What are Feynman Diagrams and how do they help explain Quantum Electrodynamics?

[vanhees71]

I don't understand what the problem in this discussion is. The evaluation of Feynman diagrams in time-position or energy-momentum representation is just a calculational issue. The physics content is completely the same. The energy-momentum framework is preferred, because the propagators are just simple algebraic functions like $D(p)=1/(p^2-m^2 + \mathrm{i} 0^+)$ for the vacuum-QFT time-ordered free propgator of a scalar field, while in time-space representation it's some more complicated Bessel function.

 

[PeterDonis]

The physics content is completely the same.

Yes, and at least one question under discussion is what is that physics content? My point is that it is not any sort of claim that "particles follow all possible paths in spacetime" in every single one of the infinite set of arbitrarily complicated Feynman diagrams that describe a particular process. The momentum space representation at least makes it obvious that that's not what is being described, whereas the spacetime representation, if not handled carefully, can lead to a belief that maybe it is.

 

[vanhees71]

Of course not. The Feynman diagrams are a notation for formulae for terms in the perturbative series for S-matrix elements, i.e., the transition-probability-density rate for scattering processes from asymptotic in states to asymptotic out states.

This pop-sci formulation about "particles follow all possible paths in spacetime" is a failed attempt to explain how to calculate such transition matrix elements in non-relativistic quantum mechanics in terms of Feynman's path integrals. These transition amplitudes are given as functional integrals over all phase-space trajectories connecting two given points in configuration space within a specified time interval. In many cases you can simplify it also to a version of the path integral, where you integrate over all paths in configuration space (because the Hamiltonian in non-relativistic point-particle mechanics is quadratic in the momenta, and you can do the path integral over the momenta exactly in that case).

In QFT you integrate of course not over such phase-space trajectories of point particles but over field configurations.

 

[Husserliana97]

Hi !

Well, after all these interventions, I believe I have a lot to think about/work on.
I just wanted to clarify a few things, and thus point out my ultimate concerns.

Firstly, regarding Witten's text; I am obviously not in a position to explain the context in which this development takes place, let alone what he does with it. On the other hand, this construction of the diagrams in the space of the coordinates (against what is generally done in QFT), with as a consequence the integration on all the possible positions of the vertices, and on all the possible trajectories; that seems to me to be a recurring point with him. For example, the PDF I attach; the first slides are enough, and the lecture is on Youtube (the first 10 minutes). I could also quote other texts.

The point is that the evaluation of the diagrams in one or the other representation is not, in my opinion, only a matter of calculation. I mean, it's first and foremost that, but the choice of picture is also important, if only for intuition.

First of all, if we agree that in the case of the free propagation of a photon, the description, admittedly simplistic, of a particle propagating on several routes at the same time (basically, in the manner of a wave - of course, a not quite "classical" wave); then I am not sure why we should not prejudge that the same is true for other more complicated cases. I understand that in the latter cases, one cannot produce beautiful interference experiments, such as those described by Feynman in QED, and that one cannot therefore attest to the existence of such a "superposition of states" for the photon(s). But on the other hand, why would these photons lose this quantum state, why would they cease to propagate in the manner of (non-classical) waves? This change of state could only be brought about by decoherence or some "objective reduction" (if one adheres to the latter) -- at least it seems to me! Therefore, to prejudge that the linear combination of trajectories is still valid in the case of more complicated diagrams seems to me to be a conjecture, but a reasonable one!

Finally, regarding the fact that :

In QFT you integrate of course not over such phase-space trajectories of point particles but over field configurations.

I would like to share with you the answer (to a similar question) of a string theorist, Lubos Motl (on Quora). I believe it gives us food for thought regarding the "complementarity" of the two pictures :"The normal path integral that Feynman started with was the sum over all histories of FIELDS. You know, you have all the configurations phi(x,y,z,t) and you calculate the action for each such configuration - the precise choice of the values of fields at all spacetime fields is a configuration of fields. With the weight of exp(iS/hbar) for each configuration of fields, you need to integrate over configurations.(...)

But Witten refers to another “sum over histories” which is not really about configurations of fields, it is a sum over configurations of particles. You may also imagine that a QFT has the “states” composed of many particles at points, and they wiggle through spacetime, over all possible trajectories, and have a one-particle-like action for each particle - which is basically mass times the proper length (time) of the trajectory in the spacetime.

With these histories composed of many point-like particles, the propagators arise as the sum over histories with 1 particle at the beginning as well as the end, but you must also allow the vertices which are pointwise mergers or splits of the pointlike particles, so you integrate over all points where these interactions may take place, and this is how you add the factors to the Feynman diagram from the vertices.

When done properly, you get the exact same expressions (integrals) for every Feynman diagram, whether you use the Gaussian-like integral over configurations of fields; or the split-and-join sum over possible propagation of individual point-like particles in the spacetime, with the splitting and joining allowed!"

 

[PeterDonis]

the choice of picture is also important, if only for intuition.

To the extent that this is true, it is because "intuition" here is just shorthand for "trying to guess the answer without calculating it", and the spacetime picture is much more likely, as far as I can tell, to lead to wrong guesses.

if we agree that in the case of the free propagation of a photon, the description, admittedly simplistic, of a particle propagating on several routes at the same time (basically, in the manner of a wave - of course, a not quite "classical" wave); then I am not sure why we should not prejudge that the same is true for other more complicated cases.

