Gaussian number

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The __Gaussian numbers__, sometimes (unnecessarily) called the __Gaussian rational numbers__, are the elements of the [[number field]] $\mathbb{Q} + \mathrm{i}\mathbb{Q}$, where $\mathbb{Q}$ is the field of [[rational numbers]].  In other words, a [[complex number]] is Gaussian (and a fortiori [[algebraic number|algebraic]]) iff both its [[real part|real]] and [[imaginary parts]] are rational.  The __Gaussian integers__ are the [[algebraic integers]] in the Gaussian numbers, which happen to be simply the elements of the [[integral domain]] $\mathbb{Z} + \mathrm{i}\mathbb{Z}$, where $\mathbb{Z}$ is the integral domain of (rational) [[integers]].

[[!redirects Gaussian number]]
[[!redirects Gaussian numbers]]
[[!redirects Gaussian rational number]]
[[!redirects Gaussian rational numbers]]
[[!redirects Gaussian rational]]
[[!redirects Gaussian rationals]]

[[!redirects Gaussian integer]]
[[!redirects Gaussian integers]]


Gaussian probability distribution

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#### Measure and probability theory
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# Contents
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## Definition

A [[probability distribution]] on a [[Cartesian space]] $\mathbb{R}^n$ is called **Gaussian** or a **normal distribution** if it is of the form

$$
  \mu_A \;\colon\; \vec x \mapsto \frac{\sqrt{det A}}{(2\pi)^{n/2}}
  \exp\left(-\tfrac{1}{2} \left\langle \vec x , A \vec x\right\rangle\right)
  \,d x^1 \cdots d x^n
  \,.
$$

where $A$ is some $n \times n$ [[matrix]] such that $\langle -, A-\rangle$ is a positive definite [[bilinear form]]. Here $\det A$ denotes the [[determinant]] and $\langle -,-\rangle$ is the canonical [[bilinear form]] on $\mathbb{R}^n$.

Since $\sqrt{\det A}$ is the coordinate of the [[volume element]] $vol_A$ associated with $A$, we may equivalently write this as

$$
  \mu_A \;\colon\; \vec x \mapsto \frac{1}{(2\pi)^{n/2}}
  \exp\left(-\tfrac{1}{2} \left\langle \vec x , A \vec x\right\rangle\right)
  \,vol_A  \,.
$$

The [[mean]] of this distribution is $\vec{0}$; for a distribution with mean $\vec{c}$, replace $\langle{\vec{x},A \vec{x}}\rangle$ with $\langle{\vec{x} - \vec{c},A \vec{x} - A \vec{c}}\rangle$.

The matrix $A$ is the [[inverse matrix|inverse]] of the [[covariance matrix]].  In particular, for $n = 1$, we may write $x^2/\sigma^2$ (or $(x-c)^2/\sigma^2$ for mean $c$) in place of $\langle{\vec{x},A \vec{x}}\rangle$, where $\sigma$ is the [[standard deviation]]; similarly, $\sqrt{\det A}$ becomes $1/\sigma$.  This gives the form

$$
  \mu_\sigma \;\colon\; x \mapsto \frac{1}{\sigma \sqrt{2\pi}}
  \exp\left(-\frac{(x-c)^2}{2\sigma^2}\right)
  \,d x
  \,,
$$

which may be more familiar to some readers.

## Related concepts

* [[Wick's theorem]]

* [[Feynman diagram]]

* [[statistical significance]]

## References

See also

* Wikipedia, _[Normal distribution](https://en.wikipedia.org/wiki/Normal_distribution)_

* Wikipedia, _[Standard deviation](https://en.wikipedia.org/wiki/Standard_deviation)_

* Good Calculators, _[Standard deviation calculator](https://goodcalculators.com/standard-deviation-calculator/)_

[[!redirects Gaussian probability measure]]
[[!redirects Gaussian probability measures]]

[[!redirects Gaussian distribution]]
[[!redirects Gaussian distributions]]

[[!redirects Gaussian probability distribution]]
[[!redirects Gaussian probability distributions]]

[[!redirects normal distribution]]
[[!redirects normal distributions]]

[[!redirects multivariate normal distribution]]
[[!redirects multivariate normal distributions]]

[[!redirects standard deviation]]
[[!redirects standard deviations]]


Gavin Bierman

* [personal page](https://labs.oracle.com/pls/apex/f?p=labs:bio:0:2044)

## Selected writings

On [[categorical semantics]] for the [[exponential modality]] in [[linear logic]]:

* {#BBP92} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], *Term assignment for intuitionistic linear logic*, Technical report 262, Computer Laboratory, University of Cambridge (August 1992) [[pdf](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-262.pdf), [[BentonBiermanDePaiva-TermAssignment.pdf:file]]]

  > (published abridged as [BBPH92](#BBPH92))

* {#BBPH92} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], [[Martin Hyland]], *Linear $\lambda$-Calculus and Categorical Models Revisited*, in *Computer Science Logic. CSL 1992*, Lecture Notes in Computer Science **702**, Springer (1993) [[doi:10.1007/3-540-56992-8_6](https://doi.org/10.1007/3-540-56992-8_6)]

* {#Bierman94} [[Gavin Bierman]], *On Intuitionistic Linear Logic*, Cambridge (1994)  [[[Bierman-LinearLogic.pdf:file]], [pdf](https://www.dropbox.com/s/hdxgubjljb96rmf/Biermanthesis.pdf?dl=0)]

A [[natural deduction]]-formulation of [[modal logic]] ([[S4 modal logic|S4]]/[[S5 modal logic|S5]]) and making explicit the [[necessity]] [[modal operator]] as a [[comonad]]:

* {#BiermanDePaiva96} [[Gavin M. Bierman]], [[Valeria de Paiva]], *Intuitionistic necessity revisited*, School of Computer Science research reports-University of Birmingham CSR (1996) [[researchgate](https://www.researchgate.net/publication/2810611_Intuitionistic_Necessity_Revisited), [[Bierman-dePaiva-NecessityRevisited.pdf:file]]]

* [[Gavin M. Bierman]], [[Valeria de Paiva]], _On an Intuitionistic Modal Logic_ Studia Logica **65** (2000) 383–416 [[https://doi.org/10.1023/A:1005291931660](https://doi.org/10.1023/A:1005291931660), [pdf] (https://www.researchgate.net/profile/Valeria-De-Paiva/publication/226515897_On_An_Intuitionistic_Modal_Logic/links/00b4951ed416906ccc000000/On-An-Intuitionistic-Modal-Logic.pdf)]

see also:

* {#BentonBiermanPaiva98} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], *Computational Types from a Logical Perspective*, Journal of Functional Programming **8** 2  (1998) 177-193 [[doi:10.1017/S0956796898002998](https://doi.org/10.1017/S0956796898002998), [web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.5778)]

Introducing [[polymonads]]:

* Nataliya Guts, [[Michael Hicks]], Nikhil Swamy, Daan Leijen, [[Gavin Bierman]], _Polymonads_, Extended version of POPL'13 submission [[pdf](https://www.cs.umd.edu/~mwh/papers/polymonadsTR.pdf), [[GHSLB-Polymonads.pdf:file]]]

* [[Michael Hicks]], [[Gavin Bierman]], Nataliya Guts, Daan Leijen, Nikhil Swamy, *Polymonadic Programming*, EPTCS **153** 79-99 (2014) [[arXiv:1406.2060](https://arxiv.org/abs/1406.2060), [doi:10.4204/EPTCS.153.7](https://doi.org/10.4204/EPTCS.153.7)]

category: people

[[!redirects Gavin M. Bierman]]


Gavin J. Seal

* [institute page](https://people.epfl.ch/gavin.seal)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=13939)

## Selected writings

On the [[Eilenberg-Moore categories]] of [[monoidal monads]]:

* {#Seal12} [[Gavin J. Seal]], Section 3 of: *Tensors, monads and actions*, Theory and Applications of Categories **28** 15 (2013) 403-434.  [[arXiv:1205.0101](http://arxiv.org/abs/1205.0101), [tac:28-15](http://www.tac.mta.ca/tac/volumes/28/15/28-15abs.html)]

category: people

[[!redirects Gavin Seal]]


Gavin K. Brennen

* [GoogleScholar entry](https://scholar.google.com/citations?user=_3tVObcAAAAJ&hl=en)

## Selected writings

On [[topological quantum computation]] by [[braiding]] of [[anyon]] [[worldlines]]:

* [[Gavin K. Brennen]], [[Jiannis K. Pachos]], *Why should anyone care about computing with anyons?*, Proc. R. Soc. A **464** (2008) 1-24 $[$[arXiv:0704.2241](https://arxiv.org/abs/0704.2241), [doi:10.1098/rspa.2007.0026](https://doi.org/10.1098/rspa.2007.0026)$]$

On [[holographic tensor network]] [[quantum error correcting codes]]:

* [[Robert J. Harris]], [[Nathan A. McMahon]], [[Gavin K. Brennen]], [[Thomas M. Stace]], *Calderbank-Steane-Shor Holographic Quantum Error Correcting Codes*, Phys. Rev. A 98, 052301 (2018) ([arXiv:1806.06472](https://arxiv.org/abs/1806.06472))

* [[Nathan A. McMahon]], [[Gavin K. Brennen]], [[Thomas M. Stace]], [[Robert J. Harris]], Elliot Coupe, *Decoding Holographic Codes with an Integer Optimisation Decoder* ([arXiv:2008.10206](https://arxiv.org/abs/2008.10206))

category: people

[[!redirects Gavin Brennen]]


Gavin Wraith

I retired from a readership in mathematics at Sussex University (UK) in 1999. I am just putting a gingerly toe into the waters of this wiki for the first time - patience please! I got courage for this from seeing here the names of some old mathematical colleagues. The trouble with hermetic retirement is that one not only loses touch with old friends but also misses the opportunity to make new ones. One misses the gossip, too!

