Haag's Theorem, Perturbation, Existence and QFT.

[Son Goku]

Hello, I just reading and learning QFT and there is something I've been wondering, hopefully somebody here can help me.

Let's say we have an interacting Quantum Field Theory, such as Quantum Electrodynamics if we want to compute an amplitude such as two electrons scattering off each other, then we take the following steps:

1. We simulate the full interacting theory by making a perturbation about the free theory.

2. Since we're doing a scattering amplitude we can approximate the incoming and outgoing states as free field states

3. We sum up the perturbation terms, using all our renormalization and regularization techniques and we get the amplitude for the process.

Now, I have heard that the perturbation series does not converge for Quantum Electrodynamics and I was wondering what is the reason for this.

Is it because the perturbation series cannot fully represent the full nonperturbative theory, kind of like what happens for the expansion of \frac{1}{1+x^2} for x > 1. This is what I would have guessed.
However I've also read it more so has to do with the interacting Hamiltonian not being a well defined operator (not densely defined) on the Hilbert space of the free field. Which would make sense since the perturbation series is nothing more (in my understanding) than an attempt to simulate evolution in the interacting field Hilbert space due to the full Hamiltonian, by acting repeatedly on the free field Hilbert space with the interacting Hamiltonian H_{I}

I've also heard that full nonperturbative Quantum Electrodynamics has not even been proven to exist. What does this mean?
Also what is the content of Haag's Theorem? I've read it's statement, but I'd like to hear from others in case my understanding is wrong.

Sorry for all the questions, just want to get it right. Fascinating stuff.

 

[meopemuk]

Also what is the content of Haag's Theorem? I've read it's statement, but I'd like to hear from others in case my understanding is wrong.

The central quantity defined in any quantum field theory is the "free quantum field". For example, the free field associated with spin-zero massive particles is

\phi(\mathbf{r}, t) = \int \frac{d^3p}{(2 \pi \hbar)^{3/2}\sqrt{2 E_p}}(e^{-\frac{i}{\hbar}(\mathbf{pr} - E_pt) }a_{p} + e^{\frac{i}{\hbar}(\mathbf{pr} - E_pt) }a^{\dag}_{p})

These fields are used to construct interaction Hamiltonians V (or, more generally, an interacting representation of the Poincare group in the Fock space) and to calculate the S-matrix, as I briefly described in https://www.physicsforums.com/showpost.php?p=1382473&postcount=231
The resulting theory is relativistically invariant and agrees very well with experimental observations.

The time evolution of free quantum fields is controlled by the free Hamiltonian H_0

\phi(\mathbf{r}, t) = e^{- \frac{i}{\hbar}H_0 t} \phi(\mathbf{r}, 0) e^{\frac{i}{\hbar}H_0 t}

and they have covariant transformation laws with respect to the non-interacting representation U_0 of the Poincare group. For example, if \Lambda is a boost, then

U_0(\Lambda)\phi(x) U_0^{-1}(\Lambda)= \phi(\Lambda x)

where x \equiv (\mathbf{r}, t).

In some textbooks you can also find so-called "interacting fields" whose time evolution is governed by the full interacting Hamiltonian H = H_0 + V. I will denote the interacting field by capital \Phi

\Phi(\mathbf{r}, t) = e^{- \frac{i}{\hbar}H t} \phi(\mathbf{r}, 0) e^{\frac{i}{\hbar}H t}

The statement of Haag's theorem is that this interacting field cannot have a covariant transformation law with respect to the interacting representation of the Poincare group U, i.e.,

U(\Lambda)\Phi(x) U^{-1}(\Lambda) \neq \Phi(\Lambda x)

Is this a disaster or not-a-big-deal? It depends on your philosophical views. If you believe that interacting quantum fields are fundamental ingredients of nature, and that relativistic invariance implies covariant transformation laws, then Haag's theorem has a disastrous effect.

However, if you think (see e.g., vol. 1 of Weinberg's "The quantum theory of fields") that the primary constituents of nature are particles and that free quantum fields are just formal mathematical objects whose only role is to help in construction of relativistic Hamiltonians as in
https://www.physicsforums.com/showpost.php?p=1382473&postcount=231
then Haag's theorem is just an insignificant curiosity.

Eugene

 

[olgranpappy]

Dyson wrote a really cute paper about divergences in pert. theory... I think the correct reference is:

Phys. Rev. 85, 631 - 632 (1952)

Basically, the idea is that the perturbation series in the coupling has to be an asymptotic series because were it convergent then the function which it converges to is analytic near zero, but that means the coupling constant (e^2 in QED or lambda in phi-fourth theory) could be taken not to be small and positive, but rather to be small and *negative.* And this is ridiculous on physical grounds.

 

[Son Goku]

Thank you very much.

Now all I need to know is what are these existence questions about. I have heard of the Wightman axioms, among others, but what does it mean for a QFT to exist?

 

[meopemuk]

Thank you very much.

Now all I need to know is what are these existence questions about. I have heard of the Wightman axioms, among others, but what does it mean for a QFT to exist?

I've never understood this talk. A non-existent theory cannot predict electron's magnetic moment to 13 significant digits.

Eugene.

 

[Klaus_Hoffmann]

Don't know if this will help, if the problem is that series is divergent you could use its 'Borel sum' attached to it, the Borel resummation can give consistent results even with asymptotic series evaluation amazing ¡¡

 

[olgranpappy]

Another reference is the book by Wightman and Streater:

PCT, Spin, Statistics and all that.

It discusses Haag's theorem.

 

[Son Goku]

I want to come back to this for a second. I’ve been learned QFT from several books, but I’ve been reading Weinberg’s book for the last month and it has really clicked with me. His philosophy seems to be “Free quantum fields are objects used to construct relativistic Hamiltonians for particles, so that the Hamiltonian commutes itself at spacelike distances and gives a Lorentz invariant S-matrix.” So he takes the view that particles are the basic object of consideration.
Could you explain in a little more detail why Haag’s Theorem is okay if you take the particle interpretation?
Also in this view, why does the perturbation series diverge?
(Thanks for Dyson’s paper by the way, the main idea of the proof was actually rather simple.)

Also, I know Baez and Segal wrote a book that goes into the existence stuff a bit, does anybody know the name?

I must get PCT, Spin, Statistics and all that, it constantly gets referenced.

