Understanding the Motivation Behind Quantizing Fields in Quantum Field Theory

[marlon]

Nor do I ! I have to say that unfortunately, I think that marlon has quite some potential to contribute here, and I regret that his replies are often more of a rather aggressive nature than of an explanatory one, which is unfortunate, because it renders them quite useless as an information source, which is the main reason to post here.
So I hope that he will learn that not everybody is supposed to know exactly what he knows (otherwise there's no reason for him to be here !), that we all would like to learn from it, but also that other people might know things that he doesn't know. Il faut que jeunesse se passe !

cheers,
patrick.

hahaha, tu as raison mon cher Patrick

First of all the reason why I gave the site (which you call euuhh whatever) was as a reply to the statements and questions made by nrqed on quarks. This is called explaining things, not making assumptions as you keep on doing.
The only thing I see you do is "dreaming" about basical and already well established facts concerning fields and QFT. Yet this is not doing science, this is waisting your time.

I have taken the effort to explain my views in several posts here so don't come over with the arguments it is not explanatory just because i don't follow your hollow assumptions. I say hollow (let me EXPLAIN) because i asked to you for several times how you got to these ideas yet I have never received a (polite) answer, just assumptions once again. You only use words like imaginary or just wondering and so on... There is nothing wrong with that, but do please make the effort to explain yourself.

You should take some lessons from nrqed who indeed has taken the effort to explain himself just as i did.

trust me, I will contribute a lot to this thread since it appears to be very interesting, even with the "assuming-nature" of it. I think you would better post your assumtions in the Theory Development forum.


Ah, and i final remark. I get mails from the QFT-forum from you. I must say that some of your solutions to certain exercises are eeuuuhh of speculative nature ??...

regards
marlon

 

[marlon]

I haven't followed much of this thread due to time constraints, but I'd like to reiterate my old point, also found in Weinberg.

The problem with dealing with relativistic quantum mechanics without fields, is precisely the fact that the Smatrix will become ill defined in many body interactions.

Two formalisms remedy the problem, one is Dirac hole theory, the other is QFT. The former has other problems, and was relegated to the historical waste bin.

Again, see chapter 5 of Weinberg..

The idea is the Smatrix will be lorentz invariant if the Hamiltonian is a lorentz scalar, satisfying the usual transformation laws. In order to satisfy cluster decomposition, you need H to be built out of creation and annihilation operators.

Now, under lorentz transformations each of those C and A operators carries momentum matrices, and is perfectly untrivial how to combine them into a lorentz scalar. This is where Weinberg motivates the field concept, and indeed that is suitably general enough to solve the problem. (Incidentally, you could think of even more abstract mathematical objects that would work as well.. but then I think there is a principle of minimality somewhat unspoken here)

There is another little caveat here, and perhaps its simpler to see.. The hamiltonian MUST satisfy a commutation relation.

[H(x), H'(x)] = 0 for (x-x')^2 > 0.. Required for the lorentz invariance of the Smatrix (and indeed this leads to causality in QED). With a little bit of math, you can convince yourself the only way you can construct the interaction density, is by making it out of *linear combinations* of the creation and annihilation operators now as fields (or else that would lead to absurdities)

Hi Haelfix,

nice post you wrote here.


marlon

 

[nrqed]

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.

That's interesting and it would be nice if you would provide more info. I am obviously not a QCD expert. All I am doing right now is some simulations on the lattice using effective field theories of QCD. SO my knowledge is extremely limited. My thesis adviser is an expert on lattice gauge theory and I am sure he knows about that model but somehow he never discussed it with me or mentioned that we should simulate that model. We always talk about simulating the boring usual QCD.

There are several questions I would like to ask (for example: what are the
assumptions behind the model? Is it meant to be a *model* of the usual QCD or is it different, etc) but I am afraid to be simply answered that it's obvious and that everybody knows it and that I would not be asking these questions if I knew even a bit of QFT.

here is a site explaining the model

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

Ok, thanks.

