Understanding the Motivation Behind Quantizing Fields in Quantum Field Theory

[nrqed]

This is indeed, to me, a big mystery too! I guess pure physicists have less trouble with it because they are raised with Lagrangians. But I started out as an electromechanical engineer, where Lagrangians are not of much use, because most engineering systems are nonlinearly dissipative (like braking forces that go to the speed power 2.6 or things like that)...
So I find it simply amazing that ALL of modern physics comes down to writing lagrangians

You make there a very interesting observation.

I have less difficulties with this. True, historically, we derived the KG and the Dirac equation as false attempts of a quantum wave equation. However, special relativity puts such huge constraints on the kinds of classical field equations that you can write down, that I think that NO MATTER HOW YOU PROCEED, if you're going to write down a differential equation and you're going to use special relativity, you'll end up with one of the known equations (K-G, Dirac, EM, proca...)

I agree with you, *once* we accept that we the correct way to go is to quantize classical fields (here I go again ). Then I agree that the possible equations are quite restricted.

But again, it's the starting point which bugs me. It's a bit like saying "ok guys, you have learn QM. Now we are going to build a formalism which satisfies SR as well. First step: let's build classical field theories which are consistent with SR. Don't worry about what they represent physically. For now, this is a purely formal exercise. Then we'll quantize them and intrpret them"

My question, as always, is : why that starting point?

I will get my hands on a copy of Weinberg's first volume this weekend and hopefully I will stop bugging you guys

I have to say that I too had quite some difficulties with a _second quantization_ (and why not a third one, once we're at it ?) and I felt it as a revelation that these were *classical* field equations...

I know that many books say that "second quantization" is a misnomer, but in some sense I feel that it's a good reflection of the thought process involved in the standard presentations. For example, first we use the p \rightarrow -i \hbar {\partial \over \partial x} prescription in order to get to a wave equation, and then we say, forget quantization, let's treat this as a classical equation. Then we say, let's quantize the fields.

So I feel "second quantization" does reflect the thought process involved. But of course, there is only one quantization involved.

That I didn't know these fields before was not really a problem: after all, once you have a mass term, you find, through quantization, why you don't notice the classical field, but that you think it are particles. Probably neutrinos act a lot more as true classical fields. The next one in the row, electrons, are already too heavy for us to notice them as a field. I guess that to notice a quantum field as a classical field, you need to have spatial resolution of the wavelength when the particles are already ultrarelativistic, so that you can create and destroy them by zillions and have coherent modes.

I agree completely. But to emphasize this point, the only logical way to introduce quantum field theory is through the quantization of the EM field. And then one should explain carefully the correspondence to classical fields through coherent states etc. And then one should explain carefully how different things are with massive modes and how the correspondence to classical fields is not as direct, etc etc. But that would still require a leap of faith: that this process (through fields) that worked for photons will still work for everything else (IMHO). In any case, that's a line of thought that I would much prefer to the standard presentations.

So I'm still enjoying the high dopamine levels from my Aha experience of "it are classical fields, not wave equations!", and I won't let you bring them down yet :Tongue2:.

Far from me the idea of taking this away from you. I also recall being bothered by the "second quantization" expression and wondering what was really going on. Until, like you, I realized that were just quantizing a classical system "once", but we were quantizing classical fields instead of point particles. Then, like you, I went AAHHHH! And I felt happy at the simplicity and beauty of the idea...for about one minute. Then the slef-doubts began. But why, oh why?!? I thought, there must be a simple motivation, but this book does not present it. Then I went out and read all the books introducing QFT I could find (that was before Weinberg was in print or even P&S even though P&S would still have left me unsatisfied). And I did not find what I was looking for anywhere. And it has been like this since then, which explains why I am depressed and cranky all the time

However, you're further in your understanding than I am, so you've had that and now you want to go back to "particles". way of doing things.

I'm just giving you my actual understanding, which gives me peace of mind and high dopamine levels.

I think your understanding is (at least) as good as mine! It's more a question of "taste" and "beauty" and "naturalness of presentation" which are all extremely subjective criteria. I still have to find the presentation that I would find natural. You have found yours. Everybody has his own.

