Strange thought

interpretation of the quantized Schrödinger field

Begin with

i∂tΨ(x,t) = HxΨ(x,t) ... [1] ,

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

HxφE(x) = EφE(x) ... [2] .

Since the eigenfunctions φE(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 aE(t)φE(x) ... [3] ,

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

idaE(t)/dt = EaE(t) ,

so that

aE(t) = e-iEt aE ... [4] .

Substituting this result back into [3] then gives

Ψ(x,t) = ∑E aEφE(x)e-iEt ... [5a] ,

and therefore,

Ψ(x,t) = ∑E aEφE*(x)eiEt ... [5b] .
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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') [aE,aE'] = δEE' , [aE,aE'] = [aE,aE'] = 0 ;

(b') {aE,aE'} = δEE' , {aE,aE'} = {aE,aE'} = 0 .

Thus, the operators aE and aE can be interpreted as creation and annihilation operators with respect to the eigenfunction "modes" φE(x) of Hx, 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 Hx,

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

and

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

Ψ(x,0)|0> = ∑EEE*(x) ,

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

Ψ(x,0)|0> = ( ∑EE><φE| ) |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

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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
Eye_in_the_Sky said:
A suitable choice of Lagrangian density is

L = ψ*(i∂t - Hx)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 momemtum 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...

Hx = -∂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

Hclass = -i ∫dx πHxψ .
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___________________

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:devil:
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!:tongue2:


regards

Patrick
 
nrqed said:
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) HN ,

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):
nrqed said:
(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 - Hx)ψ + (δψ)(-i∂t - Hx)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 - Hx)ψ + ψ*(i∂t - Hx)(δψ) ]

= ∫dt ∫dx δ [ ψ*(i∂t - Hx)ψ ] .
 

vanesch

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nrqed said:
Ok. And this is where I will play devil's advocate:devil:
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

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vanesch said:
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.
:biggrin: Well, *I* think that fields are a nuisance :smile: 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 :smile:
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

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Eye_in_the_Sky said:
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 - Hx)ψ + (δψ)(-i∂t - Hx)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 - Hx)ψ + ψ*(i∂t - Hx)(δψ) ]

= ∫dt ∫dx δ [ ψ*(i∂t - Hx)ψ ] .
My question is obviously: why not take

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

Regards

patrick
 
nrqed said:
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?
________________
nrqed said:
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) HN ,

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?
nrqed said:
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 - Hx)ψ + (δψ)(-i∂t - Hx)ψ* ] ,

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

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

= ∫dt ∫dx δ [ ψ*(i∂t - Hx)ψ ] ,

you asked:
... why not take

δS = ∫dt ∫dx [ (δψ)(i∂t - Hx)ψ + (δψ*)(-i∂t - Hx)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.
 
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nrqed said:
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 [tex]M^4[/tex] .

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):
[tex]\phi : M^{n} \rightarrow F[/tex]

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

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

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"

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

So when you quantize the "fields" (X,P), you get a "QFT" on the affine time "spacetime". This is just QM. Quantizing the fields [tex]\left( \phi ,\pi \right)[/tex] will give us QFT on [tex]M^{n}[/tex] .
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
 
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nrqed

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Eye_in_the_Sky said:
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

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samalkhaiat said:
nrqed said:
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
 
nrqed said:
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 an(t) φn(x)

with respect to some orthonormal basis {φn(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 an(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 aE and aE. 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?
 
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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 ΠHxΨ ,

into

[Ψ,H] = HxΨ
[Π,H] = -HxΠ .

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

i∂tΨ = HxΨ
i∂tΠ = -HxΠ .

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.
 
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