Abstract
So far we have only discussed quantum spin systems, defined on a discrete lattice. In this chapter we will also look at continuous systems, defined on Minkowski space-time. It turns out that many of the same techniques can be used in this setting. This resemblance is particularly clear in what is called algebraic quantum field theory (AQFT). This is an attempt to formulate quantum field theory in a mathematically rigorous way, using C ā-algebraic techniques. One of the first formulations is due to Haag and Kastler (J Math Phys 5, 848ā861, 1964). Their goal was to give a purely algebraic description of quantum field theory, that is, without reference to any Hilbert space.
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Notes
- 1.
Naimarkās name has not always been transliterated consistently from Russian: except for Naimark, āNeumarkā is also seen.
- 2.
- 3.
In June 2009 the 50 year anniversary of the birth of the theory was celebrated with a conference in Gƶttingen, where Haag, one of the founders of the subject, recollected some of the successes and problems of algebraic quantum field theory [19]. The reader might also be interested in Haagās personal recollection of this period [20].
- 4.
In fact, in practice one usually realises the net as a net of von Neumann algebras acting on some fixed Hilbert space. Under physically reasonable assumptions the algebras \(\mathfrak{A}(\mathcal{O})\) are Type III factors. These are a specific type of von Neumann algebras. See [33] for a discussion of the physical significance of this.
- 5.
Recall that this can always be achieved by taking the GNS representation of a suitable state Ļ on \(\mathfrak{A}\).
- 6.
There are some additional technical conditions necessary on the representation Ļ 0, however, the most important one being Haag duality.
- 7.
This is somewhat like the case where you would have an electron and a positron, making total electric charge zero. The difference is of course that electrons and positrons are different particles, that is, they are not their own conjugate.
- 8.
A category is a collection of objects and maps between these objects, together with an associative composition operation of maps. That is, if f:āĻāāāĻ and g:āĻāāāĻ are two maps, then there is a map g ā f:āĻāāāĻ. What a map is depends on the context, and maps are not necessarily functions. In addition, for each object Ļ there is a map 1 Ļ which acts as the identity for composition. Standard examples are the category of sets with maps between them, the category of finite groups with group homomorphisms, or the category of topological spaces with continuous maps between them. Here we will use category theory merely as a bookkeeping device. We refer to [23] for a more in-depth discussion of tensor categories, which we need here.
- 9.
In the local quantum physics community, the term plektons is also used for non-abelian anyons, but this does not seem to have caught on outside of this community.
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Naaijkens, P. (2017). Local Quantum Physics. In: Quantum Spin Systems on Infinite Lattices. Lecture Notes in Physics, vol 933. Springer, Cham. https://doi.org/10.1007/978-3-319-51458-1_5
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