The claymath 4-d QFT problem and virtual particles (as an example)

[billtodd]

I am trying to understand how would one opt to solve this open problem?, if there are some objects in the non-constructive-axiomatic QFT which mathematically are ill-defined.
One such ill defined notion is of virtual particles.

I tried to understand what constitutes a virtual particle. For example we have photons that are real which don't have mass as opposed virtual photons that do have mass. And then there are gluons which appear to be all of them virtual, and none are "real"; What does that mean?
Ok, I think I understand it now; It seems according to Wiki, that a virtual particle has mass that it's off-shell vs a real particle which its mass is on-shell, i.e. one satisfies the following relation: $E^2=(pc)^2-(mc^2)^2$ which is on shell and the other doesn't satisfy this.

Well I was under the impression that on-shell satsifies: $m^2=-k^2$ and off-shell doesn't, or vice versa.

Anyway, why can't a gluon also have a real partner, I read in Google that gluons only come as virtual particles and they have zero mass; But then are they on-shell or off-shell?

 

[weirdoguy]

Try this insights:
https://www.physicsforums.com/insights/misconceptions-virtual-particles/
https://www.physicsforums.com/insights/what-are-virtual-particles-intro/
https://www.physicsforums.com/insights/physics-virtual-particles/

 

[PeterDonis]

I read in Google

Where? Please give a specific reference.

 

[billtodd]

Where? Please give a specific reference.

Try searching in Google:"are gluons virtual or real?"
https://www.quora.com/Are-gluons-on...ticles like,virtual while gluons only virtual.

 

[PeterDonis]

Try searching in Google:"are gluons virtual or real?"

That's not an acceptable reference. First, a Google search is not a specific reference. Second, a Google search will turn up all kinds of nonsense.

https://www.quora.com/Are-gluons-only-virtual-particles-or-real-particles#:~:text=Gluons are massless particles like,virtual while gluons only virtual.

If this is the article you are actually referencing, it's a questionable reference, and it references Wikipedia, whose reliability is also questionable (some Wikipedia articles are quite reliable, but it's hard to tell which ones unless you already know the subject matter). You really need to be looking at a textbook or peer-reviewed paper.

But in any case, note that the Quora answer does not say "gluons" period are always virtual, it says "gluons in protons", which is not the only place gluons can be found.

 

[PeterDonis]

Moderator's note: Thread level changed to "I".

 

[billtodd]

That's not an acceptable reference. First, a Google search is not a specific reference. Second, a Google search will turn up all kinds of nonsense.


If this is the article you are actually referencing, it's a questionable reference, and it references Wikipedia, whose reliability is also questionable (some Wikipedia articles are quite reliable, but it's hard to tell which ones unless you already know the subject matter). You really need to be looking at a textbook or peer-reviewed paper.

But in any case, note that the Quora answer does not say "gluons" period, it says "gluons in protons", which is not the only place gluons can be found.

Which other places can gluons be in?
And are there gluons which are real i.e. on-shell?

 

[PeterDonis]

are they on-shell or off-shell?

"On shell" and "off shell" only make sense in the context of perturbation theory, but the strong interaction cannot be properly treated using perturbation theory, at least not in any context relevant for addressing the questions you are asking about gluons. So this question is unanswerable because it's not well posed.

 

[billtodd]

"On shell" and "off shell" only make sense in the context of perturbation theory, but the strong interaction cannot be properly treated using perturbation theory, at least not in any context relevant for addressing the questions you are asking about gluons. So this question is unanswerable because it's not well posed.

So how do you define a virtual vs real particle in non-perturbative QCD?
This begins to be interesting.
I read in the Wiki page that of virtual particles, that in Lattice Gauge Theory there's no use of this distinction between virtual and real.

P.S
Believe me if I'll stay alive long enough I'll read the other books on non-perturbative QCD, but I first need to finish the red book on perturbative QCD.

 

[PeterDonis]

how do you define a virtual vs real particle in non-perturbative QCD?

You don't. The concept of virtual vs. real particle only makes sense in perturbation theory.

 

[billtodd]

You don't. The concept of virtual vs. real particle only makes sense in perturbation theory.

It's interesting, and I thought that split-personality of physics appears only in the problem of quantum gravity in high energies, while you say that there this dichotomy between perturbation theory and non-perturbation and they don't coincide at least in this regard of virtual vs real.
I found an interesting and quite dated debate here in PF:
https://www.physicsforums.com/threads/non-perturbative-qft-without-virtual-particles.485597/

 

[PeterDonis]

there this dichotomy between perturbation theory and non-perturbation and they don't coincide at least in this regard of virtual vs real

This "dichotomy" is purely in our theoretical models. There is no such dichotomy in reality. The gluons don't know that they're not supposed to be doing "perturbation theory".

