Quantum Observable Generalized Orthoalgebras

Abstract

Let $\mathcal{S}(\mathcal{H})$ denote the set of all self-adjoint operators (not necessarily bounded) on a Hilbert space $\mathcal{H}$, which is the set of all physical quantities on a quantum system $\mathcal{H}$. We introduce a binary relation $\perp$ on $\mathcal{S}(\mathcal{H})$. We show that if $A\perp B$, then $A$ and $B$ are affiliated with some abelian von Neumann algebra. The relation $\perp$ induces a partial algebraic operation $\oplus$ on $\mathcal{S}(\mathcal{H})$. We prove that $(\mathcal{S}(\mathcal{H}),\perp ,\oplus ,0)$ is a generalized orthoalgebra. This algebra is a generalization of the famous Birkhoff\,--\,von Neumann quantum logic model. It establishes a mathematical structure on all physical quantities on $\mathcal{H}$. In particular, we note that $(\mathcal{S}(\mathcal{H}),\perp ,\oplus ,0)$ has a partial order $⪯$, and prove that $A⪯B$ if and only if $A$ has a value in $\Delta$ implies that $B$ has a value in $\Delta$ for every Borel set $\Delta$ not containing $0$. Moreover, the existence of the infimum $A\wedge B$ and supremum $A\vee B$ for $A,B\in \mathcal{S}(\mathcal{H})$ (with respect to $⪯$) is studied, and it is shown at the end that the position operator $Q$ and momentum operator $P$ in the Heisenberg commutation relation satisfy $Q\wedge P=0$.