bit flip code

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### Context
#### Quantum systems
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#### Computation
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[[!include constructivism - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[quantum information theory]] a *bit flip code* is a [[quantum error correcting code]] to correct for [[quantum noise]] caused by [[bit flip channels]].

## Details

### 3-qbit flip code

The following basic example of a bit flip code encodes single [[logical qbits]] in [[triples]] of [[physical qbits]].

#### Idea

Here the [[Hilbert space]]

$$
  LogicalQBit
  \;\coloneqq\;
  QBit \otimes QBit \otimes QBit
$$

of a single logical qbit has [[complex number|complex]] [[dimension of a vector space|dimension]] $2^3 \,=\, 8$. Inside this is the space of undisturbed logical qbits

\[
  \label{PureLogicalQBit}
  \begin{array}{ccccc}
    QBit
    \xrightarrow[\sim]{encode}
    DisturbedQBit_{0}
    &\overset{\phantom{---}}{\hookrightarrow}&
    LogicalQBit
    &\xrightarrow{
      P_{s = 0}
      \;\coloneqq\;
      \left\vert 0,0,0 \right\rangle
      \left\langle 0,0,0 \right\vert
      +
      \left\vert 1,1,1 \right\rangle
      \left\langle 1,1,1 \right\vert
    }&
    DisturbedQBit_0
    \\
    q_0 \left\vert 0 \right\rangle
    +
    q_1 \left\vert 1 \right\rangle
      &\mapsto&
    q_0 \left\vert 0, 0, 0 \right\rangle
    +
    q_1 \left\vert 1, 1, 1 \right\rangle
  \end{array}
\]

as well as three further orthogonal subspaces which are the images of this one under one of the three *single* bit flips, respectively:

$$
  \begin{array}{ccc}
    DisturbedQBit_{1}
    &\overset{\phantom{---}}{\hookrightarrow}&
    LogicalQBit
    &\xrightarrow{
      P_{s=1}
      \;\coloneqq\;
      \left\vert 1,0,0 \right\rangle
      \left\langle 1,0,0 \right\vert
      +
      \left\vert 0,1,1 \right\rangle
      \left\langle 0,1,1 \right\vert
    }&
    DisturbedQBit_1
    \\
    q_0 \left\vert 0 \right\rangle
    +
    q_1 \left\vert 1 \right\rangle
    &\mapsto&
    q_0 \left\vert 1, 0, 0 \right\rangle
    +
    q_1 \left\vert 0, 1, 1 \right\rangle
    \\
    {\phantom{A}}
    \\
    DisturbedQBit_{2}
    &\overset{\phantom{---}}{\hookrightarrow}&
    LogicalQBit
    &\xrightarrow{
      P_{s=2}
      \;\coloneqq\;
      \left\vert 0,1,0 \right\rangle
      \left\langle 0,1,0 \right\vert
      +
      \left\vert 1,0,1 \right\rangle
      \left\langle 1,0,1 \right\vert
    }&
    DisturbedQBit_2
    \\
    q_0 \left\vert 0 \right\rangle
    +
    q_1 \left\vert 1 \right\rangle
    &\mapsto&
    q_0 \left\vert 0, 1, 0 \right\rangle
    +
    q_1 \left\vert 1, 0, 1 \right\rangle
    \\
    {\phantom{A}}
    \\
    DisturbedQBit_{3}
    &\overset{\phantom{---}}{\hookrightarrow}&
    LogicalQBit
    &\xrightarrow{
      P_{s=3}
      \;\coloneqq\;
      \left\vert 0,0,1 \right\rangle
      \left\langle 0,0,1 \right\vert
      +
      \left\vert 1,1,0 \right\rangle
      \left\langle 1,1,0 \right\vert
    }&
    DisturbedQBit_3
    \\
    q_0 \left\vert 0 \right\rangle
    +
    q_1 \left\vert 1 \right\rangle
    &\mapsto&
    q_0 \left\vert 0, 0, 1 \right\rangle
    +
    q_1 \left\vert 1, 1, 0 \right\rangle
  \end{array}
$$

The 4 labels of these three subspaces are called the "error syndrome", which we collect in the 4-element set

$$
  Synd \;\coloneqq\; \{0,1,2,3\}
$$

to express the logical q-bit space alternatively in this linear basis:

$$
  QBit
  \xhookrightarrow{ encode }
  \underset{
    LogicalQBit
  }{
    \underbrace{
      QBit \otimes QBit \otimes QBit
    }
  }
  \xrightarrow[\sim]{
    (P_s)_{s \colon Synd}
  }
  \underset{
    s \colon Synd
  }{\bigoplus}
  DisturbedQBit_{s}
$$

\begin{remark}
Beware that there is no room to record the further possible errors of 2 qbits flipping at once. Instead, this error process takes pure qbits to one of the subspaces reserved for single bit flips. Similarly, if all three bits flip at once, the initial pure state remains in the subspace reserved for pure states, but now with its phases mixed up. Therefore, the following error correction code fails on flips of more than one of the 3-qbits and will reduce the overall error probability only if single bit flips are sufficiently more likely than double and triple spin flips.
\end{remark}

The error correction now proceeds by:

1. Performing a [[quantum measurement]] of the above error syndrome $s$ on the logical Qbit space. While this measurement [[quantum state collapse|collapses]] the quantum state onto the subspace $DisturbedQBit_s$ the particular logical qbit states of the pure form (eq:PureLogicalQBit) lose no information under this collapse, in that there is a [[unitary operator]] on $DisturbedQBit_s$ which turns any such collapse state back into the original qbit state.

1. applying this correcting unitary $X_s$, which is evidently the "[[Pauli matrix|Pauli X]]-" (or "[[NOT]]-")[[quantum gate]] $X \,\colon\, QBit \to QBit$ acting on the $s$-th qbit:

   $$
     \begin{array}{cccc}
       X_0 &\coloneqq& id \otimes id \otimes id
       \\
       X_1 &\coloneqq& X \otimes id \otimes id
       \\
       X_2 &\coloneqq& id \otimes X \otimes id
       \\
       X_3 &\coloneqq& id \otimes id \otimes X
     \end{array}
     \;\;\;\colon\;\;\;
     \underset{
       LogicalQBit
     }{
       \underbrace{
         QBit \otimes QBit \otimes QBit
       }
     }
     \multimap
     \underset{
       LogicalQBit
     }{
       \underbrace{
         QBit \otimes QBit \otimes QBit
       }
     }
   $$

1. re-preparing a logical QBit from the recovered Qbit.

#### Circuit diagram and pseudo-code
 {#ThreeBitFlipCodeCircuitDiagramsAndPseudoCode}

In terms of [[quantum circuit diagrams]] and the [[schreiber:QS]]-[[pseudo-code]], the 3-bit flip error correcting code looks as follows (following the notation from *[[quantum circuits via dependent linear types]]*):

\begin{imagefromfile}
    "file_name": "QS_3BitFlipCodeIngredients-221125.jpg",
    "width": "720",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

\begin{imagefromfile}
    "file_name": "QS_3BitFlipCode-221125.jpg",
    "width": "700",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

## Related concepts

* [[Æ code]]

## References

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], §10.1.1 in: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

* {#DevittNemotoMunro09} [[Simon J. Devitt]], Kae Nemoto, William J. Munro, §IV of: *Quantum Error Correction for Beginners*, Rep. Prog. Phys. **76** (2013) 076001 [[arXiv:0905.2794](https://arxiv.org/abs/0905.2794), [doi:10.1088/0034-4885/76/7/076001](https://iopscience.iop.org/article/10.1088/0034-4885/76/7/076001)]

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bit-flip quantum channel

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### Context
#### Quantum systems
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[[!include quantum systems -- contents]]
=--
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum information theory]], by the *bit flip channel*
one means the [[quantum channel]] on a [[quantum system]] represented by a single [[q-bit]] whose effect is to "flip" the [[linear basis|basis]] [[q-bit]]  [[quantum states]] $\left\vert 0 \right\rangle \leftrightarrow \left\vert 1 \right\rangle$ with some [[probability]] $p$ or else leave the system unchanged with probability $1 - p$.

