What Is the Connection Between Quantum Mechanics and Quantum Field Theory?

[TriTertButoxy]

Ok, so we all know the following...

Elementary quantum mechanics is constructed by postulating the canonical commutation relation (CCR) between the coordinate of a particle and its conjugate momentum.

Quantum Field Theory, on the other hand, is constructed by postulating the exitence of a field and by imposing the same sort of CCR on field at every space-time point (as though each point on the field were an elementary quantum mechanical particle).

Doesn't it seem strange that the particles that emerge as a result of applying CCR to fields themselves obey the CCR imposed by ordinary quantum mechanics, (in the low energy limit)? I think its kind of spooky.

 

[CarlB]

I think the best introduction to the concept of the difference between QM and QFT is "Quantum Field Theory in a Nutshell: Books: A. Zee". Just reading the first chapter will likely reduce some of the spookiness.

Carl

 

[nrqed]

Ok, so we all know the following...

Elementary quantum mechanics is constructed by postulating the canonical commutation relation (CCR) between the coordinate of a particle and its conjugate momentum.

Quantum Field Theory, on the other hand, is constructed by postulating the exitence of a field and by imposing the same sort of CCR on field at every space-time point (as though each point on the field were an elementary quantum mechanical particle).

Doesn't it seem strange that the particles that emerge as a result of applying CCR to fields themselves obey the CCR imposed by ordinary quantum mechanics, (in the low energy limit)? I think its kind of spooky.

This is an excellent question, TriTertButoxy.

I have always been annoyed by the standard introduction to quantum field theory in which one imposes those commutation relations on classical fields. This always seemed very strange to me. What the heck are those classical fields that one quantizes? (no one has ever seen a classical approximation of a meson field or of a fermion field!)

As far as I know, only Weinberg presents things in a way that really makes sense, in which the starting point is not the quantization of classical fields but the need to allow the number of particles to vary, so the need to introduce annihilation and creation operators. Then , imposing Lorentz invariance, one is led to quantum fields obeying the usual commutation relations (postulated from the start in standard presentations).

Going back to your question, and as far as I understand, the commutation relations (CRs) on the quantum fields have nothing to do with the commutation relations on X and P imposed in nonrelativistic quantum mechanics! In the sense that there is no way to start from the CR of quantum fields and do some approximation to get back to the QM CRs (one way to see this clearly is that there is no analogue of the position operator X in QFT).

I have to warn you that my views on QFT are non standard (a couple of years a go I launched a thread that led to a very long discussion on this and it was clear that I think differently than almost everyone else!). The only book hat seemed to have address things the way I needed to see them addressed is Weinberg's book. Before reading his introduction to quantum fields, I had always been very confused and annoyed with the standard presentation of QFT. It just did not make sense to me. It seemed to involve such a huge leap of faith with no justification at all (we start with classical fields and we quantize them...as if this should be obvious as the thing to do!). And book after book after book keeps starting from exactly the same point, without justifying why this must be the way to go. Until finally Weinberg did it the "right" way (in my point of view, of course). The only problem with Weinberg is that it is so dense that it is difficult to follow for a beginner. I wish that there would be an introductory book on QFT that would present things this way!

Anyway, just my two cents.

Regards

Patrick

 

[Eye_in_the_Sky]

My understanding of Quantum Field Theory is rather rudimentary. Nevertheless, I feel I may have some useful thoughts to offer in this thread.

I have always been annoyed by the standard introduction to quantum field theory in which one imposes those commutation relations on classical fields.

... And so, was born the MISNOMER "Second Quantization".

This always seemed very strange to me. What the heck are those classical fields that one quantizes? (no one has ever seen a classical approximation of a meson field or of a fermion field!)

By way of analogy to the photon, those classical fields would just be the fields which one is to envision in the (hypothetical) case where a COHERENT source produces an extremely LARGE number of the particles in question.

Of course, with regard to photons it is possible (and, moreover, quite reasonable) to expect to be able to DISCOVER the relevant field equations on the basis of classical principles alone. After all ... historically, this is precisely how it happened.

But on the contrary -- say for example, with regard to the electron -- it seems completely unreasonable to think that one could come up with the Dirac Equation by way of only classical physics principles. ... How the heck can you do that?!

This difference, however, is only a technical one. On the conceptual level,

"the Dirac Equation is to the electron" as "the free-field Maxwell Equations are to the photon".

Therefore, it makes just as much sense to take the Dirac Field and quantize it, as it does to take the Maxwell Field and quantize it. ... Wouldn't you say?
__________________________________

EDIT: This post is only part 1. The next part will appear below, some time later ... I hope.

 

[nrqed]

My understanding of Quantum Field Theory is rather rudimentary. Nevertheless, I feel I may have some useful thoughts to offer in this thread.

... And so, was born the MISNOMER "Second Quantization".

By way of analogy to the photon, those classical fields would just be the fields which one is to envision in the (hypothetical) case where a COHERENT source produces an extremely LARGE number of the particles in question.

Of course, with regard to photons it is possible (and, moreover, quite reasonable) to expect to be able to DISCOVER the relevant field equations on the basis of classical principles alone. After all ... historically, this is precisely how it happened.

But on the contrary -- say for example, with regard to the electron -- it seems completely unreasonable to think that one could come up with the Dirac Equation by way of only classical physics principles. ... How the heck can you do that?!

This difference, however, is only a technical one. On the conceptual level,

"the Dirac Equation is to the electron" as "the free-field Maxwell Equations are to the photon".

Therefore, it makes just as much sense to take the Dirac Field and quantize it, as it does to take the Maxwell Field and quantize it. ... Wouldn't you say?

I hear what you are saying. And I agree that this is the standard point of view.

Two comments.

First, this is not the way it is presented in any textbook on QFT I know (if it is done in in some textbook, I would love to hear about it). If one was serious about presenting things this way, one would have to show in depth the connection between a coherent quantum EM field and the classical field. I am still amazed (and a bit annoyed) by the fact that this is never done in details in textbooks. I could not even find the words "coherent states" in the index of Peskin and Schroeder!
(and I must admit that I still don't feel I really understand very well this concept and the connection with definite phase and amplitude and so on)

If this is the best way to introduce quantume fields, then one should start by showing very clearly the connection between a coherent state of an EM quantum field and classical fields.

On the other hand, I find that it is much more natural to start with the observation that any particle is observed with a discrete nature (as "packets" of energy and momentum) and that is is natural to
build a formalism of varying number of particles (because of special relativity) and to see the concept of field as emerging from allowing the number of particles to vary *and* imposing Lorentz invariance (this is what Weinberg does). This starts from the *observation* that particles are observed and *leads* to quantum fields as a "bookkeeping" device. I find this much more intuitive than to start from postulated classical fields (which do not exist, and which have never been observed even as an approximation of a quantum field!).

That's just my point of view, which I seem to share only with Weinberg!

Patrick

 

[vanesch]

Hi Pat,

I agree with most of what you write here ; especially the rather obscure voodoo at the start of most QFT books. But this is usually how many subjects are pedagogically introduced: "listen, I'm not going to justify everything, but take my word for it now, and let's get going..." ; because otherwise 3/4 of the course time is devoted to the first chapter, with still some questions remaining, and no elementary application can be done.

I have to say Weinberg's approach was an eye-opener to me, but I didn't feel lousy about the "field" approach either. In fact, Weinberg is a bit of a party-pooper because with the equivalence of both viewpoints (variable particle count + relativity <==> quantized relativistic fields) it's now impossible to see one approach as more fundamental over the other.

The way I saw QFT before was: the quantum "prescription" applied to a specific classical model, namely the model of relativistic fields, in the same way as the quantum prescription was applied to Newtonian/hamiltonian point particle mechanics, and resulted in non-relativistic quantum mechanics.
You set up a configuration space, and derive from it a hilbert space.
QFT was then just a model, where classical fields where taken as "source of inspiration", in the same way as NRQM was just a model, where Newtonian mechanics was taken as a source of inspiration.
In what way this was "justified" was only to be determined by empirical success or failure.
The "miracle" for me was that this model of QFT generated the appearance of particle-like things, all by itself. Moreover, these particle-like things, in the right limits, behave as particles in NRQM. Is this a coincidence, or something deep ?

In another limit of QFT, of course, we find back our "source of inspiration", namely the classical field (in the same way as we find back something that looks like Newtonian mechanics in NRQM). The only field for which this really works out are massless boson fields (EM). But this is much less of a surprise, because we put it in from the start. The true "miracle" is that particles come out of the "quantized classical field" model.

Now, Weinberg turns things around, and asks: what kind of model can spit out particle-like stuff and be in agreement with relativity ? And he finds that such a model must look like a quantized classical field. So the miracle here is that the field comes out. We put in "particles" by hand, plus relativity, and we get out "fields".
So what came first ? The field, or the particles ? Weinberg shows that the notions are equivalent.

 

[vanesch]

I hear what you are saying. And I agree that this is the standard point of view.

Two comments.

First, this is not the way it is presented in any textbook on QFT I know (if it is done in in some textbook, I would love to hear about it). If one was serious about presenting things this way, one would have to show in depth the connection between a coherent quantum EM field and the classical field. I am still amazed (and a bit annoyed) by the fact that this is never done in details in textbooks. I could not even find the words "coherent states" in the index of Peskin and Schroeder!
(and I must admit that I still don't feel I really understand very well this concept and the connection with definite phase and amplitude and so on)

If this is the best way to introduce quantume fields, then one should start by showing very clearly the connection between a coherent state of an EM quantum field and classical fields.

A book that does this much better than any "high energy physics" QFT book is "optical coherence and quantum optics" by Mandel and Wolf.

 

[nrqed]

Hi Pat,

I agree with most of what you write here ; especially the rather obscure voodoo at the start of most QFT books. But this is usually how many subjects are pedagogically introduced: "listen, I'm not going to justify everything, but take my word for it now, and let's get going..." ; because otherwise 3/4 of the course time is devoted to the first chapter, with still some questions remaining, and no elementary application can be done.

hi Patrick... I have enjoyed very much all the exchanges we have had over this issue

I see what you are saying but what I find annoying is the following. Ok, we have learned about this mysterious theory of QM where we start from classical hamiltonian and replace X and P by operators and so on.

And now QFT is introduced. The "field theory" way of approaching things is to again impose strange commutation relations (which are *different* than the already strange CR of QM and are just justified by analogy) now defined on even weirder things than classical momenta and positions: they are defined on classical fields which are unobservable to start with (except for the EM case)!

So it's a different type of quantization, applied to new and strange uobservable "classical" fields.

So there are two leaps of faith here: accepting those new CRs and then accepting those weird classical fields as a starting point.

However, the "Weinberg approach" in my view is *so* much more physical and intuitive. The only thing that is needed here is to accept special relativity. That because of E=mc^2, the number of particles may vary and that equations must be Lorentz covariant, etc. *This* is much more intuitive to me than the field approach!

There are additional bonuses from this approach, too. First, the field operator Phi (for a scalar case, let's say) is clearly secondary in Weinberg's approach. It is just a way to combine creation and annihilation operators with a certain Lorentz symmetry. It is never thought of as an observable. In the standard field theory approach, because one *starts* with that field, it feels like it is a key physical operator or something. It's annoying that it is the central quantity in the field approach and yet nobody says at the get go that this is not an observable, that we will never care about its eigenvalues and so on. In the Weinberg approach, it clearly has a secondary role. For example, Weinberg applies that the *Hamiltonian* at spacelike points commute and derive conditions from there. The standard approach is to impose this on the field Phi even though it's not an observable (and it is left unsaid that this is a sufficient condition because physical observables will be built out of fields Phi).

So to me it is annoying and confusing to put the field Phi at the forefront when in fact it is not even an observable.

Another point is that the usual field approach gives the impression that the CRs of the fields is similar to the CRs of X and P in QM. But actually, the CRs of the fields have nothing whatsoever to do with position vs momentum. They arise because of the varying number of particles and the CRs of the annihilation/creation operators. I do know that one can turn the tables around and see the annihilation/creation operators arising from the CRs of the field (and the conjugate momentum) but these CRs are totally unrelated to a Heisenberg-like uncertainty in position vs momentum of a quantum particle. So it seems to me that the analogy used to get the CRs of quantum fields from the CRs of QM obfuscates their meaning.

I have to say Weinberg's approach was an eye-opener to me, but I didn't feel lousy about the "field" approach either. In fact, Weinberg is a bit of a party-pooper because with the equivalence of both viewpoints (variable particle count + relativity <==> quantized relativistic fields) it's now impossible to see one approach as more fundamental over the other.

I have to admit that I, on the other hand, *did* feel lousy about the field approach .
I can see that they are equivalent, but the field approach to me has much more the feel of a "trick" to organize things in a very efficient way, whereas the Weinberg approach makes much more sense physically. There might be something deeper, physically, about the field approach but I haven't seen it.