This medium of mathematical converse was a pipedream for me for many years. It will probably seem pretty clunky to the next generation, who will no doubt be able to express themselves by a mere act of will, with all the digital _hardware_ incarnated in human flesh. Caro electronica facienda sit.

Hitherto I have not used wikis much, but have preferred to restrict myself to a medium over which I have nearly complete control, my [home pages](http://www.wra1th.plus.com/). Because of my emotional attachment to an operating system, RISC OS, that is almost extinct, I have missed out on a good deal of the software necessary for modern mathematical communication. The falling price of notebook computers has at last tempted me to come out of my shell.

## Selected writings

On [[indexed categories]]:

* [[P. T. Johnstone]], [[R. Paré]], [[R. D. Rosebrugh]], [[D. Schumacher]], [[R. J. Wood]], [[G. C. Wraith]], _Indexed Categories and Their Applications_, Lecture Notes in Mathematics **661** (1978) [[doi:10.1007/bfb0061362](https://doi.org/10.1007/bfb0061362)]

category: people

[[!redirects GavinWraith]]
[[!redirects Gavin W]]
[[!redirects GavinW]]
[[!redirects Gavin C. Wraith]]
[[!redirects G. C. Wraith]]


Gaëtan Borot

* [webpage](http://people.mpim-bonn.mpg.de/gborot/home)

### Related entries

* [[topological recursion]]

category: people


Gaëtan Gilbert

* [GitHub page](https://github.com/SkySkimmer)

## Selected writings

On formalizing the [[real numbers]] in [[homotopy type theory]]:

* [[Gaëtan Gilbert]], *Formalising Real Numbers in Homotopy Type Theory*, in *CPP 2017: Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs*  (2017) 112–124 [[arXiv:1610.05072](https://arxiv.org/abs/1610.05072), [doi:10.1145/3018610.3018614](https://doi.org/10.1145/3018610.3018614)]

category: people

[[!redirects Gaetan Gilbert]]


GCD domain

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## Definition ##

\begin{definition}
A [[GCD ring]] $R$ is a **GCD domain** if $R$ is also an [[integral domain]].
\end{definition}

## Properties

Every [[GCD domain]] of dimension at most 1 is a Bézout domain.

## Examples

* The ring of integers $\mathbb{Z}$

* For any [[GCD domain]] $R$, the [[polynomial ring]] $R[x]$ on one generator

* The ring of [[entire function|entire]] [[holomorphic functions]] on the [[complex plane]]

## See also ##

* [[GCD ring]]

* [[integral domain]]

* [[Bézout domain]]

* [[unique factorization domain]]

* [[Euclidean domain]]

* [[field]]

## References ##

* [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

[[!redirects GCD domains]]


GCD monoid

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## Definition

A [[monoid]] (or [[semigroup]]) $M$ is a __left GCD monoid__ if for all elements $m,n \in M$, there is a unique minimal left [[ideal of a monoid|ideals]] in $M$ generated by $m$ and $n$. Similarily, __right GCD monoid__ if for all elements $m,n \in M$, there is a unique minimal right [[ideal of a monoid|ideals]] in $M$ generated by $m$ and $n$. Finally, $M$ is a __two-sided GCD monoid__ if it is both a left GCD monoid and a right GCD monoid, and $M$ is a __GCD monoid__ if $M$ is a two-sided GCD monoid and for all elements $m,n \in M$ the minimal left and right ideals generated by $m$ and $n$ coincide.

## Related concepts

* [[ideal of a monoid]]

* [[principal ideal of a monoid]]

* [[principal ideal monoid]]


GCD ring

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## Definition ##

Given a [[commutative monoid]] $(M, \cdot, 1)$, we say that [[element]] $a \in M$ *divides* $b \in M$ ($a \vert b$) if there exists an element $c \in M$ such that $a \cdot c = b$ and $c \cdot a = b$. If the commutative monoid has an [[absorbing element]] $0$, then for all $a \in M$, $a \vert 0$.

\begin{definition}
A [[commutative ring]] $R$ is a **GCD ring** if for every [[element]] $a \in R$ and $b \in R$, there is an element $c \in R$ such that $c \vert a$ and $c \vert b$, and for every other element $d \in R$ such that $d \vert a$ and $d \vert b$, $d \vert c$.
\end{definition}

\begin{definition}
A [[commutative ring]] $R$ is a **GCD ring** if there is a function $\gcd:R \times R \to R$ such that for every [[element]] $a \in R$ and $b \in R$, $\gcd(a, b) \vert a$ and $\gcd(a, b) \vert b$, and for every other function $f:R \times R \to R$ such that $f(a, b) \vert a$ and $f(a, b) \vert b$, $f(a, b) \vert \gcd(a, b)$.
\end{definition}

\begin{definition}
A [[commutative ring]] $R$ is a **GCD ring** if there are functions $\gcd:R \times R \to R$, $q_0:R \times R \to R$, and $q_1:R \times R \to R$ such that for every [[element]] $a \in R$ and $b \in R$, $q_0(a, b) \cdot \gcd(a, b) = a$ and $q_1(a, b) \cdot \gcd(a, b) = b$, and for every other triple of functions $f:R \times R \to R$, $r_0:R \times R \to R$, $r_1:R \times R \to R$ such that $r_0(a, b) \cdot f(a, b) = a$ and $r_1(a, b) \cdot f(a, b) = b$, there is a function $s:R \times R \to R$ such that $s(a, b) \cdot f(a, b) = \gcd(a, b)$.
\end{definition}

\begin{definition}
A [[commutative ring]] $R$ is a **GCD ring** if there are functions $\gcd:R \times R \to R$, $q_0:R \times R \to R$, and $q_1:R \times R \to R$ such that for every [[element]] $a \in R$ and $b \in R$, $q_0(a, b) \cdot \gcd(a, b) = a$ and $q_1(a, b) \cdot \gcd(a, b) = b$, and $\gcd:R \times R \to R$ is a [[semilattice]] with unit element $0$.
\end{definition}

The last definition implies that GCD rings are [[algebraic theory|algebraic]].

## See also ##

* [[commutative ring]]

* [[Bézout ring]]

* [[unique factorization ring]]

* [[GCD domain]]

* [[Euclidean ring]]

## References ##

* [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

[[!redirects GCD rings]]


gebra theory

Gebras denote algebras, coalgebras and their combinations. Common frameworks include properads.

[[Bourbaki]] complained that in Arabic 'al' is an article so classical 'bi-' for two is incompatible with it. Thus Bourbaki and much of the French school makes it a point to say 'cogebra, bigebra' instead of 'coalgebra, bialgebra'.

In modern [[algebra]], dealing with co-operations $A\to A^{\otimes n}$ instead of only with operations $A^{\otimes n}\to A$ for some monoidal product $\otimes$, is a specialty of a group of people who are often centered around the study of [[Hopf algebras]], but the subject is much wider including internal co(al)gebras, cogroups in cartesian categories, Tannaka reconstruction, [[PROP]]s (which have even operations taking $m$ arguments to $k$ arguments, hence generalizing both operations and co-operations), co-operads, bi(al)gebroids, Hopf algebroids, entwining structures, mixed [[distributive law]]s and so on. Common tools are graphical calculations (extending also to [[planar algebra]]s and alike), [[Sweedler notation]] and so on.

We may call this area the theory of __gebras__ as in the article

* [[Jean-Pierre Serre]], _Gèbres_ (in French; Engl. Gebras) Enseign. Math. (2) __39__ (1993), no. 1-2, 33--85, [MR1225256](http://www.ams.org/mathscinet-getitem?mr=1225256) (94h:16074)

A monograph into Hopf algebra generalizations i

* [[Gabriella Böhm]], _Hopf algebras and their generalizations from a category theoretical point of view_,  Lec. Notes Math. 2226, Springer  2018

The entries about or closely related to gebras and co-operations in $n$lab so far include [[bialgebra]], [[Hopf algebra]], [[comonoid]], [[bimonoid]], [[coalgebra]], [[comonad]], [[coring]], [[Sweedler coring]], [[cocategory]], [[coaction]], [[corepresentation]], [[coinvariant]], [[comodule algebra]], [[cotensor product]], [[induced comodule]], [[module algebra]], [[Hopf module]], [[crossed product algebra]], [[Hopf-Galois extension]], [[measuring]], [[cleft extension]], [[quantum group]], [[matrix Hopf algebra]], [[Hopf envelope]], [[bialgebroid]], [[Hopf algebroid]], [[distributive law]], [[convolution algebra]], [[Heisenberg double]], [[noncommutative thin scheme]], [[Lie bialgebra]], [[Tannaka duality]], [[quasi-Hopf algebra]],
[[Takeuchi product]], [[cooperad]]...

category: algebra
[[!redirects gebra]]
[[!redirects gebras]]


Gelfand duality

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## Idea

_Gelfand duality_ is a [[Isbell duality|duality between spaces and their algebras of functions]] for the case of [[compact space|compact]] [[topological space]]s and [[commutative C-star-algebra|commutative]] [[C-star algebras]]:

> Every (nonunital) commutative $C^\ast$-algebra $A$ is equivalent to the $C^\ast$-algebra of [[continuous functions]] on the [[topological space]] called its _[[Gelfand spectrum]]_ $sp(A)$.

This theorem is the basis for regarding non-commutative $C^\ast$-algebras as formal duals to spaces in [[noncommutative geometry]].

## Definitions

The statement of Gelfand duality involves the following [[categories]] and [[functors]].

+-- {: .num_defn}
###### Definition

Write

* $C^\ast Alg$ for the category of unital [[C-star algebras]];

* $C^\ast Alg_{nu}$ for the category of nonunital $C^\ast$-algebras;

* $C^\ast Alg_{nu,nd}$ for the sub category of nonunital $C^\ast$-algebras
with morphisms being nondegenerate [[$\ast$-homomorphisms]],
i.e., the two-sided ideal generated by the image is dense in the codomain.