 

[billtodd]

This "dichotomy" is purely in our theoretical models. There is no such dichotomy in reality. The gluons don't know that they're not supposed to be doing "perturbation theory".

I wonder how to reconcile this mathematically.
Someone said that practically we may never know if PT is just a computational aid for computation, and there might be no need for such "virtual" vs "real" terminology, but then again someone can say that PT appears also in QM; and I don't think anyone thinks of dispensing with PT over there.
Are we bound to use PT in general (in both applied maths and theoretical physics) because of computer's limitations? i.e. there's no real rigorous mathematical justification of using PT in our problems?

I must confess that part of me of trying to write a thesis on PT in ODE or PDE is that I thought there's a rigorous justification for this theory; though my adviser warned me that there might not be such justification.
Otherwise I think I would have tried doing something on mathematical logic or something else in mathematical analysis.

 

[PeterDonis]

I wonder how to reconcile this mathematically.

Reconcile what?

Someone said

Who? You need to give specific references.

there's no real rigorous mathematical justification of using PT in our problems?

Physicists generally use tools that work, in the sense of making predictions that are accurate enough for their purpose, without worrying very much about whether there is a "rigorous mathematical justification". For example, they will happily compute the first few terms of a perturbation theory infinite series without worrying about whether the series converges, as long as the computation gives a result that matches experiment closely enough.

 

[jbergman]

I wonder how to reconcile this mathematically.
Someone said that practically we may never know if PT is just a computational aid for computation, and there might be no need for such "virtual" vs "real" terminology, but then again someone can say that PT appears also in QM; and I don't think anyone thinks of dispensing with PT over there.
Are we bound to use PT in general (in both applied maths and theoretical physics) because of computer's limitations? i.e. there's no real rigorous mathematical justification of using PT in our problems?

I must confess that part of me of trying to write a thesis on PT in ODE or PDE is that I thought there's a rigorous justification for this theory; though my adviser warned me that there might not be such justification.
Otherwise I think I would have tried doing something on mathematical logic or something else in mathematical analysis.

I believe that PT theory has been put on a rigorous footing. See this paper for more discussion. I'm talking out of my depth, here but I believe the problem with PT approaches are they only work if you treat QFT as an effective theory with energy cutoffs.

I believe the desire of the Yang Mills Clay problem is to construct a full non-effective QFT. Take what I've said with a large grain of salt.

 

[Adrian59]

I am trying to understand how would one opt to solve this open problem?, if there are some objects in the non-constructive-axiomatic QFT which mathematically are ill-defined.
One such ill defined notion is of virtual particles.

I tried to understand what constitutes a virtual particle. For example we have photons that are real which don't have mass as opposed virtual photons that do have mass. And then there are gluons which appear to be all of them virtual, and none are "real"; What does that mean?

My understanding is that a virtual particle is never detected on its own, as it is involved with the transmission of a force, so it is absorbed by the destination particle. As it is not detected it is deemed virtual. Of course the force carriers for the electromagnetic and weak forces have been detected independently but not the colour (UK spelling) force.
The issue of mass is not required for the argument as the mass given to a virtual photon is a fictitious mass which is only to prevent the integral having a divergence, and then this fictitious mass can be removed in the full renormalization calculation. Since the virtual photon is never detected itself, this infinitesimal mass is of no observational significance. I other words the fictitious mass is only a mathematical device.

 

[A. Neumaier]

So how do you define a virtual vs real particle in non-perturbative QCD?

In non-perturbative QCD, only real particles exist (gluons, mesons, baryons, and perhaps more complex bound states). Virtual particles are an artifact of perturbation theory.

 

[billtodd]

Another related question that I have.
If I want to know how were all the "particles" created in the beginning of the universe, when something triggered the vacuum to make the "virtual" into "real" particles? Is there a known mechanism that we know how to control that makes "virtual" particles into "real" particles.

I think I once heard in a PBS or was it a pop sci book, that in the very beginning there was only vacuum with virtual particles popping in and out of existence until something changed sort to speak the vacuum state into the next level state (I guess someone operated on $|0>$ with $\exp(-iHt/\hbar)$ on $|0>$.