The only situation in which I *would* start with the field approach would be in Condensed Matter type of situations, to explain phonons, say. Then it makes sense to me to do it that way.

The way I saw QFT before was: the quantum "prescription" applied to a specific classical model, namely the model of relativistic fields, in the same way as the quantum prescription was applied to Newtonian/hamiltonian point particle mechanics, and resulted in non-relativistic quantum mechanics.
You set up a configuration space, and derive from it a hilbert space.
QFT was then just a model, where classical fields where taken as "source of inspiration", in the same way as NRQM was just a model, where Newtonian mechanics was taken as a source of inspiration.
In what way this was "justified" was only to be determined by empirical success or failure.
The "miracle" for me was that this model of QFT generated the appearance of particle-like things, all by itself. Moreover, these particle-like things, in the right limits, behave as particles in NRQM. Is this a coincidence, or something deep ?

For me it was an annoyance to wonder about this question (miracle or not)! It always felt like a miracle to me (before I read Weinberg), and I was wondering what was new in the CRs of the classical fields as opposed to the CRs of QM. What annoyed me was that nobody was stating cleraly :"This is a miracle, there is something deep here"! OR "There is nothing deep here, and here's the reason why". I have been so annoyed for years because I wanted to find out which it was!

In another limit of QFT, of course, we find back our "source of inspiration", namely the classical field (in the same way as we find back something that looks like Newtonian mechanics in NRQM). The only field for which this really works out are massless boson fields (EM). But this is much less of a surprise, because we put it in from the start. The true "miracle" is that particles come out of the "quantized classical field" model.

Now, Weinberg turns things around, and asks: what kind of model can spit out particle-like stuff and be in agreement with relativity ? And he finds that such a model must look like a quantized classical field. So the miracle here is that the field comes out. We put in "particles" by hand, plus relativity, and we get out "fields".
So what came first ? The field, or the particles ? Weinberg shows that the notions are equivalent.

Very true. But pedagogically speaking, I find that it makes much more sense to accept that there are quantized packets of energy called mesons and electrons, etc and that because of relativity their number may vary and to work from that than to start from inexistent classical fields and to quantize them using new CRs that are inspired by analogy with QM.

I guess that if I was teaching QFT now, I would never be able to get myself to try to get the students to swallow the field theory approach (I mean, I am pretty sure that most students *would* be willing to accept it and by the time we would be doing applications, they would have forgotten about their initial misgivings, which is I think what happens in almost every QFT class...But as the instructor I would feel very uneasy about that approach.

But hey, you know that this is a pet peeve of mine

I appreciate your comments, Patrick (as always).

 

[Eye_in_the_Sky]

WOW !

Initially, I felt I may have some insights to share in this thread. But now it seems that any of my current thoughts on these matters can at best be not much more than trivial to the two of you Pats.

I do have some questions, though.

First of all, I would like to verify whether or not I have sufficiently comprehended Pat's (that is, nrqed's) grievances against the "field" approach and also whether or not we see eye to eye on some of the basic physics and math. Please bear with me.

Consider a world which respects Galilean invariance and in which the Schrödinger equation is exact. Then, Pat (nrqed), am I correct in the following assertions?

a) You would have no complaints against construing the Schrödinger field as a 'classical' field, since it does have physical meaning in terms of probability amplitudes.

b) Nevertheless, you would harbor complaints against its direct quantization as a field, because a quantized probability-amplitude makes no physical sense.

c) You would not assert that the CRs of the fields have nothing to do with CRs of X and P, because X and P can be 'fashioned' from the field operator.

 

[nrqed]

Hi Pat,

I agree with most of what you write here ; especially the rather obscure voodoo at the start of most QFT books. But this is usually how many subjects are pedagogically introduced: "listen, I'm not going to justify everything, but take my word for it now, and let's get going..." ; because otherwise 3/4 of the course time is devoted to the first chapter, with still some questions remaining, and no elementary application can be done.

I have to say Weinberg's approach was an eye-opener to me, but I didn't feel lousy about the "field" approach either. In fact, Weinberg is a bit of a party-pooper because with the equivalence of both viewpoints (variable particle count + relativity <==> quantized relativistic fields) it's now impossible to see one approach as more fundamental over the other.

The way I saw QFT before was: the quantum "prescription" applied to a specific classical model, namely the model of relativistic fields, in the same way as the quantum prescription was applied to Newtonian/hamiltonian point particle mechanics, and resulted in non-relativistic quantum mechanics.
You set up a configuration space, and derive from it a hilbert space.
QFT was then just a model, where classical fields where taken as "source of inspiration", in the same way as NRQM was just a model, where Newtonian mechanics was taken as a source of inspiration.
In what way this was "justified" was only to be determined by empirical success or failure.
The "miracle" for me was that this model of QFT generated the appearance of particle-like things, all by itself. Moreover, these particle-like things, in the right limits, behave as particles in NRQM. Is this a coincidence, or something deep ?

In another limit of QFT, of course, we find back our "source of inspiration", namely the classical field (in the same way as we find back something that looks like Newtonian mechanics in NRQM). The only field for which this really works out are massless boson fields (EM). But this is much less of a surprise, because we put it in from the start. The true "miracle" is that particles come out of the "quantized classical field" model.

Now, Weinberg turns things around, and asks: what kind of model can spit out particle-like stuff and be in agreement with relativity ? And he finds that such a model must look like a quantized classical field. So the miracle here is that the field comes out. We put in "particles" by hand, plus relativity, and we get out "fields".
So what came first ? The field, or the particles ? Weinberg shows that the notions are equivalent.

Just a few thoughts that crossed my mind while driving today. I am basically repeating myself so everryone is welcome to ignore this post but it will help *me* clarify my point of view.

If I were to compare side by side the "field" approach vs Weinberg's approach, I would say the following:

In the field approach, the starting point is the introduction of mysterious classical fields that have never been observed (except the EM field but again, I am talking about the usual presentation where the EM field is discussed much later because of all its complications) and the introduction of mysterious CRs which are only justified by analogy with QM and are applied to fields which are not observables in the first place!

(And again, the CRs of the fields and corresponding conjugate momenta have nothing to do, in the end, with commutation relations with any sort of position and momentum operators, they arise from the CRs of the annihilation and destruction operators! So the fact that they are "justified" using an analogy with the CRs of QM, is a huge cheat, IMNSHO )

So in the field approach, in the foreground are those mysterious fields and those mysterious CRs.


*Then*, one obtains as a side result that there are quantized packest of energy and momenta. And that their number is not conserved.


In Weinberg's approach, however, what is placed in the foreground is that a) there are things which act like discrete bundles of energy and momentum (like electrons ) and that b) from E=mc^2 one expects their number not to be conserved.

Now, by analogy with the harmonic oscillator formalism one introduces annihilation and creation operators, one builds operators, one imposes causality, Lorentz covariance, etc etc...And everything comes out working nicely, with quantum fields now a *byproduct* of the whole thing. Two huge bonuses (IMHO): first, no need to even *ever* introduce the weird classical fields that are the starting point of the field approach! They are gone! Second bonus: no need to postulate the weird CRs between those weird quantum fields and their conjugate momentum!
They *follow* from the CRs of the annihilation/creation operators and *their* CRs are obvious to justify since they must raise or lower the energy by an amount equal to the energy of a particle.


Now, I do understand that the field approach is an efficient way to organize things and to build in Lorentz covariance, etc. But as way to *teach* the subject, I find that they are incredibly obscure and confusing. Well, maybe it's because I am not too smart but they sure confused me for years and years until I finally came to peace with them because of Weinberg (whom I could never thank enough for writing this book!)


Anyway, sorry for the babbling... Had to get this off my chest

Pat

 

[nrqed]

WOW !

Initially, I felt I may have some insights to share in this thread. But now it seems that any of my current thoughts on these matters can at best be not much more than trivial to the two of you Pats.

Don't feel that way!
I highly appreciate your comments!
It is very useful to me to hear comments from others (especially if they are starting to learn QFT) because it gives me much needed perspective. I like to know how other people think, what they find natural, etc. I learn a lot from that.

I do have some questions, though.

First of all, I would like to verify whether or not I have sufficiently comprehended Pat's (that is, nrqed's) grievances against the "field" approach and also whether or not we see eye to eye on some of the basic physics and math. Please bear with me.

Consider a world which respects Galilean invariance and in which the Schrödinger equation is exact. Then, Pat (nrqed), am I correct in the following assertions?

a) You would have no complaints against construing the Schrödinger field as a 'classical' field, since it does have physical meaning in terms of probability amplitudes.

I think I would have a problem already at this level. Because a probability wave is already in itself completely different from any classical wave (for example the E or B field in classical E&M).

But I see your point so let's say that we are ok with treating this as a "classical" wave.

b) Nevertheless, you would harbor complaints against its direct quantization as a field, because a quantized probability-amplitude makes no physical sense.

Yes, that would definitely bother the heck out of me!

But not only because of the quantization of a *probability* amplitude. The quantization of a classical field in itself bothers me because it is an additional "leap of faith", on top of the usual things we have to make when doing QM (the existence and role of the wavefunction, the replacement of classical quantities by operators, measurements, etc). Now we have to accept a new type of commutation relations, on top of those from NQRM.

( ASIDE: I know that everybody says that "second quantization" is a misleading term, but I personally think that it's a perfectly appropriate expression in the context of the usual presentations on QFT.
(in one type of presentation, people say that one is rewriting the wavefunction as an operator..then it *is* litterally a second quantization (i.e. a quantization on *top* of another quantization). On the other hand, when one quantizes classical fields, the expression "second quantization" is still appropriate because the quantization used there is totally different in nature from the QM quantization, so it's a second quantization in the sense now of "a different type of quantization").)

c) You would not assert that the CRs of the fields have nothing to do with CRs of X and P, because X and P can be 'fashioned' from the field operator.

I am not sure what you mean. I think we would have to be more explicit to discuss this. I don't know what you have in mind by saying that they can be "fashioned: from the field operator. In NRQM, one cannot write X or P in terms of \psi.

In NRQM, one promotes P_x and X to operators. So in the case of this "wave", one would expect to impose commutation relations on p_x \psi and x \psi. That's the only way in which I could see a "natural" generalization to impose commutation relations on. However, if you write the Lagrangian leading to Schrodinger equation, the quantities unto which one imposes the QFT commutation relations are obviously not the above expressions! Far from it!


Thanks for the interesting discussion btw!

Regards
Pat

 

[Eye_in_the_Sky]

From your response to item a), I see that for you a necessary feature of a "classical" field is:

The field amplitude can, under appropriate conditions, be measured.

This condition is too strong for my tastes. I prefer the much weaker condition:

The field amplitude exists.

"What? There are things which exist but cannot be measured?"

Well, not exactly ... but maybe.

It could be that I have never been able to create the appropriate conditions. Or it could be that I have not yet found the right device which couples to it. ... Or maybe it just cannot be measured. Or maybe it exist only as a representation within space of the influence of something which resides outside of space.

Pardon my voodoo ... but I'm serious!

Perhaps it exists as an agent which contributes to phenomena that I can measure, yet it itself cannot be measured directly.
__________________

Next ... my item b), it turns out, wasn't quite the right probe I was looking for. I'll have to think about that one some more and see if I can hone it down.

The larger part of your reply, though, concerning that additional "leap of faith" over and above a heap of such leaps (and again in the ASIDE that that additional leap renders appropriate an expression which by your estimation was already fine, but not by mine) ... all of that, I would like to put aside until after we part with Galileo for something relativity more intricate.
__________________

Regarding c), now I'll be more explicit. Quantization of the Schrödinger field leads to a field operator Ψ(x,t) such that the object

Ψ^†(x,0)|0>

is in fact the ket |x> on the single-particle subspace, so that

∫xΨ^†(x,0)|0><0|Ψ(x,0)dx

is the single-particle position operator X. Since we have |x>, we can get |p> by means of a Fourier-type transformation, and with that construct P, conjugate to X.

This is what I meant when I wrote X and P can be 'fashioned' from the field operator.

So, item c) amounts to: "Do you accept the above to be true? And if so, would this be this be an instance in which you are inclined to retract your statement that the CRs of the fields have nothing to do with the CRs of X and P?"

 

[masudr]

um, I'm like a 2nd year undergraduate, so I probably don't have a clue what I'm going on about. But here I go anyway

nrqed said that the CRs of NRQM were strange enough, let alone the CRs of QFT. But aren't they just they standard "quantization" procedure (which is admittedly mysterious, but one can accept) applying to the Poisson brackets of Hamiltonian mechanics? And couldn't one show the same Poisson brackets for fields and conjugate momenta of classical fields? And then apply the same (admittedly mysterious) quantization procedure?

My point is that probably the CCRs for NRQM are just as mysterious as those for QFT, not any more mysterious or any less.

