causally local net of observables

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###Context###
#### Algebraic Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]] (or [[classical field theory]]) a _causally local net of observables_ is a system (a [[co-presheaf]]) of [[algebras of observables]] assigned to regions of [[spacetime]],  such that this satisfies a few basic properties, such as notably _[[causal locality]]_, saying that observables whose spacetime [[support]] is [[spacelike]]-separated (graded-)commute with each other ([[Poisson bracket|Poisson]]-commute, in the case of classical field theory).

(Beware that the meaning of "net" here is vaguely similar to, but different from, the concept of _[[net]]_ (as in: generalized [[sequences]] of points) as used in [[topology]]. Better terminology might be "causally local system of spacetime-localized observables". But "local net" is traditional and has become standard.)

In the context of [[algebraic quantum field theory]] the structure of the local net of quantum observables is used as the very [[axiom|axiomatization]] of what a [[quantum field theory]] actually is ("[[Haag-Kastler axioms]]"). This may be thought of as a formalization of a spacetime-localized form of the [[Heisenberg picture]] of [[quantum physics]] (or rather the [[interaction picture]] in the case of [[perturbative AQFT]]); as opposed to the formalization of the [[Schrödinger picture]] in [[FQFT]], where instead the [[state]]-propagation is used as the basic axiom.

Traditionally local nets of observables are assumed to take values in [[C*-algebras]], but the basic form of the axioms does not actually refer to [[topology|topological]] structure on the algebras, and makes sense more generally.

In particular in [[perturbative quantum field theory]] made precise via [[causal perturbation theory]], the [[algebras of quantum observables]] are taken to be [[formal power series algebras]] (reflecting the [[infinitesimal]] nature of [[perturbation theory]]) and one _derives_ from [[causal additivity]] of the [[S-matrix]] that the perturbative [[quantum observables]] form a local net of formal power series algebras (see at _[S-matrix -- Causal locality and Quantum observables](S-matrix#CausalLocality)_). Accordingly, this infinitesimal/perturbative version of [[AQFT]] is called _[[perturbative AQFT]]_.

Other variants may be considered. For [[AQFT on curved spacetimes]] one generalizes from observables associated with regions of [[Minkowski spacetime]] to observables associated with more general [[globally hyperbolic spacetimes]]. Combining this with perturbation theory is then called _[[locally covariant perturbative AQFT]]_.

Moreover, if [[gauge theory]] with nontrivial global [[gauge field]] configurations is to be considered ([[instantons]]) then one may show that one needs to consider some kind of [[homotopy theory|homotopy theoretic]] local nets of _[[homotopical algebras]]_. See at _[[homotopical algebraic quantum field theory]]_.

## Definition

In the literature there is a certain variance and flexibility of what precisely the axioms on a local net of observables are, though the core aspects are always the same: it is a [[copresheaf]] of ([[C-star algebra|C-star]] [[associative algebra|algebra]] s) on pieces of [[spacetime]] such that algebras assigned to [[causal locality|causally disconnected]] regions commute inside the algebra assigned to any joint neighbourhood.

Historically this was first formulated for [[Minkowski spacetime]] only, where it is known as the [[Haag-Kastler axioms]]. Later it was pointed out ([BrunettiFredenhagen](#BrunettiFredenhagen)) that the axioms easily and usefully generalize to arbitrary [[spacetime]]s.

We give the modern general formulation first, and then comment on its restriction to special situations.

### Basic general definition

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

Write $LorSp$ for the [[category]] whose

* [[object]]s are [[spacetime]] manifolds; ([[Lorentzian manifold]]s equipped with a time-orientation);

* [[morphism]] are _causal_ [[isometry|isometric]] [[embedding]].

Here we say a morphism $f : X \hookrightarrow Y$ is a **causal embedding** if for every two points $x_1,x_2 \in X$ we have that $f(x_1)$ is in the [[future]] of $f(x_2)$ in $Y$ only if $x_1$ is in the future of $x_2$ in $X$.

=--

Write $Alg$ for a suitable [[category]] of [[associative algebra]]s. Usually this is taken to be the category of [[C-star algebra]]s or that of [[von Neumann algebra]]s. Write

$$
  Alg_{inc} \hookrightarrow Alg
$$

for the [[subcategory]] on the [[monomorphism]]s.

