Talk:Haag's theorem

I am not very happy with this article. Haag's 1955 paper does not actually prove that which is referred to as Haag's theorem, and there are in any case different versions. The proofs seem to be Hall/Wightman (1957), Greenberg (1959), Federbush/Johnson (1960) and Jost/Schroer (1961). The definitive reference is Streater/Wightman (2000), which is not even mentioned, and although I have not managed to get to look at the Arageorgis thesis (how easy is it to get hold of, anyway?), I question whether it actually adds anything to what was known in 1961.

I think that it would be a good idea to rewrite this article based on the information in Streater/Wightman. I am not necessarily volunteering, but if no-one else wants to do it, I just might. Cgoakley 20:56, 29 October 2006 (UTC)

I have clarified things a little, but the article still needs to be rewritten by someone more knowledgable than myself Cgoakley 18:53, 6 February 2007 (UTC)

The original formulation on the main page (which is missing an edit button) needs editing as the word "in" in the phrase "in the canonical..." seems to be wrong--the sentence doesn't parse. If we remove "in" it makes sense; but is it the sense the author(s) intended? (comment by "YouRang?" which should be on this page). Personally, I do not see anything wrong with the sentence, although it could be broken into multiple shorter sentences. Cgoakley (talk) 20:18, 5 May 2008 (UTC)

Maybe someone can add why using the interaction picture is still successful in perturbative quantum field theories despite having a proven theorem that the picture does not exist? Calling the theorem "inconvenient" is amusing but doesn't really explain anything. —Preceding unsigned comment added by 79.210.193.198 (talk) 22:40, 3 August 2008 (UTC)

No, what is "amusing" is that anyone should imagine that the inconsistent mess wrongly called "perturbative quantum field theory" should satisfy any axioms at all.Cgoakley (talk) 07:33, 4 August 2008 (UTC)

I have tried to sketch a new writeup of that article, loosely following the line of argumentation offered by Earman and Fraser (Ref. 4). The entire subject is somewhat opaque, however, and I am not sure whether or not I found a proper balance with my selection of topics and references. Additional input is needed for sure. Merlitz (talk) 20:40, 10 October 2010 (UTC)

Thank-you, Dr. Merlitz for expanding this article. However I do not understand the idea, which you attribute to Reed/Simon that Haag's theorem should apply in the case of free field theory as obviously V = 0 and therefore the inequivalency of the interacting and non-interacting Hilbert spaces is not a relevant question. Could you provide more explanation? Cgoakley (talk) 12:58, 12 October 2010 (UTC)

Hi Dr. Oakley, the proof for that is sketched in [4] (can be downloaded) on p. 18: Imagine two free neutral scalar fields f1 and f2, with masses m1 and m2 respectively. Now we may assume m2 = m1 + dm and then, formally, formulate one of the fields as an interacting field. The authors of [4] refer to Reed/Simon but I did not follow up that link to check for the details. I would guess that, heuristically, a shift in mass is nothing else but a different choice of the vacuum energy, and here the old problems of having several vacuum energies are arising. But you are right, I should have written 'different masses' instead of 'non-vanishing masses', this should be clarified in the article. Apparently, you are very familiar with QFT and the Haag business, you are even writing a book on that? Please feel free to modify the text wherever suitable, I am no QFT practitioner, just came across the Haag theorem, was surprised to learn about its details (and the fact that I never heard about all that during my studies), and found the Wiki article in a pity state, so I decided to add some input - as much a I could do within my limitations. Merlitz (talk) 06:46, 13 October 2010 (UTC)

Sure, I would expect that there is no unitary equivalence between free fields of different masses, but I would not call the result "Haag's theorem". Also it should be mentioned at some stage that special relativity is required for the normal proof. Cgoakley (talk) 09:57, 13 October 2010 (UTC)

We could call it Haag-type theorem in the context of free fields, as suggested by Earman et al. - intentionally collecting a more general class of theorems under the umbrella of Haag like theorems. After all, these various works of different authors come to similar results although certain details in their assumptions and sample systems may differ. Regarding relativistic invariance, it is still an open question whether or not this is an essential requirement in forcing the use of inequivalent representations (p. 13 in [4]). Wightman believed that it is not, and that we are dealing with a more fundamental phenomenon that appears as soon as a theory is euclidean invariant and has vacuum polarization (p. 12 in [4]). When it comes to judging about all these details, I am already out of my field, I hope somebody who is more competent with these matters can clarify the details. Merlitz (talk) 12:26, 13 October 2010 (UTC)

I am not sure that what is written in the section "Formal description of Haag's theorem" does capture the actual statement. The existence of unitarily inequivalent representations of the CCR is just the failure of von Neumann's uniqueness theorem and is a general feature of systems with an infinite number of degrees of freedom, i.e. of the Weyl algebra constructed for an infinite-dimensional symplectic space. It is also not true that there are is no unitary equivalence between representations at all in relativistic quantum field theory. It is just that not all representations are equivalent. Also the choice problem has found something like a resolution in algebraic quantum field theory founded by Haag, Kastler and others, i.e. do not make the choice and consider all representations on equal grounds. AFAIK, Haag's theorem really states the unitary equivalence of representations of time-zero CCR at some time implies unitary equivalence at all times, i.e. it is not possible to describe the change from free to interacting dynamics by a unitary if the two representations are supposed to the unitarily equivalent (i.e. the fields and symmetries are intertwined correctly) at some early time ruling out a naive interaction-picture approach to scattering theory. (With the possible generalizations mentioned in the introduction.) — Preceding unsigned comment added by 93.217.243.102 (talk) 08:21, 4 August 2012 (UTC)

