Q:

In QM would the conjectured virtual force particles [bosons] experience a Doppler shift if the two interacting particles have relative motion?? [assume non-relativistic speeds]
Thanks

- Anonymous

- Anonymous

A:

You're absolutely right, although we must qualify what that means a bit and give some examples.

The first example is the easiest, but the force carriers are real, not virtual. If an atom emits a photon, it can propagate through space and be absorbed by another atom, effectively conveying a force. The frequency of the photon of course obeys the normal reference frame rules for Doppler shifting. A beautiful example of the effects of Doppler shifting of photons with very slow relative velocities between the emitter and absorber is the Mossbauer effect, which takes advantage of a wonderful property of solids in that a the recoil momentum against the photon is picked up by the entire crystal lattice and not just by the emitting atom. Photons are quite real, not just "conjectured".

If the force is carried by "virtual" photons (or other force carriers like W bosons, Z bosons or gluons), then there is no single frequency which characterizes the state of these particles. Instead, they occupy localized regions of space and have wavefunctions that fall off exponentially with distance. You still can doppler-shift them, but the rules are now a little more complicated. Fourier showed that any reasonably well-behaved function can be expressed as a sum of sine and cosine functions. So you can take these arbitrarily shaped wavefunctions for virtual particles and express them as sums of waves, each with a different wavelength. Then the frequencies of each component will obey the usual Doppler-shifting rules and you have to add up the shifted components back to get the total. If you want to do it all relativistically (which is the only way it makes consistent sense, even with just photons in the mix), you have to do a Lorentz transformation to view each of the components in the frame you'd like to work in.

I always think it's easiest to do all the physics in one frame of reference, but if you want to look at fields in other frames of refernece, then the Lorentz transformation describes how to do this.

Tom

The first example is the easiest, but the force carriers are real, not virtual. If an atom emits a photon, it can propagate through space and be absorbed by another atom, effectively conveying a force. The frequency of the photon of course obeys the normal reference frame rules for Doppler shifting. A beautiful example of the effects of Doppler shifting of photons with very slow relative velocities between the emitter and absorber is the Mossbauer effect, which takes advantage of a wonderful property of solids in that a the recoil momentum against the photon is picked up by the entire crystal lattice and not just by the emitting atom. Photons are quite real, not just "conjectured".

If the force is carried by "virtual" photons (or other force carriers like W bosons, Z bosons or gluons), then there is no single frequency which characterizes the state of these particles. Instead, they occupy localized regions of space and have wavefunctions that fall off exponentially with distance. You still can doppler-shift them, but the rules are now a little more complicated. Fourier showed that any reasonably well-behaved function can be expressed as a sum of sine and cosine functions. So you can take these arbitrarily shaped wavefunctions for virtual particles and express them as sums of waves, each with a different wavelength. Then the frequencies of each component will obey the usual Doppler-shifting rules and you have to add up the shifted components back to get the total. If you want to do it all relativistically (which is the only way it makes consistent sense, even with just photons in the mix), you have to do a Lorentz transformation to view each of the components in the frame you'd like to work in.

I always think it's easiest to do all the physics in one frame of reference, but if you want to look at fields in other frames of refernece, then the Lorentz transformation describes how to do this.

Tom

*(published on 10/22/2007)*