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

[DeltaForce]

TL;DR Summary

Can anyone explain this in layman's words?

I'm new to QED, so I want to have a general grasp of what's going on. I just want to understand it conceptually. Can anyone explain it in a way so a layman can understand?

 

[DarMM]

They are pictures to help remember the form of integrals we need to compute probabilites in QED. They are not in any way directly showing what is happening.

The Feynman diagrams are pictures/graphs you can draw with two types of lines, photon and electron with one rule for joining lines: photon lines can only be joined with two electron lines. We call joinings verticies.

Each line and vertex is connected with certain mathematical functions and the number of loops tells you how many integrations you have to perform.

The more verticies there are the less important the effect of that graph, and usually by about five or so verticies the effect is so small we often ignore it.

 

[DeltaForce]

Yeah. But what IS QED anyways? I'm not really talking about Feynman diagrams. I just want to get a gist of QED and how it relates to Feynman diagrams.

 

[dextercioby]

QED is the only acceptable theory at this moment and at the proper level of energies of the particles (*) in which the classical theory of charged particles and the electromagnetic fields (textbooks of Barut or JD Jackson come to mind) generated by them can be brought in agreement with the "quantum/discreteness hypothesis" made by Max Planck in 1900.

(*) In the sense that if the energies are too high, then QED is no longer sufficient to describe phenomena, but the Glashaw-Salam-Weinberg electro-weak theory is the proper one.

 

[DeltaForce]

So... QED is a better, more detailed explanation of electromagnetic forces? It replaces the field theory? I just want to know what QED is, not what limitation it has.

 

[dextercioby]

It IS the field theory but with special fields for the classical pointlike particles and special (complicated but to me extremely beautiful) mathematical features.

So from a high-school perspective: small magnets and electrons presented as points of no dimension and mass 9.10^-31 Kg and charge -10^-19 C are replaced by abstract (fictive if you want) "fields" of the same type of field with which the electric and magnetic fields generated by these classical objects are replaced.

QED is a quantum field theory. Stress on Quantum, and stress on "field", but this field is not something you can directly measure.

 

[PeterDonis]

QED is the quantum field theory of electrons and photons alone, including no other particles or interactions. As such it is obviously incomplete, since we know there are in fact other particles and interactions, but for many applications those other particles and interactions can either be ignored, or included within an approximation based on QED.

 

[DarMM]

It's a quantum theory of electromagnetism and the aspects of the behavior of matter related to electromagnetism. It uses very abstract objects called quantum fields in its mathematical structure. Its exact description of any process is usually too complex to handle, so Feynman diagrams are a trick we use to approximate the exact description.

 

[DeltaForce]

As an extension question,
I read parts of the QED lecture. It said that light can take an infinitely different route, but some routes are more likely for light to take. My question is that how does that idea relate to the Feynman diagrams (where 2 electrons deflect off each other because of a photon)

 

[Mentz114]

As an extension question,
I read parts of the QED lecture. It said that light can take an infinitely different route, but some routes are more likely for light to take.

Yes, (if I remember correctly) the probability of a route is the square of the sum of the amplitudes of all possible routes

My question is that how does that idea relate to the Feynman diagrams (where 2 electrons deflect off each other because of a photon)

It is also a sum of amplitudes but the lines on Feyman diagrams are not routes. The diagram shows all the ways in which given initial states can become (proposed ) output states. To calculate the probability of this one must sum the amplitudes for all the ways over all space. It is very technical as this paper ( 'Feynman diagrams for beginners') shows. The diagram that you think is an electron anihilating with a being deflected is not that. It is probably an electron anihilating with a positron.

https://arxiv.org/pdf/1602.04182.pdf

 

[PeterDonis]

It said that light can take an infinitely different route, but some routes are more likely for light to take. My question is that how does that idea relate to the Feynman diagrams (where 2 electrons deflect off each other because of a photon)

When we talk about light "taking different routes", we are basically talking about one Feynman diagram: the one that has just one photon line coming in and one photon line going out, with nothing happening in between. In other words, this one Feynman diagram represents an infinite set of possibilities: all of the possible paths that a photon could take between two fixed points in spacetime. Integrating over all those paths gives the probability amplitude that QED, or more precisely the pure photon part of QED (i.e., no electrons or interactions, just photon propagation), predicts for a photon that is emitted from the first point in spacetime to be detected at the second point.

