Understanding the Motivation Behind Quantizing Fields in Quantum Field Theory

[nrqed]

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 ([itex] \int {d^3k \over (2 \pi)^3 2 E} a e^{-i k \cdot x} + \ldots [/itex]) 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]

Excellent question : maybe the most important one about QFT

The discussion can be found for instance in Peskin & Schroeder's book. An internet link entitled Why fields.

There are several reasons : check this thread 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...
I don't doubt better will come soon.

 

[Haelfix]

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.

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.

 

[Rothiemurchus]

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]

The discussion can be found for instance in Peskin & Schroeder's book. An internet link entitled Why fields.

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 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...
I don't doubt better will come soon.

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



Pat

 

[humanino]

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 ...)

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

EDIT : thank to you for opening this great thread.

 

[humanino]

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]

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 [itex] d^3p a_p e^{i p \cdot x} \ldots [/itex]. 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]

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

EDIT : thank to you for opening this great thread.

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



Pat

 

[Doctordick]

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]

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]

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 ), which presents QFT from exactly this point of view.

 

[Rothiemurchus]

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]

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)

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

cheers,
patrick.

 

[nrqed]

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

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

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

After all, the field [itex] \phi [/itex] (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 [itex] \phi [/itex] and [itex] \pi [/itex]? 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 really 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




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




Pat

 

[Rothiemurchus]

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]

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

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]

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

Vern

 

[Fredrik]

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

I wouldn't go quite that far, since

[tex]\phi(x)\lvert 0\rangle[/tex]

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. 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]

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 . 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 . 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]

I wouldn't go quite that far, since

[tex]\phi(x)\lvert 0\rangle[/tex]

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 [itex]\phi[/itex] itself which is quantized and in itself it has nothing do with a position.

You really need to read Weinberg. 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 . 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

Thanks,

Pat

 

[vanesch]

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

cheers,
Patrick.

 

[vanesch]

After all, the field [itex] \phi [/itex] (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 [itex]\phi[/itex]. 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 [itex]\phi(x,y,z)[/itex], which we call [itex]\pi(x,y,z)[/itex]. 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 going to 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]

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]

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 [itex]\phi[/itex]. 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 [itex]\phi(x,y,z)[/itex], which we call [itex]\pi(x,y,z)[/itex]. 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 .

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 ) . 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 ?

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 going to 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]

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

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

I 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]

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]

In the light of this, it appears that Pat (nrqed) should really take a look at Weinberg's first volume (even maybe buy it ), 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]

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]

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

 

[nrqed]

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

You make there a very interesting observation.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

cheers,
Patrick.

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

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

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

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

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

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

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

Pat

 

[nrqed]

Hi Patrick,

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

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

Hi Marlon,

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

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

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

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

regards
marlon

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


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

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

Regards

Pat

 

[nrqed]

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

regards
marlon

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

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

Regards

Pat

 

[vanesch]

Hi Marlon,

Far from me the idea of talking for Patrick,

Hi Pat,

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

thanks and cheers,
Patrick.

 

[marlon]

Hi Pat

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

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

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

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


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


regards
marlon