 

[Haelfix]

I completely disagree with Eugene on this point. Usually Haags theorem is taken as a nogo for the particle concept in QFT (a sensible definition of an interacting Fock space), rather than the field concept.

Its worth pointing out that Haags theorem can be circumvented for canonical QFT in the following cases

1) Axiomatic field theory
2) Introduction of a volume cutoff
3) Renormalized canonical QFT

2 and 3 can be shown to be equivalent in some asymptotic sense with respect to Smatrix elements when you take suitable limits.

 

[meopemuk]

I want to come back to this for a second. I’ve been learned QFT from several books, but I’ve been reading Weinberg’s book for the last month and it has really clicked with me. His philosophy seems to be “Free quantum fields are objects used to construct relativistic Hamiltonians for particles, so that the Hamiltonian commutes itself at spacelike distances and gives a Lorentz invariant S-matrix.” So he takes the view that particles are the basic object of consideration.

Exactly! In my opinion, Weinberg's philosophy is the only one that makes sense (at least to me).

Could you explain in a little more detail why Haag’s Theorem is okay if you take the particle interpretation?

In Weinberg's approach, you only need *free* fields in order to build the S-matrix. The *interacting* fields are not needed at all. So, it is irrelevant for calculations and for the relativistic invariance of the theory whether the interacting fields transform by covariant formulas or not.

I also strongly recommend this paper: http://www.arxiv.org/abs/math-ph/0703021

Eugene.

 

[strangerep]

[...] I’ve been reading Weinberg’s book for the last month and it has really clicked with me. His philosophy seems to be “Free quantum fields are objects used to construct relativistic Hamiltonians for particles, so that the Hamiltonian commutes itself at spacelike distances and gives a Lorentz invariant S-matrix.”

The trouble is that this approach works reasonably well to get QED,
but cannot produce full electroweak or QCD theory. For that, Weinberg
reverts to the "postulate-a-Lagrangian" approach.

So he takes the view that particles are the basic object of consideration.

I find the term "particles" a bit ambiguous/misleading. "Irreducible
representations of the Poincare group" is probably better (when
shortened to "Poincare irreps" ).

Baez and Segal wrote a book that goes into the existence stuff a bit,
does anybody know the name?

Are you talking about Baez, Segal & Zhou "Introduction to
Algebraic and Constructive QFT"? You can download it from
http://math.ucr.edu/home/baez/papers.html

 

[strangerep]

I completely disagree with Eugene on this point. Usually Haags theorem is taken as a nogo for the particle concept in QFT (a sensible definition of an interacting Fock space), rather than the field concept.

That depends on how you define the "interacting Fock space", and the
details get rather tricky. There's a little-known book by G. Barton
"Introduction to Advanced Field Theory" which talks about Haag's
theorem in a broader perspective. Here's an extract from Barton:

The following assumptions have been introduced:
1) Lorentz covariance of the theory
2) positive-definite norm
3) local commutativity
4) existence of an invariant and normalizable vacuum
5) positive-definite energy spectrum
6) Canonical equal-time commutation rules, plus
completeness of the canonical variables
7) absence of improper unitary transformations, i.e: non-occurrence of
inequivalent representations of the CCRs.
8) The vector space in which the field theory operates also accommodates
a representation of the free field operator.

Subject to the assumptions (1)-(8), we have shown that no field
theory exists which differs from that of a free field. This is the
statement popularly known as Haag's theorem.

One further subtlety (IMHO) lies in precisely what one means by
"Lorentz covariance" in (1) above. In AQFT, it's typically defined in
conjunction with notions that the field operators are defined over Minkowski
spacetime, with field operators parameterized in terms of x,t.
This has well-known problems stemming from the fact that
these (x,t) are merely parameters, not position observables.

However, one might alternatively try to construct an interacting
representation of the Poincare group directly without the crutch
of a Minkowski spacetime picture. In that case, the detailed meaning
of "Lorentz covariance" is distinctly different from the one used in AQFT.
I think this is a source of truly vast confusions/misunderstandings
between orthodox QFT proponents and alternative approaches such
as Kita, Shirokov, Stefanovich, and others.

The bottom line here seems to be that Haag's theorem is
a no-go for certain attempted-theories only within the overall
framework of the orthodox QFT axioms (eg Haag-Kastler).

It's worth pointing out that Haags theorem can be circumvented
for canonical QFT in the following cases
1) Axiomatic field theory
[...]

Are you alluding to the LSZ formalism and the more rigorous
Haag-Ruelle scattering theory? (Igot the impression that AQFT
is not much good for local gauge theories and the physically-useful
interactions. So if even it can evade Haag's theorem I'm not sure
what help that would be.)

 

[meopemuk]

However, one might alternatively try to construct an interacting
representation of the Poincare group directly without the crutch
of a Minkowski spacetime picture. In that case, the detailed meaning
of "Lorentz covariance" is distinctly different from the one used in AQFT.
I think this is a source of truly vast confusions/misunderstandings
between orthodox QFT proponents and alternative approaches such
as Kita, Shirokov, Stefanovich, and others.

I would like to mention Weinberg as well. Weinberg's particle-based approach (vol. 1 of "The quantum theory of fields") diverges from AQFT axioms in a few important aspects. In his approach, only free quantum fields are necessary, which are just convenient mathematical objects used for the sole purpose of building interaction Hamiltonians, and void of any physical interpretation. The properties of free fields

1) Lorentz covariance
3) local commutativity
6) Canonical equal-time commutation rules, plus completeness of the canonical variables

do not have any deep physical significance as well. They are introduced for the convenience of building an interacting representation of the Poincare group in the Fock space.

However, as you correctly pointed out, Weinberg sticks to the particle-based program only in first chapters of his vol. 1. In order to "derive" interaction Hamiltonians he needs to switch completely to field Lagrangians, gauge symmetry, etc. in vol. 2. Unfortunately, currently there is no alternative way to write down realistic interactions. This is the greatest challenge for the particle-based approach in QFT.

Eugene.

 

[Haelfix]

Are you alluding to the LSZ formalism and the more rigorous
Haag-Ruelle scattering theory? (Igot the impression that AQFT
is not much good for local gauge theories and the physically-useful
interactions. So if even it can evade Haag's theorem I'm not sure
what help that would be.)