* $C^\ast Alg_{com}$, $C^\ast Alg_{com,nu}$, $C^\ast Alg_{com,nu,nd}$
for the full subcategories of the above three categories consisting
of commutative C\*-algebras.

And

* [[Top]]${}_{Haus}$ for the category of [[Hausdorff topological spaces]]

* $Top_{cpt}$ for the [[full subcategory]] of [[Top]]${}_{Haus}$ of the [[compact topological spaces]];

* $*/Top_{cpt}$ for the category of [[pointed topological space|pointed topological]] [[compact Hausdorff spaces]], i.e. the [[pointed objects]] in $Top_{cpt}$;

* $Top_{lcpt,proper}$ for the category of Hausdorff and [[locally compact topological spaces]] with morphisms being the [[proper maps]] of topological spaces.

* $Top_{lcpt,inf}$ for the category of Hausdorff and [[locally compact topological spaces]] with morphisms being the [[continuous maps]] that vanish at infinity.

=--

The duality itself is exhibited by the following [[functors]]:

+-- {: .num_defn #FunctorsOfFunctionAlgebras}
###### Definition

Write
$$
  C \;\colon\; Top_{cpt} \to C^\ast Alg_{com}^{op}
$$
for the functor which sends a [[compact topological space]] $X$ to the algebra of [[continuous function]]s $C(X) = \{f : X \to \mathbb{C} | f \; continuous\}$, equipped with the structure of a $C^\ast$-algebra in the evident way (...).

Write
$$
  C_0 \;\colon\; */Top_{cpt} \to C^\ast Alg_{com,nu}^{op}
$$
for the functor that sends a [[pointed topological space|pointed topological]] [[compact Hausdorff space]] $(X,x_0)$ to the algebra of [[continuous function]]s $f : X \to \mathbb{C}$ for which $f(x_0) = 0$.

=--

+-- {: .num_defn}
###### Definition

Write
$$
  sp : C^\ast Alg_{com}^{op} \to Top_{cpt}
$$
for the [[Gelfand spectrum]] functor: it sends a commutative $C^\ast$-algebra $A$ to the set of _[[character]]s_ -- non-vanishing $C^\ast$-algebra [[homomorphisms]] $x : A \to \mathbb{C}$ -- equipped with the [[spectral topology]].

Similarly write
$$
  sp : C^\ast Alg_{com,nu}^{op} \to Top_{lcpt}
  \,.
$$

=--

## Statement

+-- {: .num_theorem #GelfandDualityTheorem}
###### Theorem
**(Unital Gelfand duality theorem)**

The pair of functors
$$
  C^\ast Alg_{com}^{op} \stackrel{\overset{C}{\leftarrow}}{\underset{sp}{\to}} Top_{cpt}
$$
is an [[equivalence of categories]].

=--

Here $C^\ast Alg^{op}_{\cdots}$ denotes the [[opposite category]] of $C^\ast Alg_{\cdots}$.

+-- {: .num_cor}
###### Corollary

On non-unital $C^\ast$-algebras the above induces an [[equivalence of categories]]
$$
  C^\ast Alg_{com,nu}^{op} \stackrel{\overset{C_0}{\leftarrow}}{\underset{sp}{\to}} */Top_{cpt}
  \,.
$$

=--

+-- {: .proof}
###### Proof

The operation of [[unitalization]] $(-)^+$ constitutes an [[equivalence of categories]]
$$
  C^\ast Alg_{nu}
    \stackrel{\overset{ker}{\leftarrow}}{\underset{(-)^+}{\to}}
  C^\ast Alg / \mathbb{C}
$$
between non-unital $C^\ast$-algebras and the [[over-category]] of $C^\ast$-algebras over the [[complex number]]s $\mathbb{C}$.

Composed with the equivalence of theorem \ref{GelfandDualityTheorem} this yields
$$
  C^\ast Alg_{com,nu}^{op}
    \underoverset{\simeq}{(-)^+}{\to}
  (C^\ast Alg_{com}/\mathbb{C})^{op}
    \underoverset{\simeq}{C}{\to}
  * / Top_{cpt}
  \,.
$$

The weak inverse of this is the composite functor
$$
  C_0 : */Top_{cpt}
    \underoverset{\simeq}{sp}{\to}
  (C^\ast Alg_{com}/\mathbb{C})^{op}
    \underoverset{\simeq}{ker}{\to}
  C^\ast Alg_{com,nu}^{op}
$$
which sends $(* \stackrel{x_0}{\to} X)$ to
$ker(C(X) \stackrel{ev_{x_0}}{\to} \mathbb{C})$, hence to
$\{f \in C(X) | f(x_0) = 0\}$. This is indeed $C_0$ from
def. \ref{FunctorsOfFunctionAlgebras}.

=--

+-- {: .num_remark #ForLocallyCompactTopologicalSpaces}
###### Remark

Since [[locally compact Hausdorff spaces are equivalently open subspaces of compact Hausdorff spaces]], via the construction that sends a locally compact Hausdorff space $X$ to its [[one-point compactification]], and since a [[continuous function]] on the compact Hausdorff space $X^\ast$ which vanishes at the extra point is equivalently a continuous function on $X$ which [[vanishing at infinity|vanishes at infinity]], the above induces a contravariant equivalence
$$Top_{lcpt,inf} \leftrightarrows C^\ast Alg_{com,nu}$$
between the category of locally compact Hausdorff spaces and continuous maps vanishing at infinity
and the category of commutative nonunital C\*-algebras.

The duality also works with [[real numbers]] instead of [[complex numbers]] ([Johnstone 82, chapter IV](#Johnstone82))

For an overview of other generalizations see also [this MO discussion](http://mathoverflow.net/a/82960/381).

=--

If one uses nondegenerate morphisms of C\*-algebras instead, the duality works for [[locally compact topological spaces]] and [[proper maps]]. See for instance ([Brandenburg 07](#Brandenburg07)).

\begin{theorem}
The is an equivalence of categories
$$Top_{lcpt,proper} \leftrightarrows C^\ast Alg_{com,nu,nd}$$
between the category of [[locally compact]] [[Hausdorff spaces]] and [[proper maps]]
and the category of commutative nonunital [[C*-algebras]] with nondegenerate \*-homomorphisms as morphisms.
\end{theorem}

## Generalizations

### In constructive mathematics

Gelfand duality makes sense in [[constructive mathematics]] hence [[internalization|internal]] to any [[topos]]: see [[constructive Gelfand duality theorem]].

### By horizontal categorification

Gelfand duality can be extended by [[horizontal categorification]] to define the notion of [[spaceoid|spaceoids]] as formal duals of commutative $C^*$-[[C-star-category|categories]].

## Related concepts

* [[Isbell duality]]

  * [[Gelfand-Kolmogorov theorem]]

  * **Gelfand duality**, [[constructive Gelfand duality theorem|constructive Gelfand duality]]

    * [[C-star algebra]], [[topological space]]

    * [[Gelfand-Naimark theorem]]

    * [[Gelfand spectrum]]

    * [[Serre-Swan theorem]]

    * [[noncommutative topology]]

* [[Stone duality]]

The analogous statement in [[differential geometry]]:

* [[embedding of smooth manifolds into formal duals of R-algebras]]

[[!include Isbell duality - table]]

## References

The original reference is

* [[Israel Gelfand]], _Normierte Ringe_, Recueil Mathématique 9(51):1 (1941), 3–24.

Formulation of Gelfand duality in terms of [[category theory]] ([[adjoint functors]], [[monads]] ("triples") and [[adjoint equivalences]]) originates with

* {#Negropontis71} Joan W. Negrepontis, _Duality in analysis from the point of view of triples_, Journal of Algebra, 19 (2): 228–253, (1971) (doi, ISSN 0021-8693, MR 0280571)

Quick exposition is in

* Ivo Dell'Ambrogio, _Categories of $C^\ast$-algebras_ ([pdf](http://www.math.uni-bielefeld.de/~ambrogio/exercise_C_algebras.pdf))

Textbook accounts include

* {#Johnstone82} [[Peter Johnstone]], section IV.4 of _[[Stone Spaces]]_, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press 1982. xxi+370 pp. MR85f:54002, reprinted 1986.

* [[Nicolaas Landsman|N. P. Landsman]], _Mathematical topics between classical and quantum mechanics_, Springer Monographs in Mathematics 1998. xx+529 pp. [MR2000g:81081](http://www.ams.org/mathscinet-getitem?mr=1662141) [doi](http://dx.doi.org/10.1007/978-1-4612-1680-3)

* [[Gerald B. Folland]], *A course in abstract harmonic analysis*, Studies in Advanced Mathematics. CRC Press (1995) [[pdf](https://sv.20file.org/up1/1415_0.pdf), [gBooks](http://books.google.com/books?hl=en&lr=&id=0VwYZI1DypUC)]

Careful discussion of the duality for the more general case of [[locally compact topological spaces]] includes

* {#Brandenburg07} [[Martin Brandenburg]], _Gelfand-Dualität ohne 1_, 2007 ([web](http://www.matheplanet.com/matheplanet/nuke/html/article.php?sid=1111))

{#ReferencesCenterOfAction} Discussion of Gelfand duality as a [[fixed point equivalence of an adjunction]] between includes

* {#PorstTholen91} [[Hans-E. Porst]], [[Walter Tholen]], section 4-c of _Concrete Dualities_ in H. Herrlich, [[Hans-E. Porst]] (eds.) _Category Theory at Work_, Heldermann Verlag 1991 ([pdf](http://www.heldermann.de/R&E/RAE18/ctw07.pdf))

following

* {#DubucPorta71} [[Eduardo Dubuc]], [[Horacio Porta]], _Convenient categories of topological algebras_, Bull. Amer. Math. Soc., Volume 77, Number 6 (1971), 975-979 [euclid](https://projecteuclid.org/euclid.bams/1183533170)

Some other generalized contexts for Gelfand duality:

* [[Hans Porst]], Manfred B. Wischnewsky, _Every topological category is convenient for Gelfand duality_, Manuscripta mathematica __25__:2, (1978) pp 169-204

* [[Chris Heunen]], [[Klaas Landsman]], [[Bas Spitters]], Sander Wolters, _The Gelfand spectrum of a noncommutative $C^\ast$-algebra_, J. Aust. Math. Soc. __90__ (2011), 39–52 [doi](http://dx.doi.org/10.1017/S1446788711001157) [pdf](http://www.math.ru.nl/~landsman/LandsmanCarey60.pdf)

*  [[Christopher J. Mulvey]], _A generalisation of Gelfand duality_, J. Algebra 56, n. 2, (1979) 499–505 doi

* {#Parzygnat17} [[Arthur Parzygnat]], _Discrete probabilistic and algebraic dynamics: a stochastic commutative Gelfand-Naimark Theorem_ ([arXiv:1708.00091](https://arxiv.org/abs/1708.00091))

category: analysis, geometry, noncommutative geometry

[[!redirects Gelfand duality]]
[[!redirects Gel'fand duality]]

[[!redirects Gelfand-Naimark duality]]

[[!redirects Gelfand duality theorem]]


Gelfand spectrum

+-- {: .rightHandSide}
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### Context
#### Operator algebra
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The _Gelfand spectrum_ (originally Гельфанд) of a [[commutative C*-algebra]] $A$ is a [[topological space]] $X$ such that $A$ is [[generalized the|the]] algebra of complex-valued [[continuous functions]] on $X$. (This "[[Gelfand duality]]" is a special case of the general [[Isbell duality|duality between spaces and their algebras of functions]].)

## Definition

+-- {: .num_defn}
###### Definition

Given a unital, not necessarily [[commutative C*-algebra|commutative]], complex [[C*-algebra]] $A$, the set of its _[[characters]]_, that is: continuous nonzero linear [[homomorphisms]] into the [[field]] of [[complex numbers]], is canonically equipped with what is called the __spectral topology__ which is [[compactum|compact Hausdorff]]. (If applied to a nonunital $C^*$-algebra, then it is only [[locally compact space|locally compact]].) This correspondence extends to a [[functor]], called the __Gel'fand spectrum__  from the [[category]] [[C*Alg]] of unital $C^*$-algebras to the category of [[Hausdorff topological spaces]].

=--

+-- {: .num_remark}
###### Remark

A [[character]] on a unital [[Banach algebra]] is automatically a [[continuous function]] (with [[Lipschitz constant]] 1).

=--

+-- {: .num_remark}
###### Remark

The Gelfand spectrum functor is a [[full and faithful functor]] when restricted to the subcategory of commutative unital $C^*$-algebras.

=--

+-- {: .num_remark}
###### Remark

The [[kernel]] of a character is clearly a codimension-$1$ closed subspace, and in particular a closed [[maximal ideal]] in $A$; therefore the Gel'fand spectrum is a topologised analogue of the [[maximal spectrum]] of a discrete algebra.

=--

+-- {: .num_remark}
###### Remark

For noncommutative $C^*$-algebras the spaces of equivalence classes of irreducible representations (i.e., the spectrum) and their kernels (i.e., the primitive ideal space) are more important than the character space.

=--

+-- {: .num_remark}
###### Remark

The Gelfand spectrum is also useful in the context of more general commutative Banach algebras.

=--

## Related concepts

* [[spectrum (geometry)]]

* [[Isbell duality]]

  * [[Gelfand duality]]

    * [[C-star algebra]], [[topological space]]

    * **Gelfand spectrum**

    * [[constructive Gelfand duality theorem]]

    * [[spectral presheaf]], [[Bohr topos]]

    * [[Alfsen-Shultz theorem]]

## References

* [[Israel Gelfand]], _Normierte Ringe_, Recueil Mathématique 9(51):1 (1941), 3–24.

category: analysis

[[!redirects Gelfand spectrum]]
[[!redirects Gelfand spectrums]]
[[!redirects Gelfand spectra]]
[[!redirects Gel'fand spectrum]]
[[!redirects Gel'fand spectrums]]
[[!redirects Gel'fand spectra]]
[[!redirects Gel'fand spectrum]]
[[!redirects Gel'fand spectrums]]
[[!redirects Gel'fand spectra]]
[[!redirects Гельфанд spectrum]]
[[!redirects Гельфанд spectrums]]
[[!redirects Гельфанд spectra]]

[[!redirects spectral topology]]
[[!redirects spectral topologies]]


Gelfand triple

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#Contents#
* table of contents
{:toc}

## Idea

A _Gel'fand triple_ is a structure that equips a [[Hilbert space]] with a dense [[topological vector space|topological vector subspace]]  of good "test" functions, so that the dual of the subspace of test functions enhaces the Hilbert space by embedding it into a larger [[TVS]] whose elements can be considered as generalized [[eigenvector]]s for the continuous [[spectrum of an operator|spectrum]] of [[normal operator|normal]] (possibly [[unbounded operator|unbounded]]) [[linear operator]]s.

## Definition

A __Gel'fand triple__ is datum of the form

$$
B\stackrel{J}\hookrightarrow H \stackrel{K}\hookrightarrow B^*
$$

where $H$ is a [[separable Hilbert space]], $B$ is a [[Banach space]] (or more general [[topological vector space]] (TVS)), $B^*$ is a dual TVS of $B$, $J:B\hookrightarrow H$ is an injective [[bounded operator]] with dense image, and $K$ is the composition of the canonical isomorphism $H\cong H^*$ determined by the inner product (i.e. given by Riesz theorem) and of the Banach transpose (dual) $J^*: H^*\to B^*$ of the operator $J$. The fact that $J(B)$ is dense in $H$ implies that $J^*$ (hence also $K$)  is injective as well.

## Basic examples

A typical example is $B = \mathcal{S}(\mathbb{R}^n)$ (Schwarz space), $H = L^2(\mathbb{R}^n)$ and $B^* = \mathcal{S}'(\mathbb{R}^n)$ (the space of tempered (Schwarz) distributions). One of the basic facts on Fourier transform is that this Gel'fand triple is preserved by the Fourier transform.

Another natural example is $\mathcal{l}^1\hookrightarrow \mathcal{l}^2\hookrightarrow \mathcal{l}^\infty$.

#### A long example

Let $H$ be the Hilbert space $L^2(\mathbb{R}, d x)$ consisting of square integrable functions $f$ with respect to [[Lebesgue measure]]. There is an unbounded self-adjoint operator

$$m_x: H \to H: f \mapsto x \cdot f$$

where $(x \cdot f)(y) := y f(y)$. This operator is not defined on all of $H$, but it is defined on a dense subspace of $H$. For example, if $S$ is the Schwartz space consisting of smooth functions $f$ on $\mathbb{R}$ all of whose derivatives $f^{(n)}(x)$ decay rapidly at infinity (more rapidly than any negative power of $|x|$), then there is a dense inclusion map $i: S \to H$, and $m_x$ is defined globally on $S$.

Meanwhile the Schwartz space $S$ carries its own topology (as described in the article [[distribution]]), stronger than the topology it inherits from $H$, and the space of tempered distributions $S^*$ is defined to be the continuous dual of the [[topological vector space|TVS]] $S$. Since the continuous inclusion $i: S \to H$ is dense, it follows that any continuous functional

$$f: S \to \mathbb{C}$$

has at most one extension to a continuous functional $H \to \mathbb{C}$. In other words, the adjoint map

$$i^*: H^* \to S^*$$

is injective. In addition, the topology on $S$ is such that the operator $m_x: S \to S$ is continuous.

In this example, there is a dense inclusion $S \to S^*$ defined by the inner product pairing, and the operator $m_x$ extends uniquely to an operator $S^* \to S^*$, called $m_x$ by abuse of notation. Again, in this example, the operator $m_x: S^* \to S^*$ has an eigenvector $s_{\xi}$ for each $\xi \in \mathbb{R}$:

$$m_x(s_{\xi}) = \xi s_{\xi}$$

(under construction)

## Isomorphisms

An isomorphism of Gelfand triples $(B_1,H_1,B^*_1)\to (B_2,H_2,B^*_2)$ is a unitary isomorphism $H_1\to H_2$ which restricts to an isomorphism of Banach spaces $B_1\to B_2$, and which extends to a weak$*$- and norm-preserving continuous isomorphism $B_1^*\to B_2^*$.

## Terminology and references

#### Terminological variants and history

Usually, $B$ and $B^*$ are Banach spaces, when we say *Banach Gel'fand triple*, there are some other variants involving more general [[topological vector space]]s. In some cases one also uses the terminology *rigged Hilbert space*, following articles by Roberts and others since mid 1960-s. Nuclear Gel'fand triples are very common and then the notion is well behaved. In Russian literature the term *enriched* Hilbert space is used (оснащенное гильбертово пространство), sometimes translated also as *equipped* Hilbert space. The enriched word here is the same as in the phrase [[enriched category]].

#### Early references

Gel'fand triples were introduced by Gel'fand school about 1955 and quickly incorporated into the theory of generalized functions.

* [[I. M. Gelʹfand, A. G. Kostyučenko, _Expansion in eigenfunctions of differential and other operators) (Russian), Dokl. Akad. Nauk SSSR (N.S.) __103__ (1955), 349–352, [MR73136](http://www.ams.org/mathscinet-getitem?mr=73136)

* [[I. M. Gel'fand]], N. Ja. Vilenkin, _Generalized functions_, vol. 4. Some applications of harmonic analysis. Equipped Hilbert spaces, Fizmatgiz, Moscow, 1961 [MR146653](http://www.ams.org/mathscinet-getitem?mr=146653), English transl. Acad. Press 1964 [MR173945](http://www.ams.org/mathscinet-getitem?mr=173945)

Related early works include

* Ciprian Foiaş, _Décompositions intégrales des familles spectrales et semi-spectrales en opérateurs qui sortent de l'espace hilbertien_, Acta Sci. Math. Szeged __20__, 1959, 117–-155. [MR115092](http://www.ams.org/mathscinet-getitem?mr=115092)

John Roberts continued the study in the context of [[quantum mechanics]] (in his thesis work suggested by [[Paul Dirac]]), changing the name to rigged.