The full theory of QED does the same sort of thing, but with more possible diagrams. For example, if we include electrons and the possibility of electrons interacting with photons, then even the simple photon propagator--i.e., the amplitude for a photon emitted at one spacetime point to be detected at another spacetime point--has to include more than just the one simple diagram described above, because now things can happen to the photon in between the emission and detection events. For example, the photon could turn into an electron-positron pair, which could then annihilate each other and turn back into a photon. The Feynman diagram for this would have one photon line going in, ending at a loop of one electron line, and another photon line beginning on the other side of the electron loop and going out. (The technical name for this diagram is "vacuum polarization".) This diagram also represents an infinite number of possibilities, since the photon could go anywhere in spacetime before turning into the electron-positron pair and then back into a photon again. And there are further possibilities when we start considering that other photon lines could start and end on the electron loop (staying completely internal to the diagram), and those other photon lines could then spawn other electron loops, etc., etc., etc.

In practice, because each photon-electron vertex in the diagram gets a factor of $\sqrt{\alpha}$ (where $\alpha$ is the fine structure constant), which is a fairly small number, the amplitudes for these different possibilities get rapidly smaller as the diagrams get more complicated. So we can calculate reasonable predictions by taking into account just a few of the simplest diagrams. (For example, the Lamb shift can be reasonably estimated just with the vacuum polarization diagram I described above.)

 

[PeterDonis]

Btw, if you want a good layman's presentation of QED, I recommend Feynman's QED: The Strange Theory of Light and Matter.

 

[DeltaForce]

Thank you! That cleared up a lot of things.

 

[DeltaForce]

What's the mechanism behind how an electron emits (or absorbs) a photon? How does that work?

 

[PeterDonis]

What's the mechanism behind how an electron emits (or absorbs) a photon? How does that work?

Nobody knows. We don't even know if the question is meaningful. Electrons and photons aren't little machines. They're just electrons and photons. There might be no answer to the question of how they do what they do, other than "that's what they do". At some point, the search for a "mechanism", which basically means explaining something in terms of something else, has to stop; there has to be something fundamental, that just does what it does without being explainable in terms of something else.

 

[DeltaForce]

Ok, I accept that they just do it. But do electrons just go off an emit photons randomly? Or only when a circumstance happens? Is there a pattern to it?

 

[PeterDonis]

Do electrons just go off an emit photons randomly? Or only when a circumstance happens? Is there a pattern to it?

We don't measure electrons emitting single photons. The "electron emits a photon" process is a virtual process; it's part of the theoretical model shown in Feynman diagrams. It's not something we directly measure.

 

[DeltaForce]

We don't measure electrons emitting single photons. The "electron emits a photon" process is a virtual process; it's part of the theoretical model shown in Feynman diagrams. It's not something we directly measure.

So how did Feynman arrive at "electron emits a virtual photon?" And it carries momentum which causes electrons to recoil and deflect off each other.

 

[PeterDonis]

how did Feynman arrive at "electron emits a virtual photon?"

From the math of QED, which contains terms that, when translated into Feynman diagrams, can be interpreted that way. But the interpretation is not important; the predictions are. QED would make the same predictions if you didn't even try to interpret Feynman diagrams at all, but just did the integrals they told you to do and got numbers out. All the talk about electrons emitting and absorbing photons is just a way many physicists like to describe the diagrams in ordinary language to help them think about the calculations.

it carries momentum which causes electrons to recoil and deflect off each other.

We don't observe this either. We observe electrons appearing to repel each other via an "electromagnetic force", but we don't observe individual photons traveling between them and pushing on them. The photons are virtual photons. And the virtual photons appearing in the Feynman diagrams for things like the static Coulomb repulsion between electrons aren't even on shell, meaning they don't obey the relativistic energy-momentum relation for photons (another way to put it is that they don't travel at the speed of light).

 

[DeltaForce]

So sorry that I'm bombarding you with questions. I may be repeating myself here.
Coming back to the basic scenario of two electrons shot at each other and bounce off. That is what the most basic Feynman diagram represents I think.

In classical physics, the electrons are surrounded by an electric field; so they they push each other apart. In quantum physics, those electric field supposedly are made of individual discrete photons (i think).
Does that information have to do with these "virtual photons" we're talking about?
If so, how?

I also read the book QED The Strange Theory of Light and Matter. In the 2nd chapter, it talks about light and how it is most likely to take the path of least time required to travel and it also talks about how when the paths of light squeezed(blocked) it is more likely to bend and twist. When the paths are more opened, light is more likely to travel in a direct straight line.
If any of that information relevant to the electron situation I mentioned above?

Sorry again, I'm still a novice to these quantum physics stuff. I don't really know what I'm talking about. I'm trying my best to piece together information to form a big picture.