Yea amongst others. It gets into myriad subfields of physics and mathematics that I don't know much about. Afaik they have some rigorous interacting examples that are well defined in two dimensions, but for the gauge theories of physical interest, they've stalled for over thirty years in their attempts.

Its worth pointing out that effective field theories bypass the crux of the Haag theorem, as they often introduce cutoffs that break covariance explicitly (eg Lattice QfT) or alternatively, they make the math illdefined so the theorem can't be proved =) Its only when you take the point of view of QFT as truly fundamental rather than merely asymptotic solutions that you run into the deep mathematical problems.

 

[Son Goku]

Are you talking about Baez, Segal & Zhou "Introduction to
Algebraic and Constructive QFT"? You can download it from
http://math.ucr.edu/home/baez/papers.html

Yes, thanks. I'm so glad it's free.

The trouble is that this approach works reasonably well to get QED,
but cannot produce full electroweak or QCD theory. For that, Weinberg
reverts to the "postulate-a-Lagrangian" approach.

Okay I'm going to give a short run down on what I've learned so far. Feel free to correct me if anything is incorrect.

Okay firstly, we can divide free (non-interacting) particles into Irreducible
representations of the Poincare group. (If you get what I mean, as in a sense they are "Poincare irreps".) From a big list of principles (given in Weinberg), the dynamics of these free particles can be found using fields of creation and annihilation operators.
(I know the dynamics are trivial for free particles and the fields are really introduced to treat interactions, but I'm just taking slow.)

Okay, now we turn to the case of interacting particles. Again we need to satisfy Weinberg's conditions. To do this we need to construct its Hamiltonian out of creation and annihilation operators, e.t.c.

Now after all this is done we actually turn to calculating things.
Firstly, finding the eigenkets of the full Hamiltonian and evolving them with e^{-iHt} is too difficult.
To get around this, we first split the Hamiltonian into H = H_{0} + H_{I}. What we basically do is use the Free Hamiltonian, it's eigenkets and the evolution it generates to define an orthonormal set of kets. Then the interaction Hamiltonian is used on these states.
This only requires free fields, a lá Weinberg.

Now we run into two "problems".
First of all Haag's theorem basically implies that it is inconsistent/mathematically unsound to simulate the full evolution under the true Hamiltonian in this manner. In other words, globally, the method of using interaction evolution on free field states does not work as the interaction operator is not well defined on the free particle Fock space.
(I've basically come to this interpretation of Haag's theorem from reading several papers and looking for an overall theme).

Secondly in order to get real world Hamiltonians we have to return to postulating classical Lagrangians and quantising them. Although this isn't a problem exactly.

 

[meopemuk]

Okay I'm going to give a short run down on what I've learned so far. Feel free to correct me if anything is incorrect.

Okay firstly, we can divide free (non-interacting) particles into Irreducible
representations of the Poincare group. (If you get what I mean, as in a sense they are "Poincare irreps".) From a big list of principles (given in Weinberg), the dynamics of these free particles can be found using fields of creation and annihilation operators.
(I know the dynamics are trivial for free particles and the fields are really introduced to treat interactions, but I'm just taking slow.)

Okay, now we turn to the case of interacting particles. Again we need to satisfy Weinberg's conditions. To do this we need to construct its Hamiltonian out of creation and annihilation operators, e.t.c.

Now after all this is done we actually turn to calculating things.
Firstly, finding the eigenkets of the full Hamiltonian and evolving them with e^{-iHt} is too difficult.
To get around this, we first split the Hamiltonian into H = H_{0} + H_{I}. What we basically do is use the Free Hamiltonian, it's eigenkets and the evolution it generates to define an orthonormal set of kets. Then the interaction Hamiltonian is used on these states.
This only requires free fields, a lá Weinberg.

That, basically, coincides with my understanding.

Now we run into two "problems".
First of all Haag's theorem basically implies that it is inconsistent/mathematically unsound to simulate the full evolution under the true Hamiltonian in this manner. In other words, globally, the method of using interaction evolution on free field states does not work as the interaction operator is not well defined on the free particle Fock space.

That's not how I understand the essence of Haag's theorem. Yes, there are problems with describing the interacting time evolution in renormalized QFT. These problems are related to the presence of infinite counterterms in the Hamiltonian. However, Haag's theorem refers to a problem of a different kind. This theorem would be valid even if the Hamiltonian was finite and well-defined. This theorem simply says that "interacting field" \Phi, which is defined as

\Phi(\mathbf{r}, t) = e^{-\frac{i}{\hbar}Ht} \phi (\mathbf{r}, 0) e^{\frac{i}{\hbar}Ht}

cannot have a covariant transformation law with respect to boosts

e^{\frac{ic}{\hbar} \mathbf{K} \vec{\theta}} \Phi (x) e^{-\frac{ic}{\hbar} \mathbf{K} \vec{\theta}} \neq \Phi (\Lambda x)

where \mathbf{K} is interacting boost generator in the Fock space, and boost \Lambda is characterized by the rapidity \vec{\theta}.

Eugene.

 

[Hans de Vries]

I found Haag's original paper, here:

http://doc.cern.ch/yellowrep/1955/1955-008/p1.pdf

Regards, Hans

 

[meopemuk]

I found Haag's original paper, here:

http://doc.cern.ch/yellowrep/1955/1955-008/p1.pdf

Regards, Hans

Great! Thanks!
I didn't know this paper is on the web. This journal is not easy to find in libraries.

Eugene.

 

[strangerep]

[Haag's] theorem simply says that "interacting field" \Phi, [...] cannot have a covariant transformation law with respect to boosts

e^{\frac{ic}{\hbar} \mathbf{K} \vec{\theta}} \Phi (x) e^{-\frac{ic}{\hbar} \mathbf{K} \vec{\theta}} \neq \Phi (\Lambda x)

where \mathbf{K} is interacting boost generator in the Fock space, and boost \Lambda is characterized by the rapidity \vec{\theta}.

To avoid any confusion between what is/isn't mainstream, it should be
clarified that this is not what Haag's theorem "says" in its usual forms.
Rather it is one approach for evading the theorem.

 

[meopemuk]

To avoid any confusion between what is/isn't mainstream, it should be
clarified that this is not what Haag's theorem "says" in its usual forms.
Rather it is one approach for evading the theorem.