My criterion is: if I were to rederive everything from scratch, is this the way I would do it? (Of course, I would not be smart enough to work out myself all the mathematical tricks and I would get stuck on many technical points, but I mean, conecptually, is this what I would have thought about trying?).

Of course there are some ideas that you learn and you go "this is brilliant, but I would never have thought about this myself". For example, this is what I felt when I studied GR. But this is different because *after* I understand the idea, I go "ok, I would never have though about this on my own, but now that I know it it makes perfect sense". I feel ok with those kind of ideas. It just shows that I am not a genius, but that's ok, I already know that

On the other hand, there are some ideas that *even* after I learn them, I go "it does not even make sense to me!". And quantizing classical fields is one of them.

As I said before, I think this is less of an assumption. We could say: hey, there's at least ONE classical field we know of, namely EM. So fields play a role in nature. But sometimes it behaves particle-like. What if other particles were simply also the manifestation of other classical fields ? But we don't know other classical fields (well, except for gravity, but that's another story).
So what fields are thinkable ? Then we write down all partial differential equations that are compatible with special relativity, and find that there aren't so many alternatives. Moreover, we seem to be able to write their differential equations as deduced from a variational principle, so we know how to quantize.
We try each of them starting from the simplest ones, and lo and behold, each time they produce particles we know of ! So fields ARE really interesting entities to study.

cheers,
Patrick.

Good, I do like this approach much better than what most books do (including P&S), as I said above. And I would be less of a pain in the neck for you guys if most books would emphasize this. At least the leap of faith is made clear. But it's still an important leap of faith, because there is no clear reason why even massive particles should be associated to fields. Especially that these fields can be treated as classical as a starting point! I mean, the transition from the photon picture to classical fields is subtle and it's quite a leap (IMHO, again) to say that it could be done for massive particles. It could be that the transition to a classical field picture is not possible at all except for massless states, in which cae the starting point itself is inn jeopardy. I think think this whole issue would need to be carefully addressed before one could even *start* the program of quantizing classical fields. And this is why I find this approach awkward.

On the other hand, following "my" approach, the starting point would be: partciles can be created/annihilated. That would be the *only* requirement. Well, there would be other requirements but these would be quite acceptable to everybody (causality, Lorentz invariance, cluster decomposition, etc).

I personnaly would find it more "pleasing" to use as starting point that particle numbers is not conserved rather than postulating that a transition to classical fields is possible for massive particles.

If I had it my way, I would start only with creation/annihilation operators and not only would the idea of fields "falls off" from other requirements but even the wave equations themselves would come out as a by product!

This way, I would all what I consider "leaps of faith" in the traditional approach to be eliminated. So, form my point of view, the conceptual gain would be major.

When we started the QFT study group on superstringtheory.com, all those questions came back to me and I started focusing on them and trying to rebuild things myself (that's part of the reasons, together with my classes, buying a house, etc, that rendered me useless as a group leader). But I am no Weinberg so I got stuck on several technical points. I do hope that he does it the way I am thinking because then everything will fall into place and I will be able to answer why we need fields using a language that is 100% satisfactory to my stubborn mind.

Thanks again for all the input. It does make me think in new ways.

Pat

 

[nrqed]

Hi Patrick,

What do you mean in your first point in the above extract ? A third quantization ? This would mean the quantization of a particle ?

I see you are having difficulties with the concept of fields, right ? Correct me if I am assuming things here...

Hi Marlon,

Far from me the idea of talking for Patrick, but I do know that he understands very well QFT and he has no conceptual difficulties with fields.

When he talked about "third quantization" he was poking fun at the traditional expression "second quantization" which is misleading. He was basically saying that when we hear this expression for the first time, we may go "what the heck does that mean? Why not a third quantization and so on?"

Could you please explain to me what you mean by your statement on mass and it being some kind of parameter through which you do not notice the classical field.

Besides why are you always talking about them classical fields. In QFT everything is relativistic in the most general way.

regards
marlon

He is talking about the *classical* fields that are quantized in QFT. You seem to oppose the notion of "relativistic" to the notion of "classical fields"! They are not exclusive! In QFT we quantize relativistic classical field theories.