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

I'd like to ask what "abelian gauge" is but that's certainly common knowledge, so I'll try to pick it up from papers.

regards
marlon

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

Well, I got my PhD in that field, yes. But I obviously haven't learned even the fundamentals.

 

[nrqed]

fields give us the possibility to work with infinite degrees a freedom. Why are you always complaining about the number of particles ?

I don't know what you mean. I am starting to wonder if you have read my posts carefully.

I mean ain't you familiar with such things as the Hartree-Fock-states and so on ? Just look how they are constructed and you will see the complete analogy with fields in QFT !

Keep in mind that fields are used out of a certain historical evolution that i was trying to point out in the previous post.

Particles arise as excitations of the fields we are using. Now let me ask YOU a question : can you make any assumptions on the number of particles before you perform the (second) quantization ? And once you performed it, why would you even want to make a difficulty out of these questions.

No, I don't want to assume anything! I don't know where you get this idea.

I am thinking that this whole point you are trying to make does not really relate to QFT, but to your view on fields. Maybe it would be a nice thing for you to look at them the other way around. I mean, starting from a positive view. This is a bit analogous as how we should look at the Higgs-field in my opinion. This is the reason why I ask you these two questions.

regards
marlon

I just hope you are not teaching physics because I think that it woul dbe very discouraging for a student to come ask you conceptual questions. First you would not make any effort to see their point of view and then you would just say that all this is common knowledge and that they are *making difficulties* by asking questions.

Pat

 

[nrqed]

I am starting to read Weinberg and it's great!

just a quote:

Traditionally in qft one begins with such field equations, or with the Lagrangian from which they are derived, an done uses them to derive the expansion of the fields in terms of one-particle annihilation and creation operators. In the approach followed here, we start with the particles and derive the fields according to the dictate of Lorentz invariance, with the field equations arising almost incidentally as a byproduct of this construction.

yahooooooo!

 

[humanino]

OK guys, let us keep a cool head. We are between gentlemen. I believe this is nitpicking here. We all agree basically. Haelfix made a very good post, partly repeating some things, but summurazing well some motivations for the introduction of fields.

nrqed : I am aware sometime people have to explain me twice, because I do not always pay enough attention. I apologize for that. I just wanted to say : to some extent, it applies to all of us.

Maybe we should (re-)read the beginning of the Weinberg, and come back to the discussion after that.

 

[nrqed]

I haven't followed much of this thread due to time constraints, but I'd like to reiterate my old point, also found in Weinberg.

The problem with dealing with relativistic quantum mechanics without fields, is precisely the fact that the Smatrix will become ill defined in many body interactions.

Two formalisms remedy the problem, one is Dirac hole theory, the other is QFT. The former has other problems, and was relegated to the historical waste bin.

Again, see chapter 5 of Weinberg..

The idea is the Smatrix will be lorentz invariant if the Hamiltonian is a lorentz scalar, satisfying the usual transformation laws. In order to satisfy cluster decomposition, you need H to be built out of creation and annihilation operators.

Now, under lorentz transformations each of those C and A operators carries momentum matrices, and is perfectly untrivial how to combine them into a lorentz scalar. This is where Weinberg motivates the field concept, and indeed that is suitably general enough to solve the problem. (Incidentally, you could think of even more abstract mathematical objects that would work as well.. but then I think there is a principle of minimality somewhat unspoken here)

There is another little caveat here, and perhaps its simpler to see.. The hamiltonian MUST satisfy a commutation relation.

[H(x), H'(x)] = 0 for (x-x')^2 > 0.. Required for the lorentz invariance of the Smatrix (and indeed this leads to causality in QED). With a little bit of math, you can convince yourself the only way you can construct the interaction density, is by making it out of *linear combinations* of the creation and annihilation operators now as fields (or else that would lead to absurdities)

That's great Haelfix! I am indeed reading Weinberg now and that's what I neede to see. If you have made this point before and I missed it, I apologize (I recall you bringing up cluster decomposition but I replied that this only imposed the need to write things in terms of C and A operators. I don't recall you getting in as much details to motivate the introduction of fields but again I might have missed some posts).