Haag's theorem exists in many different equivalent forms, and I thought that my form is equivalent to others. For example, it seems that it is equivalent to the following more canonical formulation (that can be found in the book by Streater & Wightman and in other places):

1. Suppose that we are having two fields: the free field \phi(\mathbf{r},t}) which is covariant with respect to non-interacting boosts

U_0(\Lambda) \phi(x) U_0^{-1} (\Lambda) = \phi(\Lambda x)

and the interacting field \Phi(\mathbf{r},t}) which is covariant with respect to interacting boosts

U(\Lambda) \Phi(x) U^{-1} (\Lambda) = \Phi(\Lambda x) (1)

2. Suppose that these fields are connected by a unitary operator W (t) at all times

\Phi(\mathbf{r},t}) = W(t) \phi (\mathbf{r},t}) W^{-1}(t)

(normally, it is implied that W (t) = \exp(\frac{i}{\hbar}Ht) \exp(-\frac{i}{\hbar}H_0t), where H=H_0 + V is the interacting Hamiltonian)

3. Assume some extra standard conditions: both fields have usual transformations with respect to translations and rotations; canonical commutation relations; the existence of a unique vacuum vector; the Hamiltonian is bounded from below,...

Then, Haag's theorem states that both fields \Phi(\mathbf{r},t}) and \phi(\mathbf{r},t}) are equivalent, i.e., that W (t) = 1, or that H = H_0.

In other words, this theorem derives the non-existence of the interacting Hamiltonian from the covariant transformation law of the interacting field (1). In my formulation, I simply exchanged places of these two statements. I derived the non-existence of the covariant transformation law (1) from the existence of the interacting Hamiltonian H.

I think that both formulations are equivalent. Or I am missing something?...

The reason why I chose to formulate Haag's theorem in the non-traditional form is that I have no reason to doubt the existence of interactions, however, I am very doubtful about the physical meaning and usefulness of "interacting fields". From this point of view, the theorem becomes completely harmless.

Eugene.

 

[gptejms]

In other words, this theorem derives the non-existence of the interacting Hamiltonian from the covariant transformation law of the interacting field (1). In my formulation, I simply exchanged places of these two statements. I derived the non-existence of the covariant transformation law (1) from the existence of the interacting Hamiltonian H.

So, can we conclude that there is no covariant theory for interacting Hamiltonians?I read that the problem is circumvented by LSZ formula--how?--by dealing with the free field only?

 

[meopemuk]

So, can we conclude that there is no covariant theory for interacting Hamiltonians?

Yes, I think this is correct. However, this doesn't mean that non-covariant theories are "bad or "wrong". Let me explain what I mean. There are two completely different notions that often get mixed up with terrible consequences. There is the notion of relativistic invariance and there is the notion of manifest covariance. It is important to distinguish them.

Relativistic invariance means that the Hilbert (or Fock) space of the physical system carries an unitary representation of the Poincare group. This is the fundamental requirement that must be satisfied by any realistic relativistic quantum theory. The importance of this requirement is well explained in vol. 1 of Weinberg's book series. All traditional quantum field theories, such as QED, satisfy this requirement, which guarantees, in particular, that observables have correct transformation laws with respect to boosts and other inertial transformations. The relativistic invariance (the Poincare commutators of interacting generators) of QED has been proven in

S. Weinberg, "Photons and gravitons in perturbation theory: Derivation of Maxwell's and Einstein's equations", Phys. Rev. 138B (1965), 988.

Manifest covariance means that interacting fields transform by certain simple formulas with respect to boosts. For example, the covariant transformation law of the scalar field is assumed to be

U(\Lambda) \Phi(x) U^{-1} (\Lambda) = \Phi(\Lambda x)...(1)

These transformation laws are often assumed as something self-evident. However, I've never seen a good explanation of why they should be valid. (Wenberg's book explains why they should be valid for free fields, and I understand that. But interacting fields is a different matter.)

I think that in a broader sense, the meaning of Haag's theorem is to indicate the incompatibility of these two notions: the relativistic invariance and the manifest covariance. If I am forced to choose between the two, I'll definitely choose the relativistic invariance. I am sure that if somebody manages to construct interacting fields corresponding to QED (this is not an easy task), she will find that they do not transform covariantly. So what? Actually, there is an explicit example of an interacting fully relativistic theory in the Fock space, for which one can exactly construct the interacting fields and verify that their boost transformations are different from (1)

H. Kita, "A non-trivial example of a relativistic quantum theory of particles without divergence difficulties" Progr. Theor. Phys. 35 (1966), 934.

This contradiction between relativistic invariance and manifest covariance is not specific for quantum field theory. A theorem resembling Haag's theorem can be proved in relativistic classical mechanics of particles as well:

D. G. Currie, T. F. Jordan, E. C. G. Sudarshan, "Relativistic invariance and Hamiltonian theories of interacting particles" Rev. Mod. Phys. 35 (1963), 350.

This theorem basically says that if worldlines of interacting particles transform covariantly (by usual Lorentz formulas), then interaction between particles must be absent.

My conclusion is that the idea of manifest covariance should be abandoned in relativistically invariant theories. Of course, this is not a "mainstream" point of view. But you can judge for yourself whether it makes sense.

I read that the problem is circumvented by LSZ formula--how?--by dealing with the free field only?

I am not so familiar with LSZ formalism. Perhaps, strangerep knows better, as he mentioned LSZ in his post #12.


Eugene.

 

[meopemuk]

I read that the problem is circumvented by LSZ formula--how?--by dealing with the free field only?

It looks obvious to me that "interacting fields" are not needed in S-matrix calculations. Feynman diagrams and propagators are formulated using "free fields" only. Is there any application of QFT where "interacting fields" are absolutely necessary?

Eugene.

 

[Haelfix]

QCD, the electroweak sector, etc etc. Turn off interactions, and you have absolute nonsensical physics (like having absolutely massless particle spectrums, or unconfined quarks)

But look i'll concede that the exact mathematics behind interacting *gauge* field theories is quite incomplete and unknown. Otoh we know the standard formalism is correct in some sense (probably as a really good approximation to something), b/c it fits experiment so well, ergo there is a way to make it rigorous. This is the subject of the Clay Millenium prize, and indeed it is important for physical reasons too. We'd like to understand confinement and the mass gap analytically somehow.