 

[nrqed]

um, I'm like a 2nd year undergraduate, so I probably don't have a clue what I'm going on about. But here I go anyway

nrqed said that the CRs of NRQM were strange enough, let alone the CRs of QFT. But aren't they just they standard "quantization" procedure (which is admittedly mysterious, but one can accept) applying to the Poisson brackets of Hamiltonian mechanics? And couldn't one show the same Poisson brackets for fields and conjugate momenta of classical fields? And then apply the same (admittedly mysterious) quantization procedure?

My point is that probably the CCRs for NRQM are just as mysterious as those for QFT, not any more mysterious or any less.

Hi Masudr. Thanks for your input!

I will try to clarify my position (of course this is all an issue of perosnal taste and is very subjective. I can say that I don't find something nature=al or pedagogically sound and if someone else disagrees, that's the end of the discussion. So I am not trying to prove anything or to convince anyone of anything, I am just expressing my opinions. But I am learning a lot by seeing how others think of those things and that's why I like to have those discussions).

Two points.

A) In QM we apply the quantization rules to position and momenta of point particles, quantities which have a clear classical meaning and which we are used to work with classically. However in QFT, we apply quantization to "classical fields" that we have to postulate from the start and which have never been observed at all (like the Dirac field and so on). The only exception is the EM field, but there is no such thing as observations of classical scalar of spin 1/2 fields. So we have to "ivent" those "classical" fields before quantizing. And that's one of the two things that "bothers" me. In the "Weinberg approach", the quantum fields are a *byproduct* of the derivation and there is no need to even introduce classical fields which is great from my point of view since they have never been observed (again, except for the EM field).

B) I agree with you that one may take the position that the key idea about quantization is to replace the classical Poisson brackets by quantum CRs. However, I still see this step applied to fields as being distinct to this step applied to point particles. One can argue by "analogy" to "justify" the CR of fields from the CRs of point particles, but in the end, there is *still* a leap involved there. The CRs for the fields do not mathematically follow from the CRs for point particles. They are *analogue* but one cannot *mathematically* go from the CRs of point particles to the CRs of the fields (the only case where this can be done is if we seriously think of the field as being made of a large number of particles, i.e. a physical string. But the waves that we quantize in particle physics are not of that nature) . I agree that the analogy is strong but it is still a *new* quantization procedure that must be taken as an extra axiom on top of the axiom for the CRs of point particles. So in that sense, this is *really* a "second quantization". It's the quantization applied to fields as opposed to applied to point particles of NRQM.

There is nothing wrong with adding a new axiom to QM, this is not what I am saying. But my point is first that it sohould be recognized as such: a new axiom. Despite analogies with point particles, it is still a new axiom.
My second point is that in Weinberg's approach, those CRs on the *fields* are not postulated at all! They are a byproduct! Where do they come from? The staring point is the assumption that the number of particles is not constant (because of mc^2). Then it is natural to introduce annihilation and creation operators which then have an obvious set of CRs. So, in the field approach, there is an axiom giving the CRs of "classical fields" (which again, are not even observed) whereas in Weinberg's approach this axiom is replaced by the "axiom" that particle number is not conserved. I personally much prefer the second approach!

Does this make sense to you? I would appreciate yoru feedback!

Thanks again for you comment.

Regards

Patrick

 

[masudr]

In QM we apply the quantization rules to position and momenta of point particles, quantities which have a clear classical meaning and which we are used to work with classically.

I must disagree. What is the classical notion of spin? You could tell me that infinitesimal rotations don't commute, but that's just as bad (for me, at least) as saying that we will quantize a spin-1/2 field.

However in QFT, we apply quantization to "classical fields" that we have to postulate from the start and which have never been observed at all (like the Dirac field and so on). The only exception is the EM field, but there is no such thing as observations of classical scalar of spin 1/2 fields. So we have to "ivent" those "classical" fields before quantizing. And that's one of the two things that "bothers" me.

Interesting... the reason it doesn't bother me is because I see it like this. We have, the rules of QM. Then we have different Hamiltonians. So, we have a classical field we know: EM. We apply QM to it, and we get all these results. Then we can have different Lagrangians (e.g. the Dirac field Lagrangian etc.)

In the "Weinberg approach", the quantum fields are a *byproduct* of the derivation and there is no need to even introduce classical fields which is great from my point of view since they have never been observed (again, except for the EM field).

In all fairness, I haven't seen the Weinberg approach, and it does sound quite cool. I shall remember to give his text a look when I study QFT in my 4th year.

I agree with you that one may take the position that the key idea about quantization is to replace the classical Poisson brackets by quantum CRs. However, I still see this step applied to fields as being distinct to this step applied to point particles. One can argue by "analogy" to "justify" the CR of fields from the CRs of point particles, but in the end, there is *still* a leap involved there.

There's no analogy involved here. We have, say, the simple harmonic oscillator. We apply the rules of QM to it, i.e. represent states by those that satisfy the TDSE, observables by linear operators that obey the CRs etc. We now have a field with Lagrangian given by some expression. We apply QFT to it, so we have these position valued operators, etc. etc.

So we have Lagrangians and Hamiltonians for fields, and we quantize the fields just like we quantize particles. Your objection is similar to objecting that the rules that apply for classical particles shouldn't apply to classical fields, unless I have missed something.

The CRs for the fields do not mathematically follow from the CRs for point particles. They are *analogue* but one cannot *mathematically* go from the CRs of point particles to the CRs of the fields

I would agree with this.

(the only case where this can be done is if we seriously think of the field as being made of a large number of particles, i.e. a physical string. But the waves that we quantize in particle physics are not of that nature).

But even the classical EM field isn't particles oscillating on a string. The EM field is something quite mysterious as it is. We can get the field equations for a string by saying classical field theory is the study of continuous/infinite classical particles. But we can't really say that for the EM field. Again, what we do is set up the formalism, and then apply the rules to a brand new Lagrangian, and find that it works.

I agree that the analogy is strong but it is still a *new* quantization procedure that must be taken as an extra axiom on top of the axiom for the CRs of point particles. So in that sense, this is *really* a "second quantization". It's the quantization applied to fields as opposed to applied to point particles of NRQM. There is nothing wrong with adding a new axiom to QM, this is not what I am saying. But my point is first that it sohould be recognized as such: a new axiom. Despite analogies with point particles, it is still a new axiom.

Is it really applied on top of the rest of the axioms? We impose the CCRs for the field operators, but don't the ones for the particles of the field emerge from that? Or do we separately impose that too?

My second point is that in Weinberg's approach, those CRs on the *fields* are not postulated at all! They are a byproduct! Where do they come from? The staring point is the assumption that the number of particles is not constant (because of mc^2). Then it is natural to introduce annihilation and creation operators which then have an obvious set of CRs. So, in the field approach, there is an axiom giving the CRs of "classical fields" (which again, are not even observed) whereas in Weinberg's approach this axiom is replaced by the "axiom" that particle number is not conserved. I personally much prefer the second approach!

Does this make sense to you? I would appreciate yoru feedback!

As you have said, it is all a matter of taste and opinion, and of course, that is fair enough. I don't happen to have that much of a problem with the "standard" approach, although I must say, I haven't looked at the standard approach in much detail, nor the Weinberg approach, so I'm not really in a position to comment.

Thanks again for you comment.

No worries.

Masud.

 

[Eye_in_the_Sky]

Whoops !

Quantization of the Schrödinger field leads to a field operator Ψ(x,t) such that the object

Ψ^†(x,0)|0>

is in fact the ket |x> on the single-particle subspace, so that

∫xΨ^†(x,0)|0><0|Ψ(x,0)dx

is the single-particle position operator X. Since we have |x>, we can get |p> by means of a Fourier-type transformation, and with that construct P, conjugate to X.

This is what I meant when I wrote X and P can be 'fashioned' from the field operator.

So, item c) amounts to: "Do you accept the above to be true? And if so, would this be this be an instance in which you are inclined to retract your statement that the CRs of the fields have nothing to do with the CRs of X and P?"

The answers are: yes, no.

Nowhere in the above have I invoked the CRs of the fields to say what I say about the particle. According what is written there, the single-particle X operator will mathematically follow from no more than the existence of Ψ(x,0). And then, constructing P from X, well again that too has nothing to do with the CRs of the fields.

Besides, the CRs of the fields are connected to spin, whereas those of particle have no such connection.

Yes, yes. I see it now: an ADDITIONAL LEAP OF FAITH, of course(!), is involved when one invokes the CRs of the fields.

In NRQM, invoking the CRs (or anti-CRs) of the fields is the same as POSTULATING symmetric (or antisymmetric) subspaces for the many-Boson (or many-Fermion) system.

In hindsight, this is painfully obvious.

... Whoops !

 

[nrqed]

Hi Masudr.

Again, this is all quite subjective and a matter of taste, to a large extent. My arguments are more about pedagogy in teaching the subject (and writing textbooks). What is more "logical" as a way to introduce the subject. As a student I was bewildered by the standard presentation and did not know what was "profound", what was supposed to be analogies vs derivations, what were new axioms vs derived results, etc. And I was bewildered by the starting point of it all. To be honest, I could never get myself to teach the subject the standard way if I were to teach this class tomorrow.

Again, here's briefly my objection:

In the standard field approach, one starts with those unobserved classical field theories (like the KG field or Dirac field). One then quantizes them by postulating the CRs for fields. The presence of particle-like excitations comes out as a byproduct. (By the way, I might feel much better about this approach if they would also apply this approach to *noncovariant* field theories and show what happens then. And show clearly what is the difference and why there is a difference with applying this technique to invariant (scalar) vs non-invariant Lagrangian densities.)

(Of course, the EM field case is different because we do observe E and B fields classically. But then textbooks should start with EM fields, quantize them, show clearly the connection with classical fields through coherent states, the connection with the classical amplitude and phase and on and on. Then, *after* a solid intuition has been built on the connection between classical field theories, quantum fields and particle excitations, books should discuss why coherent states of massive particles are not observed, and so on. And *then* it would make sense to go on to quantizing KG and Dirac fields. But things are never presented this way On the other hand, Weinberg's approach bypasses all the pedagogical difficulties of the field approach, IMHO)


In Weinberg's approach, one starts with the idea that particles can be created or annihilated. Then one has many-body theories. If in addition one imposes Lorentz invariance, one is forced to introduce quantum fields.
The fields and their CRs are a *byproduct*. No need to postulate strange classical fields to quantize. No need to postulate CRs on the fields. The difference between Lorentz invariant and non-invariant theories is clear.

I find this second approach much more satisfying. Seems to be so much more logical than the field approach. And a *much* better way to *teach* the subject. Of course, after all is absorbed and the field connection is made, then the field approach should be shown too.

To be honest, I am still not totally sure whether there is something "deep" about the field approach. I still have to understand exactly how the CR between the fields and their conjugate momenta arise out of Weinberg's approach vs how they arise from the usual Lagrange/Hamiltonian approach.

I must disagree. What is the classical notion of spin? You could tell me that infinitesimal rotations don't commute, but that's just as bad (for me, at least) as saying that we will quantize a spin-1/2 field.

I did not mean to say that everything we quantize in NQRM has a clear classical analogue. I agree with you that it's not the case. I was focusing on the fundamental CR, between X and P. My point is that we impose CRs on these quantities. And then when we get to a KG field or a Dirac field, we impose a *new* set of CRs on the field and conjugate field momenta, which are themselves quantity unobservable classically (and this is a key difference even with the spin case. Spin has no classical analogue, but you get a Stern-Gerlach apparatus and you can measure it. If books want to start with a classical KG or Dirac field, theyshould explain how one would go about measuring their phase or amplitude )

On the other hand, Weinberg's approach does not require to introduce those unobservables classical fields and does not require to postulate their CRs. What *is* postulated is the need for a formalism with a vraying number of particles. The CRs are the CRs of annihilation/creation operators which are quite natural from the example of the harmonic oscillator where the energy levels different by integer multiple of an energy quantum.

Interesting... the reason it doesn't bother me is because I see it like this. We have, the rules of QM. Then we have different Hamiltonians. So, we have a classical field we know: EM. We apply QM to it, and we get all these results. Then we can have different Lagrangians (e.g. the Dirac field Lagrangian etc.)

I understand. But QFT books never show clearly the correspondence between the quantum EM fields and the classical fields we all know and love (not just showing that photons quanta arise..but showing what's the connection with classical phase and amplitude, talking about measurements of E and B fields, etc). If the field approach is to be taken as a starting point, it seems to me that the first chapter of any QFT book should be an in-depth discussion of EM fields. But QFt books sometimes don't even mention coherent states Or why we don't observe classical limits of the KG or Dirac field, etc.

In all fairness, I haven't seen the Weinberg approach, and it does sound quite cool. I shall remember to give his text a look when I study QFT in my 4th year.

I highly recommend doing so. Unfortunately, its is very dense so it takes a lot of work to see the basic ideas through all the notation. If I would teach QFT tomorrow, I would unfortunately not feel that I could use that as a textbook. I wish there was a lower level book on QFT that would introduce the ideas this way.

