Schwartz-Bruhat function

## Idea

A Schwartz-Bruhat function is a certain type of complex-valued function on a general [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] [[topological abelian group|abelian group]], generalizing the familiar notion of [[Schwartz space|Schwartz function]] on a space given as a finite product of copies of the real line, of the circle, and a finitely generated abelian group.

## Definitions

The notion of Schwartz-Bruhat function is constructed in stages that parallel developments in the general structure theory of locally compact (Hausdorff) abelian groups.

Recall the notion of *compactly generated topological group* $G$: it means there is a compact neighborhood of the identity which generates $G$ as a group. The structure of a compactly generated abelian Lie group is well-known: it is a product of type $K \times \mathbb{R}^m \times \mathbb{Z}^n$ where $K$ is a [[compactum|compact]] [[abelian group|abelian]] [[Lie group]] (thus of the form $T^p \times F$ where $F$ is a finite abelian group and $T = \mathbb{R}/\mathbb{Z}$ is a circle group). These are often called *elementary Lie groups*.

+-- {: .num_defn #def1}
###### Definition
Let $G$ be an elementary Lie group of type $K \times \mathbb{R}^m \times \mathbb{Z}^n$ where $K$ is a compact abelian Lie group. A *Schwartz-Bruhat function* on $G$ is an [[smooth function|infinitely differentiable function]] $f: G \to \mathbb{C}$ that is *rapidly decreasing*: applications of any polynomial differential operator to $f$ is uniformly bounded in the $\mathbb{R}$- and $\mathbb{Z}$-variables, in the sense that

$$
  \underset{\alpha \in \mathbb{N}^n}{\forall}\;\;
  \underset{\beta, \gamma \in \mathbb{N}^m}{\forall}\;\; \underset{K_{\alpha, \beta, \gamma} \gt 0}{\exists}\;
  \left(
  \underset{(j, x) \in \mathbb{Z}^n \times \mathbb{R}^m}{sup} {\Vert j^\alpha x^\beta \partial_{\gamma} f(x, j)  \Vert} \lt K_{\alpha, \beta, \gamma}
  \right)
$$

using the usual notations for multi-indices $\alpha, \beta, \gamma$.
=--

Next, any locally compact abelian group is canonically a [[filtered colimit]] of the system of its open compactly generated subgroups and open inclusions between them. In particular, any abelian Lie group is canonically a filtered colimit of its open elementary Lie subgroups. In fact, an abelian Lie group is of the form $A \times \mathbb{R}^m \times T^p$, where $A$ is a discrete abelian group. We may reckon $A$ as a filtered colimit of its finitely generated subgroups $A_\alpha$; taking the product with the locally compact group $\mathbb{R}^m \times T^p$, any abelian Lie group is a filtered colimit of elementary Lie subgroups $A_\alpha \times \mathbb{R}^m \times T^p$.

+-- {: .num_defn #def2}
###### Definition
A *Schwartz-Bruhat function* on an abelian Lie group $G$ is a continuous function $f: G \to \mathbb{C}$ that is supported on an open elementary Lie subgroup $H$, and whose restriction $f|_H: H \to \mathbb{C}$ is Schwartz-Bruhat in the sense of Definition \ref{def1}. (Thus $f$ is identically zero on the complement of $H$, which is a union of open cosets $g + H$.)
=--

Let $\mathcal{S}(G)$ denote the [[topological vector space|TVS]] of Schwartz-Bruhat functions on an abelian Lie group $G$. We obtain a functor $\mathcal{S}(-): AbLie^{op} \to TVS$.

Finally, the character group of a compactly generated locally compact abelian group is an abelian Lie group. By applying [[Pontryagin duality]] to the statement that a locally compact abelian group is canonically a filtered colimit of compactly generated subgroups, we see that any locally compact abelian group $G$ is canonically an inverse limit of a cofiltered diagram of abelian Lie groups $G_\alpha$:

$$G \cong \underset{\longleftarrow}{\lim}_\alpha G_\alpha.$$

We may apply the contravariant functor $\mathcal{S}(-)$ to this cofiltered diagram to produce a filtered diagram of Schwartz-Bruhat spaces $\mathcal{S}(G_\alpha)$ of abelian Lie groups. In this notation,

+-- {: .num_defn}
###### Definition
For a locally compact abelian group $G$, the Schwartz-Bruhat space $\mathcal{S}(G)$ is the colimit of the filtered diagram of spaces $\mathcal{S}(G_\alpha)$ defined according to Definition \ref{def2}.
=--

In other words, a Schwartz-Bruhat function on $G$ is one that factors through one of its Lie quotients as

$$G \twoheadrightarrow G_\alpha \stackrel{g}{\to} \mathbb{C}$$

where $g: G_\alpha \to \mathbb{C}$ is Schwartz-Bruhat in the sense given for Lie groups, Definition \ref{def2}.

## References

The extension of Schwartz functions and [[tempered distributions]] on Euclidean spaces $\mathbb{R}^n$ to more general locally compact abelian groups was given by Bruhat:

* F. Bruhat, *Distributions sur un groupe localement compact et applications
à l’étude des représentations des groupes p-adiques*, Bull. Soc. Math. France 89 (1961), 43-75. ([pdf](http://www.numdam.org/article/BSMF_1961__89__43_0.pdf))

References to the fact that Schwartz-Bruhat spaces can be presented as direct limits of [[topological vector spaces]] frequently appear in the literature, e.g.,

* A. Wawrzyńczyk, *On tempered distributions and Bochner-Schwartz theorem on arbitrary locally compact Abelian groups*, Colloquium Mathematicae Volume 19 Issue 2 (1968), 305-318. ([link](https://eudml.org/doc/263023))

(However, the precise categorical details seem to be hard to come by, or at least treated in somewhat cavalier fashion.)

Some useful background material on the structure of locally compact Hausdorff abelian groups used in the description above can be found here:

* Dikran Dikranjan, Luchezar Stoyanov, *An elementary approach to Haar integration and Pontryagin duality in locally compact abelian groups*, Topology and its Applications 158 (2011), 1942–1961. ([pdf](https://ac.els-cdn.com/S0166864111002604/1-s2.0-S0166864111002604-main.pdf?_tid=b5b2e8fe-134e-11e8-b733-00000aab0f26&acdnat=1518809121_303788c33fd1d38a829198fb1dd2741a))

[[!redirects Schwartz-Bruhat functions]]
[[!redirects Schwartz-Bruhat space]]
[[!redirects Schwartz-Bruhat spaces]]


Schwarzian derivative

## Definition

The Schwarzian derivative is an operator on complex functions that is invariant under fractional linear transformations:

$$
(S f)(z) := \left( \frac{f''(z)}{f'(z)}\right)^' - \, \frac{1}{2} \left( \frac{f''(z)}{f'(z)}\right)^2 = \frac{f'''(z)}{f'(z)} - \frac{3}{2}\left( \frac{f''(z)}{f'(z)}\right)^2
$$

In fact the Schwarzian derivative of a fractional linear transformation, considered as a function from $\mathbb{C}$ to $\mathbb{C}$, is zero.