(This would more properly be called the *q-bit flip channel*, but this is not commonly used terminology.)

The [[tensor product]] of $N$ bit flip channels model the analogous process on $N$ [[q-bits]], where every single one of them may flip with probability $p$, independently of all the others. In this form, the spin flip channel serves as a simple but important model for [[quantum noise]]. Basic examples of [[quantum error correction codes]] may be used to provide partial correction of such bit flip errors, see at *[[bit flip code]]*.

## Definition

Fix a [[real number]] $p \in [0,1]$ serving as a measure of the [[probability]] for a spin flip to occur when data is sent through the channel.

We write:

* $QBit \,\simeq\, \mathbb{C}\cdot \left\vert 0 \right\rangle \,\oplus\, \mathbb{C} \cdot \left\vert 1 \right\rangle$ for the [[space of quantum states]] of a [[q-bit]], regarded as a [[Hilbert space]] ([[Hermitian inner product space]]) with $\big\{\left\vert 0 \right\rangle, \left\vert 1 \right\rangle \big\}$ being an [[orthonormal linear basis]];

* $States(QBit) \subset Herm(QBit)$ for the space of [[density matrices]] on this Hilbert space, hence the subspace of [[positive semi-definite bilinear form|positive semi-definite]] [[Hermitian operators]] $QBit \multimap QBit$ of unit [[trace]];

* $X \;\colon\; QBit \multimap QBit$ for the "[[Pauli matrix|Pauli X]]" ("[[NOT]]") [[quantum logic gate]] given by $X \left\vert 0 \right\rangle \,=\, \left\vert 1 \right\rangle$, and $X \left\vert 1 \right\rangle \,=\, \left\vert 0 \right\rangle$.

Now, a common way to define the spin flip channel is by the following expression (e.g. [Chuang (2011), p. 296](#Chuang11)):

$$
  \begin{aligned}
    flip
    &\colon\;
    States(QBit)
    \longrightarrow
    States(QBit)
    \\
    flip(\rho)
    &\equiv\;
    (1-p) \cdot \rho \,+\, p \cdot (X \rho X)
  \end{aligned}
$$

Since $X$ is a [[unitary operator]], this makes the bit flip channel an example of a [[mixed unitary quantum channel]].

One readily checks that on [[pure states]] in the [[q-bit]]-[[linear basis|basis]] this acts as expected:

$$
  \begin{array}{ccl}
    States(QBit)
    &\longrightarrow&
    States(QBit)
    \\
    \vert 0 \rangle \langle 0 \rangle
      &\mapsto&
    \mathrlap{
      (1-p) \cdot
      \left\vert 0 \right\rangle
      \left\langle 0 \right\vert
      \;+\;
      \;\;\;\;\;\;\;\;\;
      p \cdot
      \left\vert 1 \right\rangle
      \left\langle 1 \right\vert
    }
    \\
    \vert 1 \rangle \langle 1 \rangle
      &\mapsto&
    \mathrlap{
      \;\;\;\;\;\;\;\;\;
      p \cdot
      \left\vert 0 \right\rangle
      \left\langle 0 \right\vert
      \;+\;
      (1-p) \cdot
      \left\vert 1 \right\rangle
      \left\langle 1 \right\vert
    }
  \end{array}
$$

## Properties

### Equivalent formulations

\begin{example}
The bit flip channel may be expressed as a [[unitary quantum channel]] followed by a [[quantum measurement channel]]:

$$
  flip
  \;\colon\;
  \rho
  \;\mapsto\;
  P_0 \, U_{flip} \, \rho \, U_{flip} \, P_0
  \;+\;
  P_1 \, U_{flip} \, \rho \, U_{flip} \, P_1
  \,,
$$

where the [[unitary operator]] $U_{flip}$ is:

$$
  U_{flip}
  \;=\;
  \left[
  \array{
    + \sqrt{1-p} & \sqrt{p}
    \\
    \sqrt{p} & - \sqrt{1-p}
  }
  \right]
  \;\colon\;
  QBit \multimap QBit
$$
\end{example}

\begin{example}
  The bit-flip quantum channel has an alternative presentation as a *uniformly* [[mixed unitary quantum channel]] and hence as a *[[unistochastic quantum channel]]*, see the discussion [there](unistochastic+quantum+channel#BitFlipAsUnistochasticChannel).
\end{example}

## Related concepts

* [[bit flip code]]

## References

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], §8.1 and §8.3.3 in: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

* {#Chuang11} [[Isaac L. Chuang]], *Quantum error correction*, Chapter 7 in: *Quantum Machines: Measurement and Control of Engineered Quantum Systems* Lecture Notes of the Les Houches Summer School **96** (2011) 273–320  [[doi:10.1093/acprof:oso/9780199681181.003.0007](https://doi.org/10.1093/acprof:oso/9780199681181.003.0007)]

* [[Stéphane Attal]], §6.1.6.1 in: *Quantum Channels*, Lecture 6 in:  *Lectures on Quantum Noises* [[pdf](http://math.univ-lyon1.fr/~attal/Quantum_Channels.pdf), [webpage](http://math.univ-lyon1.fr/~attal/chapters.html)]

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bitcoin

Bitcoin is the first [[blockchain]] and the respective [[cryptocurrency]], it is public and using proof of work. Its sequel, [[Ethereum]] is a more modern blockchain platform enabled for [[smart contract]]s and distributed applications from the start. Bitcoin is largely criticized for energy consuming proof of work, big latency and small number of transaction per second; its crypticurrency, also called bitcoin is rather unstable and often used for speculation and even money laundering, rather than for transactions/settlement. Now more efficient third generation distributed ledger projects exist, see [[high performance distributed ledger]].

* Satoshi Nakamoto, Bitcoin: A peer-to-peer electronic cash system, [pdf](https:
//bitcoin.org/bitcoin.pdf) 2008.

Description from [coindesk](https://www.coindesk.com/price/bitcoin):

> Bitcoin is a [[cryptocurrency]] which isn’t managed by a bank or agency but in which transactions are recorded in the blockchain that is public and contains records of each and every transaction that takes place. The cryptocurrency is traded by individuals with cryptographic keys that act as wallets. Bitcoin was first invented in 2009 by an anonymous founder known as Satoshi Nakamoto. Bitcoins are moved in blocks every 10 minutes on a decentralized ledger that connects blocks into a coherent chain dating back to the first genesis block. It was originally described as a peer-to-peer electronic cash but the technology has evolved to emphasize being a settlement layer rather than a payment network. This has left integrated second layer solutions, like Lightning Network, to prioritize that use case. It has remained the largest cryptocurrency by market cap.

Also in $n$Lab: [[blockchain]], [[high performance distributed ledger]], [[Ethereum]], [[smart contract]], [[virtual machine]]

* zoranskoda: [[zoranskoda:distributed consensus]], [[zoranskoda:blockchain regulation]], [[zoranskoda:blockchain security]], [[zoranskoda:privacy on blockchain]],  [[zoranskoda:Cardano]], [[zoranskoda:EOS]], [[zoranskoda:hyperledger]], [[zoranskoda:cryptocurrency]].

