Contents

Idea

Algebraic Quantum Field Theory or Axiomatic Quantum Field Theory or AQFT for short is a formalization of quantum field theory that axiomatizes the assignment of algebras of observables to patches of parameter space that one expects a quantum field theory to provide.

As such, the approach of AQFT is roughly dual to that of FQFT, where instead spaces of states are assigned to boundaries of cobordisms and propagation maps between state spaces to cobordisms themselves.

One may roughly think of 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 .

The axioms of traditional AQFT encode the properties of a local net of observables and are called the Haag-Kastler axioms. They are one of the oldest systems of axioms that seriously attempt to put quantum field theory on a solid conceptual footing.

From the nPOV we may think of a local net as a co-flabby copresheaf of algebras on spacetime which satisfies a certain locality axiom with respect to the Lorentzian structure of spacetime:

• locality: algebras assigned to spacelike separated regions commute with each other when embedded into any joint superalgebra.

This is traditionally formulated (implicitly) as a structure in ordinary category theory. More recently, with the proof of the cobordism hypothesis and the corresponding (∞,n)-category-formulation of FQFT also higher categorical versions of systems of local algebras of observables are being put forward and studied. Three structures are curently being studied, that are all conceptually very similar and similar to the Haag-Kastler axioms:

• factorization algebras

• topological chiral homology

• blob homology.

On the other hand, all three of these encode what in physics are called Euclidean quantum field theories, whereas only the notion of local net so far really incorporates crucially the fact that the underlying spacetime of a quantum field theory is a smooth Lorentzian space.

In the context of the Haag-Kastler axioms there is a precise theorem, the Osterwalder-Schrader theorem, relating the Euclidean to the Lorentzian formulation: this is the operation known as Wick rotation.

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Axioms

• Wightman axioms

• Haag-Kastler axioms

Theorems

• Reeh-Schlieder theorem

• Osterwalder-Schrader theorem

• PCT theorem

• Bisognano-Wichmann theorem

• spin-statistics theorem

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References

A good account of the mathematical axiomatics of Haag-Kastler AQFT is

• Hans Halvorson, Michael Müger, Algebraic Quantum Field Theory (arXiv)

This is, among other things, the ideal starting point for pure mathematicians who have always been left puzzled or otherwise unsatisfied by accounts of quantum field theory, even those tagged as being “for mathematicians”. AQFT is truly axiomatic and rigorously formal.

An account written by mathematicians for mathematicians is this:

• Hellmut Baumgärtel, Manfred Wollenberg: Causal nets of operator algebras. Berlin: Akademie Verlag 1992 (ZMATH entry)

and this:

• Hellmut Baumgärtel: Operator algebraic Methods in Quantum Field Theory. A series of lectures. Akademie Verlag 1995 (ZMATH entry)

There is much more literature one should point to here, eventually. For instance for the connection between the AQFT axioms and the perturbative Feynman-integral techniques much used in quantum field theory, see

• Romeo Brunetti, Michael Duetsch, Klaus Fredenhagen, Perturbative Algebraic Quantum Field Theory and the Renormalization Groups (arXiv)

Out of laziness for the moment I point for further references and more background to the introductory section of

• Urs Schreiber, AQFT from $n$-functorial QFT , Comm. Math. Phys., (arXiv)