That makes me much happier. It's the need to combine the C and A operators into Lorentz scalars that is the key point, from my point of view (I had actually kind of guessed that in one of my early posts but it was handwavy).

If QFT books would say: look we need to combine C and A operators in Lorentz scalars and the way to do it is the following way and, by the way, notice that this is the result we would have got by starting with a classical field and treated the amplitude of the modes as C and A operators, then I would be much happier and I would never had started this thread in the first place :-)

Pat

 

[vanesch]

I think you would better post your assumtions in the Theory Development forum.

Well, we could go on and start insulting each other but as that is not a very creative activity, I will also try to EXPLAIN what I was doing.

Pat posted an interesting question, which was: "why do we want to quantize classical fields we've never even heard of?" and after some missings myself, I think I understood finally what he was aiming at. Because I saw others (including you) post answers next to the point I tried to help convey the point Pat was making, and in doing so, I was thinking loud: when walking around in the standard model, and other quantum field theories, what quantum fields would have a hope of showing their classical behaviour. It is true that my "arguments" which was more a thought process, was handwaving, but I do not think they were fundamentally wrong.

I pointed out that a first obstacle in going from a quantum field to a classical field would be the mass of the particle. You attacked that idea, but I think it is right. On the other hand, you contributed an important point, which I bluntly forgot, that is the fermionic nature of fermions, which will also make it difficult to go to a classical field. So if mass is out, and fermions are out, there's not much that remains: there is EM of course, there are the gluons of QCD and that's it. With QCD, I know there is confinement, so that will be a major obstacle to show classical field behaviour. And then I asked the question if confinement was still there if we took away the quarks (meaning: are colored fermions essential to confinement). You waved that one away with a "QCD without quarks is science-fiction". I then tried to explain why I thought that quarks _could_ play a role in confinement, and you told me that I was wrong, again. The ONLY reason I wanted to know that is that if somehow QCD without quarks was not confined, it might lead to a classical field. There's nothing "speculative" in all the above, it was just a written-down way of a process of reasoning with the idea of stimulating readable answers from knowledgeable people.

Nevertheless, the conclusion seems clear (no matter how "wrong" and "speculative" I've been according to you) that apart from EM, no quantum field will ever give rise to a classical field, in any approximation. This is news to me, honestly! I only realized that during this thread. This makes Pat's point much stronger than it was before: why work with classical fields which we will quantize in the first place ?

Now from what I read here, I take it that Weinberg shows that if you postulate a multiparticle theory and you somehow want to incorporate special relativity, that you can always construct a quantum field that is the quantized version of a corresponding classical field as a bookkeeping device. I assume that he does because I only started reading it, but he says something of the kind in his introduction. But that doesn't take away the interpretative issue: are we finally talking about quantized classical fields, or about a multiparticle theory ? And I think it is an interesting issue, which should be made clear (and which isn't made clear at all) in many QFT texts.

So I'm quite happy because I learned something in this thread.

Ah, and i final remark. I get mails from the QFT-forum from you. I must say that some of your solutions to certain exercises are eeuuuhh of speculative nature ??...

This is the kind of useless aggressive behaviour I was advising you against. The thing that triggered my remark (which was not meant to be nasty, btw) was nothing of what you said to me, but the fact that you asked someone here who has a PhD, a postdoc and years of experience in the field if he was a student in QFT. The problem is that if each time someone advances something he doesn't know completely or is worded in a way you do not understand immediately, you start by questioning if he knows how to count to 10, the discussion stops, or turns into a showing off what one knows and how smart one is, which is not productive, because no-one will dare to truly ask questions and as such expose him/herself to your inquisition.

I don't mind anybody pointing out where I make errors. On the contrary in fact, that's the best way to learn. However, the discussion must remain constructive and respectful, and that's what you apparently have difficulties with. Look at your remark above: don't you think it would have been more helpful to explain me nicely where I went wrong in my solutions than to say something of the kind ?
BTW, as the exercises are still conceptually very basic (although instructive and inducing modesty !) I would really be surprised that there is something "speculative" (except for a sign error or so) in my solutions, which means that if you can point that out to me I would learn a lot :-)
If you are not talking about the few times I posted about the exercises, but about the other posts, if you read them well (which is also not an evident skill) you will notice that they are invitations to discussion, with exactly the aim that I stand corrected if what I write is wrong.

