Why quantizing fields? [Archive]

nrqed

Sep8-04, 04:09 PM

Hi everyone.

I have been using quantum field theory for a long time and yet there is a basic question that has always been bugging me. When I was a student I thought that the answer would become clear when I would understand more the subject but now, several years later, I still can't find an answer that I find really satisfactory. I have looked at almost all QFT books that are out there and I never quite found what I was looking for (the only one that I haven't read and which may be more helpful is the field Quantization by Greiner). Unfortunately, I now work in a small town where I can't find other particle physicists to discuss with, so I hope some people here will be willing to exchange their point of view.

My question concerns the reasoning behind "second quantization", i.e. the idea of quantizing classical fields in order to get quantum theories that are relativistically sound.

In a nutshell, I'm looking for an explanation that would be satisfying to someone starting in the field, with only a good background in QM. Also, I am not looking for equations or derivations. I have some background in QFT and I have seen all the standard derivations. What I am more interested is a conceptual explanation. Or if there is a step that is a completely wild guess, then I would like it to be made clear. Also, I know that the idea of quantizing fields is rooted into the quantization of E&M so there is an historical motivation for quantizing classical fields. But what I want is to see if there is a way to motivate this approach without referring to this fact.

The typical textbook will start directly with the quantization of fields, with little motivation . Some books work a bit harder in order to provide some motivation. Often books will introduce the Dirac equation as a mean to avoid negative probability densities of the KG equation, and then they show that there are still problems in the Dirac theory because it's not possible to decouple "negative energies" states completely (I am thinking about the Klein paradox, for example). But eventually, they still go back to the need for quantizing fields.

Now, I understand of course the idea that when energies become important, particles may be created so that the number of particles is not conserved and we need a theory which allows for a varying number of particles and we need a many body theory, and that leads to the need of many degrees of freedom and yaddi yaddi yadda. Then they talk about the normal modes of a field and Voila! that's the reason you need to quantize a field.

That leaves me dissatisfied. I am not sure whether the step from needing a varying number of particles to the step of quantizing a classical field is trivial (in the sense that it's the only thing to do) or whether it's profound! I usually think it's trivial but I can't quite convince myself.

My point is this. Let's say you wanted a many body theory. You could think of many particles (of the same mass) each obeying the KG equation (let's say). Now you could use the occupation number representation to represent states with different number of particles. Now you would be led in a natural way to introduce operators that change the occupation numbers. Those are the usual creation/annihilation operators. Then you would want to obtain their commutation relations, find the energy of an arbitrary state, add interactions, etc etc. There is no mention of field so far.

Now, maybe we could simply write an arbitrary linear combination of creation/ annihilation operators times their corresponding (free) wavefunctions in the usual form ( \int {d^3k \over (2 \pi)^3 2 E} a e^{-i k \cdot x} + \ldots ) and we could call this a "field" but there is no real motivation to do this at this point, it seems to me.

On the other hand, the traditional presentation is to start with a classical field and to quantize it and then to interpret the normal modes as "particles".

I don't see why this is a natural thing to do. When we quantize a classical field, we write the field as an expansion over its normal modes, each of different energy. And now we treat as operators the *amplitudes* of those modes. Later we discover that these operators have an interpretation as creation/annihilation operators so that the amplitude of these abstract quantum fields is related to the number of particles in each mode.

So that's my question: the first approach sounds natural to me (whereby one goes to an occupation number representation, one introduces operators that change the number of particles, etc etc) and the second sounds ad hoc to me (quantizing a classical field). I can see the similarities (the normal modes of the classical fields are infinite in number and have different energies) and I can work out the math but I am not sure I see why the two approaches are equivalent. As I said above, I guess that the equivalence is trivial but I don't see it. It's especially the step about turning the amplitudes of the normal modes of the classical field into operators that I don't find natural.

I know that my question is fairly vague and I do apologize for this. I hope some will share their opinion.

humanino

Sep8-04, 04:37 PM

The discussion can be found for instance in Peskin & Schroeder's book. An internet link entitled Why fields (http://www.ph.utexas.edu/~gleeson/httb/section1_3_7_1.html).

There are several reasons : check this thread (http://www.physicsforums.com/showthread.php?t=41728) where it is explained why, in trying to use ordinary QM and imposing as in SR same treatement of time as for the space variables (wheras in QM time is not an operator), one gets an energy spectrum which must be both continuous and unbounded from below.

Of course, you also want to quantize classical fields. Especially for locality. It seems rather difficult to respect locality and causality without fields.

Other reason : creation and annihilation of particles : in QM, the number of particles is fixed.

Yet another : statistics. Alas, you will have to check the Peskin & Schroeder for this one, because I don't remember the argument.

I hope this provides a beginning of answer... :uhh:
I don't doubt better will come soon.

Haelfix

Sep8-04, 05:38 PM

Rothiemurchus

Sep8-04, 05:38 PM

As far as the gravitational field is concerned it has been mentioned on sci.physics.research that nobody even knows for sure what it means to quantize gravity!

nrqed

Sep8-04, 05:43 PM

The discussion can be found for instance in Peskin & Schroeder's book. An internet link entitled Why fields (http://www.ph.utexas.edu/~gleeson/httb/section1_3_7_1.html).

Thanks for your reply. I have P&S but that does not really address my questions. For example, on the first 2 pages of chap 2, all they say is that one needs a multiparticle theory because the number of particles is not conserved, on one hand, and causality requires antiparticles, on the other hand. But all this is saying is that one needs a multiparticle theory. It does not say a word about why quantizing a classical field is the right way to do it. They say: "we need a multiparticle theory. Let's quantize fields." That's exactly what leaves me dissatisfied.

