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In Quantum Field Theory one often needs to compute the $n$-point correlation functions $\omega_n(x_1,\dots,x_n)$ which are traditionally given as vacuum expectation values

$$\omega_n(x_1,\dots,x_n) = \langle \Omega | T\{\phi(x_1)\cdots\phi(x_n)\}|\Omega\rangle.$$

In order to do this there are, as far as I know, two main approaches with two main issues which actually invalidate these two approaches mathematically:

  1. The first approach is work out a relation between the interacting fields $\phi(x)$, the vacuum $|\Omega\rangle$ and their free theory counterparts $\phi_0(x)$ and $|0\rangle$. This is achieved by switching to the interaction picture. The issue here is: Haag's theorem states that this cannot be done - the unitary transformation that leads to the interaction picture doesn't exist.

  2. The second approach is to derive the Schwinger-Dyson equation. This is done in Matthew Schwartz book. This doesn't use the interaction picture, but still makes a false assumption: it assumes the canonical commutation relations are obeyed for the interacting fields, which as far as I know isn't true.

Now, Algebraic Quantum Field Theory seems a nice approach to QFT. In this article Robert Wald gives an introduction to the topic, and it seems at first that those problems aren't there at all.

My question here is: concerning the issue with the interaction picture and Haag's theorem, or the Schwinger-Dyson equation counterpart based on the canonical commutation relation, Algebraic Quantum Field Theory gives any workaround to deal with $n$-point functions rigorously?

Does AQFT allow to relate the $n$-point functions to the perturbative expansion in terms of the free fields pictured as Feynman diagrams in a rigorous way? Or it gives an entirely different way to compute the $n$-point functions?

What AQFT actually solves in this matter?

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  • $\begingroup$ (2.) is kinda wrong: the problem with the DS approach is not about the CCR (which are fine for interacting fields), but about the fact that the DS equations are derived from the equations of motion for the fields, which are themselves ill-defined (for the usual reason: they include products of distributions). $\endgroup$ – AccidentalFourierTransform May 7 '17 at 7:51
  • $\begingroup$ @AccidentalFourierTransform actually one of the books I was reading (Quantum Field Theory and the Standard Model by Schwartz) uses this approach before the interaction picture one. I've read his derivation in order to find out where is the non-rigorous assumption, and my only guess was the CCR (which he says is valid because at any fixed time the Hilbert space for the interacting system is the same as the one for the free system). On the other hand, Robert Wald seems to solve the distribution product issue in the article I linked (page 39). There probably is a catch I didn't notice though $\endgroup$ – user1620696 May 7 '17 at 15:45
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Algebraic quantum field theory (AQFT) "deals" with the interaction picture only insofar that its axioms are strong enough to not need the notion of interaction picture to formally define a QFT.

The first problem is that almost no actual four-dimensional interacting QFT we have is known to have a formalization as an AQFT, be it in the sense of Haag-Kastler or in the sense of Wightman.

The second problem is that AQFT does not inherently contain any procedure to compute the Wightman functions (aka n-point functions). You have a very nice list of axioms, and one even can show the Wightman reconstruction theorem that sets of fields that obey the Wightman axioms correspond bijectively with sets of Wightman functions, that is, the correlation functions determine the theory completely. But what you cannot do is actually write down a procedure to compute the Wightman functions from the axioms. One can show a bunch of neat properties of the Wightman functions, such as well-definedness of Wick rotation (called the Osterwalder-Schrader theorem), CPT symmetry, cluster decomposition and more, but the list of axioms does not suffice to determine the Wightman functions - which is not really a surprise since it's supposed to be an axiomatisation of a class of theories, so there must be some input left to specialize to a specific QFT.

And that's where we come full circle: In many cases, the only way we know of obtaining a QFT is by quantizing the classical field theory, which in turn leads us to all the issue with the canonical formalism and/or the path integral all over again.

In the end, I would submit that the question is somewhat misguided: The issue is not that we do not have axiomatizations of QFT that avoid being mathematically non-rigorous. We have, both AQFT and FQFT do that. The problem is that we do not actually know how to produce physical theories that fulfill these axioms1. Yes, AQFTs deal with Wightman functions rigorously. That doesn't do you any good when trying to understand e.g. the Standard Model because we can't show that the Standard Model fulfills any variant of the AQFT axioms.


1Okay, that's hyperbole - free theories fulfill them, and the work of Glimm and Jaffe contains a lot of well-defined interacting QFTs in two and three dimensions via a well-defined notion of the path integral in these cases. Unfortunately, this program of constructive field theory seems currently unable to be extended into higher dimensions, or to more "uncomfortable" theories like non-Abelian gauge theories.

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  • $\begingroup$ Thanks for the answer @ACuriousMind. I might be missing something, but even though one does not know how to compute exactly these $n$-point functions, the perturbative approach combined with renormalization can be made rigorous right? Or not even the perturbative approach can be made rigorous? Because, the fact that an exact solution is not known happens in quite a lot of other problems, but usually one approximative approach can be put in rigorous terms. What I want to know is if the perturbative approach, even if needing renormalization, can be made rigorous with AQFT or something else. $\endgroup$ – user1620696 May 7 '17 at 15:51
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    $\begingroup$ @user1620696 I'm not sure what you mean by "the perturbative approach". Yes, you can compute n-point functions in the "standard" non-rigorous approach to QFT. But there's no known proof that the fields you're working with in these QFTs fulfill the Wightman (or Haag-Kastler, or whatever) axioms, so that formalism hasn't got anything to do with AQFT, I don't know what you think AQFT can "do" here. AQFT is not a magic wand you can wave to make things rigorous, it takes hard work to prove that a QFT fulfills the axioms (see, once again, Glimm/Jaffe). $\endgroup$ – ACuriousMind May 7 '17 at 15:56
  • $\begingroup$ "we can't show that the Standard Model fulfills any variant of the AQFT axioms" - more like "we can't show that there is a model satisfying those axioms and yet giving Wightman functions to which perturbative correlators of SM are a good approximation" $\endgroup$ – Prof. Legolasov May 8 '17 at 0:39

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