 

[meopemuk]

QCD, the electroweak sector, etc etc. Turn off interactions, and you have absolute nonsensical physics (like having absolutely massless particle spectrums, or unconfined quarks)

But look i'll concede that the exact mathematics behind interacting *gauge* field theories is quite incomplete and unknown. Otoh we know the standard formalism is correct in some sense (probably as a really good approximation to something), b/c it fits experiment so well, ergo there is a way to make it rigorous. This is the subject of the Clay Millenium prize, and indeed it is important for physical reasons too. We'd like to understand confinement and the mass gap analytically somehow.

OK, I agree that gauge field theories have made a huge empirical progress. However, I think, it is also correct to say that nobody understands the physical meaning of gauge fields. (If somebody understood that, (s)he would have written a clear explanation without usual handwavings). I think that current stagnation in theoretical HEP is an indication that already in 1970's we exhausted all luck that we could get by playing with gauge groups. Now, we either need to obtain a solid understanding of what the gauge fields are, or replace them with something completely different.

This is just my uneducated opinion.
Eugene.

 

[strangerep]

[...]
In other words, [Haag's] theorem derives the non-existence of the interacting Hamiltonian from the covariant transformation law of the interacting field (1). In my formulation, I simply exchanged places of these two statements. I derived the non-existence of the covariant transformation law (1) from the existence of the interacting Hamiltonian H.

I think that both formulations are equivalent. [...]

Suppose we have a theorem which says "if A then B". We can't use it to
automatically infer "if B then A". I.e: simply exchanging the places of
the two statements is not valid without supplying a detailed proof of the
reverse direction. It's like a maths exam question that asks the student
to prove "A if and only if B". If the student only proved the forward
direction "if A then B", but didn't also prove "if B then A", then
(s)he would not receive full marks.

The reason why I chose to formulate Haag's theorem in the non-traditional form is that I have no reason to doubt the existence of interactions, however, I am very doubtful about the physical meaning and usefulness of "interacting fields". From this point of view, the theorem becomes completely harmless.

That's why I said what you wrote is one approach for "evading"
the theorem. I.e: if we don't like the consequences of a theorem, then we
must abandon one or more of its required pre-conditions. In your case,
you're advocating abandonment of the usual form of the boost transformation
law which had been phrased with a Minkowski spacetime picture in mind.
Personally, I think this is an interesting research direction to pursue, especially
in view of the well-known problems about Lorentz covariance of trajectories of
interacting particles discussed elsewhere.

 

[strangerep]

So, can we conclude that there is no covariant theory for interacting Hamiltonians?I read that the problem is circumvented by LSZ formula--how?--by dealing with the free field only?

I think all we can conclude is that the orthodox axioms for QFT lead to
logical problems when trying to tackle the physical interactions we're
most interested in.

The LSZ formalism is not entirely rigorous, so I'll just say a little bit about
its more rigorous cousin known as "Haag-Ruelle scattering theory". In the
latter, one talks about asymptotic in/out fields which are assumed to be free,
since they're what we measure in scattering experiments. One also talks
about "interpolating" fields, representing what goes on during the
interaction process. The theory discusses how the interpolating fields
approach the in/out fields in a certain precise sense. Unfortunately, all
this extra effort is not much help in practical scattering calculations.

There is a brief elementary introduction to some aspects of
non-perturbative QFT in Ch7 of Peskin & Schroeder. Other books like
Streater & Wightman, as well as Haag's "Local Quantum Physics"
are rather more difficult.

 

[meopemuk]

Suppose we have a theorem which says "if A then B". We can't use it to
automatically infer "if B then A". I.e: simply exchanging the places of
the two statements is not valid without supplying a detailed proof of the
reverse direction. It's like a maths exam question that asks the student
to prove "A if and only if B". If the student only proved the forward
direction "if A then B", but didn't also prove "if B then A", then
(s)he would not receive full marks.

I don't think I violated rules of logic. In a shorthand notation (the traditional form of) Haag's theorem establishes the following implication

"covariant transformations of fields" ==> "trivial interaction"

Following rules of Aristotle-Boole logic, I am allowed to take negation of both sides and invert the direction of implication. So, I obtain

"non-trivial interaction" ==> "non-covariant transformations of fields"

i.e., Haag's theorem in my formulation.

Nevertheless, I agree with you that this is a hand-waving, and that a serious approach to the theorem requires a clear formulation of all conditions and a rigorous proof. I haven't done that.

That's why I said what you wrote is one approach for "evading"
the theorem. I.e: if we don't like the consequences of a theorem, then we
must abandon one or more of its required pre-conditions. In your case,
you're advocating abandonment of the usual form of the boost transformation
law which had been phrased with a Minkowski spacetime picture in mind.
Personally, I think this is an interesting research direction to pursue, especially
in view of the well-known problems about Lorentz covariance of trajectories of
interacting particles discussed elsewhere.

Yes, I agree with you. Haag's theorem is a serious challenge to our understanding of QFT. This paradox, indeed, requires us to abandon some of the conditions. The question is: which one? There could be different opinions.

One common idea is that free and interacting Hamiltonians may act in different Hilbert spaces, and even not share the same vacuum vector. This is one method to "evade" Haag's theorem. However, this method looks very suspicious to me. It contradicts almost everything I know about quantum theory.

Another idea is that the covariant transformations of interacting fields is an ill-justified statement. Maybe we can just abandon it, without any serious consequences for the theory?

Are there other ideas how Haag's theorem can be "evaded"?

Eugene.

 

[strangerep]

I don't think I violated rules of logic. In a shorthand notation (the traditional form of) Haag's theorem establishes the following implication

"covariant transformations of fields" ==> "trivial interaction"

Following rules of Aristotle-Boole logic, I am allowed to take negation of both sides and invert the direction of implication. So, I obtain

"non-trivial interaction" ==> "non-covariant transformations of fields"

i.e., Haag's theorem in my formulation.

You're ignoring the other pre-conditions of Haag's thm. Let me re-phrase my
previous illustration: given a theorem that says "if (A & B) then C", we can
do the negation as you said, to get "if not(C) then not(A&B)", the latter part
of which expands to "... not(A) or not(B)".

But let us not hijack this thread away any further from the issues
the OP was specifically interested in.

