This question is somewhat a headache for someone. The point is that we know how to treat quantum field theory by perturbation techniques but the only typical case we have is a free theory. In this case the generating functional is Gaussian and all functional derivatives are easily computed. So, whenever we are able to reduce to this latter situation we can draw sharp conclusions about the solution of the theory.

Indeed there are two aspects one should manage for a quantum field theory and are exactly the same as for the solution of a quantum mechanics problem where one has to determine both the eigenstates and the spectrum. In a quantum mechanics problem one knows that the poles of the Green function are the points of the spectrum of the quantum system. So, we should expect the same in quantum field theory. But we cannot be sure as the Kaellen-Lehman representation is not always expected to hold and this conclusion is not so straightforward to be drawn. Indeed, formulations of quantum field theories exist that extend the validity of the Kaellen-Lehman representation to the case where positivity is missing (e.g. see the works of Franco Strocchi as here) but currently there is no common agreement about and different points of view are met, mostly due to the fact said above that solvable quantum field theories are quite rare. The other aspect relies on the form of the states. Indeed, using Green functions and asymptotic states one is able to compute scattering sections and decay times through Lehman-Symanzik-Zimmermann reduction formula. The problem is what are these asymptotic states for some theories as confined ones.

So, with our present knowledge, a sharp definition of an exactly solved quantum field theory does not exist and also the way to get the spectrum can be a source of debate. What we are really missing is some example of an exactly solved non-linear quantum field theory. Anyhow, any other point of view about is welcome.

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9 Responses to When is a quantum field theory exactly solved?

I’ve heard it said that Fivel gave a solution of the Lee model in

D.I.Fivel, “Solution of the Lee Model in All Sectors by Dynamical Algebra”,
J. Math. Phys., Vol 11, No.2, Feb 1970, pp699-705.

However, on closer examination, it looks like he only solves the
V-\theta sector in closed form. But perhaps there are some useful
ideas in that paper nevertheless. đź™‚

The point is that nobody is able to state clearly when a quantum field theory is exactly solved. I have had difficulty to let some people believe that a spectrum is derived from the poles of the Green function. I do not dare to go beyond this…

If a quantum field were presented in terms of a Lie field formalism, as I present it in J. Math. Phys. 48, 122302 (2007), alternatively arXiv:0704.3420v2 [quant-ph], or similarly, I would take it that the quantum field had been solved — in the sense that a Lie field presentation of the quantum field had been given. The Lie field approach is far from mature, however. I take an essential feature to be that the Lie field formalism is downstream from renormalization, as algebraic quantum field theory is generally and as the Kaellen-Lehman formalism is, whereas Lagrangian and Hamiltonian formalisms are upstream of renormalization. Although a Lie field has to be a classical random field, the positive frequency scattering theory is identical to the scattering theory of a quantum field. If the approach I describe seems bizarre, my apologies.

When you posted about Lie fields over on physicsforums, you quoted someone else’s comments
anonymously, in particular:

” I looked at your paper but I could not find an answer that is of utmost importance for me. What is your proposal for the function \xi in Eq.(5) for photons? I could not find an answer to this question in Section V where you discuss the electromagnetic field. Your proposal may become experimentally testable only if you propose a specific form of the deformation for the commutation relations […]”

I had the same feeling when I read your paper, but did not feel I could add anything helpful
at the time. In your comment above it reads as if you’re claiming that you have “solved the
quantum field” because a “Lie field presentation has been given”. Is that really what you meant
to say, or was it just poor wording?

(Marco: apologies if this comment intrudes on your hospitality. If it’s not something you’re
also interested in pursuing here, I will of course stop.)

Strangerep: My wording was led by the particular discussion in Marco’s post, which suggested a reason that I hadn’t thought of for why Lie fields might be useful or interesting. A small part of what caused the connection for me was the mention of the Kaellen-Lehman representation.

A Lie field presentation of a quantum field theory is (or, would be) perhaps comparable to an integral equation “solution” of a differential equation, however the comparison is not too close: a Lie field is more of an intermediate form insofar as it’s closer to the empirical because it’s downstream of renormalization. So far, the conceptual place a Lie field is at is different enough that I have no idea how to construct a Lie field presentation from a Lagrangian presentation, say, of a quantum field, so I haven’t *solved* anything.

The point is perhaps that \xi is *not* very limited: there is a functional continuum of possible Lorentz invariant Lie fields (and I have now employed a different ansatz that I believe allows more models to be constructed than can be constructed using the ansatz in the JMP paper — but that needs peer review), so assuredly most instances are non-renormalizable. That’s good for empirical models, lots of possibilities; that’s bad for fundamental theory, too many possibilities, it’s not explanatory. A Lie field approach seems to me more likely to be an intermediary, something that lets us get away from the presentations of quantum fields that we’ve become used to, preparatory for something else, in which, I suppose, a limited class of choices of Lie field structure might be natural.

I suppose that experimental testability is a way off, and this isn’t strong theory, so no Phys.Rev. papers expected soon. Ruling out a large natural class of models is more difficult than ruling out a particular model. I note, however, that I wouldn’t claim that Lie fields as I currently discuss them can model any experimental data whatsoever, particularly at the Planck scale because I use a background Minkowski space, so Lie fields are falsifiable in principle (indeed, false in principle, but perhaps curious or useful).

Marco: I’m finding a Lie field approach to be *very* different, so I warn you that it’s slow going. What Strangerep said, it’s your blog.

