electric charge

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## Idea

In [[electromagnetism]] the [[electromagnetic field]] is modeled by a degree 2 [[differential cohomology|differential cocycle]] $\hat F \in H(X, \mathbb{Z}(2)_D^\infty)$ (see [[Deligne cohomology]]) with curvature characteristic 2-form $F \in \Omega^2(X)$.

With $\star$ denoting the [[Hodge star]] operator with respect to the corresponding pseudo-Riemannian metric on $X$, the right hand of

$$
  d \star F = j_{el} \in \Omega^3(X)
$$

is the [[conserved current]] called the **electric current** on $X$. Conversely, with $j_{el}$ prescribed this equation is one half of [[electromagnetic field|Maxwell's equations]] for $F$.

If $X$ is globally hyperbolic and $\Sigma \subset X$ is any spacelike hyperslice, then

$$
  Q_{el} := \int_\Sigma j_{el}
$$

is the [[charge]] of this current: the **electric charge**  encoded by this configuration of the [[electromagnetic field]].

Notice that due to the above equation $d j_{el} = 0$, so that $Q$ is independent of the choice of $\Sigma$. When unwrapped into separate space and time components, the expression $d j_{el} = 0$ may be expressed as

$$div j + \frac{\partial\rho}{\partial t} = 0$$

which is a statement of the physical phenomenon of _charge conservation_ .

## Remarks

* While electric current is modeled by just a differential form, magnetic charge has a more subtle model. See [[magnetic charge]] .

* The above has a straightforward generalization to higher abelian [[gauge field]]s such as the [[Kalb-Ramond field]] and the [[supergravity C-field]]: for a field modeled by a degree $n$ [[Deligne cohomology|Deligne cocycle]] $\hat F$ the electric current $j_{el}$ is the right hand of
  $$
    d \star F = f_{el} \in \Omega^{n+1}(X)
    \,.
  $$

## Related concepts

* [[Gauss law]]

* [[hypercharge]]

* [[conserved current]], [[charge]]

* **electric charge**, [[magnetic charge]]

* [[color charge]]

* [[flux]]

* [[dyon]]

* [[Hodge-Maxwell theorem]]

* [[higher electric background charge coupling]]

[[!redirects electric charges]]

[[!redirects electric current]]
[[!redirects electric currents]]


electric dipole moment

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## Idea

(...)

## Related concepts

* [[magnetic dipole moment]]

  * [[anomalous magnetic moment]]

## References

See also

* Wikipedia, _[Electric dipole moment](https://en.wikipedia.org/wiki/Electric_dipole_moment)_

Limits from [[experiment]] on the electric dipole moment of the [[electron]]:

* ACME Collaboration, _Improved limit on the electric dipole moment of the electron_, Nature Volume 562, pages355–360 (2018) ([doi:10.1038/s41586-018-0599-8](https://doi.org/10.1038/s41586-018-0599-8))

and of [[hadrons]]:

* Nodoka Yamanaka, _Hadronic EDM and New physics beyond standard model_ ([arXiv:1606.09587](https://arxiv.org/abs/1606.09587))

See also

* [[Andreas Crivellin]], Martin Hoferichter, Philipp Schmidt-Wellenburg, _Combined explanations of $(g-2)_{\mu,2}$ and implications for a large muon EDM_,  	Phys. Rev. D 98, 113002 (2018) ([arXiv:1807.11484](https://arxiv.org/abs/1807.11484))

[[!redirects electric dipole moments]]


electric eigenbrane

## Idea

A [[brane]] that is taken to a tensor-multiple of itself under the action of a  [[Wilson line operator]].

[[!include geometric Langlands QFT -- table]]

## References

* {#KapustinWitten06} [[Anton Kapustin]], [[Edward Witten]], section 8 of _Electric-Magnetic Duality And The Geometric Langlands Program_ , Communications in number theory and physics, Volume 1, Number 1, 1–236 (2007) ([arXiv:hep-th/0604151](http://arxiv.org/abs/hep-th/0604151))

[[!redirects electric eigenbranes]]


electric field

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## Idea

In [[physics]], the [[electromagnetic field]] may be split, over a choice of [[coordinate chart]] of [[spacetime]], into two components: the **electric field** and the [[magnetic field]].

## Related oncepts

* [[field (physics)]]

* [[electric charge]]

* [[electric dipole moment]]

[[!redirects electric fields]]


electric-magnetic duality

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## Idea

Electric-magnetic duality is a lift of [[Hodge star operator|Hodge duality]] from [[de Rham cohomology]] to [[ordinary differential cohomology]].

## Description

Consider a [[circle n-bundle with connection]] $\nabla$ on a space $X$. Its [[higher parallel transport]] is the [[action functional]] for the [[sigma-model]] of  $(n-1)$-dimensional objects ($(n-1)$-branes) propagating in $X$.

For $n = 1$ this is the coupling of the [[electromagnetic field]] to particles. For $n = 2$ this is the coupling of the [[Kalb-Ramond field]] to strings.

The [[curvature]] $F_\nabla \in \Omega^{n+1}(X)$ is a closed $(n+1)$-form. The condition that its image $\star F_\nabla$ under the [[Hodge star operator]]  is itself closed

$$
  d_{dR} \star F_\nabla = 0
$$

is the [[Euler-Lagrange equation]] for the standard (abelian [[Yang-Mills theory]]-[[action functional]] on the space of circle n-bundle with connection.

If this is the case, it makes sense to ask if $\star F_\nabla$ itself is the curvature $(d-(n+1))$-form of a circle $(d-(n+1)-1)$-bundle with connection $\tilde \nabla$, where $d = dim X$ is the [[dimension]] of $X$.

If such $\tilde \nabla$ exists, its [[higher parallel transport]] is the gauge interaction [[action functional]] for $(d-n-3)$-dimensional objects propagating on $X$.

In the special case of ordinary [[electromagnetism]] with $n=1$ and $d = 4$ we have that electrically charged 0-dimensional particles couple to $\nabla$ and magnetically charged $(4-(1+1)-2) = 0$-dimensional particles couple to $\tilde \nabla$.

In analogy to this case one calls generally the $d-n-3$-dimensional objects coupling to $\tilde \nabla$ the **magnetic duals** of the $(n-1)$-dimensional objects coupling to $\nabla$.

## Generalizations

For $d= 4$ EM-duality is the special abelian case of [[S-duality]] for [[Yang-Mills theory]]. Witten and Kapustin argued that this is governed by the [[geometric Langlands correspondence]].

## Examples

* In [[N=2 D=4 super Yang-Mills theory]] electric-magnetic duality is studied as _[[Seiberg-Witten theory]]_.

* In [[heterotic string theory]] one considers 1-dimensional objects in $d=10$-dimensional spaces electrically charged (under the [[Kalb-Ramond field]]). Their magnetic duals are 5-dimensional objects (fivebranes), studied in [[dual heterotic string theory]].

* [[!include electric-magnetic duality -- table]]

* [[M2-brane]] and [[M5-brane]]: the 4-form $G_4$ which the M2-brane couples to is the Hodge-dual of the 7-form $G_7$ which the M5-brane couples to

$$
  G_7 = \star G_4
$$

as imposed by the equations of motion for 11d [[supergravity]] (see [here](https://ncatlab.org/nlab/show/torsion+constraints+in+supergravity#Examples11dSuGra)). The associated [[C3-field]] and [[C6-field]] are electric-magnetic duals.

## Related concepts

[[duality in physics]], [[duality in string theory]]

* [[parent action functional]]

* [[S-duality]]

  * **electro-magnetic duality**

    * [[Montonen-Olive duality]]

    * [[dual graviton]]

    * [[dual photon]]

  * [[Seiberg duality]]

  * [[geometric Langlands correspondence]]

  * [[quantum geometric Langlands correspondence]]

## References

Detailed review is in

* [[José Figueroa-O'Farrill]], _Electromagnetic Duality for Children_ (1998) ([[FigueroaElectromagneticDuality.pdf:file]])

It was originally noticed in

* {#GNO77} [[Peter Goddard]], [[Jean Nuyts]], [[David Olive]], *Gauge Theories And Magnetic Charge*, Nucl. Phys. B **125** (1977) 1-28 [doi:10.1016/0550-3213(77)90221-8, [spire:111435](https://inspirehep.net/literature/111435)]

that where [[electric charge]] in [[Yang-Mills theory]] takes values in the [[weight lattice]] of the [[gauge group]], then [[magnetic charge]] takes values in the lattice of what is now called the [[Langlands dual group]].

This led to the electric/magnetic duality conjecture formulation in

* [[Claus Montonen]], [[David Olive]], _Magnetic Monopoles As Gauge Particles?_ Phys. Lett. B72 (1977) 117-120 ([spire:121372](https://inspirehep.net/literature/121372), doi:10.1016/0370-2693(77)90076-4)

According to ([Kapustin-Witten 06, pages 3-4](#KapustinWitten06)) the observation that the Montonen-Olive dual charge group coincides with the [[Langlands dual group]] is due to

* [[Michael Atiyah]], private communication to [[Edward Witten]], 1977

See also the references at _[[S-duality]]_.

The insight that the Montonen-Olive duality works more naturally in [[super Yang-Mills theory]] is due to

* [[David Olive]], [[Edward Witten]], _Supersymmetry Algebras That Include Topological Charges_, Phys. Lett. B78 (1978) 97-101.

and that it works particularly for [[N=4 D=4 super Yang-Mills theory]] is due to

* H. Osborn, _Topological Charges For $N = 4$ Supersymmetric Gauge Theories And Monopoles Of Spin 1_, Phys. Lett. B83 (1979) 321-326.

The observation that the $\mathbb{Z}_2$ electric/magnetic duality extends to an $SL(2,\mathbb{Z})$-action in this case is due to

* [[John Cardy]], E. Rabinovici, _Phase Structure Of Zp Models In The Presence Of A Theta Parameter_, Nucl. Phys. B205 (1982) 1-16;

* [[John Cardy]], _Duality And The Theta Parameter In Abelian Lattice Models_, Nucl. Phys. B205 (1982) 17-26.

