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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeJun 17th 2011
• (edited Jun 17th 2011)

I did not change anything, I would not like to do it without Urs’s consent and some opinion. The entry AQFT equates algebraic QFT and axiomatic QFT. In the traditional circle, algebraic quantum field theory meant being based on local nets – local approach of Haag and Araki. This is what the entry now describes. The Weightman axioms are somewhat different, they are based on fields belonging some spaces of distributions, and 30 years ago it was called field axiomatics, unlike the algebraic axiomatics. But these differences are not that important for the main entry on AQFT. What is a bigger drawback is that the third approach to axiomatic QFT if very different and was very strong few decades ago and still has some followers. That is the S-matrix axiomatics which does not believe in physical existence of observables at finite distance, but only in the asymptotic values given by the S-matrix. The first such axiomatics was due Bogoliubov, I think. (Of course he later worked on other approaches, especially on Wightman’s. Both the Wightman’s and Bogoliubov’s formalisms are earlier than the algebraic QFT.)

I would like to say that axiomatic QFT has 3 groups of approaches, and especially to distinguish S-matrix axiomatics from the “algebraic QFT”. Is this disputable ?

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeJun 17th 2011

I am now completely sure, but with Urs’s emphasis that

AQFT as being a formalization of what in basic quantum mechanics textbooks is called the Heisenberg picture of quantum mechanics. On the other hand FQFT axiomatizes the Schrödinger picture

one could maybe say that S-matrix formalism with emphasis on the interaction in finite time while not axiomaticisting it emphasises on S-matrix whose meaningful values are exctracted typically in interaction picture. In and out states are eigenstates of the free Hamiltonian, while interaction happens in between and one uses the interaction picture in writing various series for matrix elements.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeJun 17th 2011
• (edited Jun 17th 2011)

In those days there were even statements like paper title

Connections between Field Theory and Axiomatic S-Matrix Theory (M. Dresden 1963)

and the following statement in the paper of HP Stapp, Axiomatic S-matrix theory (aps):

The idea that field theory be abandoned in favor of analyticity requirements on the S matrix has, of course, been pushed vigorously in the past several years, particularly by Landau and Chew.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJun 17th 2011

Right, so the entry really means to be about that particular form of axiomatization of QFT that is called algebraic QFT .

The problem of course is that all these terms have been badly chosen by those who introduced and established them. They should have used much better descriptive terms. Even “algebraic QFT” is still a very ambiguous term.

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeJun 17th 2011

Yes, Haag likes emphasis on “local” more than on “algebraic”.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 17th 2011

Unfortunately “local QFT” is also a very ambiguous term. Every axiomatization of QFT will be local in the end. FQFT can be called local with exactly the same justification (and actually often is, nowadays), so that the terminology then becomes quite useless for actually distinguishing the different axiom systems.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeSep 15th 2011

Added brief sentence in new Properties-section of AQFT in order to record the reference provided here.

• CommentRowNumber8.
• CommentAuthorzskoda
• CommentTimeSep 15th 2011

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeSep 15th 2011

Ah. I am not sure if I can change this. This is a forum in “beta phase”. I’ll see what I can do. But the whole thing will become public in 6 days, as far as I understand.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeApr 19th 2012

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeApr 20th 2012
• (edited Apr 20th 2012)

Thanks to input from Igor Khavkine here I have added a section References – Examples.

• CommentRowNumber12.
• CommentAuthorDavid_Corfield
• CommentTimeAug 25th 2015

I see David R mentioning Theo Johnson-Freyd’s Heisenberg-picture quantum field theory. Since the FQFT and AQFT approaches are suggested to be dual presumably we should expect to see analogues of the FQFT constructions Urs has been developing.

Is there something that makes constructing duals here difficult? Something more than just Isbell duality?

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeAug 25th 2015

In his example 1.4 and above his example 2.5 TJF writes that his goal is to generalize the relation that I gave in 2009 to the case that the FQFT is not “unitary” in physics speak, i.e. does not necessarily have time evolution by invertible linear maps.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeApr 9th 2019
• (edited Apr 9th 2019)

which claims to finally construct examples of non-perturbative interacting AQFTs in dimension $d \geq 4$ (local nets of observables at least, existence of sensible quantum states remains the open issue).

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeApr 9th 2019

also added pointer to today’s lecture notes:

• CommentRowNumber16.
• CommentAuthorStephen Pena
• CommentTimeNov 2nd 2019

Updated the page to include recent developments on factorization algebras.