Dear All, I just want to mention that the reference Hall, D. and Wightman, A.S.: A theorem on invariant analytic functions with applications to relativistic quantum field theory, Matematisk-fysiske Meddelelser, 31, 1 (1957) is actually 31, 5, see also http://www.sdu.dk/en/Bibliotek/E-hotel/MatFys.aspx — Preceding unsigned comment added by 149.217.1.5 (talk) 10:35, 20 August 2012 (UTC)

I don't like the latest version of the subsection "Ignorance on the part of the QFT practitioner". In my opinion it constitutes an unwarranted attack on QFT and violates the wikipedia policy of impartiality. I suggest one of the 3 options: 1) This subsection is completely removed. 2) It is completely re-written in an impartial way (ie including the view of a QFT practitioner) 3) In the worst case, just revert it to the previous version which was far from ideal, yet less offending. (Bilingsley (talk) 15:24, 23 September 2012 (UTC))


 * This paragraph is certainly not supposed to be interpreted as an attack on QFT - Haag's theorem is challenging the mathematical foundations of QFT, but not its success in practical applications. I would suggest to work out your option 2, but this would require further input from those practitioners who actually know about Haag's theorem and its implications, and there seem to be few of them. When I composed the paragraph in question, I was on the line of reasoning provided by Paul Teller in his "An interpretive introduction to quantum field theory", Princeton 1995. This is one of the few text books on QFT which at least makes an attempt to comment on Haag's theorem. Teller writes:


 * Experts wildly disagree in their attitudes toward Haag's theorem. Barton (1963, p.157) expresses one extreme when he writes that ... the statement popularly known as Haag's theorem ... shows that no field theory exists which differs from that of a free field. At the other extreme, most practitioners simply dismiss the issue. ... Everyone must agree that as a piece of mathematics Haag's theorem is a valid result that at least appears to call into question the mathematical foundation of interacting quantum field theory, and agree that at the same time the theory has proved astonishingly successful in application to experimental results (p. 115).


 * He continues:


 * These few words on the ramifications of Haag's theorem do not do the slightest justice to this important topic. But the subject commands no consensus, and I return to my exposition on common practice, which simply puts the whole issue aside (p. 124).


 * There exists no final consensus, hence this is definitely an unsolved issue which should be critically accounted for inside the present article on Haag's theorem. Please help to develop it further.--Merlitz (talk) 11:19, 3 November 2012 (UTC)

(The order of the comments here is forward chronological - the last two comments were incorrectly placed at the top, so I have moved them down). I do not see anything controversial in the last section. Haag's theorem hardly gets a mention in QFT text books, and when it does, the authors never seem to be bothered about it. They also freely acknowledged that QFT and QED in particular are not axiomatic and, once again, this does not seem to be the cause of any sleepless nights. Cgoakley (talk) 14:45, 24 October 2012 (UTC)

I hope I have found a neutral formulation of the last paragraph, now titled Conflicting reactions of the practitioners of QFT. I have added Teller's book as well as an interesting article by T. Lupher which I have recently found. The previous paragraph Workarounds has now got a short mention of the LSZ formalism and its connection to Haag-Ruelle.--Merlitz (talk) 08:44, 11 November 2012 (UTC)
 * I approve this version and I think it addresses all my previous concerns. Thanks for editing. Bilingsley (talk) 12:16, 17 November 2012 (UTC)

question?
Isn't anything missing in the present formulation of the theorem? like infinite dimensional CCR? otherwise, isn't that in contracdiction with von Neumann's theorem on unicity of the representation of a Weyl algebra?

Ed Seidewitz
I don't remember exactly what the policies are, but shouldn't we wait before citing a recently written paper? Specially from the amateur part. --181.31.86.55 (talk) 00:57, 3 February 2015 (UTC)

That sentence reeks of grandstanding and I haven't seen this paper get any attention in the physics community. I'm removing it until it becomes notable. 150.135.210.59 (talk) 23:40, 29 March 2015 (UTC)

I agree. I have not properly read the paper but where is the evidence that any of the ideas have been accepted by the physics community? I recommend removing the reference until such can be provided. Cgoakley (talk) 12:24, 28 July 2015 (UTC)

Postulated?
The current opening sentence
 * Rudolf Haag postulated that the interaction picture does not exist in an interacting, relativistic quantum field theory (QFT), something now commonly known as Haag's theorem.

is a contradiction in terms: if you postulate something it becomes an axiom, not a theorem. Contributions/95.199.143.153 (talk) 09:05, 12 June 2019 (UTC)


 * If you postulate something, it becomes a postulate, not an axiom. But yes, I too would be grateful for enlightenment as to how this became "Haag's Theorem" when the cited 1955 paper contains no proof. Was there some additional, perhaps unpublished work that I am not aware of? Cgoakley (talk) 13:04, 15 June 2019 (UTC)