* A. Shapere and [[Frank Wilczek]], _Selfdual Models With Theta Terms_, Nucl. Phys. B320 (1989) 669-695.

and specifically the embedding of this into [[string theory]] [[S-duality]] originates in

* [[Ashoke Sen]], _Dyon - Monopole Bound States, Self-Dual Harmonic Forms on the Multi-Monopole Moduli Space, and $SL(2,\mathbb{Z})$ Invariance in String Theory_ ([arXiv:hep-th/9402032](http://arxiv.org/abs/hep-th/9402032))

The understanding of this $SL(2,\mathbb{Z})$-symmetry as a remnant conformal transformation on a 6-dimensional [[principal 2-bundle]]-theory -- the [[6d (2,0)-superconformal QFT]] -- compactified on a torus is described in

* {#Witten95a} [[Edward Witten]], pages 4-5 of _Some Comments On String Dynamics_, Proceedings of _String95_ ([arXiv:hepth/9507121](http://arxiv.org/abs/hepth/9507121))

* {#Witten95b} [[Edward Witten]], _On S-Duality in Abelian Gauge Theory_ ([arXiv:hep-th/9505186](http://arxiv.org/abs/hep-th/9505186))

* [[Cumrun Vafa]], _Geometric Origin of Montonen-Olive Duality_, Adv.Theor.Math.Phys.1:158-166, 1998 ([arXiv:hep-th/9707131](https://arxiv.org/abs/hep-th/9707131))

* {#Witten07} [[Edward Witten]], _[[Conformal field theory in four and six dimensions]]_, in [[Ulrike Tillmann]], _Topology, Geometry and Quantum Field Theory: Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal_,  London Mathematical Society Lecture Note Series (2004) ([arXiv:0712.0157](http://arxiv.org/abs/0712.0157))

The relation of S-duality to [[geometric Langlands duality]] was understood in

* {#KapustinWitten06} [[Anton Kapustin]], [[Edward Witten]], _Electric-Magnetic Duality And The Geometric Langlands Program_ , Communications in number theory and physics, Volume 1, Number 1, 1–236 (2007) ([arXiv:hep-th/0604151](http://arxiv.org/abs/hep-th/0604151))

Exposition of this is in

* [[Edward Frenkel]],  _What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common?_ KITP talk 2011 ([web](http://online.kitp.ucsb.edu/online/bblunch/frenkel/))

A perspective unifying EM-duality of [[higher gauge theories]]  with [[non-abelian T-duality]]:

* [[Ján Pulmann]], [[Pavol Ševera]], [[Fridrich Valach]], _A non-abelian duality for (higher) gauge theories_ ([arXiv:1909.06151](https://arxiv.org/abs/1909.06151))

An arithmetic version of EM-duality:

* [[David Ben-Zvi]], [[Yiannis Sakellaridis]], [[Akshay Venkatesh]], *Relative Langlands Duality* ([pdf](https://www.math.ias.edu/~akshay/research/BZSVpaperV1.pdf))

[[!redirects electric-magnetic dualities]]

[[!redirects electro-magnetic duality]]
[[!redirects electro-magnetic dualities]]

[[!redirects electromagnetic duality]]
[[!redirects electromagnetic dualites]]

[[!redirects electromagnetic dual]]
[[!redirects electromagnetic duals]]


electric-magnetic duality -- table

[[electric-magnetic duality]] of [[D-branes]]/[[RR-fields]] in [[type II string theory]]:

| [[electric charge]] | [[magnetic charge]] |
|---------------------|---------------------|
| [[D0-brane]]        | [[D6-brane]]        |
| [[D1-brane]]        | [[D5-brane]]        |
| [[D2-brane]]        | [[D4-brane]]        |
| [[D3-brane]]        | [[D3-brane]]        |


electromagnetic field

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## Idea

The **electromagnetic field** is is a [[gauge field|gauge]] [[field (physics)|field]] which unifies the [[electric field]] and the [[magnetic field]]. A configuration of the **electromagnetic field** on a space $X$ in the _absence_ of [[magnetic charge]] is modeled by a [[cocycle]] $\hat F$ in degree 2 [[ordinary differential cohomology]].

This may be realized in particular equivalently by

* a $U(1)$-[[principal bundle]] [[connection on a bundle|with connection]] on $X$;

* a degree 2 [[Deligne cohomology|Deligne cocycle]] on $X$.

In the presence of [[magnetic charge]] the electromagnetic field is modeled by a cocycle in differential [[twisted cohomology]], where the twist is given by the differential 3-cocycle that models the [[magnetic current]].

The analogous field modeled by a degree 3 [[Deligne cohomology|Deligne cocycle]] is the [[Kalb-Ramond field]].

## History

...historical section eventually goes here..

...electricity and magnetism were discovered independently,
[[Maxwell's equations]] in classical vector analysis which allows the formulation as a tensor $F$ as below, and "magnetism is a consequence of electrostatics and covariance, hence the composite noun electromagnetism"...

(...)

## Mathematical model from physical input

The electromagnetic field is modeled by a [[circle bundle]] with [[connection on a bundle|connection]].

We describe how this identification arises from experimental input.

The input is two-fold

1. [Maxwell's equations](#MaxwellEquations) say that (using the experimentally observed absence of net [[magnetic charge]]) the [[field strength]] of electromagnetism is a closed [[differential form|differential 2-form]] on [[spacetime]];

1. The _Dirac charge quantization argument_ shows that in order for the electromagnetic field to serve as the [[background gauge field]] to which a charged [[quantum mechanics|quantum mechanical]] [[particle]] couples (for instance an [[electron]]), it must be true that this 2-form is the [[curvature]] 2-form of a [[circle bundle]] with [[connection on a bundle|connection]].

We say this now in more detail.

### Maxwell's equations
 {#MaxwellEquations}

We first discuss in [Field strength as a closed 2-form](#FieldStrengthAsClosed2Form) how [[Maxwell's equations]] state that the electromagnetic [[field strength]] is a closed [[differential form|differential 2-form]] on [[spacetime]]

#### Field strength is a closed 2-form
 {#FieldStrengthAsClosed2Form}

In modern language, the insight of ([Maxwell, 1865](#Maxwell)) is that locally, when physical [[spacetime]] is well approximated by a patch of its tangent space, i.e. by a patch of 4-dimensional [[Minkowski space]] $U \subset (\mathbb{R}^4, g = diag(-1,1,1,1))$, the electric field $\vec E = \left[ \array{E_1 \\ E_2 \\ E_3} \right]$ and magnetic field $\vec B = \left[ \array{B_1 \\ B_2 \\ B_3} \right]$ combine into a differential [[differential form|2-form]] -- the **Faraday tensor**

$$
  \begin{aligned}
    F & \coloneqq E \wedge d t + B
      \\
      &
      \coloneqq
        E_1 d x^1 \wedge d t +
        E_2 d x^2 \wedge d t +
        E_3 d x^3 \wedge d t
     \\
      & +
        B_1 d x^2 \wedge d x^3 +
        B_2 d x^3 \wedge d x^1 +
        B_3 d x^1 \wedge d x^2
  \end{aligned}
$$

in $\Omega^2(U)$ and the [[electric charge]] density and current density combine to a differential 3-form

$$
\begin{aligned}
    j_{el} & \coloneqq j\wedge dt - \rho d x^1 \wedge d x^2 \wedge d x^3 \\
& \coloneqq
        j_1 d x^2 \wedge d x^3 \wedge d t +
        j_2 d x^3 \wedge d x^1 \wedge d t +
        j_3 d x^1 \wedge d x^2 \wedge d t
        -
        \rho \; d x^1 \wedge d x^2 \wedge d x^3
\end{aligned}
$$

in $\Omega^3(U)$ such that the following two equations of differential forms are satisfied

$$
  \begin{aligned}
    d F = 0
    \\
    d \star F  = j_{el}
  \end{aligned}
  \,,
$$

where $d$ is the de Rham differential operator and $\star$ the [[Hodge star]] operator. If we decompose $\star F$ into its components as before as

$$
\begin{aligned}
\star F
&= -D + H\wedge dt \\
&= -D_1 \; d x^2 \wedge d x^3
   -D_2 \; d x^3 \wedge d x^1
   -D_3 \; d x^1 \wedge d x^2
     \\
     & +
     H_1 \; d x^1 \wedge d t
     +
     H_2 \; d x^2 \wedge d t
     +
     H_3 \; d x^3 \wedge d t
  \end{aligned}
$$

then in terms of these components the field equations -- called **Maxwell's equations** -- read as follows.

$d F = 0$

* **magnetic Gauss law**: $div B = 0$

* **Faraday's law**: $\frac{d}{d t} B + rot E = 0$

$d \star F = 0$

* **Gauss' law**: $div D = \rho$

* **Ampère's law** $- \frac{d}{d t} D + rot H = j_{el}$

#### Vector potential -- as a Cech-Deligne 2-cocycle

By the [above](#FieldStrengthAsClosed2Form), the electromagnetic [[field strength]] (in the absence of net [[magnetic charge]]) on [[spacetime]] $X$ is given by a 2-form $F \in \Omega^2(X)$ and an [[electric current]] 3-form $j_{el} \in \Omega^3(X)$ satisfying Maxwell's equations

$$
  d F = 0
$$

$$
  d \star F = j_{el}
$$

The first equation with the [[Poincare lemma]] implies that one may find

* on a [[good open cover]] of $X$ by [[open subsets]] $\{U_i \to X\}$

* a collection $(A_i \in \Omega^1(U))_i$ of [[differential form|differential 1-forms]], such that

  $$
    d A_i = F|_{U_i}
  $$

* and a collection $(\lambda_{i j} \in C^\infty(U_i \cap U_j), \mathbb{R})_{i,j}$ of real valued functions on double overlaps such that

  $$
     A_j|_{U_i \cap U_j} -  A_i|_{U_i \cap U_j} = d \lambda_{i j}
     \,.
  $$

The forms $\{A_i\}$ are called a **vector potential** or the **[[electromagnetic potential]]** for the electromagnetic field.

Notice that it follows that on triple overlaps $U_i \cap U_j \cap U_k$ we have

$$
  d \lambda_{i j} + d \lambda_{j k} = d \lambda_{i k}
$$

which means that on that overlap the function

$$
  \lambda_{i j} + \lambda_{k j} - \lambda_{i k}
  :
  U_i \cap U_j \cap U_k \to \mathbb{R}
$$

is constant. If one requires these constants all to be inside a discrete subgroup $\Gamma \hookrightarrow \mathbb{R}$, then the data $(\{A_i\}, \{\lambda_{i j} mod \Gamma\})$ defines a degree 2-[[cocycle]] in [[Cech cohomology|Cech]]-[[Deligne cohomology]] on $X$ with coefficients in $\mathbb{R}/\Gamma$. [Below](#AsBackgroundGaugeField) we see that experiment demands that such a subgroup exists and is given by the additive group of [[integer]]s.

