Why doesn't the free vacuum transform under Poincare?

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In summary: Sorry, I didn't make myself clear. The Fock space would still be constructed in the usual way. The only difference is how the free vacuum transforms under Poincare transformations:Before, we have U(\Lambda)|0\rangle = |0\rangleNow, we would some non-trivial transformation U(\Lambda)|0\rangle = \sum c_i |i\ranglewhere the coefficients of this transformation are chosen such that the interacting vacuum is invariant.
  • #1
Riposte
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My understanding of Haag's theorem (see link below) is that there is a mismatch between the Hilbert spaces of free and interacting particles. The argument seems to be that we require both the free and interacting vacuum states to be invariant under Poincare transformations. Now since the entire Hilbert space of the free particle is built by acting on the vacuum with creation operators which are not invariant under Poincare transformations, then our only choice for the interacting vacuum is to have it be proportional to the free vacuum. Here we get a contradiction, since the free vacuum cannot be an eigenstate of both the free Hamiltonian and the full Hamiltonian.

http://philsci-archive.pitt.edu/archive/00002673/01/earmanfraserfinalrevd.pdf" [Broken])

All that's good and well, except for one thing. Why do we require the free vacuum to be invariant under Poincare transformations? The free fields aren't the physical ones, the interacting ones are. I see no reason why the non-physical free vacuum can't transform non-trivially under a translation or a rotation.

If, instead, we require only that the interacting vacuum be invariant under Poincare transformations, the mismatch between the Hilbert spaces disappears.
 
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  • #2
Riposte said:
[...] Why do we require the free vacuum to be invariant under Poincare transformations? The free fields aren't the physical ones, the interacting ones are. I see no reason why the non-physical free vacuum can't transform non-trivially under a translation or a rotation.

If, instead, we require only that the interacting vacuum be invariant under Poincare transformations, the mismatch between the Hilbert spaces disappears.

Your final sentence above seems like a sweeping statement, imho...

If you abandon the principle that the (free) vacuum is not annihilated
by the (free) Poincare generators, then... how do you construct a Fock
space? What Lie algebra do you start from?? And how do you construct
the interacting Hilbert space?
 
  • #3
Riposte said:
My understanding of Haag's theorem (see link below) is that there is a mismatch between the Hilbert spaces of free and interacting particles.
There is no such a mismatch in Atomic Physics or non relativistic QM. So the "mismatch" in QFT is due to wrong interaction term, first of all.
The argument seems to be that we require both the free and interacting vacuum states to be invariant under Poincare transformations. Now since the entire Hilbert space of the free particle is built by acting on the vacuum with creation operators which are not invariant under Poincare transformations, then our only choice for the interacting vacuum is to have it be proportional to the free vacuum. Here we get a contradiction, since the free vacuum cannot be an eigenstate of both the free Hamiltonian and the full Hamiltonian.
If the full Hamiltonian "coincides" with the free one in asymptotic, initial and final states, then the interaction transforms the state vectors within the same Hilbert space, so there is no problem with it. Your statement about non-observability of the free states is mostly dictated with infinities that are "hidden" into the "bare" parameters in course of renormalizations, isn't it? This, conceptual problem is due to self-action interaction term. The theory can be reformulated without self-action (only with the true interaction) and in such a formulation there is no "corrections" to the masses and charges. Think of non relativistic QM as an example where the Haag's theorem fails.
 
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  • #4
strangerep said:
If you abandon the principle that the (free) vacuum is not annihilated
by the (free) Poincare generators, then... how do you construct a Fock
space? What Lie algebra do you start from?? And how do you construct
the interacting Hilbert space?

Sorry, I didn't make myself clear. The Fock space would still be constructed in the usual way. The only difference is how the free vacuum transforms under Poincare transformations:

Before, we have [tex]U(\Lambda)|0\rangle = |0\rangle [/tex]
Now, we would some non-trivial transformation [tex]U(\Lambda)|0\rangle = \sum c_i |i\rangle[/tex]
where the coefficients of this transformation are chosen such that the interacting vacuum is invariant.
 
  • #5
Bob_for_short said:
So the "mismatch" in QFT is due to wrong interaction term, first of all.

Wrong interaction term? What do you mean by this?

