\phi^{4} theory rigorously

A. Neumaier

From the thread, Rigorous Quantum Field Theory:

I only saw the first (2D) part of this promise delivered. Since I found it a quite interesting summary, I would be interested in the remaining parts on the 3D and 4D case.

strangerep

So would I, but DarMM seems to have been absent from PF for a while now.

You could perhaps try sending him a PM... :-)

DarMM

Hello all,

My apologies for the enormous delay! I will immediately proceed to gather my notes and make a post on the three dimensional case. Unless there are comments left over from the two-dimensional case.

A. Neumaier

I am looking forward what you have to say about the 3D case. This case should be more interesting, since one here has for the first time a nontrivial renormalization (beyond normal ordering) to make the field theory even well-defined. I hope that you'll able to give an easily understandable introduction of the main lines without drowning us in details (the original literature seems fairly opaque in this respect).

If then issues in understanding appear that can already be illustrated in the 2D setting, you'd come back to the latter.

Later, if the 3D case has been clarified, I'd like to know your synopsis of the 4D situation. For example, I don't believe that Phi^4_4 theory is trivial, but this is frequently claimed, so I'd like to know your position in this issue, and how you arrived at it.

If you know something about the status of rigorous Phi^6_3, that would also be interesting!

So these are my hopes about your input!

DarMM

(Okay, I realised I was delaying too long and I had already enough to deal with the first stage, so here goes.)

Alright, let us deal with the first step of [tex]\phi_{3}^{4}[/tex].

The major starting point for this model is the paper:
James Glimm "Boson Fields with the [tex]:\phi^{4}:[/tex] Interaction in Three Dimensions", Comm. Math. Phys. 10, 1-47.
Glimm only deals with the finite volume theory.

This is the first model with nontrivial renormalisations. It has one single mass renormalization which is not equivalent to a Wick ordering.

In the two dimensional case Wick ordering removed all ultraviolet divergences and the finite volume theory was well-defined on Fock space. Here things are more difficult. Wick ordering does not remove all divergences because in more than two dimensions Wick ordered fields of power greater than 3 do not produce operators after smearing.

Taking inspiration from perturbation theory Glimm cuts of the Hamiltonian and adds the first order and second order mass counterterms to the Lagrangian. This should produce a Hamiltonian with energy levels that no longer diverge with the cutoff.

However by checking the overlap of the new Hamiltonian's vacuum with any element of Fock space, one sees that it is becoming orthogonal to every Fock vector.

To analyse the model, a dressing transformation is constructed between the states of the free field and the states of the interacting field. However the full dressing transformation is not needed as one only cares about controlling the passage to the new Hilbert space. So one removes from the dressing transformation the parts associated with all momenta confined to a bounded region and those pieces corresponding to the components of [tex]:\phi^{4}:[/tex] with less than four creation operators. What remains is an approximate dressing transformation that contains only the part that maps out of Fock space.

One then finds that the norm of every vector is diverging in Fock space by a term:
[tex]\left(T_{\kappa}\Omega,T_{\kappa}\Omega\right)[/tex], where [tex]T_{\kappa}[/tex] is the dressing transformation of the cutoff model and [tex]\Omega[/tex] the Fock vacuum.

Since every vector is related to an interacting vector by the dressing transformation, one then contructs the interacting product by:
[tex]\frac{\left(T_{\kappa}\Psi,T_{\kappa}\Psi\right)}{\left(T_{\kappa}\Omeg a,T_{\kappa}\Omega\right)}[/tex]
Glimm then proves that in the limit [tex]\kappa \rightarrow \infty[/tex], this space of [tex]T_{\kappa}\Psi[/tex] vectors is a Hilbert space complete with respect to the [tex]\kappa \rightarrow \infty[/tex] norm.

Then returning to the Hamiltonian, one places the cutoff Hamiltonian with the counterterms [tex]H^{ren}_{\kappa}[/tex] into the inner product and one can demonstrate that every matrix element converges and the Hamiltonian remains symmetric in the limit.

So one has constructed a symmetric Hamiltonian on a new (non-Fock) Hilbert space.

However one has not shown that it is self-adjoint or semi-bounded (non-negative energy in physical terms). This will be dealt with in the next post.

So in short:
(1) Hamiltonian diverges
(2) Cutoff the Hamiltonian and add counterterms
(3) Energy levels no longer diverge, but eigenstates become orthogonal to Fock space.
(4) "Follow" the eigenvalues by constructing a norm that moves with them. This norm uses the dressing transform of the vacuum, this is because all vectors move out of Fock space at the same rate: the rate at which the vacuum moves out.
(5) Place the Hamiltonian in the norm to find out how it behaves as one moves to the new Hilbert space.
(6) One discovers that the vectors move to a well-defined Hilbert space and the Hamiltonian is symmetric on that space.