* [[John Roberts]], _The Dirac bra and ket formalism_, J. Mathematical Phys. 7 (1966), 1097--1104, [MR216836](http://www.ams.org/mathscinet-getitem?mr=216836), _Rigged Hilbert spaces in quantum mechanics_ , Comm. Math. Physics __3__, n. 2 (1966), 98--119, [MR228243](http://www.ams.org/mathscinet-getitem?mr=228243), [euclid](http://projecteuclid.org/euclid.cmp/1103839388)

#### Modern references and links

* [[rigged Hilbert space]]

* wikipedia [rigged Hilbert space](http://en.wikipedia.org/wiki/Rigged_Hilbert_space)

* Р. А. Минлос, [Оснащенное гильбертово пространство](http://dic.academic.ru/dic.nsf/enc_mathematics/3760/%D0%9E%D0%A1%D0%9D%D0%90%D0%A9%D0%95%D0%9D%D0%9D%D0%9E%D0%95), online article from (Soviet) Matem. enc.

* MathOverflow question: [good-references-for-rigged-hilbert-spaces](http://mathoverflow.net/questions/43313/good-references-for-rigged-hilbert-spaces)

* Feichtinger, _A Banach Gelfand triple framework for
regularization and approximation_, [slides pdf](http://www.univie.ac.at/nuhag-php/dateien/talks/1086_eucetifeislides.pdf)

A unification of various inequivalent approaches is claimed in

* M. Gadella, F. Gómez, _A unified mathematical formalism for the Dirac
formulation of quantum mechanics_  Foundations of Physics __32__, No. 6, (2002)

* S. Wickramasekara, A. Bohm, _Symmetry representations in the rigged Hilbert space formulation of quantum mechanics_, [math-ph/0302018](http://arxiv.org/abs/math-ph/0302018)

[[!redirects Gelfand triple]]
[[!redirects Gel'fand triple]]
[[!redirects enriched Hilbert space]]
[[!redirects equipped Hilbert space]]


Gelfand-Fuks cohomology

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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

__Gel'fand-Fuks__ cohomology is the [[cochain cohomology]] of continuous alternating chains on the topological algebra of smooth [[vector fields]] on [[smooth manifolds]], where the topology on the algebra is given by [[uniform convergence]] of all (higher) [[partial derivative]] on [[compact subsets]] (sometimes called $C^\infty$-topology on that algebra).

## References

* [[I. M. Gel'fand]], D. B. Fuks, _The cohomology of the Lie algebra of vector fields on a smooth manifold_, J. Funct. Analysis __33__, 1969, 194--210, II, J. Funct. Anal. __4__ (1970) 110-116; _The cohomology of the Lie algebra of formal vector fields_, Izv. AN SSR 34 (1970), 110-116
* Shigeyuki Morita, _Geometry of characteristic classes_, Transl. Math. Monographs 199, AMS 2001

* [[Jean Celeyrette]], *Catégories internes et fibrations* & *Cohomologie de Gel'fand-Fuks*, PhD thesis, Paris (1975) [[pdf](/nlab/files/celeyrette-thesis-1975.pdf)]

[[!redirects Gel'fand-Fuks cohomology]]
[[!redirects Gelfand-Fuchs cohomology]]


Gelfand-Kolmogorov theorem

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### Context
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

Consider the [[functor]]

$$
  C(-,\mathbb{R})
  \;\colon\;
  Top \longrightarrow \mathbb{R}Alg^{op}
$$

from the [[category]] [[Top]] of [[topological spaces]] to the [[opposite category]] of [[associative algebras]] over the [[real numbers]] given by forming [[algebras of functions|algebras of]] [[continuous functions]] with values in the [[real numbers]].

If $X,Y \in Top_{cptHaus} \hookrightarrow Top$ are two [[compact Hausdorff spaces]], then every [[morphism]] in $\mathbb{R}Alg$ ([[algebra homomorphism]]) of the form $C(Y,\mathbb{R}) \overset{\simeq}{\to} C(X,\mathbb{R})$ arises as pre-composition with a [[morphism]] in [[Top]] ([[homeomorphism]]) of the form $f \;\colon\; X \overset{\simeq}{\to} Y$.

In other words, the [[restriction]]

$$
  C(-,\mathbb{R})
  \;\colon\;
  Top_{cptHaus} \overset{\phantom{AAAA}}{\hookrightarrow} \mathbb{R}Alg^{op}
$$

is a [[fully faithful functor]].

It follows in particular that [[compact Hausdorff spaces]] $X$ and $Y$ are [[homeomorphism|homeomorphic]] precisely if the algebras $C(X,\mathbb{R})$ and $C(Y,\mathbb{R})$ are isomorphic.

In the literature, it is often this corollary alone which is referred to as the _Gelfand-Kolmogorov theorem_, but its proof by Gelfand-Kolmogorov, using a lemma by Stone, actually shows the full faithfulness of $C(-,\mathbb{R})$.

## Related statements

[[!include Isbell duality - table]]

## References

Original reference:

* [[И. М. Гельфанд]], [[А. Н. Колмогоров]], _О кольцах непрерывных функций на топологических пространствах_, Доклады Академии наук СССР 22:1 (1939), 11–15.

Reprinted in:

* [[А. Н. Колмогоров]], _Избранные труды.  Математика и механика_, Наука, 1985, 264–269.

English translation:

* [[I. M. Gel'fand]], [[A. N. Kolmogorov]], _On rings of continuous functions on topological spaces_, In: Selected Works of A. N. Kolmogorov.  Volume I.  Mathematics and Mechanics.  Edited by V. M. Tikhomirov, translated from the Russian by V. M. Volosov.  Kluwer, 1991.

See also

* {#GarridoJaramillo02} M. Garrido, J. Jaramillo , theorem 18 in _Variations on the Banach-Stone theorem_, 2002 ([web](http://eprints.ucm.es/21378/))


Gelfand-Mazur theorem

The **Gel'fand--Mazur theorem** states: the only complex [[Banach algebra]] which is also a [[field]] is the algebra of [[complex number]]s $\mathbb{C}$.

The proof is a very simple consequence of the [[spectrum (geometry)|spectral]] theory of elements in a unital complex Banach algebra. It is a basic result of spectral theory that the spectrum $sp(A)$ of any element $a$ in a Banach algebra $A$ (which is by definition the set of complex numbers $\lambda$ such that $a- \lambda 1$ is not invertible) is a **nonempty compact subset** of $\mathbb{C}$. Now if the algebra is a field (or even a [[skewfield]]) then the only noninvertible element is $0$, hence every point in the spectrum of an arbitrary element $a\in A$ provides $\lambda$ such that $\lambda 1 = a$. Therefore the algebra can be identified with a unital complex subalgebra of $\mathbb{C}$, hence it is $\mathbb{C}$.

[[!redirects Gel'fand-Mazur theorem]]
[[!redirects Gel'fand-Mazur theorem]]
[[!redirects Gelfand–Mazur theorem]]
[[!redirects Gel'fand–Mazur theorem]]
[[!redirects Gel'fand–Mazur theorem]]
[[!redirects Gelfand--Mazur theorem]]
[[!redirects Gel'fand--Mazur theorem]]
[[!redirects Gel'fand--Mazur theorem]]


Gelfand-Naimark-Segal construction

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#Contents#
* table of contents
{:toc}

## Idea

The _Gelfand-Naimark-Segal construction_ ("GNS construction") represents a [[state on a star-algebra]] over the [[complex numbers]] --- which *a priori* is defined purely algebraically as a non-degenerate positive linear function

$$
  \rho \;\colon\; \mathcal{A} \longrightarrow \mathbb{C}
  \,,
$$

--- by a [[vector]] $\psi \in \mathcal{H}$ in a [[complex numbers|complex]] [[Hilbert space]] $\mathcal{H}$ as the "[[expectation value]]"

$$
  \begin{aligned}
    \rho(A)
    & = \langle \psi \vert \, A \, \vert \psi \rangle
    \\
    & \coloneqq \langle \psi, \pi(A) \psi \rangle
  \end{aligned}
$$

with respect to some [[star-representation]]

$$
  \pi \;\colon\; \mathcal{A} \longrightarrow End(\mathcal{H})
$$

of $\mathcal{A}$ on (a [[dense subspace]] of) $\mathcal{H}$; where $\langle -,-\rangle$ denotes the [[Hermitian form|Hermitian]] [[inner product]] on the [[Hilbert space]].

Originally and typically by default this is considered for [[C*-algebras]] and [[C*-representations]] ([Gelfand & Naimark 1943](#GelfandNaimark43), [Segal 1947](#Segal47)), see for instance ([Schmüdgen 1090](#Schmuedgen90)), but the construction applies to general [[unital algebra|unital]] [[star algebras]] $\mathcal{A}$ ([Khavkine& Moretti 2015](#KhavkineMoretti15)) as well as to other [[coefficient]] [[rings]], such as to  [[formal power series algebras]] over $\mathbb{C}[ [\hbar] ]$ ([Bordemann & Waldmann 1996](#BordemannWaldmann96)).

The GNS-construction plays a central role in [[algebraic quantum field theory]] (cf. [Haag 1996](#Haag96), [Moretti 2017](#Moretti17), [Khavkine & Moretti 2015](#KhavkineMoretti15)), where $\mathcal{A}$ plays the role of an [[algebra of observables]] and $\rho \colon \mathcal{A} \to \mathbb{C}$ the role of an actual [[state]] of a [[physical system]] (whence the terminology) jointly constituting the "[[Heisenberg picture]]"-perspective of [[quantum physics]]; so that the GNS-construction serves to re-construct a corresponding [[Hilbert space|Hilbert]] [[space of states]] as in the [[Schrödinger picture]] of quantum physics. In this context the version for [[C*-algebras]] corresponds to [[non-perturbative quantum field theory]], while the generalization to [[formal power series algebras]] corresponds to [[perturbative quantum field theory]].

