Understanding Exchange of Particles as a Force

[NanakiXIII]

I'm trying to understand how exchange of particles can constitute a force. I read a chapter on this in Zee's Quantum Field Theory in a Nutshell, which covers it very briefly, presumably only to make it plausible to the reader, but there is something I'm not content with.

To illustrate, he places two time-independent delta functions ($J(x) = \delta(\vec{x} - \vec{x}_a) + \delta(\vec{x} - \vec{x}_b)$) on a scalar field to represent two massive particles that couple to the field. Then he claims that these particles generate disturbances in the field, propagating from one particle to the other to constitute a force. But how can a time-independent disturbance create a propagating (i.e. time-dependent) particle? In his jumping-on-a-mattress analogy, this doesn't make sense.

He illustrates that placement of two delta functions causes a decrease in energy and that the energy is lowered even further if you put them closer together, but that doesn't clarify to me how any exchanged particles are generated or involved in the mechanism.

Another thing I don't understand is why any disturbance caused by one massive particle would propagate only towards the other particle. Shouldn't it propagate in all directions?

 

[DrDu]

See my post #33 in the thread https://www.physicsforums.com/showthread.php?t=401974

 

[NanakiXIII]

Your idea does restore the analogy, but I still don't quite understand. You speak of these virtual particles as simply being the Fourier components of the deformation in the mattress caused by the particles that are coupled to it. But it seems that would mean that the distribution in $k$ is dictated by the coupled particles, while Zee speaks of a resonance at $k = m$, i.e. there is a prescription for at least one property of the distribution which comes from the field itself.

Also, if these virtual particles are in fact the Fourier components of this deformation, then they apparently "propagate" only in the spatial dimensions. Also, you're describing flat, monochromatic waves, which isn't exactly what came to mind when I read of an exchange of particles (i.e. localized things).

 

[DrDu]

Well, I think this speak of "exchange of virtual particles" is often somewhat misleading.
Mathematically, the interaction is described by a propagator of the field. The propagator describes the reaction at some space time point to a perturbation at another point (creation and destruction of a "particle" due to coupling to a source). As it is a forced reaction, there is usually no restriction on the allowed frequency and wave vectors. That's what is meant with "virtual". Especially it describes also the static interaction where omega=0 and the superposition of k values gives rise to a Coulombic potential.

I was mainly interested in working out when and how this leads to attraction an when it leads to repulsion in the link I gave you. That's why I restricted to considering static (time independent) sources. However, it should be clear that the vibration of a matress will lead to a retarded interaction due to their finite speed of propagation as, e.g. the lattice vibrations do in a superconductor.

 

[NanakiXIII]

I think I understand then, I'm only a little disappointed. I was expecting something more ground-breaking.

One thing still remains unclear, though, which is the "mass shell" story. Since these particles aren't really localized particles, I guess I shouldn't actually think of it as mass. But I still don't see how the resonance at $k = m$ would arise from just considering the two sources.

 

[DrDu]

I don't have Zee here at the moment.
But it should be clear that a propagator or Greensfunction is a solution of an inhomogeneous differential equation. "On the mass shell", the differential equation also has a solution in the homogeneous case, that is in the limit of vanishing sources, namely the free field.