Now that I'm preaching, I can just as well continue

I can give you more or less exactly the level at which I know QFT: it is Peskin and Schroeder, except for the last few chapters which I studied less thoroughly. This is however a few years ago so I don't have everything ready to be exploited and may have forgotten things. I also know some superficial phenomenology (style monte-carlo generators used in experimental particle physics). This means that I lack a lot of deep knowledge, and it is my pleasure to try to fill in these holes at a leasurely pace. But it also means that I should know enough of it so that it is possible to explain me certain aspects of QFT. If I cannot make sense at all of an explanation I take it therefore that the explanation has a problem, not me.

cheers,
Patrick.

 

[Haelfix]

I am not an expert, but I think condensed matter has quantum fields that have well behaved classical limits.

Anyway, its true there are many examples (the Dirac field immediately springs to mind) that are intrinsically quantum with no suitable classical limit.

 

[marlon]

The thing that triggered my remark (which was not meant to be nasty, btw) was nothing of what you said to me, but the fact that you asked someone here who has a PhD, a postdoc and years of experience in the field if he was a student in QFT. Patrick.

Hallo Patrick,

You are right, let's just continue on the QFT and set our euuhh conflicts behind us. I did not mean to insult you or anybody else and if I gave you that impression I apologize...

I did read your posts thoroughly in order to try to understand what you are trying to say. I did not mean to insult the other person by asking the above question. I was really wondering about that because of the questions. I Find it difficult to believe that someone with a PhD in the field would make such remarks as "i don't know the abelian higgs model and all i am doing are QCD lattice simulations "

But obviously i made the mistake here and I apologize

regards
marlon

 

[vanesch]

I Find it difficult to believe that someone with a PhD in the field would make such remarks as "i don't know the abelian higgs model and all i am doing are QCD lattice simulations "

But obviously i made the mistake here and I apologize

Ok, and I apologize to you and all the others if I'm too picky ! I'm here essentially to learn, you know. In fact, most of what I've learned of QFT I did it on my own because I had a very bad professor in QFT in that he didn't believe in QFT and hence refused to teach it, instead he worked on "multi particle dirac equations" or something of the kind. (the guy is gone now and replaced with a hotshot string theorist) I never understood his course very well ; it was one of the reasons I went into experimental physics instead of theory, which interests me more in fact.

But you'll see that you will meet many people who have quite different backgrounds, and it is not because they don't know exactly what *you* know very well, that they are ignorant of a whole field! There are lots of specialisations all over the place.
So let's shake hands and forget about it, ok ?

cheers,
patrick.

 

[nrqed]

...I Find it difficult to believe that someone with a PhD in the field would make such remarks as "i don't know the abelian higgs model and all i am doing are QCD lattice simulations " ...


marlon

To be honest, my PhD was not in QCD but in high precision QED calculations. I collaborated on a two loops calculation of a certain subset of diagrams contributing to the hfs and decay rate of positronium. It's only lately that I decided I would like to do lattice gauge theory stuff and went back to see my adviser and got involved in a project. Maybe that explains better my deep ignorance concerning QCD and all the models used in that field (because, as I understand, you are discussing a *model* of QCD). By the way, do you know NRQCD or NRQED of potential QCD? These are now the standard in computing nonrelativistic bound state properties (which, for QCD, is applicable to heavy quark systems). These are not models, but effective field theories. If I were rude I could telle everybody that is not familiar with the details of these theories that they don't know QFT. But that would be rude.

Regards

Pat

 

[marlon]

So let's shake hands and forget about it, ok ?

cheers,
patrick.

You from Belgium right ? Me too...

We shouldn't be fighting because so few of us...

