# Haag's Theorem in QFT

So I'm currently studying QFT, and I got to the point where I realized that the S operator, initially assumed to be unitary, is not unitary anymore, since it is assumed to act between t0 = - infinity and tf = infinity. The author of the book I'm using says this is due to Haag's Theorem, so I gave that a search, and found a rather interesting bunch on information.

http://en.wikipedia.org/wiki/Haag's_theorem

It seems that the Interaction Picture does not exist in QFT. What is the importance of this theorem and what are its other implications?

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George Jones
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See also section 10.5, "How to stop worrying about Haag's theorem", in The Conceptual Framework of Quantum Field Theory by Anthony Duncan (Oxford, 2012).
Also, the interaction Hamiltonian doesn't exist as a genuine mathematical object. From Folland's "Quantum Field Theory for Mathematicians: A Tourist's Guide for Mathematicians" (a book that I highly recommend) page 123

"... the ##H_I##'s that we shall need are too singular."

Too singular, because they products of distributions. From page 180

"A precise mathematical construction of the interacting fields that describe actual fundamental physical processes in 4-dimensional space-time is still lacking and may not be feasible without serious modifications to the theory. Similarly, we have no way to define the Hamiltonian ##H## in a mathematically rigorous way as a self-adjoint operator. It was presented as the sum of the free Hamiltonian ##H_0##, which is well-defined and the interaction Hamiltonian ##H_I##; but the latter was presented as the integral of a density consisting of products of fields, which are operator-valued distributions rather than functions."

atyy
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Most physical QFT is not rigourous. You can roughly think of it as a many-body system on a lattice and QFT emerges at low energies. This sort of thinking works for QCD, but there is some problem for chiral gauge theories. Regardless the idea is that physical QFTs are only low energy effective theories, and may not exist at all energies. http://quantumfrontiers.com/2013/06/18/we-are-all-wilsonians-now/

In rigourous QFT, the aim is to construct a Lorentz invariant QFT that exists for all energies. Haag's theorem applies to such theories. I've found these comments about the theorem useful.

http://www.rivasseau.com/resources/book.pdf, p15: "The Gell-Mann Low formula .... is difficult to justify because the usual argument, based on the so called "interaction picture" is wrong, by a theorem of Haag ... Euclidean formulation ... theorems which allow us to go back from Euclidean to Minkowski space ..."

Also, https://webspace.utexas.edu/lupher/www/papers/KronzLupherPreprint.pdf [Broken], p7: "If representations are unitarily inequivalent, one is left to wonder whether they are at least empirically equivalent in some sense. If a reasonable notion of empirical equivalence cannot be found, then it will be necessary to introduce criteria for representation selection. To resolve this selection problem, Haag and Kastler (1964) introduced the notion of physical equivalence, which is related to Fell’s (1960) notion of weak equivalence, and a theorem proved by Fell, which they reformulate in terms of physical equivalence."

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Avodyne
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In rigourous QFT, the aim is to construct a Lorentz invariant QFT that exists for all energies. Haag's theorem applies to such theories.
In 3+1 dimensions, most QFTs are believed not to exist at all (only asymptotically free theories are believed to exist). So any attempt at a rigorous construction of a non-asymptotically free theory in 3+1 dimensions should fail.

See also section 10.5, "How to stop worrying about Haag's theorem", in The Conceptual Framework of Quantum Field Theory by Anthony Duncan (Oxford, 2012).

Short version: QFT requires an ultraviolet regulator (such as a lattice), and Haag's theorem does not apply when the regulator is in place.

atyy
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In 3+1 dimensions, most QFTs are believed not to exist at all (only asymptotically free theories are believed to exist). So any attempt at a rigorous construction of a non-asymptotically free theory in 3+1 dimensions should fail.

See also section 10.5, "How to stop worrying about Haag's theorem", in The Conceptual Framework of Quantum Field Theory by Anthony Duncan (Oxford, 2012).

Short version: QFT requires an ultraviolet regulator (such as a lattice), and Haag's theorem does not apply when the regulator is in place.
3+1? You must be a squalid state physicist who can't add. All HEP people say 4 :)

(I'm a biologist.)

dextercioby
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For people seeing physics beyond the Standard Model, then 3+1 instead of 4 is a must.

George Jones
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Stating the signature 3 + 1 is essential.

For people seeing physics beyond the Standard Model, then 3+1 instead of 4 is a must.
An for people working with models in 1 + 1 and 2 + 1.

In 3+1 dimensions, most QFTs are believed not to exist at all (only asymptotically free theories are believed to exist). So any attempt at a rigorous construction of a non-asymptotically free theory in 3+1 dimensions should fail.

See also section 10.5, "How to stop worrying about Haag's theorem", in The Conceptual Framework of Quantum Field Theory by Anthony Duncan (Oxford, 2012).

Short version: QFT requires an ultraviolet regulator (such as a lattice), and Haag's theorem does not apply when the regulator is in place.
First of all, what is an "asymptotically free theory" ? Second, I don't know how you apply a lattice to QFT. From what wikipedia says, "Every lattice in $$ℝ^n$$ can be generated from a basis for the vector space by forming all linear combinations with integer coefficients." So taking linear combinations of the eigenbasis of a Hilbert space, with integer coefficients gives you a lattice?

Avodyne
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atyy
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One formal possibility in addition to asymptotic freedom is asymptotic safety. However, as Avodyne says, currently no 3+1 theory has been shown to be asymptotically safe without being asymptotically free. It is being researched if 3+1 gravity might be asymptotically safe, even though it is not asymptotically free.