* [bitcoin.stackexchange](https://bitcoin.stackexchange.com)
* [blockchainlibrary.org/2017/10/most-cited-bitcoin-publications](https://blockchainlibrary.org/2017/10/most-cited-bitcoin-publications)
* [bitcoinwiki](https://en.bitcoinwiki.org/wiki/Main_Page)
* wikipedia [blockchain](https://en.wikipedia.org/wiki/Blockchain) [Блокчейн](https://ru.wikipedia.org/wiki/%D0%91%D0%BB%D0%BE%D0%BA%D1%87%D0%B5%D0%B9%D0%BD); [CAP theorem](https://en.m.wikipedia.org/wiki/CAP_theorem), [smart contract](https://en.wikipedia.org/wiki/Smart_contract), [public-key cryptography](https://en.wikipedia.org/wiki/Public-key_cryptography), [elliptic curve cryptography](http://en.wikipedia.org/wiki/Elliptic_curve_cryptography), [lattice cryptography](https://en.wikipedia.org/wiki/Lattice-based_cryptography)
* list.wiki: [blockchain](http://list.wiki/Blockchain), [cryptocurrencies](http://list.wiki/Cryptocurrencies), [tokens](http://list.wiki/Tokens), [TON](http://list.wiki/Telegram_Open_Network_%28TON%29)
* (popularization) [How bitcoin works in 5 minutes](https://www.youtube.com/watch?v=l9jOJk30eQs)
* Krzysztof Okupski, _Bitcoin developer reference_, Working Paper, Last changes 30th July 2016 [pdf](https://www.lopp.net/pdf/Bitcoin_Developer_Reference.pdf)
* Ittay Eyal, Emin Gün Sirer, _Majority is not enough: Bitcoin mining is vulnerable_, in: Financial Cryptography and Data Security 436–454, Springer 2014 [pdf](http://fc14.ifca.ai/papers/fc14_submission_82.pdf)
* Ayelet Sapirshtein, Yonatan Sompolinsky, Aviv Zohar, _Optimal selfish mining strategies in bitcoin_, [arxiv/1507.06183](https://arxiv.org/abs/1507.06183)

Bitcoin is quite centralized, see [top-100-richest-bitcoin-addresses](https://bitinfocharts.com/top-100-richest-bitcoin-addresses.html).

* SegWit [wikipedia](https://en.wikipedia.org/wiki/SegWit), [bitcoinwiki](https://en.bitcoinwiki.org/wiki/Segregated_Witness)

#### Bitcoin lightning network

* [lightning.network](https://lightning.network)
* Joseph Poon, Tadge Dryja, _Lightning Network_, [pdf](https://lightning.network/
lightning-network-paper.pdf) Mar 2015.
* A van Wirdum _Understanding the lightning network [1](https://bitcoinmagazine.com/articles/understanding-the-lightning-network-part-building-a-bidirectional-payment-channel-1464710791) [2](https://bitcoinmagazine.com/articles/understanding-the-lightning-network-part-creating-the-network-1465326903)


bitensoring

## Idea

For the moment see at
_[operad for bimodules - relation to bitensoring](operad+for+bimodules#RelationToBitensoring)_.

## Related concepts

* [[tensoring]]


bitopological space

# Bitopological space
* table of contents
{: toc}

## Definitions

Recall that a [[topological space]] is a [[set]] $X$ equipped with a [[topological structure]] $\mathcal{T}$.  Well, a __bitopological space__ is simply a set equipped with *two* topological structures $(X, \mathcal{T}, \mathcal{T}^*)$.  Unlike with [[bialgebras]], no compatibility condition is required between these structures.

A __bicontinous map__ is a [[function]] between bitopological spaces that is [[continuous map|continuous]] with respect to each topological structure.

Bitopological spaces and bicontinuous maps form a [[category]] $BiTop$.

## Separation axioms for topologies

Let $Cl$ denote the [[closure operator]] with respect to $\mathcal{T}$ and let $Cl^*$ denote the closure operator with respect to $\mathcal{T}^*$.

\begin{proposition}\label{implications}
Let $(X, \mathcal{T}, \mathcal{T}^*)$ be a bitopological space.
Consider the following properties of this space:

1. {#regular1} for each point $x$ there is a $\mathcal{T}$-[[neighborhood base]] consisting  of $\mathcal{T}^*$-closed sets;

1. {#regular2} for all $x\in X$ and all $\mathcal{T}$-opens $U$ containing $x$ there is a $\mathcal{T}^*$-closed $\mathcal{T}$-neighborhood $V$ of $x$ such that $ V \subset U$;

1. {#coupled1} $Cl^*(O) \subset Cl(O)$ for each $\mathcal{T}^*$-open $O$;

1. {#coupled2} for all $x\in X$ and all $\mathcal{T}$-neighborhoods $U$ of $x$ the closure $Cl^*(U)$ is a $\mathcal{T}^*$-neighborhood;

1. {#cotopology2} for each point $x$ and each $\mathcal{T}$-closed $\mathcal{T}$-neighborhood $V$ of $x$ in $X$ there exists a $\mathcal{T}^*$-closed $\mathcal{T}$-neighborhood $U$ of $x$ in $X$ such that $U$ is contained in $V$.

There are the following implications among these properties

\begin{centre}
  \begin{tikzcd}
	 & (2)	& (3) \\
	(5) & (1) & (4)
    \arrow[Rightarrow, from=2-1, to=1-2, "\text{if $\mathcal{T}$ regular}"]
    \arrow[Leftrightarrow, from=1-2, to=2-2]
    \arrow[Leftrightarrow, from=1-3, to=2-3]
    \arrow[Rightarrow, from=2-2, to=2-1]
    \arrow[Rightarrow, from=1-3, to=1-2, "\text{if $\mathcal{T}$ regular}"]
    \arrow[Rightarrow, from=2-2, to=2-3]
\end{tikzcd}
\end{centre}

Especially, all properties are equivalent if $\mathcal{T}$ is [[regular space|regular]].
\end{proposition}

\begin{proof}
__[(1)](#regular1) $\iff$ [(2)](#regular2):__
Given a neighborhood base for a point $x$ as guaranteed by the first property. When you spell out the properties of this neighborhood base, you end up with the second property. For the reverse direction start with an arbitrary $\mathcal{T}$-neighborhood base of a point $x$ consisting of open. Apply the second property to every element of this neighborhood base to the desired neighborhood base.

__[(3)](#coupled1) $\iff$ [(4)](#coupled2):__
Suppose property [(3)](#coupled1), and let $U$ be a $\mathcal{T}$-neighborhood of an arbitrary point $x$. Then the [[complement]] $ \widetilde{Cl^*(U)} $ is in $\mathcal{T}^*$, so that $ Cl^*(\widetilde{Cl^*U}) \subset Cl(\widetilde{Cl^* U}) $ by the first property. Hence $ \widetilde{Cl(\widetilde{Cl^*U})} \subset \widetilde{Cl^*(\widetilde{Cl^* U})} $ for the complements.
Since $U$ is a $\mathcal{T}$-neighborhood of $x$, $x$ does not belong to $Cl(\widetilde{Cl^*U})$.
Moreover, $\widetilde{Cl^*(\widetilde{Cl^*U})} $ is $\mathcal{T}^*$-open and a subset of $ Cl^*(U)$.
Hence $Cl^*(U)$ is a $\mathcal{T}^*$-neighborhood of $x$.

For the converse suppose property [(4)](#coupled2). Let $O$ be a nonempty $\mathcal{T}^*$-open set and $x$ an element of $Cl^*(O)$. Then if $U$ is any $\mathcal{T}$-neighborhood of $x$, some point $y \in O$ belongs to $Cl^*(U)$ due to the second property. Hence, as $O$ is a $\mathcal{T}^*$-neighborhood of $y$, some point of $U$ belongs to $O$. Thus $x \in Cl(O)$, and therefore $Cl^*(G) \subset Cl(O)$.