So how did all this quarks and leptons popped into existence?
What sort of "stories" are there for this interesting question of mine. (perhaps it's not well defined, but assuming there are such real particles I guess one is bound to ask how were they created from the vacuum?

 

[PeterDonis]

how were all the "particles" created in the beginning of the universe, when something triggered the vacuum to make the "virtual" into "real" particles?

No such thing occurred.

I once heard in a PBS or was it a pop sci book

Neither of these are good sources for learning actual science. You need to be looking at textbooks and peer-reviewed papers.

how did all this quarks and leptons popped into existence?

Nothing "popped into existence".

In inflation models, at the end of inflation, there is a large transfer of energy from the inflaton field to the Standard Model fields, as the inflaton field transitions from a "false vacuum" to a "true vacuum" state. That might be what you are vaguely groping towards here, but I can't tell for sure. Nothing "pops into existence" in this scenario; it's just a straightforward energy transfer.

What sort of "stories" are there for this interesting question of mine.

Physics is not about "stories". It is about building models that make accurate predictions.

 

[billtodd]

No such thing occurred.


Neither of these are good sources for learning actual science. You need to be looking at textbooks and peer-reviewed papers.


Nothing "popped into existence".

In inflation models, at the end of inflation, there is a large transfer of energy from the inflaton field to the Standard Model fields, as the inflaton field transitions from a "false vacuum" to a "true vacuum" state. That might be what you are vaguely groping towards here, but I can't tell for sure. Nothing "pops into existence" in this scenario; it's just a straightforward energy transfer.


Physics is not about "stories". It is about building models that make accurate predictions.

The models are sort of "stories".
So do you say that the quarks and leptons always existed? (I know that in cosmology there are two possible regimes: radiation epoch without matter and matter-radiation epoch; but in both of them there are particles which are excitations of fields).
You know either the particles always existed in the universe or they somehow popped into existence,
So I wonder which of these two possbilities had happened in the beginning if there were such a beginning to begin with. (I know this sounds a bit philosophical, but cosmolgy started as metaphysics really before making predictions; and the difference between being metaphysical and being physical lies in the end in numerical evidence, so to speak).

 

[weirdoguy]

when something triggered the vacuum to make the "virtual" into "real" particles?

So again - do you remember what is the definition of a virtual particle? If you did, you wouldn't ask that question, because it's meaningless, per definition of virtual particles. It's just a name for an internal lines of Feynman diagrams, which by itself are just a graphical tool to make calculations easier. There is nothing more to that, even if pop-science makes you think otherwise.

 

[billtodd]

So again - do you remember what is the definition of a virtual particle? If you did, you wouldn't ask that question, because it's meaningless, per definition of virtual particles. It's just a name for an internal lines of Feynman diagrams, which by itself are just a graphical tool to make calculations easier. There is nothing more to that, even if pop-science makes you think otherwise.

Well, aren't "real" particles only a mathematical tool too?

Anyway, I would like to know how do particles real or virtual came into existence?
Isn't this a legitimate question?
I know that in field theory they assume that fields always existed even if there aren't no particles around. But then how did those particles start existing? what triggered the particles' (real or virtual) existence?
I understand that the fields cause their existence is those excitations; but I once heard that there's a possible universe where there are no particles and only fields, which begs the question what triggered those quantum fluctuations? Or they also always evident in the fabric of reality, i.e. quarks and leptons always existed.

 

[haushofer]

Well, aren't "real" particles only a mathematical tool too?

Anyway, I would like to know how do particles real or virtual came into existence?
Isn't this a legitimate question?
I know that in field theory they assume that fields always existed even if there aren't no particles around. But then how did those particles start existing? what triggered the particles' (real or virtual) existence?
I understand that the fields cause their existence is those excitations; but I once heard that there's a possible universe where there are no particles and only fields, which begs the question what triggered those quantum fluctuations? Or they also always evident in the fabric of reality, i.e. quarks and leptons always existed.

The "story" depends on the initial stage of the universe. One such story is inflation: the inflaton field dumped its energy into spacetime, such that space extended enormously and the quantum fields got excited such that particles were produced.

What we call "real particles" are, in the QFT, idealizations of quantum field states "at infinity" (hence: where the interactions can be ignored) which can be detected by particle detectors. So in a sense they're also "mathematical tools". We call these states "particles" because they're localized in momentum, can be counted and obey Einstein's energy relation. The virtual particles on the other hand don't have these properties. They're particle-like interpretations of quantum field fluctuations causing interactions, and pop up in perturbation theory because we can't solve those interactions analytically. The idea that "virtual particles can become real" can be interpreted as a resonance effect: just like swinging a pendulum with it's natural frequency enhances its amplitude enormously, pumping a certain energy into the quantum field can enhance the probability of a detectable (hence "real") particle to pop up.