There are several reasons : check this thread (http://www.physicsforums.com/showthread.php?t=41728) where it is explained why, in trying to use ordinary QM and imposing as in SR same treatement of time as for the space variables (wheras in QM time is not an operator), one gets an energy spectrum which must be both continuous and unbounded from below.

Ok Thanks, I'll look at it. You did not mention fields, so I'll see if the thread explains why one must quantize fields.

Of course, you also want to quantize classical fields. Especially for locality. It seems rather difficult to respect locality and causality without fields.

That's an interesting direction, and maybe that's closer to what would convince me. But then the question is: why is it difficult to build in causality and locality using a bunch of separate annihilation/creation operators without referring to a field? Maybe explaining this would help me appreciate more the field approach.

Other reason : creation and annihilation of particles : in QM, the number of particles is fixed.

Yet another : statistics. Alas, you will have to check the Peskin & Schroeder for this one, because I don't remember the argument.

Again, if you remember my post, I talk about defining a bunch od annihilation/creation operators acting on states in occupation number representation. There is no need to introduce a field in order to get annihilation/creation operators, whether they commute or anticommute.

I have the gut feeling that at a deep level, there is no need to quantize fields, one could build in all the results just working the way I described. My gut feeling is that the field approach just makes things simpler. But I have never seen any book saying this clearly.

I guess, what I am asking is: is the field approach just a convenient way to to incorporate locality, micro-causality, unitarity, etc? or is there something deeper to it (I am pretty sure this answer is that this all there is to it).

Then, why not simply defined creation/annihilation operators and impose locality, etc etc. Is there a step where things would be very difficult to do properly or it would be as easy?

But, the most important question is this: let's say you knwe only about quantum mechanics and SR and you knew about the need for a multiparticle theory, causality, etc etc. But you had never heard of quantum field theory (and you had never heard of the quantization of E&M). My question is: would you ever think about quantizing a classical field?? If yes, what would be your reasoning??

That's the question I would really like to see answered. If somebody can answer this and explain his/her rationale, then it would clear things up. I have to admit that, personally, I would never have thought about this approach on my own, because I, obviously, don't understand it at a deep level. I had to accept the starting point (the idea of quantizing classicla fields) and then I can work out the steps and see that it works *a posteriori*. But I don't really understand the first step so I have to say that I don't understand QFT.

I hope this provides a beginning of answer... :uhh:
I don't doubt better will come soon.

Thanks a lot for the input. :smile: I hope I made my concerns more clear.

Best regards,

Pat

humanino

Sep8-04, 05:52 PM

humanino

Sep8-04, 05:55 PM

As far as the gravitational field is concerned it has been mentioned on sci.physics.research that nobody even knows for sure what it means to quantize gravity!
You are right. But that would be the next step !
Besides, some respectable scientists like Carlo Rovelli claim that they are very near success, without making any new assumption such as extended objects (strings) or new (super)symmetry (supergravity).

nrqed

Sep8-04, 07:03 PM

I'd like to point out one thing that may be helpful. I too found the historical progression slightly bizarre when I studied this.. There are of course various things that are typically left out of textbooks that make it seem more natural when studied in depth.

However, I'd like to point out all of this is secondary in the modern framework.

relativistic quantum mechanics really requires two things, that are purely physical and pretty much *force* you into a field theoretic description. At this point you see second quantization is a completely natural description, and will leave you with the desirable results.

1) The cluster decomposition principle.

If you choose to use products of sums of creation and annihilation operators, the S Matrix will automatically satisfy this requirement, namely that distant experiments provide uncorrelated results. This is related to what was said above, namely the idea of locality. The problem with making a theory nonlocal, is that for n>2 n-body interactions, the Lippmann-Schwinger equations will contain anonomalous products of delta functions, that will either violate Lorentz invariance or violate cluster decomposition. You can play around with those equations, but in general this has not been very successful

2) Unitarity of the operators

This is almost for free when you work in this formalism, and removes the historical problem of negative energy states when combined with a careful second quantization description

Thanks a lot for your input. That's stimulating and it's heading in the right direction but I still have a few questions (btw, thanks for the reference, I did not think about checking Weinberg. That's an obvious reference to consider given how thorough he is. Unfortunately, I don't have volume 1, just volume 2! And no library has his books in the area....I'll have to wait until I drive to Montreal to check it out ).

I understand your points, but it does not get me to fields yet. I agree that cluster decomposition necessitate using products of a, a^dagger. But that's what I was saying in my first post: I would construct my theory (let's say the action) using those operators in the first place so that would be satisfied without involving fields.

As for unitarity, it seems to me that all I have to do is to ensure that all terms in the action are real. Again, that seems as easy to do with my operators as with fields.

So I still don't see the need for fields....

Anyway, coupled with lorentz invariance, and a bit of math one can clearly see these 2 requirements need a field theoretic framework, as outlined for instance in Weinberg chapter 5. The idea is basically that coupling operators in such a way so as to make a desired lorentz scalar, requires the hamiltonian to be made from fields.

Ok, the crux of the argument is probably there (I can't wait to put my hands on the book). The argument must boil down to something fairly simple...is it possible to see at what point he must introduce a field?

Now that I am typing, I can maybe see.....If one constructs the lagrangian density, it must be made of things that are function of x,t. But the annihilation/creation operators act in momentum space. So I must write some type of Fourier transform and write down the terms in my Lagrangian in terms of this Fourier transformed quantity. And this quantity would look like an integral of d^3p a_p e^{i p \cdot x} \ldots . And that what we usually write as our quantum fields...

