nLab
observable

Context

Physics

, ,

Surveys, textbooks and lecture notes

  • ,


,

, ,

    • , , , ,

      • ,

      • ,

      • ,

        • ,

      • and
    • Axiomatizations

          • ,
        • -theorem

    • Tools

      • ,

        • ,

        • ,
    • Structural phenomena

    • Types of quantum field thories

        • ,

        • , ,

        • examples

          • ,
          • ,
        • , , , ,

        • , ,

Contents

Idea

In physics and in the theory of dynamical systems (deterministic, stochastic, quantum, autonomous, nonautonomous, open, closed, discrete, continuous, with finite or infinite number of degrees of freedom…), an observable is a quantity in some theoretical framework whose value can be measured and observed in principle. Any good theoretical framework of physical phenomena should come with carefully established notion of an observable.

In classical physics

In classical mechanics an observable is any smooth function on the phase space of the system, and of time. The value of the observable is just the value of the function for fixed argument.

In quantum physics

In quantum mechanics an observable is a Hermitean operator on the physical Hilbert space of the theory. See quantum observable for more details.

In this case, one distinguishes the concepts of the expectation value of the observable and the concept of the measured value; they are evaluated in some state of the system. The expectation value can be taken in any state of the system, while the measured value is always in some eigenstate of the observable operator. The process of measurement results in the quantum mechanical collapse or reduction, in which the system passes to an eigenstate of the measured operator. The probability of taking a given eigenstate depends on the the transition matrix element from the previously prepared state to the given eigenstate.

In quantum field theory

In relativistic quantum mechanics and relativistic quantum field theory the question of observables is more complicated: issues like causality and superselection sectors are involved.

In the AQFT approach to quantum field theory the system of quantum observables localized in given spacetime regions are the very foundation of the theory, called a local net of observables (the Haag-Kastler axioms for QFT).

In non-perturbative quantum field theory the algebras of observables are meant to be C*-algebras, while in perturbative quantum field theory (perturbative AQFT) they are formal power series algebras.

of in :

local field linear microcausal polynomial general regular \array{ && \text{local} \\ && & \searrow \\ \text{field} &\longrightarrow& \text{linear} &\longrightarrow& \text{microcausal} &\longrightarrow& \text{polynomial} &\longrightarrow& \text{general} \\ && & \nearrow \\ && \text{regular} }

between and

A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg op\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg fin op\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}
A\phantom{A} A\phantom{A}
A\phantom{A}A\phantom{A}
A\phantom{A}()A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}
A\phantom{A} (“”)

in :

A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}-A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}A\phantom{A}A\phantom{A}A\phantom{A}

References

Careful discussion of local gauge invariant observables in gravity/general relativity is in

showing that, while there are no globally defined local gauge invariant observables, they do exist on an open cover of the space of field configuration and form something like a sheaf of observables (but, hence, one without global sections).

Careful discussion of observables in abelian gauge theory (electromagnetism) is in

Last revised on January 10, 2018 at 07:41:02. See the history of this page for a list of all contributions to it.