If you use the phrase "I once heard" here, people will ask you for sources, by the way. It depends on what you mean by "possible universe". You can think of a static, flat eternal universe with only quantum fields and no particle excitations, sure. But quantum fluctuations are not "triggered"; they're inherent properties of the quantum fields. It's like asking what "triggered" the charge of the electron. We don't have a mechanism explaining such properties, and a physicist could ask a philosopher if such a question would even make sense. It's not like quantum fluctuations are "oscillations in time", like a simple pendulum; they're fluctuations influencing measuring outcomes, and we can only measure these influences. We cannot directly measure quantum fluctuations like you can measure the amplitude of a swinging pendulum as a function of time.

 

[haushofer]

Physics is not about "stories". It is about building models that make accurate predictions.

Maybe for researchers. Meanwhile there's also an enormous lay public fascinated by physics without the mathematical training researchers have. For them, science most definitely is about stories. In my humble opinion it's a great mistake for scientists to just shove away the need for "stories". Jonathan Gosthaldt uses the term "Homo fictus" in his book The Storytelling Animal. It's a very important part of what makes us human. And if scientists are not able to convey these stories because "science is not about stories", the gap between academia and the lay public will only increase. I don't think I have to explain why that's a problem, because everyone understands that fundamental research needs funding and pseudoscience (and even more extreme ways of reasoning like conspiracy thinking) is all around us. And frankly, I also think this attitude doesn't add to what this subforum tries to do. But as I said, just my humble opinion as a popular science writer.

 

[weirdoguy]

Well, aren't "real" particles only a mathematical tool too?

Have you read the links I gave you? The most basic thing is that real particles have states, virtual don't. And virtual are not even a thing in non-perturbative methods.

 

[weirdoguy]

it's a great mistake for scientists to just shove away the need for "stories".

Can everything in physics be turned into "stories"? I think that it's not the case. Sometimes the stories are so far away from what they are trying to represent, that they are just a waste of time - both for those who tell them and those who read them. E.g. can you express that:

The idea that "virtual particles can become real" can be interpreted as a resonance effect: just like swinging a pendulum with it's natural frequency enhances its amplitude enormously, pumping a certain energy into the quantum field can enhance the probability of a detectable (hence "real") particle to pop up.

in terms of math? How can a propagator (virtual particle) become a state (real particle)? I think I know enough QFT to handle everything that is needed.

Besides, quite often those stories bacome a basis of pseudscience, including conspiracies.

 

[billtodd]

The "story" depends on the initial stage of the universe. One such story is inflation: the inflaton field dumped its energy into spacetime, such that space extended enormously and the quantum fields got excited such that particles were produced.

What we call "real particles" are, in the QFT, idealizations of quantum field states "at infinity" (hence: where the interactions can be ignored) which can be detected by particle detectors. So in a sense they're also "mathematical tools". We call these states "particles" because they're localized in momentum, can be counted and obey Einstein's energy relation. The virtual particles on the other hand don't have these properties. They're particle-like interpretations of quantum field fluctuations causing interactions, and pop up in perturbation theory because we can't solve those interactions analytically. The idea that "virtual particles can become real" can be interpreted as a resonance effect: just like swinging a pendulum with it's natural frequency enhances its amplitude enormously, pumping a certain energy into the quantum field can enhance the probability of a detectable (hence "real") particle to pop up.

If you use the phrase "I once heard" here, people will ask you for sources, by the way. It depends on what you mean by "possible universe". You can think of a static, flat eternal universe with only quantum fields and no particle excitations, sure. But quantum fluctuations are not "triggered"; they're inherent properties of the quantum fields. It's like asking what "triggered" the charge of the electron. We don't have a mechanism explaining such properties, and a physicist could ask a philosopher if such a question would even make sense. It's not like quantum fluctuations are "oscillations in time", like a simple pendulum; they're fluctuations influencing measuring outcomes, and we can only measure these influences. We cannot directly measure quantum fluctuations like you can measure the amplitude of a swinging pendulum as a function of time.

Do we know for sure there is only one unique initial state?
If the multiverse is indeed the reality of nature then there are infinite number of initial states, each universe with its own initial state.
Perhaps one can build a sort of Einstein-Rosen Bridge between different universes IDK.


