Re: QFT question

In article <1116964615.756223.183730@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Zigoteau <zigoteau@xxxxxxxxx> wrote:
>
>
>Hi, Greg,
>
>
>> They're the field represented in a basis of momentum eigenstates.
>>
>> I can't think of a simple way to say that. But quantum field theory is
>> still a field theory. It doesn't replace the field with virtual photons,
>> it represents the field with them. They represent the momentum transfers
>> that the field might give to a particle passing through. A virtual photon
>> going to the right might cause a particle to move to the left, which is
>> fine because the field could be attractive and virtual photons do what
>> the field does; they're not little billiards balls.
>
>
>Thanks for giving your definition. I was pondering doing so, but
>thought that putting down my thoughts in answer to Guy was just too
>much like hard work. I don't tend to think using the concept of virtual
>particles, but it gets mentioned all the time, and confuses a fair few
>people. I think it is worthwhile talking around the point a bit, if you
>don't mind.
>
>I have not heard the word 'virtual' used in the way you have defined
>it. However I may be wrong, and I would like to hear about how you use
>it and how it appears naturally in the solution to problems.

I was confused about virtual particles for a long time. And I still might
be. But from layman's literature we have the picture of little billiard
balls popping in and out of existence. It doesn't seem to clear up much
even in graduate school-- in a homework session one students asked the
prof how virtual particles can mediate an attractive force, and the prof
said something about uncertainty in where the virtual particle appears.
The student said, "That makes sense. Wait, no it doesn't!" Maybe the
professor kicked himself later that night when he thought of a better
explanation, but at the time I think they must both have been thinking
in terms of little billiard balls colliding as billiard balls do.

Turning to Peskin & Shcroeder doesn't help much. At one point they
introduce the photon propagator, in a notation that doesn't make the
connection to Green's functions in classical field theory very obvious,
and without talking much about it as a basis. The student
has probably never solved the analogous scattering problem in classical
field theory anyway, so he probably comes away thinking all of that
mathematical machinery is unique to the quantum problem.

And if we were all confused at that level, you can imagine what the
layman must think. And so I think the common "particles popping in and
out of existence" line is a horrible picture. But I can't think of a
better way to express it.

If it helps, there's a strong correlation between photons of the
electromagnetic field and phonons in a crystal. Strong enough that
Greiner, I believe it was, showed how to derive excitations of a field by
starting with those in a lattice and going to the limit of a continuum.
And a phonon isn't a localized disturbance, it's a normal mode. A phonon
stretches all the way from one end of the crystal to the other. You might
ask what is pseudoparticle-like about it at all. And then ask what is
particle-like about a photon.

And interesting follow-on would be the nature of a quantum fluid. That
is, not a fluid made of a bunch of particles, but "stuff" (not a field)
that really is a continuum, and how it might apply to aether theories.
That's going in a pretty loopy direction, but it's at least useful in
making the distinction between the fundamental mechanics and the physical
assumptions we apply the mechanics to.

>
>In my experience, the word 'virtual' is used in connection with
>perturbation solutions. For example, if you apply an electric field to
>an atom or molecule, it becomes polarized. In a perturbation treatment,
>the natural basis for expressing the orbitals of the polarized molecule
>is the set of orbitals of the unpolarized molecule. It turns out that
>the HOMO of the polarized molecule has a significant component of the
>LUMO of the unpolarized molecule. The dipole moment of the polarized
>molecule can be considered to be a static version of the oscillating
>dipole you get when you have a superposition of ground and first
>excited states. So, when you're talking about the ground state of the
>molecule in the electric field, you can say there is virtual LUMO there
>(since I don't use this turn of phrase myself, I'm not quite sure how
>to say it: I can't get it to come out right).
>
>I think that this is the sense in which you can say, in connection with
>Lamb-Rutherford splitting of the 2s and 2p orbitals of hydrogen, that
>there are virtual photons associated with the electron in a hydrogen
>atom. Sort of, in a first approximation you have no photons there, but
>to a better approximation the state you are talking about is not quite
>orthogonal to an 'unperturbed' photon mode. I am saying this badly, but
>I hope you can see where I'm coming from, and can correct it if
>possible.
>
>I would say that the uniform electric field between the plates of a
>capacitor is composed of virtual photons. It's certainly not a
>propagating EM wave, i.e. not a photon in the sense of something moving
>at the speed of light. However it's not quite 'orthogonal' to a 'pure'
>photon mode.
>
>Is this garbage, and can you quote chapter and verse? If not, can you
>put your finger more exactly on what a virtual particle is?

Three references have been very helpful for me. One is Barut's book,
available as a skinny Dover reprint, on classical field theory. In it, he
presents the classical analogs of the machinery commonly developed in
quantum field theory. In the context of virtual particles, the Green's
function methods, towards the end, were enlightening. The virtual
particle is a Green's function-- it's an expression of the effect at point
A due to a source at point B.

Bjorken & Drell's book on quantum field theory, the first volume. They're
pretty old-fashioned, but that's a good thing. One of the applications
they worked out is the scattering of an electron in a Coulomb field. It
wasn't particle-particle scattering, but scattering from an assumed field.
The solution involved Fourier transforming the field, just as Barut did
when he introduced Green's functions. Go on to first-order perturbation
theory where it's assumed the particle moves in a straight line, interacts
once, and moves away in a straight line. The key difference between the
classical and quantum theories is DeBroglie's relation-- in the classical
theory a Fourier component w is allowed to transfer just a little bit of
momentum, while in the quantum theory it's a kick or it's nothing. The
frequency of a Fourier coefficient tells you how much momentum is
transferred, and the coefficient tells you the likelihood.

A real treasure is Greiner's book, "Field Quantization". In it, he
deliberately delayed his choice of a basis for as long as he could. By
doing so, he made it clear first of all that photon propagators are a
basis, and second that it's an arbitrary decision. Momentum eigenstates
are useful because you can relate them to the momenta of particles coming
in to and out of a scattering event, but that's not a necessary choice.
And so I'd wondered if perturbation problems like you've mentioned above
could be usefully worked in a basis other than plane waves, something
that takes better advantage of the symmetry of the system. But I haven't
had the gumption to try working that out. Another invaluable insight that
Greiner gives is that quantum field theory is just another field theory.
He finds e.g. the energy and angular momentum from the stress-energy
tensor, which he works out from the Lagrangian, and generally does as
much as he can in a generic, quantum-agnostic manner. It's a powerful
antidote for the student who can't imagine his theoretical apparatus being
applied in any other way.

Wald's book, Quantum Field Theory in Curved Spacetime and Black Hole
Thermodynamics. I haven't gone through it in detail, but I was sure to
look up all references of virtual particles in the index. And Wald
stresses that quantum field theory is a theory of fields, and that there's
a natural particle interpretation in flat spacetimes but particles just
don't generalize to arbitrary spacetimes. He gives an example problem
that shows that the concept of a particle is never actually needed. Some
words about paradigms are probably due here, but I'll just note that
photons are as much a particle as electrons are, and electrons are
excitations of the Dirac field.


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