Qm Uncertainty And Compatibility

(a) Setting. Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the choice hypotheses of the spectral theorem. Let $H$ be a nonzero complex Hilbert space (Hilbert space) and let $A$ and $B$ be self-adjoint operators on $H$ with domains $D(A)$ and $D(B)$ (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators). By the spectral theorem each of them has a unique spectral projection-valued measure on the Borel sigma-algebra of $\mathbb{R}$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure, The Borel sigma-algebra of a topological space); write $E_A$ for the spectral projection-valued measure of $A$ and $E_B$ for that of $B$.

(b) Strong spectral commutation. The operators $A$ and $B$ strongly commute, or are spectrally compatible, when their spectral projection-valued measures commute on every pair of Borel sets: $E_A(\mathrm{\Lambda })E_B(\mathrm{\Sigma })=E_B(\mathrm{\Sigma })E_A(\mathrm{\Lambda })\text{for all Borel }\mathrm{\Lambda },\mathrm{\Sigma }\subseteq \mathbb{R}\mathrm{.}$ This is the relation that the term "compatible sharp observables" abbreviates on the physical pages of the category. It is a condition on the two families of projections, not on a single vector or on a formal difference of operator products.

(c) Not to be confused with a commutator identity on a dense invariant domain. Let $D$ be a dense linear subspace of $D(A)\cap D(B)$ such that $AD\subseteq D$ and $BD\subseteq D$. Then $ABu$ and $BAu$ are defined for every $u\in D$. Strong spectral commutation implies $ABu=BAu$ on $D$: Teschl's bounded Borel calculus for a strongly commuting pair gives, for each bounded Borel function $h$, $h(B)D(A)\subseteq D(A)$ and $Ah(B)x=h(B)Ax$ for $x\in D(A)$ (Section 4.2, Lemma 4.5). Apply this with $h_n(\lambda )=\lambda {\mathbf{1}}_{[-n,n]}(\lambda )$. For $u\in D$, both $u$ and $Au$ belong to $D(B)$, so the spectral theorem gives $h_n(B)u\to Bu$ and $h_n(B)Au\to BAu$. Since $A{h}_n(B)u=h_n(B)Au$ and $A$ is closed, it follows that $Bu\in D(A)$ and $ABu=BAu$. The converse can fail even for a dense common invariant domain: the examples page records a pair of self-adjoint extensions for which the commutator vanishes on such a domain but the spectral projections do not commute.

For bounded everywhere-defined self-adjoint $A$ and $B$, if the commutator vanishes on a dense $D$, boundedness extends $AB=BA$ to all of $H$; the bounded Borel calculus then makes this equivalent to strong spectral commutation (Teschl, Section 4.2, Corollary 4.7). No graph-norm core property is needed in the comparison above. Clause (b), rather than a domain commutator identity, is the definition governing joint sharp outcomes.

(d) Status. Clauses (a)-(c) fix a definition and a comparison of conditions. They assert no physical claim, no joint measurement, no simultaneous sharp value and no dynamical statement. All further content about strong spectral commutation used by the category -- the bounded-calculus consequences, the joint projection-valued measure criterion and the physical readings -- is established as separate results with their own hypotheses.

• Why projections and not commutators. For unbounded $A$ and $B$ the products $AB$ and $BA$ may be defined on different domains, so the expression $AB-BA$ need not be an operator at all; the spectral projections $E_A(\mathrm{\Lambda })$ and $E_B(\mathrm{\Sigma })$ are bounded everywhere-defined projections, so their products are always defined and clause (b) makes literal sense. This is why the definition is stated at the level of the projection-valued measures.
• Symmetric role of the two operators. Clause (b) is manifestly symmetric in $A$ and $B$; the notation $E_A(\mathrm{\Lambda })E_B(\mathrm{\Sigma })=E_B(\mathrm{\Sigma })E_A(\mathrm{\Lambda })$ is a single condition on each pair of Borel sets, not two conditions.