__[(1)](#regular1) $\implies$ [(4)](#coupled2):__
Given $x\in X$ and a $\mathcal{T}$-neighborhood $U$ by property [(1)](#regular1) there is a $\mathcal{T}^*$-open $O \subset U$ containing $x$. Hence $O \subset Cl^*(U)$, and $Cl^*(U)$ is a $\mathcal{T}^*$-neighborhood.

__[(3)](#coupled1) and $\mathcal{T}$ regular $\implies$ [(2)](#regular2):__
Let $x\in X$ and $U$ be a $\mathcal{T}$-open containing $x$.
By regularity of $\mathcal{T}$ we can find disjoint $\mathcal{T}$-opens $V' \ni x$ and $U' \supset \tilde{U}$ ($\tilde{U}$ denotes the [[complement]]). Set $V \coloneqq Cl^*(V')$. This set is obviously a $\mathcal{T}^*$-closed $\mathcal{T}$-neighborhood of $x$. Due to property [(3)](#coupled1) $V \subset Cl(V')$. Since also $Cl(V') \subset \widetilde{U'}$, we have $V \subset U$. This is to say that $V$ is the $\mathcal{T}$-neighborhood we sought.

__[(5)](#cotopology2) and $\mathcal{T}$ regular $\implies$ [(2)](#regular2):__
Let $x\in X$ and $U$ be a $\mathcal{T}$-open containing $x$. By regularity of $\mathcal{T}$ we can find disjoint $\mathcal{T}$-opens $V' \ni x$ and $U' \supset \tilde{U}$ ($\tilde{U}$ denotes the [[complement]]). Due to property [(5)](#cotopology2) the closed set $\widetilde{U'}$ contains a $\mathcal{T}^*$-closed neighborhood of $x$. This is the neighborhood we sought.

__[(1)](#regular1) $\implies$ [(5)](#cotopology2):__
Given some $\mathcal{T}$-closed $\mathcal{T}$-neighborhood $V$ of some point $x$ choose a neighborhood base according to property [(1)](#regular1) and take an element $U$ therein that is contained in $V$.
\end{proof}

\begin{definition}\label{regular}
Let $(X, \mathcal{T}, \mathcal{T}^*)$ be a bitopological space.
The topology $\mathcal{T}$ is __regular with respect to__ $\mathcal{T}^*$ if  one of the two equivalent conditions [(1)](#regular1) and [(2)](#regular2) from proposition \ref{implications} holds.
A bitopological space $(X, \mathcal{T}, \mathcal{T}^*)$ is called __pairwise regular__ if $\mathcal{T}$ is regular with respect to $\mathcal{T}^*$ and vise versa.
\end{definition}

\begin{definition}\label{coupled}
Let $(X, \mathcal{T}, \mathcal{T}^*)$ be a bitopological space.
The topology $\mathcal{T}^*$ is __coupled to__ $\mathcal{T}$ if one of the two equivalent conditions [(3)](#coupled1) and [(4)](#coupled2) from proposition \ref{implications} holds.
\end{definition}

Not that is if $\mathcal{T}^*$ is coupled to a finer topology $\mathcal{T} \supset \mathcal{T}^*$ then $\mathcal{T}^*$ is coupled to every topology coarser than $\mathcal{T}$ due to property [(3)](#coupled1). Moreover in this case also $\mathcal{T}$ is coupled to $\mathcal{T}^*$ (again a direct consequence of property [(3)](#coupled1)).

\begin{definition}\label{cotopology}
Let $(X, \mathcal{T}, \mathcal{T}^*)$ be a bitopological space.
The topology $\mathcal{T}^*$ is called a __[[cotopology]] of__ $\mathcal{T}$ if $\mathcal{T}^* \subseteq \mathcal{T}$ and property [(5)](#cotopology2) from proposition \ref{implications} holds.
The space $(X, \mathcal{T}^*)$ is also called a __cospace__ of $(X, \mathcal{T}$.
\end{definition}

## Remarks

It is interesting and perhaps surprising that many advanced topological notions can be described using bitopological spaces, even when you would not naively think that there are two topologies around.  (At least, that's my vague memory of what they were good for.  I think that this was in some article by Isbell.)

## Pointfree analogues

Expressed in the opposite category of [[frames]],
there are several pointfree analogues of bitopological spaces:
[[D-frames]], [[biframes]], and [[finitary biframes]].
The latter has the best properties of all three.
For an overview, see [Suarez](#Suarez).

## Related Articles

* [[cotopology]]
* [[topologically complete space]]

## References

* [Wikipedia entry](http://en.wikipedia.org/wiki/Bitopological_space)

* [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], _Abstract and Concrete Categories: The Joy of Cats_, Dover New York 2009. ([pdf](http://katmat.math.uni-bremen.de/acc/acc.pdf)) pp. 59-60, 278

* B. Dvalishvili, _Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures and Applications_, Elsevier Amsterdam 2005.

* [[Peter Johnstone]], _Collapsed Toposes as Bitopological Spaces_, pp. 19-35 in _Categorical Topology_, World Scientific Singapore 1989.

* O. K. Klinke, A. Jung, A. Moshier, _A bitopological point-free approach to compactications_ (2011). ([preprint](http://www.cs.bham.ac.uk/~axj/pub/papers/compactifications.pdf))

* R. Kopperman, _Asymmetry and duality in topology_, Topology Appl. **66** no. 1 (1995) pp. 1-39.

The idea naturally appeared first in the context of quasi-metric spaces

* Wallace Alvin Wilson, _On Quasi-Metric Spaces_, American Journal of Mathematics (1931), vol. 53, no. 3, pp. 675-684.

The notions of separation axioms were introduced in

* J. D. Weston, _On the comparison of topologies_ 1956, Journal of the London Mathematical Society, vol. s1-32 no. 3, pp. 342-354,

* J. C. Kelly, _Bitopological spaces_, Proc. London Math. Soc. **13** no.3 (1963) pp. 71-89.

Only Kelly introduced the concept in its nowadays formulation of a set equipped with two topologies. The Russian school contributed the following comprehensive overviews of this and related topics

* A. A. Ivanov, _Problems of the Theory of Bitopological Spaces_, 1990, Journal of Soviet Mathematics, vol. 52, Issue 1, pp. 2759-2790. Originally published as _Проблематика теории битопологических пространств_ in Zap. Nauchn. Sem. POMI, 1988, vol. 167 ([Russian version](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=znsl&paperid=5560&option_lang=en)).

* A. A. Ivanov, _Problems of the Theory of Bitopological Spaces 2_, 1996, Journal of Math. Sciences, vol. 81, Issue 2. Originally publishes as _Проблематика теории битопологических пространств. 2_ in Zap. Nauchn. Sem. POMI, 1993, Volume 208, pp. 5–67 ([Russian version](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=znsl&paperid=5828&option_lang=eng)).

* A. A. Ivanov, _Problems of the Theory of Bitopological Spaces 3_, 1998, Journal of Math. Sciences, vol. 91, Issue 6, pp 3339–3364. Originally published as _Проблематика теории битопологических пространств. 3_ in Zap. Nauchn. Sem. POMI, 1995, Volume 231, pp. 9–54 ([Russian version](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=znsl&paperid=3739&option_lang=eng)).

as well as a more introductory text book

* A. A. Ivanov, N. V. Khmylko, _Битопологические пространства_, 1997. Исследования по топологии. 9, Zap. Nauchn. Sem. POMI, 242, editor A. A. Ivanov ([Russian version](http://www.mathnet.ru/php/archive.phtml?jrnid=znsl&wshow=issue&bshow=contents&series=0&year=1997&volume=242&issue=&option_lang=rus&bookID=335)).