Two pieces of information:
1. Glimm calls the division by the dressed vacuum norm "wave function renormalisation". This is not "wave function renormalisation" as known by physicists. It is a completely nonperturbative effect associated with the change of Hilbert space, which is not something dealt with by perturbative QFT.

2. In [tex]\phi^{4}_{4}[/tex] part of the difficulty comes from the fact that all the vectors move out of Fock space at a different rate, so dividing by the norm of the dressed vacuum is not enough. Of course this is only the beginning of the difficulties for the four-dimensional case.

A. Neumaier

Thanks! Let me check my understanding:

- I guess you meant: follow the eigenvectors...
- One now knows that on the new Hilbert space, H is defined as a quadratic form.
- This H will turn out (by different arguments, not yet given) to be self-adjoint and bounded below.
- Thus we have such an H for every bounded region in space.
- These must now be used to define an infinite volume limit, I presume, which needs again new arguments that you'll summarize for us.
- Then one must proceed to verify that the limiting theory is Lorentz invariant and nontrivial.

I just noticed a paper by Brydges et al. in Comm. Math. Phys. 91 (1983), 141-186,
which I haven't read yet but which promises to give an ''extremely simple proof'' for the existence and nontriviality of Phi^4_3 theory in Euclidean space. Is this a very different approach, or just (because of the Wick rotation) a slicker handling of the same technicalities as those you presented/will present?

DarMM

Thanks for pointing that out, indeed "follow the eigenvectors".

Perfectly correct. Arguing that it is bounded below is the truly difficult part.
Correct, this will play an important role later. This is similar to the 2D case, before the infinite volume limit. However in the 2D case all the Hamiltonian for each region are all the "same" (in a sense I will define later). Not so for the 3D case.

Yes, however the arguments for the infinite volume limit are significantly more difficult than for the 2D case. For reasons I will explain.

Yes, in some ways the most difficult step. It is difficult to establish that a limit exists and is well-defined, but far more difficult to prove Lorentz invariance of that limit, since that is property of only the limit itself, not of any of the cutoff models. Hence it can only be proven with estimates on the limit itself, not using the cutoff models and uniformity arguments. Also showing the theory is nontrivial requires quite advanced techniques since you must obtain estimates on the correlation functions.

It's a very different approach. Constructive Field Theory has two main branches, Hamiltonian and Functional Integral. One proves existence of quantum field theories by constructing the Hamiltonian on the correct Hilbert space. The other shows convergence of the Euclidean path integral.

However Brydges et. al, manage to prove the existence of a QFT via perturbation theory, something which was shocking at the time. This is because for scalar super-renormalizable models, the path integral on a infinite lattice can be rewritten as a kind of diffusion process which allows you to obtain three very powerful bounds on the four point function. Then those bounds, together with the Schwinger Dyson equation for the propogator and standard perturbation theory are enough to show the convergence and nontriviality of the theory in the continuum limit.

Basically the Schinger-Dyson equation and perturbation theory allow you to express the propagator as a function of itself, the free propogator and the four point function. The bounds on the four point function replace the four point function with expressions involving the propagator. So now one only has the propagator as a function of itself and the free propagator. The free propagator can be bounded easily since we have an explicit form for it. So finally one has the propagator bounded as a polynomial of itself.
Continuity in the coupling constant shows this polynomial is bounded above for small values of the coupling. So the propagator is bounded above and hence cannot diverge.

You can then show that the difference between the propagator and the free propagator is bounded away from zero. Also going back to the estimates on the four point function show it is bounded away from zero, so the theory is nontrivial.

However the method has some drawbacks:
1. Lorentz/Euclidean invariance is very difficult to show. It has never been done with this method.
2. It will only work for scalar theories, since to get the estimates on the 4-point function required writing the theory as a diffusion process. This can't be done for other theories like those containing fermions or gauge fields.
3. It only works for super-renormalizable models because the estimates, although still true, are useless for a renormalizable theory. They don't tell you anything you can use.
4. The interaction must be quartic.

Beyond that, it is so slick that you don't see what is happening "behind the scenes" in the construction of a quantum field theory. So for pedagogy I choose to leave it alone. Of course if you want to understand a complete proof of the existence of a QFT it's quite good, if you can tolerate the absence of a proof of Euclidean invariance. I will mention another simple proof, which proves everything, in a later post as well.

A. Neumaier

But this is, of course, the key property. Without Lorentz invariance, the Wightman setting would be worthless. After all, there are lots of local QFTs that lack this property.

Thanks for the outline.

Euclidean field theory, though it has some successes, is for me like trying to attack fluid dynamics problems by analytic continuation to elliptic equations, simply because the latter have a nicer theory. It is against the spirit....