Is that it? This is the step where I would go (in my approach) from my creation/annihilation operators to a linear combination of them and I would end up with what people get when they start from classical fields and write them as sum over modes and promote the amplitudes to operators....

Of course, after that I still have to impose lorentz invariance and so on but that's basically the same as the usual approach once I have introduced the above "fields".

There's still a detail bugging me... But I think that must be the gist of it.

Does the above sound right?

Thanks again

Pat

nrqed

Sep8-04, 07:10 PM

ok, the other thread will only tell you that you cannot easily make time an operator. The next argument is then : if quantisizing position and time does not work directely, what else could I quantize, except functions of position and time ! How could I impose Lorentz invariance without having something related to position and time ? (not even considering causality or locality, or unitarity, or continuity ...)

Ah, ok, I see what you are saying. That adds a very interesting spin to the discussion. I have to think about this.

I ended up partly answering my question by going in a quite different direction (see my other post) and that's why I haven't had the time to read the thread you mentioned but I will for sure.

So I guess, yes, after having the argument of the previously mentioned thread, I would be lead to quantize field. Obviously I think I would fail, but that is a motivation :wink:

EDIT : thank to you for opening this great thread.

Well, thanks a lot for all the input. It's very stimulating!

Best regards,

Pat

Doctordick

Sep8-04, 10:27 PM

Hi Pat,

I have just read your thread and get the distinct impression that you are worrying over the same issue that bothered me forty years ago when I was a graduate student. In my humble opinions, fields are a way of displaying data, not a fundamental characteristic of reality. In many cases, the approach is very valuable but those who try to explain everything from the perspective that "it is all fields" are just not facing reality.
Maybe explaining this would help me appreciate more the field approach. [and later] So I still don't see the need for fields.... Now I am clearly not the one to give you reasons for the "field approach" but I think you sound like someone open to an approach which does not involve "fields" except as a final representation of solutions (in the cases where they are valid).
ok, the other thread will only tell you that you cannot easily make time an operator.Again, in my opinion, that is exactly the center of the problem confronting the "field theorists" and is, in fact, a central problem of modern physics. I would suggest you take a look at some of my writings: check out my paper at

http://home.jam.rr.com/dicksfiles/Explain/Explain.htm

If you can follow that presentation, go to chapter II of my book which you will find at

http://home.jam.rr.com/dicksfiles/reality/CHAP_II.htm
Well, thanks a lot for all the input. It's very stimulating!I hope that my stuff doesn't "over stimulate" you. I do not know what happens but I seem to run people off; to date, no scientifically competent person has ever made any attempt to analyze what I have done.

Have fun -- Dick

vanesch

Sep9-04, 05:45 AM

I would construct my theory (let's say the action) using those operators in the first place so that would be satisfied without involving fields.

Ah, NOW I'm understanding some discussions we've had in the good old days which seemed incomprehensible to me :-))
Let me summarize:
You take as essential entities "particles" which are described by creation and annihilation operators. You see "quantum fields" as a kind of bookkeeping device.

Traditional QFT takes as essential entities "fields" and we apply quantum theory to their classical configuration space. The resulting "lumpiness" in the form of particles is simply a consequence. I have to say I personally always had the last view without ever understanding your point of view.

As hinted here, I suppose that both formalisms will turn out to be equivalent, if you add the necessary assumptions. If this is the case, it is just a matter of interpretation, whatever you like best, or whichever picture allows more easily to go to the next step (whatever that may be).

cheers,
Patrick.

humanino

Sep9-04, 09:14 AM

You take as essential entities "particles" which are described by creation and annihilation operators. You see "quantum fields" as a kind of bookkeeping device.

In the light of this, it appears that Pat (nrqed) should really take a look at Weinberg's first volume (even maybe buy it :wink: ), which presents QFT from exactly this point of view.

Rothiemurchus

Sep9-04, 11:06 AM

I don't like the idea of virtual particles and the idea of "off-shell and on-shell"
bosons.I think physics would be a lot more understandable and representative of reality if it dispensed with imaginary numbers and terminology like:
"virtual particles are only an aid to calculation." Why can't force mediators be real
like EM waves - how can quantum field theory represent reality when it is qualitatively different to theories that we know work in the classical world which do not use particles that are believed to be only aids in calculation?If you took the uncertainty principle out of physics then a lot of things would have to change such as paricles being allowed to "borrow" energy for a small period of time.

vanesch

Sep9-04, 12:43 PM

I think physics would be a lot more understandable and representative of reality if it dispensed with imaginary numbers

Well, I think physics would even be more understandable if it dispensed itself with all quantities except for the 3 first natural numbers (1, 2 and 3) :biggrin:

Ok, this is not nice, but it is a logical application to an extreme of the reasoning you present here.

cheers,
patrick.

nrqed

Sep9-04, 03:22 PM

Ah, NOW I'm understanding some discussions we've had in the good old days which seemed incomprehensible to me :-))

:tongue2: :smile: :biggrin:

hehehe... And all those years (ok, maybe months) you thought that I was ready for a straight jacket :biggrin:

I'm kidding. I have enjoyed all the discussions we've had and I have learned much from them. I probably had not made myself too clear when we discussed this.

I recall asking similar questions when I was a student and it looked to me as if the idea of quantizing classical fields seemed pretty obvious to everybody I would talk to. To the point that they did not even seem to understand why I was bothered. So I decided that there was something very obvious that I was missing entirely and that one day I would finally get it. But I never did.