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H$ be a complex Hilbert space with inner product linear in the first argument (Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space), let $A,B$ be symmetric densely defined operators on $H$ (Symmetric, self-adjoint and essentially self-adjoint operators, Adjoint of a densely defined operator), and let $u\in D(A)\cap D(B)$ with $\mathrm{\parallel }u\mathrm{\parallel }=1$ satisfy in addition $u\in D(AB)\cap D(BA)$, that is $Bu\in D(A)$ and $Au\in D(B)$. Put $a:=⟨Au,u⟩,b:=⟨Bu,u⟩,x:=(A-a)u,y:=(B-b)u,\mathrm{\Delta }A:=\mathrm{\parallel }x\mathrm{\parallel },\mathrm{\Delta }B:=\mathrm{\parallel }y\mathrm{\parallel },$ and write $\{A,B\}u:=ABu+BAu$ and $[A,B]u:=ABu-BAu$ for the pairings at $u$. Then:

(a) $a,b\in \mathbb{R}$, $x,y\in H$, and $\mathrm{Re}⟨x,y⟩=\frac{1}{2}⟨\{(A-a),(B-b)\}u,u⟩=\frac{1}{2}⟨\{A,B\}u,u⟩-ab,⟨[A,B]u,u⟩=-2i\mathrm{Im}⟨x,y⟩;$

(b) (Robertson bound) $\mathrm{\Delta }A\mathrm{\Delta }B\ge \frac{1}{2}\mid ⟨[A,B]u,u⟩\mid$;

(c) (Schroedinger-Robertson bound) $(\mathrm{\Delta }A{)}^2(\mathrm{\Delta }B{)}^2\ge (\frac{1}{2}⟨\{(A-a),(B-b)\}u,u⟩{)}^2+\frac{1}{4}\mid ⟨[A,B]u,u⟩{\mid }^2,$ that is, with the expanded anticommutator pairing $(\mathrm{\Delta }A{)}^2(\mathrm{\Delta }B{)}^2\ge (\frac{1}{2}⟨\{A,B\}u,u⟩-ab{)}^2+\frac{1}{4}\mid ⟨[A,B]u,u⟩{\mid }^2$;

(d) equality holds in (c) if and only if $x$ and $y$ are linearly dependent; in particular the Robertson bound (b) follows from (c) because the first square is nonnegative;

(e) if $A$ and $B$ are bounded everywhere-defined self-adjoint operators, then the product-domain hypothesis $u\in D(AB)\cap D(BA)$ holds for every unit $u\in H$ and clauses (a)-(d) apply to every unit $u$.

No energy-time inequality and no apparatus statement is made.

(a) Setting. Adopt the state and composition postulate P1 and the observable and probability postulate P2 in the readings of The state space of the framework: preparations, pure rays and density operators and Assigned sharp observables and the outcome law of the framework (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes), and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Let $(A,u_A)$ be an assigned sharp observable with declared unit $u_A$ and outcome law ${\mu }_{\rho }^A$ in a preparation $\rho$, in the sense of Assigned sharp observables and the outcome law of the framework clause (b).

(b) Preparation variance. When the mean $m(\rho ):={\int }_{\mathbb{R}}\lambda d{\mu }_{\rho }^A(\lambda )$ and the second moment ${\int }_{\mathbb{R}}{\lambda }^{2}d{\mu }_{\rho }^A(\lambda )$ of the outcome law are finite, the preparation variance of $(A,u_A)$ in $\rho$ is ${\mathrm{Var}}_{\rho }(A):={\int }_{\mathbb{R}}(\lambda -m(\rho ){)}^{2}d{\mu }_{\rho }^A(\lambda ),$ read in the square $u_A^2$ of the declared unit, and the preparation spread is ${\mathrm{\Delta }}_{\rho }A:={\mathrm{Var}}_{\rho }(A{)}^{1\mathrm{/}2}\ge 0$, read in the unit $u_A$. When either quantity is not finite the variance and the spread of the pair in $\rho$ are not defined; the definition does not extend them by $+\mathrm{\infty }$.

