Why quantizing fields?

1. Sep 8, 2004

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 ($\int {d^3k \over (2 \pi)^3 2 E} a e^{-i k \cdot x} + \ldots$) 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.

2. Sep 8, 2004

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

3. Sep 8, 2004

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.

Last edited: Sep 8, 2004
4. Sep 8, 2004

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!

5. Sep 8, 2004

nrqed

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.

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.

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.

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.

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

Best regards,

Pat

6. Sep 8, 2004

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.

7. Sep 8, 2004

humanino

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).

8. Sep 8, 2004

nrqed

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....

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

9. Sep 8, 2004

nrqed

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.

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

Best regards,

Pat

10. Sep 8, 2004

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.
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).
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
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

11. Sep 9, 2004

vanesch

Staff Emeritus
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.

12. Sep 9, 2004

humanino

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.

13. Sep 9, 2004

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.

Last edited: Sep 9, 2004
14. Sep 9, 2004

vanesch

Staff Emeritus
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.

15. Sep 9, 2004

nrqed

:tongue2:

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 :surprised

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.

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!

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

16. Sep 9, 2004

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.

Last edited: Sep 9, 2004
17. Sep 9, 2004

humanino

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.

18. Sep 9, 2004

Vern

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

Vern

19. Sep 9, 2004

Fredrik

Staff Emeritus
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.

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.

20. Sep 9, 2004

nrqed

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