 

[gptejms]

One common idea is that free and interacting Hamiltonians may act in different Hilbert spaces, and even not share the same vacuum vector. This is one method to "evade" Haag's theorem. However, this method looks very suspicious to me. It contradicts almost everything I know about quantum theory.

Please elaborate your statement--how does this contradict almost everything you know about quantum theory.

 

[meopemuk]

Please elaborate your statement--how does this contradict almost everything you know about quantum theory.

In quantum mechanics, every physical system is described by a Hilbert space. States of the system are described as unit vectors in this Hilbert space, and observables are described as Hermitian operators there. If we know the state vector and the operator of observable, we can calculate the probabilities of measurements of different values of this observable in this state and compare them with experiments.

In addition, we would like to know how the state vactor is transformed when the observer changes (the time dynamics is one example of such a transformation). This knowledge is provided by the interacting representation of the Poincare group acting in the Hilbert space of the system. The interacting Hamiltonian H is one (out of ten) generator of this representation. Also, one can always define the non-interacting representation of the Poincare group in the same Hilbert space with the time-translation generator H_0. Both H and H_0 act in the same Hilbert space.

The same principles work in quantum field theory. The only difference is that QFT deals with systems in which the number of particles is not conserved, so the Hilbert space is, actually, the Fock space with variable number of particles. This is how I understand quantum mechanics and QFT.


I don't know how one can justify introduction of two different Hilbert spaces (one for the interacting Hamiltonian and vacuum, and another for the non-interacting Hamiltonian and vacuum). I think these ideas (AKA "inequivalent representations of the canonical commutation relations") go against letter and spirit of quantum mechanics. These ideas are presented in many places (e.g., Umezawa "Thermo fields dynamics and condensed states") but they don't make sense to me.

Eugene.

 

[gptejms]

.

Also, one can always define the non-interacting representation of the Poincare group in the same Hilbert space with the time-translation generator H_0. Both H and H_0 act in the same Hilbert space.

The same principles work in quantum field theory. The only difference is that QFT deals with systems in which the number of particles is not conserved, so the Hilbert space is, actually, the Fock space with variable number of particles. This is how I understand quantum mechanics and QFT.

Eugene.

Starting with a state of the free hamiltonian,say, we switch on the interaction--the state initially lies in Hilbert space with fixed no. (say n) of particles and later evolves into a state in the Fock space with variable number of particles.So do H and H_0 really act on the same Hilbert space?
Regarding vacuum, one can insist that vacuum should remain vacuum in the presence of interactions--but the two vacua(don't know if this is the right word!) are not really the same.

 

[meopemuk]

Starting with a state of the free hamiltonian,say, we switch on the interaction--the state initially lies in Hilbert space with fixed no. (say n) of particles and later evolves into a state in the Fock space with variable number of particles.So do H and H_0 really act on the same Hilbert space?

Yes. In both cases the Hilbert space is, actually, the Fock space, which is built as a direct sum of tensor products of n-particle spaces. For example, in the world where only one particle type exists, and the 1-particle Hilbert space is H, the Fock space has the form

F = |0 \rangle \oplus H \oplus (H \otimes H) \oplus (H \otimes H \otimes H) \ldots

where |0 \rangle is the vacuum vector, and tensor products H \otimes H \ldots imply symmetrization/antisymmetrization, as appropriate.

The Fock space allows you to describe all kinds of time evolutions, including those in which the number of particles can change.

Regarding vacuum, one can insist that vacuum should remain vacuum in the presence of interactions--but the two vacua(don't know if this is the right word!) are not really the same.

It is true that for most Hamiltonians in QFT the original (bare) vacuum state |0 \rangle is not an eigenstate. The same is true for bare 1-particle states |1 \rangle. They are not eigenstates of the interacting Hamiltonian. The usual solution is to say that the vacuum and one particle states change in the presence of interaction, so that they become "physical" vacuum |vac \rangle and "physical" particles |one \rangle, respectively, which are rather complex linear combinations of n-particle bare states. In popular books this is also described as "virtual particles", which fill vacuum and form "clouds" surrounding real particles.

In physical applications, we are not interested in properties of bare particles and bare vacuum. All real physical processes (scattering, formation of bound states, etc.) involve physical states |vac \rangle and |one \rangle. So, the fact that the Hamiltonian is normally expressed in terms of creation and annihilation operators of bare particles is very inconvenient.

Luckily, there is an alternative formulation of QFT in which bare particles do not appear at all. The Hamiltonian is expressed directly in terms of creation and annihilation operators of physical particles. The vacuum state is a state with no physical particles, as expected. This "dressed particle" approach was first suggested in

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

It appears that traditional quantum field theories (including QED) can be recast into the "dressed particle" form without losing any of their predictive power. The characteristic feature of interacting Hamiltonians in dressed particle theories is that they yield zero when acting on physical vacuum and one-particle states. So, these states do not depend on whether the interaction is present or absent. Intuitively, this is understandable: by definition, interaction can take place only when there are two or more particles. One (or zero) particle has nothing to interact with. In other words, the "dressed particle" approach eliminates the notion of self-interaction in the vacuum and one-particle states. One can also say that even if the self-interaction is present (as in usual QFT theories), it can be always incorporated into the definition of physical vacuum and physical particles and "forgotten".

I've written a bit more about these ideas in

https://www.physicsforums.com/showpost.php?p=1391726&postcount=6

Eugene.

 

[gptejms]

Yes. In both cases the Hilbert space is, actually, the Fock space, which is built as a direct sum of tensor products of n-particle spaces. For example, in the world where only one particle type exists, and the 1-particle Hilbert space is H, the Fock space has the form

F = |0 \rangle \oplus H \oplus (H \otimes H) \oplus (H \otimes H \otimes H) \ldots

where |0 \rangle is the vacuum vector, and tensor products H \otimes H \ldots imply symmetrization/antisymmetrization, as appropriate.

The Fock space allows you to describe all kinds of time evolutions, including those in which the number of particles can change.

Ok,so H_0 acts on a subspace while (operator)H acts on the complete (Fock)space.

In other words, the "dressed particle" approach eliminates the notion of self-interaction in the vacuum and one-particle states. One can also say that even if the self-interaction is present (as in usual QFT theories), it can be always incorporated into the definition of physical vacuum and physical particles and "forgotten".