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

A **causally local net of observables** is a [[functor]]

$$
  \mathcal{A} : LorSp \to Alg_{inc} \to Alg
$$

such that

* ([[causal locality]]) whenever $X_1 \coprod X_2 \hookrightarrow X$ is a causal embedding, def. \ref{CatOfSpacetimeEmbeddings}, we have that $\mathcal{A}(X_1) \subset \mathcal{A}(X)$ commutes with $\mathcal{A}(X_2) \subset \mathcal{A}(X)$.

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

The [[local quantum field theory|locality]] [[axiom]] encodes the the physical property known as _[[Einstein-causality]]_ or _micro-causality_, which states that physical effects do not propagate faster that the speed of light.

=--

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

Many _auxiliary_ [[linear operator|operators]] in [[quantum field theory]] do _not_ satisfy causal locality: for instance operators associate to [[current]]s in [[gauge theory]]. The idea is that those operators that actually do qualify as _[[observables]]_ do satisfy the axiom, however, i.e. in particular those that are [[gauge symmetry|gauge invariant]].

=--

### Extra axioms

#### Einstein locality
 {#EinsteinLocalities}

Commutativity of spacelike separated observables can be argued to capture only part of [[causal locality]].

A natural stronger requirement is that [[spacelike]] separated regions of [[spacetime]] are literally [independent quantum subsystems](quantum%20mechanics#Subsystems) of any larger region. By the formalization of _independent subsystem_ in [[quantum mechanics]] this means the following:

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

A local net $\mathcal{A}$ satisfies **Einstein locality** if for every causal embedding $X_1 \coprod X_2 \to X$ the [subsystems](http://ncatlab.org/nlab/show/quantum%20mechanics#Subsystems)

$$
  \mathcal{A}(X_1) \hookrightarrow \mathcal{A}(X)
$$

and

$$
  \mathcal{A}(X_2) \hookrightarrow \mathcal{A}(X)
$$

are _independent_ in that the algebra $\mathcal{A}(X_1) \vee \mathcal{A}(X_2) \in \mathcal{A}(X)$ which they generate is [[isomorphism|isomorphic]] to the [[tensor product]] $\mathcal{A}(X_1) \otimes \mathcal{A}(X_2)$.

=--

This appears as ([BrunettiFredenhagen, 5.3.1, axiom 4](#BrunettiFredenhagen)).

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

A local net  is Einstein local precisely if it is a [[monoidal functor]]

$$
  \mathcal{A} : (LorSp, \coprod) \to (Alg, \otimes)
  \,.
$$

=--

This appears as ([BrunettiFredenhagen, 5.3.1, theorem 1](#BrunettiFredenhagen)).

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

Einstein locality implies [[causal locality]], but is stronger.

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

Other properties implied by Einstein locality are sometimes extracted as separate axioms. For instance the condition that for $X_1 \coprod X_2 \to X$ a causal embedding, we have

$$
  \mathcal{A}(X_1) \cap \mathcal{A}(X_2) = \mathbb{C}
  \,.
$$

=--

#### Strong locality
 {#StrongLocality}

In ([Nuiten 11](#Nuiten11)) the following variant of [[causal locality]] was considered and shown to be equivalent to a [[descent]] condition for the system of [[Bohr toposes]] associated with a local net of observables

+-- {: .num_defn #StrongLocality}
###### Definition
**([[strong causal locality]])**

A net of observables is **[[strong causal locality|strongly (causally) local]]** if it is microlocal (i.e. [[causal locality|causally local]]) in that algebras $A_1 = A(O_1)$ and $A_2 = A(O_2)$ associated with  spacelike separated regions commute with each other, and in addition for all commutative subalgebras $C_1 \subset A_1$ and $C_2 \subset A_2$ the algebra $C_1 \vee C_2 \subset A(O_1 \vee O_2)$ satisfies

1. $(C_1 \vee C_2) \cap A_1 = C_1$

1. $(C_1 \vee C_2) \cap A_2 = C_2$.

=--

This is ([Nuiten 11, def. 14](#Nuiten11)).

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

It is clear that Einstein locality implies strong locality, def. \ref{EinsteinLocality}

$$
   Einstein\;locality \;\;\Rightarrow  \;\; Strong\;locality
  \,.
$$

In fact strong locality is strictly weaker than Einstein locality in that there are strongly locally embedded subalgebras which are not Einstein locally embedded. More discussion of this is in ([Wolters 13, section 6.3.3](#Wolters13)).

