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.}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.}

arXiv:1508.07386?context=math-phAug 29, 20150 citations

Self-adjoint Operators as Functions I

Andreas Döring, Barry Dewitt

Communications in Mathematical PhysicsApr 11, 201414 citations

Self-adjoint extensions of phase and time operators (4 pages)

F. C. Khanna, Gilad Gour

Physical Review AJan 1, 20040 citations

The Consistency of a Bounded, Self-Adjoint Time Operator Canonically Conjugate to a Hamiltonian with Non-empty Point Spectrum

Eric A. Galapon

Pauli's well known theorem (W. Pauli, Hanbuch der Physik vol. 5/1, ed. S. Flugge, (1926) p.60) asserts that the existence of a self-adjoint time operator canonically conjugate to a given Hamiltonian implies that the time operator and the Hamitlonian posses completely continuous spectra spanning the entire real line. Thus the conclusion that there exists no self-adjoint time operator conjugate to a Hamiltonian with a spectrum which is a proper subspace of the real line. But we challenge this conclusion. We show rigourously the consitency of assuming a bounded, self-adjoint time operator conjugate to a Hamiltonian with an unbounded, or semibounded, or finitely countable point spectrum. Pauli implicitly assumed unconditionally that the domain of the Hamiltonian is invariant under the action of $U_\\beta=\\exp(-i\\beta T)$, where $T$ is the time operator, for arbitrary real number $\\betaA$ . But we show that the $\\beta$ 's are at most the differences of the eigenvalues of the Hamiltonian. And this happens, under some other conditions, when the Hamiltonian has a non-empty point spectrum extending from negative to positive infinity. For a Hamiltonian with a simibounded or finitely countable point spectrum, we show that no $U_\\beta$. We demonstrate our claim by giving an explicit example.Pauli's well known theorem (W. Pauli, Hanbuch der Physik vol. 5/1, ed. S. Flugge, (1926) p.60) asserts that the existence of a self-adjoint time operator canonically conjugate to a given Hamiltonian implies that the time operator and the Hamitlonian posses completely continuous spectra spanning the entire real line. Thus the conclusion that there exists no self-adjoint time operator conjugate to a Hamiltonian with a spectrum which is a proper subspace of the real line. But we challenge this conclusion. We show rigourously the consitency of assuming a bounded, self-adjoint time operator conjugate to a Hamiltonian with an unbounded, or semibounded, or finitely countable point spectrum. Pauli implicitly assumed unconditionally that the domain of the Hamiltonian is invariant under the action of $U_\\beta=\\exp(-i\\beta T)$, where $T$ is the time operator, for arbitrary real number $\\betaA$. But we show that the $\\beta$'s are at most the differences of the eigenvalues of the Hamiltonian. And this happens, under some other conditions, when the Hamiltonian has a non-empty point spectrum extending from negative to positive infinity. For a Hamiltonian with a simibounded or finitely countable point spectrum, we show that no $U_\\beta$. We demonstrate our claim by giving an explicit example.

Aug 11, 19993 citations

Interpreting Self-Adjoint Operators as q-Functions

Cecilia Flori

Lecture notes in physicsJan 1, 20180 citations

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

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

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.}

T

is the time operator, for arbitrary real number

\\betaA

. But we show that the

\\beta

's are at most the differences of the eigenvalues of the Hamiltonian. And this happens, under some other conditions, when the Hamiltonian has a non-empty point spectrum extending from negative to positive infinity. For a Hamiltonian with a simibounded or finitely countable point spectrum, we show that no $U_\\beta$. We demonstrate our claim by giving an explicit example.