There's no analogy involved here. We have, say, the simple harmonic oscillator. We apply the rules of QM to it, i.e. represent states by those that satisfy the TDSE, observables by linear operators that obey the CRs etc. We now have a field with Lagrangian given by some expression. We apply QFT to it, so we have these position valued operators, etc. etc.

So we have Lagrangians and Hamiltonians for fields, and we quantize the fields just like we quantize particles. Your objection is similar to objecting that the rules that apply for classical particles shouldn't apply to classical fields, unless I have missed something.



I would agree with this.

You agreed that the CRs of the quantum fields cannot be derived from the CRs of point particles. My point is that the CRs of fields must be seen as a new axiom. Sure, the basic principle is similar to what we do for point particles so it's easy to accept/ But it is still a new axiom. And on top of that it is applied to fields which are not even observed classically.

So the field approach says : look, we have all those particles around (electrons, mesons, etc). To do calculations, we will introduce those "classical fields" (that we have never observed) and we will quantize them, based on an analogy with point particles NRQM and assuming that it makes sense to take the continuum limit for these unobservable fields. To *me* this seems like voodoo. And my definition of voodoo is this: I would personally would have never thought about doing this if someone had not told me that this was the way to go.

Sure, you will say that it's a natural way to go given EM fields. But then why not work out in details the EM fields and the quantum field/classical field correspondence limit in great depth before jumping to those abstract KG and Dirac fields?

On the other hand, Weinberg's approach says: look, we have those particles around. Because of relativity, we expect particle number to change. Let's go from there. That's it

But even the classical EM field isn't particles oscillating on a string. The EM field is something quite mysterious as it is. We can get the field equations for a string by saying classical field theory is the study of continuous/infinite classical particles. But we can't really say that for the EM field. Again, what we do is set up the formalism, and then apply the rules to a brand new Lagrangian, and find that it works.

I agree! The point that I had made is that strictly speaking, the correspondence between the CRs of point particles and of QFTs can only be made explicit (i.e. one can derive one from the other) in the case of a vibrating string. I agree completely that even in the EM case, there is a "leap of faith" involved in defining the CRs. In the sense that it's a new axiom. And the correspondence quantum field/classical field is far from obvious and trivial in the EM case.

So my point is that if one is serious about presenting the field apporach, one shoudl devote some time discussing the EM field in depth.

On the other hand, Weinberg's approach does not require to postulate those CRs for quantum fields as a starting point. One only need to introduce creation/annihilation operators. And one can build any field theory (KG, Dirac, EM) from this principle. I personally find this much more satisfying.

Is it really applied on top of the rest of the axioms? We impose the CCRs for the field operators, but don't the ones for the particles of the field emerge from that? Or do we separately impose that too?

As far as I can tell, they are completely separate (although another poster suggested an idea that I still have to look at).
If you (or anyone) could show me how to start from the field CRs and recover the position-momentum CRs for a single particle, I would be very interested. The CRs of the field arise from the annihilation/destruction operators CRs, which don't say anything about the X,P CRs of a single particle state.

This is one of my main arguments. One talks about the field and its conjugate momentum and use an analogy with the position and momentum of a single particle in order to "justify" the CRs for the fields. But it seems to me to be very decieving, because in the end, the CRs of the field and conjugate field momentum have nothing to do with position/momentum CRs. They are connected instead to the varying number of particles. So it feels like a cheat to me. It uses an analogy to get the right CR but later one realizes that the analogy has no basis. This is one of the main things that bothers me about the usual approach!
(again, in the case of an actual string, then there really is a connection between the CRs of the position and momenta of each particle and the CRs of the field, but as you pointed out, even EM is not a quantized string!)

As you have said, it is all a matter of taste and opinion, and of course, that is fair enough. I don't happen to have that much of a problem with the "standard" approach, although I must say, I haven't looked at the standard approach in much detail, nor the Weinberg approach, so I'm not really in a position to comment.

Sometimes I feel like an alien
Because from the moment I first read about QFT I was bothered about what appeared to me to be huge leaps of faith with no logical basis. I do know that some leaps of faith are required in QM but at least see how experiments and observations led to those leaps fo faith. Then I would read about QFT and the very first thing they would say is "relativistic wavefunction euqations have problems. So "obviously" what we will do is to consider classical fields and quantize them It's the biggest non-sequitur that I have seen in physics . I mean, special relativity involves a weird leap of faith, but at least I could see why it made sense to make this leap. I surely would never had been able to do it myself, but after I learned the theory, it did make sense to me. Same thing for GR. But the way textbooks present QFT, it did not make any sense to me.

I understand that historically, quantization of the EM field played a major role. But then why don't textbooks work this out in details first?

But again, now that I have read Weinberg, it makes even more sense to me to start this way (with the idea of varying number of particles) and to build from there. No non-sequitur involved there.

Again, just a question of taste.

egards

patrick

 

[nrqed]

The answers are: yes, no.

Nowhere in the above have I invoked the CRs of the fields to say what I say about the particle. According what is written there, the single-particle X operator will mathematically follow from no more than the existence of Ψ(x,0). And then, constructing P from X, well again that too has nothing to do with the CRs of the fields.

Right!

Besides, the CRs of the fields are connected to spin, whereas those of particle have no such connection.

well, I see what you mean. Whether we have CR or anti-CR depend on the spin, yes. But I would say that the Crs of the fields are connected to the *varying number of particles*. In weinberg's approach , one *starts* for the need for annihilation and creation operators and then one *derives* the CRs for the fields. I find it so much more satisfying to start from the need to have varying number of particles (because of relativity) than to have to postulate those weird unobserved classical fields and to postuale field CRs. But apparently, I am alone feeling this way (with Weinberg)

Yes, yes. I see it now: an ADDITIONAL LEAP OF FAITH, of course(!), is involved when one invokes the CRs of the fields.

Right! In Weinberg's approach, there *is* a leap of faith, but that's the need for varying number of particles. I would have much rather learned QFT this way than to have to accept as a starting point those weird classical fields and then to accept the field CRs as a new axiom.

In NRQM, invoking the CRs (or anti-CRs) of the fields is the same as POSTULATING symmetric (or antisymmetric) subspaces for the many-Boson (or many-Fermion) system.

Right. But again, I think you are focusing on the distinction between CRs vs anti-CRs. But even if we consider only bosonic excitations, the question can still be asked: what is the principle behind the need to invoke CRs of the fields and their conjugate momenta? The answer is the need to have a *many-body* system. It has nothing to do with position/momentum CR of a point particle.

And this is one of my pet peeves. One invokes the analogy with the position/momentum CR of NRQM to justify the axiom of the CRs of quantum fields. But later one realizes that the CRs of the quantum fields have nothing to do with position/momentum CRs. They have to do with varying number of particles. So it feels like the analogy used to postulate the CRs in the first place is a cheat.

(to be honest, I am still wondering if there is something deep going on here that I am missing. On one hand the CRs can be derived simply by considering varying number of particles. On the other hand, they can be derived by assuming a continuum analogue of position/momentum CRs on fields. Is the fact that the two approaches lead to the same result "deep" or not? I am still unclear about this. And I really want to understand this because I feel that once I can answer this to my satisfaction, no matter what the answer is, I will finally understand QFT at more than a very shallow level)

In hindsight, this is painfully obvious.

... Whoops !

I hope this helps make my point of view sound a bit less crazy

Regards

Patrick

 

[nrqed]

A book that does this much better than any "high energy physics" QFT book is "optical coherence and quantum optics" by Mandel and Wolf.

Thank you for the suggestion, Patrick. I will try to get my hands on a copy. I always found it so strange that QFT books don't cover the crucial topic of the correspondence between the quanized EM fields and classical EM. This seems to me to be one way to make the "quantization of classical fields" less of a huge non sequitur.

Thanks again.

Pat

 

[masudr]

Hello again (after a bit of a break)

I was focusing on the fundamental CR, between X and P. My point is that we impose CRs on these quantities. And then when we get to a KG field or a Dirac field, we impose a *new* set of CRs on the field and conjugate field momenta...

Well, we don't impose a new set of CCRs: it's really the old set. But this time we are applying them to a new system.

That is:

• In NRQM we quantize hypothetical (or not) classical particle systems with any Hamiltonian. So, we upgrade observables to operators, and upgrade the Poisson brackets of position and conjugate momenta (from Hamiltonian particle mechanics) to commutators on those operators.
• In QFT, we quantize hypothetical (or not) classical fields with any Hamiltonian. So, we upgrade observables to operators, and upgrade the Poisson brackets of position and conjugate momenta (from Hamiltonian field mechanics) to commutators on those operators.

So it's not an additional imposition: it's just the standard quantization procedure (which I assume you are happy with for particles). Luckily for us, the Hamiltonian for some fields ends up looking like the Hamiltonian for an oscillator...

Masud

 

[nrqed]

Hello again (after a bit of a break)



Well, we don't impose a new set of CCRs: it's really the old set. But this time we are applying them to a new system.

That is:

• In NRQM we quantize hypothetical (or not) classical particle systems with any Hamiltonian. So, we upgrade observables to operators, and upgrade the Poisson brackets of position and conjugate momenta (from Hamiltonian particle mechanics) to commutators on those operators.
• In QFT, we quantize hypothetical (or not) classical fields with any Hamiltonian. So, we upgrade observables to operators, and upgrade the Poisson brackets of position and conjugate momenta (from Hamiltonian field mechanics) to commutators on those operators.

So it's not an additional imposition: it's just the standard quantization procedure (which I assume you are happy with for particles). Luckily for us, the Hamiltonian for some fields ends up looking like the Hamiltonian for an oscillator...

Masud

Thanks for your input. Discussing this with knowledgeable people helps me deepen my understanding.

I want to make two comments.

First, I don't see it as a trivial extension from the point particle case.
For a point particle, we impose a commutation relation between the momentum and position of the point particle.

For a field, we impose a commutation relation between the conjugate momentum (which is *not* the total momentum of the field!) and the field itself.

Now, for a point particle, the conjugate momentum and the actual momentum are the same, but it's not that clearcut for a field.

My question is this: you are teaching QFT, say, and you get to imposing the commutation relation saying that it's exactly the same as for a point particle but of course extended to the case of a continuous field.
And then a student stops you and says "but the momentum carried by the field is not the conjugate momentum \pi but the integral over all space of T^{0,i} so why don't we impose the CRs on this latter quantity instead of on the conjugate momentum?"

How would you justify from the point particle case only that the usual procedure is indeed the correct way to go? I don;t see why. And it's in that sense that I feel that there is an additional nontrivial step there.


My second comment is that *even* if it was the "same" quantization as for a point particle, the fact that those pesky "classical fields" are not observable is a nontrivial point, imho. One has observed the pointlike nature of electrons. One has never observed anything like a meson field or a Dirac field.

If one looks at QM textbooks and then QFT textbooks, it feels to me as if there is a huge conceptual gap there. It's totally discontinuous, from a logical point of view. It's a bit as if one would teach SR and then, when getting to GR, one would start with "now let's assume that the metric is a dynamical quantity and that it depends on the stress-energy tensor in the following way.

Sure, it works, but it would sound like voodoo physics. It would sound like someone got up one morning and had a completely unmotivated crazy idea. Of course, the way to teach the subject is to start with the equivalence of inertial and gravitational mass, the fact that gravity is therefore a kind of fictitious force akin to those observed in noninertial frames and on and on. As for QFT textbooks, the logical reasoning between NRQM and QFT is absent. There is no conceptual continuity from one to the other. What they say is essentially "we will now quantize imagined classical fields in a way that is a nontrivial extension of the quantization for point particles and you will see that it makes sense because it works!"

Patrick

 

[masudr]

For a field, we impose a commutation relation between the conjugate momentum (which is *not* the total momentum of the field!) and the field itself.

Now, for a point particle, the conjugate momentum and the actual momentum are the same, but it's not that clearcut for a field.

Well that's not all true. We actually impose the CCRs on the conjugate momenta, even for particles. The conjugate momentum is not the momentum. Instead, it is given by

\pi_i=p_i = \frac{\partial L}{\partial \dot{q}_i}

where L is the Lagrangian of the system, and the "q"s are the generalized co-ordinates. The momenta conjugate to Cartesian co-ordinates, x, y, z are m\dot{x}, m\dot{y}, m\dot{z} respectively. A simple case where the conjugate momentum isn't mv is for angular co-ordinates; in fact it's

p_\theta=mr^2\dot{\theta} - \frac{\partial V}{\partial \dot{\theta}},

since the Lagrangian is

L = \frac{1}{2}m\left(\dot{r}^2 + r^2\dot{\theta}^2\right) - V(r, \theta, \dot{r}, \dot{\theta})

Side-issue: this misunderstanding often arises from studying QM without properly studying Lagrangian and Hamiltonian classical mechanics. This is one of my pet-peeves with the standard physics undergrad courses! The reason this issue rarely surfaces is because it is the conjugate momentum that becomes -i\hbar\partial / \partial x not the plain old momentum. So when studying a particle in an external EM field in NRQM, we never have to worry that p_x=m\dot{x} - qA_x.