I've written a bit more about these ideas in

https://www.physicsforums.com/showpost.php?p=1391726&postcount=6

Eugene.

So the virtual particles are artefacts of using bare states as our basis in QFT--we should get rid of this approach and use physical states to avoid confusion.

Coming to Haag's theorem--if you acknowledge that physical vacuum is different from bare vacuum,what implications does it have on the interpretation of Haag's theorem?

 

[meopemuk]

Ok,so H_0 acts on a subspace while (operator)H acts on the complete (Fock)space.

No, H_0 also acts on the complete Fock space. Its expression in the entire Fock space is

H_0 = \int d^3p \sqrt{m^2c^4 + p^2c^2} a^{\dag}_p a_p

From this formula one can obtain more explicit and physically transparent formulas in each individual sector. For example, in the 2-particle sector

H_0 = \sqrt{m^2c^4 + p_1^2c^2} + \sqrt{m^2c^4 + p_2^2c^2}

where p_1 and p_2 are momenta of the two particles.

So the virtual particles are artefacts of using bare states as our basis in QFT--we should get rid of this approach and use physical states to avoid confusion.

Yes, I believe so. That's what Greenberg-Schweber "dressed particle" approach does.

Coming to Haag's theorem--if you acknowledge that physical vacuum is different from bare vacuum,what implications does it have on the interpretation of Haag's theorem?

That's an interesting question. I haven't thought it through. I am not sure whether Haag's theorem requires both interacting and non-interacting theories to have the same vacuum vector. This requirement would exclude all traditional relativistic quantum field theories known to man.

On the other hand in "dressed particle" theories the bare in interacting vacua are exactly the same, and Haag's theorem doesn't apply, because interacting fields do not transform by traditional formulas

U(\Lambda) \psi(x) U^{-1} (\Lambda) = \psi(\Lambda x)

You can find more discussions in

M. I. Shirokov, "Dressing" and Haag's theorem, http://www.arxiv.org/math-ph/0703021

Eugene.

 

[gptejms]

From this formula one can obtain more explicit and physically transparent formulas in each individual sector. For example, in the 2-particle sector

H_0 = \sqrt{m^2c^4 + p_1^2c^2} + \sqrt{m^2c^4 + p_2^2c^2}

where p_1 and p_2 are momenta of the two particles.

What I meant was that once an H_0 is defined thus(in a partiular sector),it continues to act only in this sector or subspace.

That's an interesting question. I haven't thought it through. I am not sure whether Haag's theorem requires both interacting and non-interacting theories to have the same vacuum vector. This requirement would exclude all traditional relativistic quantum field theories known to man.

Why would it exclude all traditional relativistic quantum field theories known to man?You may like to read the following from an article on Haag's theorem that I have downloaded.This is as the author says 'a gist of the heuristic version of Haag's original theorem'.

'This argument takes the form of a reductio. Suppose that we are trying to describe both a free scalar field and a self-interacting scalar field using the same Hilbert space H. Suppose that we demand of the vacuum state that it be the unique (up to phase) normalized state |0> \epsilon H that is invariant under Euclidean translations. And suppose that the vacuum state is the ground state in that it is an eigenstate of the Hamiltonian with eigenvalue 0.These suppositions are fullfilled in the case of the free scalar field with mass m > 0, the usual no-particle state |0_F> (bare vacuum), and the free field Hamiltonian H_F .
Since the vacuum state |0_I> of the interacting field (dressed vacuum or physical vacuum) should also be invariant under Euclidean translations,it follows from the stated assumptions that |0_I> = c|0_F>, |c| = 1, and since |0_I> is annihilated by the Hamiltonian H for the interacting field, it follows that H|0_F> = 0. But the typical Hamiltonians for interacting fields take the form H_F +H_I , where H_I describes the interaction of the field with itself, and such Hamiltonians do not annihilate |0_F> (H polarizes the vacuum)'.

If you want to read more google for 'Haag's Theorem and Its Implications for the Foundations of Quantum Field Theory'.

 

[meopemuk]

From this formula one can obtain more explicit and physically transparent formulas in each individual sector. For example, in the 2-particle sector

H_0 = \sqrt{m^2c^4 + p_1^2c^2} + \sqrt{m^2c^4 + p_2^2c^2}

where p_1 and p_2 are momenta of the two particles.

What I meant was that once an H_0 is defined thus(in a partiular sector),it continues to act only in this sector or subspace.

Yes, we agree on this point. Each sector of the Fock space is invariant with respect to the action of the free Hamiltonian H_0. In other words, free time evolution conserves the number of particles, as expected.

Why would it exclude all traditional relativistic quantum field theories known to man?You may like to read the following from an article on Haag's theorem that I have downloaded.This is as the author says 'a gist of the heuristic version of Haag's original theorem'.

'This argument takes the form of a reductio. Suppose that we are trying to describe both a free scalar field and a self-interacting scalar field using the same Hilbert space H. Suppose that we demand of the vacuum state that it be the unique (up to phase) normalized state |0> \in H that is invariant under Euclidean translations. And suppose that the vacuum state is the ground state in that it is an eigenstate of the Hamiltonian with eigenvalue 0.These suppositions are fullfilled in the case of the free scalar field with mass m > 0, the usual no-particle state |0_F> (bare vacuum), and the free field Hamiltonian H_F .
Since the vacuum state |0_I> of the interacting field (dressed vacuum or physical vacuum) should also be invariant under Euclidean translations,it follows from the stated assumptions that |0_I> = c|0_F>, |c| = 1, and since |0_I> is annihilated by the Hamiltonian H for the interacting field, it follows that H|0_F> = 0. But the typical Hamiltonians for interacting fields take the form H_F +H_I , where H_I describes the interaction of the field with itself, and such Hamiltonians do not annihilate |0_F> (H polarizes the vacuum)'.

If you want to read more google for 'Haag's Theorem and Its Implications for the Foundations of Quantum Field Theory'.

Thank you for the reference. I downloaded this article, and started to read it. It looks interesting.