(c) Values and hypotheses. The variance and its hypotheses are exactly those of the moment theorem: for a pure preparation of a unit vector $\psi \in D(A)$ the spread is ${\mathrm{\Delta }}_{\psi }A=\mathrm{\parallel }(A-⟨A\psi ,\psi ⟩I)\psi \mathrm{\parallel }$, equal to $(⟨A^2\psi ,\psi ⟩-⟨A\psi ,\psi {⟩}^2{)}^{1\mathrm{/}2}$ when $\psi \in D(A^2)$; for a preparation $\rho$ the identities and existence criteria are those of Mean, second moment and variance of an assigned observable in a preparation and Mean, second moment and variance of the scalar spectral law. In particular, for an orthonormal density expansion $\rho ={\sum }_n{w}_n{T}_{u_n,u_n}$, the mean is finite exactly when ${\sum }_{n:w_n>0}{w}_n{\int }_{\mathbb{R}}\mathrm{\mid }\lambda \mathrm{\mid }d{E}_{A,u_n}(\lambda )<\mathrm{\infty },$ by Mean, second moment and variance of the scalar spectral law clause (d); zero-weight indices are omitted before evaluating their component integrals. For a pure unit vector the corresponding criterion is membership in $D(\mathrm{\mid }A{\mathrm{\mid }}^{1\mathrm{/}2})$ by clause (b). Componentwise domain membership alone does not imply convergence of the weighted sum, and finite moments are not automatic.

(d) Products of spreads. For two assigned sharp observables $(A,u_A)$ and $(B,u_B)$ in a common preparation $\rho$ in which both spreads are defined, the product ${\mathrm{\Delta }}_{\rho }A{\mathrm{\Delta }}_{\rho }B$ is a nonnegative real number read in the product unit $u_A{u}_B$; for a pure preparation of a unit vector in $D(A)\cap D(B)$ it is $\mathrm{\parallel }(A-a)\psi \mathrm{\parallel }\mathrm{\parallel }(B-b)\psi \mathrm{\parallel }$ with the real means $a=⟨A\psi ,\psi ⟩$ and $b=⟨B\psi ,\psi ⟩$.

(e) Reading and scope. The spread ${\mathrm{\Delta }}_{\rho }A$ is a property of the pair (preparation, assigned observable): it is the standard deviation of the outcome law that P2 assigns. It is not an apparatus resolution, not a measurement error and not a disturbance of the preparation; those are further objects that require a declared instrument or apparatus model and are not introduced here. Clause (b) makes no claim that a preparation realizes a specific laboratory state, that a laboratory procedure measures $A$ with resolution $u_A$, or that the spread is observed in any experiment; the identification of an apparatus with the pair $(A,u_A)$ is a tested identification under the preparation and apparatus postulate P7. Dimensions, SI unit conventions and quantity types of the framework fixes the units in which the displays are read.

• Vanishing spread and eigenstates. For a pure preparation of a unit vector $\psi \in D(A)$, clause (c) shows ${\mathrm{\Delta }}_{\psi }A=0$ exactly when $A\psi =⟨A\psi ,\psi ⟩\psi$, that is exactly when $\psi$ is an eigenvector of $A$; this is the framework's reading of the textbook statement that the uncertainty of an observable vanishes in an eigenstate. The identification of a laboratory state with such a preparation is again a tested identification, not a consequence of the definition.
• The product is not a joint distribution. Clause (d) forms a product of two spreads of the same preparation; it does not introduce a joint outcome law of $A$ and $B$, and the product can be defined even when no joint sharp observable of the pair exists. Joint sharp observables are a separate notion with its own criterion.