Stone duality for bitopological spaces with applications to domain theory is studied in

* Achim Jung and M. Andrew Moshier: [On the bitopological nature of Stone duality](https://www.cs.bham.ac.uk/~axj/pub/papers/Jung-Moshier-2006-On-the-bitopological-nature-of-Stone-duality.pdf). Technical Report CSR-06-13. School of Computer Science, University of Birmingham, 2006.

The pointfree analogues of bitopoligal spaces are reviewed in

* {#Suarez} Anna Laura Suarez,
_The category of finitary biframes as the category of pointfree bispaces_,
[arXiv:2010.04622](https://arxiv.org/abs/2010.04622).

[[!redirects bitopological space]]
[[!redirects bitopological spaces]]
[[!redirects Bitop]]
[[!redirects BiTop]]
[[!redirects Bi Top]]


bivariant algebraic K-theory

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* table of contents
{:toc}

## Idea

A [[bivariant cohomology theory|bivariant]] version of [[algebraic K-theory]].

## Related concepts

* [[K-motive]]

* [[KK-theory]]

## References

* [[Guillermo Cortiñas]], [[Andreas Thom]], _Bivariant algebraic K-theory_. J. Reine Angew. Math. 510 (2007), 71–124. ([arXiv:math/0603531](http://arxiv.org/abs/math/0603531))

* [[Grigory Garkusha]], Ivan Panin, _K-motives of algebraic varieties_ ([arXiv:1108.0375](http://arxiv.org/abs/1108.0375))

* [[Grigory Garkusha]], _Algebraic Kasparov K-theory. II_ ([arXiv:1206.0178](http://arxiv.org/abs/1206.0178))


bivariant cohomology theory

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## Idea

Where a [[homology theory]] is a [[covariant functor]] and a [[cohomology theory]] is a [[contravariant functor]] on some [[category]] of [[spaces]], a _bivariant cohomology theory_ is a [[bifunctor]], hence a functor of two variables, contravariant in the first, and covariant in the second.

Examples:

* In [[noncommutative topology]]: [[KK-theory]].

* In [[noncommutative algebraic geometry]]: [[bivariant algebraic K-theory]], [[noncommutative motives]].

## Axiomatization in homotopy theory
 {#AxiomatizazionInHomotopyTheory}

Here are some notes on a proposal for how to usefully formalize bivariant cohomology theory in [[stable homotopy theory|stable]] [[homotopy theory]]. (This is in generalization of the structure of [[KK-theory]], while the original axioms of ([Fulton-MacPherson 81](#FultonMacPherson81)) are a little different[^footnote].  Aspects of the following appear in ([Nuiten 13](#Nuiten13), [Schreiber 14](#Schreiber14)). See also at _[[dependent linear type theory]]_ the section on _[secondary integral transforms](dependent+linear+type+theory#SecondaryIntegralTransforms)_).

$\,$

Let $E$ be an [[E-∞ ring]], write $GL_1(E)$ for its [[∞-group of units]]. With $\mathbf{H}$ the ambient [[(∞,1)-topos]], write $\mathbf{H}_{/\mathbf{B}GL_1(E)}$ for the [[slice (∞,1)-topos]] over the [[delooping]] of this [[abelian ∞-group]]. This is the [[(∞,1)-category]] of [[spaces]] equipped with [[(∞,1)-module bundle|(∞,1)-line bundles]] over $E$. Consider an [[(∞,1)-functor]]

$$
  \Gamma^\ast \;\colon \; \mathbf{H}_{/\mathbf{B}GL_1(E)} \to E Mod
$$

to the [[(∞,1)-category of (∞,1)-modules]] over $E$, which form $E$-modules of co-sections of $E$-[[(∞,1)-module bundles]] (generalized [[Thom spectra]]).

This is well understood for $\mathbf{H} = $ [[∞Grpd]] in which case $\Gamma \simeq \underset{\to}{\lim} \circ i$ is the [[(∞,1)-functor]] [[homotopy colimits]] in $E Mod$ under the canonical embedding $\mathbf{B} GL_1(E) \simeq E Line \hookrightarrow E Mod$. But one can consider similar constructions $\Gamma$ for more general ambient [[(∞,1)-toposes]] $\mathbf{H}$.

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

For $\chi_i \colon X_i \to \mathbf{B}GL_1(E)$ two [[objects]] of $\mathbf{H}_{/\mathbf{B}GL_1(E)}$, the _$(\chi_1,\chi_2)$-[[twisted cohomology|twisted]] bivariant $E$-[[cohomology theory|cohomology]]_ on $(X_1,X_2)$ is

$$
  E^{\bullet + \chi_2 - \chi_1}(X_1,X_2)
  \;\coloneqq\;
  Hom_{E Mod}\left(\Gamma^\ast_{X_1}\left(\chi_1\right), \Gamma^\ast_{X_2}\left(\chi_2\right)\right)
  \in
  E Mod
  \,.
$$

=--

+-- {: .num_example}
###### Example

By the general discussion at [[twisted cohomology]], following ([ABG, def. 5.1](#ABG)) we have

* for $X_2 = \ast$ the point, the above bivariant cohomology is the $\chi_1$-twisted $E$-cohomology of $X_1$;

  $$
    E^{\bullet + \chi_1}(X_1, \ast) \simeq E^{\bullet + \chi_1}(X_1)
    \,.
  $$

* for $X_1 = \ast$ the point, the above bivariant cohomology is the $\chi_2$-twisted $E$-[[generalized homology|homology]] of $X_2$;

  $$
    E^{\bullet + \chi_2}(\ast, X_2) \simeq E_{\bullet + \chi_2}(X_2)
    \,.
  $$

=--

+-- {: .num_example}
###### Example

[[KK-theory]] is a model for bivariant twisted [[topological K-theory]] over [[differentiable stacks]] (hence 1-truncated suitably representable objects in $\mathbf{H} = $ [[Smooth∞Grpd]], see [Tu-Xu-LG 03](#TuXuLG03)). According to ([Joachim-Stolz 09, around p. 4](#JoachimStolz09)) the category $KK$ first of all is naturally an [[enriched category]] $\mathbb{KK}$ over the category $\mathcal{S}$ of [[symmetric spectra]]  and as such comes with a symmetric [[monoidal functor|monoidal]] [[enriched functor]]

$$
  \mathbb{KK} \to KU Mod
  \,.
$$

This sends an object to its [[operator K-theory]] spectrum, hence to the $E$-[[dual object|dual]] of the $E$-module of co-sections.

=--

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

Generally, one may want to consider in def. \ref{BivariantCohomologyByHomsOfCoSections} the dualized co-section functor

$$
  \Gamma = [\Gamma^\ast(-), E] \;\colon\; \left(\mathbf{H}_{/\mathbf{B}GL_1(E)}\right)^{op} \to E Mod
  \,.
$$

=--

+-- {: .num_example}
###### Example

A [[correspondence]] in $\mathbf{H}_{/\mathbf{B}GL_1(E)}$

$$
  \array{
    && Q
    \\
    & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}}
    \\
    X_1 && \swArrow_{\xi} && X_2
    \\
    & {}_{\mathllap{\chi_1}}\searrow && \swarrow_{\mathrlap{\chi_2}}
    \\
    && \mathbf{B}GL_1(E)
  }
$$

is a morphism of "twisted $E$-[[pure motive|motives]]" in that it is a [[correspondence]] in $\mathbf{H}$ between the [[spaces]] $X_1$ and $X_2$ equipped with an $(i_1^\ast \chi_1, i_2^\ast \chi_2)$-twisted bivariant $E$-cohomology [[cocycle]] $\xi$ on the correspondence space $Q$. Under the co-sections / [[Thom spectrum]] functor this is sent to a [[correspondence]]

$$
  \Gamma_{X_1}(\chi_1)
  \stackrel{\xi}{\rightarrow}
  \Gamma_Q(i_2^\ast \chi_2)
  \stackrel{i_2^\ast}{\leftarrow}
  \Gamma_{X_2}(\chi_2)
$$

in $E Mod$. If the wrong-way map of this is [[orientation in generalized cohomology|orientable]] in $E$-cohomology then we may form its [[dual morphism]]/[[Umkehr map]] to obtain the corresponding "[[index]]"

$$
  \Gamma_{X_1}(\chi_1)
   \stackrel{(i_2)_! \xi}{\to}
  \Gamma_{X_2}(\chi_2)
$$

in $E Mod$. Identifying correspondences that yield the same "[[index]]" this way yields a presentation of bivariant cohomology by [[pure motive|motive]]-like structures.
This is how (equivariant) [[bivariant K-theory]] is presented, at least over manifolds, see at _[KK-theory -- References -- In terms of correspondences](KK-theory#ReferencesInTermsOfCorrespondences)_.