My question is this: you are teaching QFT, say, and you get to imposing the commutation relation saying that it's exactly the same as for a point particle but of course extended to the case of a continuous field.
And then a student stops you and says "but the momentum carried by the field is not the conjugate momentum \pi but the integral over all space of T^{0,i} so why don't we impose the CRs on this latter quantity instead of on the conjugate momentum?"

I would refer him to the analytical classical mechanics course that our institution runs In all seriousness, I would probably quickly demonstrate the difference between the momentum T^{0,i}, and the conjugate momentum p_i = \partial L / \partial \dot{q}_i.

One has observed the pointlike nature of electrons. One has never observed anything like a meson field or a Dirac field.

I don't really see why that should be an objection! Of course one has never observed a classical Dirac field. But that's because the world is quantum (apparently), and not classical! The only reason we do observe EM classically is because it happens to allow coherent state excitations (I'm sure there's a whole bunch of other reasons...)

Conversely, we have never observed the pointlike nature of quarks, nor quark-antiquark creation/annihilation processes etc. but we still formulate a field theory based on those ideas. Your argument (which I believe says that only observed ideas should be taught) would imply that we couldn't teach QCD via the Weinberg approach. For that approach says we have varying particle numbers and SR, and this leads to QFT. But we have never seen varying numbers of quarks; in fact no quarks at all!

As we have both agreed, this is all a matter of taste. But perhaps (I think you said) the Weinberg approach takes longer to get to the same point? I guess that must be the real reason why most courses/texts adopt to choose the quantization route.

It's funny how we are discussing the merits to merely an approach to physics.

Masud.

PS. I definitely agree with you that the emergence of the classical EM field from QED should be explained in detail in major QFT texts.

 

[nrqed]

Well that's not all true. We actually impose the CCRs on the conjugate momenta, even for particles. The conjugate momentum is not the momentum. Instead, it is given by

\pi_i=p_i = \frac{\partial L}{\partial \dot{q}_i}

where L is the Lagrangian of the system, and the "q"s are the generalized co-ordinates. The momenta conjugate to Cartesian co-ordinates, x, y, z are m\dot{x}, m\dot{y}, m\dot{z} respectively. A simple case where the conjugate momentum isn't mv is for angular co-ordinates; in fact it's

p_\theta=mr^2\dot{\theta},

since the Lagrangian is

L = \frac{1}{2}m\left(\dot{r}^2 + r^2\dot{\theta}^2\right) - V(r, \theta)

Side-issue: this misunderstanding often arises from studying QM without properly studying Lagrangian and Hamiltonian classical mechanics. This is one of my pet-peeves with the standard physics undergrad courses! The reason this issue rarely surfaces is because it is the conjugate momentum that becomes -i\hbar\partial / \partial x not the plain old momentum. So when studying a particle in an external EM field in NRQM, we never have to worry that p_x=m\dot{x} + qA_x.

We agree completely on this. It's one of my pet peeves as well!
My point was that this distinction between quantizing conjugate momentum vs "physical" momentum is not done in most undergraduate textbook. So that when one gets to field quantization, most people are not aware of the issue (*I* certainly wasn't when I tried to learn QFT. It was only much later, even after my first QFT class that I understood the issue). I guess my complaint was that QFT books don't make this clear, given the background that most students have.

But after reading your post, I realized that you are right. This should not be the job of the QFT books, but the job of the undergraduate QM books. You made a good point.

I would refer him to the analytical classical mechanics course that our institution runs In all seriousness, I would probably quickly demonstrate the difference between the momentum T^{0,i}, and the conjugate momentum p_i = \partial L / \partial \dot{q}_i.



I don't really see why that should be an objection! Of course one has never observed a classical Dirac field. But that's because the world is quantum (apparently), and not classical! The only reason we do observe EM classically is because it happens to allow coherent state excitations (I'm sure there's a whole bunch of other reasons...)

Conversely, we have never observed the pointlike nature of quarks, nor quark-antiquark creation/annihilation processes etc. but we still formulate a field theory based on those ideas. Your argument (which I believe says that only observed ideas should be taught) would imply that we couldn't teach QCD via the Weinberg approach. For that approach says we have varying particle numbers and SR, and this leads to QFT. But we have never seen varying numbers of quarks; in fact no quarks at all!

Well, if you jump right ahead to QCD then yes, I would have to concede that we must use the lagrangian approach to do things in a reasonably quick way. But it's not fair to consider the most difficult application as a way to decide how to learn (one might as well use the formalism of differntial geometry when teaching for the first time what a vector is!)

I think that in order to motivate the whole QFT formalism, I see only two good ways: either

A) quantize classical EM thoroughly, showing how photons arise and showing very explicitly the correspondence between quantum EMfields and classical EM fields and then making the bold step of assuming that it makes sense to quantize unobserved classical fields like the Dirac field and so on and to justify after the fact that it works.

OR: B) following Weinberg's approach. I don't think it's hard to accept that electrons or mesons are seen as discrete "packets" of energy and momentum. And I don't think it's hard to accept that we need a formalism in which their number is varying when we take into account relativity.

As we have both agreed, this is all a matter of taste. But perhaps (I think you said) the Weinberg approach takes longer to get to the same point? I guess that must be the real reason why most courses/texts adopt to choose the quantization route.

Maybe, but the speed at which one gets to applications is not a good criterion in learning a technique, imho. For example, someone could teach directly the rules of differentiation in calculus without proving them from the limit approach in order to "get faster" to applications. But the students would always be left feeling that even though differentiation has a lot of applications, it's all a bit magical.



The standard approach is indeed much faster. But it's because there is a lot of "why?"'s left aside. It's a bit as if one would teach GR by saying "hey! Let's make the metric a variable and let's assume it obeys this specific equation!" That would "cut to the chase". But it would leave everyone scartching their head about why this works.

It's funny how we are discussing the merits to merely an approach to physics.

Masud.

True. But I personally think that the way things are taught has a huge impact on students' comprehension and grasp of a topic. If an approach is used because it leads to applications as fast as possible to the detriment of another approach which shows how things can be built on previous knowledge, then I think it is a bad choice.

But again, it's my personal opinion. If I was teaching QFT, I could not get myself to say "so you know basic QM. Now we want to take into account relativity. Of course, the way to do this is to quantize relativistic *classical field theories* !"

I mean, nobody has ever observed a classical Dirac field, but everybody is aware of the effects of electrons as discrete packets of energy and momentum. The classical Dirac field is a trick used to get to the quantized Dirac field which is then used to get to what are actually observed, the electrons (we observe the effects of the electrons of course but you know what I mean). So it makes more sense to me to start from electrons than to start from an imaginary (not in the sense of complex numbers :-) ) classical Dirac field. It's a neat *mathematical* shortcut to quantization but it obscures completely the physics , imho.

At least, it bothered *me* for a long time until I read Weinberg which really clarified for me what the fields were (a bookeeping device, not a fundamental quantity). It's only then that QFT really made sense for me.

But I have never met anyone else (on the forums or in real life) who felt that going from NQRM to quantizing imaginary classical fields was a non sequitur.

So I am starting to think that my brain is just wired completely wrong!

Thank you for your interesting post!

Pat

 

[reilly]

In fact, quantization of fields is simply a rewrite of conventional quantum formalisms. That is, one does not need creation and destruction operators to do QFT. See any discussion of "second quantized" many body theory. Don't forget that the idea, due originally to V. Fock, is, more or less, to replace states with operators, which turns out to be very convenient when dealing with systems that do not conserve particle number.

The E&M field turns out to be a very tricky case because of gauge invariance, which is why its treatment is often deferred. But, re a Lagrangian approach, the E&M field is blessed with momentum, which can be deduced from the reasoning of Poynting's Thrm, and can be seen from the standard lagrangian approach as discussed in Jackson, and many E&M and mechanics texts. Momentum + imagination says CCR are worth checking out; then a bit of work indicates that the CCR approach is nothing more than a conventional Fock space approach to keeping track of photons.

That is, for example, a quantized Dirac field is a operator constructed of well chosen (creation and destruction) operators that allow easy construction of states and interactions, just as is done in NR theory. One could do QED without "2nd quantization", but it would be very tedious indeed.

Regards,
Reilly

 

[Eye_in_the_Sky]

quantization of the Schrödinger field

A suitable choice of Lagrangian density is

L = ψ*(i∂_t - H_x)ψ ,

where

H_x = -∂_x²/(2m) + V(x) .

The momentum conjugate to ψ is

π = ∂L/∂(∂_tψ) = iψ* ,

while that conjugate to ψ* vanishes, since

∂L/∂(∂_tψ*) = 0 .

The associated field Hamiltonian is then found to be

H_class = -i ∫dx πH_xψ .
___________________
___________________

QUANTIZATION

Promote the fields to field operators:

ψ → Ψ ,

π = iψ* → Π = iΨ^† .

The associated field Hamiltonian becomes

H = -i ∫dx ΠH_xΨ ,

and the equations of motion are given by

i∂_tΨ = [Ψ,H]

and

i∂_tΠ = [Π,H] .

We are at liberty to choose for the fields either one of:

(a) equal-time commutation relations

[Ψ(x,t),Π(x',t)] = iδ(x - x') ,

[Ψ(x,t),Ψ(x',t)] = [Π(x,t),Π(x',t)] = 0 ;

or

(b) equal-time anticommutation relations

{Ψ(x,t),Π(x',t)} = iδ(x - x') .

{Ψ(x,t),Ψ(x',t)} = {Π(x,t),Π(x',t)} = 0 .

In both cases the formulation is self-consistent and the equations of motion for the fields turn out to be

i∂_tΨ = H_xΨ ,

and

-i∂_tΠ = H_xΠ .
___________________

Using Π = iΨ^† in the above, the quantization formulation reduces to:

i∂_tΨ = H_xΨ ,

and either one of

(a) [Ψ(x,t),Ψ^†(x',t)] = δ(x - x') ,

[Ψ(x,t),Ψ(x',t)] = [Ψ^†(x,t),Ψ^†(x',t)] = 0 ,

or

(b) {Ψ(x,t),Ψ^†(x',t)} = δ(x - x') ,

{Ψ(x,t),Ψ(x',t)} = {Ψ^†(x,t),Ψ^†(x',t)} = 0 .

 

[Eye_in_the_Sky]

interpretation of the quantized Schrödinger field

Begin with

i∂_tΨ(x,t) = H_xΨ(x,t) ... [1] ,

and assume (for ease of notation) that H_x has a discrete, nondegenerate spectrum. Let φ_E(x) denote the corresponding eigenfunctions, so that

H_xφ_E(x) = Eφ_E(x) ... [2] .

Since the eigenfunctions φ_E(x) form a complete set, the field operator Ψ(x,t) can be expanded in terms of them. That is, we can write

Ψ(x,t) = ∑_E a_E(t)φ_E(x) ... [3] ,

which when substituted into [1] in connection with [2] yields

ida_E(t)/dt = Ea_E(t) ,

so that

a_E(t) = e^-iEt a_E ... [4] .

Substituting this result back into [3] then gives

Ψ(x,t) = ∑_E a_Eφ_E(x)e^-iEt ... [5a] ,

and therefore,

Ψ^†(x,t) = ∑_E a_E^†φ_E*(x)e^iEt ... [5b] .
___________________

Consider the two cases (as explained in the previous post):

(a) [Ψ(x,t),Ψ^†(x',t)] = δ(x - x') , [Ψ(x,t),Ψ(x',t)] = [Ψ^†(x,t),Ψ^†(x',t)] = 0 ;

(b) {Ψ(x,t),Ψ^†(x',t)} = δ(x - x') , {Ψ(x,t),Ψ(x',t)} = {Ψ^†(x,t),Ψ^†(x',t)} = 0 .

In light of the expressions [5a] and [5b] for Ψ(x,t) and Ψ^†(x,t), the above two cases are seen to be equivalent to:

(a') [a_E,a_E'^†] = δ_EE' , [a_E,a_E'] = [a_E^†,a_E'^†] = 0 ;

(b') {a_E,a_E'^†} = δ_EE' , {a_E,a_E'} = {a_E^†,a_E'^†} = 0 .

Thus, the operators a_E^† and a_E can be interpreted as creation and annihilation operators with respect to the eigenfunction "modes" φ_E(x) of H_x, where case (a) corresponds to bosonic excitations, and case (b) corresponds to fermionic excitations.

In particular we can write, for all E in the spectrum of H_x,

a_E|0> = 0 ... [6a] ,

and

a_E^†|0> = |φ_E> ... [6b] .
___________________

Consider the object Ψ^†(x,0)|0>. From equations [5b] and [6b], we have

Ψ^†(x,0)|0> = ∑_E |φ_E>φ_E*(x) ,

and upon writing φ_E*(x) = <φ_E|x>, the last relation becomes

Ψ^†(x,0)|0> = ( ∑_E |φ_E><φ_E| ) |x> ;

that is,

Ψ^†(x,0)|0> = |x> .