No, I want to see the Hamiltonian approach. But can't one make use there of field equations and perturbation theory, too? Has it been tried and found wanting? (I had difficulties even to find papers on field equations in the rigorous approach - found only very few...)


I don't want to tolerate that. It is needed to be able to go back from the Euclidean regime to the Wightman setting.

DarMM

Indeed, luckily it can be proved. However it usually proven in a separate paper to other results because it requires such different estimates.

Yes, perhaps there is a better way of doing things, but so far perturbation theory hasn't been much use in the Hamiltonian approach, except as a guide to what renormalizations are necessary.
The field equations have conventionally been something that is checked in the model after it is constructed. For example Schrader has a paper "Local Operator Products and Field Equations in [tex]\mathcal{P}(\phi)_{2}[/tex] Theories", Fortschritte der Physik 22, p611. Where he verifies that the real field on the physical Hilbert space obeys:
[tex](\partial^{2} - m^{2})\phi = :\mathcal{P}^{'}(\phi):[/tex], where the Wick ordering is with respect to the vacuum of the true Hilbert space.

There have been some use of the field equations as a tool of construction in the finite volume case, but more on that later.

A. Neumaier

Since climbing the ladder seems to be contingent on a very time-consuming process of gathering notes , let me ask some interim questions which can perhaps be answered without notes:

I know that QED in 1+1 dimensions exists but is somewhat trivial, and that it is very difficult to say something about QED in 1+3 dimensions. But I haven't heard anything about the status of QED in 1+2 dimensions. It should be super-renormalizable and hence easier to handle than the 4D case. Is there any work on this?

DarMM

Okay, thus far we have shown that the Hamiltonian is defined and symmetric on a new Hilbert space.
Now we need to demonstrate that the Hamiltonian is semi-bounded and self-adjoint.

Firstly, that the Hamiltonian is self-adjoint is handled in Glimm's paper. One simply uses the
Friedrich's extension, which gives a self-adjoint operator from a symmetric one. However this leaves
Glimm with two uniqueness problems:
(1) Is the Friedrich's extension the only possible self-adjoint Hamiltonian
(2) The new Hilbert space is constructed by approximating the dressing transformation,
Will different Hilbert spaces result from different approximations.

Both problems were solved by Eckmann and Osterwalder in:
On the Uniqueness of the Hamiltonian and of the Representation of the CCR for the Quartic Boson Interaction in three Dimensions. Helv. Phys. Acta 44, 884-909

One essentially analyses the Hilbert spaces from an abstract C*-algebra point of view. You can prove then that all alternative truncations of the dressing transformation result in unitarily equivalent Hilbert spaces, and that the Friedrich's extension is unique.

This leaves only the problem of proving that the Hamiltonian is semi-bounded.

Solving this problem resulted in, in my opinion, the most difficult paper ever published rigorous field theory.
J. Glimm and A. Jaffe, Positivity of the [tex]\phi^{4}_{3}[/tex] Hamiltonian, Fortschr. Phys. 21, 327–376.

I will only give a brief idea of this paper. Essentially the authors consider [tex]e^{-tH}[/tex] as an operator and
separate out its action on degrees of freedom on different scales, trying to see which degrees of freedom dominate others
which will show if the Hamiltonian is a positive operator. This involves studying the action of the Hamiltonian on kets that correspond to states localised in different regions of spacetime or, to put it another way, spatial modes of the Hamiltonian localised in different areas and showing that they are independent in certain well-defined way.
To be more explicit one splits up the action of the Hamiltonian into phase-cell regions, (phase = "phase space") of length [tex]R^{i}[/tex] in position space and [tex]R^{-i}[/tex] in Momentum space. One then "flows" inductively from [tex]i[/tex] to [tex]i + 1[/tex], obtaining estimates on each degree of freedom. Degrees of freedom at lower momenta dominate those at higher momenta, something true for all super-renormalizable models. This allows you to estimate the Hamiltonian itself and show that it is semi-bounded. (Of course the "flow" is related to the renormalization group.)

In the original draft of Glimm and Jaffe's paper everything was carried through using operators. However in the published version functional integral representations of [tex]e^{-tH}[/tex] are used to simplfy the proofs, which shortened the paper by a factor of two.

So the Hamiltonian is self-adjoint and semi-bounded in a finite volume, hence the continuum limit is dealt with. All that remains is the infinite volume limit.

A. Neumaier

Thanks for the new installment. I know it is a lot of work to make this sort of abstract stuff reasonably intelligible.
I haven't yet read the paper you cited (this will take time!), but doesn't forming this operator already assume the semi-boundedness, or at least some bounds on the spectral density of very negative eigenvalues of H? What is the theorem employed that guarantees that e^{-tH} exists?
And I guess, this is what goes wrong in the 4D case?
I am looking forward to Part III!