I'm very thankful that you're taking time to answer my (probably) dumb questions.

 

[PeterDonis]

Coming back to the basic scenario of two electrons shot at each other and bounce off. That is what the most basic Feynman diagram represents I think.

There are an infinite number of Feynman diagrams for this case. The simplest one has two electron lines coming in, two electron lines going out, and a single internal photon line between them. But there are more complicated diagrams that have more internal lines. (Actually, even that's not the simplest possible diagram--see below.)

Note that even the simplest diagram is not correctly described as "two electrons shot at each other and bounce off". The electrons don't interact with each other directly. They interact by exchanging a photon (a virtual photon).

In quantum physics, those electric field supposedly are made of individual discrete photons (i think).

Made of virtual photons, which we never directly measure.

Also, this description is based on the description of QED in terms of Feynman diagrams, which actually are only one possible way of viewing QED. This way, which is more general than just QED (it can be applied to any quantum field theory) is called "perturbation theory", because each of the Feynman diagrams containing internal lines can be viewed as a perturbation, or correction, to the most basic diagram that just has the external lines with nothing happening in between.

For example, remember that I said in an earlier post that "light taking different routes" refers to the simplest possible Feynman diagram for a single photon, where it just goes in and comes out and nothing happens in between. All of the more complicated diagrams with one photon line going in and one photon line coming out are perturbations to this.

For the case of two electrons--i.e., two electron lines going in and two electron lines coming out--the simplest possible Feynman diagram is actually one with just two electron lines--two going in and two coming out--with nothing happening in between. All of the other diagrams, including the one with just a single internal photon line between the two electrons, are perturbations to that. So basically QED on this perturbation theory view is just adding up all the possible perturbations to nothing happening at all!

In the 2nd chapter, it talks about light and how it is most likely to take the path of least time required to travel

This is actually another way to view the pure photon part of QED; the "path of least time" described in that chapter is the same as the "path of greatest amplitude" when you calculate the integral over all possible paths for the simplest one-photon Feynman diagram.

If any of that information relevant to the electron situation I mentioned above?

Not really, because, as I said above, when we are talking about electrons repelling each other, the photons they are exchanging are virtual photons; whereas when we are talking about light propagation and the various phenomena described in QED chapter 2, we are talking about real photons that we actually observe (more precisely, we observe the light propagation phenomena being described).

 

[PeterDonis]

I'm very thankful that you're taking time to answer my (probably) dumb questions.

You're welcome! (Your questions aren't dumb; all of us started out not knowing about this stuff.)

 

[Husserliana97]

Hi !

I hope that reopening a three year old conversation is not prohibitive; my apologies if it is.

But PeterDonis ; in your description of the "vacuum polarization" diagram, concerning the fact that it always envelops (=as does the "propagation of a photon without interaction" diagram) an infinite number of possible trajectories in space-time: can we say that this is true both at the entrance of the diagram and at the exit ? As you said, "the photon could go anywhere in spacetime before turning into the electron-positron pair" ; but could it be the same after this "transformation" ? It seems to me that, when turning back into the photon, its transition amplitude still remains a sum of an infinite number of contributions, associated with all the possible paths.

In short; that each line of the diagram must be read as a sum of several possible lines between two points or two vertices? All these possibilities contribute to the total process.

 

[vanhees71]

You should not think about photons in terms of particles. That's almost always misleading. Photons don't even have a position observable to begin with, i.e., they cannot be localized in a small region of space. It's better to think in terms of electromagnetic waves and take the (normalized) energy density of the electromagnetic wave as the probability density for detecting a photon at a given position of the detector.

 

[PeterDonis]

PeterDonis ; in your description of the "vacuum polarization" diagram

I assume you mean what I said about it in post #11; but you should explicitly quote what you are referring to.

concerning the fact that it always envelops (=as does the "propagation of a photon without interaction" diagram) an infinite number of possible trajectories in space-time: can we say that this is true both at the entrance of the diagram and at the exit ?

Sort of. If we interpret the diagram as a spacetime diagram, the starting and ending events are fixed. What happens in between could happen anywhere. You could break "what happens in between" down into: the photon can go anywhere from the fixed starting event; it could then have any of an infinite number of possible intermediate processes happen to it; and then it has to go from whatever middle point it went to, to the fixed ending event.

We should note, though, that most physicists interpret Feynman diagrams as diagrams in momentum space, which is more abstract to the ordinary person, but matches up better to how the diagrams are actually dealt with in practice. In momentum space we would say that the starting and ending momentum of the photon are fixed (and are the same--and here "momentum" is relativistic momentum, so it means the photon's starting and ending energy-momentum 4-vector are fixed and are the same), but the intermediate processes in between can involve exchanges of arbitrary amounts of momentum.