### Kirchhoff's laws

[[Kirchhoff's laws]] are a kind of coarse graining of Maxwell's equations, where instead of infinitesimal quantities one considers actual macroscopic [[current]] and [[voltage]].

### Background gauge field for charge quantum
 {#AsBackgroundGaugeField}

The exponentiated [[action functional]] for the [[sigma model]] describing a [[particle]] on [[target space]] $X$ charged under an electromagnetic [[background gauge field]] has to satisfy two properties:

1. It must be given by the [[holonomy]] of the 1-forms $A$ (this encodes, via [[variational calculus]], the [[Lorentz force]] exerted by the electromagnetic field on the particle ):

   $$
     \exp(i S(-)) :
     (S^1 \stackrel{\gamma}{\to} X)
     \mapsto
     \exp(i S_{kin}(\gamma)) hol(\gamma)
   $$

1. It must take values in the [[circle group]] $U(1)$ (this is a basic rule of the [[quantum mechanics]] describing the particle):

   $$
     \exp( i S ) : C^\infty(S^1, X) \to U(1)
     \,.
   $$

Therefore the above data is subject to the additional constraint that it induces well-defined $U(1)$-valued [[holonomy]] -- this is **Dirac's quantization condition**, a necessary requirement for the existence of quantum mechanical particles on $X$ that are charged under the background electromagnetic field.

   Concretely: for any smooth curve $\gamma : S^1 \to X$ and any [[cover]] $\{V_i \to S^1\}$ of $S^1$ refining the pullback of the cover $U$ to $S^1$, and for every triangulation $\{v, e\}$ of $S^1$ subordinate to $\{U_i \to X\}$, i.e. such that there is an index map $\rho$ such that $\gamma(e) \subset U_{\rho(e)}$ and $\gamma(v) \subset U_{\rho(v)}$

   the expression

   $$
     hol(\gamma)
     \coloneqq
     \prod_v \exp(i \int_{v} \gamma^* A_{\rho(v)})
     \prod_{e} \exp(i (-1)^{\sigma_{e,v}}
       \gamma^*\lambda_{\rho(e) \rho(v)}
      (v)
   $$

   (where $\sigma_{e,v} = 1$ if $v$ is the final vertex of $e$ and $-1$ otherwise)

 has to be a well defined element in $U(1)$ (independent of all the choices made).

   This implies in particular that cancelling from the triangulation an edge $e_i$ of vanishing length must have no effect on the formula, which in turn means that for all $i,j,k$ we have

   $$
     \exp(i \gamma^*\lambda_{i j}(v_{i j}))
     \exp(i \gamma^*\lambda_{j k}(v_{j k}))
     =
     \exp(i \gamma^*\lambda_{i k}(v_{i k}))
   $$

   and hence

   $$
     \lambda_{i j} + \lambda_{j k} =
     \lambda_{i k} mod 2\pi
     \,.
   $$

In short: the [[holonomy]] of the constant path on a point $x \in X$ must be $1 \in U(1)$, but if that path sits in a triple intersection $U_i \cap U_j \cap U_k$ then the holonomy is equivalently given as the exponentiated sum of the three transition functions. This forces the sum $\lambda_{i j} + \lambda_{j k} - \lambda_{i k}$ to land in $\mathbb{Z} \hookrightarrow \mathbb{R}$.

In total this says precisely that the data

$$
  (A_i, \lambda_{i j} mod \mathbb{Z})
$$

is a [[Čech cohomology|Čech cocycle]] with coefficients in the degree 2 [[Deligne cohomology|Deligne complex]] whose [[curvature]]2-form is the given Maxwell [[curvature]] 2-form.

## Charge quantization
 {#ChargeQuantization}

See at _[[Dirac charge quantization]]_.

### Dirac's original argument
 {#DiracArgument}

> under construction

Dirac originally presented the following reasoning, which captures the main point of the above considerations.

He considered $X$ to be $\mathbb{R}^3$ without the origin,
$$
  X \coloneqq \mathbb{R}^3 \backslash \{0\}
  \,,
$$

which is a manifold of the topology of ([[weak homotopy equivalence|weakly homotopy equivalent to]]) the 2-[[sphere]] $S^2$.

He imagined a situation with a magnetic charge supported on the point located at the origin and removed that point in order to keep the field strength $F$ to be a _closed_ 2-form on all of $X$.

(Indeed,  if one does not remove the support of magnetic charge, the argument becomes much more sophisticated and involves higher differential cocycles given by [[bundle gerbe]]s. This was not understood before Dan Freed's
[Dirac charge quantization and generalized differential cohomology](http://arxiv.org/abs/hep-th/0011220).)

Then he considered a single coordinate patch

$$
  U \coloneqq \mathbb{R}^3 \backslash \{x^1 \geq 0\} \subset X
$$

given by $X$ minus the right half of the first coordinate axis.

Traditionally physicist try to give that half-line a physical interpretation by imagining that it is the body of an idealized infinitely-thin and to one side infinitely-long solenoid. Indeed, such a solenoid would have a magnetic monopole charge on each of its ends, so if the one end is imagined to have disappeared to infinity, then the other one is the magnetic charge that Dirac imagines to sit at the origin of our setup.

In this context the half-line $\{x^1 \geq 0\}$ is called a **Dirac string**. While there is the possibility to sensibly discuss the idea that this Dirac string actually models a physical entity like an idealized solenoid, its main purpose historically is to confuse physics students and keep them from understanding the theory of [[fiber bundles]] (see at _[[fiber bundles in physics]]_). Therefore here we shall refrain from talking about Dirac strings and consider $U \coloneqq \mathbb{R}^3 \backslash \{x^1 \geq 0\} \subset X$ as exactly what it is, by itself: an open subset that is part of a [[cover]] of $X$.  Unfortunately, of course, Dirac didn't mention the other open subsets in that [[cover]] (at least one more is needed for a decent discussion), so that the Dirac string keeps haunting physicists.

> ...running out of time...just quickly now...

...Dirac effectively considered the overlap cocycle condition $A_j - A_i = something$, found that by the requirement that $A$ has well defined holonomy it follows that there must be $g_{ij}$ a function with values in $U(1)$ such that  $A_j - A_i = d log g_{ij}$, then did away with the $j$-patch (considering a kind of limit as we encircle the half axis) and concluded that $A$ must be the log-differential of a $U(1)$-valued function, whose winding number around the half-axis he identified with the magnetic charge, which in terms of bundles one identifies with the Chern-class of the bundle in question ...

> ...have to run...

In modern terms:

The [[clutching construction]] gives $U(1)$-principal bundle ob $S^2$ by covering with two hemispheres $U_0$ and $U_1$ and picking a transition function $g \colon S^1 \to U(1)$
on the overlap $U_0 \cap U_1 \simeq S^1\times (0,\epsilon)$. The integral [[winding number]] of $g$ represents the [[first Chern class]] of the line bundle.

By [the standard formula](connection+on+a+bundle#ExistenceOfConnections) for existence of principal connections on given principal bundles, given a choice of [[partition of unity]] $\{\rho_i\}$ then the connection on $U_0$ is given by

$$
  A_0 = \rho_1 \mathbf{d} log g
$$

If we think (as we may) of $U_0 = S^2 - \{\ast\}$ as covering most of the 2-sphere except one point and of $U_1$ the $\epsilon$-open neighbourhood of that point, then this $A_0$ vanishes on most of the sphere and close to the point taken out ("the Dirac string") it becomes non-vanishing and equal to $g^{-1}\mathbf{d}g$.

### Electric-magnetic charge quantization

Suppose we had a [[circle bundle]] with [[connection on a bundle]] representing an electromagnetic field with net [[magnetic charge]] $Q$ given by some magnetic [[current]] $j_{mag}$

$$
  d F = j_{mag}
$$

that is supported in some compact spatial region $U \times \mathbb{R} \subset X$ with boundary sphere $\partial U \simeq S^2$.

$$
  q = \int_U j_{mag} = \int_U dF =\int_{\partial U} F = \int_{S^2} F
$$

by the [[Stokes theorem]]. It follows from the fact that $F$ is the curvature 2-form on a circle bundle that $q$ is integral: it is given by the first [[Chern class]] of the bundle.

(...)

For $\gamma : S^1 \to X$ a closed but contractible trajectory of an electrically charged particle, the action functional is

$$
  \exp(i e \int_{S^1} \gamma^* A )
  =
  \exp(i e \int_{D^2} \gamma^* F)
$$

by the [[Stokes theorem]], for some 2-disk cobounding the circle. If now $\gamma$ approaches a constant path and the 2-disk is taken to wrap the 2-cycle $\partial U$, then this becomes

$$
  1 = \exp(i e \int_{S^2} F) = \exp(i e q)
  \,.
$$

Which implies that with the magnetic charge being quantized, also the electric charge is.

(...)

## Related concepts

* [[fiber bundles in physics]]

* [[Dirac charge quantization]]

* [[dipole moment]]

* [[Hodge-Maxwell theorem]]

* [[quantum electrodynamics]], [[Lamb shift]]

* [[Yang-Mills field]]

* [[Born-Infeld theory]]

* [[B-field]], [[C-field]]

* [[ordinary differential cohomology]], [[circle n-bundle with connection]]

* [[geometric origin of inhomogeneous media]]

## References

[[Maxwell's equations]] originate in

* [[James Clerk Maxwell]], _[A Dynamical Theory of the Electromagnetic Field](http://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field),_ Philosophical Transactions of the Royal Society of London 155, 459--512 (1865).
 {#Maxwell}

[[Dirac's charge quantization]] argument appeared in

* [[P.A.M. Dirac]], _Quantized Singularities in the Electromagnetic Field_,  Proceedings of the Royal Society, A133 (1931) pp 60--72.

Review is for instance in

* [[Theodore Frankel]], section 5.5 of _[[The Geometry of Physics - An Introduction]]_

Discussions of the basic geometry behind Maxwell equations can be found in

* Hong-Mo Chan, Sheung Tsun Tsou, _Some elementary gauge theory concepts_, [gBooks](http://books.google.hr/books?id=BdB3SqCq33wC)
* Gregory L. Naber, _Topology, geometry, and gauge fields: interactions_
* [Differential Forms in Electromagnetic Theory](http://eceformsweb.groups.et.byu.net/forms-home.html)

For undergraduate lectures including experimental material see

* Walter Lewin, Electricity and magnetism, 2002, [MIT opencourseware](http://ocw.mit.edu/courses/physics/8-02-electricity-and-magnetism-spring-2002)

The [[Lorentz group|Lorentz]]-[[invariants]] of the [[electromagnetic field]]:

* C. A. Escobar, L. F. Urrutia, _The invariants of the electromagnetic field_, Journal of Mathematical Physics 55, 032902 (2014) ([arXiv:1309.4185](https://arxiv.org/abs/1309.4185))

[[!redirects electromagnetic field]]
[[!redirects electromagnetic fields]]

[[!redirects Faraday tensor]]
[[!redirects Faraday tensors]]


electromagnetic field strength

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## Idea

The [[field strength]] of the [[electromagnetic field]].

