunction and evolves in time accordingly (Schrodinger Equation), is this clear?

 

[Demystifier]

Can the phase velocity (or whatever) ever be faster than c?

Yes.

 

[Demystifier]

One interesting but not well-known fact is the following. The velocity of wave propagation is not the group velocity but the front velocity. For tachyon waves (Klein-Gordon equation with the opposite sign of the mass term) the group velocity is larger than c but the front velocity is not.

 

[muscaria]

The velocity of wave propagation is not the group velocity but the front velocity.

Thanks for this! I didn't know the name "front velocity". I guess "velocity of wave propagation" depends on what we mean, right? E.g. If a density perturbation was introduced in a condensate say (through some localised potential), then the velocity of "node creation" would be the front velocity, whereas the velocity of sound waves would be that of the low wavelength modes excited in the system. Isn't that so?

 

[Azurite]

Yes.

You agreed the phase velocity can be faster than c...
Here, we see in pilot wave a set of phase waves moving faster than the speed of light, c, creating and directing a wave group moving at a speed slower than light which, in turn, is directing a positive mass particle which is also traveling at vp = vg< c. , so what must be operating in nature to allow such waves to interact with a particle across the relativistic light barrier at v=c?

 

[PeterDonis]

You agreed the phase velocity can be faster than c...

Yes, but he didn't say the phase velocity being faster than c means anything physically.

Here, we see in pilot wave a set of phase waves moving faster than the speed of light, c, creating and directing a wave group moving at a speed slower than light which, in turn, is directing a positive mass particle which is also traveling at vp = vg< c. , so what must be operating in nature to allow such waves to interact with a particle across the relativistic light barrier at v=c?

The pilot wave interpretation is non-relativistic. There is no working relativistic version of it that I'm aware of.

However, there is also no "relativistic light barrier". In quantum field theory, it is perfectly possible for particles to have a non-zero amplitude to move faster than light. The way QFT enforces causality is by ensuring that all measurements at spacelike separated events commute, so that the results of the measurements do not depend on the order in which they are performed. This does not require any "relativistic light barrier".

 

[Demystifier]

In quantum field theory, it is perfectly possible for particles to have a non-zero amplitude to move faster than light.

Can you elaborate on that? The probability that measured velocity (here I exclude weak measurements) will be larger than c is exactly zero.

 

[Azurite]

Demystifier. For real waves.. why is the velocity of the guided particle zero? And where is the complex wave above?

Doesn't the pilot wave moves in 3D since the particle is in 3D?

 

[Demystifier]

@Azurite see e.g. https://arxiv.org/abs/quant-ph/0611032 Eq. (1).

 

[PeterDonis]

Can you elaborate on that?

I was doing my best given that this is a "B" level thread. A better way to put it might be that you cannot exclude Feynman diagrams that have virtual particles propagating outside the light cone. But of course you can't measure the velocity of virtual particles (and the "virtual particle" language itself has issues).

 

[Azurite]

I read in stacks that "In both crystal and de-Broglie formalism, momentum is define as p=hk. This creates a mapping between a pure wave with wavenumber k and a point in momentum space. If a wavefunction is a superposition of, say, three waves, its momentum representation will consist of three points."

I assumed the "de-Broglie formalism" referred to his matter wave thesis... but did this also apply to the later 1927 de Broglie pilot wave in configuration space? can this also create a mapping between a pure wave with wavenumber k and a point in momentum space?

 

[PeterDonis]

I read in stacks

Can you give a specific reference?

 

[Azurite]

Can you give a specific reference?

See bottom of

https://physics.stackexchange.com/q...s-fourier-space-called-as-momentum-space?rq=1

Again my question was.. I assumed the "de-Broglie formalism" referred to his matter wave thesis... but did this also apply to the later 1927 de Broglie pilot wave in configuration space? can this also create a mapping between a pure wave with wavenumber k and a point in momentum space?

 

[PeterDonis]

See bottom of

This is a statement about the math of Fourier Transforms, not about physics.

did this also apply to the later 1927 de Broglie pilot wave in configuration space? can this also create a mapping between a pure wave with wavenumber k and a point in momentum space?

No, because the pilot wave itself is not a Fourier transform, it's a wave function--more precisely, it's a particular interpretation of what the wave function means. Of course you can apply a Fourier transform to the wave function as represented in position space, to get the corresponding wave function represented in momentum space. But that's just the same math you can do with any wave function. It doesn't say anything in particular about the pilot wave interpretation as opposed to other interpretations.

 

[slow]

It has been 3 weeks since the beginning of the thread, but phase speed is a recurring theme. That's why I add this note. Could the following didactic material serve?

http://www.physics.usyd.edu.au/~jbh/share/PHYS1901/chapter7-Guided-Wave-Optics.pdf

The text shows that the fundamental mode can be interpreted in terms of geometric optics, as two symmetrical elementary waves that zigzag between the walls of the guide. The Waveguide Modes item explains the group and phase speeds. In the guide also the group speed is the speed of the movement of energy, less than $C$. And the phase velocity is also an abstract term, which is greater than $C$. The case is analogous to a long wall built on the shore of the sea. When a wave of water that falls obliquely on the wall begins wetting a point of the wall and little by little wet the rest. If the front of the water wave strikes parallel to the wall, all the points of the wall get wet in the same place. The phase velocity is calculated by dividing the length of the wall by the time it takes for the wave to dip from the first point to the last. When the front impacts parallel to the wall, there is no difference in time between the wetting of the first point and the wetting of the last. Then the denominator is zero and the phase velocity becomes infinite. The same thing happens in the guide. If the front impacts obliquely, the phase velocity is finite and may well be greater than $C$. If the vacuum has been made within the guide, the group speed, the phase velocity and C comply with the following relationship.
$$v_g.v_p=C^2$$