Peter, strangerep, please, keep on posting. Maybe it is not an answer to my post but can be interesting anyhow. Of course, if you get a plausible answer to define solved a quantum field theory it will be welcome here.

> I have now employed a different ansatz that I believe allows
> more models to be constructed than can be constructed using
> the ansatz in the JMP paper â€” but that needs peer review […]

Is this a new paper on the arxiv? (I’m happy to take a look at it
in advance of peer review if you want.)

Sadly not even written yet, just in moderately neat calculations on paper. I feel fairly confident about the approach, but writing up will of course lead to complications. I get only a little time for research in the Summer, so an arXiv posting will be the end of September. Many thanks for the interest!

I’ve heard it said that Fivel gave a solution of the Lee model in

D.I.Fivel, “Solution of the Lee Model in All Sectors by Dynamical Algebra”,

J. Math. Phys., Vol 11, No.2, Feb 1970, pp699-705.

However, on closer examination, it looks like he only solves the

V-\theta sector in closed form. But perhaps there are some useful

ideas in that paper nevertheless. đź™‚

The point is that nobody is able to state clearly when a quantum field theory is exactly solved. I have had difficulty to let some people believe that a spectrum is derived from the poles of the Green function. I do not dare to go beyond this…

Marco

If a quantum field were presented in terms of a Lie field formalism, as I present it in J. Math. Phys. 48, 122302 (2007), alternatively arXiv:0704.3420v2 [quant-ph], or similarly, I would take it that the quantum field had been solved — in the sense that a Lie field presentation of the quantum field had been given. The Lie field approach is far from mature, however. I take an essential feature to be that the Lie field formalism is downstream from renormalization, as algebraic quantum field theory is generally and as the Kaellen-Lehman formalism is, whereas Lagrangian and Hamiltonian formalisms are upstream of renormalization. Although a Lie field has to be a classical random field, the positive frequency scattering theory is identical to the scattering theory of a quantum field. If the approach I describe seems bizarre, my apologies.

Dear Peter,

It does not seem bizarre. It just needs some time for us to be digested.

Marco

Re Peter Morgan’s comment…

When you posted about Lie fields over on physicsforums, you quoted someone else’s comments

anonymously, in particular:

” I looked at your paper but I could not find an answer that is of utmost importance for me. What is your proposal for the function \xi in Eq.(5) for photons? I could not find an answer to this question in Section V where you discuss the electromagnetic field. Your proposal may become experimentally testable only if you propose a specific form of the deformation for the commutation relations […]”

I had the same feeling when I read your paper, but did not feel I could add anything helpful

at the time. In your comment above it reads as if you’re claiming that you have “solved the

quantum field” because a “Lie field presentation has been given”. Is that really what you meant

to say, or was it just poor wording?

(Marco: apologies if this comment intrudes on your hospitality. If it’s not something you’re

also interested in pursuing here, I will of course stop.)

Strangerep: My wording was led by the particular discussion in Marco’s post, which suggested a reason that I hadn’t thought of for why Lie fields might be useful or interesting. A small part of what caused the connection for me was the mention of the Kaellen-Lehman representation.

A Lie field presentation of a quantum field theory is (or, would be) perhaps comparable to an integral equation “solution” of a differential equation, however the comparison is not too close: a Lie field is more of an intermediate form insofar as it’s closer to the empirical because it’s downstream of renormalization. So far, the conceptual place a Lie field is at is different enough that I have no idea how to construct a Lie field presentation from a Lagrangian presentation, say, of a quantum field, so I haven’t *solved* anything.

The point is perhaps that \xi is *not* very limited: there is a functional continuum of possible Lorentz invariant Lie fields (and I have now employed a different ansatz that I believe allows more models to be constructed than can be constructed using the ansatz in the JMP paper — but that needs peer review), so assuredly most instances are non-renormalizable. That’s good for empirical models, lots of possibilities; that’s bad for fundamental theory, too many possibilities, it’s not explanatory. A Lie field approach seems to me more likely to be an intermediary, something that lets us get away from the presentations of quantum fields that we’ve become used to, preparatory for something else, in which, I suppose, a limited class of choices of Lie field structure might be natural.

I suppose that experimental testability is a way off, and this isn’t strong theory, so no Phys.Rev. papers expected soon. Ruling out a large natural class of models is more difficult than ruling out a particular model. I note, however, that I wouldn’t claim that Lie fields as I currently discuss them can model any experimental data whatsoever, particularly at the Planck scale because I use a background Minkowski space, so Lie fields are falsifiable in principle (indeed, false in principle, but perhaps curious or useful).

Marco: I’m finding a Lie field approach to be *very* different, so I warn you that it’s slow going. What Strangerep said, it’s your blog.

Peter, strangerep, please, keep on posting. Maybe it is not an answer to my post but can be interesting anyhow. Of course, if you get a plausible answer to define solved a quantum field theory it will be welcome here.

Marco

Peter,

> I have now employed a different ansatz that I believe allows

> more models to be constructed than can be constructed using

> the ansatz in the JMP paper â€” but that needs peer review […]

Is this a new paper on the arxiv? (I’m happy to take a look at it

in advance of peer review if you want.)

Cheers.

Sadly not even written yet, just in moderately neat calculations on paper. I feel fairly confident about the approach, but writing up will of course lead to complications. I get only a little time for research in the Summer, so an arXiv posting will be the end of September. Many thanks for the interest!