## Details

Over [[Minkowski space]] $\mathbb{R}^{4}$:

the [[electromagnetic potential]]

$$
  A = \phi \mathbf{d}t + A_1 \mathbf{d}x^1 + A_2 \mathbf{d}x^2 + A_3 \mathbf{d}x^3
$$

then [[field strength]] is the [[de Rham differential]]

$$
  F
    \coloneqq
  \mathbf{d}A
    =
   E_1 \mathbf{d}t \wedge \mathbf{d}x^1
    +
   E_2 \mathbf{d}t \wedge \mathbf{d}x^2
    +
   E_3 \mathbf{d}t \wedge \mathbf{d}x^3
    +
    B_1 \mathbf{d}x^2 \wedge \mathbf{d}x^3
    +
    B_2 \mathbf{d}x^3 \wedge \mathbf{d}x^1
    +
    B_3 \mathbf{d}x^1 \wedge \mathbf{d}x^2
$$

with

$$
  E_i = \frac{\partial \phi}{\partial x^i}
$$

the _electric field strength_

and

$$
  B_1 = \frac{\partial A_2}{\partial x^3} - \frac{\partial A_3}{\partial x^2}
$$

etc

the _magnetic field strength_.

The field strength is closed, $\mathbf{d} F = 0$

this are the first 2 of 4 [[Maxwell equations]]

## Related concepts

* [[electromagnetic potential]]

* [[Aharonov-Bohm effect]]

* [[Dirac charge quantization]]


electromagnetic potential

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## Definition

The [[electromagnetic field]] on a [[spacetime]] $X$ is mathematically modeled by a [[circle bundle with connection]] $\nabla$ on $X$. If the underlying bundle is trivial, or else on local [[coordinate patches]] $\mathbb{R}^n \hookrightarrow X$ over which it is so, this [[connection on a bundle|connection]] is equivalently a [[differential 1-form]] $A \in \Omega^1(\mathbb{R}^n)$.

This is then called the **electromagnetic potential** of the [[electromagnetic field]] (sometimes: "vector potential" or "[[gauge potential]] of the electromagnetic field").

Its [[de Rham differential]]

$$
  F \coloneqq \mathbf{d}A
$$

is the actual [[field strength]] of the electromagnetic field.

On a 4-dimensiona [[Minkowski spacetime]] with its canonical [[coordinates]] $\{t,x^1, x^2, x^3\}$, the electromagnetic potential $A$ is naturally expanded into corredinate components, traditionally written as

$$
  A = \phi \mathbf{d}t + A_1 \mathbf{d}x^1 + A_2 \mathbf{d}x^2 + A_3 \mathbf{d}x^3
  \,.
$$

Here

* $\phi$ is the **electric potential**

* $\vec A = [A_1, A_2, A_3]$ is the **magnetic potential**

(for this choice of [[coordinates]]).

## Related concepts

* [[gauge potential]]

* [[Aharonov-Bohm effect]]

## References

The identification of [[gauge potentials]] with [[connections]] on [[fiber bundles]] is due to:

* {#WuYang75} [[Tai Tsun Wu]], [[Chen Ning Yang]], *Concept of nonintegrable phase factors and global formulation of gauge fields*, Phys. Rev. D **12** (1975) 3845 [[doi:10.1103/PhysRevD.12.3845](https://doi.org/10.1103/PhysRevD.12.3845)]

See also at *[[fiber bundles in physics]]*.

History:

* A. C. T. Wu, [[Chen Ning Yang]], *Evolution of the concept of vector potential in the description of the fundamental interactions*,  International Journal of Modern Physics A **21** 16 (2006) 3235-3277 [[doi:10.1142/S0217751X06033143](https://doi.org/10.1142/S0217751X06033143)]

* [[Chen Ning Yang]], *The conceptual origins of Maxwell’s equations and gauge theory*, Phyics Today **67** 11 (2014) [[doi:10.1063/PT.3.2585](https://doi.org/10.1063/PT.3.2585), [pdf](http://home.ustc.edu.cn/~lxsphys/2021-3-18/The%20conceptual%20origins%20of%20Maxwell%27s%20equations%20and%20gauge%20theory.pdf)]

Review:

* [[Theodore Frankel]], Section 5 of: _[[The Geometry of Physics - An Introduction]]_

* [[geometry of physics]], _[The electromagnetic field strength](geometry+of+physics#ElectromagneticFieldStrength)_

* Wikipedia, *[Magnetic vector potential](https://en.wikipedia.org/wiki/Magnetic_vector_potential)*

[[!redirects electromagnetic potentials]]

[[!redirects vector potential]]

[[!redirects vector potentials]]


electromagnetic radiation

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## Idea

The __electromagnetic radiation__ of some localized body is an [[electromagnetic field]] caused by (the dynamics of) that body in the limit of a large distance from that body. See also [[radiation]], [[electromagnetic waves]].

Sometimes one refers to electromagnetic radiation through some medium as simply the presence of an externally caused electromagnetic waves passing through that medium, not specifying any particular source. Sometimes the source of a radiation may be distributed matter in [[thermal equilibrium]] (e.g. [[black body radiation]]) and the radiation
occurs within the medium containing the matter in equilibrium.

Theoretical treatment of a radiation caused by a moving charged and or magnetized body at large distances from the sources involves theoretical issues like multipole expansions, asymptotics of expressions for fields, retarded potentials and so on.

## related concepts

* [[black body radiation]]

* [[hydrogen line]]

## References

* wikipedia [Electromagnetic radiation](http://en.wikipedia.org/wiki/Electromagnetic_radiation)

Wikipedia says:

> Electromagnetic radiation is a particular form of the more general electromagnetic field (EM field), which is produced by moving charges. Electromagnetic radiation is associated with EM fields that are far enough away from the moving charges that produced them, that absorption of the EM radiation no longer affects the behavior of these moving charges. These two types or behaviors of EM field are sometimes referred to as the near and far field. In this language, EMR is merely another name for the far-field.

* Jackson, _Electrodynamics_
[[!redirects EM radiation]]
[[!redirects em radiation]]


electromagnetism

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## Idea

__Electromagnetism__ or __electrodynamics__ is the [[gauge theory]] whose [[field (physics)|field]] is the [[electromagnetic field]], see there for more details.

The corresponding [[quantum field theory]] is [[quantum electrodynamics]].

## Related concepts

* [[Maxwell's equations]]

* [[dielectric medium]], [[pre-metric electromagnetism]]

* [[electromagnetic radiation]]

  * [[light]]

* [[Hall effect]], [[quantum Hall effect]]

  [[spin Hall effect]], [[quantum spin Hall effect]]

* [[topological photonics]]

* [[magnetohydrodynamics]]

* [[Dirac charge quantization]]

* [[uncertainty of fluxes]]

* [[Einstein-Maxwell theory]]

* [[Born-Infeld theory]]

* [[vector meson dominance]]

* [higher Maxwell-type gauge theory](higher+gauge+field#HigherGaugeTheoryOfMaxwellType)

* [[quantum field theory]]

  * [[gauge theory]]
    * [[classical electrodynamics]]
    * **electromagnetism**

    * [[Yang-Mills theory]]

    * [[electroweak field]]

  * [[higher U(1)-gauge theory]]

    * [[B-field]]

    * [[supergravity C-field|C-field]]

## References

### Classical electromagnetism

(Pre-)history:

* [[Edmund T. Whittaker]]:

  *A History of the Theories of Aether and Electricity*

  * Vol. 1: *The Classical Theories*

    *From the age of Descartes to the close of the Nineteenth century*

    (1910, 1951) [[pdf](https://ia804708.us.archive.org/32/items/historyoftheorie00whitrich/historyoftheorie00whitrich.pdf)]

  * Vol. 2: *The Modern Theories 1900-1926*

    (1953) [pdf]

  reprinted by Humanities Press (1973)

  [Wikipedia entry]

Textbook accounts of [[classical field theory|classical]] electrodynamics:

* V. C. A. Ferraro, *Electromagnetic Theory*, The Athlone Press (1954, 1956)

* [[Lev Landau]], [[Evgeny Lifshitz]], *Electrodynamics of Continuous Media*, volume 8 of: *[[Course of Theoretical Physics]]*, Pergamon Press (1960, 1984) [[ISBN:9780750626347](https://shop.elsevier.com/books/electrodynamics-of-continuous-media/koch/978-0-08-057060-0), [archive](https://archive.org/details/ost-physics-landaulifshitz-electrodynamicsofcontinuousmedia/mode/2up),  [pdf](https://www.aldebaran.cz/studium/books/1984_Landau-Electrodynamics.pdf)]

* [[Julian Schwinger]] et al., *Classical Electrodynamics*, based on Schwinger's 1976 lecture, Routledge (1999) [[ISBN:9780738200569](https://www.routledge.com/Classical-Electrodynamics/Schwinger-Jr-Milton-Tsai/p/book/9780738200569), [pdf](http://www.astrosen.unam.mx/~aceves/Metodos/ebooks/schwinger_EM.pdf)]

* [[John D. Jackson]], *Classical Electrodynamics*, Wiley (1962, 1975, 1998) [[ISBN:978-0-471-30932-1](https://www.wiley.com/en-us/Classical+Electrodynamics%2C+3rd+Edition-p-9780471309321), Wikipedia entry]

* [[Friedrich W. Hehl]], [[Yuri N. Obukhov]], *Foundations of Classical Electrodynamics -- Charge, Flux, and Metric*, Progress in Mathematical Physics **33**, Springer (2003) [[doi:10.1007/978-1-4612-0051-2](https://doi.org/10.1007/978-1-4612-0051-2)]

  > (from the [[pre-metric electromagnetism|pre-metric]] perspective)

* [[Walter Greiner]], *Classical Electrodynamics*, Springer (1998) [[doi:10.1007/978-1-4612-0587-6](https://doi.org/10.1007/978-1-4612-0587-6)]

* [[Mikio Nakahara]], Section 10.5 of: _[[Geometry, Topology and Physics]]_, IOP (2003) [[doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf]

* [[Robert Wald]], *Advanced Classical Electromagnetism*, Princeton University Press (2022) [[ISBN:9780691220390](https://press.princeton.edu/books/hardcover/9780691220390/advanced-classical-electromagnetism)]

[[!include electromagnetism in terms of differential forms -- references]]

### Phase space and Poisson brackets
 {#ReferencesPhaseSpaceAndPoissonBrackets}

* {#BlaschkeGieres21} [[Daniel N. Blaschke]], [[François Gieres]], *On the canonical formulation of gauge field theories and Poincaré transformations*, Nuclear Physics B **965** (2021) 115366 [[doi:10.1016/j.nuclphysb.2021.115366](https://doi.org/10.1016/j.nuclphysb.2021.115366), [arXiv:2004.14406](https://arxiv.org/abs/2004.14406)]

* [Corichi 1998, §III](#Corichi98)

For more see the references at *[Yang-Mills theory -- References -- Phase space and canonical quantization](Yang-Mills+theory#ReferencesPhaseSpaceAndCanonicalQuantization)*.