In short; that each line of the diagram must be read as a sum of several possible lines between two points or two vertices?

For purely internal lines, in the spacetime interpretation, this would be the case, yes. For lines that are external legs, the external part is fixed.

In the momentum space interpretation, which, as I noted above, is the one mostly used in practice, the momentum of any internal line can be anything, subject to the constraint that momentum has to be conserved at each vertex. A given diagram actually represents integration over all possible momenta for each internal line in the diagram.

 

[Husserliana97]

PeterDonis Thank you for your reply, which was particularly helpful ! I would just have three questions :

For purely internal lines, in the spacetime interpretation, this would be the case, yes. For lines that are external legs, the external part is fixed.

About these external lines (in the space-time picture): the parts that are fixed, if I understand correctly, are the entry points ("fixed starting event", as you said) and the arrival points.
1) But if I understand the first part of your message correctly: all "trajectories" are possible, between these two points, the photon can go anywhere. However, as we are not dealing here with free propagation, we have to split this propagation into: any path between the entry point and the vertex / any path between the vertex and the arrival point.
This is already a first question I ask (sorry if this seems trivial or obvious): when you say that it goes "from whatever middle point it went to, to the fixed ending event", it is implicitly assumed that it can take all possible routes between this point and the fixed ending event, right?

2) But furthermore, concerning the randomness of this "middle point", of the vertex; isn't it necessary to integrate (sum over) all possible vertex positions, in addition to all possible trajectories?

3)Finally, about the fact that momentum space (with quadrivectors, relativity obliges) is generally privileged; does this apply to all theoretical frameworks of physics? It seemed to me that string theorists preferred to draw their Feynman diagrams in the space of coordinates (thus the diagram would describe the propagation of individual point-like particles in the spacetime, with all the splitting and joining allowed)...perhaps because this image makes it easier to derive their sum over 2D histories (Riemann surfaces) of string world sheets in the spacetime ?

 

[PeterDonis]

if I understand correctly, are the entry points ("fixed starting event", as you said) and the arrival points.

In the spacetime picture, they are the fixed starting and ending events. In the momentum space picture, they are the fixed starting and ending momenta.

when you say that it goes "from whatever middle point it went to, to the fixed ending event", it is implicitly assumed that it can take all possible routes between this point and the fixed ending event, right?

concerning the randomness of this "middle point", of the vertex; isn't it necessary to integrate (sum over) all possible vertex positions, in addition to all possible trajectories?

In practice, as I have said, the integrals are actually done in momentum space. This actually makes things a lot easier, because in momentum space a given line does not "go anywhere"; it has the same momentum until it meets a vertex. And momentum has to be conserved at each vertex. So all you have to integrate over are the possible momenta for each internal line.

about the fact that momentum space (with quadrivectors, relativity obliges) is generally privileged; does this apply to all theoretical frameworks of physics?

If you mean theoretical frameworks of quantum field theory (string theory is a type of quantum field theory, just of strings instead of point particles), I don't know. However:

It seemed to me that string theorists preferred to draw their Feynman diagrams in the space of coordinates (thus the diagram would describe the propagation of individual point-like particles in the spacetime, with all the splitting and joining allowed)

No. In string theory the diagrams refer to different possible string configurations. String theory is a quantum field theory of strings, not point particles. I'm not sure string configurations have a direct analogue to either the spacetime or the momentum representation for point particles.

 

[Husserliana97]

Thanks again!
Regarding my question about string theory, your answer saves me from falling into other pitfalls.
Now, regarding this:

In practice, as I have said, the integrals are actually done in momentum space. This actually makes things a lot easier, because in momentum space a given line does not "go anywhere"; it has the same momentum until it meets a vertex. And momentum has to be conserved at each vertex. So all you have to integrate over are the possible momenta for each internal line.

I understand that in practice, momentum space is preferred, hence your answer. But from a purely theoretical point of view (without prejudging its applicability): if the diagram is constructed in space-time, can we indeed argue that each line "goes anywhere", between the conditions (initial and ending events + vertices)?
It seems to me that it can, and that you also argued this above. Even if, once again, this spacetime image is not the one that is favoured by physicists, including string theorists.

 

[PeterDonis]

from a purely theoretical point of view (without prejudging its applicability): if the diagram is constructed in space-time, can we indeed argue that each line "goes anywhere", between the conditions (initial and ending events + vertices)?