I shake your hand and apologize again to you.
I admit i came on a bit too strong , my fault

regards
en België boven
marlon

 

[marlon]

By the way, do you know NRQCD or NRQED of potential QCD? These are now the standard in computing nonrelativistic bound state properties (which, for QCD, is applicable to heavy quark systems).

Regards

Pat

I have heard of these, but i don't know them very well , i admit that

you see, i am not always rude

could you provide me with some more info on this matter, i would like to see and learn.

regards
marlon, let's smoke the peace-pipe (is it ok to say that in english like that?)

 

[nrqed]

I have heard of these, but i don't know them very well , i admit that

you see, i am not always rude

could you provide me with some more info on this matter, i would like to see and learn.

regards
marlon, let's smoke the peace-pipe (is it ok to say that in english like that?)

No, it does not sound right in English. But I am French speaking so I know what you mean.

And sure, let's go back to trying to learn more.

Cheers,

Pat

 

[nrqed]

Ok, and I apologize to you and all the others if I'm too picky ! I'm here essentially to learn, you know. In fact, most of what I've learned of QFT I did it on my own because I had a very bad professor in QFT in that he didn't believe in QFT and hence refused to teach it, instead he worked on "multi particle dirac equations" or something of the kind. (the guy is gone now and replaced with a hotshot string theorist)

I am wondering if it's because he kept asking himself : "By why do we quantize classical fields?"

hehehe...

I never understood his course very well ; it was one of the reasons I went into experimental physics instead of theory, which interests me more in fact.

cheers,
patrick.

That's a shame, really. You sound like someone with all the right assets to make a very good theorist.

Pat

 

[nrqed]

Weinberg's preface

I am just starting to have some time to look at Weinberg and I am ecstatic

Just the preface already makes me happy

The traditional approach...has been to take the existence of fields for granted, relying for justification on our experience with electromagnetism and "quantize them"...This is certainly a way of getting rapidly into the subject, but it seems to me that it leaves the reflective reader with too many unanswered questions...why should we adopt the simple field equations and Lagrangians that are found in the literature? For that matter WHY HAVE FIELDS AT ALL?

(emphasis mine!)



I could not believe my own eyes, to tell you the truth. And all those years I thought that I was kinda dumb for wondering this myself!

Thanks guys for pointing this reference to me. As I have said before, I should have read it but I had read so many QFT books before Wienberg's was published that basically always repeated the same things that I had figured it would still be the same old stuff. This was counting without Weinberg's deep ingeniosity and originality (he does everything his own special way). Of course now I will buy it, even if it's just to thank him for writing it !

He later writes

The point of view of this book is that quantum field theory is the way it is because...it is the only way to reconcile the principles of quantum mechanics...with those of special relativity.

And from the little I have read so far, the concepts of fields, the field equations and the Lagrangians all *follow* from the above principles instead of being taken for granted from the start. In particular fields arise out of imposing Lorentz invariance on combination of creation/annihilation operators, as pointed out by Haelfix and as I have seen in chap 5. That's much more satisfying to me.


Often people say "of course you must quantize fields because you need an infinite number of degrees of freedom". That has never made much sense to me. I can write a bunch of creation/annihilation operators and introduce as many degrees of freedom as I want without ever introducing fields. Now I see that it's not a question of degrees of freedom, it's a question of Lorentz invariance.


Haelfix has written, talking about Weinberg's presentation:

Now, under lorentz transformations each of those C and A operators carries momentum matrices, and is perfectly untrivial how to combine them into a lorentz scalar. This is where Weinberg motivates the field concept, and indeed that is suitably general enough to solve the problem. (Incidentally, you could think of even more abstract mathematical objects that would work as well.. but then I think there is a principle of minimality somewhat unspoken here)

Indeed. Now I will try to really understand how this works (when teaching will allow me to take some time off this week). If it's highly nontrivial as Haelfix suggests, then this would mean that it's also highly untrivial to simply say that "quantizing classical field is the obvious way to go", a statement that most people make (explicitly, or implicitly when they go when I ask about the meaning of quantizing classical fields).

