Local quantum physics

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Local quantum physics is the Haag-Kastler framework for quantum field theory, also known as AQFT (for Algebraic Quantum Field Theory, see Axiomatic quantum field theory#Haag-Kastler axioms).

Formalism

Let Mink be the category of open subsets of Minkowski space with inclusion functions as morphisms.

We have a covariant funtor. ${mathcal {A}}$ from Mink a uC*alg, the category of C* algebras with enemy unity, such that every morphism in Mink mapping a monomorphism in uC*alg (isotony).

The Poincaré group acts continuously in Mink. There is a pullback of this action, which is continuous in the topology of the norm ${displaystyle {mathcal {A}}(mathbb {R} ^{4})}$ (Poincaré covariance).

Minkowski's space has a causal structure. If an open set V is in the causal complement of an open set U, then the image of the functions: ${displaystyle {mathcal {A}}(i_{U,Ucup V})}$ and ${displaystyle {mathcal {A}}(i_{V,Ucup V})}$ switch (space type computing). Yeah. ${displaystyle {bar {U}}}$ is the causal completion of an open set U, then the ${displaystyle {mathcal {A}}(i_{U,{bar {U}}})}$ is an isomorphism (primitive healing).

A state with respect to a C^♪-algebra is a positive linear functional on it with a unitary standard. If we have a state over ${displaystyle {mathcal {A}}(M)}$ , we can take the "partial job" to get states associated with ${displaystyle {mathcal {A}}(U)}$ for each set open via network monomorphism. It is easy to prove that states on open assemblies form a structure of prey. According to the GNS construction, for each state, we can associate a Hilbertian representation of ${displaystyle {mathcal {A}}(M)}$ . Pure states correspond to irreducible representations and mixed states correspond to reductionable representations. Each irrep (except equivalence) is called the super-selection sector. We assume that there is a pure state called void such that the space of Hilbert associated with it is a unitary representation of the group of Poincaré compatible with the covariance of Poincaré of the network such that if we look at the algebra of Poincaré, the specter with respect to the Energy-Impulse Tensor, associated with translational symmetry in space-time, lies in the cone of positive light. This is the vacuum sector.

More recently, the approach has been developed to include an algebraic version of quantum theory in curved spacetime. In fact, the local quantum physics point of view is particularly apt to generalize the renormalization procedure of quantum field theory developed in the curved context. Many rigorous results from quantum field theory in the presence of black holes have been reproduced within this approach.

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