Quantum observable generalized orthoalgebras

Shang Yun, Yongming Li

Eissa D. Habil

Bernd S. W. Schröder

Giacomo Mauro D’Ariano

Qiang Lei, Junde Wu

In quantum mechanics, many observables for a quantum system are represented as self-adjoint linear operators (possibly unbounded) on a complex Hilbert space $\mathcal{H}$. In this paper, we introduce a binary relation $\perp$ on $S(\mathcal{H})$ of all self-adjoint linear operators (including unbounded) on $\mathcal{H}$. We obtain that $A\perp B$ implies that $A,B$ are affiliated with some abelian von Neumann algebra at the same time. Depending on $\perp$$, S(\mathcal{H})$ can be constructed to be a generalized orthoalgebra. Then we present the orthoalgebra order $\preceq$ and the traditional order $\leq$ on $S(\mathcal{H})$ and give some properties about these two orders. We prove that for $A,B\in S(\mathcal{H})$, $A\preceq B$ 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, we study the existence of the infimum $A\wedge B$ and supremum $A\vee B$ for $A,B\in S(\mathcal{H})$ with respect to $\preceq$. Finally, we prove that $Q\wedge P=0$ where $Q$ and $P$ are the significant quantum observables corresponding to the position and momentum.}

Anatolij Dvurečenskij, Mária Kuková