## Details

> under construction

### For $C^\ast$-algebras
  {#ForCStarAlgebras}

+-- {: .num_theorem}
###### Theorem

Given

1. a [[C*-algebra]] $\mathcal{A}$;

1. a [[state on a star-algebra|state]] $\rho \;\colon\; \mathcal{A} \to \mathbb{C}$ in the sense of

   1. a [[linear map]] on the [[underlying]] [[vector space]]

      \[
        \label{UnderlyingLinearMapOfStateExpectationValue}
        \rho \,\colon\, \mathcal{A} \to \mathbb{C}
        \,,
      \]

   1. which takes star-involution to [[complex conjugation]]

      \[
        \label{StateRespectsStar}
        A \in \mathcal{A}
        \;\;\;
           \vdash
        \;\;\;
        \rho(A^\ast) = \overline{\rho(A)}
        \,,
      \]

   1. and satisfies "[positivity](state+on+a+star-algebra#PositivityCondition) in the sense that

      \[
        \label{StateSatisfiesPositivity}
        A \in \mathcal{A}
        \;\;\;
          \vdash
        \;\;\;
        \rho(A^\ast A) \in \mathbb{R}_{\geq 0} \subset \mathbb{C}
       \,,
      \]

there exists

1. a [[C*-representation]]

   $$
     \pi
     \;\colon\;
     \mathcal{A}
      \longrightarrow
      End(\mathcal{H})
   $$

   of $\mathcal{A}$ on some [[Hilbert space]] $\mathcal{H}$

1. a [[cyclic vector]] $\psi_\rho \in \mathcal{H}$

such that $\rho$ is the [[pure state]] corresponding to $\psi_\rho$, in that

$$
  \begin{aligned}
    \rho(A)
      & = \langle \psi_\rho \vert\, A \, \vert \psi_\rho \rangle
      \\
      & \coloneqq
      \langle \psi_\rho , \pi(A) \psi_\rho \rangle
  \end{aligned}
$$

for all $A \in \mathcal{A}$.

=--

\begin{proof}\label{ProofOfGNS}
Consider on the underlying [[complex vector space]] of $\mathcal{A}$ the [[sesquilinear form]] (inner product)

\[
  \label{InnerProductBeforeDevidingOutNullVectors}
  \langle A,B \rangle_\rho
  \;\coloneqq\;
  \rho
  \big(
    A^\ast B
  \big)
  \,.
\]

By the "[positivity](state+on+a+star-algebra#PositivityCondition)"-condition (eq:StateSatisfiesPositivity) on $\rho$, the pairing $\langle-,-\rangle$ is [positive semi-definite](sesquilinear+form#Definity), and therefore it satisfies the [[Cauchy-Schwarz inequality]]:

\[
  \label{CauchySchwarzBeforeDividingOutNullVectors}
  {\big\vert
    \rho\big( A^\ast B \big)
  \big\vert}^2
    \;\leq\;
  \rho\big(A^\ast A\big)
  \,
  \rho(B^\ast B)
  \,.
\]

However, (eq:InnerProductBeforeDevidingOutNullVectors) will in general not be definite, in that there may be a non-trivial [[linear subspace]] of 0-norm elements:

\[
  \label{TheNullIdeal}
  \mathcal{N}
    \;\coloneqq\;
  \big\{
    A \in \mathcal{A}
    \,\vert\,
    \rho(A^\ast A) = 0
  \big\}
  \,,
\]

and hence we have to quotient out by this null-space in order to produce the desired Hilbert space.

Observe that this null space (eq:TheNullIdeal) is not just a [[linear subspace]] but a left [[ideal]] for the algebra structure, as follows by the [[Cauchy-Schwarz inequality]] (eq:CauchySchwarzBeforeDividingOutNullVectors):

\[
  \label{NullSpaceIsLeftIdeal}
  \left.
  \begin{array}{l}
    N \in \mathcal{N}
    \\
    A \in \mathcal{A}
  \end{array}
  \right\}
  \;\;\;
  \vdash
  \;\;\;
  \left\{
  \begin{array}{l}
    \big\vert
    \rho\big(
      (A N)^\ast
      A N
    \big)
    \big\vert^2
    \\
    \;=\;
    \big\vert
    \rho\big(
      (N^\ast A^\ast A) N
    \big)
    \big\vert^2
    \\
    \;\leq\;
    \rho\big(
      (N^\ast A^\ast A)
      (N^\ast A^\ast A)^\ast
    \big)
    \,
    \underset{
      = 0
    }{
    \underbrace{
    \rho\big(
      N^\ast N
    \big)
    \,.
    }
    }
  \end{array}
  \right.
\]

Similarly, the [[Cauchy-Schwarz inequality]] (eq:CauchySchwarzBeforeDividingOutNullVectors) implies that any inner product with a null vector vanishes:

\[
  \label{InnerProductWithNullVectorIsNullVector}
  \left.
  \begin{array}{l}
    N \in \mathcal{N}
    \\
    A \in \mathcal{A}
  \end{array}
  \right\}
  \;\;
  \Rightarrow
  \;\;
  \rho\big(
    A^\ast N
  \big)
  \;=\;
  0
  \,.
\]

Therefore $\langle-,-\rangle$ first of all descends to the [[quotient vector space]]

\[
  \label{HilbertSpaceAsQuotientOfStarAlgebraByNullSpace}
  \mathscr{H}
  \;\coloneqq\;
  \mathcal{A}/\mathcal{N}
  \,,
\]

because

$$
  \begin{array}{l}
    \big\langle
      A_1
      ,\,
      A_2 + N_2
    \big\rangle
    \\
    \;\equiv\;
    \rho\big(
      A_1^\ast
      (A_2 + N_2)
    \big)
    \\
    \;=\;
    \rho\big(
      A_1^\ast
      A_2
    \big)
    +
    \rho\big(
      A_1^\ast
      N_2
    \big)
    \\
    \;=\;
    \rho\big(
      A_1^\ast
      A_2
    \big)
    \,,
  \end{array}
$$

where in the last step we used (eq:InnerProductWithNullVectorIsNullVector), and from this:

$$
  \begin{array}{l}
    \big\langle
      A_1 + N_1
      ,\,
      A_2
    \big\rangle
    \\
    \;=\;
    \overline{
    \big\langle
      A_2
      ,\,
      A_1 + N_1
    \big\rangle
    }
    \\
    \;=\;
    \overline{
    \big\langle
      A_2
      ,\,
      A_1
    \big\rangle
    }
    \;=\;
    \big\langle
      A_1
      ,\,
      A_2
    \big\rangle
    \,,
  \end{array}
$$

where in the first and last step we used (eq:StateRespectsStar).

Now descended to the quotient, the pairing $\langle-,-\rangle$ becomes [positive definite](sesquilinear+form#Definity) by construction, so that it defines a [[Hermitian form|Hermitian]] -- by (eq:StateRespectsStar) -- [[inner product]] on $\mathscr{H}$, making it the desired [[Hilbert space]].

Finally, again by the left-ideal property (eq:NullSpaceIsLeftIdeal), the left multiplication [[action]] of $\mathcal{A}$ on itself also descends to an action on the quotient Hilbert space (eq:HilbertSpaceAsQuotientOfStarAlgebraByNullSpace):

$$
  \array{
    \mathcal{A} \otimes \mathcal{H}
    &
    \overset{
      \;\;\;\;\;
      \pi
      \;\;\;\;\;
    }{
      \longrightarrow
    }
    &
    \mathcal{H}
    \\
    (A, [\psi]) &\mapsto& [A \cdot \psi]
    \,,
  }
$$

and so the cyclic vector in question is that represented by the [[unit element]] $1 \in \mathcal{A}$:

$$
  \psi_\rho
    \;\coloneqq\;
  [1]
  \;\in\;
  \mathscr{H}
  \,.
$$

Hence on this Hilbert space $\mathscr{H}$, the original  operator-algebraic state $\rho$ is now represented by the tautological [[density matrix]]

$$
  \left\vert
    \psi_\rho
  \right\rangle
  \left\langle
    \psi_\rho
  \right\rangle
  \;=\;
  {\big\vert [1] \big\rangle}
  {\big\langle [1] \big\vert}
  \,.
$$
\end{proof}

Notice in summary that this GNS construction constitutes a kind of [[operator-state correspondence]] modulo null operators:

$$
  \begin{array}{rcl}
    \mathcal{A}/\mathcal{N}
    &\xleftrightarrow{\phantom{---}}&
    \mathscr{H}
    \\
    [A]
      &\mapsto&
    A \cdot \psi_\rho
  \end{array}
$$

### For $C^\ast$-categories
 {#ForCStarCategories}

The GNS construction for $C^\ast$-algebras is a special case of a more general construction of [Ghez, Lina & Roberts 1985, Prop. 1.9](#GhezLinaRoberts85) applied to [[C*-categories]] ([[horizontal categorification]] of $C^\ast$-algebras):

+--{: .num_theorem}
###### Theorem

Let $\mathcal{C}$ be a [[C-star category|$C^\ast$-category]]. Fix an object $A \in \operatorname{Ob}\mathcal{C}$ and let $\sigma$ be a state on the $C^\ast$-algebra $\mathcal{C}(A,A)$. Then there exists a $*$-representation
$$
\rho_\sigma \colon \mathcal{C} \to \mathbf{Hilb}
$$
together with a cyclic vector $\xi \in \rho_\sigma(A)$ such that for all $x \in \mathcal{C}(A,A)$,
$$
\sigma(x) = \langle \xi, \rho_\sigma(x)\xi \rangle.
$$
=--

A [[C*-algebra]] $\mathcal{A}$ is a $C^\ast$-category with a single [[object]] $\bullet$, where we make the identification $A = \mathcal{A}(\bullet,\bullet)$. In this case the theorem reduces to the classical GNS construction.

