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∗-algebraic techniques. One of the first formulations is due to Haag and Kastler (J Math Phys 5, 848–861, 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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