One-Parameter Group of Transformations and Stone’s Theorem

Abstract

In this chapter we will define the topos notion of a one-parameter group taking values in the complex number object and in the real number objects. To this end we first of all need to upgrade the monoids $\underline{\mathbb{C}}^{\leftrightarrow}$ and $\underline{\mathbb {R}}^{\leftrightarrow}$ to groups. This can be done using a standard method called Grothendieck k-Construction already mentioned in Doring and Isham (2008). Having done that the construction of a one-parameter group can be defined. This in turn allows us to define the topos analogue of the Stone’s theorem which uniquely associates to each self adjoint operator $\breve{\delta}(\hat{A}): {\underline{\varSigma}\rightarrow \underline{\mathbb{R}}}$ a one parameter group $\underline{Q}^{\hat{A}}$. This is of particular importance in the view of defining a unique time evolution. In fact, given a Hamiltonian operator $\underline{H}$, the topos analogue would be $\breve{\delta}(\hat {H})$ with associated the unique one-parameter group of transformations $\underline{Q}^{\hat{H}}$. This group would represent the group of time evolutions in topos quantum theory. The detailed analysis of such a group and how it acts had not yet been carried out but it would be of particular interest to do so.