As for the mass, he is pointing out that we *do* easily observe the classical limit of quantum field associated to the photons, that's just the EM field already studied by Faraday, Maxwell etc. But we don't observe in normal conditions the classical limit of the electron field, for example. Why is that? It's because the photon is massless so that under normal conditions there are always tons of photons present when we excite the EM field. On the other hand, under normal circumstances ( for example in the Stern-Gerlach experiment or even in an ordinary circuit) we see a fixed number of electrons, so we don't see the classical limit of the quantum field.

Of course each individual electron exhibits a wave/particle duality. I am talking about the classical limit of the quantum field, which means that there must be enough particles to create a coherent state type of description. This is easy to accomplish with massless states, such as photons. But not with massive states.

Regards

Pat

 

[nrqed]

As an addendum. Due to the particle/wave-duality one can not say that an electron for example is either a particle or a wave (excitation of a field). So noticing electrons as particles in stead of fields because they are to heavy is something you cannot say. Both the ways to look at the electron are valid at all time. They are dual, you know, in that aspect that you describe the same thing but you use a different language. None of the two languages can be preferred over the other in some way...

regards
marlon

Each electron exhibits a particle/wave duality. But he was talking about a classical limit of the quantum field associated to the electron. He is talking about coherent states of the quantum field! Why was the EM field first thought as a wave (as opposed to a collection of photons) and the electron first discovered as a particle? Because we don't observe coherent states of electrons under normal conditions because they are massive.

It's important to distinguish the wave nature of individual electrons (already present in nonrelativistic QM) and the classical limit of the *quantum fields* which is easy to see for the EM quantum field not not for the electron quantum field. See my other post also.

Regards

Pat

 

[vanesch]

Hi Marlon,

Far from me the idea of talking for Patrick,

Hi Pat,

You can always talk for me, you do it better than I do

thanks and cheers,
Patrick.

 

[marlon]

Hi Pat

First of all thanks for your extensive reply. I don't want to be too difficult but to be honest i must say that your description of the influence of mass on the presence of particles is quite vague and in may opinion even untrue.

I mean you say that because electrons are massive (and always a fixed number of them present, i agree with that) you don't see the classical limit of the field theory. Let me be honest : what do you mean by that.

I don't understand the motivation you are using in order to back this up ? Photons and electrons are totally different particles. Making a distinction between them based upon mass is something new to me. (though i may say QFT is not new to me it is my major.)

remeber that in every QFT the particles are massless, yet their properties (fermionic or bosonic and so on ) are already determined before the Higgs-mechanism "gives" those particles their mass. Mass is just to be seen as some sort of coupling constant that expresses the strength of the interaction of them elementary particles with the Higgs-field.


Again sorry, but i just don't see the evidence for what you are saying. Perhaps i am not getting you, in that case i apologize and ask you friendly to explain.


regards
marlon

 

[marlon]

Just another thought.

I read somewhere that you were questioning these Lagrangians from which we start in QFT in order to construct a field theory. Keep in mind that this is done by trial and error basically.
Just look at how the Yang Mills Lagnrangian was constructed for QCD by making the SU(3)-colour group LOCAL.

The gauge-fixing terms as welll as the ghost-terms of this Lagrangian were not there from the beginning ofcourse.

For example when starting from a Lagnrangian whithout ghost-term we found non-physical properties of particles after the variation of the corresponding functional (just like the variational principle yields the Euler-Lagrange-equations). these properties were things like negative expectation values or integer spin for Grassmann-variables (anti-commuting variables describing the fermions in QFT). In order to get rid of these "sick" things extra particles were added (ie them Fadeev-Popov ghosts) in order to annihilate the unphysical degrees of freedom. Another solution was to "adapt" the basic equations of motion into the Gupta-Bleuler-equations...

regards
marlon

 

[Rothiemurchus]

And the Higgs potential is arbitrary though it is expected to work because it restores symmetry to other fields in the standard model.
The use of trial and error in filed theory is what makes it unsatisfactory.

 

[marlon]

And the Higgs potential is arbitrary though it is expected to work because it restores symmetry to other fields in the standard model.
The use of trial and error in filed theory is what makes it unsatisfactory.

What are you saying, my dear friend Rothiemurchus ?

ps thanks for your support

marlon

 

[humanino]

... as well as the BRS transformation necessary to have the complete healthy QCD, and the Slavnov-Taylor identity ensuring that evolution does not take physical states to non-physical ones. But that is technical more than fundamental.