=--

## Related concepts

* [[noncommutative motive]]

* [[motivic quantization]]

## References

A general introduction to bivariant cohomology theories is in

* {#FultonMacPherson81} [[William Fulton]], [[Robert MacPherson]], _Categorical framework for the study of singular spaces_, Memoirs of the AMS, 243, 1981

A general construction of bivariant theories on [[smooth manifolds]] from [[cohomology theories]] by geometric cycles, generalizing the construction of [[K-homology]] by [[Baum-Douglas geometric cycles]], is in

* Martin Jakob, _Bivariant theories for smooth manifolds_, Applied Categorical Structures 10 no. 3 (2002)

A similar construction for PL manifolds is in

* S. Buoncristiano, C. P. Rourke and B. J. Sanderson, _A geometric approach to homology theory_, Cambridge Univ. Press, Cambridge, Mass. (1976)

A study of bivariant theories in the context of motivic stable homotopy theory, and more generally in the broader framework of Grothendieck six functors formalism is in

* F. Déglise, _Bivariant theories in motivic stable homotopy_, ([arXiv:1705.01528] (https://arxiv.org/abs/1705.01528))

References related to the discussion in _[Axiomatization in homotopy theory](#AxiomatizazionInHomotopyTheory)_ above include the following

* {#ABG} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics [vol 81](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))

* {#TuXuLG03} [[Jean-Louis Tu]], [[Ping Xu]], [[Camille Laurent-Gengoux]], _Twisted K-theory of differentiable stacks_ ([arXiv:math/0306138](http://arxiv.org/abs/math/0306138))

* {#JoachimStolz09} [[Michael Joachim]], [[Stephan Stolz]], _An enrichment of $KK$-theory over the category of symmetric spectra_ Münster J. of Math. 2 (2009), 143–182 ([pdf](http://www3.nd.edu/~stolz/KKenrich.pdf))

* {#Nuiten13} [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_, master thesis, August 2013

* {#Schreiber14} [[Urs Schreiber]], _[[schreiber:Quantization via Linear homotopy types]]_ ([arXiv:1402.7041](http://arxiv.org/abs/1402.7041))

[^footnote]: Thanks to [[Thomas Nikolaus]] for patiently emphasizing this.

[[!redirects bivariant cohomology theories]]

[[!redirects bivariant homology theory]]
[[!redirects bivariant homology theories]]

[[!redirects bivariant cohomology]]

[[!redirects twisted bivariant cohomology]]
[[!redirects bivariant twisted cohomology]]


bivector

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{:toc}

## Idea

Where a [[vector]] specifies a _direction_ and a _magnitude_, a _bivector_ specifies a _plane_ and a magnitude.

A _bivector_ is a [[multivector]]_ of degree 2.

## Definition

For $V$ a $k$-[[vector space]], a _bivector_ in $V$ is a decomposable element $b = b_1\wedge b_2 \in \wedge^2 V$ of the second [[exterior power]]  of $V$. If the dimension of $V$ is $3$ or less every element in $\Lambda^2 V$ is decomposable, hence bivectors form the vector space, namely $\Lambda^2 V$ itself. The plane associated to a nonzero bivector $b_1\wedge b_2$ is $Span\{b_1,b_2\}$.

Alternatively, a bivector is a
class of equivalence of pairs $(b_1,b_2)\in V\times V$
by the smallest equivalence relation for which $(b_1,b_2)\sim (b_1+\lambda b_2,b_2)$ for any $\lambda\in k$, $(b_1,b_2)\sim (b_1,\lambda b_1+b_2)$ for any $\lambda\in k$, and $(b_1,b_2)\sim(\frac{b_1}\mu,\mu b_2)$ for any
$\mu\in k\backslash \{0\}$. This axiomatics is parallel (and related) to the axiomatic definition of a $2\times 2$ determinant.

Some authors (including English wikipedia) by a bivector mean any element
$\Sigma_i b_{1 i}\wedge b_{2 i}\in\Lambda^2 V$, regardless the decomposability property (and in particular, regardless the dimension of $V$ which guarantees decomposability it $dim V\leq 3$), while they call the decomposable
elements "simple bivectors".
This terminology may be inconvenient in analytic geometry
as only the nonzero _simple_ bivectors define a 2-plane
and most standard constructions which are used in the Euclidean geometry with  bivectors need the restriction, for example the notion of a vector being parallel
to a bivector (vector $v$ is parallel to
a (simple) bivector $b$ iff $b = v\wedge w$ for some vector $w$, or equivalently
when $v$ is in the span of some pair $b_1,b_2\in V$
such that we can write $b = b_1\wedge b_2$).

A bivector is canonically identified with an element of degree 2 in the [[Grassmann algebra]] $\wedge^\bullet V$.

## Properties

If $V$ is equipped with a non-degenerate [[inner product]] then the space $\Lambda^2 V$ is also canonically identified with a subspace of the [[Clifford algebra]] $Cl(V)$.

If we write $\gamma_v \in Cl(V)$ for the Clifford algebra element corresponding to a vector $v \in V$, then this identification is given by the map

$$
  v \wedge w \mapsto
  \frac{1}{2}\left(\gamma_{v} \cdot \gamma_w - \gamma_w \cdot \gamma_v\right)
  \,.
$$

(The [[inverse]] of this map is called the [[symbol map]].)

Under the [[commutator]] in the Clifford algebra bivectors go to bivectors and hence form a [[Lie algebra]]. This Lie algebra is the [[special orthogonal Lie algebra]] $\mathfrak{so}(V)$ of $V$.

## References

Discussion of [[Clifford algebra]] and [[exterior algebra]] that amplifies the role of bivectors is notably in the references at _[[Geometric Algebra]]_ .

See also

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

[[!redirects bivectors]]


Bjorn Poonen

* [webpage](http://www-math.mit.edu/~poonen/)

category: people


Bjørn Ian Dundas

* [webpage](http://www.uib.no/People/nmabd/)

## Selected writings

Interpretation of [[algebraic K-theory]] of [[topological K-theory]] ([[iterated algebraic K-theory]]) as the [[K-theory]] of [[BDR 2-vector bundles]]:

* [[Nils Baas]], [[Bjørn Ian Dundas]], [[John Rognes]], _Two-vector bundles and forms of elliptic cohomology_, London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, 2004 ([arXiv:0909.1742](https://arxiv.org/abs/0909.1742), [doi:10.1017/CBO9780511526398.005](https://doi.org/10.1017/CBO9780511526398.005))

* [[Nils Baas]], [[Bjørn Ian Dundas]], [[Birgit Richter]], [[John Rognes]], _Stable bundles over rig categories_, Journal of Topology, Volume 4, Issue 3, September 2011, Pages 623–640 ([doi:10.1112/jtopol/jtr016](https://doi.org/10.1112/jtopol/jtr016))

Divisibility of a [[gerbe]] on the [[3-sphere]] seen as a [[BDR 2-vector bundle]] is in