But from [5a] and [6a], we have

Ψ(x,0)|0> = 0 .

From these last two relations, in conjunction with the equal-time commutation or anticommutation relations for the field (i.e. cases (a) or (b) above), it follows that:

Ψ^†(x,0) and Ψ(x,0) can be interpreted as creation and annihilation operators for a particle at the position x.

 

[nrqed]

Hi Eye in the Sky,
That's an interesting couple of posts. I am not sure what your targeted audience is, but just in case you had me in mind, I am familiar with the formalism, applied to relativistic fields as well as to nonrelativistic systems (including Schroedinger's equation). My questions were more about the motivation concerning quantizing fields. To me it has always seemed as an "ad hoc" starting point, with no logical connection with NRQM and pecial relativity. I know that it does work, I just feel that there is no logic in doing this after learning NRQM and SR. It makes it sound like magic. If I was teaching QFT I would not feel comfortable about presenting things this way.

I will just make a couple of short comments

A suitable choice of Lagrangian density is

L = ψ*(i∂_t - H_x)ψ ,

This is unrelated to my other comments but: how did you get this Lagrangian starting from Schroedinger's equation? I am asking because something has bothered me about this step (and that's what I discuss in the thread "What's wrong with this Lagrangian"). It seems to me possible to write a lagrangian which contains separate terms in Psi and Psi^* and which does lead to the eqs fro Psi and Psi^*. And yet this Lagrangian leads to problems when getting to canonical quantization because the momentum conjugate to Psi is Psi itself. So I am wondering what the rules are for constructing a Lagrangian besides obtaining the correct eoms from the Euler-Lagrangian equations!

Going back to my central comments...

H_x = -∂_x²/(2m) + V(x) .

The momentum conjugate to ψ is

π = ∂L/∂(∂_tψ) = iψ* ,

while that conjugate to ψ* vanishes, since

∂L/∂(∂_tψ*) = 0 .

The associated field Hamiltonian is then found to be

H_class = -i ∫dx πH_xψ .
___________________
___________________

QUANTIZATION

Promote the fields to field operators:

ψ → Ψ ,

π = iψ* → Π = iΨ^† .

The associated field Hamiltonian becomes

Ok. And this is where I will play devil's advocate
This is always the step that felt like a stumbling block to me when I learned this and it really cleared up only when I finally reda Weinberg.

Playing the devil's advocate, I could ask the following

"Professor, *why* do we promote the wavefunction to an operator??"
"What are we trying to do here anyway??"

You would probably answer "We want to be able to describe a many-body system", right? (I don't want to "put words in your mouth"!)

But the the students would say "Why is promoting the wavefunction to an operator accomplishing this?? I don't see at all the connection"

And you would probably reply "Wiat, we will go through all the consequences of this and you will see that doing this does end up leading to a many-body system of particles which obey the N-body Schroedinger's equation!"

Sure enough, after some work this comes out, but the students may feel that it's a lot of "hocus-pocus" . It *does* work, but what is the logical connection between promoting the wavefunction to the status of an operator, introducing the CRs between the field and its conjugate momentum, etc etc... and obtaining a many-body system?!? It sounds quite mysterious (at least to me).

I mean, if I had never ver heard of QFT and I knew NRQM, and I waned to describe a many-body system, I would never go "but that's obvious, I just have to treat the wavefunction as an operator! And I will defined its conjugate momentum by treating the whole system as a classical field theory (with *complex waves*, mind you) and then impose CRs and all that is obviously the way to get a many-body system.
I would never have thought that! But of course, maybe this is an obvious thing to do for some other people.

On the other hand, Weinberg's approach does not involve huge leaps of faith like this, besides the need to have a many-body system. The fields become completely secondary to the many-body aspect. After a few simple systems, it becomes clear that the Lagrangian approach is more efficient in building relativistically covariant quantum systems, but it is by now clear that this is all a bookkeeping "trick".

A last question from a nagging student:
The student asks
"Professor, we are promoting the wavefunction to the status of an operator. So this *really* is a "second quantization"! We quantized first when we promoted X and P of classical physics to the status of operators in order to find Schrodinger's equation. Now we quantize the wavefunction so we are quantizing on top of quantizing, no? But I read all the time that "second quantization" is a misnomer, that we are really quantizing only once.!"

That nagging student!


regards

Patrick

 

[Eye_in_the_Sky]

Hi Eye in the Sky,
That's an interesting couple of posts. I am not sure what your targeted audience is ...

Those two posts are intended for all readers of this thread ... especially me.

So many points are being made in this thread, and I want to see how they stand when "relativity" is taken OUT of the picture. Once things have been properly understood there – in the nonrelativistic domain – then it makes sense to put the "relativity" back in and see just how things change.

But, in order to see things clearly (albeit, in only a nonrelativistic 'light'), I needed to work out the details. And in order to discuss those details with the group here in the forum, I felt it would be helpful to post them in a clear and concise way.

My questions were more about the motivation concerning quantizing fields. To me it has always seemed as an "ad hoc" starting point, with no logical connection with NRQM and pecial relativity.

Like I said, let's take "relativity" OUT of the picture. Then, the question becomes one about the canonical quantization of the Schrödinger field and its connection to NRQM. The "logic" of this connection is purely on the level of abstract formalism, and as far as pedagogy is concerned, I agree with you 100% – this is not the way to begin.

Let's take a closer look at what we are dealing with. We want to start from a single-particle nonrelativistic quantum theory and eventually reach a
many-particle relativistic quantum theory. From a pedagogical point of view, I think the best sequence to follow is:

1/ Put in the "many";

2/ Go back to the "one", and put in the "relativity";

3/ Put in the "many".

Right now I am thinking about #1, and that it ought to be done twice. Afterwards, a comparison needs to be made.

The first time around, one looks at the N-particle Hilbert space

H₁ (x) ... (x) H_N ,

and says that this is fine for distinguishable particles, but for indistinguishable particles the space needs to be restricted to either the "symmetric" subspace (Bosonic case) or the "antisymmetric" subspace (Fermionic case). Then, creation and annihilation operators need to be introduced as a means of 'going back and forth' from higher and lower values of N, in essence, to construct the Fock space. (Is this basically what Weinberg's approach amounts to when transferred over to the nonrelativistic domain?)

The second time around, one looks at the canonical quantization of the Schrödinger field according to the 'mysterious' method of Heisenberg.

Afterwards, the results of each of the two approaches are to be compared.

And that is one of the things I wanted to make accessible through my last two posts. In particular, I wanted to 'set the stage' to be able

to begin to probe the question of the 'mysteriousness' of field quantization (... without any of the complications of "relativity").

So, now I am able to begin to appreciate the following question, posed several posts earlier (except that in my case, I am only in the nonrelativistic domain):

(to be honest, I am still wondering if there is something deep going on here that I am missing. On one hand the CRs can be derived simply by considering varying number of particles. On the other hand, they can be derived by assuming a continuum analogue of position/momentum CRs on fields. Is the fact that the two approaches lead to the same result "deep" or not? I am still unclear about this. And I really want to understand this because I feel that once I can answer this to my satisfaction, no matter what the answer is, I will finally understand QFT at more than a very shallow level)

Tell me, does your question still hold for you with regard to the Schrödinger field?
______________________

Next, I would like to play my own devil's advocate along with yours, because as I said, as far as pedagogy is concerned, I agree with you 100%.

"Professor, *why* do we promote the wavefunction to an operator??"
"What are we trying to do here anyway??"

"We are just taking the idea of (q,p) → (Q,P) from the case of discrete coordinates and applying it to the case of fields defined on a manifold. So, we take (ψ,π) → (Ψ,Π). A priori we have no idea of what is going to happen. Let's try it and see what we get."

You would probably answer "We want to be able to describe a many-body system", right? (I don't want to "put words in your mouth"!)

No. Like the prof said, "A priori we have no idea of what is going to happen. Let's try it and see what we get."

But the the students would say "Why is promoting the wavefunction to an operator accomplishing this [i.e. a many-body system description]?? I don't see at all the connection"

"Hey, who told you that? We do not yet know that that's going to give us that. In fact since we haven't even yet learned about what a Fock space is and how to build it up from a vacuum state |0> in connection with creation and annihilation operators, when it will come to the interpretation of our quantized field, we will have to resort to pulling many things out of a hat. None of what we do will be transparent to any of you at all."

Etc ...
______________________

Your next point, I will need to think about some more:

On the other hand, Weinberg's approach does not involve huge leaps of faith like this, besides the need to have a many-body system. The fields become completely secondary to the many-body aspect. After a few simple systems, it becomes clear that the Lagrangian approach is more efficient in building relativistically covariant quantum systems, but it is by now clear that this is all a bookkeeping "trick".

______________________

A last question from a nagging student:
The student asks
"Professor, we are promoting the wavefunction to the status of an operator. So this *really* is a "second quantization"! We quantized first when we promoted X and P of classical physics to the status of operators in order to find Schrodinger's equation. Now we quantize the wavefunction so we are quantizing on top of quantizing, no? But I read all the time that "second quantization" is a misnomer, that we are really quantizing only once.!"

When we promoted the classical (x,p) to operators, we did not find the Schrödinger equation. We found operators (X,P) which then satisfied the same classical equations of motion as (x,p). And in this sense, what we are doing with the fields is the same thing: we promote the classical (ψ,π) to operators (Ψ,Π) which then satisfy the same classical equations of motion as (ψ,π).
______________________
______________________

This is unrelated to my other comments but: how did you get this Lagrangian starting from Schroedinger's equation?

Seek an action functional S[ψ,ψ*] which pays complete respect to the symmetry between ψ and ψ*. Specifically, look for S such that

δS = ∫dt ∫dx [ (δψ*)(i∂_t - H_x)ψ + (δψ)(-i∂_t - H_x)ψ* ] .

Then, 'break' that symmetry by means of a judicious integration by parts applied to the second term of the integrand (once for the (δψ)∂_tψ* subterm, and twice for the (δψ)∂_x²ψ* subterm), so that the variation of the action becomes

δS = ∫dt ∫dx [ (δψ*)(i∂_t - H_x)ψ + ψ*(i∂_t - H_x)(δψ) ]

= ∫dt ∫dx δ [ ψ*(i∂_t - H_x)ψ ] .

 

[vanesch]

Ok. And this is where I will play devil's advocate
This is always the step that felt like a stumbling block to me when I learned this and it really cleared up only when I finally reda Weinberg.

Playing the devil's advocate, I could ask the following

"Professor, *why* do we promote the wavefunction to an operator??"
"What are we trying to do here anyway??"

You would probably answer "We want to be able to describe a many-body system", right? (I don't want to "put words in your mouth"!)

But the the students would say "Why is promoting the wavefunction to an operator accomplishing this?? I don't see at all the connection"

This is something that bothered me also quite a lot during my first introductions to QFT (which were very "oldfashioned" of the Bjorken and Drell style).

The way I understood things later was different: it was: let us forget about particles, which are a nuisance anyway. Let us look at *other* classical models of systems to which we can apply quantum rules, and see if they have anything to do with nature.

And one of those other systems are "fields": functions over spacetime satisfying a certain local dynamical condition (usually in the form of a partial differential equation). The functions can be real functions, or vector functions, or other stuff.

A first approach is then to consider *free* fields, which have linear dynamics. We know of one field of course, the EM field, but the question is: what other of these kinds of beasts could exist ? What other kinds of fields, with linear dynamics, could exist ? There can be quite a lot, but the number of possibilities is strongly reduced when you require them to be relativisitically invariant, and of, at most, second order in time. In fact, you end up with only a handful of possibilities! There is the KG equation for scalar fields, the Dirac equation for spinor fields (and a few variants, such as majorana spinors and so on), there are vector fields like Maxwell's equations...

I didn't see this as "this is the way it should be" but simply as "this is another kind of classical 'thing' which we can quantize, let's see what it gives us". And what it gives us, are, o ye o ye, particle-like behaviour.

Now, is this a lesson, that if we try to get away from particles, we get pulled back to them (so we shouldn't have given up on them in the first place) ? Or should we see the entire particle stuff as some thing that comes out of the approach of quantum theory, applied to fields ? And that all we always thought as being particles, were just quantum manifestations of fields ? In the same way as we always saw the sun turn around the earth, but this is just a manifestation of the rotation of the Earth ?

It's hard to say what is the 'more fundamental' approach, especially as both are equivalent.

 

[nrqed]

This is something that bothered me also quite a lot during my first introductions to QFT (which were very "oldfashioned" of the Bjorken and Drell style).

The way I understood things later was different: it was: let us forget about particles, which are a nuisance anyway.

