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”

available free for download here.

Longo, Guido: An Algebraic Spin Statistics Theorem

My motivation is twofold:

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).

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.

Helv. Phys. Acta30, 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." $\endgroup$