Your options are numbered when you refer to them in the text, but are listed as bullet points originally. Probably they should also be numbered there!
Now we can get down to the actual physics discussion. I have a bag of fairly unrelated statements to make.
The “center of mass moves at constant velocity” thing is actually just as solid as, say, conservation of angular momentum. It’s just less famous. Both are consequences of Noether’s theorem, angular momentum conservation arising from symmetry under rotations and the center of mass thing arising from symmetry under boosts. (i.e. the symmetry that says that if two people fly past each other on spaceships, there’s no fact of the matter as to which of them is moving and which is stationary)
Even the fairly nailed down area of quantum mechanics in an electromagnetic field, we make a distinction between mechanical momentum (which appears when calculating kinetic energy) and the canonical momentum (for Heisenberg). Canonical momentum has the operator −iℏ∇ while mechanical momentum is −iℏ∇+eA.
Minkowski momentum is, I’m fairly sure, the right answer for the canonical momentum in particular. An even faster proof of Minkowski is to just note that the wavelength is scaled by 1/n and so −iℏ∇ψ gets scaled by a factor of n.
The mirror experiments are interesting in that they raise the question of what happens when we put an airgap between the mirror and the fluid. If the airgap is large, we get the vacuum momentum, ℏω/c, since the index of refraction for air is nearly 1. If the airgap gets taken to 0, then we’re back to nℏω/c. What happens in between?
I will say that overall, option 1 looks pretty good to me.
Edit: Removed redundant video link (turned out to already be in original post).
Numbering the options properly is a good idea, done.
To answer your points:
This is interesting. Symmetry under rotations gives us conservation of angular momentum. Symmetry under translations conservation of linear momentum. You are saying symmetry under boosts gives conservation of centre of mass velocity. Although in “normal” situations (billiard balls colliding) conservation of centre of mass velocity is a special case of as conservation of linear momentum—which I suppose is why I have not heard of it before. I need to look at this more as I find I am still confused. Intuitively I feel like if translation symmetry is doing momentum for us boost symmetry should relate to a quantity with an extra time-derivative in it somewhere. There is no symmetry under angular boosts, which I imagine is why fly-wheels (or gyroscopes) allow for an “internal reaction drive” for angular velocity.
I did not know that the kinetic and canonical momentum had different values in other fields. That makes option (1) more believable.
Yes, the k-vector (wavevector) certainly extends by a factor of n. So if you want your definition of “momentum” to be linear in wavevector then you are stuck with Minkowski.
I believe, that at the interface between the water and the air we will have a partial reflection of the light. The reflected component of the light has an evanescent tail associated with it that tunnels into the air gap. If we had more water on the other side of the air gap then the evanescent tail would be converted back into a propagating wave, and the light would not reflect from the first water interface in the first place. As the evanescent tail has a length of the order of a wavelength this means that random gaps between the atoms in water or glass don’t mess with the propagating light wave, as the wavelength is so much longer than those tiny gaps they do not contribute.
Applying this picture to your question, I think we would expect to interpolate smoothly between the two momentum values as the air gap size was changed, with the interpolation function an exponential with decay distance equal to the length of our evanescent wave.
Very nice post, thanks for writing it.
Your options are numbered when you refer to them in the text, but are listed as bullet points originally. Probably they should also be numbered there!
Now we can get down to the actual physics discussion. I have a bag of fairly unrelated statements to make.
The “center of mass moves at constant velocity” thing is actually just as solid as, say, conservation of angular momentum. It’s just less famous. Both are consequences of Noether’s theorem, angular momentum conservation arising from symmetry under rotations and the center of mass thing arising from symmetry under boosts. (i.e. the symmetry that says that if two people fly past each other on spaceships, there’s no fact of the matter as to which of them is moving and which is stationary)
Even the fairly nailed down area of quantum mechanics in an electromagnetic field, we make a distinction between mechanical momentum (which appears when calculating kinetic energy) and the canonical momentum (for Heisenberg). Canonical momentum has the operator −iℏ∇ while mechanical momentum is −iℏ∇+eA.
Minkowski momentum is, I’m fairly sure, the right answer for the canonical momentum in particular. An even faster proof of Minkowski is to just note that the wavelength is scaled by 1/n and so −iℏ∇ψ gets scaled by a factor of n.
The mirror experiments are interesting in that they raise the question of what happens when we put an airgap between the mirror and the fluid. If the airgap is large, we get the vacuum momentum, ℏω/c, since the index of refraction for air is nearly 1. If the airgap gets taken to 0, then we’re back to nℏω/c. What happens in between?
I will say that overall, option 1 looks pretty good to me.
Edit: Removed redundant video link (turned out to already be in original post).
Numbering the options properly is a good idea, done.
To answer your points:
This is interesting. Symmetry under rotations gives us conservation of angular momentum. Symmetry under translations conservation of linear momentum. You are saying symmetry under boosts gives conservation of centre of mass velocity. Although in “normal” situations (billiard balls colliding) conservation of centre of mass velocity is a special case of as conservation of linear momentum—which I suppose is why I have not heard of it before. I need to look at this more as I find I am still confused. Intuitively I feel like if translation symmetry is doing momentum for us boost symmetry should relate to a quantity with an extra time-derivative in it somewhere.
There is no symmetry under angular boosts, which I imagine is why fly-wheels (or gyroscopes) allow for an “internal reaction drive” for angular velocity.
I did not know that the kinetic and canonical momentum had different values in other fields. That makes option (1) more believable.
Yes, the k-vector (wavevector) certainly extends by a factor of n. So if you want your definition of “momentum” to be linear in wavevector then you are stuck with Minkowski.
I believe, that at the interface between the water and the air we will have a partial reflection of the light. The reflected component of the light has an evanescent tail associated with it that tunnels into the air gap. If we had more water on the other side of the air gap then the evanescent tail would be converted back into a propagating wave, and the light would not reflect from the first water interface in the first place. As the evanescent tail has a length of the order of a wavelength this means that random gaps between the atoms in water or glass don’t mess with the propagating light wave, as the wavelength is so much longer than those tiny gaps they do not contribute.
Applying this picture to your question, I think we would expect to interpolate smoothly between the two momentum values as the air gap size was changed, with the interpolation function an exponential with decay distance equal to the length of our evanescent wave.
Thanks for reading. Enjoy your option (1)!