# Proof of PCT theorem for Haag-Kastler nets in QFT

This question is about a theorem in the Haag-Kastler axiomatic approach to quantum field theory (QFT), also known as axiomatic or algebraic or local QFT.

PCT stands for parity, charge and time, a "PCT theorem" says roughly that if a quantum field theory describes a universe, then after reversing the parity, charges and the arrow of time in the theory, the resulting theory still describes the same universe. It is one of the fundamental symmetries of today's theoretical physics. Various versions of PCT theorems in different QFT frameworks have been around at least since the 1950ties.

My question is: What versions of the PCT theorem exist that are stated and proven using some version of the Haag-Kastler axioms?

There are three papers that I am aware of:

H.J. Borchers, J. Yngvason: On the PCT--Theorem in the Theory of Local Observables

H.J. Borchers: “On Revolutionizing of Quantum Field Theory with Tomita’s Modular Theory”

Longo, Guido: An Algebraic Spin Statistics Theorem

My motivation is twofold:

1. The "conceptional proofs" of physicists tend to be rather simple and general, but are not rigorous. The proofs in the papers above are rigorous, but rather involved and not as general as I expected (which could simply be a misunderstanding or too big expectations on my part). I would like to know if there is a proof that is either simpler (does use less mathematical machinery, e.g. does not use modular theory) or more general (does not need one of the assumptions or weakens one of the assumptions).

2. I would like to know who constructed the first proof of a PCT theorem in the Haag-Kastler approach and when.

BTW: Any information about the twin of the PCT symmetry, the spin-statistics theorem, in the Haag-Kastler approach would be welcome, too.

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R. Jost. "Eine Bemerkung zum CPT Theorem" Helv. Phys. Acta 30, 409 (1957). See the nLab ( ncatlab.org/nlab/show/PCT+theorem ) or Haag's book ( amazon.com/Local-Quantum-Physics-Theoretical-Mathematical/dp/… ), which both cite this." –  Steve Huntsman Jun 10 '10 at 14:54
Yes, the nLab page was authored by me :-) (you can see the name of the last one who edited the page as the bottom of the page). Jost's proof uses the Wightman axioms, I'm looking for a proof using the Haag-Kastler axioms. There are some additional conditions known for the Wightman axioms to allow a construction of Haag-Kastler nets and vice versa, but the philosophy of Borchers is to avoid the use of Wightman fields entirely for this specific topic (the philosophy of the Haag-Kastler axioms is that fields are of secondary importance only, if at all). –  Tim van Beek Jun 10 '10 at 19:09
Serves me right for not thinking...I should've recalled that AQFT wasn't even around in the fifties. Have you seen this? citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.44.8323 –  Steve Huntsman Jun 10 '10 at 19:16
No problem :-) Yes, seen that, cited it on the spin-statistics theorem page on the nLab but forgot to include it here - but I can cheat and edit my question :-) Thanks for the reminder. –  Tim van Beek Jun 10 '10 at 21:06

Jens Mund has some papers on spin-statistics and PCT in the case of massive particles in d=2+1 (e.g. here and here), but as far as I recall he also uses modular theory, so this might not provide a full answer to your question.

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Thank you very much! Modular theory was just a random example for "mathematical machinery that may be too sophisticated for the context". I'll definitly take a look at those papers once I can afford the time to do so :-) –  Tim van Beek Sep 28 '10 at 7:13
Modular theory is interesting in this respect since it seems to appear all the time in proofs of technical results (structure of local algebra, wedge duality or Haag duality in some models, PCT...). It would be very interesting to know if one can do without modular theory, but I don't know of any results in this direction. –  Pieter Naaijkens Sep 28 '10 at 8:56