Understanding the Motivation Behind Quantizing Fields in Quantum Field Theory

[nrqed]

Humanino wrote

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

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

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

and Patrick wrote

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

cheers,
patrick.

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


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

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

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

Regards

Pat

 

[humanino]

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

 

[nrqed]

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

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

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

...
cheers,
Patrick.

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



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

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

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

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

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

POINT 3:

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

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

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

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



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

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


So back to trying to understand fields




Pat

 

[marlon]

Here is a thought...

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

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

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

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

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


Just a thought, what do you think...

regards
marlon

 

[humanino]

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

 

[nrqed]

Here is a thought...

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

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

[snip]
regards
marlon

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

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

What am I missing here??!

Pat

 

[marlon]

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

The EM-field is NOT a classical field.

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

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

regards
marlon

 

[marlon]

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

marlon

 

[marlon]

And consider this :


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

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

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

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

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


regards
marlon