## Functorial Aspects
See *[[Functorial Aspects of the GNS Representation]]*.

## Related concepts

* [[Riesz representation theorem]]

[[!include states and observables -- content]]

## References

The original construction for [[C*-algebras]] and [[C*-representations]] is due to:

* {#GelfandNaimark43} [[Israel Gelfand]], [[Mark Naimark]], *On the imbedding of normed rings into the ring of operators on a Hilbert space*,  Matematicheskii Sbornik. 12 (2): 197–217 (1943)

  reprinted in:

  Robert Doran (ed.), *$C^\ast$-Algebras: 1943–1993*, Contemporary Mathematics **167**, AMS 1994 ([doi:10.1090/conm/167](http://dx.doi.org/10.1090/conm/167))

* {#Segal47} [[Irving Segal]], _Irreducible representations of operator algebras_, Bull. Am. Math. Soc. 53: 73–88, (1947)  ([pdf](http://www.ams.org/journals/bull/1947-53-02/S0002-9904-1947-08742-5/S0002-9904-1947-08742-5.pdf), [euclid](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-53/issue-2/Irreducible-representations-of-operator-algebras/bams/1183510397.full))

Textbook accounts:

* [[Gerard Murphy]], Section 3.4 of: _$C^\ast$-algebras and Operator Theory_, Academic Press 1990 ([doi:10.1016/C2009-0-22289-6](https://doi.org/10.1016/C2009-0-22289-6))

* {#Schmuedgen90} [[Konrad Schmüdgen]], Section 8.3 of: _Unbounded  operator  algebras  and  representation  theory_, Operator  theory, advances and applications, vol. 37. Birkhäuser, Basel (1990) ([doi:10.1007/978-3-0348-7469-4](https://link.springer.com/book/10.1007/978-3-0348-7469-4))

* [[Kehe Zhu]], Section 14 of: *An Introduction to Operator Algebras*, CRC Press 1993 ([ISBN:9780849378751](https://www.routledge.com/An-Introduction-to-Operator-Algebras/Zhu/p/book/9780849378751))

* [[Richard V. Kadison]], [[John R. Ringrose]], Theorem 4.5.2 in: *Fundamentals of the theory of operator algebras -- Volume I: Elementary Theory*, Graduate Studies in Mathematics **15**, AMS 1997 ([ISBN:978-0-8218-0819-1](https://bookstore.ams.org/gsm-15), [ZMATH] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0888.46039&format=complete))

in the context of [[algebraic quantum field theory]]:

* {#Haag96} [[Rudolf Haag]], *[[Local  Quantum  Physics -- Fields,  Particles,  Algebras]]*,  Texts  and  Monographs  in Physics. Springer (1996).

* {#Moretti17} [[Valter Moretti]], _Spectral Theory and Quantum Mechanics -- Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation_, Springer (2017) [[doi:10.1007/978-3-319-70706-8](https://doi.org/10.1007/978-3-319-70706-8)]

Review with an eye towards [[quantum probability]] and [[entropy]]:

* A. P. Balachandran, T. R. Govindarajan, Amilcar R. de Queiroz, A. F. Reyes-Lega, Section II of: _Algebraic approach to entanglement and entropy_, Phys. Rev. A 88, 022301 (2013) ([arXiv:1301.1300](http://arxiv.org/abs/1301.1300))

See also

* Wikipedia, _[Gelfand-Naimark-Segal construction](https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction)_

In the generality of [[C-star category|$C^\ast$-categories]] (and with an eye towards [[W-star category|$W^\ast$-categories]]):

* {#GhezLinaRoberts85} [[P. Ghez]], [[Ricardo Lima]], [[John E. Roberts]], Prop. 1.9 in: *$W^\ast$-categories*,  Pacific J. Math. **120** 1 (1985) 79-109 [[euclid:pjm/1102703884](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-120/issue-1/Wast-categories/pjm/1102703884.full)]

For general unital [[star-algebras]]:

* {#KhavkineMoretti15}  [[Igor Khavkine]], [[Valter Moretti]], *Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction*, Chapter 5 in: [[Romeo Brunetti]] et al. (eds.) *Advances in Algebraic Quantum Field Theory* Springer (2015) [[doi:10.1007/978-3-319-21353-8](https://doi.org/10.1007/978-3-319-21353-8)]

and in relation with the classical moment problem and the notion of [[POVM]]:

* {#DragoMoretti19} Nicolò Drago,  [[Valter Moretti]], *The notion of observable and the moment problem for $\ast$-algebras and their GNS representations*, Lett. Math. Phys. 2020 ([arXiv.org:1903.07496](https://arxiv.org/abs/1903.07496))

For [[formal power series algebras]] over $\mathbb{C}[ [ \hbar ] ]$:

* {#BordemannWaldmann96} [[Martin Bordemann]], [[Stefan Waldmann]], _Formal GNS Construction and States in Deformation Quantization_, Commun. Math. Phys. (1998) 195: 549.  ([arXiv:q-alg/9607019](https://arxiv.org/abs/q-alg/9607019), [doi:10.1007/s002200050402](https://doi.org/10.1007/s002200050402))

Discussion in terms of [[universal properties]] in ([[higher category theory|higher]]) [[category theory]] is in

* {#Jacobs10} [[Bart Jacobs]], _Involutive Categories and Monoids, with a GNS-correspondence_, Foundations of Physics, July 2012, Volume 42, Issue 7, pp 874–895 ([arXiv:1003.4552](https://arxiv.org/abs/1003.4552))

* {#Parzygnat16} [[Arthur Parzygnat]], _From observables and states to Hilbert space and back: a 2-categorical adjunction_ ([arXiv:1609.08975](https://arxiv.org/abs/1609.08975))

category: operator algebras

[[!redirects Gelfand-Naimark-Segal construction]]
[[!redirects Gelfand–Naimark–Segal construction]]
[[!redirects Gelfand--Naimark--Segal construction]]
[[!redirects GNS construction]]
[[!redirects GNS-construction]]
[[!redirects GNS representation]]
[[!redirects GNS-representation]]
[[!redirects GNS theorem]]


Gelfand-Neumark theorem

[[!redirects Gelfand-Naimark theorem]]

+-- {: .rightHandSide}
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### Context
#### Operator algebra
+-- {: .hide}
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=--
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#Contents#
* table of contents
{:toc}

## Idea

The _Gelfand–Neumark theorem_ (alternative spelling transliterated
from the Russian: Gel'fand–Naĭmark; Гельфанд–Наймарк) says that every [[C*-algebra]] is [[isomorphism|isomorphic]] to a $C^\ast$-algebra of [[bounded linear operators]] on a [[Hilbert space]].

## Related concepts

* [[Gelfand spectrum]]

* [[Gelfand duality]]

## References

* [[Israel Gelfand]], [[Mark Neumark]],
_On the imbedding of normed rings into the ring of operators in Hilbert space_,
Recueil Mathématique 12(54):2 (1943), 197–217.

* Wikipedia, _[Gelfand-Naimark theorem](https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark_theorem)_

[[!redirects Gelfand-Neumark theorem]]
[[!redirects Gelfand-Neumark theorems]]
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[[!redirects Gel'fand–Naimark theorems]]
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[[!redirects Gel'fand–Naimark theorems]]
[[!redirects Gel'fand--Naimark theorem]]
[[!redirects Gel'fand--Naimark theorems]]


Gelfand-Raikov theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

__Gel'fand-Raikov theorem__ (Гельфанд-Райков) The [[irreducible representation|irreducible]] [[unitary representation]]s of a [[locally compact topological group]] $G$ separates its points.

In other words, for any two group elements $g,h\in G$ there exist an irreducible unitary representation $\rho : G\to U(H)$ such that $\rho(g)\neq \rho(h)$.

## Related concepts

* [[state on a star-algebra]]

## References

The characterization of [[state on a star-algebra|states]] on [[group algebras]] and what came to be known as the [[Gelfand-Raikov theorem]]:

* {#GelfandRaikov43} И. М. Гельфанд, Д. А. Райков, *Неприводимые унитарные представления локально бикомпактных групп*, Матем. сб., 13(55):2–3 (1943)  301–316 [[mathnet pdf](https://www.mathnet.ru/links/728d630fdf2220b1ea8bdb2a5f6a5488/sm6181.pdf)]

  \linebreak

  [[Israel Gelfand]], [[Dmitri Raikov]], *Irreducible unitary representations of locally bicompact groups*, Recueil Mathématique. N.S., 13(55) 2–3 (1943) 301–316 [[mathnet:eng/sm6181](https://www.mathnet.ru/eng/sm6181)]

See also

* Wikipedia, *[Gelfand-Raikov theorem](https://en.wikipedia.org/wiki/Gelfand%E2%80%93Raikov_theorem)*

* [[Gerald B. Folland]], *A course in abstract harmonic analysis*, Studies in Advanced Mathematics. CRC Press (1995) [[pdf](https://sv.20file.org/up1/1415_0.pdf), [gBooks](http://books.google.com/books?hl=en&lr=&id=0VwYZI1DypUC)]


Gelfand-Tsetlin algebra > history

see *[[Gelfand-Tsetlin basis]]*


Gelfand-Tsetlin basis

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
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#### Algebra
+--{: .hide}
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=--
#### Representation theory
+-- {: .hide}
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#Contents#
* table of contents
{:toc}

## Idea

Gelʹfand-Cetlin or Gelfand-Tsetlin bases -- an early example of canonical bases on the [[linear representations]] of [[finite groups]] and [[Lie groups]].

## Related concepts

* [[seminormal representation]]

* [[Jucys-Murphy element]]

* [[state on a star-algebra]]

## References

Original articles:

* [[Israel Gelfand]], M. L. Cetlin, _Finite-dimensional representations of the group of unimodular matrices_, (Russian) Doklady Akad. Nauk SSSR (N.S.) __71__, (1950) 825–828; _Finite-dimensional representations of groups of orthogonal matrices_, Doklady Akad. Nauk SSSR (N.S.) __71__, (1950) 1017–1020.