##References

* wikipedia: [Schwarzian derivative](https://en.wikipedia.org/wiki/Schwarzian_derivative)
* Brad Osgood, _Old and new about Schwarzian derivative_, [pdf](http://www-ee.stanford.edu/~osgood/papers/oldandnew1998.pdf)
* V. Ovsienko, S. Tabachnikov, _Projective differential geometry old and new. From the Schwarzian derivative to the cohomology of diffeomorphism groups_, Cambridge Tracts in Mathematics 165 (2005) [MR2177471](http://www.ams.org/mathscinet-getitem?mr=2177471); _What is...the Schwarzian derivative_, Notices Amer. Math. Soc. Jan. 2009, [pdf](http://www.ams.org/notices/200901/tx090100034p.pdf)
* V. Ovsienko, _Lagrange schwarzian derivative and symplectic Sturm theory_, Ann.Fac.Sci.Toulouse 2:1, 73--96, 1993 [numdam]()
* MathOverflow: [Is there an underlying explanation for the magical powers of the Schwarzian derivative?](http://mathoverflow.net/questions/38105/is-there-an-underlying-explanation-for-the-magical-powers-of-the-schwarzian-deri)

category: analysis
[[!redirects Schwarz derivative]]


Schwarzschild radius

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#### Gravity
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[[!include gravity contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The [[radius]] of the [[horizon]] of a [[Schwarzschild spacetime]] [[black hole]] of [[mass]] $m$ is called the _Schwarzschild radius_,

$$
  r_m  = 2 m G/c^2
$$

where $G$ denotes the [[gravitational constant]] and $c$ denotes the [[speed of light]].

## Properties

### Relation to Compton wavelength

Another [[physical unit]] of [[length]] parameterized by a mass $m$ is the _[[Compton wavelength]]_ $\ell_m = \frac{2 \pi \hbar}{m c}$. Solving the [[equation]]

  $$
    \array{
      & \ell_m &=& r_m
      \\
      \Leftrightarrow & 2\pi\hbar / m c &=& 2 m G / c^2
    }
  $$

  for $m$ yields the _[[Planck mass]]_

  $$
     m_{P} \coloneqq \tfrac{1}{\sqrt{\pi}} m_{\ell = r} = \sqrt{\frac{\hbar c}{G}}
    \,.
  $$

  The corresponding [[Compton wavelength]] $\ell_{m_{P}}$ is given by the _[[Planck length]]_ $\ell_P$

  $$
    \ell_{P} \coloneqq \tfrac{1}{2\pi} \ell_{m_P} = \sqrt{ \frac{\hbar G}{c^3} }
    \,
  $$

## Related concepts

[[!include fundamental scales -- contents]]


Schwarzschild spacetime

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#### Gravity
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[[!include gravity contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
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#Contents#
* table of contents
{:toc}

## Idea

The [[spacetime]] that describes a [[black hole]] with a given [[mass]]. And nothing else.

Described for instance by [[Kruskal–Szekeres coordinates]].

## Related concepts

[[!include charged and rotating black holes -- table]]

## References

* Wikipedia, _[Schwarzschild metric](https://en.wikipedia.org/wiki/Schwarzschild_metric)_

In higher dimensions:

* F.R. Tangherlini, _Schwarzschild field in $n$ dimensions and the dimensionality of space problem_, Nuovo Cim. 27 (1963) 636.

* R.C. Myers and M.J. Perry, _Black holes in higher dimensional space-times_, Ann. Phys. (NY) 172 (1986) 304.

[[!redirects Schwarzschild spacetimes]]

[[!redirects Schwarzschild black hole]]
[[!redirects Schwarzschild black holes]]
[[!redirects Schwarzschild metric]]


Schwinger effect

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### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

\begin{imagefromfile}
    "file_name": "VacuumPairCreation.jpg",
    "float": "right",
    "width": 200,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": -20,
        "right": 0,
        "left": 10
    },
    "caption": "from [Hebenstreit 14](#Hebenstreit14)"
\end{imagefromfile}

What came to be called the _Schwinger effect_ ([Sauter 31](#Sauter31), [Heisenberg-Euler 36](#HeisenbergEuler36), [Schwinger 51](#Schwinger51), [Affleck-Manton 82](#AffleckManton82), [Affleck-Alvarez-Manton 82](#AffleckAlvarezManton82)) is a [[non-perturbative effect]] of [[vacuum polarization]] expected in [[quantum electrodynamics]], where a strong [[electric field]] causes [[electron]]/[[positron]]-[[pairs]] to appear out of the [[vacuum]].  The analogous effect in [[quantum chromodynamics]] would lead to [[deconfinement]] of [[quarks]] in a strong [[electric field]].

While the effect is clearly predicted by established [[theory (physics)|theory]], it has not been observed in [[experiment]] yet, since the required electric [[field strengths]] are so large. But recent experiments get close to the required intensities ([Dunne 09](#Dunne09)).

## The field strengths

We consider $(\vec E, \vec B)$ a [[constant function|constant]] [[electromagnetic field]] on 4d [[Minkowski spacetime]] in a given Lorentz frame.

Write

\[
  \label{ActualFieldStrengths}
  \begin{aligned}
    E
    & \coloneqq
    \sqrt{
      \vec E \cdot \vec E
    }
    \\
    B
    & \coloneqq
    \sqrt{
      \vec B \cdot \vec B
    }
  \end{aligned}
\]

for the [[norm]] of these field vectors. These, of course, depend on the choice of Lorentz frame. For the Schwinger effect the relevant [[Lorentz group|Lorentz]] [[invariants]] are

\[
  \label{LorentzInvariantFieldStrenghts}
  \begin{aligned}
    \mathcal{E}
     \;\coloneqq\;
     \sqrt{
      \sqrt{
        \left(
          \tfrac{1}{2}
            \big(
              \vec E \cdot \vec E
              -
              \vec B \cdot \vec B
            \big)
        \right)^2
        +
        \big(
          \vec E \cdot \vec B
        \big)^2
      }
      +
      \tfrac{1}{2}
      \big(
        \vec E \cdot \vec E
        -
        \vec B \cdot \vec B
      \big)
    }
    \\
    \mathcal{B}
     \;\coloneqq\;
    \sqrt{
      \sqrt{
        \left(
          \tfrac{1}{2}
            \big(
              \vec B \cdot \vec B
              -
              \vec E \cdot \vec E
            \big)
        \right)^2
        +
        \big(
          \vec E \cdot \vec B
        \big)^2
      }
      +
      \tfrac{1}{2}
      \big(
        \vec B \cdot \vec B
        -
        \vec E \cdot \vec E
      \big)
    }
  \end{aligned}
\]

