Interacting fields in QFT and minimum excitations....

[asimov42]

TL;DR Summary

Clarifications on interactions between fields in QFT...

Following up on @A. Neumaier's excellent series of articles on virtual particles, I'm confused about one thing (well of of several). If you pop over to the discussion of virtual particles on Matt Strassler's page, he mentions that, for example, an excitation in the photon (em) field will also induce a disturbance in the electron field, etc. This is all well and good (although he seems to illustrate things with single Feynman diagrams - not so good as he never mentions that you have an infinite number of such diagrams), but...

Since all fields are quantized, isn't the minimum possible disturbance you can induce in the electron field equivalent to an electron? Or once you have a fully interacting picture, does this change (realized that the notion of a 'particle' may not be well defined)?

 

[Heikki Tuuri]

In a Feynman diagram of QED, you have electron lines and photon lines.

People informally call internal lines "virtual particles".

There may be a virtual electron-positron pair, a "loop".

In that sense, the minimum excitation of the electron field is an electron-positron pair. The pair does not need to possesses the 1.022 MeV of energy if it is short-lived.

The minimum excitation of the photon field is a single photon. The photon does not need to possesses the hf of energy if it is short-lived.

If we have interacting classical fields f1 and f2, then the disturbance which f1 causes on f2 is usually permanent and can have an arbitrarily small magnitude.

In QED, a permanent disturbance is a real particle, and it must possesses the appropriate energy for that particle. Thus, permanent disturbances have a minimum magnitude, a real photon of a frequency f, or a real pair of an electron and a positron.

In QED, a temporary disturbance of a field does not need to possesses the appropriate energy of a particle. Can we say that a temporary disturbance can have "an arbitrarily small magnitude"? At least in the sense that its effect can be arbitrarily small on the probability amplitudes of the various outcomes from the Feynman diagram.

 

[asimov42]

I don't think the answer above matches with @A. Neumaier's articles / posts and others regarding virtual particles being non-physical.

 

[vanhees71]

Feynman diagrams are ingenious notations for formulae to evaluate S-matrix elements in perturbation theory, no more no less.

The fundmental feature that you cannot probe the em. field with test charges with a charge of less than 1 elementary charge is all built into the QFT formalism. Particularly you cannot determine all field components with arbitrary accuracy, i.e., there's no state where all components $\vec{E}$ and $\vec{B}$ take arbitrarily accurate values. They obey some Heisenberg uncertainty relations similar to the better-known uncertainty relation of position and momentum components in non-relativistic QM.

The same holds true for the fact that any attempt to localize a charged particle better than to an accuracy given by its de Broglie wavelength $h/(mc)$, which is due to the inevitable creation of particle- antiparticle pairs when squeezing the particles in ever smaller spatial locations, is also built into the formalism of QFT. It's the heuristic reason for why all successful relativistic formulations of QT are many-body theories, i.e., QFT.

 

[Heikki Tuuri]

I don't think the answer above matches with @A. Neumaier's articles / posts and others regarding virtual particles being non-physical.

I have not read those articles by Arnold Neumaier. My answer was purely based on the formalism of Feynman diagrams.

People sometimes call internal lines virtual particles.

If you encounter the term virtual particle anywhere else than in Feynman diagrams, be very suspicious. You have to check if the author has defined it in a precise way. If the author just waves his hands when writing the word virtual, then it is poor science.

EDIT:

Misconceptions about virtual particles. That virtual particles transmit the fundamental forces proves the ”existence” of virtual particles in the eyes of their afficionados. But since they lack states (multiparticle states are always composed of on-shell particles only), they lack reality in any meaningful sense. States involving virtual particles cannot be created for lack of corresponding creation operators in the theory. Thus they cannot cause anything or interact with anything. In short, virtual particles are ”virtual” particles only, as their name says.

https://www.physicsforums.com/insights/physics-virtual-particles/

Now that you mentioned it, I agree with Arnold Neumaier's article. A virtual particle only has a meaning in a Feynman diagram.

Popular science books may mention that "virtual particle-antiparticle pairs pop out of the vacuum." Are they talking about Feynman diagrams? No. I have never understood what popular science writers mean with those words. Maybe it is nonsense.

