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I am in the process of brushing up and expanding the entry local net of observables.
I have entirely reorganized the Definition-section: it starts now with the general basic definition on arbitrary spacetimes, then gives a list of all the add-on axioms that people consider.
I also finally noticed that this page coexisted with a synonymous entry that I have now moved to causal net of algebras > history, making “local nets” redirect to “causal net”. I made sure to move all the information over.
I am not quite done yet, will continue to add stuff. Maybe a little later.
I put causal net of algebras > history in the standard format.
at local net of observables I have slightly expanded the section on Einstein causality and then added below that a brief section on “strong locality” (an axiom in fact slightly weaker than Einstein locality).
Any chance that some examples could be added to the article local net?
Todd, I have been waiting for years for you to ask this question! :-)
There are the standard textbooks on AQFT, but I’d recommend this article here:
Where are the examples of local nets in that paper?
I’d be inclined to say: “In the whole article!” but maybe what you are looking for is the statement of theorems 3.10 and 3.20, which assert that what has been constructed are indeed local nets.
(The authors don’t use the term “local net of observables”, they say just “quantum field theory” for it.)
Maybe it helps if I give a quick guide to how the construction works:
One wants to functorially assigns $C^\ast$-algebras to spacetimes for a fixed free field theory. The present authors are careful and observe that generally the domain of the functor should be spacetimes equipped with vector bundles (field bundles) and with a hyperbolic operator on their sections. That gives their category $GlobHypGreen$ in def. 3.1. This may be restricted to trivial vector bundles if desired, as implicitly done in much of the literature. For instance for the standard example of scalar field theory the vector bundle would be taken to be the trivial $\mathbb{C}$-fiber bundle throughout.
The procedure of the quantization construction then goes as follows:
Produce the space of solutions of the hyperbolic operator on each spacetime, expressed in terms of its Green’s functions. This is the content of section 2 for many standard examples of field theories.
Produce symplectic structure on the space of solutions (to make them become “phase spaces” of physical states). This happens from prop. 3.4. to lemma 3.8, see top of page 14 for the conclusion.
Quantize. Since here we are dealing with free fields this is functorial, the assignment of Weyl algebras to symplectic vector spaces. Middle of p. 14 for the bosonic case with details in appendix A, and section 3.2 for the fermionic case.
Theorems 3.10 and 3.20 assert that what has been constructed thereby are indeed local nets.
The last section 4 is about extracting quantum states and quantum fields (in the sense of operator valued distributions) from the local net of observables.
Thanks! Those are useful pointers, and pretty much what I was looking for, I think.
I have expanded a little the Idea-section at local net of observables.
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