 

[Rothiemurchus]

MARLON:
What are you saying, my dear friend Rothiemurchus ?

Rothie M:
Just saying that it would be nice to have a field theory that
resembles GR and leaves you thinking
"nature must be like this really."
I don't care what the maths says, as far as I am concerned virtual particles are real
just like EM waves are real.Those path integrals people on here mention:
they are just a mathematical trick - nobody understands why they work.
I've got a lot of respect for Feynman - even buy some of his books - but I won't think his theory is right until someone explains it from a more fundamental level.

 

[humanino]

total agreement from my part. Anything having "quantum" in it, is desperately phenomenological

except maybe for LQG.

 

[marlon]

That is a strong argument Rothiemurchus.
I am not saying you cannot make it, but beware that if you want to criticize a certain very well established theory like QED you got to come up with something better you know.

Maybe the Higgs-field is indeed not yet found, it is nevertheless the best "system" for mass-generation in QFT up till now...

About them virtual particles, they are basically NOT real, really. But they can become real when enough energy is available to give them a "valid reason to exist" for a short while, conform Heisenberg-uncertainty.

They are in QED in fact used as a "trick" to calculate interactions in perturbationtheory. The best way to illustrate their use is (according to me) the following : You can make an infinite sum of powers of x in order to approximate for example cos(x), using Taylor-expansion. Now in this sum you have the index k (from 0 to infinity) to indicates a certain term in the expansion. Well them virtual particles are in QED the sam as the index k in the Taylor-expansion of a given function.

Keep in mind, this is just an analogy to give you a better understanding of their use.

regards
marlon

 

[vanesch]

I mean you say that because electrons are massive (and always a fixed number of them present, i agree with that) you don't see the classical limit of the field theory. Let me be honest : what do you mean by that.

I don't understand the motivation you are using in order to back this up ? Photons and electrons are totally different particles. Making a distinction between them based upon mass is something new to me. (though i may say QFT is not new to me it is my major.)

You're probably right that the fermionic nature also plays a fundamental role in the case of electrons. But the point I was making (and Pat seems to back it up) is that the classical field (a solution of the Dirac equation in the case of electrons, but also the solution of the KG equation in the case of, say, pions) was never observed in the same way as were solutions of the Maxwell equations.

This is the whole point of the discussion here (remember what I said about person A and person B and so on ): where do these fields come from?
We already had the Maxwell equations, so we knew there was something like an EM field. Quantizing this is not difficult. But where did the KG field come from ? Or the Dirac field ? Who ordered that one ? We never had those fields as classical objects before turning it into a quantum field. So why consider them in the first place ?

Now my (granted, intuitive) reasoning was that in order to observe a quantum field as a classical field (from analogy of what happens in the EM case) you need to build up coherent modes of many, many particles (a classical 107MHz wave in EM is made up of a lot of photons in a coherent state), and if you want to do that with a quantum field which has mass (whether this is a true mass term or an effective one such as by the Higgs mechanism doesn't matter), you are on such high energies and such short distances that you won't notice it classically (meaning on human-scale distances).

But, as you point out, the fermionic nature will of course also change that picture. This is an interesting question: is there a way, with massless fermions, to recover a classical field behaviour in the same way you find back the classical EM fields in coherent photon states ?

There are also probably other reasons why we don't see most quantum fields as classical fields (mass is one, fermionic nature is one). This brings up the question: does pure SU(3) gauge theory, but without the quarks, give rise to a classically usuful theory ? I would write "free gluon field" but it's not free of course because the non-abelian self interaction. Is QCD without quarks confined ? Marlon, QCD expert, tell me.

cheers,
Patrick.

 

[vanesch]

I read somewhere that you were questioning these Lagrangians from which we start in QFT in order to construct a field theory.

He was not questioning where a specific lagrangian for a specific field theory came from, he was questioning why you'd be able to write a lagrangian giving rise to field equations in the first place, if I understood Pat correctly. His problem is that in particle physics, well, we talk about particles. So we could build a multiparticle theory directly. Who ordered fields ?

cheers,
Patrick.

 

[vanesch]

Until, like you, I realized that were just quantizing a classical system "once", but we were quantizing classical fields instead of point particles. Then, like you, I went AAHHHH! And I felt happy at the simplicity and beauty of the idea...for about one minute.