### Quantum electromagnetism

Discussion of [[Fock space]] quantization based on careful analysis of the [[covariant phase space]]:

* {#Corichi98} [[Alejandro Corichi]], *Introduction to the Fock Quantization of the Maxwell Field*, Rev. Mex. Fis. **44**  4  (1998) 402-412 [[arXiv:physics/9804018](https://arxiv.org/abs/physics/9804018)]

Discussion of electromagnetism in the generality of [[quantum field theory on curved spacetimes]] originates with

* {#Dimock92} [[Jonathan Dimock]], _Quantized Electromagnetic Field on a Manifold_, Reviews in Mathematical Physics, Volume 04, Issue 02, 1992 ([doi:10.1142/S0129055X92000078](https://doi.org/10.1142/S0129055X92000078))

and is further developed in (e.g. regarding [[Hadamard states]]) in

* K. Sanders, [[Claudio Dappiaggi]], T.-P. Hack, _Electromagnetism, local covariance, the Aharonov-Bohm effect and Gauss’ law_, Commun. Math. Phys. 328(2), 625–667 (2014)

* [[Claudio Dappiaggi]], D. Siemssen, _Hadamard states for the vector potential on asymptotically flat spacetimes_, Rev. Math. Phys. 25, 1 (2013)

* [[Claudio Dappiaggi]], B. Lang, _Quantization of Maxwell’s equations on curved backgrounds and general local covariance_, Lett. Math. Phys. 101(3), 265–287 (2012)

Discussion of global topological effects due to [[flux quantization]]:

* [[Daniel Freed]], [[Greg Moore]], [[Graeme Segal]], *The Uncertainty of Fluxes*, Commun. Math. Phys. **271**  (2007) 247-274 [[arXiv:hep-th/0605198](http://arxiv.org/abs/hep-th/0605198), [doi:10.1007/s00220-006-0181-3](https://doi.org/10.1007/s00220-006-0181-3)]

* [[Daniel Freed]], [[Greg Moore]], [[Graeme Segal]], *Heisenberg Groups and Noncommutative Fluxes*,  Annals Phys. **322** (2007) 236-285 [[arXiv:hep-th/0605200](http://arxiv.org/abs/hep-th/0605200), [doi:10.1016/j.aop.2006.07.014](https://doi.org/10.1016/j.aop.2006.07.014)]

* [[Alexei Kitaev]], [[Gregory W. Moore]], [[Kevin Walker]], *Noncommuting Flux Sectors in a Tabletop Experiment* [[arXiv:0706.3410](https://arxiv.org/abs/0706.3410)]

[[!redirects electrodynamics]]

[[!redirects Maxwell theory]]

[[!redirects electrostatic force]]
[[!redirects electrostatic forces]]

[[!redirects electricity]]
[[!redirects magnetism]]


electromagnetism in terms of differential forms -- references

### Maxwell's equations via differential forms
 {#ReferencesForMaxwellEquationsViaDifferentialForms}

On the expression of [[classical field theory|classical]] [[electromagnetism]], and especially of [[Maxwell's equations]], in terms of [[differential forms]], the [[de Rham differential]] and [[Hodge star operators]]:

* [[Élie Cartan]], §80 in: *Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite)*, Annales scientifiques de l’É.N.S. 3e série, tome **41** (1924) 1-25 $[$[numdam:ASENS_1924_3_41__1_0](http://www.numdam.org/item?id=ASENS_1924_3_41__1_0)$]$

  > (already in [[pregeometric electromagnetism|pregeometric form]])

* [[Charles Misner]], [[Kip Thorne]], [[John Wheeler]], §3.4 and §4.3 in: *[[Gravitation]]*, W. H. Freeman, San Francisco (1973) \[ISBN:9780716703440\]

* [[Theodore Frankel]], *Maxwell's equations*, The American Mathematical Monthly **81** 4 (1974) \[doi:10.1080/00029890.1974.11993557, jstor:2318995, doi:10.2307/2318995\]

* {#Thirring78} [[Walter Thirring]], vol 2 §1.3 in: *A Course in Mathematical Physics* -- 1 *Classical Dynamical Systems* and 2 *Classical Field Theory*, Springer (1978, 1992) $[$[doi:10.1007/978-1-4684-0517-0](https://doi.org/10.1007/978-1-4684-0517-0)$]$

* [[Dominic G. B. Edelen]], §9.2 in: *Applied exterior calculus*, Wiley (1985) \[GoogleBooks\]

* {#Frankel97} [[Theodore Frankel]], §3.5 & §7.2b in: *[[The Geometry of Physics - An Introduction]]*, Cambridge University Press (1997, 2004, 2012) \[doi:10.1017/CBO9781139061377\]

* [[Gregory L. Naber]], §2.2 in: *Topology, Geometry and Gauge fields -- Interactions*, Applied Mathematical Sciences **141** (2011) $[$[doi:10.1007/978-1-4419-7895-0](https://doi.org/10.1007/978-1-4419-7895-0)$]$

* [[Masao Kitano]], *Reformulation of Electromagnetism with Differential Forms*, Chapter 2 in: *Trends in Electromagnetism -- From fundamentals to applications*, InTech (2012) 21-44 $[$[ISBN:978-953-51-0267-0](https://www.intechopen.com/books/1647), [[Kitano-ElectromagnetismWithForms.pdf:file]]$]$

* {#FBM20} Sébastien Fumeron, Bertrand Berche, Fernando Moraes, *Improving student understanding of electrodynamics: the case for differential forms*, American Journal of Physics **88**  (2020) 1083 $[$[arXiv:2009.10356](https://arxiv.org/abs/2009.10356), [doi:10.1119/10.0001754](https://doi.org/10.1119/10.0001754)$]$


electron

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## Idea

The _electron_ is one of the fundamental [[particles]]/[[matter]] [[field (physics)|fields]] in the [[standard model of particle physics]], one of the [[leptons]].

Mathematically, the electron is a [[quantum]] of a [[Dirac field]].

In [[quantum electrodynamics]] the electron [[interaction|interacts]] with the [[photon]] via the [[electron-photon interaction]].

## Properties

* [[anomalous magnetic moment]]

## Related concepts

* [[Dirac field]], [[electron propagator]]

* the [[bound state]] of an electron...

  ... with a [[proton]] is a  [[hydrogen atom]]

  ... with a [[positron]] is a [[positronium]]

* the [[bound state]] of several electrons with a [[nucleus]] is an [[atom]]

* [[Lamb shift]]

* [[quantum electrodynamics]]

* the [[antiparticle]] of the [[electron]] is the [[positron]]

\linebreak

[[!include flavors of fundamental particles -- table]]

## References

The concept of intrinsic [[spin]] of the [[electron]] (making it a "[[spinor]]") originates with:

* [[Samuel A. Goudsmit]], [[George E. Uhlenbeck]]: *Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons*, Naturwissenschaften **13** 47 (1925) 953 [pdf, [[GoudsmitUhlenbeck-ElektronSpin.pdf:file]]]

Historical recollection:

* [[Samuel A. Goudsmit]], *The discovery of the electron spin*, in: *Foundations of Modern EPR*, World Scientific (1998) [[pdf](https://www.ifsc.usp.br/~laf/quantica1/goudsmit_discovery_of_spin.pdf), [webpage](https://lorentz.leidenuniv.nl/history/spin/goudsmit.html), [doi:10.1142/3624](https://doi.org/10.1142/3624)]

* [[Richard G. Milner]], *A Short History of Spin*, [Proceedings of PSTP2013](https://pos.sissa.it/182/PSTP2013), POS **182** (2014) [[arXiv:1311.5016](https://arxiv.org/abs/1311.5016), [doi:10.22323/1.182.0003 ](https://doi.org/10.22323/1.182.0003 )]

[[!redirects electrons]]


electron propagator

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## Idea

In [[quantum field theory]], specifically in the discussion of [[quantum electrodynamics]], the [[Feynman propagator]] for the [[free field|free]] [[Dirac field]] modeling the [[electron]] is often called the _[[electron propagator]]_.

[[!include Feynman diagrams in causal perturbation theory -- summary]]

## Definition

For details see

* at _[[Feynman propagator]]_ [this def.](Feynman+propagator#FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime)

* at _[[S-matrix]]_ [this def.](S-matrix#FieldSpeciesQED)

## Related concepts

* [[photon propagator]]

## References

* [[Radovan Dermisek]], _Quantum Electrodynamics (QED)_ ([pdf](http://www.physics.indiana.edu/~dermisek/QFT_09/qft-II-5-4p.pdf), [[DermisekQED.pdf:file]])

[[!redirects electron propagators]]


electron-photon interaction

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## Idea

In [[quantum electrodynamics]] the [[interaction]] between the [[Dirac field]] $\Psi$, whose [[quanta]] are [[electrons]], and the [[electromagnetic field]] $A$, whose [[quanta]] are [[photons]], is encoded by the [[interaction]] [[Lagrangian density]]

$$
  \mathbf{L}_{int}
  \;=\;
  i (\Gamma^\mu)^\alpha{}_\beta
  \overline{\psi}_\alpha \psi^\beta a^\mu
  \,
  dvol_\Sigma
$$

(with notation as as used at _[[A first idea of quantum field theory]]_, see [this example](A+first+idea+of+quantum+field+theory#LagrangianQED)).

