# Haag's Theorem, Perturbation, Existence and QFT.

1. Jul 22, 2007

### 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.

2. Jul 22, 2007

### meopemuk

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

3. Jul 22, 2007

### 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.

4. Jul 23, 2007

### 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?

5. Jul 23, 2007

### meopemuk

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

Eugene.

6. Jul 23, 2007

### 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 ¡¡

7. Jul 23, 2007

### olgranpappy

Another reference is the book by Wightman and Streater:

PCT, Spin, Statistics and all that.

It discusses Haag's theorem.

8. Jul 27, 2007

### 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.

9. Jul 27, 2007

### 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.

10. Jul 27, 2007

### meopemuk

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

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.

11. Jul 27, 2007

### strangerep

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.

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

Are you talking about Baez, Segal & Zhou "Introduction to
http://math.ucr.edu/home/baez/papers.html

12. Jul 27, 2007

### strangerep

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
theorem in a broader perspective. Here's an extract from Barton:

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)
.

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.)

13. Jul 28, 2007

### meopemuk

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.

14. Jul 28, 2007

### Haelfix

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.

15. Jul 29, 2007

### Son Goku

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

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.

16. Jul 29, 2007

### meopemuk

That, basically, coincides with my understanding.

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.

17. Jul 29, 2007

### Hans de Vries

18. Jul 29, 2007

### meopemuk

19. Jul 30, 2007

### strangerep

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.

20. Jul 30, 2007

### meopemuk

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.

Last edited: Jul 31, 2007