(reviewed in [Dunne 04, (1.6)](#Dunne04))

Noticing that if $\vec E \cdot \vec B \neq 0$ then there is a [[Lorentz transformation]] to $(\vec E', \vec B')$ such that the electric field is strictly parallel to the magentic field $\vec E \parallel \vec B$, these invariants are more explicity given as follows:

$$
  \begin{aligned}
  \mathcal{E}
  \; = \;
  \left\{
    \array{
      \sqrt{
        \vec E \cdot \vec E - \vec B \cdot \vec B
      }
      &
      \vert
      &
      \mathrlap{
        \vec E \cdot \vec B = 0
      }
      \\
      E'
      &\vert&
      \mathrlap{
      \vec E \cdot \vec B \neq 0
      \;\;
      \Rightarrow
      \underset{
        {
          Lorentz
          \atop
          transformation
        }
        \atop
        {
          (\vec E, \vec B)
          \mapsto
          (\vec E', \vec B')
        }
      }{\exists}
        \vec E' \parallel \vec B'
      }
    }
  \right.
  \\
  \mathcal{B}
  \; = \;
  \left\{
    \array{
      \sqrt{
        \vec B \cdot \vec B
        -
        \vec E \cdot \vec E
      }
      &
      \vert
      &
      \mathrlap{
        \vec E \cdot \vec B = 0
      }
      \\
      B'
      &\vert&
      \mathrlap{
      \vec E \cdot \vec B \neq 0
      \;\;
      \Rightarrow
      \underset{
        {
          Lorentz
          \atop
          transformation
        }
        \atop
        {
          (\vec E, \vec B)
          \mapsto
          (\vec E', \vec B')
        }
      }{\exists}
        \vec E' \parallel \vec B'
      }
    }
  \right.
  \end{aligned}
$$

\linebreak

## The Schwinger effect

### For general field strength at small coupling

Assuming

* **parallel field components:**  $\vec E \cdot \vec B \neq 0$;

and

* **weak coupling**: $e$ small;

the rate of pair creation out of the [[vacuum]] of [[spinor]] [[particles]] of [[electric charge]] $e$ and [[mass]] $m$ is

$$
  \label{SchwingerEffectAtGeneralFieldStrengthForWeakCoupling}
  \Gamma_{
   {
     e\,small
   }
  }
  \;=\;
  \frac
  {
    e^2 \mathcal{E} \mathcal{B}
  }{
    8 \pi^2
  }
  \underoverset
    {n = 1}
    {\infty}
    {\sum}
  \frac{1}{n}
  \coth
  \left(
    \frac{\mathcal{B}}{\mathcal{E}} n \pi
  \right)
  \exp
  \left(
    -
    \frac{
      m^2 \pi n
    }{
      e \mathcal{E}
    }
  \right)
$$

For [[electron]]/[[positron]] pair-creation in [[electromagnetism]] this is due to [Nikishov 69](#Nikishov69), [Bunkin-Tugov 70](#BunkinTugov70), reviewed in [Dunne 04, (1.28)](#Dunne04). For [[quark]]/[[antiparticle|ani-]][[quark]] pair creation in [[quantum chromodynamics]] the analogous formula is due to [Suganuma-Tatsumin 91](#SuganumaTatsumin91), [Suganuma-Tatsumi 93](#SuganumaTatsumi93), reviewed in [Hidaka-Iritani-Suganuma 11, (2)](#HidakaIritaniSuganuma11).

In the [[limit of a sequence|limit]] $\mathcal{B} \to 0$, using that

$$
  \underset{
    \underset{
      x \to 0
    }{
      \longrightarrow
    }
  }{\lim}
  \,
  x \coth(x)
  \;=\;
  x
  \,,
$$

this reduces to

\[
  \label{ForVanishingBFieldAtWeakCoupling}
  \Gamma_{
   {
    {
      \mathcal{B} = 0
    }
   }
   \atop
   {
     e\,small
   }
  }
  \;\coloneqq\;
  \underset{
    \underset{
      B \to 0
    }{
      \longrightarrow
    }
  }{\lim}
  \,
  \Gamma
  \;=\;
  \frac
  {
    e^2 \mathcal{E}^2
  }{
    8 \pi^3
  }
  \underoverset
    {n = 1}
    {\infty}
    {\sum}
  \frac{1}{n^2}
  \exp
  \left(
    -
    \frac{
      m^2 \pi n
    }{
      e \mathcal{E}
    }
  \right)
\]

This is due to [Schwinger 51](#Schwinger51) (review in [Dunne 04, (1.25)](#Dunne04))

### For small field strength at small coupling

Assuming in addition

* **weak fields:**: $\mathcal{E}$ small compared to $m$

the expression (eq:ForVanishingBFieldAtWeakCoupling) simplifies to

\[
  \label{ForWeakFieldsAndWeakCoupling}
  \Gamma_{
   {
    {
      \mathcal{B} = 0
    }
    \atop
    {
      \mathcal{E}\, small
    }
   }
   \atop
   {
     e\,small
   }
  }
  \;\coloneqq\;
  \frac
  {
    e^2 \mathcal{E}^2
  }{
    8 \pi^3
  }
  \exp
  \left(
    -
    \frac{
      m^2 \pi
    }{
      e \mathcal{E}
    }
  \right)
\]

For [[electron]]/[[positron]] pair-creation in [[electromagnetism]] this is originally due to [Heisenberg-Euler 36](#HeisenbergEuler36), reviewed in [Dunne 04, (1.10)](#Dunne04). For [[quark]]/[[antiparticle|ani-]][[quark]] pair creation in [[quantum chromodynamics]] the analogous formula is due to [Suganuma-Tatsumin 91](#SuganumaTatsumin91), [Suganuma-Tatsumi 93](#SuganumaTatsumi93), reviewed in [Hidaka-Iritani-Suganuma 11, (3)](#HidakaIritaniSuganuma11).

### For small field strength at strong coupling

For strong coupling $e$ the expression (eq:ForWeakFieldsAndWeakCoupling) get corrected to

$$
  \Gamma_{
   {
    {
      \mathcal{E}\, small
    }
    \atop
    {
      \mathcal{B} = 0
    }
   }
  }
  \;=\;
  \frac
  {
    e^2 \mathcal{E}^2
  }{
    8 \pi^3
  }
  \exp
  \left(
    -
    \frac{
      m^2 \pi
    }{
      e \mathcal{E}
    }
    +
    \tfrac{1}{4}e^2
  \right)
$$

This was argued in [Affleck-Alvarez-Manton 82](#AffleckAlvarezManton82).