* [[Christian Ausoni]], [[Bjørn Ian Dundas]] and [[John Rognes]],  _Divisibility of the Dirac magnetic monopole as a two-vector bundle over the three-sphere_, Doc. Math. 13, 795-801 (2008). ([pdf](http://www.math.univ-paris13.fr/~ausoni/papers/ADR08.pdf))

On [[algebraic K-theory]]:

* {#DundasGoodwillieMcCarthy13} [[Bjørn Dundas]], [[Thomas Goodwillie]], [[Randy McCarthy]], *The local structure of algebraic K-theory*, Springer (2013) [[doi:10.1007/978-1-4471-4393-2](https://doi.org/10.1007/978-1-4471-4393-2)]

On [[symmetry]] and introducing the language of [[homotopy type theory]] for [[univalent foundations of mathematics]]:

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: *[[Symmetry]]* (2021) [[pdf](https://unimath.github.io/SymmetryBook/book.pdf)]

## Related entries

* [[BDR 2-vector bundle]]

* [[red-shift conjecture]]

category: people

[[!redirects Ian Dundas]]
[[!redirects Bjorn Dundas]]
[[!redirects Bjørn Dundas]]


black body radiation

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## Idea

What is called _black body radiation_ is the [[electromagnetic radiation]] emitted by an ideal non-reflecting ("black") piece of [[matter]] in [[thermal equilibrium]] with the [[electromagnetic field]] around it, at any given [[temperature]].

The [[experiment|experimentally]] observed black body radiation spectrum is a key effect and property of [[quantum physics]], governed by [[energy]]/[[frequency]] quantization controlled by [[Planck's constant]]. In contrast, in [[classical field theory]] black body radiation comes out incorrectly, with an infinite [[energy] in the high [[frequency]] modes (historically: "[[ultraviolet catastrophe]]").

## References

* Wikipedia, _[Black body radiation](http://en.wikipedia.org/wiki/Black-body_radiation)_


black brane

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{:toc}

## Idea

The theory of [[gravity]] in $3+1$ dimensions famously has [[black hole]] solutions, being the limiting configuration of a _point_ mass gravitational source. In higher dimensional gravity, and in particular in higher dimensional [[supergravity]], there are analogous solutions, which however are limiting configurations of a gravitational source that is supported on a line, or a surface, or a higher dimensional space. For a surface one might speak of _black [[membrane]]_ solutions hence generally of _black brane_ solutions.

Particularly the [[BPS states]] among the black branes in [[supergravity]], i.e. those solutions that carry [[Killing spinors]], include configurations that look like the strong-coupling version of the [[Green-Schwarz super p-branes]].

[[!include black branes in supergravity -- table]]

The [[near-horizon geometry]] of these black branes is generically that of [[anti de Sitter spacetime]] times a [[sphere]]. To the extent that the [[worldvolume]] theory of the branes is a [[superconformal QFT]], this is the origin of the [[AdS-CFT correspondence]].

## Properties

### Weak coupling correspondence

The types of black branes that can occur in theories of [[supergravity]] that are obtained from the maximal [[11-dimensional supergravity]] match precisely the types of [[D-brane]]s and [[NS-brane]]s that appear in the corresponding perturbative [[superstring theories]].

The idea is that both these brane-phenomena are aspects of one single entity:

* At low [[string coupling]] the [[D-brane]]/[[NS-brane]] description is accurate. Low [[string coupling]] implies that the coupling of [[gravity]] is weak, hence that the back-reaction of the branes on the background geometry is negligible.

* At large [[string coupling]] but low [[energy]], the [[effective field theory|effective]] [[supergravity]] description becomes accurate. Here the branes do back-react on the gravitational background and hence create the _black brane_ spacetime geometry.

This duality of the brane picture is at the heart of the [[AdS/CFT correspondence]]. See there for more details.

## Examples

* [[black hole]], [[black hole in string theory]]

* [[black string]]

* [[black membrane]]

* [[black fivebrane]]

* [[string triple junction]]

* [[membrane triple junction]]

## Related concepts

[[!include table of branes]]

## References

### Prehistory

* {#EinsteinInfeldHoffmann39} [[Albert Einstein]], [[Leopold Infeld]], B. Hoffmann, _The gravitational equations and the problem of motion_, Annals of Mathematics, Vol 39, No. 1, 1938

See also

* Wikipedia, _[Black hole electron](https://en.wikipedia.org/wiki/Black_hole_electron)_

### General

Original articles:

* [[Michael Duff]], [[Ramzi R. Khuri]], [[Jian Xin Lu]], *String and Fivebrane Solitons: Singular or Non-singular?*, Nucl.Phys. B **377** (1992) 281-294 [[arXiv:hep-th/9112023](https://arxiv.org/abs/hep-th/9112023), doi:10.1016/0550-3213(92)90025-7]

* [[Michael Duff]], [[Jian Xin Lu]], *Black and super $p$-branes in diverse dimensions*, Nucl. Phys. B **416** (1994) 301-334 [[arXiv:hep-th/9306052](http://arxiv.org/abs/hep-th/9306052), doi:10.1016/0550-3213(94)90586-X]

* {#DuffKhuriLu94} [[Mike Duff]], [[Ramzi R. Khuri]], [[Jian Xin Lu]], *String Solitons*, Phys. Rept. **259** (1995) 213-326 [[arXiv:hep-th/9412184](https://arxiv.org/abs/hep-th/9412184), doi:10.1016/0370-1573(95)00002-X]

  > ([[solitonic branes]])

* [[Michael Duff]], H. Lu, [[Christopher Pope]], _The Black Branes of M-theory_, Phys. Lett. B **382** (1996) [[arXiv:hep-th/9604052](http://arxiv.org/abs/hep-th/9604052)]

The [[M5-brane]] was maybe first found as a black brane of [[11-dimensional supergravity]] (the [[black fivebrane]]) in

* {#Gueven92} [[Rahmi Gueven]], _Black $p$-brane solutions of $D = 11$ supergravity theory, Phys. Lett. B276 (1992) 49 and in [[Mike Duff]] (ed.) _[[The World in Eleven Dimensions]]_ 135-141 ([spire:338203](http://inspirehep.net/record/338203))

The observation that black $p$-branes metric for _odd_ $p$ are completely non-singular is due to

* [[Gary Gibbons]], [[Gary Horowitz]], [[Paul Townsend]], p. 15 of  _Higher-dimensional resolution of dilatonic black hole singularities_, Class. Quant. Grav.12:297-318,1995 ([arXiv:hep-th/9410073](https://arxiv.org/abs/hep-th/9410073))

The suggestion that extremal/[[BPS state]] black branes are the strong coupling incarnation of fundamental branes originates in

* {#HullTownsend94} [[Chris Hull]], [[Paul Townsend]], _Unity of Superstring Dualities_, Nucl.Phys.B438:109-137,1995 ([arXiv:hep-th/9410167](http://arxiv.org/abs/hep-th/9410167))

* {#Witten95} [[Edward Witten]], bottom of p. 6 of _String Theory Dynamics In Various Dimensions_, Nucl.Phys.B443:85-126,1995 ([arXiv:hep-th/9503124](http://arxiv.org/abs/hep-th/9503124))

Review includes,

* {#Duff99} [[Mike Duff]], chapter 5 of _[[The World in Eleven Dimensions]]: Supergravity, Supermembranes and M-theory_, IoP 1999 ([publisher](https://www.crcpress.com/The-World-in-Eleven-Dimensions-Supergravity-supermembranes-and-M-theory/Duff/9780750306720))