Asymptotic freedom/safety are physics concepts and not rigourous. However, it is widely believed that it is not worth attempting a rigourous construction of a theory, unless it has been shown to be asymptotically free or safe. 3+1d Yang-Mills is asymptotically free, which is why its rigourous construction has been posed as a Clay Millenium problem.

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Demystifier
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See also section 10.5, "How to stop worrying about Haag's theorem", in The Conceptual Framework of Quantum Field Theory by Anthony Duncan (Oxford, 2012).

Short version: QFT requires an ultraviolet regulator (such as a lattice), and Haag's theorem does not apply when the regulator is in place.
That's exactly what I wanted to say.

kith
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Could this lattice be associated with a fundamental discreteness of spacetime in the Planck regime or are these different concepts?

Avodyne
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We have to distinguish between the real world and mathematical formulations of a particular theory. We've been discussing the latter here. In this context, the lattice is just a technical tool.

True discreteness at the Planck scale is an intriguing possibility. That would provide a real-world ultraviolet cutoff for the Standard Model. But then there are many other possibilities for an "ultraviolet completion" of the Standard Model + gravity that don't involve fundamental discreteness (e.g. zillions of string models).

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DarMM
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Short version: QFT requires an ultraviolet regulator (such as a lattice), and Haag's theorem does not apply when the regulator is in place.
QFTs do not require a regulator, although perhaps you mean this in a practical sense, i.e. to work with QFTs at our current level of mathematics requires a regulator at intermediate steps.

Also Haag's theorem is not affected by the ultraviolet cutoff, Haag's theorem applies even with an ultraviolet cutoff in place. You need an infrared cutoff to prevent Haag's theorem.

Demystifier
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Also Haag's theorem is not affected by the ultraviolet cutoff, Haag's theorem applies even with an ultraviolet cutoff in place. You need an infrared cutoff to prevent Haag's theorem.
To prevent Haag's theorem, is the IR cutoff enough, or do we need both UV and IR cutoff?

Demystifier
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QFTs do not require a regulator, although perhaps you mean this in a practical sense, i.e. to work with QFTs at our current level of mathematics requires a regulator at intermediate steps.
Can you be more specific how, at least in principle, one can work with QFT without a regulator?

BPHZ approach does not use a regulator.

Demystifier
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BPHZ approach does not use a regulator.
OK, but BPHZ method subtracts an infinite quantity from an infinite quantity, which is not well defined at the mathematical level. The Haag's theorem, on the other hand, assumes that one works with quantities which are well defined mathematically. Physically, the BPHZ method postulates that the mathematically ill defined expression is actually equal to a finite coupling constant extracted from experiments. In this way, BPHZ is also a dirty trick which replaces an infinite quantity by a finite one, so in this sense it is not very different from a "regularization".

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atyy
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To prevent Haag's theorem, is the IR cutoff enough, or do we need both UV and IR cutoff?
Can you be more specific how, at least in principle, one can work with QFT without a regulator?
As I understand it, physical QFT does not require it to exist at all energies, a cut-off is fine, and QFT is just a low energy effective theory.

I think DarMM is saying that there are nonetheless some nonlinear relativistic QFTs that exist at all energies, and in infinite volume. In my understanding, the AdS/CFT proposal assumes that the CFT exists for all energies, which is why it is conjectured to provide a non-perturbative definition of some sector of string theory.

http://www.claymath.org/sites/default/files/yangmills.pdf (p8) mentions that relativistic QFTs that exist at all energies, and in infinite volume, have been constructed in fewer than 3+1 spacetime dimensions.

George Jones
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Can you be more specific how, at least in principle, one can work with QFT without a regulator?
As I understand it, physical QFT does not require it to exist at all energies, a cut-off is fine, and QFT is just a low energy effective theory.
A cut-off is (one of the types of) regularization.

atyy
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A cut-off is (one of the types of) regularization.
Yes, physically-relevant QFT, like the standard model, requires a cut-off. However, the article I linked to goes on to say that nonlinear relativistic QFTs in fewer than 4 spacetime dimensions in infinite volume at all energies have been rigourously constructed. Yang-Mills in 4D is asymptotically free, which is why it is a candidate for rigourous construction of a 4D QFT without a cut-off.

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Avodyne
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QFTs do not require a regulator
They don't require a regulator if they exist. But it is widely believed by experts that, in 3+1D, only asymptotically-free theories exist.

Also Haag's theorem is not affected by the ultraviolet cutoff, Haag's theorem applies even with an ultraviolet cutoff in place.
I disagree, and my opinion is backed up by Duncan's book.

atyy
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I disagree, and my opinion is backed up by Duncan's book.
Duncan seems to say on his p369 that both the ultraviolet cutoff and the finite volume cut-off are needed to have the interaction picture. IIRC, sometime ago on these forums Dr Du said the same thing about condensed matter in the infinite volume limit. In condensed matter physically it can't really matter, since physically any material in the lab has a natural ultraviolet cutoff and finite volume.

DarMM
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I disagree, and my opinion is backed up by Duncan's book.
Haag's theorem is a theorem proving the non-existence of the interaction picture provided the theory is translation invariant, i.e. if there is no infrared cutoff, it does not discuss the ultraviolet cutoff.

Basically Haag's theorem proves that free and interacting theories are unitarily inequivalent unless an infrared cutoff is in place.

There is no such general result for ultraviolet cutoffs because there are counterexamples. For example ##\phi^4## without an ultraviolet cutoff (but with an infrared cutoff) is unitarily equivalent to the free theory in 1+1 dimensions and so the interaction picture does exist.

For interaction picture to exist sometimes both an ultraviolet and infrared cutoff is needed, sometimes only an infrared cutoff. Haag's theorem is the statement that an infrared cutoff is always needed for the interaction picture to exist.

Demystifier