## Properties

### Schwinger limit -- Critical electric field strength
 {#CriticalFieldStrength}

From (eq:SchwingerEffectAtGeneralFieldStrengthForWeakCoupling)
one deduces a critical [[electric field|electric]] [[field strength]] $\mathcal{E}_{crit}$ which sets the [[scale]] beyond which the [[vacuum polarization]] due to the [[Schwinger effect]] counteracts the ambient electric field and/or leads to [[vacuum decay]].

As a [[Lorentz group|Lorentz]] [[invariant]] (eq:LorentzInvariantFieldStrenghts) this _[[Schwinger limit]]_ for the [[electric field|electric]] [[field strength]] is:

\[
  \label{CriticalFieldStrengthLorentzInvariant}
  \mathcal{E}_{crit}
  \;\coloneqq\;
  \frac{
    m^2 c^3
  }{
    e \hbar
  }
\]

([Dunne 04, (1.3)](#Dunne04), [Martin 07, (40)](#Martin07))

Here

* $e$ is the [[charge]] of the [[charged particles]],

* $m$ is the [[mass]] of the [[charged particles]],

* $c$ is the [[speed of light]],

* $\hbar$ is [[Planck's constant]].

This is such that the corresponding [[Lorentz force]]

$$
  F_{crit}
  \; \coloneqq \;
  e
  \,
  \mathcal{E}_{crit}
$$

acting over the [[Compton wavelength]] $\lambda_{Comp} \;\coloneqq\; \frac{\hbar }{m c}$ equals the [[rest energy]] $m c^2$ of the given [[charged particle]]:

$$
  \begin{aligned}
    &
    F_{crit} \lambda_{Comp}
    \; = \;
    m c^2
    \\
    \Leftrightarrow
    \;\;\;
    &
    \mathcal{E}_{crit}
    \; = \;
    \frac{
      m c^2
    }{
      e \lambda_{Comp}
    }
  \end{aligned}
$$

Expressing (eq:CriticalFieldStrengthLorentzInvariant) in terms of the corresponding critical value $E_{crit}$ of the actual [[electric field|electric]] [[field strength]] (eq:ActualFieldStrengths) in the given Lorentz frame yields ([Hashimoto-Oka-Sonoda 14b, (2.17)](#HashimotoOkaSonoda14b), [check](https://www.wolframalpha.com/input/?i=solve+for+e+%3A+c++%3D+Sqrt%5B+Sqrt%5B+%281%2F4%29*%28+e%5E2+-+b%5E2+%29%5E2+%2B+%28p%5E2%29*%28e%5E2%29+%5D+%2B+%281%2F2%29*+%28e%5E2+-+b%5E2%29++%5D+)):

$$
  \mathcal{E}(E_{crit}, B)
  \;=\;
  \mathcal{E}_{crit}
  \phantom{AA}
   \Leftrightarrow
  \phantom{AA}
  E_{crit}
  \;=\;
  \mathcal{E}_{crit}
  \sqrt{
    \frac{
      \mathcal{E}_{crit}^2 + B^2
    }
    {
      \mathcal{E}_{crit}^2 + B_{\parallel}^2
    }
  }
$$

This happens to coincide with the critical field strength of the [[DBI-action]], see [there](DBI-action#CriticalFieldStrength).

### In holographic QCD
 {#InHolographicQCD}

It has been argued that in terms of [[intersecting D-brane models]] the Schwinger effect is what is reflected by the non-linearities in the [[DBI-action]] on probe branes in [[AdS/CFT]] ([Semenoff-Zarembo 11](#SemenoffZarembo11)) and on [[flavor branes]] in [[holographic QCD]] ([Hashimoto-Oka 13](#HashimotoOka13), [Hashimoto-Oka-Sonoda 14a](#HashimotoOkaSonoda14a), [Hashimoto-Oka-Sonoda 14b](#HashimotoOkaSonoda14b)). This is now referred to as the _holographic Schwinger effect_.

## References

### In quantum electrodynamics

Discussion of the [[Schwinger effect]] in [[quantum electrodynamics]]:

The original theoretical prediction:

* {#Sauter31} F. Sauter, _Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs_, Z. Physik 69, 742–764 (1931) ([doi:10.1007/BF01339461](https://doi.org/10.1007/BF01339461))

* {#HeisenbergEuler36} [[Werner Heisenberg]], [[Hans Euler]],  _Folgerungen aus der Diracschen Theorie des Positrons_, Z. Physik 98, 714–732 (1936) ([doi:10.1007/BF01343663](https://doi.org/10.1007/BF01343663))

* {#Schwinger51} [[Julian Schwinger]], _On Gauge Invariance and Vacuum Polarization_, Phys. Rev. 82, 664 (1951) ([doi:10.1103/PhysRev.82.664](https://doi.org/10.1103/PhysRev.82.664))

* {#Nikishov69} A. I. Nikishov, _Pair production by a constant external field_, Zh. Eksp. Teor. Fiz. 57 (1969) 1210-1216 ([spire:59436](http://inspirehep.net/record/59436))

* {#BunkinTugov70} F. V. Bunkin, I. I. Tugov, _Possibility of creating electron-positron pairs in avacuum by the focusing of laser radiation_, Sov. Phys. Dokl.14 (1970), 678

* {#AffleckManton82} [[Ian Affleck]], [[Nicholas Manton]], _Monopole pair production in a magnetic field_, Nuclear Physics B Volume 194, Issue 1, (1982) Pages 38-64 (doi:10.1016/0550-3213(82)90511-9

* {#AffleckAlvarezManton82} [[Ian Affleck]], [[Orlando Alvarez]], [[Nicholas Manton]], _Pair production at strong coupling in weak external fields_, Nuclear Physics B Volume 197, Issue 3 (1982) Pages 509-519 (doi:10.1016/0550-3213(82)90455-2)

Discussion via [[worldline formalism]]:

* [[Gerald Dunne]], [[Christian Schubert]], _Worldline Instantons and Pair Production in Inhomogeneous Fields_, Phys. Rev. D72 (2005) 105004 ([arXiv:hep-th/0507174](https://arxiv.org/abs/hep-th/0507174))

* [[Christian Schubert]], *The worldline formalism in strong-field QED* [[arXiv:2304.07404](https://arxiv.org/abs/2304.07404)]

More theoretical background on [[QED]] in [[background fields]]:

* Uwe Hernandez Acosta, Burkhard Kämpfer, *Strong-field QED in Furry-picture momentum-space formulation: Ward identities and Feynman diagrams* [[arXiv:2303.12941](https://arxiv.org/abs/2303.12941)]

Review:

* Walter Dittrich, Holger Gies, _Probing the Quantum Vacuum --
Perturbative Effective Action Approach in Quantum Electrodynamics and its Application_, Springer Tracts in Modern Physics, Vol. 166 ([ISBN 978-3-540-45585-1](https://www.springer.com/gp/book/9783540674283))