Just as an example: consider the usual way one imposes equal time commutation relations on fields. One makes an analogy between the position and momentum of a point particle and the field and momentum density of classical fields. That step alway makes me want to go :surprised :cry:

After all, the field \phi (let's say) has nothing to do with a position! Unless one pushes too far the analogy with a vibrating rope, one must admit that there is no relation between the fields we use (even at the classical level!!) and a direction in space. Likewise, the "momentum density" we derive from our actions have nothing to do (again, even at classical level!) with an actual momentum, even at a superficial level! So why on earth do we use the QM commutation relations between position and momentum to impose commutation relations on \phi and \pi ????? Most books use the rope analogy, where the displacement field i san actual position, to suggest that this is th eright way to do. But that's unfair, I feel. Even in the KG case, it's hard to make it believable to think of the field as a "position"!!

From my point of view, the fact that this ultimately work in the "quantize a classical field approach" is a mystery. It does ultimately give the correct answer but I would find it hard to convince students that this is a sensible thing to try. Sounds like a wild guess to me!!

Whereas in my approach, I would get the commutation relations from simple considerations of states in the number representation picture. Then my "fields" built out of those operators would automatically inherit the correct commutation relations.

You take as essential entities "particles" which are described by creation and annihilation operators. You see "quantum fields" as a kind of bookkeeping device.

Traditional QFT takes as essential entities "fields" and we apply quantum theory to their classical configuration space. The resulting "lumpiness" in the form of particles is simply a consequence. I have to say I personally always had the last view without ever understanding your point of view.

That's right. But you see, I am still not sure I even see why I need fields *even* as a book keeping device! I need to see where it would enter in my approach and why it would be needed!

My bet is that when I realy understand this, I will say "AAHHHH, that's all there is to it!?!?!". But I also bet that I will never find the field approach natural. I will probably see why it's equivalent to the "particle approach" but just "a posteriori". In other words, I will probably always think that the right way to teach the subject is to go through the particle way and *then* show that the results can be recovered starting from a "quantize a classical field" approach.But it looks as if I am the only one thinking this!

As hinted here, I suppose that both formalisms will turn out to be equivalent, if you add the necessary assumptions. If this is the case, it is just a matter of interpretation, whatever you like best, or whichever picture allows more easily to go to the next step (whatever that may be).

cheers,
Patrick.

I agree, I think it must be just a bookkeeping trick to work in terms of fields. When I will see this clearly then I guess everything will become clear and I will see directly why the usual approach (treating the amplitudes of the modes of a classical field as operators) does the same job as me building an action out of a bunch of creation/annihilation operators.

As I mentioned in another post, the key step I think would come when I would impose locality and be forced to build linear combinations of my operators (which create/annihilate states of definite momentum) to get something that is a function of x. Then I would recover the usual expressions for quantum fields. But that begs the question: why not simply build everything from operators that create/annihilate particles at a spacetime point? I guess one could do everything that way and never talk about classical fields at all (actually, it would look like a classical field theory with the fields being quantize except that it would be a superficial analogy, without any power). But in order to apply the formalism to actual experiments, there is the need to express things in terms of particles of definite momenta. So one would reexpress the operators creating a particle
at definite spacetime points in terms of operators creating/annihilating momentum states. And it's this reorganization that looks exactly like a classical field expanded in terms of modes with amplitudes being operators! After that one must still impose Lorentz invariance, etc etc. SO at this point only, someone could say: "look, what we can do is to treat the operators as classical, quantities in which case our terms look like classical fields. Let's build a classical action that satisfies Lorentz invariance, etc etc and *then* afterward we'll put back the fact that we really meant those amplitudes to be operators. Then, after doing this with a few theories, it would be clear that we might as well start with classical field theories and then quantize the fields.

I know it's a long detour to get the same result, but to me that would be more satisfying conceptually than to say "to nuild multiparticle theories, the only way to go is to quantize classical fields: which makes me go :surprised

Btw, how do you get the "Science Expert" thingy that appears beside your name?

Best regards,

Pat

Rothiemurchus

Sep9-04, 03:24 PM

Imaginary numbers yield results that can be backed by experiment but they are not intuitive. For example, what is imaginary time? I can't see it ticking away on a clock can I?
And why would the universe be a place of real and imaginary numbers.
I can square an imaginary number and get a real number, I can't square
a real number and get an imaginary number.Physics is generally based on symmetry.There doesn't seem to be symmetry here.

humanino

Sep9-04, 04:19 PM

Just as an example: consider the usual way one imposes equal time commutation relations on fields. One makes an analogy between the position and momentum of a point particle and the field and momentum density of classical fields. That step alway makes me want to go :surprised :cry:

We just impose the usual relation for conjugate variables which a concept already present in classical mechanics with Poisson brackets. In view of this the equivalence principle only amounts to promoting calssical Poisson bracket to operator commutation rules. This is no mystery, and this is very natural.

Vern

Sep9-04, 04:20 PM

If you think of space as having an electromagnetic saturation amplitude constant for all photons then quantization is demanded by that concept.

Vern

Fredrik

Sep9-04, 07:55 PM

After all, the field \phi (let's say) has nothing to do with a position!

I wouldn't go quite that far, since

\phi(x)\lvert 0\rangle

can be interpreted as the state of a particle at position x.

I will probably always think that the right way to teach the subject is to go through the particle way and *then* show that the results can be recovered starting from a "quantize a classical field" approach.But it looks as if I am the only one thinking this!