If the full Hamiltonian "coincides" with the free one in asymptotic, initial and final states, then the interaction transforms the state vectors within the same Hilbert space, so there is no problem with it.

I agree, if you have interactions which turn adiabatically on and off, then there's no problem. However, that's usually not the case.

Your statement about non-observability of the free states is mostly dictated with infinities that are "hidden" into the "bare" parameters in course of renormalizations, isn't it?

No, what I meant by calling the free vacuum non-physical is that once interactions are added to the theory the eigenstates of the theory (vacuum, 1-particle states, etc) are shifted from the free ones. The free eigenstates no longer correspond to physical states, and therefore there is no need for them to retain the expected behavior under Poincare transformations. It's like watching an electron whiz by and say "Oh look, there's a 1-particle state of QED". No matter how long you keep looking, you'll never see the free 1-particle state of a neutral electron go floating by.
 
  • #6
Riposte said:
Wrong interaction term? What do you mean by this?
A self-action term.
I agree, if you have interactions which turn adiabatically on and off, then there's no problem. However, that's usually not the case.
You can separate the projectile and the target in the experimental setup. Such a separation is described with non overlapping wave packets.
No, what I meant by calling the free vacuum non-physical is that once interactions are added to the theory the eigenstates of the theory (vacuum, 1-particle states, etc) are shifted from the free ones.
Not always. Sometimes "free" states correspond to the separated variables (elementary modes) of one compound system, so the main part of interaction is already present in them.
 
  • #7
Riposte said:
No, what I meant by calling the free vacuum non-physical is that once interactions are added to the theory the eigenstates of the theory (vacuum, 1-particle states, etc) are shifted from the free ones.

Yes, that's exactly the weird property of QFT (in its current formulation) that makes it so different from the ordinary quantum mechanics. It appears that QFT (bare) vacuum "interacts with itself" and QFT (bare) particle also "interacts with itself". This self-interaction is ultimately responsible for ultraviolet divergences.

One approach to deal with this problem is to take this difference between bare and dressed states for granted, consider different Hilbert spaces for non-interacting and interacting theories, etc, etc. Personally, I find this approach cumbersome and not appealing.

There is, however, another line of thought. We can ask ourselves, are we sure that the interaction adopted in QFT (take QED as an example) is the correct one? After all, this interaction (the "minimal" coupling between the photon field and electron current; I think this is what Bob_for_short had in mind when writing about the "wrong interaction term" in QFT.) was "derived" by using rather shaky analogies with classical Maxwell's electrodynamics. The only experimental data supporting this choice of the QED interaction is related to scattering. But it is well-known that there are many (scattering-equivalent) Hamiltonians, which produce exactly the same S-matrix. So, perhaps, we can choose QED interaction in another form, such that 1) the self-interaction is not present anymore; 2) there is no difference between free (bare) and interacting (dressed) states, just as in ordinary quantum mechanics; 3) the S-matrix remains the same (i.e., agreeing with experiment) as in renormalized QED; 4) ultraviolet divergences are not present.

This strategy (known as the "dressed particle" approach) was first formulated in a beautiful paper

O. W. Greenberg and S. S. Schweber, "Clothed particle operators in simple models of quantum field theory", Nuovo Cim. 8 (1958), 378.

It appears rather promising, as you can check by following multiple references to this work. Within this approach, Haag's theorem does not present any difficulty. See, for example

M.I. Shirokov, "Dressing" and Haag's theorem. http://www.arxiv.org/abs/math-ph/0703021
 
  • #8
meopemuk said:
Yes, that's exactly the weird property of QFT (in its current formulation) that makes it so different from the ordinary quantum mechanics. It appears that QFT (bare) vacuum "interacts with itself" and QFT (bare) particle also "interacts with itself". This self-interaction is ultimately responsible for ultraviolet divergences.
And I would add that the renormalizations remove (perturbatively) the self-action contribution.

Understanding the "free" particles as elementary excitation modes of one compound system makes it unnecessary to add the self-action term. One can write just an interaction term. For example, scattering of charges can be constructed as a potential scattering of compound systems (just like atoms) with inevitable exciting their internal degrees of freedom (photon oscillators).
 