* {#Dunne04} [[Gerald Dunne]], _Heisenberg-Euler Effective Lagrangians: Basics and Extensions_, in:  [[Misha Shifman]], [[Arkady Vainshtein]], [[John Wheater]] (eds.), _[[From Fields to Strings -- Circumnavigating Theoretical Physics]]_, pp. 445-522, World Scientific 2005 ([arXiv:hep-th/0406216](https://arxiv.org/abs/hep-th/0406216), [doi:10.1142/9789812775344_0014](https://doi.org/10.1142/9789812775344_0014))

* [[Gerald Dunne]], _The Heisenberg-Euler Effective Action: 75 years on_, Int. J. Mod. Phys. A27 (2012) 1260004 ([arXiv:1202.1557](https://arxiv.org/abs/1202.1557))

* Francois Gelis, Naoto Tanji, _Schwinger mechanism revisited_, Progress in Particle and Nuclear Physics Volume 87, March 2016, Pages 1-49 ([arXiv:1510.05451](https://arxiv.org/abs/1510.05451))

See also

* Wikipedia, _[Schwinger effect](https://en.wikipedia.org/wiki/Schwinger_effect)_

* Wikipedia, _[Schwinger limit](https://en.wikipedia.org/wiki/Schwinger_limit)_

Discussion of [[experiments]] that could/should see the [[Schwinger effect]]:

* {#Dunne09} [[Gerald Dunne]], _New Strong-Field QED Effects at ELI: Nonperturbative Vacuum Pair Production_, Eur. Phys. J. D55:327-340, 2009 ([arXiv:0812.3163](https://arxiv.org/abs/0812.3163))

* Hidetoshi Taya, _Mutual assistance between the Schwinger mechanism and the dynamical Casimir effect_ ([arXiv:2003.12061](https://arxiv.org/abs/2003.12061))

* {#Hebenstreit14} Florian Hebenstreit, _A space-time resolved view  of the Schwinger effect_, Frontiers of intense laser physics – KITP 2014 ([[HebenstreitSchwingerEffect2014.pdf:file]])

* A. Fedotov et al.,  Section 8 of: *Advances in QED with intense background fields* [[arXiv:2203.00019](https://arxiv.org/abs/2203.00019)]

* M. Pentia, C. R. Badita, D. Dumitriu, A. R. Ionescu, H. Petrascu, *Nonperturbative QED Processes at ELI-NP* [[arXiv:2307.09315](https://arxiv.org/abs/2307.09315)]

and in relation to [[magnetic monopoles]]:

* B. Acharya et al., *First experimental search for production of magnetic monopoles via the Schwinger mechanism* ([arXiv:2106.11933](https://arxiv.org/abs/2106.11933))

Discussion in [[cosmic inflation|inflationary]] [[cosmology]]:

* {#Martin07} Jerome Martin, _Inflationary Perturbations: the Cosmological Schwinger Effect_, Lect. Notes Phys. 738:193-241, 2008 ([arXiv:0704.3540](https://arxiv.org/abs/0704.3540), [doi:10.1007/978-3-540-74353-8_6](https://doi.org/10.1007/978-3-540-74353-8_6))

See also:

* Shintaro Takayoshi, Jianda Wu, Takashi Oka, _Twisted Schwinger Effect: Pair Creation in Rotating Fields_ ([arXiv:2005.01755](https://arxiv.org/abs/2005.01755))

* Prasant Samantray, Suprit Singh, _Schwinger Effect in Compact Space_ ([arXiv:2010.13453](https://arxiv.org/abs/2010.13453))

### In quantum chromodynamics

Discussion of the [[Schwinger effect]] in [[quantum chromodynamics]]:

* Asim Yildiz, Paul H. Cox, _Vacuum Behavior in Quantum Chromodynamics_, Phys. Rev. D21 (1980) 1095 ([spire:7860](http://inspirehep.net/record/7860))

* M. Claudson, Asim Yildiz, Paul H. Cox, _Vacuum behavior in quantum chromodynamics. II_, Phys. Rev. D 22, 2022 (1980) ([doi:10.1103/PhysRevD.22.2022](https://doi.org/10.1103/PhysRevD.22.2022))

* Paul H. Cox, Asim Yildiz, _$q \bar q$ pair creation: A field-theory approach_, Phys. Rev. D 32, 819 (1985) ([doi:10.1103/PhysRevD.32.819](https://doi.org/10.1103/PhysRevD.32.819))

* {#SuganumaTatsumin91} Hideo Suganuma, Toshitaka Tatsumim, _On the behavior of symmetry and phase transitions in a strong electromagnetic field_, Annals of Physics Volume 208, Issue 2, June 1991, Pages 470-508 (doi:10.1016/0003-4916(91)90304-Q)

* {#SuganumaTatsumi93} Hideo Suganuma, Toshitaka Tatsumi, _Chiral Symmetry and Quark-Antiquark Pair Creation in a Strong Color-Electromagnetic Field_, Progress of Theoretical Physics, Volume 90, Issue 2, August 1993, Pages 379–404 ([spire:318092](http://inspirehep.net/record/318092), [doi:10.1143/ptp/90.2.379](https://doi.org/10.1143/ptp/90.2.379))

* Naoto Tanji, _Dynamical view of pair creation in uniform electric and magnetic fields_, Annals Phys. 324:1691-1736, 2009 ([arXiv:0810.4429](https://arxiv.org/abs/0810.4429))

* {#HidakaIritaniSuganuma11} Yoshimasa Hidaka, Takumi Iritani, Hideo Suganuma, _Fast Vacuum Decay into Quark Pairs in Strong Color Electric and Magnetic Fields_, AIP Conference Proceedings 1388, 516 (2011) ([arXiv:1103.3097](https://arxiv.org/abs/1103.3097), [doi:10.1063/1.3647442]( https://doi.org/10.1063/1.3647442))

* Sho Ozaki, Takashi Arai, Koichi Hattori, Kazunori Itakura, _Euler-Heisenberg-Weiss action for QCD+QED_, Phys. Rev. D 92, 016002 (2015) ([arXiv:1504.07532](https://arxiv.org/abs/1504.07532))

* Koichi Hattori, Kazunori Itakura, Sho Ozaki, _Note on all-order Landau-level structures of the Heisenberg-Euler effective actions for QED and QCD_ ([arXiv:2001.06131](https://arxiv.org/abs/2001.06131))

* Patrick Copinger, Pablo Morales, _Schwinger Pair Production in $SL(2,\mathbb{C})$  Topologically Non-Trivial Fields via Non-Abelian Worldline Instantons_ ([arXiv:2011.12526](https://arxiv.org/abs/2011.12526))