You really need to read Weinberg. :smile: The first Lagrangian appears in chapter 7, after he has covered one-particle states, many-particle states, the S-matrix, the cluster decomposition principle, quantum fields and even the Feynman rules.

nrqed

Sep9-04, 08:51 PM

We just impose the usual relation for conjugate variables which a concept already present in classical mechanics with Poisson brackets. In view of this the equivalence principle only amounts to promoting calssical Poisson bracket to operator commutation rules. This is no mystery, and this is very natural.

Hi humanino.

I know. I understand that the algorithm is to identify the conjugate variables and promote the classical PB to quantum commutators and to introduce h bar, etc. But, it does not sound natural to me to do this in this context. Well, ok, after reading it hundreds of times, it started to sound "natural" until I decided that I would try to reconstruct the whole formalism of QFT on my own, following what *I* found natural instead of following what I had tricked myself as accepting as natural just because I had read the same thing over and over again. ( Don't misinterpret my words, I am talking only for myself here, I am not saying that you or anyone else posting here is tricking themselves. I am convinced that many many people have a much deeper grasp of QFT than I do . All I am saying is that I am trying to build a picture that will be the most natural to me.)

Of course, since this is quantum physics, there are necessarily some steps that are strictly wild guesses. Only afterward can one check if things agree with experiment. They can be wild educated guesses but they are still pretty wild :smile: . But I think it's nevertheless important to point to the steps that are wild guesses and to argue why this seems like the correct thing to do. Whenever we say that something is "natural", we are really saying that it's a guess but that we can understand the motivation behind the guess. Since it's an entirely subjective criterion, it will obviously vary from person to person.

So I'll tell you why it does not sound natural to me. Ok, I write down the KG equation. Then, after playing with it and realizing it has problems, I decide to develop a multiparticle theory.

The first step that I don't find natural at all is to decide to quantize the field (for the reasons I have explained before).

Then there is the choice of commutation relations. Now, I have only NRQM as my guide so I look at that. I agree with you that I can think of p as being the conjugate variable to x and if I think about classical PB, I can start to get some urge to decree that the transition to the quantum world is accomplished by promoting classical PB to commutators proportional to h bar, etc etc. But it is still a wild guess, and given that we only have one example to rely on (x and p), I find this quite a wild guess.

I can live with this (now that I have heard and read it so many times) but I would have a hard time convincing a bright student who knows QM that this is the right thing to do. And in the end, *that's* what is my ultimate criterion to decide if something is natural to me or not.

But then, something even more bothersome comes up: we write equal-time commutation relations which are clearly not covariant equation. We started from the idea that we wanted to build a relativistically invariant theory and I am already starting to confuse things by writing non covariant equations. I *know* that it works out in the end, but that's quite a lot to swallow right at the beginning of laying down the formalism!

In any case, maybe I should not have get into this pet peeve I have about the field commutation relations. My biggest problem is the motivation for quantizing the fields in the first place, so I will focus on this issue for now. I am sorry if I am questioning everything :redface: . It's already difficult to get people to even entertain this type of discussion, I should focus on one thing at a time .

I do appreciate greatly having your feedback!

regards

Pat

nrqed

Sep9-04, 08:59 PM

I wouldn't go quite that far, since

\phi(x)\lvert 0\rangle

can be interpreted as the state of a particle at position x.

I understand, but the x here is simply a label and is not quantized. It's \phi itself which is quantized and in itself it has nothing do with a position.

You really need to read Weinberg. :smile: The first Lagrangian appears in chapter 7, after he has covered one-particle states, many-particle states, the S-matrix, the cluster decomposition principle, quantum fields and even the Feynman rules.

Wow. Well, you are not the only one who has said this so I am becoming convinced now. I have only bought volume 2 because I had figured that the first volume would be a repeat of the same old stuff. That was without counting on Weinberg's very personal approach to pretty much everything.

It will certainly be neat to see someone doing it the way I want to see it done, for a change :biggrin: . And to finally see the fields relegated to an afterthought!

And if Weinberg shares my preference of presentation, I will consider myself in good company :wink:

Thanks,

Pat

vanesch

Sep9-04, 11:11 PM

Btw, how do you get the "Science Expert" thingy that appears beside your name?

I have no clue. It appeared a few days ago. The engine on this site must be very clever indeed :biggrin: :biggrin:

cheers,
Patrick.

vanesch

Sep9-04, 11:41 PM

After all, the field \phi (let's say) has nothing to do with a position! Unless one pushes too far the analogy with a vibrating rope, one must admit that there is no relation between the fields we use (even at the classical level!!) and a direction in space.

Usually when person A tries to explain to person B something "obvious", and person B doesn't see the "obvious", it is A who misses the whole point. So I'll play the role of A and you, B.

But the way I naively saw things was as follows.

Let us first consider an electrical circuit with capacitors and selfs. There is a finite number of degrees of freedom, say the charges on the capacitors. You can write out the lagrangian and you will find the conjugate momenta to be the currents (if I remember well). Now you could think of currents as moving charges, but we deal here with a circuit from an engineering point of view, not knowing that there are electrons moving. So currents are to be seen as abstract dynamical quantities in our system.
You can then quantize this system, by applying the canonical quantization relations. Nothing to it.

Let us now consider a scalar field \phi. In a classical sense, this means, to me, that there is an entity out there which has a degree of freedom at each point in space. The configuration space is then the set of functions over space. If the entity obeys a certain dynamics, it will trace out a parametrised curve in this configuration space, parametrised in time. If that dynamics can be described by a Langrangian formalism, then it turns out that we can associate a canonical momentum to each degree of freedom \phi(x,y,z), which we call \pi(x,y,z). This canonical momentum has nothing to do with moving in space, just as our currents were not related to things moving.
We then turn the crank of quantum theory, namely you take each degree of freedom, and it has to obey [q,p] = i hbar when q and p are conjugate, and [q,p] = 0 when they aren't. Because of the continuity of the labeling of the indices (in space coordinates instead of integer indices) this gives us the equal-time commutation relations.