Well, *I* think that fields are a nuisance I think that the physical effects little packets of energy that we called electrons are more dierctly observable than a classical approximation to the Dirac field

Let us look at *other* classical models of systems to which we can apply quantum rules, and see if they have anything to do with nature.

And one of those other systems are "fields": functions over spacetime satisfying a certain local dynamical condition (usually in the form of a partial differential equation). The functions can be real functions, or vector functions, or other stuff.

A first approach is then to consider *free* fields, which have linear dynamics. We know of one field of course, the EM field, but the question is: what other of these kinds of beasts could exist ? What other kinds of fields, with linear dynamics, could exist ? There can be quite a lot, but the number of possibilities is strongly reduced when you require them to be relativisitically invariant, and of, at most, second order in time. In fact, you end up with only a handful of possibilities! There is the KG equation for scalar fields, the Dirac equation for spinor fields (and a few variants, such as majorana spinors and so on), there are vector fields like Maxwell's equations...

I didn't see this as "this is the way it should be" but simply as "this is another kind of classical 'thing' which we can quantize, let's see what it gives us". And what it gives us, are, o ye o ye, particle-like behaviour.

Now, is this a lesson, that if we try to get away from particles, we get pulled back to them (so we shouldn't have given up on them in the first place) ? Or should we see the entire particle stuff as some thing that comes out of the approach of quantum theory, applied to fields ? And that all we always thought as being particles, were just quantum manifestations of fields ? In the same way as we always saw the sun turn around the earth, but this is just a manifestation of the rotation of the Earth ?

It's hard to say what is the 'more fundamental' approach, especially as both are equivalent.

I agree with you. But you know my feelings about this... A classical "wavefunction field"? A classical KG field? a calssical Dirac field? Who ordered that to paraphrase Rabi. It sounds to me like a purely formal exercise... a "solution in search of a problem". Until...lo and behold!, one gets particles out of it! Mystery...hocus-pocus...voodoo ??
Of course, if we do that on those imaginary classical fields, why don't we *really* go all the way. Why not quantizing *any* classical field equation we may think of (not just relativistic..after all if we start the program with no specific goal in my mind, there should be no specific limit). Why not trying to quantize the Navier-Stokes eqs...or the equations of thermodynamics (we might have to work a bit at seeing if it makes sense to take about generalized momenta, etc, but hey, we are just trying things here). Why not quantize Schrodinger's equation with no "i" factor and a classical field? The point, it seems to me, is that it does not make sense to quantize any old classical field equation.

I know that the goal being to do relativistic physics we should work with Lorentz covariant field equations. But why trying to run before walking? If the idea is just to quantize fields, one should start with nonrelativistic equations, and then we are back to a huge list of eqs, including the few I mentioned above.

Not only that. If our approach is simply "we have seen how to quantize things by promoting generalized coordinates and momenta to the status of operators in NQRM, now let's apply this to everything we can think of", then an obvious thing that should be done before even considering relativistic classical wave equations would be to consider a relativistic point particle (again, walking before running)!


On the other hand, if there is a way to start with a well-motivated (we want to describe a many-body theory) problem and we work from there and see the fields coming out as a byproduct, I find this more satisfying.

I find that the "particle" approach is so much more natural. One can do it for a nonrelativistic system of identical particles obeying Schroedinger's equation, on is naturally led to introduce a Fock space, operators that change the number of particles, operators that create particles of a specific energy or at a specific spacetime point (which are therefore field operators) etc etc. Going to a relativistic system imposes that the creation/annihilation operators must be combined in a certain way, and on and on.

Then, *after* having seen how things work in details for a nonrelativistic system, for the KG field, for the Dirac field, and the connection with a Lagrangian approach, I would see how it would make sense to switch to the lagrangian approach. But at least by then there would be a more solid understanding of what those weird quantum fields are. Imho.

I agree that the *maths* is the same in both approach. But I personally feel that the physical meaning of what is going on and the meaning of the quantum fields and of expressions like <0|Phi|1_k> and so on mean would be more solid.

But I realize it's a question of taste to some extent.

Regards

 

[nrqed]

Those two posts are intended for all readers of this thread ... especially me.

Ok! I actually have wanted to present the same results as you are obtaining but using my point of view (starting from the many-body aspect) for a while. I kow it would help *me* clarify my thoughts, even if nobody else reads me!

So many points are being made in this thread, and I want to see how they stand when "relativity" is taken OUT of the picture. Once things have been properly understood there – in the nonrelativistic domain – then it makes sense to put the "relativity" back in and see just how things change.

I agree 100%!

But, in order to see things clearly (albeit, in only a nonrelativistic 'light'), I needed to work out the details. And in order to discuss those details with the group here in the forum, I felt it would be helpful to post them in a clear and concise way.

yes, it is a good idea. As I sadi above, I have wanted to do this myself for a while now, using the approach I prefer.

Like I said, let's take "relativity" OUT of the picture. Then, the question becomes one about the canonical quantization of the Schrödinger field and its connection to NRQM. The "logic" of this connection is purely on the level of abstract formalism, and as far as pedagogy is concerned, I agree with you 100% – this is not the way to begin.

Oh! Well, it's nice to hear you say that! Because my point is almost exclusively about pedagogy. I am not disputing that the traditional approach is flawed in any way, of course. I am mostly complaining about the way textbooks (whose first goal is to *teach*) almost invariably start with "well, let's quantize classical fields that have never been observed now".

I will get back to some of the things you mentioned in your post, but for now I just wanted to ask one thing..

Seek an action functional S[ψ,ψ*] which pays complete respect to the symmetry between ψ and ψ*. Specifically, look for S such that

δS = ∫dt ∫dx [ (δψ*)(i∂_t - H_x)ψ + (δψ)(-i∂_t - H_x)ψ* ] .

Then, 'break' that symmetry by means of a judicious integration by parts applied to the second term of the integrand (once for the (δψ)∂_tψ* subterm, and twice for the (δψ)∂_x²ψ* subterm), so that the variation of the action becomes

δS = ∫dt ∫dx [ (δψ*)(i∂_t - H_x)ψ + ψ*(i∂_t - H_x)(δψ) ]

= ∫dt ∫dx δ [ ψ*(i∂_t - H_x)ψ ] .

My question is obviously: why not take

δS = ∫dt ∫dx [ (δψ)(i∂_t - H_x)ψ + (δψ*)(-i∂_t - H_x)ψ* ?

Regards

patrick

 

[Eye_in_the_Sky]

Ok! I actually have wanted to present the same results as you are obtaining but using my point of view (starting from the many-body aspect) for a while. I kow it would help *me* clarify my thoughts, even if nobody else reads me!

If you are talking about a presentation of Weinberg's approach, then I (for one) would definitely be interested in reading about that. Early on in this thread, I wanted to ask if that approach uses the advanced tools of quantum field theory, and if not, whether you would be able to give us some of the relevant details. But then I decided to wait a little bit and see how this thread would develop. Now I see the time is ripe to ask:

Does an understanding of Weinberg's approach require a knowledge of the advanced tools of QFT? If not, would you like to present some of the details?
________________

I will get back to some of the things you mentioned in your post ...

There are two main points which I am hoping you will come to address.

1) What does Weinberg's approach amount to when transferred to the nonrelativistic domain? Is it basically what I said in my post regarding the construction of the Fock space?

... one looks at the N-particle Hilbert space

H₁ (x) ... (x) H_N ,

and says that this is fine for distinguishable particles, but for indistinguishable particles the space needs to be restricted to either the "symmetric" subspace (Bosonic case) or the "antisymmetric" subspace (Fermionic case). Then, creation and annihilation operators need to be introduced as a means of 'going back and forth' from higher and lower values of N, in essence, to construct the Fock space.

2) Does your question (below) hold for you even with regard to the Schrödinger field?

On one hand the CRs can be derived simply by considering varying number of particles. On the other hand, they can be derived by assuming a continuum analogue of position/momentum CRs on fields. Is the fact that the two approaches lead to the same result "deep" or not? I am still unclear about this.

________________
________________

Regarding the action S[ψ,ψ*], for which one makes the "guess"

δS = ∫dt ∫dx [ (δψ*)(i∂_t - H_x)ψ + (δψ)(-i∂_t - H_x)ψ* ] ,

and finds that the "guess" actually 'pays off' since then (by integrating by parts)

δS = ∫dt ∫dx [ (δψ*)(i∂_t - H_x)ψ + ψ*(i∂_t - H_x)(δψ) ]

= ∫dt ∫dx δ [ ψ*(i∂_t - H_x)ψ ] ,

you asked:

... why not take

δS = ∫dt ∫dx [ (δψ)(i∂_t - H_x)ψ + (δψ*)(-i∂_t - H_x)ψ* ?

The only way a "guess" of this kind can possibly 'pay off' is if the form

(δu)v + u(δv)

can be achieved by means of appropriate integrations by parts. And, in order for that to be at all possible, each one of the two terms in the integrand of the candidate δS must contain both ψ and ψ*. This condition then rules out the candidate you suggest.

 

[samalkhaiat]

I have always been annoyed by the standard introduction to quantum field theory in which one imposes those commutation relations on classical fields. This always seemed very strange to me. What the heck are those classical fields that one quantizes? (no one has ever seen a classical approximation of a meson field or of a fermion field!)

As far as I know, only Weinberg presents things in a way that really makes sense, in which the starting point is not the quantization of classical fields but the need to allow the number of particles to vary, so the need to introduce annihilation and creation operators. Then , imposing Lorentz invariance, one is led to quantum fields obeying the usual commutation relations (postulated from the start in standard presentations).

Hi Patrick,

20 years ago I was introduced ,for the first time, to the classical field theories and almost immediately I asked my professor the following question:
"what do Dirac wavefunctions have in common with the classical fields of EM and gravity?"
My reason to ask about fermions and not about bosons was (I thought) obvious, because in principle (Pauli) only fermions have no classical limit.
Any way, His answer was:

1) it (Dirac spinor) is a C-number, smooth function over spacetime M^4 .

2) it has a definite transformation law with respect to Poincare' group.

3) a scalar action can be constructed out of Dirac spinors.

4) (1)+(2) define a relativistic classical field, (a general classical field is defined to be a C-number map from spacetime to some target space):
\phi : M^{n} \rightarrow F

5) classical field theory is nothing but (1)+(2)+(3), i.e the pair (M^{n}, F) together with \delta S[\phi] = 0 .

Defined this way, even classical mechanics is a field theory on (1+0)-dimensional spacetime (affine time):
(t=M^{1}, \math R^{3} )
\delta S[x] = 0

The point is this; In the linear GR, the classical field is given by the symmetric 2nd rank tensor representation of Poincare' group. We also use the vector representation to describe the classical EM field, eventhough it (the vector potential) is not an observable.
So, at least mathematically, why not include the other irreducible representations of Poincare' group, that is the scalar and the spinor representations, especially when the aim is formulating QFT using the quantization "law"

{A,B} \rightarrow i \left[\hat{A},\hat{B}\right]

So when you quantize the "fields" (X,P), you get a "QFT" on the affine time "spacetime". This is just QM. Quantizing the fields \left( \phi ,\pi \right) will give us QFT on M^{n} .
Notice that the above mathematical structure has no room for (the often misleading) 2nd quantization.

Let me summarise by saying that the canonical formalism consists of:

i) postulating a global symmetry group G.

ii) working out all irreducible representations of G.

iii) constracting a real, G-scalar Lagrangian density out of these IRR's and their 1st order derivatives at the same point. And
iv) making this Lagrangian invariant under the local G group. This gives rise to interactions.
Up to this point, everything is classical and the theory is CFT.

Now, bringing in Hilbert space together with { , } --> i[ , ] will automatically produce a G-invariant as well as Poincare'-invariant interactive QFT. So you almost have everything. you see, there are good reasons for the saying:

"give me the Lagrangian, I give you everything"

Regarding what you called "the Weinberg's approach" :

a) I would like to remined you that Weinberg (in the first few chapters of his book) was following the so-called constractive (axiomatic) QFT, also known as the S-matrix formalism. I know 2 old but classic books on this subject:

BOGOLUBOV, LOGUNOV & TODOROV, "INTRODUCTION TO AXIOMATIC QUANTUM FIELD THEORY", 1969.

JOST, "THE GENERAL THEORY OF QUANTIZED FIELDS", 1965.

and one relatively new "good" book

JAN TOPUSZANSKI, "AN INTRODUCTION TO SYMMETRY AND SUPERSYMMETRY IN QUANTUM FIELD THEORY", 1991.

b) the current form of axiomatic QFT shows that we still can not go too far without a Lagrangian. In chapter 7, where Weinberg is forced to introduce the canonical formalism, he explains the main difficulty of the approach, namely, guessing the form of the Hamiltonian;
(read the last 2 paragraphs on page 292).

c) while the Lagrangian formalism is always possible, the constractive approach is absolutely hopeless in each of the cases where;
1) the pair (M^n , F) is equipped with local (Lie or graded-Lie) algebraic structure; as in the local (symmetric or supersymmetric) non-abelian theories.