For $g_{sw} \in C^\infty_{cp}(\Sigma)$ a [[bump function]] on [[spacetime]] thought of as an [[adiabatic switching|adiabatically switched]] [[coupling constant]], the corresponding [[interaction]] [[action functional]] is the [[local observable]]

$$
  \begin{aligned}
    S_{int}
    & \coloneqq
    i
    \underset{\Sigma}{\int}
      g_{sw}(x)
      \,
      (\Gamma^\mu)^\alpha{}_\beta
      \,
      \overline{\mathbf{\Psi}}_\alpha(x)
      \cdot
      \mathbf{\Psi}^\beta(x)
      \cdot
      \mathbf{A}_\mu(x)
      \,
     dvol_\Sigma(x)
     \\
     & =
    i
    \underset{\Sigma}{\int}
      g_{sw}(x)
      \,
      (\Gamma^\mu)^\alpha{}_\beta
      \,
      :
      \overline{\mathbf{\Psi}}_\alpha(x)
      \mathbf{\Psi}^\beta(x)
      \mathbf{A}_\mu(x)
      :
      \,
     dvol_\Sigma(x)
  \end{aligned}
  \,,
$$

where in the first line we have the [[integral]] over a pointwise product ([this def.](A+first+idea+of+quantum+field+theory#Observable)) of three [[field observables]] ([this def.](A+first+idea+of+quantum+field+theory#PointEvaluationObservables)), which in the second line we write equivalently as a [[normal ordered product]], by the discusssion at _[[Wick algebra]]_ ([this def.](Wick+algebra#NormalOrderedProductNotation)).

(e.g. [Scharf 95, (3.3.1)](#Scharf95))

The corresponding [[Feynman diagram]] is

<img src="https://ncatlab.org/nlab/files/InteractionVertexOfQED.jpg" width="200">

The square of the [[coupling constant]]

$$
  \alpha \coloneqq \tfrac{1}{4 \pi} g^2
$$

is called the _[[fine structure constant]]_.

## Related concepts

* [[Lamb shift]]

* [[phi^n interaction]]

## References

Discussion in the context of [[causal perturbation theory]] is in

* {#Scharf95} [[Günter Scharf]], section 3.3 of _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

[[!redirects electron-photon interactions]]

[[!redirects electron-photon vertex]]
[[!redirects electron-photon vertices]]


electronic band structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

By [[Bloch-Floquet theory]] the available [[energies]] of [[electron]]-excitations in [[crystals]] depends smoothly on the [[momentum]]/[[wave number]] $k$ of the excitation, taking values in the [[Brillouin torus]], and otherwise only on a [[discrete space|discrete]] index set labelling the remaining "[[quantum numbers]]" (the [[electron]]'s [[orbitals]], [[spin]]. etc.). Fixing the latter values (and jointly denoting them $\mathbf{n}$, say), the given function $k \mapsto E_{\mathbf{n}}(k)$ (or rather its [[graph of a function|graph]]) is called the *$\mathbf{n}$th electronic band* (or just *$\mathbf{n}$th band*, for short) of the material.

Often, the values of $E_{\mathbf{n}}(-)$ are closely spaced as $\mathbf{n}$ varies in certain subsets of its allowed values. For instance if $\mathbf{n}$ includes the [[spin]] of the electrons and if there is a (typically small) [[spin-orbit coupling]] and no sizeable external [[magnetic field]], then the energies $E_{n,\uparrow}(-)$ and $E_{n, \downarrow}$ differ (only) slightly. In these cases the [[graph of a function|graphs]] of these values jointly look approximately like a single but thickened curve (as such shown in the following schematic graphics), which is where the name "band" originates from.

The band geometry around the electron [[chemical potential]] of a material controls its electrical conductivity:

(graphics from [[schreiber:Anyonic topological order in TED K-theory|SS 22]])

|  |  |
|--|--|
| [[metal]]/[[conductor]] | the electron [[chemical potential]] is inside the valence band |
| [[insulator]] | the electron chemical potential is inside a *large* gap between (what is then) the valence- and conduction-band |
| [[semi-conductor]] | the electron chemical potential is inside a *small* gap between valence and conduction band |
| [[semi-metal]] | there is a *large* gap between valence and conduction band, except over a [[codimension]]$\geq 2$ locus, where the gap closes right at the chemical potential |

(graphics from [[schreiber:Anyonic topological order in TED K-theory|SS 22]])

## Related concepts

* [[Bloch-Floquet theory]]

* [[Berry connection]]

* [[dispersion relation]]

* energy band structure is also observed in "[[metamaterials]]" such as:

  * [[photonic crystals]]

  * [[phononic crystals]]

## References

### General

Textbook accounts:

* [[Karlheinz Seeger]], Section 2 of: *Semiconductor Physics*, Advanced texts in physics, Springer (2004) $[$[doi:10.1007/978-3-662-09855-4](https://doi.org/10.1007/978-3-662-09855-4)$]$

* [[Sheng San Li]], *Energy Band Theory*, in S. S. Li, (ed.): *Semiconductor Physical Electronics*, Springer (2006)  61-104 $[$[doi:10.1007/0-387-37766-2_4](https://doi.org/10.1007/0-387-37766-2_4)$]$

Lecture notes:

* [[David Tong]], Chapter 2 of: *Lectures on solid state physics* (2017) $[$[pdf](http://www.damtp.cam.ac.uk/user/tong/aqm/solidstate.pdf), [webpage](http://www.damtp.cam.ac.uk/user/tong/solidstate.html)$]$

Account with focus on [[topological phases of matter]] ([[topological insulators]], [[semimetals]] etc.):

* {#Vanderbilt18} [[David Vanderbilt]],  *Berry Phases in Electronic Structure Theory -- Electric Polarization, Orbital Magnetization and Topological Insulators*, Cambridge University Press (2018) ([doi:10.1017/9781316662205](https://doi.org/10.1017/9781316662205))

* [[Jérôme Cayssol]], [[Jean-Noël Fuchs]], *Topological and geometrical aspects of band theory*, J. Phys. Mater. **4** (2021) 034007 ([arXiv:2012.11941](https://arxiv.org/abs/2012.11941), [doi:10.1088/2515-7639/abf0b5](https://doi.org/10.1088/2515-7639/abf0b5))

See also:

* Wikipedia, *[Electronic band structure](https://en.wikipedia.org/wiki/Electronic_band_structure)*

Discussion of bands of [[metamaterials]] over [[hyperbolic spaces]] by a hyperbolic variant of [[Bloch's theorem]]:

* Joseph Maciejko, Steven Rayan, *Hyperbolic band theory*, Science Advances **7** 36 (2021) [[doi:10.1126/sciadv.abe9170](https://doi.org/10.1126/sciadv.abe9170)]

* Adil Attar, Igor Boettcher, *Selberg trace formula in hyperbolic band theory*, Phys. Rev. E **106** 034114 (2022) [[arXiv:2201.06587](https://arxiv.org/abs/2201.06587), [doi:10.1103/PhysRevE.106.034114](https://doi.org/10.1103/PhysRevE.106.034114)]

### Examples

The electronic band structure of [[graphene]] was predicted in

* [[Philip Russel Wallace]], *The Band Theory of Graphite*, Phys. Rev. **71** (1947) 622 ([doi:10.1103/PhysRev.71.622](https://doi.org/10.1103/PhysRev.71.622))

* Selberg trace formula in hyperbolic band theory
Adil Attar, Igor Boettcher

### For interacting electrons

* Yuejin Guo, Jean-Marc Langlois and William A. Goddard , *Electronic Structure and Valence-Bond Band Structure of Cuprate Superconducting Materials*, New Series, **239** 4842 (1988) 896-899 $[$[jstor:1700316](https://www.jstor.org/stable/1700316)$]$

* Jingsan Hu, Jianfei Gu, Weiyi Zhang, *Bloch’s band structures of a pair of interacting electrons in simple one- and two-dimensional lattices*, Physics Letters A **414** (2021) 127634 $[$[doi:10.1016/j.physleta.2021.127634](https://doi.org/10.1016/j.physleta.2021.127634)$]$

[[!redirects electronic band structures]]

[[!redirects electron band structure]]
[[!redirects electron band structures]]

[[!redirects electron band theory]]
[[!redirects electron band theories]]

[[!redirects electronic band theory]]
[[!redirects electronic band theories]]

[[!redirects electronic energy band]]
[[!redirects electronic energy bands]]

[[!redirects electron energy band]]
[[!redirects electron energy bands]]

[[!redirects electron band]]
[[!redirects electron bands]]

[[!redirects energy band]]
[[!redirects energy bands]]

[[!redirects energy band structure]]
[[!redirects energy band structures]]


electronvolt

A [[physical unit]] of [[energy]].

also:

* $10^{-3} \; eV$ = 1 [[meV]]

* $10^0 \; eV$ = 1 eV

* $10^3 \; eV$ = 1 [[KeV]]

* $10^{6} \; eV = $  1 [[MeV]]

* $10^9 \; eV = $ 1 [[GeV]]

* $10^{12} \; eV = $ 1 [[TeV]]

## References

* Wikipedia, _[Electronvolt](https://en.wikipedia.org/wiki/Electronvolt)_
[[!redirects electron volt]]

[[!redirects eV]]

[[!redirects meV]]
[[!redirects KeV]]


electroweak field

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The [[standard model of particle physics]] is a [[Yang-Mills theory|Yang-Mills]] [[gauge theory]] with [[gauge group]] (a [[discrete group|discrete]] [[quotient]] of) $SU(3) \times SU(2) \times U(1)$. Here the first factor is the gauge group of [[QCD]] while the $(SU(2) \times U(1))$-[[gauge field]] is that transmitting what is called the **electroweak force**.

The name is due to the fact that this gauge field unifies the [[electromagnetic field]] and that transmitting the [[weak nuclear force]]: the [[Higgs mechanism]] induces a [[spontaneous symmetry breaking]] of the electroweak field to these two fields. See at _[[electroweak symmetry breaking]]_.

