The so-called topos approach provides a radical reformulation of quantum
theory. Structurally, quantum theory in the topos formulation is very similar
to classical physics. There is a state object, analogous to the state space of
a classical system, and a quantity-value object, generalising the real numbers.
Physical quantities are maps from the state object to the quantity-value object
-- hence the `values' of physical quantities are not just real numbers in this
formalism. Rather, they are families of real intervals, interpreted as `unsharp
values'. We will motivate and explain these aspects of the topos approach and
show that the structure of the quantity-value object can be analysed using
tools from domain theory, a branch of order theory that originated in
theoretical computer science. Moreover, the base category of the topos
associated with a quantum system turns out to be a domain if the underlying von
Neumann algebra is a matrix algebra. For general algebras, the base category
still is a highly structured poset. This gives a connection between the topos
approach, noncommutative operator algebras and domain theory. In an outlook, we
present some early ideas on how domains may become useful in the search for new
models of (quantum) space and space-time.Comment: 32 pages, no figures; to appear in Proceedings of Quantum Field
Theory and Gravity, Regensburg (2010