  • #9
Riposte said:
Why do we require the free vacuum to be invariant under Poincare transformations? The free fields aren't the physical ones, the interacting ones are. I see no reason why the non-physical free vacuum can't transform non-trivially under a translation or a rotation.

We need a vacuum state (for the free theory too), which we define to be that state which is annihilated by lowering operators. The Hamiltonian should be a bounded operator in the theory. If your state doesn't satisfy the annihilation property, it cannot be the ground state and you have to continue till you find it or have defined it. So where is yours?
 
  • #10
Riposte said:
The Fock space would still be constructed in the usual way.
OK, so have the usual (free) a/c operators, i.e., [itex]a_k^*[/itex],
etc, satisfying the usual CCRs, and these operators constitute an
irreducible set, (meaning that any operator on the Fock space can be
expressed as a polynomial or function of the a/c ops).

That means you still have the usual (free) representation
of the Poincare generators, so there are still operators
[itex]U(\Lambda)[/itex] such that [itex]U(\Lambda)|0\rangle =|0\rangle[/itex].

Then...

The only difference is how the free vacuum transforms under
Poincare transformations:

Before, we have [itex]U(\Lambda)|0\rangle = |0\rangle [/itex]
Now, we would some non-trivial transformation
[itex]U(\Lambda)|0\rangle = \sum c_i |i\rangle [/itex]
where the coefficients of this transformation are chosen such that the
interacting vacuum is invariant.

I believe this is pretty much the same thing as constructing an "interacting
representation of the Poincare group". Let's call these operators [itex]W(\Lambda)[/itex]
and let's call the interacting vacuum [itex]|\Omega\rangle[/itex],
where [itex]W(\Lambda) |\Omega\rangle = 0[/itex],
but [itex]W(\Lambda) |0\rangle \ne 0[/itex].

But the original (free) a/c ops constitute an irreducible set, so [itex]W[/itex]
must be expressible as function of them (if indeed the free and interacting
Hilbert spaces coincide). One would also expect to find "interacting" a/c operators,
i.e., [itex]\alpha_k^*[/itex], etc, corresponding to single-particle states in the interacting
theory (which are eigenstates of the interacting Hamiltonian). These must satisfy
[itex]\alpha_k |\Omega\rangle = 0[/itex].

The task is then to express the [itex]\alpha_k[/itex] ops in terms of the [itex]a_k^[/itex] ops.
That's pretty much what the "dressed particle" approach involves (also known under the
phrase "unitary dressing transformation"). As well as the references mentioned by
meopemuk, there's also this review article:

Shebeko, Shirokov:
"Unitary Transformations in Quantum Field Theory and Bound States"
Available as nucl-th/0102037

Unfortunately, one generally encounters infinite quantities when
perturbatively constructing the transformation, which must be
"renormalized" in a manner reminiscent of standard renormalization.
One thus encounters a different variation of Haag's theorem in that
the interacting representation cannot live in the Fock space constructed
from the free a/c ops.
 
  • #11
strangerep said:
Unfortunately, one generally encounters infinite quantities when
perturbatively constructing the transformation, which must be
"renormalized" in a manner reminiscent of standard renormalization.
One thus encounters a different variation of Haag's theorem in that
the interacting representation cannot live in the Fock space constructed
from the free a/c ops.

Yes, it is true that the perturbative unitary dressing transformation is rather messy. However, there are good reasons to believe that this difficulty is technical rather than fundamental. It is easy to imagine how a full interacting theory can be constructed in a single Fock space, without self-interactions, ultraviolet divergences, and Haag's theorem problems.

First, one can define the usual "Fock space constructed from the free a/c ops". Then one automatically gets the non-interacting representation of the Poincare group there. The next step is to construct the interaction part of the Hamiltonian (the Poincare generator of time translations) as a normally-ordered polynomial in "a/c ops". Of course, in order to be physically admissible, this polynomial must satisfy a few conditions:

1. Each term in the polynomial must have at least two annihilation operators and at least two creation operators. This is necessary to avoid self-interactions in the vacuum and 1-particle states, i.e., to make sure that these states are low-energy eigenvectors of the interacting Hamiltonian.
2. The momentum-dependent coefficient functions in each term must vanish sufficiently rapidly away from the "energy shell". This is necessary to ensure that all loop integrals encountered in S-matrix calculations are finite, so that there are no ultraviolet divergences.
3. In parallel with the above construction of the interacting energy one needs to build the "interacting boost" operator having similar properties, and making sure that commutation relations of the Poincare Lie algebra remain intact. This would guarantee the relativistic invariance of the theory.
4. Finally, one needs to make sure that the S-matrix calculated with the above Hamiltonian is exactly the same as the S-matrix of the renormalized QFT. This can be done by properly adjusting coefficient functions in each perturbation order. Then the new theory is guaranteed to agree with all existing experiments.