How else would you quantize the dynamics of a field ?

I'm not defending here the fact that we should use fields, I'm just saying that *IF* we're gonna study fields, taken as fundamental dynamical entities, then I don't see how we could quantize their dynamics differently.
I thought that quantum field theory was, well, eh, the study of how one should quantize fields. This is a different issue of WHY we should quantize fields in the first place of course, but to me it was sufficient that people told me: well, IF you do it, you get neat results and even particles come out naturally.
To me a much bigger mistery is: why should all this stuff still be described by a theory based on a lagrangian formulation ?

cheers,
patrick.

vanesch

Sep10-04, 03:41 AM

For example, what is imaginary time? I can't see it ticking away on a clock can I?

You can't see real numbers either on a clock. The bulk of it you cannot even write down on a piece of paper that would fit in the visible universe.

cheers,
Patrick.

nrqed

Sep10-04, 06:52 AM

Usually when person A tries to explain to person B something "obvious", and person B doesn't see the "obvious", it is A who misses the whole point. So I'll play the role of A and you, B.

But the way I naively saw things was as follows.

Let us first consider an electrical circuit with capacitors and selfs. There is a finite number of degrees of freedom, say the charges on the capacitors. You can write out the lagrangian and you will find the conjugate momenta to be the currents (if I remember well). Now you could think of currents as moving charges, but we deal here with a circuit from an engineering point of view, not knowing that there are electrons moving. So currents are to be seen as abstract dynamical quantities in our system.
You can then quantize this system, by applying the canonical quantization relations. Nothing to it.

Let us now consider a scalar field \phi. In a classical sense, this means, to me, that there is an entity out there which has a degree of freedom at each point in space. The configuration space is then the set of functions over space. If the entity obeys a certain dynamics, it will trace out a parametrised curve in this configuration space, parametrised in time. If that dynamics can be described by a Langrangian formalism, then it turns out that we can associate a canonical momentum to each degree of freedom \phi(x,y,z), which we call \pi(x,y,z). This canonical momentum has nothing to do with moving in space, just as our currents were not related to things moving.
We then turn the crank of quantum theory, namely you take each degree of freedom, and it has to obey [q,p] = i hbar when q and p are conjugate, and [q,p] = 0 when they aren't. Because of the continuity of the labeling of the indices (in space coordinates instead of integer indices) this gives us the equal-time commutation relations.

How else would you quantize the dynamics of a field ?

Hi Patrick,

Thanks. I like it. Maybe I sounded a bit dumb by raising the question so let me push a bit my point of view. Promise me to tell me when I say something stupid :smile: .

I agree with what you are saying, but let me play the devil's advocate a bit more. We agree that we should not think about momentum in the usual sense but as the abstract conjugate momentum of Lagrangian mechanics. Here's my problem: (again, I am just saying what I would have said when I was learning the stuff if I had not been scared to look too dumb. I now am old enough to not care :smile:) . Ok, then we need this "momentum" density thingy. How do we get it? Well, people will say: simply get the Lagrangian and proceed as usual. I would reply: how do we know we can write down a Lagrangian in the first place? That's an assumption? People would say "of course you can write down the Lagrangian, just find the Lagrangian whose eom gives the Klein Gordon equation" (say). I would then say: but that's a bit strange, we wrote down the KG by replacing the usual operator forms for P_mu into the relativistic energy equation, so this is not a classical equation to start with. People would say "no, we obtain it in a nonclassical way, but now just go ahead and treat it as a classical equation obeyed by a classical field. Don't worry, keep going". Then I would say "ok, so I am making up this new classical field, whose meaning I know nothing about and I will now quantize it.... weird".

You see, my point is that there are two ways I would find treating the field and its conjugate momentum as the "natural" things to impose commutation relations on. First, if they were actual positions and momenta in the NRQM sense (maybe in the case of a solid where the field could represent the actual position of particles). Then imposing those commutation relations is a natural extension of NRQM. The second case is if I was actually starting from a classical field (E&M comes to mind). Then I would find natural to think about configuration space, etc. But in QFT applied to particle physics, there is an extra step whic is what leaves me dissatisfied. We first get an eom already using quantum ideas, and now we "make up" this imaginary classical field which we assume is associated to a Lagrangian which allows us to derive this momentum density to use in the commutation relations. This "classical" field is associated to a configuration space which has no classical picture to start with! It seems to me to be an assumption to say that we should be able to have a conjugate momentum and a configuration space to quantize in the first place since the meaning of this field is not defined. So it seems to me that already the introduction of those classical fields that we'll later quantize is already a major "educated" guess! So the field approach requires two important leaps of faith: first the introduction of these mysterious classical fields which must be associated to some lagrangians and some configuration space, and *then* the quantization of these fields which somehow solve all the problems.

Does that make some sense to you, or am I crazy :rofl: ?

I understand the idea of introducing quantum effects by imposing commutation relations between conjugate variables and so on. But these fields we are introducing (even before quantizing!) leave me a bit queasy.

I guess that if I just accept that they can be associated to some lagrangian there is no problem. But in the "particle" approach sounds more natural to me. We say we must have multiparticle theories and we run with that. In the classical field approach, we get an equation of motion (using quantum ideas!), we introduce those weird classical fields and create this abstract classical configuration space, and *then* we quantize!