A MINUTE ??!

You're way too wasteful with those moments ! I'm tripping on it for years now :-)

When we started the QFT study group on superstringtheory.com, all those questions came back to me and I started focusing on them and trying to rebuild things myself (that's part of the reasons, together with my classes, buying a house, etc, that rendered me useless as a group leader). But I am no Weinberg so I got stuck on several technical points. I do hope that he does it the way I am thinking because then everything will fall into place and I will be able to answer why we need fields using a language that is 100% satisfactory to my stubborn mind.

And then, to make up for the past, will you do Weinberg ??


cheers,
Patrick.

 

[marlon]

This brings up the question: does pure SU(3) gauge theory, but without the quarks, give rise to a classically usuful theory ? I would write "free gluon field" but it's not free of course because the non-abelian self interaction. Is QCD without quarks confined ? Marlon, QCD expert, tell me.

cheers,
Patrick.

Hallo Patrick,

What you say about QCD sounds like science fiction to me? What do you want to achieve here ?

QCD withou quarks is like EM without charged particles. QCD is there espacially in order to describe the properties of them quarks.

Gluons themselves can be confined once they carry an electrical colour charge. But the Abelian Higgs model also predicts two colour neutral (ie abelian) gluons that propagate like free particles ! So basically gluons themselves can be confined (6 of them) and two of them are not !

regards
marlon

 

[marlon]

Who ordered fields ?

Good question and the answer is HISTORY

I am convinced you are familiar with the way EM would explain the interaction of an electron with a photon : The EM wave (photons) exerts a Lorentzforce onto the electron. This electron accelerates and the momentum goes from p to p''. Part of the momentum of the EM wave is being absorbed by the electron (Poynting vector). Because the electron is accelerated it will emit an EM wave with wavelength lambda'. So basically the incident photon has a wavelength that goes from lambda to lambda'. The momentum of the electron them changes into p'. So we have p --> p'' --> p'

This EM way of thinking is a local fieldtheory because their is no activity of forces "on a distance". EM-forces are being carried over by fields that fill the entire "space" and they interact with charges positioned at a specific place. This is a big difference with the Newton-way of thinking.

It is in complete accordance with the local fieldequations of Maxwell that electrons are pointcharges. This is a consequence of the fact that Maxwell equations need to be relativistically invariant. The only question remains as to why the entire EM-wave is absorbed as one single quantum. The answer to that question is ofcourse the wave-mechanics of Schrödinger...as we all know...(first quantization)

So we have fields, and particles and it is the second quantization that gives us force carriers (viewed at as particles not as waves) and the fermionic matterfields (like the Diracfield being the general solution to the Dirac-equation)that yield the elementary massless particles of the Standard Model

So, first fields then particles.

A second way to look at things is fields that are needed in the canonical quantization of a system with infinite amount of degrees of freedom. Just look at the Euler-Lagrange equations for fields and the way they are built...
regards
marlon

 

[vanesch]

What you say about QCD sounds like science fiction to me? What do you want to achieve here ?

QCD withou quarks is like EM without charged particles. QCD is there espacially in order to describe the properties of them quarks.

Let me explain. That f*ing b*st*rd of a Pat has injected a slowly working poison in me that slowly takes its toll... I'm more and more questioning the utility of fields

Indeed, I'm wondering if there is ANY circumstance in which ANY quantum field theory gives rise to a classical field, except for EM !
After your remark I realized that it's going to be damn difficult to have a classical dirac field with "anti-commuting numbers" (he, an old demon rises its ugly head ) Then I thought about the gluon field, the closest thing I can think of to EM. But there is confinement. So I was wondering if confinement is only due to the presence of quarks or not.

Didn't you ever wonder what it would be to have a semiclassical QCD, with a quantized SU(3) gluon field, but with classical sources (the J_mu A^mu term) ? This is how you look upon the transition from a quantum field to a classical field in EM. But of course if confinement still holds (I think so, but I don't know, hence my question) even the pure gluon field will never be classical.

So what remains of our ansatz of starting with classical fields and quantizing them ? Damn Pat !

So the only thing that remains is what I think is in Weinberg: there are no classical fields to be quantized! It's all just bookkeeping of creation and annihilation operators.