DarMM

The Glimm and Jaffe paper is very hard going, I can confidently say it is the paper that took me the longest period of time to understand. Instead of reading it entirely, I would recommend the exposition of phase space analysis found in Vincent Rivasseau's "From Perturbative to Constructive Renormalization". You can understand the method presented there and read the Glimm and Jaffe paper much more easily bearing in mind that their method is an "older" version of what is in Rivasseau's book.
Of course I should mention that many people, including Jaffe himself consider the older, more subtle method found in their original paper to be deeper than later developments and it is possible that the method is more powerful than was realised at the time.

Yes, that is true. I should have said that they analyse [tex]e^{-tH_{\kappa}}[/tex], where [tex]H_{\kappa}[/tex] is the cutoff renormalized Hamiltonian. By this phase space analysis you can bound from below its spectrum in a way that shows that the continuum Hamiltonian acting on Glimm's Hilbert space is semi-bounded. This involves slowly removing the cutoff and showing that the new higher modes introduced as the cutoff is raised are dominated by the lower modes. The difficult part of the analysis comes from the fact that the momentum cutoff has an effect back in coordinate space and you have to show that playing around with the momentum cutoff doesn't effect spatial independence. So you need a technique of controlling both momentum and spatial degrees of freedom, hence phase space analysis.

Exactly. In the 4D case all modes are equally important, none dominate obviously over others. So it is necessary to obtain the best estimates you can in order to show that the Hamiltonian remains bounded. Folk knowledge is that it is probably necessary to obtain the optimal estimates in order to complete the proof.

The extra difficulties of the four dimensional case are:
(i) The importance of all modes, this is the nonperturbative realisation of the fact that all orders of perturbation theory need to be renormalised.
(ii) The coupling constant is renormalised. In the three-dimensional case [tex]\lambda[/tex] is just a parameter labelling interactions and doesn't really effect the phase space analysis too much. In a theory where the coupling constant is renormalised, the coupling is different on every scale, so different modes interact at different strengths.

There are other difficulties associated with the infinite volume limit of the four dimensional theory, but I'll mention them in my post about the three dimensional infinite volume limit.

DarMM

Not very much. Both QED and the Higgs model have been shown to exist in two dimensions. For QED in three dimensions, Guy Battle has shown that it has a well behaved perturbation theory, however I'm not aware of any work on the non-perturbative existence of the theory. People have shown that scalar and fermion interactions work in three and two dimensions and abelian gauge field and fermion interactions work in two dimensions. I think people became more interested in showing Yang-Mills existed and nobody choose to work particularly with 3d QED because it involved an extension of what was already known to a case where nobody expected many surprises but would still require a lot of work.

A. Neumaier

Do you refer to this paper?
Guy Battle,
Ondelettes: The spinor QED_3 connection
Annals of Physics 201 (1990), 117-151

It is strange that Google Scholar lists not a single paper citing it!
Well, it would at least make a good thesis topic for an excellent student, whereas constructing Yang-Mills would be too risky to give as task to even the best graduate student.

Also I don't quite agree that nothing interesting is to be expected. Isn't 2D QED too degenerate to give much insight into higher dimensional gauge field theories? Does it have a nontrivial infrared problem?

On the other hand, with Yang-Mills in mind, what is known about non-abelian gauge theories in the easier dimensions 2 and 3?

DarMM

Yes, that's it. I've read the paper and it's good work, but I think it has no citations because the book:
J. Feldman, T. Hurd, L. Rosen, J. Wright, QED: A Proof of Renormalizability
shows that QED is renormalizable and people might not really be interested in a "more constructive" approach to the same result in lower dimensions. What Battle does is to go beyond the book above in the three dimensional case and obtain good estimates on the perturbative series.


I agree.
I definitely agree!
There would be a lot to learn about the theory in the three dimensional case, for instance the infra-red issues of defining the asymptotic states and constructing the S-matrix. I would be very interested to see exactly what QED is like in a non-trivial setting, outside the generalities of algebraic field theory. Unfortunately QED's infra-red issues are not believed to help with the infra-red issues of Yang-Mills and people are very interested in Yang-Mills now, so it has escaped people's minds a bit.
In the the two dimensional case, baring some minor technicalities, the theory is pretty much known to exist. I think after this thread is completed I will start another one on Yang-Mills to describe this work.
In the three-dimensional case Balaban and others have produced hundreds of pages of extremely technical estimates which have brought a proof of the continuum limit into sight. However Balaban's work "passes beyond human comprehension" to paraphrase Vincent Rivasseau.

Apparently Balaban has proven more than appears in his papers, but it has become so technical that he prefers to wait for a simpler method. Of course this is just "Folk knowledge" but it is interesting none the less.

A. Neumaier

Me too!
Are there rational grounds for this belief?