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#### Time-slice axiom

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###### Definition

A local net is said to satisfy the **[[time slice axiom]]** if whenever

$$
  i : X_1 \to X_2
$$

is a causal embedding of [[globally hyperbolic]] [[spacetime]]s such that $X_1$ contains a [[Cauchy surface]] of $X_2$, then

$$
  \mathcal{A}(i) : \mathcal{A}(X_1) \stackrel{\simeq}{\to}
   \mathcal{A}(X_2)
$$

is an [[isomorphism]].

=--

#### Duality

See [[dual net of von Neumann algebras]]

#### Positive energy condition

(...)

#### Spectrum condition

(...)

### Special cases and variants

#### Minkowski nets / Vacuum representation

* [[Haag-Kastler vacuum representation]]

#### Conformal nets

The notion of local net in the context of [[conformal field theory]] is a [[conformal net]].

## Examples

see for instance ([Bär-Ginoux 11](#BarGinoux11))

## Related concepts

* [[quantum field theory]]

  * [[local quantum field theory]], [[local prequantum field theory]]

  * [[causal locality]], [[Einstein locality]]

  * [[AQFT]]

    * [[Haag-Kastler axioms]]

    * **local net of observables**

      * [[net of C-star systems]] ([[global gauge group]], [[actions]])

      * [[field net]]

      * [[time slice axiom]], [[split property]]

      * [[cohomology of local net of observables]]

      * [[factorization algebra of observables]], [[chiral algebra]], [[topological chiral homology]], [[blob complex]].

  * [[FQFT]]

## References
 {#References}

For historical references see at *[[AQFT]]*.

In particular:

Early suggestion of a notion of [[non-abelian cohomology]] of local nets of observables with [[coefficients]] in [[infinity-categories|$\infty$-categories]] (cf. history recalled in [Street (2010), p. 9-10](strict+omega-category#Street09)):

* {#Roberts79} [[John E. Roberts]], *Mathematical Aspects of Local Cohomology*, in: *Algèbres d’opérateurs et leurs applications en physique mathématique*, Colloques Internationaux du Centre National de la Recherche Scientifique (C.N.R.S) **274**, Paris (1979) 321–332 [ISBN:2-222-02441-2, [pdf](https://dmitripavlov.org/scans/roberts-mathematical-aspects-of-local-cohomology.pdf), [[Roberts-LocalCohomology.pdf:file]]]

### General

The axioms of local nets on general spacetimes were first articulated in

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Quantum Field Theory on Curved Backgrounds_ ([arXiv:0901.2063](http://arxiv.org/abs/0901.2063))
 {#BrunettiFredenhagen}

A comprehensive review, with plenty of background information, is in

* {#BaerFredenhagen} [[Christian Bär]], [[Klaus Fredenhagen]], (eds.) _Quantum field theory on curved spacetime_ , Lecture notes in physics, Springer (2009)

Discussion of Einstein locality of a net of observables equivalently as a [[descent]] condition on the system of [[Bohr toposes]] induced by the [[algebras of observables]] is in

* {#Nuiten11} [[Joost Nuiten]], _[[schreiber:bachelor thesis Nuiten|Bohrification of local nets of observables]]_, Proceedings of [QPL 2011](http://qpl.science.ru.nl/), [EPTCS 95, 2012](http://rvg.web.cse.unsw.edu.au/eptcs/content.cgi?QPL2011), pp. 211-218 ([arXiv:1109.1397](http://arxiv.org/abs/1109.1397))

A review of this with some further discussion is in section 6 of

* {#Wolters13} Sander Wolters, _Quantum toposophy_, PhD Thesis 2013 ([pdf](http://www.math.ru.nl/~landsman/Wolters.pdf))

Discussion of properly co-stacky nets of local observables in [[gauge theory]] is in

* {#BeniniSchenkelSzabo15} [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], _Homotopy colimits and global observables in Abelian gauge theory_ ([arXiv:1503.08839](http://arxiv.org/abs/1503.08839))

An [[operad]] for local nets of observables in [[AQFT]] is considered in

* {#BeniniSchenkelWoike17} [[Marco Benini]], [[Alexander Schenkel]], [[Lukas Woike]], _Operads for algebraic quantum field theory_ ([arXiv:1709.08657](https://arxiv.org/abs/1709.08657))

The [[Boardman-Vogt resolution]] of the [[operad]] for [[local nets of observables]] ([Benini-Schenkel-Woike 17](#BeniniSchenkelWoike17)), lifting it to [[homotopy AQFT]], is considered in

* {#Yau18} [[Donald Yau]], _Homotopical Quantum Field Theory_ ([arXiv:1802.08101](https://arxiv.org/abs/1802.08101))

### Examples

Discussion of standard example includes

* {#BarGinoux11} [[Christian Bär]], N. Ginoux, _Classical and quantum fields on lorentzian manifolds_, in _Global Differential Geometry_, Springer Proceedings in Math. 17, Springer-Verlag (2011), 359-400  ([arXiv:1104.1158](http://arxiv.org/abs/1104.1158))

### In perturbation theory

The observation that in [[perturbation theory]] the [[renormalization|Stückelberg-Bogoliubov-Epstein-Glaser]] local [[S-matrix|S-matrices]] yield a [[local net of observables]] was first made in

* V. Il'in, D. Slavnov, _Observable algebras in the S-matrix approach_ Theor. Math. Phys. **36** , 32 (1978) ([spire](http://inspirehep.net/record/135575), [doi](http://dx.doi.org/10.1007/BF01035870))

which was however mostly ignored and forgotten. It is taken up again in

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_  Commun.Math.Phys.208:623-661 (2000) ([arXiv](http://arxiv.org/abs/math-ph/9903028))

(a quick survey is in section 8, details are in section 2).

For more on this see at _[[S-matrix]]_ and at _[[pAQFT]]_.

[[!redirects causally local nets of observables]]

[[!redirects causally local net of quantum observables]]
[[!redirects causally local nets of quantum observables]]

[[!redirects local net]]
[[!redirects local nets]]

[[!redirects Haag-Kastler net]]

[[!redirects net of observables]]
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[[!redirects local nets of observables]]

[[!redirects local net of algebras of observables]]
[[!redirects local nets of algebras of observables]]

[[!redirects net of quantum observables]]
[[!redirects nets of quantum observables]]
[[!redirects local net of quantum observables]]
[[!redirects local nets of quantum observables]]

[[!redirects causal net]]
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[[!redirects causal net of algebras]]
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[[!redirects causally local net]]
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causet

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The concept of _causal set_ -- or *causet*, for short -- is a [[concept with an attitude]]: In itself it is just a [[partially ordered set]] (or *poset*, for short), but meant to be understood as a set of [[spacetime]] [[events]] subject to the relation of [[causality]].

As such, causal sets play a role in:

* **computer science**, where causal sets model [[concurrency|concurrent]] [[computations]]: there exists a morphism $a \to b$ in the poset if $a$ and $b$ are states of some machine and if operating the machine can take it from state $a$ to state $b$. A simple example would be type of computation which can be done in a single step $in \to out$, but which needs to be done on _two_ inputs of the same kind. The causal set modelling this situation is

  $$
    \left\lbrace
    \array{
       (in_1, in_2) &\to& (out_1, in_2)
       \\
       \downarrow && \downarrow
       \\
       (in_1, out_2) &\to& (out_1,out_2)
    }
    \right\rbrace
  $$

* **[[relativistic field theory]]**, where every [[globally hyperbolic Lorentzian manifold|globally hyperbolic Lorentzian]] [[structure]] equips the [[underlying]] [[set]] of points in a [[manifold]] naturally with the structure of a causal set: there is a morphism from the point $x$ to the point $y$ in the manifold precisely if $y$ is in the _future_ of $x$, i.e. precisely if there exists a smooth path from $x$ to $y$ whose tangent vector is everywhere non-spacelike with respect to the Lorentzian metric.
Moreover, the Lorentzian metric on a manifold can essentially  (need to dig out the details here, see discussion at [[smooth Lorentzian space]]) be reconstructed from this poset structure and from a _measure_. This has led to some attempts to use posets as a foundational concept for relativistic physics.

  For more see [[smooth Lorentzian space]]

The only technical distinction between the notion of posets and that of causal sets is that for a causal set the [[interval|under-over category]]
$x\darr X\darr y$ for all objects $x$ and $y$ in the poset
(the category of "two-step time evolution paths" from $y$ to $x$)
is required to be _finite_. This means these are required to have a [[finite set]] of objects (and hence necessarily, being a poset, a finite set of morphisms).

## Definition

A causal set, or **causet** is a [[partial order|poset]] $X$ such that for all objects $x,y$ the [[interval]] $[x,y] = \{z \mid x \le z \le y\}$ is finite.

## Related concepts

* [[causal structure]]

* [[causal order]]

* [[causal complement]], [[causal closure]]

* [[causal locality]], [[causal additivity]]

* [[causal perturbation theory]]

* [[time-ordered product]]

* [[S-matrix]]

## References

Discussion in [[relativistic field theory]]:

* Nomaan X, *Quantum Field Theory On Causal Sets*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2306.04800](https://arxiv.org/abs/2306.04800)]

* Stav Zalel, *Covariant Growth Dynamics*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2302.10582](https://arxiv.org/abs/2302.10582)]

See also:

* Wikipeida, *[Causal set](http://en.wikipedia.org/wiki/Causal_sets)*

* [[Tim Porter]], _[[PorterDipath.pdf:file]]_

[[!redirects causets]]
[[!redirects causal set]]
[[!redirects locally finite poset]]

[[!redirects causal sets]]


Cayley distance

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### Context
#### Group Theory
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[[!include group theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The *Cayley distance* between two [[permutations]] is the [[minimum]] [[natural number|number]] of [[transpositions]] of elements needed to turn one into the other.

(This is in contrast to the [[Kendall distance]] which counts the minimum number of [[transpositions]] of *adjacent* pairs of elements.)

## Definition

\begin{definition}\label{CayleyDistance}
**(Cayley distance)**
For $n \in \mathbb{N}$ consider the [[symmetric group]] $Sym(n)$ with its [[mathematical structure|structure]] as a [[finitely generated group]] exhibited by the [[finite set]] of [[generators]] given by all [[transposition permutations]]

$$
  Gen
  \;\coloneqq\;
  \big\{
    \sigma_{i,j}
    \,\vert\,
    1 \leq i, j \lt n
  \big\}
  \;\subset\;
  Sym(n)
  \,.
$$

Then the *Cayley distance* (e.g [Diaconis 88, p. 112](#Diaconis88)) is the [[distance]] on $Sym(n)$ which is the [[graph distance]] of the corresponding [[Cayley graph]], hence the [[function]]

$$
  d_C(-,-)
  \;\colon\;
   Sym(n) \times Sym(n)
 \longrightarrow
 \mathbb{N}
    \hookrightarrow
  \mathbb{R}
$$

which sends any [[pair]] of [[permutations]] $\sigma_1, \sigma_2 \in Sym(n)$ to the [[minimum]] [[number]] $d(\sigma_1, \sigma_2) \in \mathbb{N}$ of [[transpositions]] $\sigma_{i_1, j_1}, \sigma_{i_2, j_2}, \cdots, \sigma_{i_d, j_d}$ such that

$$
  \sigma_2
  \;=\;
  \sigma_{i_d, j_d}
  \circ
  \cdots
  \circ
  \sigma_{i_2, j_2}
  \circ
  \sigma_{i_1, j_1}
  \circ
  \sigma_1
  \,.
$$
\end{definition}

## Properties

\begin{prop}\label{CayleyMetricIsRightInvariant}
**(Cayley metric is right invariant)**
  The Cayley metric is an [[right invariant metric]] in that for all $\sigma \in Sym(n)$ we have

$$
  \sigma^\ast d_C(-,-)
  \;\coloneqq\;
  d_C
  \big(
    (-) \circ \sigma
    ,\,
    (-) \circ \sigma
  \big)
  \;=\;
  d_C
  \big(
    -
    ,\,
    -
  \big)
 \,.
$$
\end{prop}
\begin{proof}
  This is immediate from the definition.
\end{proof}

\begin{prop}\label{CayleyObservation}
**(Cayley's formula)**
  The Cayley distance (Def. \ref{CayleyDistance}) between $\sigma_1, \sigma_2 \,\in\, Sym(n)$ equals $n$ minus the [[natural number|number]] of [[permutation cycles]] in $\sigma_1 \circ \sigma_2^{-1}$:

$$
  d_C(\sigma_1, \sigma_2)
  \;=\;
  n
  -
  \left\vert
     Cycles
     \big(
       \sigma_1
         \circ
       \sigma_2^{-1}
     \big)
  \right\vert
  \,.
$$
\end{prop}
(e.g. [Diaconis 88, p. 118](#Diaconis88))
\begin{proof}
  By right invariance (Prop. \ref{CayleyMetricIsRightInvariant}) it is sufficient to see the statement for the case that $\sigma_2 = e$ is the [[neutral element]] of $Sym(n)$, hence the [[identity function|identity]] [[permutation]].