I don't have a full mathematical proof that this construction is possible, but there are many indirect indications that there are no fundamental obstacles on this path.
 
  • #12
meopemuk said:
Yes, it is true that the perturbative unitary dressing transformation is rather messy. However, there are good reasons to believe that this difficulty is technical rather than fundamental. It is easy to imagine how a full interacting theory can be constructed in a single Fock space, without self-interactions, ultraviolet divergences, and Haag's theorem problems.

This raises the following question: "How does the dressed electron look like in QED?" considered in another thread.

meopemuk said:
...
4. Finally, one needs to make sure that the S-matrix calculated with the above Hamiltonian is exactly the same as the S-matrix of the renormalized QFT. This can be done by properly adjusting coefficient functions in each perturbation order. Then the new theory is guaranteed to agree with all existing experiments.

I don't have a full mathematical proof that this construction is possible, but there are many indirect indications that there are no fundamental obstacles on this path.

Apart from UV divergences there are IR ones in QED an in other QFTs with massless (thresholdless) excitations. In fact it is the inclusive cross section that is well comparable with the experiments, not S-matrix itself. Unfortunately, working with the creation/annihilation operators in each perturbative order does not reveal how this problem can be resolved. In order to resolve it one needs a better physical idea and the corresponding mathematical construction for interacting fields. I tried to propose something constructive in my publications which I am not allowed to cite here.
 
  • #13
Bob_for_short said:
This raises the following question: "How does the dressed electron look like in QED?" considered in another thread.

In the approach I am describing here the electron is just a point (structureless) particle characterized by measured values of mass, spin, and charge. Nothing fancy.
 
  • #14
According to my experience, any interaction smears quantum mechanically the charge. Probably your interacting (dressed) electron is still point-like because you have not completed the dressing, isn't it?
 
  • #15
Bob_for_short said:
According to my experience, any interaction smears quantum mechanically the charge. Probably your interacting (dressed) electron is still point-like because you have not completed the dressing, isn't it?


The point-like character of particles and the spread of their wave functions are two separate issues. In QM, the stationary wave function of a free electron is a plane wave, i.e., it is completely delocalized. However, this does not prevent us from talking about the point-like electron.
 
  • #16
Bob_for_short said:
Apart from UV divergences there are IR ones in QED an in other QFTs with massless (thresholdless) excitations.

You are right that infrared divergences is a difficult problem. I don't think it has been studied in the "dressed particle" approach. However, this problem has been successfully solved within the standard renormalized QED. I believe that a similar solution could be found in the "dressed particle" approach as well.
 
  • #17
meopemuk said:
The point-like character of particles and the spread of their wave functions are two separate issues. In QM, the stationary wave function of a free electron is a plane wave, i.e., it is completely delocalized. However, this does not prevent us from talking about the point-like electron.

Yes, it prevents. We speak of de Broglie waves, probability amplitude instead of point-like particle. And I speak of the electron coupled to the quantized EMF. Its charge should be smeared, just like in an atom: the compound system has a center of inertia and relative motion wave functions. Both are smeared quantum mechanically. Such a smearing is different from a classical one but it is sufficient to eliminate divergences.
 
  • #18
Bob_for_short said:
Yes, it prevents. We speak of de Broglie waves, probability amplitude instead of point-like particle. And I speak of the electron coupled to the quantized EMF. Its charge should be smeared, just like in an atom: the compound system has a center of inertia and relative motion wave functions. Both are smeared quantum mechanically. Such a smearing is different from a classical one but it is sufficient to eliminate divergences.

