Let ${ \cal S}(\mathcal{ H})$ 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 $\bot$ on ${ \cal S}(\mathcal{ H})$. We show that if $A\bot B$, then $A$ and $B$ are affiliated with some abelian von Neumann algebra. The relation $\bot$ induces a partial algebraic operation $\oplus$ on ${ \cal S}(\mathcal{ H})$. We prove that $({ \cal S}({ \mathcal{ H}}), \bot, \oplus, 0)$ 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 $({ \cal S}({ \mathcal{ H}}), \bot, \oplus, 0)$ has a partial order $\preceq$, and prove that $A\preceq B$ 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 $\preceq$) is studied, and it is shown at the end that the position operator $Q$ and momentum operator $P$ in the Heisenberg commutation relation satisfy $Q\wedge P=0$.
Categories: math-ph math.MP