The mathematical structure of quantum real numbers

The mathematical structure of the sheaf of Dedekind real numbers $\text{RsubD}(X)$ for a quantum system is discussed. The algebra of physical qualities is represented by an $O^{\ast }$ algebra $\mathcal{M}$ that acts on a Hilbert space that carries an irreducible representation of the symmetry group of the system. $X=\text{EsubS}(\mathcal{M})$, the state space for $\mathcal{M}$, has the weak topology generated by the functions $a_Q(\cdot )$, defined for $\hat A \in \mathcal M_{sa} $and$\forall \hat \rho \in \EsubS(\mathcal M) $,by$ a_{Q}(\hat \rho) = Tr \hat A \hat \rho $\mathrm{.}Foranyopensubset$W$of$\EsubS(\mathcal M)$,thefunction$ a_{Q}|_{W}$ is the numerical value of the quality $\widehat{A}$ defined to the extent $W$. The example of the quantum real numbers for a single Galilean relativistic particle is given.