So I started reading Weinberg...

cheers,
Patrick.

 

[marlon]

Confinement is "caused" by the presence of colour-charges (quarks, some gluons). So confinement will always be THE fundamental part of QCD, you cannot get rid of it. unless on an extremely short distance-scale...

regards
marlon

 

[nrqed]

You're probably right that the fermionic nature also plays a fundamental role in the case of electrons. But the point I was making (and Pat seems to back it up) is that the classical field (a solution of the Dirac equation in the case of electrons, but also the solution of the KG equation in the case of, say, pions) was never observed in the same way as were solutions of the Maxwell equations.

This is the whole point of the discussion here (remember what I said about person A and person B and so on ): where do these fields come from?
We already had the Maxwell equations, so we knew there was something like an EM field. Quantizing this is not difficult. But where did the KG field come from ? Or the Dirac field ? Who ordered that one ? We never had those fields as classical objects before turning it into a quantum field. So why consider them in the first place ?

I could not have said it better. Patrick and I ar on the same wavelength (maybe our brains have become entangled!)

Now my (granted, intuitive) reasoning was that in order to observe a quantum field as a classical field (from analogy of what happens in the EM case) you need to build up coherent modes of many, many particles (a classical 107MHz wave in EM is made up of a lot of photons in a coherent state), and if you want to do that with a quantum field which has mass (whether this is a true mass term or an effective one such as by the Higgs mechanism doesn't matter), you are on such high energies and such short distances that you won't notice it classically (meaning on human-scale distances).

But, as you point out, the fermionic nature will of course also change that picture. This is an interesting question: is there a way, with massless fermions, to recover a classical field behaviour in the same way you find back the classical EM fields in coherent photon states ?

That's an interesting discussion, and I agree that the fermionic aspect brings in another layer of subtlety. I have read somewhere something about impossibility of a classical field limit for fermions because of the exclusion principle.


But to get back to Patrick (and my) point, we can focus on bosons. For example the pion. It's a boson and yet we don't observe the classical field limit of the pion field, we "see" its particle nature first. Exactly the opposite is true for photons. As Patrick said, this is because of the mass.


To Marlon: Patrick and I are discussing the classical limit of quantum fields in the sense of coherent states. Do you see what we mean? We can observe easily this limit for the photon by simply shining light on two slits. But this is not so for massive particles.

Regards

Pat

 

[nrqed]

Let me explain. That f*ing b*st*rd of a Pat has injected a slowly working poison in me that slowly takes its toll... I'm more and more questioning the utility of fields

Lol! Actually, let me tell you the truth: I am the devil incarnated and I am only doing this to make you and other physicists doubt and question their faith in the almighty field concept!

Indeed, I'm wondering if there is ANY circumstance in which ANY quantum field theory gives rise to a classical field, except for EM !
After your remark I realized that it's going to be damn difficult to have a classical dirac field with "anti-commuting numbers" (he, an old demon rises its ugly head ) Then I thought about the gluon field, the closest thing I can think of to EM. But there is confinement. So I was wondering if confinement is only due to the presence of quarks or not.

Didn't you ever wonder what it would be to have a semiclassical QCD, with a quantized SU(3) gluon field, but with classical sources (the J_mu A^mu term) ? This is how you look upon the transition from a quantum field to a classical field in EM. But of course if confinement still holds (I think so, but I don't know, hence my question) even the pure gluon field will never be classical.

Very interesting and I think this subject would deserve a whole thread by itself. There are several considerations that make other particles qualtitatively different from the photons. There's the fermion/boson distinction. There's mass. There's also confinement as you pointed out (for QCD). But there's also stability of the particle (for example, the W's and Z_0 will decay to other stuff). That's why I used the pions (let's say the neutral pion) as my example in another post. That's the best one I can think of to put aside all these issues and talk about the coherent states/classical field limit of a quantum field. But of course, it's not a fundamental particle, so someone might object on this ground.

So what remains of our ansatz of starting with classical fields and quantizing them ? Damn Pat !

Lol! So you don't think I'm nuts anymore?

So the only thing that remains is what I think is in Weinberg: there are no classical fields to be quantized! It's all just bookkeeping of creation and annihilation operators.