Here we need to see that for $\sigma = \sigma_1$ any permutation, the minimum number of transpositions whose product yields $\sigma$ equals $n$ minus the number of [[permutation cycle|cycles]] in $\sigma$.

Now observe that:

1. a [[cyclic permutation]] of $k$ elements is the composite of no less than $k-1$ transpositions:

   \[
     \label{CyclicPermutationAsProductOfTranspositions}
     \left(
       \array{
         1 & 2 & 3 & \cdots & k
         \\
         k & 1 & 2 & \cdots & k-1
       }
     \right)
     \;=\;
     \underset{
        k-1 \; factors
     }{
     \underbrace{
     \sigma_{k-1,k-2}
     \circ
     \cdots
     \circ \sigma_{3,2} \circ \sigma_{2,1} \circ \sigma_{1,k}
     }
     }
   \]

1. Any $\sigma$ is the product of elements of the form (eq:CyclicPermutationAsProductOfTranspositions), one for each of its [[permutation cycles]].

Since it is $k-1$ transposition factors in (eq:CyclicPermutationAsProductOfTranspositions) for cycles of $k$ elements, each cycle reduces the number of transposition needed by one.
\end{proof}

\begin{prop}\label{CayleyDistancePreservedByInclusionOfSymmetricGroups}
  Cayley distance is preserved under the canonical inclusions of [[symmetric groups]]

\[
  \label{CanonicalInclusionsOfSymmetricGroups}
  Sym(n)
  \overset{
    i
  }{
    \hookrightarrow
  }
  Sym(n+1)
  \hookrightarrow
  Sym(n+2)
  \hookrightarrow
  \cdots
\]

in that

$$
  d_C( \sigma_1, \sigma_2 )
    \; = \;
  d_C\big(
    i(\sigma_1), i(\sigma_2)
  \big)
  \,.
$$

In other words, when regarding the [[metric space]] given by the set of permutations in $Sym(n)$ with their Cayley distance function between them

* as an $(\mathbb{R}_{\geq 0}, \geq)$0-[[enriched category]] (see [here](metric+space#LawvereMetricSpace)), then the [[functors]] induced by the inclusions (eq:CanonicalInclusionsOfSymmetricGroups) are [[fully faithful functor|fully faithful]] and hence are [[full subcategory]] inclusions.

* as a [[matrix]], then the inclusions (eq:CanonicalInclusionsOfSymmetricGroups) correspond to  [[principal submatrices]].

\end{prop}
\begin{proof}\label{ProofOfCayleyDistancePreservedByInclusionOfSymmetricGroups}
  Observing that
  $$
    \left\vert
    Cycles
    \big(
      i(\sigma_1) \circ i(\sigma_2)^{-1}
    \big)
    \right\vert
    \;=\;
    \left\vert
    Cycles
    \big(
      \sigma_1 \circ \sigma_2^{-1}
    \big)
    \right\vert
    \;+\;
    1
  $$
  the claim follows by Cayley's formula (Prop. \ref{CayleyObservation}).
\end{proof}

## Examples

\begin{example}\label{CayleyDistanceOnSym3}
**([[Cayley distance on Sym(3)]])** \linebreak

The [[Cayley graph]] of the [[symmetric groups]] on 3 elements, $Sym(3)$, with [[edges]] corresponding to any [[transposition]] (not necessarily adjacent) looks like this:

\begin{tikzcd}[row sep=-6pt, column sep=14pt]
  123
  \ar[rrrr,-]
  \ar[ddddd,-]
  \ar[ddrrr,-]
  &&
  &&
  132
  \ar[ddddd,-]
  \\
  {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A}}}}}}}}
  \\
  &&&
  \scalebox{.8}{$321$}
  \ar[dddr,-]
  \\
  &
  \scalebox{1.2}{231}
  \ar[ddl,-]
  \ar[uuurrr,-, crossing over]
  \ar[urr,-]
  \\
  {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A}}}}}}}}
  \\
  213
  \ar[rrrr,-]
  &&&&
  312
\end{tikzcd}

from which one readily reads off the Cayley distance: Ordering the 6 [[permutations]] as