## Related concepts

* [[QED]], [[QCD]]

* [[standard model of particle physics]]

## References

### General

The electroweak interaction was introduced in:

* [[Sheldon Glashow]], *The renormalizability of vector meson interactions*, Nucl. Phys. **10** (1959) 107-117 [doi:10.1016/0029-5582(59)90196-8]

* [[Abdus Salam]], [[John Clive Ward]], *Weak and electromagnetic interactions*, Nuovo Cimento. **11** 4 (1959) 568–577 [[doi:10.1007/BF02726525](https://doi.org/10.1007/BF02726525)]

* [[Sheldon Glashow]], *Partial-symmetries of weak interactions*, Nuclear Physics **22** 4 (1961) 579-588 [doi:10.1016/0029-5582(61)90469-2]

* {#Weinberg67} [[Steven  Weinberg]], *A model of Leptons*, Phys. Rev. Lett. **19** (1967) 1264 [[doi:10.1103/PhysRevLett.19.1264](https://doi.org/10.1103/PhysRevLett.19.1264)]

The history of [Weinberg 67](#Weinberg67) is recounted in:

* [[Steven Weinberg]], *The Red Camaro*, in *George* (1997), reprinted in: [[Steven Weinberg]]: *Facing Up: Science and its Cultural Adversaries*, Harvard University Press (2001) [[pdf](https://www.ictp.it/media/1867057/100reasons-Weinberg.pdf), [[Weinberg-RedCamaro.pdf:file]], [ISBN:9780674011205](https://www.hup.harvard.edu/catalog.php?isbn=9780674011205)]

* Frank Close, _[Birth of a Symmetry](http://cerncourier.com/cws/article/cern/70137)_, Cern Courier, Oct 13 2017

Textbook account:

* [[Robert E. Marshak]], Chapter 6 of: *Conceptual Foundations of Modern Particle Physics*, World Scientific 1993 ([doi:10.1142/1767](https://doi.org/10.1142/1767))

See also

* [[Michael Peskin]], _Precision Theory of Electroweak Interactions_ ([arXiv:2003.05433](https://arxiv.org/abs/2003.05433))

* Wikipedia, _[Electroweak interaction](https://en.wikipedia.org/wiki/Electroweak_interaction)_

Discussion in [[causal perturbation theory]] is in

* Andreas Aste, [[Michael Dütsch]], [[Günter Scharf]], _Perturbative gauge invariance: electroweak theory II_, Annalen Phys.8:389-404,1999 ([arXiv:hep-th/9702053](http://arxiv.org/abs/hep-th/9702053))

* {#Scharf01} [[Günter Scharf]], chapter 4 of _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

### In relation to flavour anomalies

In relation to [[flavour anomalies]]:

* Lina Alasfar, Aleksandr Azatov, Jorge de Blas, Ayan Paul, Mauro Valli, _B anomalies under the lens of electroweak precision_ ([arXiv:2007.04400](https://arxiv.org/abs/2007.04400))

[[!redirects electroweak]]
[[!redirects electroweak field]]
[[!redirects electroweak fields]]
[[!redirects electroweak force]]
[[!redirects electroweak forces]]
[[!redirects electroweak theory]]
[[!redirects electroweak theories]]
[[!redirects electroweak model]]
[[!redirects electroweak models]]

[[!redirects EW]]


electroweak symmetry breaking

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A core ingredient of the [[standard model of particle physics]] is _electroweak symmetry breaking_, the [[spontaneous symmetry breaking]] of the [[electroweak field]] via the [[Higgs mechanism]].

## Related concepts

* [[composite Higgs boson]]

## References

Surveys include

* S. Dawson, _Introduction to Electroweak Symmetry Breaking_, Lectures given at the 1998 Summer School in High Energy Physics and Cosmology, Trieste, Italy, June 29-uly 17, 1998 ([arXiv:hepph/9901280](http://arxiv.org/abs/hepph/9901280))

* Gautam Bhattacharyya,  _A Pedagogical Review of Electroweak Symmetry Breaking Scenarios_, Rept. Prog. Phys. 74:026201, 2011 ([arXiv:0910.5095](https://arxiv.org/abs/0910.5095))

* {#Brambilla14} Brambilla et al., Section 9.2. of: _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981  ([arXiv:1404.3723](https://arxiv.org/abs/1404.3723), [doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))

  (in the context of [[holographic QCD]])

* [[Antonio Pich]], _Electroweak Symmetry Breaking and the Higgs Boson_, [Acta Phys. Pol. B 47, 151](https://www.actaphys.uj.edu.pl/index_n.php?I=R&V=47&N=1#151) (2016) ([arXiv:1512.08749](https://arxiv.org/abs/1512.08749))

See also

* Wikipedia, _[Higgs mechanism](https://en.wikipedia.org/wiki/Higgs_mechanism)_

Discussion of [[electroweak symmetry breaking]] by [[effective field theory]] and in analogy to [[chiral perturbation theory]] of [[quantum hadrodynamics]]:

* A. Dobado, J. R. Pelaez, _On the Equivalence Theorem in the $\chiPT$ Description of the Symmetry Breaking Sector of the Standard Model_, Nucl. Phys. B425:110-136, 1994; ERRATUM-ibid. B434:475, 1995 ([arXiv:hep-ph/9401202](https://arxiv.org/abs/hep-ph/9401202))

* [[Gino Isidori]], _Effective theories of electroweak symmetry breaking_, PoS CD09:073, 2009 ([arXiv:0911.3219](https://arxiv.org/abs/0911.3219))

[[!redirects electro-weak symmetry breaking]]


elegant Reedy category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

An **elegant Reedy category** is a [[Reedy category]] $R$ such that the following *equivalent* conditions hold

* For every [[monomorphism]] $A\hookrightarrow B$ of [[presheaves]] on $R$ and every $x\in R$, the induced map $A_x \amalg_{L_x A} L_x B \to B_x$ is a monomorphism.

* Every [[span]] of codegeneracy maps in $R_-$ has an [[absolute colimit|absolute]] [[pushout]] in $R_-$.

* Every element of a presheaf $R$ is a degeneracy of some nondegenerate element in a unique way.

=--

## Properties

The principal theorem about elegant Reedy categories is that the [[Reedy model structure]] on [[presheaves]] (i.e. contravariant diagrams) over an elegant Reedy category coincides with the [[injective model structure]].  This is not true for presheaves valued in *any* model category, only well-behaved ones.  We clarify the necessary conditions by building up to this theorem in stages, adding hypotheses on the codomain of the presheaves as necessary.

### Degeneracies are split

+-- {: .num_lemma #DegSplit}
###### Lemma
If $R$ is elegant, then every codegeneracy map (i.e. morphism in $R_-$) is a [[split epimorphism]].
=--
+-- {: .proof}
###### Proof
Let $f:x\to y$ be a codegeneracy map; then the span $y \xleftarrow{f} x \xrightarrow{f} y$ has an absolute pushout, consisting of say $g:y\to z$ and $h:y\to z$ with $g f = h f$.  This absolute pushout is preserved by $R(z,-)$, so $1_z\in R(z,z)$ must be the image under $g$ or $h$ of some map $s:z\to y$; WLOG say it is $g$, so we have $1_z = g s$.  Now we have $s h:y\to y$ and $1_y$ such that $g s h = h = h 1_y$, and our absolute pushout is preserved by $R(y,-)$, so there must be a zigzag of elements in $R(y,z)$ relating $s h$ to $1_y$.  At one end of that zigzag, there must be a $t:y\to x$ such that $f t = 1_y$; hence $f$ is split epi.
=--

+-- {: .num_lemma #MonoPreservesDeg}
###### Corollary
For every monomorphism $A\hookrightarrow B$ of presheaves on $R$, every nondegenerate element of $A$ remains nondegenerate in $B$.
=--
+-- {: .proof}
###### Proof
Let $a$ be a nondegenerate element of $A_x$, for some $x\in R$, $f : x \to y$ a codegeneracy map, and $b\in B_y$ such that $B_f b = a$. We have to show that $f = \id$. By Lemma \ref{DegSplit}, $f$ has a section $s: y \to x$, hence $B_s a = B_s B_f b = b$, which implies that $b \in A_y$. Since $a$ is nondegenerate, it follows that $f = \id$.
=--

### Degeneracies in subobjects

+-- {: .num_lemma #DegSub}
###### Lemma
Let $R$ be elegant and $f:x\to y$ a codegeneracy in $R$.  Let $M$ be any category, and $\mu:A\to B$ a monomorphism in $M^{R^{\mathrm{op}}}$.  Then the following naturality square is a pullback:
$$\array{ A_y & \xrightarrow{A_f} & A_x \\
  {}^{\mu_y}\downarrow & & \downarrow^{\mu_x} \\
  B_y & \xrightarrow{B_f} & B_x  }$$
=--
+-- {: .proof}
###### Proof
This depends only on the fact that $f$ is split epi in $R$.  Let $s:y\to x$ be a section of it, and let $P$ be the pullback of $B_f$ and $\mu_x$, with projections $p:P\to A_x$ and $q:P\to B_y$ with $\mu_x p = B_f q$, and an induced map $\phi:A_y \to P$ such that $p \phi = A_f$ and $q\phi = \mu_y$.

We claim that $A_s p : P \to A_y$ is an inverse of $\phi$, making it an isomorphism.  On the one hand we have $A_s p \phi = A_s A_f = A_{f s} = 1$.  On the other, to show that $\phi A_s p = 1$ it suffices to show that $p \phi A_s p = p$ and $q \phi A_s p = q$.  For the first, since $\mu_x$ is monic, it suffices to show $\mu_x p \phi A_s p = \mu_x p$, and for that we have
$$ \mu_x p \phi A_s p = B_f q \phi A_s p = B_f \mu_y A_s p = B_f B_s \mu_x p = B_{f s} \mu_x p = \mu_x p. $$
And for the second, we have
$$q \phi A_s p = \mu_y A_s p = B_s \mu_x p = B_s B_f q = B_{f s} q = q. $$
=--

+-- {: .num_lemma #LatchSub}
###### Lemma
Let $R$ be elegant, $M$ a category with pullback-stable colimits, and $\mu:A\to B$ a monomorphism in $M^{R^{\mathrm{op}}}$.  Then for any object $x\in R$, the following square is a pullback, where $L_x$ denotes the Reedy latching object at $x$:
$$\array{ L_x A & \to & A_x \\
  {}^{L_x \mu}\downarrow & & \downarrow^{\mu_x} \\
  L_x B & \to & B_x.  }$$
=--
+-- {: .proof}
###### Proof
The map $L_x B \to B_x$ is by definition the colimit in $M/B_x$ of a diagram whose objects are morphisms of the form $B_f : B_y \to B_x$, for $f$ a codegeneracy.  By the Lemma \ref{DegSub}, each of these pulls back along $\mu_x$ to $A_f : A_y \to A_x$, forming the corresponding diagram whose colimit is $L_x A \to A_x$, and by assumption the pullback preserves the colimit.
=--

### All presheaves are "Reedy monomorphic"

+-- {: .num_lemma #ReedyCofibrant}
###### Lemma
Let $R$ be elegant and let $M$ be an [[infinitary-coherent category]].  Then for any $x\in R$ and $A\in M^{R^{\mathrm{op}}}$, the map $L_x A \to A_x$ is a monomorphism.
=--
+-- {: .proof}
###### Proof
We use the terminology from the page [[∞-ary exact category]].  Consider the [[sink]] with target $A_x$ consisting of all morphisms $A_f : A_y \to A_x$ indexed by nonidentity codegeneracies $f$ with domain $x$.  By assumption, for any two such $f:x\to y$ and $f':x\to y'$ there is an absolute pushout $g:y\to z$ and $g':y'\to z$.  By absoluteness, $A_z$ is the pullback $A_y \times_{A_x} A_y$.  Thus, the images of these absolute pushouts form the *kernel* of this sink.