So I started reading Weinberg...

Hehehe... I just got my hands on the first volume. I can't wait to read it.



PAt
cheers,
Patrick.[/QUOTE]

 

[humanino]

I could not have said it better. Patrick and I ar on the same wavelength (maybe our brains have become entangled!)

I think classical QCD is already dealt with in specialized texts, it is just not physical because of confinement and scales at which quantum effect operate. However, instantons for instance (oops) are classical solutions of the pure glue field.

 

[nrqed]

Hallo Patrick,

What you say about QCD sounds like science fiction to me? What do you want to achieve here ?

QCD withou quarks is like EM without charged particles. QCD is there espacially in order to describe the properties of them quarks.

Gluons themselves can be confined once they carry an electrical colour charge. But the Abelian Higgs model also predicts two colour neutral (ie abelian) gluons that propagate like free particles ! So basically gluons themselves can be confined (6 of them) and two of them are not !

regards
marlon

Patrick is just wondering about "toy models" (like pure glue) in order to understand the classical field limit of QFT (in the sense of coherent states). It might sound like science-fiction but then most of physics research is done this way!

It's a legitimate question to inquire about QCD without matter fields. And in that case there can be glueballs, i.e. pure glue bound states. But is confinement still a property of pure glue is what Patric was asking.


I am not sure what you mean by your last paragraph! What do you mean by the "colour neutral gluons"? (I assume you are talking about the Standard Model, if not please tell us exactly what model you are discussing).

The gauge bosons which do not carry colour charge are not gluons, by definition. And there are 4 of them. SO I am not sure if you are talking about the Standard Model. If not, tell us what are the gauge groups you have in mind and in what representation (fundamental, etc) you are using for all the particles.

Regards

Pat

 

[nrqed]

Who ordered fields ?

Good question and the answer is HISTORY

I am convinced you are familiar with the way EM would explain the interaction of an electron with a photon : The EM wave (photons) exerts a Lorentzforce onto the electron. This electron accelerates and the momentum goes from p to p''. Part of the momentum of the EM wave is being absorbed by the electron (Poynting vector). Because the electron is accelerated it will emit an EM wave with wavelength lambda'. So basically the incident photon has a wavelength that goes from lambda to lambda'. The momentum of the electron them changes into p'. So we have p --> p'' --> p'

This EM way of thinking is a local fieldtheory because their is no activity of forces "on a distance". EM-forces are being carried over by fields that fill the entire "space" and they interact with charges positioned at a specific place. This is a big difference with the Newton-way of thinking.

It is in complete accordance with the local fieldequations of Maxwell that electrons are pointcharges. This is a consequence of the fact that Maxwell equations need to be relativistically invariant. The only question remains as to why the entire EM-wave is absorbed as one single quantum. The answer to that question is ofcourse the wave-mechanics of Schrödinger...as we all know...(first quantization)

Hi Marlon,

Yes, all you wrote is totally right and I am convinced that this is all pretty clear to Patrick.

So we have fields, and particles and it is the second quantization that gives us force carriers (viewed at as particles not as waves) and the fermionic matterfields (like the Diracfield being the general solution to the Dirac-equation)that yield the elementary massless particles of the Standard Model

So, first fields then particles.

Well, this is the step that I have been complaining about since the very start of this thread! This field connection is clear in the case of EM. But it's a *huge* leap of faith to star from a *classical* field theory for the electron! That's the step that I have been questioning in this thread.

A second way to look at things is fields that are needed in the canonical quantization of a system with infinite amount of degrees of freedom. Just look at the Euler-Lagrange equations for fields and the way they are built...
regards
marlon

True (and Patrick knows that too). But it's not clear at all (at least to me) why this is the correct way to build in a theory which must account for a varying number of particles! After all there is also a "field" in NRQM, the wavefunction, so an infinite number of df's is also there but it has a very different meaning. The fact that quantizing these df's will necessarily lead to a multiparticle theory is not at all obvious to me although it seems obvious to you and many others.

It took me years to convey my point of view to Patrick, who is very smart and knowledgeable. So I am not surprised if other smart people don't understand what my concerns are right away.

Cheers

Pat

 

[vanesch]

To Marlon: Patrick and I are discussing the classical limit of quantum fields in the sense of coherent states. Do you see what we mean? We can observe easily this limit for the photon by simply shining light on two slits. But this is not so for massive particles.