 

[A. Neumaier]

Following up on @A. Neumaier's excellent series of articles on virtual particles, I'm confused about one thing (well of several). If you pop over to the discussion of virtual particles on Matt Strassler's page, he mentions that, for example, an excitation in the photon (em) field will also induce a disturbance in the electron field, etc. This is all well and good (although he seems to illustrate things with single Feynman diagrams - not so good as he never mentions that you have an infinite number of such diagrams), but...

Diagrams with a single vertex only represent not only standard Feynman diagrams for Feynman integrals but also the possible interactions in the Lagrangian description of a field theory. In quantum electrodynamics (QED), this is a single diagram with a single photon leg and two electron legs, all three meeting at the vertex.

Since all fields are quantized, isn't the minimum possible disturbance you can induce in the electron field equivalent to an electron? Or once you have a fully interacting picture, does this change (realized that the notion of a 'particle' may not be well defined)?

There is just one electron field, unevenly filling the universe, and its state contains contributions from an arbitrarily large number of electrons. Similarly, there is just one electromagnetic field, unevenly filling the universe, and its state contains contributions from an arbitrarily large number of photons. These fields change all the time by much more than the minimal allowed disturbance.

For conceptual analysis, one can pretend that these fields are very feeble and contain no electron and photon. Then you have the (of course fictitious) vacuum state. The elementary excitations of this state are the 1-electron states and the 1-photon states.

If you consider a tiny, microscopic piece of the universe, such as a hydrogen atom with its single electron in an excited state, it will (due to interactions with the remainder of the universe) spontaneously emit a single (real, not virtual) photon. This photon ''is'' a change in the state of the electromagnetic field, first close to the position of the atom considered and then radially extending from there at the speed of light. Conversely, a photon may be absorbed by an electron, which slightly changes both fields. This is probably meant when referring to minimal disturbances (which is not a technically well-defined expression).

 

[asimov42]

Thanks @A. Neumaier! I think my confusion is with some of the other language on Matt Strassler's page - for example, when referring to a single Feynman diagram, he says:

"Another example involves the photon itself. It is not merely a ripple in the electromagnetic field, but spends some of its time as an electron field disturbance, such that the combination remains a massless particle. The language here is to say that a photon can turn into a virtual electron and a virtual positron, and back again; but again, what this really means is that the electron field is disturbed by the photon..."

This seems like imprecise language - but I am still unclear on whether a photon can 'disturb' the electron field at all - since the electron field is quantized, isn't it the case that a regular low energy photon cannot transfer energy to the electron field? (i.e., it does not have sufficient energy to excite the electron field) Strassler's page associates these 'disturbances', as he calls them, with virtual particles (off-shell and all), which just makes things much more confusing...

 

[PeterDonis]

the electron field is quantized

No, the electron field is not quantized. Particular observables on the electron field, such as particle number, are quantized.

isn't it the case that a regular low energy photon cannot transfer energy to the electron field?

No. What is the case is that if you have a regular low energy photon, you will not observe, for example, electron-positron pair creation. But "observe" means the electron-positron pair is real, not virtual. Strassler was talking about virtual particles; he was simply saying that the photon propagator includes contributions from Feynman diagrams in which the photon turns into a virtual electron-positron pair that then annihilates back into a photon. (Note that if only a single photon is involved, there is no "real" process corresponding to this diagram; real electron-positron pair creation requires at least two photons in order to satisfy energy-momentum conservation. Virtual processes can violate conservation laws since virtual particles don't have to be on the mass shell.) And that means the propagator for the photon field involves the electron field as well. (And conversely, the propagator for the electron field involves the photon field as well.) In other words, even when you think there are only photons propagating, the electron field is involved.

 

[asimov42]

Ah thanks @PeterDonis - indeed that's what I was getting at, I believe. So when one thinks about the physical process involved in photon propagation, this process does indeed involve 'perturbations' in all interacting fields (not just the EM field). Here I'm using the term 'perturbation' in the context of a field state different from the ideal vacuum state - not in the context of the perturbation theory.