It appears that we use different definitions of "point-like" or "localizable" particles. In my terminology, a particle is "point-like" if there is a well-defined position operator whose eigenfunctions are Dirac delta functions. The particle can be always prepared in such localized states. According to this definition, the electron is "localizable". However, this does not mean that only delta-functions are allowed as descriptions of electron's states. All kinds of delocalized functions (e.g., plane waves or atomic orbitals) are permissible as well. In most potentials, electron's stationary wave functions are pretty much delocalized, but the electron itself is considered a point-like particle.
 
  • #19
meopemuk said:
It appears that we use different definitions of "point-like" or "localizable" particles. In my terminology, a particle is "point-like" if there is a well-defined position operator whose eigenfunctions are Dirac delta functions. The particle can be always prepared in such localized states. According to this definition, the electron is "localizable". However, this does not mean that only delta-functions are allowed as descriptions of electron's states. All kinds of delocalized functions (e.g., plane waves or atomic orbitals) are permissible as well. In most potentials, electron's stationary wave functions are pretty much delocalized, but the electron itself is considered a point-like particle.

So what is electron de-localization due to dressing in the "dressed" particle approach? Is it different from a "free" electron de-localization and in what respect?
 
  • #20
Bob_for_short said:
So what is electron de-localization due to dressing in the "dressed" particle approach? Is it different from a "free" electron de-localization and in what respect?

I am not sure I understand the question, but let me try to explain how I understand the issue of electron localization in QFT.

In the standard textbook QFT, the "bare" electron is a normal point-like particle. As I said above, its wave function can be either localized (delta function) or delocalized (plane wave), but the important point is that one can easily define the position operator corresponding to the single electron, so the "bare" electron is "localizable", just as in ordinary quantum mechanics.

When the interaction is turned on in QFT, the "bare" electron states are no longer eigenstates of the Hamiltonian. So, they cannot correspond to states of "physical" or "dressed" electrons. One can try to build states of such "physical" electrons as linear combinations of "bare" particle states. Such linear combinations must involve contributions of infinite number of "bare" states like (1 electron) + (1 electron + 1 photon) + (1 electron + 1 electron-positron pair) + ... This is a rather complicated wave function. I don't think it is possible to write it down in a complete form. I am also not sure how one can talk about the "localizability" of such states.

The above was the situation in the standard QFT. The difference between "bare" and "dressed" electrons (and the difficulty in explicit definition of "physical" states) was determined by the fact that QFT interaction operator had a nontrivial action on the "bare" 1-particle states (and on the "bare" vacuum states).

Now, the "dressed particle" approach says: The interaction operator chosen in QFT is not good. The correct interaction between "bare" particles must have a trivial action (i.e., yield zero) on the vacuum and 1-particle states. If such a correct interaction is chosen, then "bare" states remain eigenstates of the full interacting Hamiltonian, and there is no difference between "bare" and "physical" particles. In this case the localizability of "physical" particles is not different from the localizability of "bare" particles discussed above. "Physical" electron is a point-like particle. One can easily write down a position operator whose eigenvectors represent localized states of the "physical" electron in the "dressed particle" approach.

So, in answering your question, I can say that there is no "de-localization due to dressing in the "dressed" particle approach", or, at least, I don't understand what you mean by that.
 
  • #21
meopemuk said:
I am not sure I understand the question, but let me try to explain how I understand the issue of electron localization in QFT.

In the standard textbook QFT, the "bare" electron is a normal point-like particle. As I said above, its wave function can be either localized (delta function) or delocalized (plane wave), but the important point is that one can easily define the position operator corresponding to the single electron, so the "bare" electron is "localizable", just as in ordinary quantum mechanics.

As soon as we describe the electron with help of a wave function or coordinate operators rather than just coordinates, it is quantum mechanical with its inevitable smearing. I could understand if you would invoke the r-dependence of the interaction potential to say: "See, it depends on distance or position, thus the electron is point-like", but even then the QM smearing is involved in calculations. The simplest example is a charge form-factor: is is determined with the wave function.
Now, the "dressed particle" approach says: The interaction operator chosen in QFT is not good. The correct interaction between "bare" particles must have a trivial action (i.e., yield zero) on the vacuum and 1-particle states. If such a correct interaction is chosen, then "bare" states remain eigenstates of the full interacting Hamiltonian, and there is no difference between "bare" and "physical" particles. In this case the localizability of "physical" particles is not different from the localizability of "bare" particles discussed above. "Physical" electron is a point-like particle. One can easily write down a position operator whose eigenvectors represent localized states of the "physical" electron in the "dressed particle" approach.
Then my argumentation about non point-like electron is applicable again.
 