Review:

* [[Anatoly Vershik]], *Gel'fand-Tsetlin algebras, expectations, inverse limits, Fourier analysis*, pp. 619 in: [[Pavel Etingof]], [[Vladimir S. Retakh]], [[Isadore Singer]] (eds.) *The Unity of Mathematics* In Honor of the Ninetieth Birthday of [[I. M. Gelfand]], Birkhäuser 2006 ([arXiv:math/0503140](https://arxiv.org/abs/math/0503140), [ISBN:978-0-8176-4467-3](https://www.springer.com/gp/book/9780817640767))

* David Grabovsky, *The Gelfand-Tsetlin Basis*, 2018 ([pdf](http://www.math.columbia.edu/~ums/Finite%20Group%20Rep%20Theory2.pdf), [[GrabovskyGelfandTsetlinBasis.pdf:file]])

See also:

* [[Victor Guillemin]], [[Shlomo Sternberg]], _The Gel'fand-Cetlin system and quantization of the complex flag manifolds_, Journal of Functional Analysis 52 (1): 106–128, 1983 doi [MR705993](http://www.ams.org/mathscinet-getitem?mr=705993)

* D. P. Želobenko, _Compact Lie groups and their representations_

* A. I. Molev, _Gelfand-Tsetlin bases for classical Lie algebras_, [arxiv/0211289](http://arxiv.org/abs/math/0211289)

* Terry Tao, [Gelfand obituary](http://terrytao.wordpress.com/tag/gelfand-tsetlin-patterns)

* Megumi Harada, _The symplectic geometry of the Gel'fand-Cetlin-Molev basis for representations of Sp(2n,C)_, published [pdf](http://intlpress.com/JSG/p/2006/4-1/JSG-4-1-001-041.pdf), [math.SG/0404485](http://arxiv.org/abs/math/0404485)

* Grigori Olshanski, _Projections of orbital measures, Gelfand-Tsetlin polytopes, and splines_, [arxiv/1302.7116](http://arxiv.org/abs/1302.7116)

* wikipedia [Gelfand-Tsetlin integrable system](http://en.wikipedia.org/wiki/Gelfand%E2%80%93Zeitlin_integrable_system)

* [[Bertram Kostant]], Nolan Wallach, _Gelfand-Zeitlin theory from the perspective of classical mechanics. I_, Studies in Lie theory, Progr. Math. __243__, Birkhäuser 2006, pp. 319–364

[[!redirects Gelfand-Tsetlin bases]]

[[!redirects Gel'fand-Tsetlin basis]]
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[[!redirects Gel'fand-Cetlin basis]]
[[!redirects Gel'fand-Cetlin bases]]

[[!redirects Gelfand-Tsetlin algebra]]
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[[!redirects Gel'fand-Tsetlin algebra]]
[[!redirects Gel'fand-Tsetlin algebras]]


Gelfond's theorem

If $a$ is an [[algebraic number]] different from $0$ or $1$, and $b$ [[irrational number|irrational]], then $a^b$ is transcendental (in particular irrational).

* [[A. O. Gelfond]], _Transcendental and algebraic numbers_, Dover Publications, New York 1960, translation from 1st Russian edition by Leo F. Boron.

[[!redirects Gelfond theorem]]


Gell-Mann principle

+-- {: .rightHandSide}
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#### Philosophy
+-- {: .hide}
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=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

While the inherently [[probability|probabilistic]] nature of [[quantum physics]] means, roughly, that as soon as there is more than one possibility, no [[quantum measurement]] outcome can be known with certainty; conversely this also means that all outcomes are possible with some probability. This converse statement had been stated by [Gell-Mann 1956](#GellMann56) for the case of [[quantum hadrodynamics]] (p. 859) in this form:

> "Principle of Compulsory Strong Interactions". Among baryons, antibaryons, and mesons, any process which is not forbidden by a conservation law actually does take place with appreciable probability.

The statement is recalled by [Bunge 1976, p. 31](#Bunge76) in the form:

> Anything that is not forbidden is compulsory.

> A more careful wording of the above principle is this: All possible repetitive chance events (in particular all those consistent with the conservation laws) are likely to occur in the long run.

and by [Israel 1996](#Israel96) in the form:

> What is not forbidden is compulsory.

For better or worse, [Gell-Mann 1956](#GellMann56) on that same page 859 found it helpful to add (about the [[contrapositive]] statement) that:

> this is related to the state of affairs that is said to prevail in a perfect totalitarian state. Anything that is not compulsory is forbidden.

and some historians of science swallowed this red herring and ever since refer to Gell-Mann's "principle of quantum compulsion"  instead as the "totalitarian principle" (eg. [Jaeger 2017](#Jaeger17), [Wikipedia](https://en.wikipedia.org/wiki/Totalitarian_principle), cf. [Kragh 2019a](#Kragh19a)). See also the "principle of plenitude" (cf. [Kragh 2019b](#Kragh19b)).

Notice that Gell-Mann's principle goes against the grain of the *classical* implication (see [[modal logic]]), which is the implication in the opposite direction:

$$
  \array{
    necessary && actual && possible
    \\
    \Box (-) &\rightarrow& (-) &\rightarrow& \lozenge (-)
  }
$$

That in [[quantum physics]] this implication may in fact be reversed can be understood as [[ambidextrous adjunction|ambidexterity]]
$$
  \lozenge \mathscr{H}_\bullet
  \;\simeq\;
  \Box \mathscr{H}_\bullet
$$

of finite-dimensional [[dependent linear types]] $\mathscr{H}_\bullet$, see at *[[quantum circuits via dependent linear types]]*.

## References

The original article:

* {#GellMann56} [[Murray Gell-Mann]], *The interpretation of the new particles as displaced charge multiplets*,  Nuovo Cim **4** 2 (1956) 848–866 [[doi:10.1007/BF02748000](https://doi.org/10.1007/BF02748000)]

Further discussion:

* {#Bunge76} [[Mario Bunge]], p. 31 in: *Possibility and Probability*, in: *Foundations of Probability Theory, Statistical Interference, and Statistical Theories of Science*  Reidel (1976) 17-34 [[doi:10.1007/978-94-010-1438-0_2](https://doi.org/10.1007/978-94-010-1438-0_2)]

* {#Israel96} Werner Israel, p. 607 of: *Imploding stars, shifting continents, and the inconstancy of matter*, Foundations of Physics **26** (1996) 595–616 [[doi:10.1007/BF02058234](https://doi.org/10.1007/BF02058234)]

* {#Jaeger17} Gregg Jaeger, p. 5 of: *Quantum randomness and unpredictability*, Fortschr. Phys. **65** 6-8 (2017) [[doi:10.1002/prop.201600053](https://doi.org/10.1002/prop.201600053)]

* {#Kragh19a} Helge Kragh, *Physics and the Totalitarian Principle* [[arXiv:1907.04623](https://arxiv.org/abs/1907.04623)]

* {#Kragh19b} Helge Kragh, *Plenitude Philosophy and Chemical Elements*, International Journal for Philosophy of Chemistry [**25** 1](http://hyle.org/journal/issues/25-1/) (2019) [[pdf](http://hyle.org/journal/issues/25-1/kragh.pdf)]

[[!redirects totalitarian principle]]

[[!redirects principle of plenitude]]
[[!redirects plenitude principle]]

[[!redirects quantum compulsion]]
[[!redirects principle of quantum compulsion]]
[[!redirects Gell-Mann principle of quantum compulsion]]


Gell-Mann-Low renormalization cocycle > history

> see [[renormalization group flow]]


genera and partition functions - table

**[[partition functions]] in [[quantum field theory]] as [[index|indices]]/[[genera]]/[[orientation in generalized cohomology|orientations]] in [[generalized cohomology theory]]**:

| $d$ | [[partition function]] in $d$-dimensional QFT | [[supercharge]]  | [[index]] in [[cohomology theory]] | [[genus]] | logarithmic coefficients of [[Hirzebruch series]] |  |
|--|--|---|---|--|--|--|
| 0 |    |  |  push-forward in [[ordinary cohomology]]: [[integration of differential forms]]  |  |  | [[orientation]] |
| 1 | [[spinning particle]] | [[Dirac operator]] | [[KO]][[K-theory|-theory]] [[index]] | [[A-hat genus]]  | [[Bernoulli numbers]] | [[Atiyah-Bott-Shapiro orientation]] $M Spin \to KO$  |
|   | endpoint of [[2d Poisson-Chern-Simons theory]] string | [[Spin^c Dirac operator]] twisted by [[prequantum line bundle]] | [[space of quantum states]] of boundary [[phase space]]/[[Poisson manifold]] | [[Todd genus]] |  [[Bernoulli numbers]] | [[Atiyah-Bott-Shapiro orientation]] $M Spin^c \to KU$ |
|   | endpoint of [[type II superstring]] | [[Spin^c Dirac operator]] twisted by [[Chan-Paton gauge field]] |  [[D-brane charge]] |  [[Todd genus]] | [[Bernoulli numbers]] | [[Atiyah-Bott-Shapiro orientation]] $M Spin^c \to KU$ |
| 2 | [[type II superstring]] | [[Dirac-Ramond operator]] |  [[superstring]] [[partition function]] in NS-R sector | [[Ochanine elliptic genus]] |   | [[SO orientation of elliptic cohomology]]  |
|  | [[heterotic superstring]] | [[Dirac-Ramond operator]] |  [[superstring]] [[partition function]]  | [[Witten genus]] | [[Eisenstein series]]  | [[string orientation of tmf]] |
|  | [[self-dual string]]  |   |  [[M5-brane charge]] |   |   | |
| 3|                       |   |                      |   |   | [[w4-orientation of EO(2)-theory]] |