This "heuristic" version of Haag's theorem uses the fact that all relativistic quantum field theories "known to man" polarize vacuum. This means that the lowest energy eigenstate |vac \rangle of the full interacting Hamiltonian H = H_0 + V (or H = H_F + H_I in your notation) is different from the lowest energy eigenstate |0 \rangle of the free Hamiltonian H_0. In QED, this fact is evident from the presence of tri-linear terms in the interaction Hamiltonian V. For example, there are terms with three creation operators

a^{\dag}b^{\dag}c^{\dag}...(1)

with a non-trivial action on the vector |0 \rangle (here a^{\dag}b^{\dag}c^{\dag} are creation operators for electrons, positrons, and photons, respectively). The presence of these terms implies that |0 \rangle is not an eigenvector of H.

This version of Haag's theorem does not apply to "dressed particle" theories (including the "dressed particle" version of QED), because in such theories terms of the type (1) are explicitly absent in interaction Hamiltonians and the vacuum is not "polarized".

In my previous posts I discussed the version of Haag's theorem, which is marked as "HWW theorem, Part II" in the paper you referred to. My major objection was related to the use of the covariant transformation law for interacting fields

U(\Lambda) \phi(x) U^{-1}(\Lambda) = \phi(\Lambda x)

(which, in a different notation, is given in eq. (11) for j=2). My point was that there is no empirical or theoretical evidence for the validity of this transformation law. For example, I am pretty sure that interacting fields in QED do not obey such transformations. However, this doesn't mean that QED is inaccurate or relativistically non-invariant or inappropriate for any other reason.

Eugene.

 

[strangerep]

I think I can now clarify a few more things...

Perhaps the most important thing to know about Haag's thm
is that it's formulated within the framework of (orthodox)
axiomatic QFT. This point is critical because if one
doesn't have solid axioms, one doesn't have a solid
mathematical framework, and therefore cannot prove any
worthwhile theorems.

First, here's a summary of the (orthodox) Axioms...

Axiom-1 postulates the physical existence of Minkowski
spacetime, asserting the "events happen in a 4D Minkowski
spacetime, an event being something about which it makes
sense to say that it does or does not occur".

Axiom-1 also postulates that a quantum state of a physical
system is described by a ray in a separable Hilbert
space "H". To every measurable physical quantity "a"
there corresponds a self-adjoint, generally unbounded,
operator "A" in the separable Hilbert space.

Axiom-2 postulates that we are given a finite number of
operator-valued distributions phi_i(x) over Minkowski
spacetime, called "fields". These fields, smeared out
with test functions and then denoted phi(f), define an
(in general unbounded) operators acting in the Hilbert space.

It is not until Axiom-3 that we come to the postulate that
there exists in H a unitary continuous representation of
the Poincare group. It is here that the usual expression
of the transformation rules for fields enters, expressed
in terms of how these transformations work for Minkowski
spacetime points. (This is the item about which Eugene has
frequently been asking whether a "proof" exists.
Answer: "no, it's an axiom".)

Axiom-4 deals with the spectral conditions of the mass^2
and energy operators.

Axiom-5 deals with causality, using the notion of test
functions with compact support in Minkowski space to
define "local" operator fields which either commute
or anti-commute with each other if the supports of the
respective test functions are spacelike-separated.

Axiom-6 postulates that the vacuum vector (being
the eigenstate of the energy operator with lowest
eigenvalue) is "cyclic" with respect to the field
operators phi(f). This means (loosely) that the whole
Hilbert space H can be generated by the action of the
phi(f) operators on the vacuum. It also involves the
operators being "irreducible" in H, meaning that every
operator in H can be approximated arbitrarily closely
by functions of the phi(f).

Then there's also an "Axiom-0" postulating that the
Hilbert space decomposes in coherent Hilbert spaces
corresponding to the various physically-meaningful
quantum numbers (e.g: spin, charges, etc).

----- (End of Axioms) -----

You don't have to read very far into these axioms to see
that Eugene's approach to QFT is outside this framework.
E.g: the use of Minkowski spacetime points as physical
events, and "attaching" field operators to these points,
is distinctly different from an approach which insists strictly
that physically-measurable positions must correspond
to the eigenvalues of a self-adjoint operator.

Now, about Haag's thm... It doesn't talk specifically
about "interacting" and "non-interacting" representations.
Rather it takes the 1st representation as free and assumes
the existence of a 2nd Poincare representation in the same
Hilbert space described by the above axioms (and with
the same kind of expression for Lorentz transformations
of the fields). It finds that this 2nd representation
must also be free.

Discussions then turn to what should be "done" about
Haag's thm, i.e: which pre-condition(s) or axiom(s)
should be abandoned or modified. Eugene (and Shirokov,
afaict) advocate that the problem lies in the form
of Lorentz boosts acting on the fields (which was
originally motivated by the way the fields are "attached"
to points of Minkowski spacetime). The Earman/Fraser
paper mentioned earlier (and also other authors,
e.g: Barton, as well as some others who start from
an algebraic approach to QFT) advocate the importance
of unitarily inequivalent representations (with their
different vacuum vectors and disjoint Hilbert spaces).
Ironically, this is closely related to the way Haag
originally discussed his theorem in the paper
mentioned earlier in this thread.

Also earlier in this thread, the question arose about whether
Haag's thm excluded "all traditional relativistic QFTs
known to man". I think the key point here is that the
orthodox axiomatic QFT framework does not work at all
well with modern gauge field theories such as the Standard
Model. Hence, Haag's thm is pretty much irrelevant there.
The modern approach is to use Path Integral techniques,
but even this has not yet been rigorously successful
wrt proving existence of a 4D interacting QFT.

[And BTW, the irrelevance of Haag's thm seems also to
apply to the famous Coleman-Mandula thm which motivated
supersymmetry - since the C-M thm is also formulated
under the auspices of axiomatic QFT.]

I'm not sure whether the above sufficiently addresses all the issues
previously raised in this thread, so I'll leave it at that for now and
wait to see if anyone wants to talk further.

 

[Haelfix]

Coleman Mandula is considerably stronger as it requires far less axioms than Haags theorem.. AFAICR, you basically only need finite particle species below some mass scale and analytic cross sections for elastic 2 body scattering.

Strictly speaking I seem to recall it hasn't been proved for all QFTs of interest, and the cases where you can evade the axioms are all thoroughly studied and interesting in their own right.

 

[meopemuk]

Thank you strangerep,

your summary is very good. I agree completely.

For me your analysis means that AQFT (with its present axioms) is a purely formal exercise disconnected from reality.

Eugene.