In [[quantum hadrodynamics]]:

* William R. Tavares, Sidney S. Avancini, _Schwinger mechanism in the $SU(3)$ Nambu--Jona-Lasinio model with an electric field_, Phys. Rev. D 97, 094001 (2018) ([arXiv:1801.10566](https://arxiv.org/abs/1801.10566))

[[!include holographic Schwinger effect -- references]]

[[!redirects Schwinger effects]]

[[!redirects holographic Schwinger effect]]
[[!redirects holographic Schwinger effects]]


Schwinger limit

+-- {: .rightHandSide}
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### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum electrodynamics]] and [[quantum chromodynamics]] the _Schwinger limit_ is a maximal [[scale]]

$$
  \;\approx\;
  \frac{
    m^2 c^3
  }{
    e \hbar
  }
$$

of the [[field strength]] of an [[electric field|electric]] [[background field]] beyond which the [[vacuum polarization]] caused by pair creation [[electrons]]/[[positrons]] or [[quarks]]/[[antiparticle|anti]]-[[quarks]] (of [[charge]] $e$ and [[mass]] $m$)  out of the [[vacuum]], via the [[Schwinger effect]], becomes sizeable, leading to non-linearities and/or [[vacuum decay]].

From the perspective of [[geometric engineering of QFT|geometric engineering]] of [[QCD]] in [[intersecting D-brane models]] ([[holographic QCD]]) the Schwinger limit corresponds to the limiting field strength in the [[DBI-action]] of the [[Chan-Paton gauge field]] on [[D-branes]] ("[[holographic Schwinger effect]]").

The [[Schwinger effect]] and its resulting [[Schwinger limit]] are securely prediuted by established [[quantum field theory]], but have not been observed in [[experiment]] yet. However, recent experiments are getting very close and upcoming experiments might see the effect.

## Details

From the rate

$$
  \label{SchwingerEffectAtGeneralFieldStrengthForWeakCoupling}
  \Gamma
   \;=\;
  \frac
  {
    e^2 \mathcal{E} \mathcal{B}
  }{
    8 \pi^2
  }
  \underoverset
    {n = 1}
    {\infty}
    {\sum}
  \frac{1}{n}
  \coth
  \left(
    \frac{\mathcal{B}}{\mathcal{E}} n \pi
  \right)
  \exp
  \left(
    -
    \frac{
      m^2 \pi n
    }{
      e \mathcal{E}
    }
  \right)
$$

([here](Schwinger+effect#eq:SchwingerEffectAtGeneralFieldStrengthForWeakCoupling)) of particle pair creation via the [[Schwinger effect]]  one deduces a critical [[electric field|electric]] [[field strength]] $\mathcal{E}_{crit}$ which sets the [[scale]] beyond which the [[vacuum polarization]] due to the [[Schwinger effect]] counteracts the ambient electric field and/or leads to [[vacuum decay]].

As a [[Lorentz group|Lorentz]] [[invariant]] ([here](Schwinger+effect#eq:LorentzInvariantFieldStrenghts))
 this _[[Schwinger limit]]_ for the [[electric field|electric]] [[field strength]] is:

\[
  \label{CriticalFieldStrengthLorentzInvariant}
  \mathcal{E}_{crit}
  \;\coloneqq\;
  \frac{
    m^2 c^3
  }{
    e \hbar
  }
\]

([Dunne 04, (1.3)](#Dunne04), [Martin 07, (40)](#Martin07))

Here

* $e$ is the [[charge]] of the [[charged particles]],

* $m$ is the [[mass]] of the [[charged particles]],

* $c$ is the [[speed of light]],

* $\hbar$ is [[Planck's constant]].

This is such that the corresponding [[Lorentz force]]

$$
  F_{crit}
  \; \coloneqq \;
  e
  \,
  \mathcal{E}_{crit}
$$

acting over the [[Compton wavelength]] $\lambda_{Comp} \;\coloneqq\; \frac{\hbar }{m c}$ equals the [[rest energy]] $m c^2$ of the given [[charged particle]]:

$$
  \begin{aligned}
    &
    F_{crit} \lambda_{Comp}
    \; = \;
    m c^2
    \\
    \Leftrightarrow
    \;\;\;
    &
    \mathcal{E}_{crit}
    \; = \;
    \frac{
      m c^2
    }{
      e \lambda_{Comp}
    }
  \end{aligned}
$$

## Properties

### Relation to DBI-limit

Expressing (eq:CriticalFieldStrengthLorentzInvariant) in terms of the corresponding critical value $E_{crit}$ of the actual [[electric field|electric]] [[field strength]] ([here](Schwinger+effect#eq:ActualFieldStrengths)) in the given Lorentz frame yields ([Hashimoto-Oka-Sonoda 14b, (2.17)](#HashimotoOkaSonoda14b), [check](https://www.wolframalpha.com/input/?i=solve+for+e+%3A+c++%3D+Sqrt%5B+Sqrt%5B+%281%2F4%29*%28+e%5E2+-+b%5E2+%29%5E2+%2B+%28p%5E2%29*%28e%5E2%29+%5D+%2B+%281%2F2%29*+%28e%5E2+-+b%5E2%29++%5D+)):

$$
  \mathcal{E}(E_{crit}, B)
  \;=\;
  \mathcal{E}_{crit}
  \phantom{AA}
   \Leftrightarrow
  \phantom{AA}
  E_{crit}
  \;=\;
  \mathcal{E}_{crit}
  \sqrt{
    \frac{
      \mathcal{E}_{crit}^2 + B^2
    }
    {
      \mathcal{E}_{crit}^2 + B_{\parallel}^2
    }
  }
$$

This happens to coincide with the critical field strength of the [[DBI-action]], see [there](DBI-action#CriticalFieldStrength).

## Related concepts

[[!include fundamental scales -- contents]]

## References

### General

Review:

* {#Dunne04} [[Gerald Dunne]], _Heisenberg-Euler Effective Lagrangians: Basics and Extensions_, in:  [[Misha Shifman]], [[Arkady Vainshtein]], [[John Wheater]] (eds.), _[[From Fields to Strings -- Circumnavigating Theoretical Physics]]_, pp. 445-522, World Scientific 2005 ([arXiv:hep-th/0406216](https://arxiv.org/abs/hep-th/0406216), [doi:10.1142/9789812775344_0014](https://doi.org/10.1142/9789812775344_0014))

* {#Martin07} Jerome Martin, around (40) in: _Inflationary Perturbations: the Cosmological Schwinger Effect_, Lect. Notes Phys. 738:193-241, 2008 ([arXiv:0704.3540](https://arxiv.org/abs/0704.3540), [doi:10.1007/978-3-540-74353-8_6](https://doi.org/10.1007/978-3-540-74353-8_6))

For more see the references at _[[Schwinger effect]]_.

