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.
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)!