2) both M^n and F are Riemannian manifolds; string theory and non-linear sigma model.

3) different combinations of (1) and (2) like superstring and supergravity.

4) M^n is non-commutative spacetime.

5) different combinations of (4) with the rest.

6) all the above with nontrivial topology on F; topological field theories.

regards

sam

 

[nrqed]

If you are talking about a presentation of Weinberg's approach, then I (for one) would definitely be interested in reading about that. Early on in this thread, I wanted to ask if that approach uses the advanced tools of quantum field theory, and if not, whether you would be able to give us some of the relevant details. But then I decided to wait a little bit and see how this thread would develop. Now I see the time is ripe to ask:

Does an understanding of Weinberg's approach require a knowledge of the advanced tools of QFT? If not, would you like to present some of the details?

In that case, if you are interested, I will try to post something (just presenting the main points at first; I will get more detailed if anybody is interested).

I am not sure what you mean by "advanced tools". His approach gives the same results as the standard approach (via fields) and actually, after a few examples he admits (as I do!) that the field approach makes things easier. But I find his approach, as an introduction to the concept of QFT, much more pedagogical and intuitive.

In any case, to address your question, I guess it depends what you define as "advanced tools". The end result is the same except that the starting point and the motivation are quite different. One starts from the desire of having a many-body description and the introduction of the Fock space and of annihilation/creation operators. From the basic meaning of those operators, their commutation relation is deduced. The field is introduced as a *byproduct*, not as a starting point. And there is no classical field introduced, one gets directly to the quantum fields. The CRs between the field and conjugate momenta *follow* from the Crs (or antiCRs) between the creation/annihilation operators and these are introduced very naturally (imho), being due to the very nature of the creation/annihilation operators.

________________
There are two main points which I am hoping you will come to address.

1) What does Weinberg's approach amount to when transferred to the nonrelativistic domain? Is it basically what I said in my post regarding the construction of the Fock space?

The nonrelativistic case is handled exactly the same way as the relativistic case. The only difference is the need to impose that one-particle state created by the field operator have definite transformation properties under Lorentz transformations. For example, for a real KG field, this forces one to combine annihilation and creation operators (of positive energy states) together, which was not required in the non-relativistic case (and this leads to the fact that in the KG case, the momentum conjugate to Phi is the time derivative of Phi whereas it is Phi^* in the nonrelativistic case).

2) Does your question (below) hold for you even with regard to the Schrödinger field?

My question was about the "deep" connection between the CRs of the field and conjugate momentum and the CRs of the annihilation/creation operators. My thoughts on this have evolved a bit. In Weinberg's approach, the basic CRs are simply between the annihilation/creation operators. The CRs between the fields and conjugate momenta are simply wasy to encompass this information.

Thinking in terms of independent degrees of freedom, it seems to me now that the conventional approach is simply a way to isolate the independent degrees of freedom. It is instructive to see that for an equation linear in time, the conjugate field momentum is the complex conjugate of the field (for the nonrelativistic Schrodinger case or for the Dirac equation) whereas it is the time derivative for the real KG field, and it's the complex conjugate of the time derivative of the field for the complex KG field. But in the end, th CR's between the field and the conjugate momentum always boils down to isolating the creation and annihilation operators and specifying the CRs between them (by CRs here I include antiCRs, i.e. I am talking in the most general context). So I think that the field approach is just a clever way of isolating the independent degrees of freedom and giving their CRs which, in the end, always amounts to specifiying the CRs between the annihilation and creation operators.

So, looking only at the CRs, it seems to me that the field approach is a very indirect (but very clever) way of giving the CRs between the annihilation and creation operators. The conjugate momentum keeps changing definition but in the end, the CRs between the field and conjugate momentum always ends up giving the same CRs between the annihilation/creation operators.

The real advantage of the field approach is that it automatically builds the fields with the correct Lorentz properties, whereas in the Weinberg approach, some work is required to find the right combination of annihilation and creation operators that will do that.

I still have a question about your comment concerning the action for Schroedinger's equation but I will post later.

Regards

Patrick

 

[nrqed]

Hi Patrick,

20 years ago I was introduced ,for the first time, to the classical field theories and almost immediately I asked my professor the following question:
"what do Dirac wavefunctions have in common with the classical fields of EM and gravity?"
My reason to ask about fermions and not about bosons was (I thought) obvious, because in principle (Pauli) only fermions have no classical limit.
Any way, His answer was:
...
regards

sam

Wow. Thank you Sam, this is very interesting. I have tons of questions on all of this and I really want to understand all of this. But I think that my questions will be extremely naive and misguided at first because I have no background in mathematical physics, only in phenomenology (I am still trying to understand the need for differential forms, for example!). But I really want to understand topological fiedl theories, string theory etc.

So I will try to absorb your post and will certainly have tons of questions to ask. I hope you will be patient enough to answer as much as you want.

Thansk again for a very interesting post!

Regards

Patrick

 

[Eye_in_the_Sky]

In that case, if you are interested, I will try to post something (just presenting the main points at first; I will get more detailed if anybody is interested).

As I said, I (for one) am interested. And I agree, a "generalized sketch" is an appropriate way to start. Afterwards it can be evaluated whether or not and just how to continue.

... And I see you've already begun to sketch things out:

One starts from the desire of having a many-body description and the introduction of the Fock space and of annihilation/creation operators. From the basic meaning of those operators, their commutation relation is deduced. The field is introduced as a *byproduct*, not as a starting point. And there is no classical field introduced, one gets directly to the quantum fields. The CRs between the field and conjugate momenta *follow* from the Crs (or antiCRs) between the creation/annihilation operators and these are introduced very naturally (imho), being due to the very nature of the creation/annihilation operators.

... The nonrelativistic case is handled exactly the same way as the relativistic case. The only difference is the need to impose that one-particle state created by the field operator have definite transformation properties under Lorentz transformations. For example, for a real KG field, this forces one to combine annihilation and creation operators (of positive energy states) together, which was not required in the non-relativistic case (and this leads to the fact that in the KG case, the momentum conjugate to Phi is the time derivative of Phi whereas it is Phi^* in the nonrelativistic case).

I already have some questions. But for now I am going to hold back on them.
________________

I am not sure what you mean by "advanced tools".

I mean the 'tools' which need to be introduced in connection with the non-free field.

By what you have said, it seems to me that a knowledge of those 'tools' will not be required.
________________

My question was about the "deep" connection between the CRs of the field and conjugate momentum and the CRs of the annihilation/creation operators.

So, I think then we can all see and agree that – as far the nonrelativistic case is concerned – with regard to this connection, there is nothing "deep".

If we perform a quantization of the field (see posts #25 and #26), then the CRs for Ψ and Π (in light of Π = iΨ^†) simply tell us that Ψ^† and Ψ are already themselves no more (and no less) than creation and annihilation operators. (As for their dependence on time, this merely reflects the fact that they are now in the "Heisenberg picture".)

In fact, if we go back to an arbitrary wavefunction ψ(x) which solves the time-dependent Schrödinger equation, and write it as

ψ(x) = ∑_n a_n(t) φ_n(x)

with respect to some orthonormal basis {φ_n(x)}, then a little bit of reflection shows us that "quantization of the field" is the same as invoking a direct canonical quantization upon the very coefficients a_n(t) themselves and their complex conjugates, which when promoted to operators become the 'familiar' annihilation and creation operators (now expressed in the "Heisenberg picture").

In other words, upon a little more reflection, we see that:

Annihilation and creation operators are precisely what one gets when one "quantizes" probability amplitudes and their complex conjugates.

And furthermore, from this, we can then see that the reason why, in the relativistic case, the fields Ψ^† and Ψ cease to have the character of creation and annihilation operators is, quite simply, because at the 'classical' level the field ψ itself ceases to have the character of a probability amplitude.

Next ... for the nonrelativistic case, what are the "mode" functions with respect to which creation and annihilation operators are defined? ... Well, a complete set of "modes" can be understood as nothing but the complete set of eigenfunctions of some observable on the single-particle Hilbert space. For example, if we choose X, then we get the field operators Ψ^†(x) and Ψ(x) (now, in the "Schrödinger picture"). If we choose P, we get field operators
Φ^†(p) and Φ(p). If we choose the single-particle Hamiltonian H, then we get operators a_E^† and a_E. And on it goes ...

... So, I wonder now how all of this works in the relativistic case?

Patrick, can you tell me?

My thoughts on this have evolved a bit. In Weinberg's approach, the basic CRs are simply between the annihilation/creation operators. The CRs between the fields and conjugate momenta are simply wasy to encompass this information.

Thinking in terms of independent degrees of freedom, it seems to me now that the conventional approach is simply a way to isolate the independent degrees of freedom. It is instructive to see that for an equation linear in time, the conjugate field momentum is the complex conjugate of the field (for the nonrelativistic Schrodinger case or for the Dirac equation) whereas it is the time derivative for the real KG field, and it's the complex conjugate of the time derivative of the field for the complex KG field. But in the end, th CR's between the field and the conjugate momentum always boils down to isolating the creation and annihilation operators and specifying the CRs between them (by CRs here I include antiCRs, i.e. I am talking in the most general context). So I think that the field approach is just a clever way of isolating the independent degrees of freedom and giving their CRs which, in the end, always amounts to specifiying the CRs between the annihilation and creation operators.

So, looking only at the CRs, it seems to me that the field approach is a very indirect (but very clever) way of giving the CRs between the annihilation and creation operators. The conjugate momentum keeps changing definition but in the end, the CRs between the field and conjugate momentum always ends up giving the same CRs between the annihilation/creation operators.

The real advantage of the field approach is that it automatically builds the fields with the correct Lorentz properties, whereas in the Weinberg approach, some work is required to find the right combination of annihilation and creation operators that will do that.

I look forward to hearing more from you about this, starting with a "generalized sketch" ... that is, if you aren't having 'second thoughts'.
[I know that "time" is a valuable and, often (ha!ha!), scarce resource.] Please, let me know if you are (or if you are not) going to make an attempt.

In the mean time, I want to go back and comb through all of the posts so far in this thread in order to pick out some more of the many points which have been made that can use further clarification.
_____________________________
_____________________________

I still have a question about your comment concerning the action for Schroedinger's equation but I will post later.

So, what is it?

 

[Eye_in_the_Sky]

recovering X & P from the quantized Schrödinger-field

After combing through the various posts of this thread, I have found several points which, it seems to me, could use some further clarification.

Here is one of them:

Starting from the quantized Schrödinger-field, how does one recover the single-particle description in terms of X and P? What connection, if any, exists between the CRs (or anti-CRs) of the quantized fields and the CRs of the single-particle operators?

Back in the last part of post #26, it was shown that we can make the identification

Ψ^†(x,0)|0> = |x> ,

the eigenkets of X, the single-particle position operator in the "Schrödinger picture".

A similar derivation, with the time-dependence retained in Ψ^†, shows that we can make the identification

Ψ^†(x,t)|0> = |x,t> ,

the eigenkets of X(t), the single-particle position operator expressed in the "Heisenberg picture". Therefore, we can write

X(t) = ∫ x Ψ^†(x,t)|0><0|Ψ(x,t) dx ... [1] .

This identification follows directly from the CRs (or anti-CRs) of the fields in connection with their equations of motion.

It follows in two steps:

1/ The CRs (or anti-CRs) of the fields are what allow us to compute for the dynamics of the fields; that is, they allow us to convert the CRs on the right-hand-sides of the relations (see post #25)

i∂_tΨ = [Ψ,H]
i∂_tΠ = [Π,H] ,

where H = -i ∫dx ΠH_xΨ ,

into

[Ψ,H] = H_xΨ
[Π,H] = -H_xΠ .

From here, we can write the equations of motion for the fields:

i∂_tΨ = H_xΨ
i∂_tΠ = -H_xΠ .

2/ Once those equations of motion have been given and solved for, the CRs (or anti-CRs) of the fields will then serve again. They will now tell us that Ψ^† (remember that Π = iΨ^†) and Ψ have the character of creation and annihilation operators. And from this, the identification of X(t) according to [1] will follow.

Next, once we have identified X(t) as in [1], what about P(t)?

By definition P(t) is the momentum conjugate to X(t). That is, by definition
P(t) is an operator on the single-particle Hilbert space such that

[X(t),P(t)] = i , for all t .

This relation has nothing to do with the CRs (or anti-CRs) of the fields.

But ... so what? At this point we don't even need P(t)! The only reason we ever had for the introduction of P(t) was in order to solve for the single-particle dynamics. But this we have already accomplished by having solved for the dynamics of the fields.

On the other hand, if we do decide to introduce P(t), what then will our field have to say?

... Well, from the form of the equation of motion of the field,

i∂_tΨ(x,t) = [-∂_x²/(2m) + V(x)]Ψ(x,t) ,

it will follow that an acceptable choice for P(t) is mdX(t)/dt.