$$
  \vec \sigma
  \;\coloneqq\;
  \left[
  \array{
    123
    \\
    213
    \\
    132
    \\
    321
    \\
    312
    \\
    231
  }
  \right]
$$

the [[Cayley distance]] on $Sym(3)$, regarded as a $6 \times 6$ [[matrix]]

$$
  [d_C]
  \;\coloneqq\;
  \big(
  d_C
  \big(
    \sigma_i, \sigma_j
  \big)
  \big)_{i,j}
  \;\in\;
  Mat_{6 \times 6}(\mathbb{N})
$$

is this:

$$
  [d_C]
  \;=\;
  \left[
  \array{
    0 & 1 & 1 & 1  & 2 &  2
    \\
    1 & 0 & 2 & 2 & 1 & 1
    \\
    1 & 2 & 0 & 2 & 1 & 1
    \\
    1 & 2 & 2 & 0 & 1 & 1
    \\
    2 & 1 & 1 & 1 & 0 & 2
    \\
    2 & 1 & 1 & 1 & 2 & 0
  }
  \right]
$$

\end{example}

## Related concepts

* [[Cayley distance kernel]]

* [[Kendall distance]]

* [[Cayley graph]]

## References

Textbook accounts:

* {#Diaconis88} [[Persi Diaconis]], Chapter 6B of: *Group Representations  in Probability  and  Statistics*, IMS Lecture Notes Monogr. Ser., 11: 198pp. (1988) ([jstor:i397389](https://www.jstor.org/stable/i397389), [ISBN: 0940600145](https://projecteuclid.org/ebooks/institute-of-mathematical-statistics-lecture-notes-monograph-series/Group-representations-in-probability-and-statistics/toc/10.1214/lnms/1215467407), [pdf](https://jdc.math.uwo.ca/M9140a-2012-summer/Diaconis.pdf))

See also:

* Richard G.E. Pinch, *The distance of a permutation from a subgroup of $S_n$* ([arXiv:math/0511501](https://arxiv.org/abs/math/0511501))

* Thaynara Arielly de Lima; Mauricio Ayala-Rincón, *Complexity of Cayley distance and other general metrics on permutation groups* ([doi:10.1109/ColombianCC.2012.6398020](https://doi.org/10.1109/ColombianCC.2012.6398020), [pdf](https://www.mat.unb.br/ayala/shortw4v2.pdf))

Discussion of [[character of a linear representation|characters]] for the [[symmetric group]] that depend only on Cayley distance from the neutral element ("block character"):

* {#GnedinGorinKerov11} [[Alexander Gnedin]], [[Vadim Gorin]], [[Sergei Kerov]], *Block characters of the symmetric groups*, Journal of Algebraic Combinatorics, 38, no. 1 (2013), 79-101 ([arXiv:1108.5044](https://arxiv.org/abs/1108.5044), [doi:10.1007/s10801-012-0394-9](https://doi.org/10.1007/s10801-012-0394-9))

Discussion of the [[Cayley distance kernel]]:

* [[Michael A. Fligner]], [[Joseph S. Verducci]], Section 4 of: *Distance Based Ranking Models*, Journal of the Royal Statistical Society. Series B (Methodological) Vol. 48, No. 3 (1986), pp. 359-369 ([jstor:2345433](https://www.jstor.org/stable/2345433))

* [[Persi Diaconis]], [[Phil Hanlon]], Section 4 of: *Eigen Analysis for Some Examples of the Metropolis Algorithm*, in Donald Richards (ed.) *Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications*, Contemporary Mathematics Vol. 138, AMS 1992 ([doi:10.1090/conm/138](http://dx.doi.org/10.1090/conm/138), [EFS NSF 392](https://statistics.stanford.edu/research/eigen-analysis-some-examples-metropolis-algorithm), [pdf](https://statweb.stanford.edu/~cgates/PERSI/papers/eigen-metalg.pdf))

* [[Michael A. Fligner]], [[Joseph S. Verducci]] (eds.), p. xx of: *Probability Models and Statistical Analyses for Ranking Data*, Lecture Notes in Statistics **80**, Springer 1993 ([doi:10.1007/978-1-4612-2738-0](https://link.springer.com/book/10.1007/978-1-4612-2738-0))

* {#CSS21} [[David Corfield]], [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Fundamental weight systems are quantum states]]* ([arXiv:2105.02871](https://arxiv.org/abs/2105.02871))

[[!redirects Cayley distances]]