On the other hand, if combining the creation/annihilation operators in quantities with specific Lorentz properties leading to fields is something straightforward (even though this seems unlikely given Haelfix' comments), then the obvious question will be: why don't books (and people) explain this more often ?


Anyway, I'll try to understand Weinberg's presentation in depth and I would like to summarize my understanding to you guys/gals in order to get your comments/criticism/feedback.

Pat

 

[humanino]

I feel Weinberg's presentation is the best too. Yet, it is so peculiar. One of my teacher told me it is a bad idea to read it as a first text to QFT, because it is really Weinberg's point of view. For instance, the canonical formalism is delayed to chapter 7 or so.

My opinion is that : Weinberg is one of the main contributor to QFT, and he thought in depth what would be the best presentation. Besides, the mathematical level of rigor is, if not totally satisfactory for a mathematician, quite above usual texts. I discovered QFT through these book, and I am glad.

I would like to point that a third volume has been issued, on supersymmetry, which is not well-known.

 

[vanesch]

Anyway, I'll try to understand Weinberg's presentation in depth and I would like to summarize my understanding to you guys/gals in order to get your comments/criticism/feedback.

Now THAT's a great idea ! (call it a study group :-)))))

Finally.

cheers,
Patrick.

 

[vanesch]

I discovered QFT through these book, and I am glad.

I tried and didn't manage, but that's now several years ago. I got upto page 130 or so, and then I drowned: too many new ideas at once. You have to be quite a clever guy to be able to absorb all that material from scratch! I think I'll give it a second try, if you guys can provide some coaching :-)

cheers,
patrick.

 

[nrqed]

Humanino wrote

I feel Weinberg's presentation is the best too. Yet, it is so peculiar. One of my teacher told me it is a bad idea to read it as a first text to QFT, because it is really Weinberg's point of view. For instance, the canonical formalism is delayed to chapter 7 or so.

My opinion is that : Weinberg is one of the main contributor to QFT, and he thought in depth what would be the best presentation. Besides, the mathematical level of rigor is, if not totally satisfactory for a mathematician, quite above usual texts. I discovered QFT through these book, and I am glad.

I would like to point that a third volume has been issued, on supersymmetry, which is not well-known.

and Patrick wrote

I tried and didn't manage, but that's now several years ago. I got upto page 130 or so, and then I drowned: too many new ideas at once. You have to be quite a clever guy to be able to absorb all that material from scratch! I think I'll give it a second try, if you guys can provide some coaching :-)

cheers,
patrick.

I had only bought and looked at volume II (especially because I wanted to see his presentation of effective field theories) but I also quickly found that it was too dense to my liking. I know about the SUSY volume but I just think that, given my total lack of understanding o fthe subject, a more basic book would be more useful.


And now that I look at volume I, I realize that I would probably never have been able to use it as my first introduction to QFT. Too dense. But now that I have matured a little bit, absorbed the basic ideas and notation, it's a delight to read this book because it does not "hide" anything or force the reader to accept wild claims passed as obvious

So, it seems to me, his books are good for someone who has matured a bit and has already pass through the basic concepts using more informal and "digestable" source (but less complete and satisfying, for sure).

So I still think that Peskin and Schroeder is still the best starting point (or, at a lower level, Aitchison and Hey for QFT and Griffiths for an intro to Particle Physics). My problem of course was always the same: I would get stuck on the very starting point . If only the books would have said something to the effect that "we know this sounds strange, a more in depth treatment would show that blabblabla", I would have been much happier and willing to set it aside and to keep going. But the starting point always remained clouded in mystery to me so I was never able to really learn QFT. I could *use* it, but not *understand* it.

Regards

Pat

 

[humanino]

No I am not clever. I am the worse experimental physicist ever. Yet I like math, and the maths involved in Weinberg's books are not really high-level, except for a few parts. Besides, I did not absorb it. I followed it, but probably forgot more than half of it.