  • #22
Bob_for_short said:
Then my argumentation about non point-like electron is applicable again.

It looks like we are arguing about terminology rather than substance. Of course the electron's wave function can take any shape from point-like (delta function) to completely delocalized (plane wave). So, one can say that the electron is not a point particle. How is it related to the original posting?
 
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  • #23
meopemuk said:
... So, one can say that the electron is not a point particle. How is it related to the original posting?

Directly. A real, observable electron and the quantized EMF form a compound system described quantum mechanically in the same way as an atom: with the center of inertia and relative coordinates. Such a dressed electron (I call it "electronium") is observable (not bare) and a potential interaction of such electrons is accompanied with radiation as naturally as atom excitation in atomic collisions. In this formulation there are no non-physical (non-observable) particles, vacuums, etc., and the Haag's theorem fails: the interaction does not bring problems with the Hilbert space.
 
  • #24
Bob_for_short said:
Directly. A real, observable electron and the quantized EMF form a compound system described quantum mechanically in the same way as an atom: with the center of inertia and relative coordinates. Such a dressed electron (I call it "electronium") is observable (not bare) and a potential interaction of such electrons is accompanied with radiation as naturally as atom excitation in atomic collisions. In this formulation there are no non-physical (non-observable) particles, vacuums, etc., and the Haag's theorem fails: the interaction does not bring problems with the Hilbert space.

We already discussed your proposal several times. Few important yet unresolved issues remain. What is the Hamiltonian? How it was derived? Is it relativistically invariant? Can you calculate and compare with experiment scattering amplitudes for some simple collisions?
 
  • #25
meopemuk said:
We already discussed your proposal several times. Few important yet unresolved issues remain. What is the Hamiltonian? How it was derived? Is it relativistically invariant? Can you calculate and compare with experiment scattering amplitudes for some simple collisions?
Yes, I can but these are questions to my results that belong to the Independent Research section. I cannot answer them here.
 
  • #26
Riposte said:
All that's good and well, except for one thing. Why do we require the free vacuum to be invariant under Poincare transformations? The free fields aren't the physical ones, the interacting ones are. I see no reason why the non-physical free vacuum can't transform non-trivially under a translation or a rotation.
If, instead, we require only that the interacting vacuum be invariant under Poincare transformations, the mismatch between the Hilbert spaces disappears.
There are two things to consider here.
1)It is easy to prove that the free vacuum is invariant under Poincare transformation, so there isn't really anyway around the fact that it transforms in the required way.
2)For any field theory that has actually been rigorously constructed nonperturbatively the result is that they live in a different Hilbert space than the free theory.

Haag's theorem isn't really something to be got around. Perturbatively there is no problem with doing Fock space calculations to approximate the interacting theory. All it says is that nonperturbatively you'll require heavier mathematical machinery if you want to do things rigorously.
 
  • #27
DarMM said:
...Perturbatively there is no problem with doing Fock space calculations to approximate the interacting theory.

Except for statements like free particles are not observable, they have infinite parameters, etc.
It is like developing a function in Taylor series with saying that the zeroth order approximation is non observable. => There are severe problems in perturbative calculations in actual QFTs.
 
  • #28
Bob_for_short said:
Except for statements like free particles are not observable, they have infinite parameters, etc.
I don't understand how this is a problem. If the system is not free then you shouldn't observe free particles.

Bob_for_short said:
It is like developing a function in Taylor series with saying that the zeroth order approximation is non observable. => There are severe problems in perturbative calculations in actual QFTs.
I don't see how. The expansion makes sense once you know its rigorous justification and it agrees fantastically with experiment. So I don't think there are severe problems.
 
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  • #29
DarMM said:
It don't understand how this is a problem. If the system is not free then you shouldn't observe free particles.

That is the problem: you do not understand.