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H$ be a nonzero complex Hilbert space and let $A,B$ be self-adjoint operators on $H$ with spectral projection-valued measures $E_A,E_B$ that strongly commute in the sense of Strong spectral commutation of self-adjoint operators clause (b): $E_A(\mathrm{\Lambda })E_B(\mathrm{\Sigma })=E_B(\mathrm{\Sigma })E_A(\mathrm{\Lambda })\text{for all Borel }\mathrm{\Lambda },\mathrm{\Sigma }\subseteq \mathbb{R}\mathrm{.}$ For a bounded Borel function $h$ write $h(A):=\int hd{E}_A$ and $h(B):=\int hd{E}_B$ for its operators under the bounded Borel functional calculus (Bounded borel pvm integral). Then:

(a) for all bounded Borel $f,g:\mathbb{R}\to \mathbb{C}$, $f(A)g(B)=g(B)f(A);$

(b) for every bounded Borel $f$ one has $f(B)D(A)\subseteq D(A)$, and $Af(B)x=f(B)Ax\text{for every }x\in D(A);$ the same statement holds with the roles of $A$ and $B$ exchanged, and it is also true with $f(B)$ replaced by any bounded linear combination or norm limit of bounded Borel functions of $B$;

(c) for every bounded Borel $f$ the bounded operator $f(B)$ commutes with all spectral projections of $A$: $f(B)E_A(\mathrm{\Lambda })=E_A(\mathrm{\Lambda })f(B)$ for every Borel $\mathrm{\Lambda }\subseteq \mathbb{R}$. Consequently, for real-valued bounded Borel $f$ the pair $(f(A),B)$ strongly commutes: the map $F(\mathrm{\Lambda }):=E_A(f^{-1}(\mathrm{\Lambda }))$ is a projection-valued measure on the Borel sets of $\mathbb{R}$ representing the bounded self-adjoint operator $f(A)$, and $F(\mathrm{\Lambda })E_B(\mathrm{\Sigma })=E_B(\mathrm{\Sigma })F(\mathrm{\Lambda })$ for all Borel $\mathrm{\Lambda },\mathrm{\Sigma }$; hence also $(f(A),g(B))$ strongly commutes for real-valued bounded Borel $f,g$.

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H$ be a nonzero complex Hilbert space and let $A,B$ be self-adjoint operators on $H$ with spectral projection-valued measures $E_A,E_B$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure). Then the following are equivalent:

(i) $A$ and $B$ strongly commute in the sense of Strong spectral commutation of self-adjoint operators clause (b), that is $E_A(\mathrm{\Lambda })E_B(\mathrm{\Sigma })=E_B(\mathrm{\Sigma })E_A(\mathrm{\Lambda })$ for all Borel $\mathrm{\Lambda },\mathrm{\Sigma }\subseteq \mathbb{R}$;

(ii) there is a regular projection-valued measure $G$ on the Borel sigma-algebra of ${\mathbb{R}}^2$ whose values on measurable rectangles (Measurable rectangles in a product of measurable spaces) are $G(\mathrm{\Lambda }\times \mathrm{\Sigma })=E_A(\mathrm{\Lambda })E_B(\mathrm{\Sigma })$, and whose marginals are $G(\mathrm{\Lambda }\times \mathbb{R})=E_A(\mathrm{\Lambda })$ and $G(\mathbb{R}\times \mathrm{\Sigma })=E_B(\mathrm{\Sigma })$;

(iii) $f(A)g(B)=g(B)f(A)$ for all bounded Borel functions $f,g:\mathbb{R}\to \mathbb{C}$, where the operators are those of the bounded Borel functional calculus (Bounded borel pvm integral).

If these conditions hold, the projection-valued measure $G$ of (ii) is unique, and for all bounded Borel $f,g$ one has the representation $f(A)g(B)=g(B)f(A)={\int }_{{\mathbb{R}}^2}f(a)g(b)dG(a,b)\mathrm{.}$

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Fix integers $d\ge 1$, $r\ge 1$ and $\mathrm{\hslash }>0$, let $H^{(r)}=L^2({\mathbb{R}}^d;{\mathbb{C}}^r)$ be the component-valued space with the componentwise position and momentum operators $Q_j,P_k$ of Componentwise position and momentum operators on the component-valued wave-function space clauses (a) and (b), with domains $D(Q_j)=\{f=(f_l):x_j{f}_l\in L^2\text{ for all }l\},D(P_k)=\{f=(f_l):D_k{f}_l\in L^2\text{ for all }l\},$ and let $D_k$ denote the weak partial derivative (Weak derivative of a locally integrable function). Then: