  It is well known that certain features of a quantum theory cannot be
described in the standard picture on a Hilbert space. In particular, this
happens when we try to formally frame a quantum field theory, or a
thermodynamic system with finite density. This forces us to introduce different
types of algebras, more general than the ones we usually encounter in a
standard course of quantum mechanics. We show how these algebras naturally
arise in the Schwinger description of the quantum mechanics of an infinite spin
chain. In particular, we use the machinery of Dirac-Feynman-Schwinger (DFS)
states developed in recent works to introduce a dynamics based on the modular
theory by Tomita-Takesaki, and consequently we apply this approach to describe
the Ising model.