Let us take the following experiment: an atom as a target and another atom as a fast projectile. We study scattering backwards, at large angles. We can write mechanical or quantum mechanical equations, whatever, containing all involved particles: electrons and nuclei. Then we solve these equations. Scattering backward occurs due to nucleus-nucleus (Coulomb or not) interaction because the target electrons cannot cause such a projectile deviation (i.e., at large angles). So we are tempted to neglect the electron-nucleus interaction in the zeroth approximation. At this stage we still have two different possibilities:

1) to take exactly into account a certain part of the electron-nucleus interaction in order to correctly represent the experimental initial and final conditions (neutral atoms in some states). The rest can be considered perturbatively.

2) But we can also take all electron-nucleus interactions perturbatively, so we do not have neutral atoms in the initial and final states in our theory. Nevertheless such a theory give good prediction for scattering at large angles (the Rutherford formula, for example). Then we try to take into account the remaining electron-nucleus interactions, in particular, "build" atomic wave functions perturbatively. Naturally here we encounter mathematical difficulties: divergences, etc. In order to overcome these difficulties we invent the notion of bare initial particles with infinite masses, etc. It is evident that such an invention has nothing to do with reality but is solely due to our incapability to correctly build the initial and final states (atoms) perturbatively.

On the other hand, we can use a better initial approximation (1) for our problem and obtain a correct cross section of atom-atomic scattering at large angles. In the latter approach there is no need in "bare" particle ideology at all. The atoms are not "free" but are interacting in our theory, yet there are no bare (non observable) particles with bizarre properties to cancel bizarre perturbative corrections. The initial and final states of (interacting = free) atoms are observable and the atom-atomic interaction can be considered perturbatively (remember, our projectile is fast). In this case there is no problem like the Haag's theorem.

Do you understand now where the difference comes from?
 
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  • #30
Bob_for_short said:
Do you understand now where the difference comes from?
I must admit that I do not fully understand. I my mind, this is how things work:
A quantum field theory may (basically) be characterised by its set of Green's functions. These contain all the information in the theory. Now let's say I want to study particle scattering, the information on this is contained within the S-matrix. The S-matrix itself, by the LSZ formalism can be calculated as the on-shell poles of the Green's functions.

So basically all I need to do is calculate the Green's functions and I have the scattering predictions of the theory. Now since QFT is too difficult to do calculations with, one usually needs an expansion to calculate the Green's functions. The expansion we take is the one around a theory we can actually solve, namely a free field theory.

However because the interacting theories are singular with respect to the free theory we also need terms with infinite coefficients in the expansion. This is not a problem, it is simply a fact. QFTs are highly singular mathematical objects and unlike non-relativistic quantum mechanics two theories can be so mutually singular as to live in different Hilbert spaces, so your expansion must take this into account, hence the counterterms.

This has nothing really to do with Haag's theorem which is related to nonperturbative infrared divergences, not perturbative ultraviolet ones that show up in usual calculations.

I don't see what all this talk about "bare" particles is about. The difference between the "bare" particles, by which I assume you mean the free theory Fock space particles and the "real" particles, by which I assume you mean the asymptotically free particles in the real Hilbert space doesn't really show up in scattering calculations due to the whole LSZ amputation of the external legs. Where it will show up is in finite time calculations.
 
  • #31
DarMM said:
...A quantum field theory may (basically) be characterised by its set of Green's functions. These contain all the information in the theory.
One cannot hide oneself behind Green's functions. If one knows the exact solutions, one can build the exact Green's function and vice versa.
DarMM said:
...However because the interacting theories are singular with respect to the free theory we also need terms with infinite coefficients in the expansion.
It is only so in theories with self-action. Renormalizations remove (not always though) the singular self-action contribution. So the net result belongs to another theory - with an interaction without self-action, just like in non relativistic QM. Remember, the interaction term is our guess and it can be improved. And renormalization is also our guess or try, it does not work automatically.
I don't see what all this talk about "bare" particles is about. The difference between the "bare" particles, by which I assume you mean the free theory Fock space particles and the "real" particles, by which I assume you mean the asymptotically free particles in the real Hilbert space doesn't really show up in scattering calculations due to the whole LSZ amputation of the external legs. Where it will show up is in finite time calculations.
No, the self-action gives (unnecessary) corrections to the particle constants involved into the Green's functions, so "the whole LSZ" approach is as plagued with these problems as any other (standard) QFT. These problems, including the Haag's theorem, are eliminated with a better choice of the initial approximation, just like in the atom-atomic scattering problem outlined in the previous post of mine (post #29).
 