See also

* Wikipedia, _[Schwinger limit](https://en.wikipedia.org/wiki/Schwinger_limit)_

### Experimental realization

Discussion of [[experiments]] that could/should see physics at the [[Schwinger limit]]:

* {#Dunne09} [[Gerald Dunne]], _New Strong-Field QED Effects at ELI: Nonperturbative Vacuum Pair Production_, Eur. Phys. J. D55:327-340, 2009 ([arXiv:0812.3163](https://arxiv.org/abs/0812.3163))

* Hidetoshi Taya, _Mutual assistance between the Schwinger mechanism and the dynamical Casimir effect_ ([arXiv:2003.12061](https://arxiv.org/abs/2003.12061))

* Florian Hebenstreit, _A space-time resolved view  of the Schwinger effect_, Frontiers of intense laser physics – KITP 2014 ([[HebenstreitSchwingerEffect2014.pdf:file]])

[[!include holographic Schwinger effect -- references]]


Schwinger-Dyson equation

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###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]] the _Schwinger-Dyson equation_ ([Dyson 49](#Dyson49), [Schwinger 51](#Schinger51)) equates, [[on-shell]], the [[time-ordered product]] of the [[functional derivative]] of the [[action functional]] $S$ for a [[free field theory]] and another [[observable]] $A$ with the [[time ordered product|time ordering]] of the corresponding [[functional derivative]] of just $A$ itself, times $i \hbar$ ([[imaginary unit]] times [[Planck's constant]]):

$$
 \label{EquationSchwingerDyson}
  \mathcal{T}
  \left(
    \underset{\Sigma}{\int}
      \frac{\delta S}{\delta \mathbf{\Phi}^a(x)}
      \cdot
      A^a(x)
      \, dvol_\Sigma(x)
  \right)
  \;=\;
  i \hbar
  \mathcal{T}
  \left(
    \underset{\Sigma}{\int}
    \frac{\delta A^a(x)}{\delta \mathbf{\Phi}^a(x)}
    \,
    dvol_\Sigma(x)
  \right)
  \phantom{AAA}
  \text{on-shell}
  \,.
$$

(e.q. [Henneaux-Teitelboim 92, (15.25)](#HenneauxTeitelboim92), [Rejzner 16, remark 7.7](#Rejzner16)).

This may be understood as a special case of the quantum-correction of the [[BV-differential]] by the [[BV-operator]] in [[pQFT]], which is hence also called the _[[Schwinger-Dyson operator]]_; see [there](BV-operator#SchwingerDysonEquation).

Often (eq:EquationSchwingerDyson) is displayed before spacetime-smearing of observables in terms of [[operator products]] of [[operator-valued distributions]]

$$
  A^a(x)
    \;\coloneqq\;
  \delta(x-x_0) \delta^a_{a_0}
  \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
$$

which makes the [[distribution|distributional]] [[Schwinger-Dyson equation]] read

$$
  \begin{aligned}
  & T
  \left(
    \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
      \cdot
    \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
  \right)
  \\
  &
  \underset{\text{on-shell}}{=}
  i \hbar
  \underset{k}{\sum}
  T
  \left(
    \mathbf{\Phi}^{a_1}(x_1)
      \cdots
    \mathbf{\Phi}^{a_{k-1}}(x_{k-1})
      \cdot
    \delta(x_0 - x_k) \delta^{a_0}_{a_k}
      \cdot
    \mathbf{\Phi}^{a_{k+1}}(x_{k+1})
      \cdots
    \mathbf{\Phi}^{a_n}(x_m)
  \right)
  \end{aligned}
$$

(e.q. [Dermisek 09](#Dermisek09))

In particular this means that if $(x_0,a_0) \neq (x_k, a_k)$ for all $k \in \{1,\cdots ,n\}$ then

$$
  T
  \left(
    \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
      \cdot
    \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
  \right)
  \;=\;
  0
  \phantom{AAA}
  \text{on-shell}
$$

Since (by the [[principle of extremal action]]) the equation

$$
  \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
  \;=\;
  0
$$

is  the [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] (for the [[classical field theory]]) "at $x_0$", this may be interpreted as saying that the classical equations of motion for fields at $x_0$ still hold for [[time-ordered product|time-ordered]] [[quantum theory|quantum]] [[expectation values]], as long as all other observables are evaluated away from $x_0$; while if observables do coincide at $x_0$ then there is a correction (governed by the [[BV-operator]] of the theory, see [this prop.](BV-operator#DysonSchwinger)).

## Details

For details and proof see at _[[BV-operator]]_ [this prop.](BV-operator#DysonSchwinger), following [Rejzner 16, remark 7.7](#Rejzner16), following [Henneaux-Teitelboim 92, section 15.5.3](#HenneauxTeitelboim92)

## Related concepts

* [[path integral]]

* [[Feynman diagram]]

## References

Precursor discussion for [[quantum mechanics]] (QFT in $1+0$-dimensions) goes back to

* {#Feynman48} [[Richard P. Feynman]], equation (46) in: *Space-Time Approach to Non-Relativistic Quantum Mechanics*, Rev. Mod. Phys. **20** (1948) 367 [[doi:10.1103/RevModPhys.20.367](https://doi.org/10.1103/RevModPhys.20.367), [pdf](http://www-f1.ijs.si/~ramsak/km1/feynman.pdf)]

The Dyson-Schwinger equation is named after:

* {#Dyson49} [[Freeman Dyson]], _The S Matrix in Quantum Electrodynamics_, Phys. Rev. 75: 1736 (1949) ([doi:10.1103/PhysRev.75.1736](https://doi.org/10.1103/PhysRev.75.1736))

* {#Schinger51} [[Julian Schwinger]], _On Green's functions of quantized fields I + II_, PNAS. 37: 452–459 (1951) ([doi:10.1073/pnas.37.7.452](https://doi.org/10.1073%2Fpnas.37.7.452))

The traditional informal account in terms of [[path integral]]-heuristics is reviewed for instance in

* {#HenneauxTeitelboim92} [[Marc Henneaux]], [[Claudio Teitelboim]], section 15.1.4, 15.5.1, 15.5.3 of _[[Quantization of Gauge Systems]]_, Princeton University Press 1992.