Well, that's maybe not the best example, because simple photon counting can do the trick. You can also do this with an electron beam, yet it is not a coherent beam.
I was more thinking about a radiowave, and two antennae which are at a certain distance from one another, and you look at the two signals and their phase difference with an oscilloscope.

Another point: the reason why I could think that QCD without quarks is maybe not confined (although of course still asymptotically free, because the number of flavors works in the opposite sense), is the intuitive picture: once the flux tube of two separated color charges becomes long enough, there is enough energy to create a quark-anti-quark pair that neutralises the flux tube in between. But if you haven't got such a quark anti quark pair, I was wondering if this still holds. Of course you could do something similar with gluons of opposite color.

cheers,
Patrick.

 

[nrqed]

I think classical QCD is already dealt with in specialized texts, it is just not physical because of confinement and scales at which quantum effect operate. However, instantons for instance (oops) are classical solutions of the pure glue field.

That's a very interesting point. Yes, classical field theory is used to study important properties of the corresponding quantum fields like instantons. And these are associated to "nonperturbative" results. I have to admit that I never really understood these results. Does that imply that the classical field limit is well-defined? Does that imply that results concerning the classical limit are necessarily nonperturbative in the QFT expansion in Feynman diagrams? Etc.

That's a fascinating issue and I think Patrick will find that an interesting point as well. That would deserve a separate thread!

Pat

 

[humanino]

in another thread about [thread=38964]mass-gap[/thread] you can readily find some infos on instantons. The tunneling amplitude :
$${\cal A} \sim e^{-S} = e^{-\frac{8\pi^2}{g^2} }= e^{-\frac{2\pi}{\alpha_s} }$$ makes it clear that no perturbatively designed calculation can deal with instantons. This is classical.

Oh by the way, you can post in that thread, I would appreciate if it did not disapear so somebody could eventually bring an answer

 

[marlon]

The gauge bosons which do not carry colour charge are not gluons, by definition. And there are 4 of them. SO I am not sure if you are talking about the Standard Model. If not, tell us what are the gauge groups you have in mind and in what representation (fundamental, etc) you are using for all the particles.

Regards

Pat

That is untrue. Where did you get that ?

I am referring to the best model (ofcourse up til now) that would explain the quarkconfinement. The dual abelian Higgs model. It starts from a dual QCD-vacuum and has to incorporate magnetic monopoles. It is very well known together with the glueball-model and widely established among QCD-people.

here is a site explaining the model

http://arxiv.org/PS_cache/hep-ph/pdf/0310/0310102.pdf


And i am using the SU(3)colour-symmetry (what else ?) in the abelian gauge with fundamental quark-representations...

regards
marlon

May I ask, are you a student in the field of QFT ?

 

[humanino]

Originally Posted by nrqed
The gauge bosons which do not carry colour charge are not gluons, by definition. And there are 4 of them. SO I am not sure if you are talking about the Standard Model. If not, tell us what are the gauge groups you have in mind and in what representation (fundamental, etc) you are using for all the particles.

Ooops. Escaped to me
Right, this is very bad. And there are 8 gluons, not 4.

------------
EDIT : excellent paper you refer too. Bah, this guy works partly for CEA Saclay right

 

[marlon]

once the flux tube of two separated color charges becomes long enough, there is enough energy to create a quark-anti-quark pair that neutralises the flux tube in between. But if you haven't got such a quark anti quark pair, I was wondering if this still holds. Of course you could do something similar with gluons of opposite color.

cheers,
Patrick.

What ? that is also not true, Patrick. I agree with the way this pair is created but the fuxtube is always there between two quarks ! And most certainly in the long range QCD (low energies).

It is this fluxtube that describes the interaction between the two-quarks (well, i mean the potenttal along the tube).

The fluxtube is the electrical field bound together by the magnetic monopoles that constitute the dual vacuum. These monopoles undergoe circular motions around the electrical field and thus constraining the field-lines to a tube...

But this fluxtube is always there in a quark-anti-quark-pair. The tube is just shortened into two smaller pieces (two pairs). If there were no quarks present the fuxtube would "decay" into gluons, nothing else.

regards

marlon