Virtual particle in path integral and perturbative approaches

[Shaw1950]

"Virtual particle" in path integral and perturbative approaches

The term "virtual particle" is used in path integral and perturbative approaches.

How do these "virtual particles" differ and how are they related?

[For example, static, bound states such as the hydrogen atom are solvable by path integral methods with "virtual particles" (Duru and Kleinert 1979) but perturbative solutions do not appear to work for static bound states.]

 

[alxm]

I don't quite see how you think these things relate to each other? It seems confused (or I'm the one who's confused)

The path integral formulation is a reformulation of "standard" quantum mechanics, analogous to Lagrangian classical mechanics in much the same way the Schrödinger equation is analogous to Hamiltonian classical mechanics. Perturbation theory, on the other hand, is essentially a mathematical approximation method. And "virtual particles" don't necessarily enter into it.

Let's try to sort this out: Are you talking about non-relativistic QM or QFT? "Virtual particles" are a graphical visualization or interpretation of the terms in a perturbation series in perturbative QFT.

In non-relativistic QM, the (exact) solutions for the hydrogen atom are easy to calculate in the Hamiltonian/Schrödinger picture, and that's usually done in most introductory textbooks and courses. Doing so in the path-integral (Green's function) formalism is a lot more difficult, but that's what Duru and Kleinert succeeded in doing. But they didn't use perturbation theory (they calculated the exact result), much less "virtual particles".

The path-integral formalism is seldom used for bound states, in particular bound electronic states, since it's basically unnecessarily complicated if it's a time-independent system you're interested in. It's much simpler to use/solve the time-independent Schrödinger equation in those cases. And there, http://en.wikipedia.org/wiki/M%C3%B8ller%E2%80%93Plesset_perturbation_theory" (MBPT) is routinely used to calculate bound electronic states. The helium atom ground-state is often a textbook example used to teach perturbation theory. In analogy to Feynman diagrams in QFT, you can make (Hugenholtz/Goldstone) diagrams, drawn according to a set of 'rules', which form a topological representation the perturbation series terms in MBPT. You can also use perturbation theory on bound states with a Green's function (electron propagator) formalism as well.

Perturbation theory typically involves calculating an interacting system by using a non-interacting case (where you have exact/known solutions) as a basis, and summing contributions from 'virtual' excited states. (If you consider the variational principle, obviously the non-interacting system's excited states will contribute to the interacting ground-state) In the context of QFT, where the excitations of the field are particles, then you have 'virtual particles'.

 

[Shaw1950]

Thanks, happy with all your comments. There seem to be two unresolved issues:
1) path integral: some commentators have referred to this formulation in terms of virtual "particles" - would they be more sensible to call them virtual "paths"?
2) bound states: Feynman repeatedly describes the Hydrogen atom as exchange of an infinite number of virtual photons. This does not seem to be a correct interpretation, since as far as I know perturbative methods do not converge for bound states such as Hydrogen and are therefore inapplicable.

 

[alxm]

1) path integral: some commentators have referred to this formulation in terms of virtual "particles"

Who and where? That's just wrong. As I already said, the concept of "virtual particles" is an visual interpretation of the perturbation series in (perturbative) quantum field theories. Since the path-integral formulation of quantum mechanics is applicable to non-relativistic, classical-field 'standard' quantum mechanics (as I also said) the concept of "virtual particles" obviously has no specific connection to the path-integral formalism.

would they be more sensible to call them virtual "paths"?

Since I don't know what "they" are, I can't answer that.

bound states: Feynman repeatedly describes the Hydrogen atom as exchange of an infinite number of virtual photons.

I only know of him giving such a description in his popular scientific book QED - The Strange Theory of Light and Matter. It's the aforementioned 'visual interpretation' of a perturbation series, and gives a working description of how Feynman diagrams and perturbation calculations are done. It shouldn't be confused with a description of the actual physical process though.

This does not seem to be a correct interpretation, since as far as I know perturbative methods do not converge for bound states such as Hydrogen and are therefore inapplicable.

Where'd you learn perturbation theory? The first application anyone ever learns is to apply it to a bound, time-independent state, so clearly that's possible, and in every textbook. And I already said so. The first thing you learn about it in general, is underlying assumption of perturbation theory: that the perturbation is small relative whatever your unperturbed system is. If your unperturbed states are free particles, then the series won't likely converge for bound states.

How should I interpret "as far as I know"? The tag on the right says you have a PhD in high-energy particle physics. But honestly - What you're saying sounds more like someone who's trying to grasp various QFT/QED concepts without having understood basic QM.

 

[Shaw1950]

Alxm, look I really appreciate your input, but as regards my PhD I got that 37 years ago and haven't touched a physics text until a few weeks ago. As you can imagine, one forgets a lot in 37 years, so please bear with me. Physics is still a lot of fun and your help is very much appreciated.

My question on the unperturbed Hydrogen wavefunction was perhaps not explained clearly enough. To solve for the undperturbed Hydrogen wavefunction, my understanding is that you would solve it directly as an eigenvalue problem; as such you would not use perturbation theory. Then if you want to examine small disturbances to these eigenfunctions, you would consider perturbation theory, either stationary or time-dependent, depending on the perturbation. Are you saying that this understanding is wrong?

What bothered me was Feynman's description of the unperturbed Hydrogen wavefunction (i.e. Laguerre polynomials etc.) as if it was also expressible as the sum of a perturbative series - something that surprised me, and I hadn't seen proved. But maybe I should simply ignore Feynman's comments in this case?