 

[nrqed]

Nevertheless, the conclusion seems clear ...that apart from EM, no quantum field will ever give rise to a classical field, in any approximation. This is news to me, honestly! I only realized that during this thread. This makes Pat's point much stronger than it was before: why work with classical fields which we will quantize in the first place ?

Now from what I read here, I take it that Weinberg shows that if you postulate a multiparticle theory and you somehow want to incorporate special relativity, that you can always construct a quantum field that is the quantized version of a corresponding classical field as a bookkeeping device. I assume that he does because I only started reading it, but he says something of the kind in his introduction. But that doesn't take away the interpretative issue: are we finally talking about quantized classical fields, or about a multiparticle theory ? And I think it is an interesting issue, which should be made clear (and which isn't made clear at all) in many QFT texts.

So I'm quite happy because I learned something in this thread.

...
cheers,
Patrick.

Now that I am plunging myself in Weinberg, my thinking is slowly evolving on this issue. And, as always, this brings in more questions than answers. I hope to get some feedback from people around here because I think this is all getting very interesting.



FIRST POINT : After looking at Weinberg in more details, here is the way I now think about fields in the canonical quantization approach. Starting from the need for a multiparticle theory, fields are relegated to a very secondary role. The need for *quantum* fields just pops up as a need to regroup creation/annihilation operators in combinations that have certain properties under Lorentz transformation (scalar, vector ,etc). So we go *directly* to *quantum* fields. Classical fields never appear anywhere at all. They're no needed in any way, not even to motivate a field equation or anything else for that matter. So exit the classical fields!

However, of course, once one has gone through all these long discussions on transformation properties of creation/annihilation opereators, etc etc, one may realize that a very useful *formal shortcut* would be to start with theories of classical fields and to impose ETCR on them. But that's all there is to it: it's a formal trick. The real motivation is all the stuff Weinberg goes through. It's just a shame that textbooks don't present things this way. That would have saved me countless hours of scratching my head.

POINT 2: So I was happy...for about one day . Then I started scrathcing my head again . Could there still be some meaning to these classical fields...

Then I started thinking aboutPI quantization. There, the fields are indeed classical, in a certain sense. No mention of creation/annihilation operators. Just fluctuations around "classical configurations" obeying the equations of motion (well, if one is willing to consider "classical" Grassmann numbers ). So we are back to fields.

But since the PI is equivalent to covariant quantization, maybe we should still see those classical fields as still as "formal" as in covariant quantization. Except that it's more difficult to see it now.

POINT 3:

*Except* that these classical fields *are* used to do important physics! An example was pointed out by Humanino: to obtain instanton configurations. Instanton configurations must be incorporated in the PI in order to resolve some anomaly issues, for example. The neat thing is these types of contributions are nonperturbative. So treating seriously the classical field leads to highly nontrivial physics.

So what is the meaning of these classical fields and why do they contain so nontrivial information?

First of all, I don't think they should be treated as "classical fields" in the sense of "observable classically" (in the sense that the EM field is observable classically). There is actually a recent thread on sci.physics.research in which people were arguing about classical fields and they realized that one was using "classical field" in the above sense whereas the other was using it in the sense of a stationary phase solution of the PI. It's in this second sense that "classical" fields are used in nonperturbative calculations. Maybe the equivalence is more obvious than I realize in which case I would like to hear about it.

In any case, focusing on the "stationary phase definition", it's still a bit of a mystery to me why they carry so much information (even nonperturbative information). I guess that those field configurations are related to the vev's of the corresponding quantum fields. So they are giving information about around which vacuum we are expanding. So the classical eom can be used to gather information about the vacuum structure of the theory which is nonperturbative information.



Another, completely different, issue is the one of "graviton picture" vs "classical metric obeying a differential equation" picture of GR. (Btw, I have started a few threads over the last few years asking what people means exactly when they say that a coherent state of gravitons can be used to "obtain" a curvature of spacetime in the usual, classical sense. I am still confused by the usual presentations in string theory textbooks). This is an example where one can discuss the field picture from the classical point of view, from the covariant quantization point of view (using coherent states, I've heard) and from the PI point of view (the extremization of the action yields the usual Einstein eqs). So this is a case where it sounds as if the classical field obtained as a coherent state in the operator formalism coincides with the classical field obtained from the stationary phase approach in the PI formalism which itself agrees with the "usual" classical field. So that's a case similar to E&M.