Because the cases are different. I explained why in the last part of post #31.

why would these photons lose this quantum state

You're missing the point. For "virtual" particles (like the photons traveling on any lines in a Feynman diagram that have at least one internal vertex), there is no quantum state. They are calculational conveniences. They are not particles that can even be assigned a quantum state. The only quantum states in a Feynman diagram calculation are the initial and final states, which are states that are or can be directly observed.

In other words, as soon as you start thinking of virtual particles as "like" real particles, you are already doing it wrong.

the answer (to a similar question) of a string theorist, Lubos Motl (on Quora)

Please give a link.

 

[Husserliana97]

Okay for the virtual particles. But regarding the real ones, I mean the ones symbolized by "external lines" in the diagram ; can we say that they propagate in the manner of non-classical waves, even in more complex cases than that of a freely propagating photon ?

As for the link :

https://www.quora.com/Is-there-an-a...al-and-external-lines-then-each-symbolizing-a

(the excerpt I quote can be found in his second answer)

 

[PeterDonis]

regarding the real ones, I mean the ones symbolized by "external lines" in the diagram ; can we say that they propagate in the manner of non-classical waves

I don't know what this means. I would suggest looking at the actual math. If your answer is "I don't know what the words propagate in the manner of non-classical waves would mean in terms of the actual math", that should indicate to you that your question is not answerable because it's not well defined. Physics is done in math, not vague ordinary language.

As for the link :

https://www.quora.com/Is-there-an-a...al-and-external-lines-then-each-symbolizing-a

Thanks for the link.

 

[PeterDonis]

the excerpt I quote can be found in his second answer

What second answer? I only see one by Motl.

 

[Husserliana97]

I don't know what this means.

My apologies. I mean : the wave as a superposition of probability amplitudes (one for each possible paths), constructively and destructively interfering with each other -- hence, the "non classical wave" picture.

And Lubos second answer can be found in the comment section of his first answer (the op. asks him something, to which he replies).

 

[PeterDonis]

Lubos second answer can be found in the comment section of his first answer (the op. asks him something, to which he replies).

I'm not seeing this; I'm only seeing Motl's original answer, with nothing indicating any comment section.

 

[PeterDonis]

I mean : the wave as a superposition of probability amplitudes (one for each possible paths), constructively and destructively interfering with each other -- hence, the "non classical wave" picture.

Ok, but that still doesn't help much in answering your question. If your question is "can we use path integrals to calculate answers", then of course yes, we can. But I don't know if that's the question you were asking. And if you were asking some other question, it's still quite possible that it's unanswerable, because the question I just answered is really the only question we can get out of the path integral formalism that we can be sure is answerable.

 

[Husserliana97]

I'm not seeing this; I'm only seeing Motl's original answer, with nothing indicating any comment section.

Don't you see the bubble at the end of his answer? If you click on it, you will see an additional question, and Motl's answer. But I can copy them to you.

If your question is "can we use path integrals to calculate answers", then of course yes, we can.

That's my question indeed ! Which I can split into two, namely: 1) can we use the path integral in the context of a diagram?
2) what is summed up in this way, if not precisely the possible paths of the particle (here the photon), whether between the entry point and the vertex, or between the vertex and the arrival point?

 

[PeterDonis]

Don't you see the bubble at the end of his answer?

No. I don't have a Quora account; if you do and are signed in, it's possible that you can see the bubble when I can't.

can we use the path integral in the context of a diagram?

Not a single diagram, not, because no actual process is represented by a single diagram.

what is summed up in this way, if not precisely the possible paths of the particle (here the photon), whether between the entry point and the vertex, or between the vertex and the arrival point?

Feynman diagrams are a way of representing the perturbation expansion of a process pictorially, to help keep track of the terms in the expansion and evaluate the integrals involved. Calculating answers involves adding together enough terms in the perturbation expansion to make the calculated numerical answer have the same precision as the precision of experiments, so the two can be compared.

As such, the question you are asking is unanswerable, because the question presupposes that diagrams and the "summing up" of them must represent something "real" about the process being calculated. But they don't.

 

[PeterDonis]

@Husserliana97, I would suggest reading this Insights article:

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

Since "lines in Feynman diagrams" means the same thing as "virtual particles", everything said in the Insights article about the latter applies to the former.

 

[vanhees71]

The most problems with "intuition" in quantum theory and even more so in relativistic quantum field theory is that it is suggestive to think in terms of point particles. A lot of misconceptions can be avoided when rather thinking in terms of fields. Then it becomes clear that "propagators" are just "Green's functions" which describe the field at a given point in space as a function of time due to their sources. The quantum-theoretical formalism of perturbation theory which specific propagator you need, and that's the "time-ordered propagator", which in vacuum QFT is identical with the Feynman propagator. That explains the internal lines in an intuitive way without all this confusing lingo about "virtual particles".

The vertices stand for charges and/or currents involved in the local (!) interactions between fields, which explains why they mathematically stand for coupling constants and/or other elements describing the kind of interactions (scalar, spinor, tensor, etc.).

The external lines stand for asymptotic free states, represented by solutions of the free field equations. Usually one puts plane waves in there to calculate S-matrix elements describing scattering processes of particles with specific momenta in both the in and out state (plus the "polarizations", i.e., spin (massive particles) or helicities (massless particles). This is of course somewhat problematic, because plane waves are "generalized eigenstates" of momentum (and energy), which needs some regularization or one has to put true square-integrable states, i.e., "wave packets" for the asymptotic free states. In any case you can take the limit to "plane waves" after taking the S-matrix element squared. For the wave-packet regularization strategy (most physical), see Peskin, Schroeder, Introduction to quantum field theory.