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  • #32
Bob_for_short said:
One cannot hide oneself behind Green's functions. If one knows the exact solutions, one can build the exact Green's function and vice versa.
Maybe I didn't make myself clear. I was only mentioning this as a set up to my discussion.

It is only so in theories with self-action. Renormalizations remove (not always though) the singular self-action contribution. So the net result belongs to another theory - with an interaction without self-action, just like in non relativistic QM. Remember, the interaction term is our guess and it can be improved. And renormalization is also our guess or try, it does not work automatically.
No theory has "self-action" contributions. That only results from a Feynman diagram type expression of the Wick contractions in perturbation theory. The full nonperturbative theory has no "self-actions". That is to say I can represent an interacting Green's function as a sum of convolutions of free Green's functions. These convolutions if interpreted literally would look like a particle interacting with itself, but why would you literally interpret a term in the perturbative expansion. Also renormalization is a guess as you say, however it has now been proven to work. Also a lot of non-relativistic QM systems also involve corrections to coupling constants, they're just not infinite like in QFT. In QM you will also have "bare" and "physical" coupling constants.

No, the self-action gives (unnecessary) corrections to the particle constants involved into the Green's functions, so "the whole LSZ" approach is as plagued with these problems as any other (standard) QFT. These problems, including the Haag's theorem, are eliminated with a better choice of the initial approximation, just like in the atom-atomic scattering problem outlined in the previous post of mine (post #29).
These things are not "problems", they are facts. The interacting theory lives in a different Hilbert space so the perturbative expansion is going to be very, very singular. I don't really see Haag's theorem as something to be gotten around, it makes perfect sense that free and interacting particles live in different reps because otherwise it would be possible for a real electron to evolve into a fictitious particle from a world with no photons, i.e. a free electron.
 
  • #33
I have already expressed myself on this subject and I do not want to repeat my arguments.
 
  • #34
DarMM said:
The interacting theory lives in a different Hilbert space...

I often see this statement, but I am not sure about its validity. In my opinion, this statement goes against basic postulates of quantum theory. Let me explain why I think the Hilbert space used to describe a physical system should be independent on whether the system is interacting or not.

Let us first ask why we use Hilbert spaces to describe physical systems (their states and observables) in QM? The answer is given by "quantum logic". This theory tells us that subspaces in the Hilbert space are representatives of "yes-no experiments" or logical "propositions" or experimental "questions". Meets, joins, and orthogonal complements of subspaces represent usual logical operations OR, AND, and NOT. It seems reasonable to assume that the same questions can be asked about interacting and non-interacting system. The logical relationships between these questions should not depend on the interaction as well. Therefore, the same Hilbert space (= logical propositional system) should be applied to both interacting and non-interacting system, if their particle content is the same.

So, interaction has no any effect on the Hilbert space structure. Interaction only changes the representation of the Poincare group (the group of transformations between inertial observers) acting in this Hilbert space. The non-interacting system is described by one representation of the Poincare group. The interacting system is described by a different representation. In particular, this means that generators of the two representations (e.g., the Hamiltonians) are different.
 
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  • #35
Getting back to the original question, there are a few workarounds that I can remember, and mentioned in a previous thread about this subject. Generically they all relax one or more of the axioms of the theorem

1) If you introduce a volume cutoff, this explicitly breaks the covariance condition and the theorem does not hold
2) Renormalization by formally infinite counterterms (it does so by making such a mess of
the mathematics and being so illdefined, that it violates a number of Wightman axioms to start with, so its no surprise that it also evades Haags theorem)
3) Supersymmetry. One of the assumptions of the theorem is that the fields either obey commutation relations, or anticommutation relations, but not both. So strictly speaking it does not apply to SuSY I think
4) If the interacting theory lives in a different Hilbert space (the idea Eugene hates) also strictly speaking bypasses one of the assumptions.

Anyway, the point is that perturbative QFT that we all know and measure everyday is fine. But a real, sensible, nonperturbative definition of an interacting field theory is going to be difficult.
 

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