To summarize,
What do fields represent? Why do they carry nonperturbative information? In what case is the "coherent state" picture of a classical field equivalent to the "classical field" obtained from imposing the stationary phase condition?


So back to trying to understand fields




Pat

 

[marlon]

Here is a thought...

The distinction between classical fields and QM-fields is there because of difference in the action of the phenomena described by these fields.

Classical fields (like temerature fields and so on) all have the concept of "action at a distance". This means that the interaction is NOT localized. For example if you put a stove next to ice, it will melt because of the generated heat. Now ofcourse the distance between the to has an influence on how the melting evolves, so the INTERACTION itself depends on the distance between the two...

The interaction between an EM (which fills up the entire room is continuous so a field...) and a charged particle only occurs at the specific position of the charge and only there...The "distance" between the electrical charhe an the EM-field has no meaning so this is the reason the EM-field is not classical.

Ofcourse will still have the fundamental fields describing the interactions (the EM is even not just a QM-field, it is a fundamental field.) The fundamental fields are the third kind of fields next to classical and QM-fields, in my opinion.
Temperature is an inherent property of the temperature field that you can measure, this should also be a definition of a classical field. You cannot measure gluons directly like you would measure pressure. The quantization of these fields is a necessary because of the need of a particle-like interpretation of such fields. Just look at the photo-electrical-effect that needs a particle-interpretation of photons. All fields that describe interactions which donnot follow the non-locality of the action at a distance are not classical and must be quantized in order to fulfill the needs of the "theory" used to describe experimental observations.

Basically fields are historically an extension of the physics of discrete objects that undergo interactions caracterized by the "action at a distance"


Just a thought, what do you think...

regards
marlon

 

[humanino]

Relevant remark Marlon : Weinberg might have a point about quantum fields, yet the concept of field in classical physics is much broader, and valid.

 

[nrqed]

Here is a thought...

The distinction between classical fields and QM-fields is there because of difference in the action of the phenomena described by these fields.

Classical fields (like temerature fields and so on) all have the concept of "action at a distance". This means that the interaction is NOT localized.

[snip]
regards
marlon

I am very confused by that statement. Are you saying that all classical field theories are non-local (and therefore violate SR)?!

Classical E&M is a local, Lorentz invariant classical field theory! No action at a distance here, no instantaneous transmission of forces, etc! General relativity is also a classical field theory which obviously respects SR!

What am I missing here??!

Pat

 

[marlon]

I am trying to say that in the case of the EM-field the interaction between the field and the charges occurs ONLY at the position of the charges. This is a difference with classical fields.

The EM-field is NOT a classical field.

Also in General Relativity, the curvature only occurs at the position of the massive objects.

Perhaps i was not clear enough but i wanted to say that this non-locality should be the main criterium in deciding whether a theory is classical or not, otherwise there is to much discussion possible on what is what...

regards
marlon

 

[marlon]

Another difference between classical fields and QM-fields is that the latter cannot be measured directly...classical fields can...

marlon

 

[marlon]

And consider this :


We cannot deny that everything at the atomic scale seems to follow the rules of QM. Lots of experiments (like the photo-electric-effect) back this up.

It was this consideration that lead to the believe that the "field discovered by Maxwell" is not a classical one but a QM-one.

Unlike temperature and pressure-fields in air, the electromagnetic field does not arise from the many atoms that constitute the air (this is the classical picture)

The QM-field description, once interpreted correctly, seems to be the whole story. It plays the role of the "air" from the classical picture. The quantization is necessary for experimental reasons.

QM needs fields (they provide us with the best description) and especially quantized fields...and since QM rules the atomic-scaled-events, I think the use and reason they exist is very clear and must therefor be widely accepted...
Just my opinion though


regards
marlon