* {#Dermisek09} [[Radovan Dermisek]], _Schwinger-Dyson equations_, 2009  ([pdf](http://www.physics.indiana.edu/~dermisek/QFT_09/qft-II-4-4p.pdf))

A rigorous derivation in terms of [[BV-formalism]] in [[causal perturbation theory]]/[[pAQFT]] is provided in

* {#Rejzner16} [[Katarzyna Rejzner]], remark 7.7 _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

For discussion in the context of the [[master Ward identity]], see

* {#DuetschFredenhagen02} [[Michael Dütsch]], [[Klaus Fredenhagen]], _The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory_, Commun.Math.Phys. 243 (2003) 275-314 ([arXiv:hep-th/0211242](https://arxiv.org/abs/hep-th/0211242))

See also

* Wikipedia, _[Schwinger-Dyson equation](https://en.wikipedia.org/wiki/Schwinger%E2%80%93Dyson_equation)_

Discussion of [[round chord diagrams]] organizing Dyson-Schwinger equations:

* Nicolas Marie, [[Karen Yeats]], *A chord diagram expansion coming from some Dyson-Schwinger equations*, Communications in Number Theory and Physics, 7(2):251291, 2013 ([arXiv:1210.5457](https://arxiv.org/abs/1210.5457))

* Markus Hihn, [[Karen Yeats]], *Generalized chord diagram expansions of Dyson-Schwinger equations*, Ann. Inst. Henri Poincar Comb. Phys. Interact. 6 no 4:573-605 ([arXiv:1602.02550](https://arxiv.org/abs/1602.02550))

* Paul-Hermann Balduf, Amelia Cantwell, [[Kurusch Ebrahimi-Fard]], Lukas Nabergall, Nicholas Olson-Harris, [[Karen Yeats]], *Tubings, chord diagrams, and Dyson-Schwinger equations* \[arXiv:2302.02019\]

Review in:

* Ali Assem Mahmoud, Section 3 of: *On the Enumerative Structures in Quantum Field Theory* ([arXiv:2008.11661](https://arxiv.org/abs/2008.11661))

[[!redirects Schwinger-Dyson equations]]

[[!redirects Dyson-Schwinger equation]]
[[!redirects Dyson-Schwinger equations]]


Schwinger-Keldysh formalism

__Schwinger-Keldysh formalism__ is an approach to the treatment of non-equilibrium quantum systems pioneered by Keldysh and in an archetypal case earlier by [[Julian Schwinger]].

## Literature

Historical papers:

* L. V. Keldysh, _Diagram technique for nonequilibrium processes_, JETP 20:4 (1965) [pdf](http://jetp.ras.ru/cgi-bin/dn/e_020_04_1018.pdf)
* J.S. Schwinger, _Brownian motion of a quantum oscillator_, J. Math. Phys. 2 (1961) 407 [doi]( https://doi.org/10.1063/1.1703727)

Recent papers

* Mohammad F. Maghrebi, Alexey V. Gorshkov, _Nonequilibrium many-body steady states via Keldysh formalism_, Phys. Rev. B 93, 014307 [doi](https://doi.org/10.1103/PhysRevB.93.014307)
* Paolo Glorioso, Michael Crossley, Hong Liu, _A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems_, [arXiv:1812.08785](https://arxiv.org/abs/1812.08785)

This formalism also involves the Feynman-Vernon influence functional also known as the Schwinger-Keldysh path integral.

* Yoni BenTov, _Schwinger-Keldysh path integral for the quantum harmonic oscillator_, [arXiv:2102.05029](https://arxiv.org/abs/2102.05029)

Recent connection to a case of BRST complex and related equivariant cohomology

* F.M. Haehl, R. Loganayagam, M. Rangamani, _Schwinger-Keldysh formalism I: BRST symmetries and superspace_, [arXiv:1610.01940](https://arxiv.org/abs/1610.01940) J. High Energ. Phys. __69__ (2017) doi; _Schwinger-Keldysh formalism. Part II: thermal equivariant cohomology_, J. High Energ. Phys. 2017, 70 (2017) doi

category: physics


Schwinger-Tomonaga-Feynman-Dyson

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[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The origins of [[perturbative quantum field theory]] go back to informal ideas on [[quantum electrodynamics]] due to [[Julian Schwinger]], [[Shin'ichirō Tomonaga]], [[Richard Feynman]] and [[Freeman Dyson]], culminating in the informal idea of "[[renormalization]]" due to ([Dyson 49](#Dyson49)).

While highly successful, the conceptual nature of this original formulation, in particular of the process of "removal of [[UV-divergences]]", had remained mysterious (see [Scharf 95, section 0.0](#Scharf95) for survey):

> $[$ the theory is $]$ an ugly and incomplete one ([Dirac 51](#Dirac51))

> I think that the renormalization theory is simply a way to sweep the difficulties of the divergences  of electrodynamics under the rug. I am, of course, not sure of that. ([Feynman 66, Nobel lecture](#Feynman66))

> {#Feynman85SuchABunchOfWords} ...is technically called 'renormalization.' But no matter how clever the word, it is what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory $[...]$ is self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; $[...]$ What is certain is that we do not have a good mathematical way to describe the theory of quantum electrodynamics: such a bunch of words... ([Feynman 85, Chap. 4. "Loose Ends"](#Feynman85))

These conceptual mysteries were resolved in a mathematically rigorous formulation of [[renormalization|("re"-)normalization]] in [[perturbative QFT]] by ([Epstein-Glaser 73](perturbative+quantum+field+theory#EpsteinGlaser73)), based on ([Bogoliubov-Shirkov 59](perturbative+quantum+field+theory#BogoliubovShirkov59) and [Stückelberg 51](perturbative+quantum+field+theory#Stueckelberg51)), now known as _[[causal perturbation theory]]_; laid out, together with other rigorous approaches, in the seminal Erice summer school proceedings ([Velo-Wightman 76](perturbative+quantum+field+theory#VeloWightman76)) and later developed by ([Scharf 95](#Scharf95), [Scharf 01](causal+perturbation+theory#Scharf01)) and eventually grown into _[[perturbative AQFT]]_ (see there for more).

## References

### Original articles

The [[renormalization]] process was first suggested in

* {#Dyson49} [[Freeman Dyson]], _The radiation theories of Tomonaga, Schwinger and Feynman_, Phys. Rev. 75, 486, 1949 ([pdf](http://web.ihep.su/dbserv/compas/src/dyson49b/eng.pdf))

### Review and history

* {#Dirac51} [[Paul Dirac]], Proc. Roy. Soc. A 209, 291 (1951)

* {#Feynman66} [[Richard Feynman]], Nobel lecture, reproduced in Science 153, 699 (1966)

* {#Feynman85} [[Richard Feynman]], _[QED: The Strange Theory of Light and Matter](https://en.wikipedia.org/wiki/QED:_The_Strange_Theory_of_Light_and_Matter)_, 1985

* Silvan Schweber, _QED and the men who made it: Dyson, Feynman Schwinger and Tomonaga_, Princeton Series in Physics, 1994

* {#Scharf95} [[Günter Scharf]], section 0.0 of _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

category: reference


science

## Related entries

* [[philosophy]]

* [[mathematics]]

* [[physics]]

## References

* Wikipedia, _[Science](http://en.wikipedia.org/wiki/Science)_

[[!redirects sciences]]