Now $L_x A$ is the colimit of the diagram whose objects are $A_y$ indexed by such $f:x\to y$ and whose morphisms are $A_g: A_{y'} \to A_{y}$ for $g:y\to y'$ a codegeneracy with $g f = f'$.  In this case, by the universal property of pullback, we have a unique map from $A_{y'}$ to $A_z$, where $z$ is the absolute pushout of $f$ and $f'$.  Thus, a cocone under the above kernel is also a cocone under this diagram, and the converse is easy to see.  Hence, $L_x A$ is the quotient of the above kernel.

However, in any infinitary-regular category, the quotient of the kernel of a sink is exactly the extremal-epic / monic factorization of that sink.  Therefore, the induced map $L_x A \to A_x$ is monic.
=--

### Reedy = injective

+-- {: .num_theorem #RelativeLatching}
###### Theorem
If $R$ is elegant and $M$ is a [[Grothendieck topos]], then for any $x\in R$ and monomorphism $\mu:A\to B$ in $M^{R^{\mathrm{op}}}$, the induced map $L_x B \sqcup_{L_x A} A_x \to B_x$ is monic.
=--
+-- {: .proof}
###### Proof
Since Grothendieck toposes are infinitary-coherent, by Lemma \ref{ReedyCofibrant} $L_x B\to B_x$ is monic.  By assumption $A_x \to B_x$ is monic.  And since Grothendieck toposes have pullback-stable colimits, by Lemma \ref{LatchSub} the square
$$\array{ L_x A & \to & A_x \\
  {}^{L_x \mu}\downarrow & & \downarrow^{\mu_x} \\
  L_x B & \to & B_x.  }$$
is a pullback.  In other words, $L_x A$ is the [[intersection]] of the subobjects $L_x B$ and $A_x$ of $B_x$.  But in any [[coherent category]], the pushout of two subobjects over their intersection is their [[union]], and hence in particular a subobject of their common codomain.
=--

+-- {: .un_cor #ReedyIsInjective}
###### Corollary
If $R$ is elegant and $M$ is a [[Cisinski model category]], then the [[Reedy model structure]] on $M^{R^{\mathrm{op}}}$ coincides with the [[injective model structure]].
=--
+-- {: .proof}
###### Proof
By definition, they have the same [[weak equivalences]], so it suffices to show that their classes of cofibrations coincide.  But every Reedy cofibration in any Reedy model structure is an injective (i.e. objectwise) cofibration, and the converse is Theorem \ref{RelativeLatching}.
=--

The most common application is when $M = SSet$.  Thus, for instance, every simplicial presheaf on an elegant Reedy category is Reedy cofibrant.

## Examples

* The [[simplex category]] $\Delta$ is an elegant Reedy category.

* Joyal's [[Theta category|disk categories]] $\Theta_n$ are elegant Reedy categories.

* Every [[direct category]] is a Reedy category with no degeneracies, hence trivially an elegant one.

* If $X$ is any presheaf on an elegant Reedy category $R$, then the opposite of its [[category of elements]] $(el X)^{op}$ is again an elegant Reedy category.  This is fairly easy to see from the fact that $Set^{el X}$ is equivalent to the slice category $Set^{R^{op}}/X$.

* Every [[EZ-Reedy category]] that is a strict Reedy category is elegant.

Note that unlike the notion of [[Reedy category]], the notion of elegant Reedy category is not self-dual: if $R$ is elegant then $R^{op}$ will not generally be elegant.

## Related concepts

* [[generalized Reedy category]], [[Eilenberg-Zilber category]]

* [[Reedy model structure]]

## References

* [[Julie Bergner]] and [[Charles Rezk]], *Reedy categories and the $\Theta$-construction*, [arXiv:1110.1066](http://arxiv.org/abs/1110.1066)

Elegant Reedy categories are useful to model [[homotopy type theory]].

* {#Shulman12} [[Michael Shulman]], _The univalence axiom for elegant Reedy presheaves_ ([arXiv:1307.6248](http://arxiv.org/abs/1307.6248))

* {#Shulman12} [[Michael Shulman]], _Univalence for inverse diagrams and homotopy canonicity_, Mathematical Structures in Computer Science, Volume 25, Issue 5 ( _From type theory and homotopy theory to Univalent Foundations of Mathematics_ ) June 2015 ([arXiv:1203.3253](https://arxiv.org/abs/1203.3253), [doi:/10.1017/S0960129514000565](https://doi.org/10.1017/S0960129514000565))

* Benno van den Berg and Ieke Moerdijk, *W-types in homotopy type theory*, [PDF](http://www.staff.science.uu.nl/~berg0002/papers/WinHoTT.pdf)

[[!redirects elegant Reedy categories]]


element

> This entry is about the concept in [[mathematics]], specifically in [[set theory]]/[[type theory]]/[[category theory]]. For the concept in [[chemistry]] see at [[chemical element]].

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An **element** of a [[set]] is a thing which "belongs to," or "is an element of," that set.

The circularity of this definition is unavoidable in [[foundations of mathematics|foundational]] [[set theories]] in which "set" is an undefined term.  In "definitional" set theories, where "set" is defined in terms of something else, elements are likewise defined in terms of the same "something else."

If sets (or [[setoids]]) are regarded as the [[semantics]] of some [[type theory]], then an element of a set is the interpretation of a [[term]] of some [[type]].

## Generalisations

* A [[term]] of some [[type]].
*  a [[global element]] of any object in any category or higher category with a [[terminal object]].
*  a [[generalised element]] of any object in any category or higher category;
*  a [[morphism]] in a category, a different sort of element from an object;
*  An [[object]] in a [[category]];
*  in general, $k$-[[k-morphism|morphisms]] or $k$-cells in an $\infty$-[[infinity-category|category]];

## Related concepts

[[!include logic symbols -- table]]

[[!redirects member]]
[[!redirects elements]]
[[!redirects members]]


element in abelian category > history

< [[element in abelian category]]
[[!redirects element in abelian category -- history]]


element in an abelian category

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## Idea

Not every [[abelian category]] is a [[concrete category]], such as [[Ab]] or $R$[[Mod]], hence its [[objects]] do not necessarily have underlying [[sets]] whose elements one can reason about. To circumvent this and reason by "[[diagram chasing|diagram chases]]" of elements in general abelian categories, one can instead use [[generalized elements]] in a suitable way.

This is closely related to embedding the abelian category fully-faithfully and exactly into a concrete abelian category. See at _[[abelian category]]_ the section _[Embedding theorems](abelian+category#EmbeddingTheorems)_ for more on this.

## Definition

One definition goes likes this ([MacLane, VIII.4](#MacLane) and also for instance [Gelfand-Manin, II.5](#Gelfand-Manin)):

An __element__ of an [[object]] $W$ in a given [[abelian category]] $\mathcal{A}$ is an [[equivalence class]] $[X,h]$ of pairs $(X,h)$ where $X$ is an object of $\mathcal{A}$ and $h:X\to W$ a morphism (hence a [[generalized element]]) and the equivalence is defined as follows: $[X,h] = [Y,k]$ iff there exists an object $Z$ in $\mathcal{A}$ and [[epimorphisms]] $u:Z\to X$, $v:Z\to Y$ such that $h\circ u = k\circ v : Z\to W$.

However, beware that the passage to equivalence classes does not respect the abelian group structure and hence generalized elements in this sense cannot be added or subtracted. A more natural approach is discussed in [Bergman 1974](#Bergman74) where the actual generalized elements are remembered but a refinement of their domain is allowed, much as familiar from [[topos theory]].

## References

Equivalence classes of generalized elements are considered for instance in

* [[Saunders Mac Lane]], _[[Categories Work|Categories for the working mathematician]]_, 2nd edition, Springer 1998.
 {#MacLane}

* [[Sergei Gelfand]], [[Yuri Manin]], _Methods of homological algebra_,  transl. from the 1988 Russian (Nauka Publ.) original. Springer 1996. xviii+372 pp.; 2nd corrected ed. 2002.
{#Gelfand-Manin}

Genuine generalized elements are considered in

* {#Bergman74} [[George Bergman]], _A note on abelian categories -- translating element-chasing proofs, and exact embedding in abelian groups_ (1974) [[pdf](http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf), [[Bergman-ElementChasing.pdf:file]]]

[[!redirects elements in an abelian category]]
[[!redirects elements in abelian categories]]

[[!redirects element of an abelian category]]
[[!redirects elements of an abelian category]]

[[!redirects element in abelian category]]


element of an abelian category > history

< [[element of an abelian category]]
[[!redirects element of an abelian category -- history]]


elementarily topical dagger 2-poset

[[!redirects elementarily topical dagger 2-posets]]

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## Idea ##

An elementarily topical dagger 2-poset is a dagger 2-poset whose [[category of maps]] is an [[elementary topos]].

## Definition ##

An **elementarily topical dagger 2-poset** $C$ is a [[unital dagger 2-poset|unital]] [[tabular dagger 2-poset|tabular]] [[power dagger 2-poset|power]] [[dagger 2-poset]].

## Examples ##

The dagger 2-poset [[Rel]] of [[sets]] and [[relations]] is an elementarily topical dagger 2-poset.

## See also ##

* [[dagger 2-poset]]
* [[unital dagger 2-poset]]
* [[tabular dagger 2-poset]]
* [[power dagger 2-poset]]
* [[Boolean topical dagger 2-poset]]
* [[W-topical dagger 2-poset]]
* [[ETCR]]
* [[SEAR]]