P.S
Before of hearing Susskind's paper on EPR and ER I must confess I had some thoughts about these two notions and how do they correlate to each other.
But I must attest that I was surely half serious half joking... (in a superposition :-)).


Maybe there's a universe where I am a mad professor and not the endless student... now here's a script for a sci-fi show.

 

[billtodd]

@haushofer can you give me a technical reference (book or paper) for your following paragraph:

The idea that "virtual particles can become real" can be interpreted as a resonance effect: just like swinging a pendulum with it's natural frequency enhances its amplitude enormously, pumping a certain energy into the quantum field can enhance the probability of a detectable (hence "real") particle to pop up.

I must confess that while reading books on QFT and QCD I didn't understand the difference between a virtual gluon or a real gluon.
But now I understand that the real obey Einstein SR pseudo inner product, while the virtual need a correction to satisfy the pseudo inner product.
https://en.wikipedia.org/wiki/Pseudo-Euclidean_space

P.S
If there's no paper or book, you can write the maths behind the idea that a virtual becomes a real particle; or is it described in the links weirdoguy gave?
I must confess that I don't have a lot of time to spare these coming weeks due to exams; so I dunno when the next time I'll be replying to this thread.

 

[WernerQH]

In resonance scattering, the intermediate state of the atom is often called virtual. But for photon frequencies close enough to the line centre there isn't much difference between the intermediate state and a real state.

In a sense every real photon is actually virtual if one looks over sufficiently long time scales. It is always absorbed somewhere in the universe. What charactarizes a real photon is that $k^2 \rightarrow 0$ (since it is not real at all times, by the uncertainty principle, $k^2$ is not identically = 0) and therefore the propagator $1/ k^2 \rightarrow \infty$.

(The Theory of Fundamental Processes, p. 95)

 

[PeterDonis]

The models are sort of "stories".

No, they're not, they are prediction makers. You can't make accurate predictions from "stories".

So do you say that the quarks and leptons always existed?

It depends on what you mean by "quarks and leptons". You appear to mean "particles" by these terms, but "particles" are just particular states of quantum fields, and the fields don't have to be in these states. The quantum fields described in the Standard Model always existed, but their states aren't always the same; in particular, their states before the end of inflation were very different from their states after the end of inflation.

either the particles always existed in the universe or they somehow popped into existence,

Nope. Nothing "popped into existence" because the quantum fields were always there, they just changed states. But "particles" did not always exist because the quantum fields were not always in "particle" states. So neither of your alternatives here is correct.

This is what comes of trying to understand physics using "stories" instead of looking at the actual models and the predictions they make.

 

[PeterDonis]

aren't "real" particles only a mathematical tool too?

All of our models are mathematical tools. But some entities in our models correspond to things we can directly detect in reality. Others don't. "Real particles" (with appropriate qualifications for what that term means) are examples of the former. "Virtual particles" are examples of the latter.

 

[PeterDonis]

there's also an enormous lay public fascinated by physics without the mathematical training researchers have. For them, science most definitely is about stories.

Even if this is the case, it's irrelevant here at PF, because here at PF we help people to understand the actual mainstream physics, not the pop science "stories" that are told to the lay public. People who just want the pop science "stories" can go elsewhere to discuss them.

 

[PeterDonis]

if scientists are not able to convey these stories because "science is not about stories", the gap between academia and the lay public will only increase.

This is off topic for this thread.

 

[billtodd]

So, when do we use pQCD and when do we use non-pQCD (p for perturbative) in our calculations?
I can also ask the bot, but first try the humans.

If in non-pQCD we don't use the notion of virtual particles, it seems that one should opt doing the calculations in non-pQCD, but obviously it might not be practical (I haven't yet learned non-pQCD).

On another notion, I am trying to understand the mass gap problem here:
https://en.wikipedia.org/w/index.php?title=Mass_gap&section=1&oldid=1236681154&action=edit

Does a Harmonic Oscillator have this mass gap property?
I asked the famous chatGPT, but as I have been told not to post its answer I'll let you answer.

 

[A. Neumaier]

So, when do we use pQCD and when do we use non-pQCD (p for perturbative) in our calculations?

You? Your calculations? Did you ever do calculations in quantum field theory?

Both perturbative (Schwinger-Dyson based) and nonperturbative QCD (on lattices) are approximate and have their uses.

Does a Harmonic Oscillator have this mass gap property?

Trivially yes.