nLab observable

Redirected from "algebra of observables".

Contents

See also at quantum observable.

Context

Physics

Idea
In classical physics
In quantum physics
In quantum field theory
Related concepts
References

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.

types of observables in perturbative quantum field theory:

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

Related concepts

quantum probability theory – observables and states

duality between $\;$ algebra and geometry

$\phantom{A}$ geometry $\phantom{A}$	$\phantom{A}$ category $\phantom{A}$	$\phantom{A}$ dual category $\phantom{A}$	$\phantom{A}$ algebra $\phantom{A}$
$\phantom{A}$ topology $\phantom{A}$	$\phantom{A}$ $\phantom{NC}TopSpaces_{H,cpt}$ $\phantom{A}$	$\phantom{A}$ $\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$ $\phantom{A}$	$\phantom{A}$ commutative algebra $\phantom{A}$
$\phantom{A}$ topology $\phantom{A}$	$\phantom{A}$ $\phantom{NC}TopSpaces_{H,cpt}$ $\phantom{A}$	$\phantom{A}$ $\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$ $\phantom{A}$	$\phantom{A}$ comm. C-star-algebra $\phantom{A}$
$\phantom{A}$ noncomm. topology $\phantom{A}$	$\phantom{A}$ $NCTopSpaces_{H,cpt}$ $\phantom{A}$	$\phantom{A}$ $\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$ $\phantom{A}$	$\phantom{A}$ general C-star-algebra $\phantom{A}$
$\phantom{A}$ algebraic geometry $\phantom{A}$	$\phantom{A}$ $\phantom{NC}Schemes_{Aff}$ $\phantom{A}$	$\phantom{A}$ $\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$ $\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$ commutative ring $\phantom{A}$
$\phantom{A}$ noncomm. algebraic $\phantom{A}$ $\phantom{A}$ geometry $\phantom{A}$	$\phantom{A}$ $NCSchemes_{Aff}$ $\phantom{A}$	$\phantom{A}$ $\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$ $\phantom{A}$	$\phantom{A}$ fin. gen. $\phantom{A}$ associative algebra $\phantom{A}$ $\phantom{A}$
$\phantom{A}$ differential geometry $\phantom{A}$	$\phantom{A}$ $SmoothManifolds$ $\phantom{A}$	$\phantom{A}$ $\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$ $\phantom{A}$	$\phantom{A}$ commutative algebra $\phantom{A}$
$\phantom{A}$ supergeometry $\phantom{A}$	$\phantom{A}$ $\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$ $\phantom{A}$	$\phantom{A}$ $\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$ $\phantom{A}$	$\phantom{A}$ supercommutative $\phantom{A}$ $\phantom{A}$ superalgebra $\phantom{A}$
$\phantom{A}$ formal higher $\phantom{A}$ $\phantom{A}$ supergeometry $\phantom{A}$ $\phantom{A}$ (super Lie theory) $\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$ differential graded-commutative $\phantom{A}$ $\phantom{A}$ superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$ algebra $\phantom{A}$	$\phantom{A}$ geometry $\phantom{A}$
$\phantom{A}$ Poisson algebra $\phantom{A}$	$\phantom{A}$ Poisson manifold $\phantom{A}$
$\phantom{A}$ deformation quantization $\phantom{A}$	$\phantom{A}$ geometric quantization $\phantom{A}$
$\phantom{A}$ algebra of observables	$\phantom{A}$ space of states $\phantom{A}$
$\phantom{A}$ Heisenberg picture	$\phantom{A}$ Schrödinger picture $\phantom{A}$
$\phantom{A}$ AQFT $\phantom{A}$	$\phantom{A}$ FQFT $\phantom{A}$

$\phantom{A}$ higher algebra $\phantom{A}$	$\phantom{A}$ higher geometry $\phantom{A}$
$\phantom{A}$ Poisson n-algebra $\phantom{A}$	$\phantom{A}$ n-plectic manifold $\phantom{A}$
$\phantom{A}$ En-algebras $\phantom{A}$	$\phantom{A}$ higher symplectic geometry $\phantom{A}$
$\phantom{A}$ BD-BV quantization $\phantom{A}$	$\phantom{A}$ higher geometric quantization $\phantom{A}$
$\phantom{A}$ factorization algebra of observables $\phantom{A}$	$\phantom{A}$ extended quantum field theory $\phantom{A}$
$\phantom{A}$ factorization homology $\phantom{A}$	$\phantom{A}$ cobordism representation $\phantom{A}$

References

In the algebraic formulation of quantum mechanics:

Nikolay Bogolyubov, A. A. Logunov, A. I. Oksak, I. T. Todorov, G. G. Gould, Algebra of Observables and State Space [doi:10.1007/978-94-009-0491-0_6], Chapter in: General principles of quantum field theory, Mathematical Physics and Applied Mathematics 10, Kluwer (1990) [doi:10.1007/978-94-009-0491-0]
Jonathan Gleason, The $C^*$ -algebraic formalism of quantum mechanics (2009) [pdf, pdf]
Jonathan Gleason, From Classical to Quantum: The $F^\ast$ -Algebraic Approach, contribution to VIGRE REU 2011, Chicago (2011) [pdf, pdf]

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

Igor Khavkine, Local and gauge invariant observables in gravity, Class. Quantum Grav. 32 185019, 2015 (arXiv:1503.03754, CQG+, pdf slides for talk at Operator and Geometric Analysis on Quantum Theory
Levico Terme, Italy, September 2014)

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

Marco Benini, Alexander Schenkel, Richard Szabo, Homotopy colimits and global observables in Abelian gauge theory (arXiv:1503.08839)

Last revised on December 20, 2023 at 10:23:57. See the history of this page for a list of all contributions to it.