From a "purely theoretical" point of view, Feynman diagrams themselves are just a calculational convenience; the lines and vertices depicted in them don't actually exist. So it makes no sense to ask about their properties independent of the practical use we make of them, since their only purpose is practical use: making approximate calculations of things we don't know how to calculate exactly or for which calculating them exactly would take longer than the lifetime of the universe.

 

[Husserliana97]

Sorry if I said something completely irrelevant , that was not my intention.
I just wanted to know if this "description" in space-time was still possible (if not the most used, I understand - but that does not necessarily mean that this picture is never used, does it ?).
Especially since in your previous answers (still in this same thread), you seemed to build the diagram in this space-time image.

 

[PeterDonis]

I just wanted to know if this "description" in space-time was still possible

It depends on what you mean by "possible". If you're willing to basically give up any connection to how the diagrams are actually used, you can of course imagine that each line represents an infinity of lines between two vertices, since "the photon could take any path". Nobody can stop you. But I don't see the point, since as soon as you try to actually use the diagrams for anything, which means actually writing down and evaluating the corresponding integrals, what you see in the math will not match what you have been imagining.

in your previous answers (still in this same thread), you seemed to build the diagram in this space-time image.

Yes, but only at a very simple level. For example, note what I said explicitly about the "infinite number of lines" thing:

When we talk about light "taking different routes", we are basically talking about one Feynman diagram: the one that has just one photon line coming in and one photon line going out, with nothing happening in between.

In other words, I was using that description as a (simplistic spacetime) description of one particular diagram. I was not using it as a way of describing what lines mean in any diagram whatsoever.

It's also worth looking at why that simplistic spacetime description of that one particular diagram is used. It's used because, experimentally, for that one simple process, we can actually investigate its implications. Feynman describes this in some detail in the QED book I mentioned: basically, we can test, experimentally, whether light that starts at one fixed point in spacetime and ends up at another fixed point in spacetime takes multiple paths through spacetime in between, by obstructing some of the paths and seeing if it changes the behavior (for example, Feynman describes diffraction this way--by cutting off some of the possible multiple paths, you can change the observed behavior of the light).

But for a line in an arbitrary Feynman diagram, where at most one end (if it's an external leg), or more likely neither end (if it's a purely internal line), are even accessible to experiment at all, we have no way of testing whether such a line actually, physically corresponds to "multiple paths". So if we still use the "multiple paths" description at all, it can only be as a story to tell that sounds good, without actually conveying any useful information. That's why, as more details got asked for in this thread, I backed away from the simplistic spacetime viewpoint and started talking about how diagrams actually get evaluated in the momentum space representation.

 

[Husserliana97]

Thanks again, I understand your last answers better!
I myself oscillate between an immoderate taste for talking pictures (and in fact, Feynman's QED is my bedside book), and a desire to learn the mathematical details (yet, Feynman's technical texts on the subject are still beyond my reach). Sometimes all this comes into conflict...

One last thing, though, about the possible use of the space-time image. I found this passage in a book by E. Witten (also completely out of my reach in its detail... but there is this passage):

"Now let us look at a typical Feynman diagram, as in fig. 1.5. Instead of evaluating this diagram in momentum space, as is customary, let us think of fig. 1.5 in coordinate space, so that the external particles originate at space-time points A, B, C and D, and the interactions occur at points p and q. According to the usual rules for computing an amplitude from the Feynman diagram, each line in fig. 1.5 corresponds to a propagator. With the representation (1.4.2) for the propagator, each line in the figure represents an integration over the trajectory of a particle that propagated in space-time between the indicated points. In evaluating the diagram one is also instructed to integrate over the interaction points p and q, and to include at the vertices certain factors that depend on the precise theory considered. The point is that a Feynman diagram can be viewed as an actual history of particles propagating in space-time and joining and splitting at interaction vertices." (cf the document I joigned)

What do you think of this?

 

[PeterDonis]

I found this passage in a book by E. Witten

What book?

 

[Husserliana97]

What book?

Superstring theory, Volume 1 (Green, Schwarz, Witten), p. 27-28

 

[PeterDonis]

Superstring theory, Volume 1 (Green, Schwarz, Witten), p. 27-28

Ok, thanks. I don't have that book so I don't know the context of the statement you quote; the fact that it is in Chapter 1 and that the diagram shown is extremely simple, leads me to believe that it is more of a "sounds good" statement than a description of how much more complicated cases are actually dealt with. AFAIK more complicated cases in superstring theory are still dealth with in momentum space, as they are in other quantum field theories.

 

[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.