Again, I know that it does work, but I don't find it natural.

I'm not defending here the fact that we should use fields, I'm just saying that *IF* we're gonna study fields, taken as fundamental dynamical entities, then I don't see how we could quantize their dynamics differently.
I thought that quantum field theory was, well, eh, the study of how one should quantize fields. This is a different issue of WHY we should quantize fields in the first place of course, but to me it was sufficient that people told me: well, IF you do it, you get neat results and even particles come out naturally.

Ok, I agree with you completely. But, as I pointed out above, for me, also the introduction of the classical fields to quantize is unsettling.

To me a much bigger mistery is: why should all this stuff still be described by a theory based on a lagrangian formulation ?

cheers,
patrick.

Ah! Maybe we share some concerns, then!

Anyway, thanks a lot for your input, I appreciate very much.

Pat

vanesch

Sep10-04, 09:11 AM

I would reply: how do we know we can write down a Lagrangian in the first place? That's an assumption?

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).
The electric circuit I took in my example is a very special one: a linear, non-dissipative network. There are tricks to include linear resistors into a lagrangian formulation, but once you put semiconductors in it, it's over.
So I find it simply amazing that ALL of modern physics comes down to writing lagrangians :bugeye:

I would then say: but that's a bit strange, we wrote down the KG by replacing the usual operator forms for P_mu into the relativistic energy equation, so this is not a classical equation to start with. People would say "no, we obtain it in a nonclassical way, but now just go ahead and treat it as a classical equation obeyed by a classical field. Don't worry, keep going". Then I would say "ok, so I am making up this new classical field, whose meaning I know nothing about and I will now quantize it.... weird".

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 gonna write down a differential equation and you're gonna use special relativity, you'll end up with one of the known equations (K-G, Dirac, EM, proca...)

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 guess you could fill in a Jackson on the Dirac equation instead of on the Maxwell equations and have solutions with Bessel and Elliptic functions to impossibly difficult homeworks and problems :Devil:

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.

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:.

However, you're further in your understanding than I am, so you've had that and now you want to go back to "particles". I guess I have to read Weinberg too. But I'm now busy studying Zee in Hendrik's course (originally this was meant to be a course on Weinberg) which is not lost time, because I'm much less at ease with path integrals than with the canonical way of doing things.

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

But in QFT applied to particle physics, there is an extra step whic is what leaves me dissatisfied. We first get an eom already using quantum ideas, and now we "make up" this imaginary classical field which we assume is associated to a Lagrangian which allows us to derive this momentum density to use in the commutation relations. This "classical" field is associated to a configuration space which has no classical picture to start with!

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.

humanino

Sep10-04, 09:48 AM

You guys provide here a great discussion. I have already said it earlier and I am not flattering.

OK, I just wanted to motivate the Lagrangian formalism. Indeed theoretician use it since kindergarden, so it would be very difficult for them to get rid of it I guess. The point is the following : you want to optimize a Lorentz scalar. That is all there is : the Maupertuis conviction that Nature is elegant, and acts in an optimized fashion. The least action principle, or more accurately, principle of stationnary action :

Nature is thrifty in all its actions

This has been developped by : Euler, Leibniz, Fermat, Hamilton and of course, Lagrange (to quote a few). Unlikely to ever disappear it seems to me.

nrqed

Sep11-04, 01:02 PM

In the light of this, it appears that Pat (nrqed) should really take a look at Weinberg's first volume (even maybe buy it :wink: ), which presents QFT from exactly this point of view.

Hi Humanino,

Thanks for your input. Indeed, it does sound like Weinberg does it the way I want to see it done! I will go borrow the book this weekend (I may have to drive a bit more than one hour to a nearby university in order to do so. There is a local university but there is no particle physicist and because of that, the ir library is quite poor in particle physics/QFT/string theory/etc).

Thanks again!

Pat

marlon

Sep11-04, 01:51 PM

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.

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.Patrick.

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...

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

marlon

Sep11-04, 01:54 PM

nrqed

Sep11-04, 02:23 PM

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 :bugeye:

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 gonna write down a differential equation and you're gonna 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 :wink: ). 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 :wink:

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:.

:biggrin: 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 :yuck:

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 :wink:

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

Sep11-04, 02:38 PM

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

Sep11-04, 02:44 PM

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

Sep11-04, 03:06 PM

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 :approve:

thanks and cheers,
Patrick.

marlon

Sep11-04, 04:11 PM

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 :blushing: 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. :biggrin:

regards
marlon

marlon

Sep11-04, 05:18 PM

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

Sep11-04, 05:30 PM

marlon

Sep11-04, 05:33 PM

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 :blushing: :blushing:

marlon

humanino

Sep11-04, 05:35 PM

... 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

Sep11-04, 05:43 PM

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

Sep11-04, 05:50 PM

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

except maybe for LQG. :biggrin:

marlon

Sep11-04, 05:53 PM

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 gotta 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 :smile:

vanesch

Sep11-04, 10:43 PM

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 :blushing: 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 :devil:): 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

Sep11-04, 10:49 PM

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

Sep11-04, 11:06 PM

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 ??
:tongue:

cheers,
Patrick.

marlon

Sep12-04, 05:45 AM

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

Sep12-04, 06:05 AM

vanesch

Sep12-04, 06:57 AM

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... :grumpy: I'm more and more questioning the utility of fields :cry:

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 realised that it's gonna be damn difficult to have a classical dirac field with "anti-commuting numbers" (he, an old demon rises its ugly head :devil: ) 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 ! :wink:

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

Sep12-04, 07:14 AM

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

Sep12-04, 11:29 AM

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 :devil:): 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

Sep12-04, 11:44 AM

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... :grumpy: I'm more and more questioning the utility of fields :cry:

:devil: :devil: :devil:
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 realised that it's gonna be damn difficult to have a classical dirac field with "anti-commuting numbers" (he, an old demon rises its ugly head :devil: ) 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 ! :wink:

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

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

Sep12-04, 12:01 PM

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

:rofl: :rofl: :rofl:

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

Sep12-04, 12:41 PM

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

Sep12-04, 12:53 PM

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

Sep12-04, 12:53 PM

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

Sep12-04, 12:57 PM

:rofl: :rofl: :rofl:

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

Sep12-04, 03:03 PM

in another thread about mass-gap 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 :redface:

marlon

Sep12-04, 03:20 PM

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

Sep12-04, 03:27 PM

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 :redface:
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 :wink:

marlon

Sep12-04, 03:30 PM

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 in to two smaller pieces (two pairs). If there were no quarks present the fuxtube would "decay" into gluons, nothing else.

regards

marlon

marlon

Sep12-04, 03:40 PM

Hi Marlon,

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

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.

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

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

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 wanna make a difficulty out of these questions.

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 :smile:

humanino

Sep12-04, 03:51 PM

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.

Almost no computation can be made in this fashion, that is the true problem. QFT allows to make many calculations, partly due to Feynman.

Feynman brought QFT to the masses.

vanesch

Sep12-04, 04:09 PM

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

If you read the text again, Pat was talking about the 4 non-colored gauge bosons in the standard model (photon, W+/- and Z0), not the number of gluons.

cheers,
patrick.

humanino

Sep12-04, 04:14 PM

Right, sorry again, this time for a real reason !!!

vanesch

Sep12-04, 04:32 PM

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 !!!

What is not true ? First of all, this is a very intuitive model. I'm not particularly knowledgeable of the technical details of all these models. But naively, I thought you had:

+ === - the original flux tube

and later

+ ==== -/+ ====- pair creation

and still later:

+===- .............+===-

this is my extremely naive picture of confinement.

You claim that we should have:

+===- ============== +===-

???

Also something else: the paper you cite is probably very interesting and all that, but it is a *phenomenological model* of confinement. This model building has a priori nothing to do with the more fundamental question that was addressed here in this thread. We were discussing the bare bones starting point of why one should start out, when building the standard model, with classical fields we've never ever seen before and - that's what I'm realizing only now - we're never ever gonna see !

You can take as an argument "because it works". I can live with that. But I hope you can understand that this can be considered not sufficient as an explanation.

cheers,
patrick.

humanino

Sep12-04, 04:37 PM

I agree with Patrick here. This is very intuitive and very convincing. It is even probably the true physical reason for confinement, in my opinion.

humanino

Sep12-04, 04:39 PM

And I must say : I don't see where you guys disagree !

vanesch

Sep12-04, 04:53 PM

And I must say : I don't see where you guys disagree !

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 agressive 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.

humanino

Sep12-04, 04:59 PM

rather agressive nature than of an explanatory one

No no no ! That has always been my problem with my teachers, and certainly is here too, even worse because of the language (au fait, c'est un peu stupide :wink:)

I generally seem to try to prove something, whereas I just display my reasoning. I want to be proven wrong if i am, and this requires precise statements ! When I try to explain something, I feel it is great if somebody corrects me.

Haelfix

Sep12-04, 05:36 PM

marlon

Sep12-04, 06:29 PM

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 agressive 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 recieved 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 ??... :uhh: :uhh:

regards
marlon :smile: :smile:

marlon

Sep12-04, 06:32 PM

nrqed

Sep12-04, 07:02 PM

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

Sep12-04, 07:44 PM

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 wanna 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 :smile:

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

Sep12-04, 07:51 PM

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.

:biggrin: :biggrin: yahooooooo!! :biggrin: :biggrin:

humanino

Sep12-04, 07:53 PM

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

Sep12-04, 08:01 PM

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

Sep13-04, 03:31 AM

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

Well, we could go on and start insulting eachother 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 realised 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 ??... :uhh: :uhh:

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 truely 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 apparantly 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 :biggrin:

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

Sep13-04, 05:18 AM

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

Sep13-04, 05:46 AM

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 :smile:

regards
marlon

vanesch

Sep13-04, 06:29 AM

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 :smile:

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

Sep13-04, 07:00 AM

....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

Sep13-04, 07:04 AM

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 :tongue: , my fault

regards
en België boven
marlon :cool:

marlon

Sep13-04, 07:07 AM

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 :blushing:

you see, i am not always rude :blushing:

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

Sep13-04, 08:45 AM

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

you see, i am not always rude :blushing:

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

Sep13-04, 08:48 AM

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?" :biggrin:

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

Sep13-04, 12:53 PM

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

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!)

:rofl: :surprised :smile: :biggrin: :smile:

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 :surprised 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 :grumpy: ?

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

Sep13-04, 01:21 PM

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. :approve:

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

vanesch

Sep13-04, 01:26 PM

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

Sep13-04, 01:32 PM

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

Sep13-04, 02:07 PM

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 :wink:

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 :cry: . 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

Sep13-04, 02:09 PM

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

Sep15-04, 01:33 PM

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 realised 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 :biggrin: . Then I started scrathcing my head again :redface:. 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 :surprised

Pat

marlon

Sep16-04, 07:16 AM

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