* {#Stelle98} [[Kellogg Stelle]], around section 3.3. of *BPS Branes in Supergravity*, in: *Quantum Field Theory: Perspective and Prospective*, NATO Science Series **530** (1999) 257-351 [[arXiv:hep-th/9803116](http://arxiv.org/abs/hep-th/9803116), [doi:10.1007/978-94-011-4542-8_12](https://doi.org/10.1007/978-94-011-4542-8_12)]

* [[Ofer Aharony]], S.S. Gubser, [[Juan Maldacena]], H. Ooguri, Y. Oz, section 1.3 of _Large N Field Theories, String Theory and Gravity_ ([arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111))

* {#Mohaupt00} [[Thomas Mohaupt]], _Black holes in supergravity and string theory_, Class. Quantum Grav. 17 (2000) 3429–3482 ([pdf](http://www.itp.uni-hannover.de/~giulini/papers/BlackHoleSeminar/Mohaupt_CQG_2000.pdf))

* {#BlumenhagenLustTheisen13} [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], _Brane solutions in supergravity_, chapter 18.5 in: _Basic Concepts of String Theory_, Springer (2013) [[doi:10.1007/978-3-642-29497-6](https://doi.org/10.1007/978-3-642-29497-6)]

and in the context of multiple [[M2-branes]] in the  [[BLG model]]:

* {#BaggerLambert12} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], Constantinos Papageorgakis, section 1.6 of _Multiple Membranes in M-theory_ ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546))

Further developments include

* Gerard Clement, Dmitri Gal'tsov, Cedric Leygnac, _Black branes on the linear dilaton background_, Phys. Rev. D71 (2005) 084014 ([arXiv:hep-th/0412321](http://arxiv.org/abs/hep-th/0412321))

* D. Gal'tsov, S. Klevtsov, D. Orlov, G. Clement, _More on general $p$-brane solutions_, Int.J.Mod.Phys.A21:3575-3604, 2006 ([arXiv:hep-th/0508070](http://arxiv.org/abs/hep-th/0508070))

* {#Duff08} [[Michael Duff]], _Near-horizon brane scan revived_, Nucl. Phys. B810:193-209, 2009 ([arXiv:0804.3675](http://arxiv.org/abs/0804.3675))

* Jay Armas, Joan Camps, Troels Harmark, Niels A. Obers, _The Young Modulus of Black Strings and the Fine Structure of Blackfolds_ ([arXiv:1110.4835](http://arxiv.org/abs/1110.4835))

### Blackfold approach

* [[Roberto Emparan]], [[Troels Harmark]], [[Vasilis Niarchos]], [[Niels A. Obers]], *Essentials of Blackfold Dynamics*, JHEP 1003 063 (2010) [[arXiv:0910.1601](https://arxiv.org/abs/0910.1601), doi:10.1007/JHEP03(2010)063]

* [[Roberto Emparan]], [[Troels Harmark]], [[Vasilis Niarchos]], [[Niels A. Obers]], *Blackfolds in Supergravity and String Theory*, JHEP 1108 (2011) 154 [[arXiv:1106.4428](https://arxiv.org/abs/1106.4428)doi:10.1007/JHEP08(2011)154]

[[!redirects black branes]]
[[!redirects black p-brane]]
[[!redirects black p-branes]]

[[!redirects blackfold]]
[[!redirects blackfolds]]

[[!redirects singular brane]]
[[!redirects singular branes]]


black branes in supergravity -- table

**1/2 [[BPS state|BPS]] [[black branes]] in [[supergravity]]**: [[D-branes]], [[F1-brane]], [[NS5-brane]], [[M2-brane]], [[M5-brane]]

<img width="600" src="http://ncatlab.org/nlab/files/BraneSolutionsInSupergravity.jpg" >

> (table taken from [Blumenhagen-Lüst-Theisen 13, Chapter 18.5](black+brane#BlumenhagenLustTheisen13))


black fivebrane

## Idea

The [[M5-brane]] in its incarnation as a [[black brane]] in [[11-dimensional supergravity]].

## References

* [[Rahmi Gueven]], _Black $p$-brane solutions of $D = 11$ supergravity theory, Phys. Lett. B276 (1992) 49.


black hole

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## Idea

A _black hole_ is a [[spacetime]] that solves [[Einstein equations]] of [[general relativity]] characterized by the fact that it possesses an __[[event horizon]]__ hypersurface (or several of them) which has a number of special characteristics, for example that light cannot escape from the space confined by the horizon hypersurface due to [[gravity|gravitational]] effects. Much of the theoretical considerations are about the entropy of black holes (cf. [[Bekenstein-Hawking entropy]]) and the information paradox.

Black holes are considered theoretically for [[gravity|gravitational theories]] in various number $d$ of dimension. For $d \geq 5$ a black hole [[spacetime]] may have nontrivial [[topology]], e.g. _[[black rings]]_ are possible.

## Examples

* In usual asymptotically 3+1-dimensional [[Minkowski spacetime]], the first black hole solution that was found is the [[Schwarzschild black hole]] solution; such a black hole posses a single horizon hypersurface and seems to be stable under various perturbations.

* Another solution with finite angular momentum is called the [[Kerr spacetime]], and there is a simple generalization having also the electric charge, the Newman solution or the Kerr-Newman black hole. This solution differs pretty much from the Schwarzschild solution and its structure is unstable under various physical mechanisms and perturbations; it possesses two horizons, inner and outer.

## Properties

### Topology

Hawking's _Theorem of Black Hole topology_ asserts that the in case of $d = 4$ asymptotically flat stationary black holes satisfying the suitable [[dominant energy condition]], the cross sections of the [[event horizon]] are spherical.

[Galloway and Schoen](#GallowaySchoen) extended this theorem to higher dimensions; they showed that the cross sections of event horizon (stationary case) and the outer (apparent) horizon (general case) are of Yamabe type.

### Entropy

* [[Bekenstein-Hawking entropy]]

* [[generalized second law of thermodynamics]]

## Empirical observation

See

* [[observing black holes]].

Some candidate [[astrophysics|astrophysical]] objects which seem to point to black hole have been observed.

## Examples

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

$\,$

* [[black hole in anti-de Sitter spacetime]]

## Related concepts

* [[supernova]]

* [[Schwarzschild radius]]

* [[Kruskal–Szekeres coordinates]]

* [[Bekenstein-Hawking entropy]], [[black hole radiation]]

* [[Penrose-Hawking singularity theorem]], [[cosmic censorship hypothesis]]

* [[extremal black hole]]

* [[generalized second law of thermodynamics]]

* [[black hole information paradox]]

* [[black ring]]

* [[black brane]], [[black holes in string theory]]

* [[firewall problem]]

## References

### Observation

Resolved image of the direct vicinity of the [[event horizon]] of the [[black hole]] in the center of the [[galaxy]] [[Messier 87]]:

* {#EHT19} [[Event Horizon Telescope Collaboration]], _First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole_,  The Astrophysical Journal Letters, Volume 875, Number 1, 2019 ([doi:10.3847/2041-8213/ab0ec7](https://iopscience.iop.org/article/10.3847/2041-8213/ab0ec7))

### Review

* {#FrolovZelnikov11} [[Valeri Frolov]], Andrei Zelnikov, _Introduction to black hole physics_, Oxford 2011

* Wikipedia, *[Black hole](http://en.wikipedia.org/wiki/Black_hole)*

* [[Croatian black hole school]], 2010

* Barrett O'Neill, _The geometry of Kerr black holes_

* S. Chandrasekhar, _The mathematical theory of black holes_

* G. T. Horowitz, A. Strominger, _Counting states of near-extremal black holes_, Phys. Rev. Lett. 77 (1996) 2368–2371, [hep-th/9602051](http://arxiv.org/abs/hep-th/9602051).

* Gregory Galloway, Richard Schoen, _A Generalization of Hawking's Black Hole Topology Theorem to Higher Dimensions_ Commun. Math. Phys. (2006) ([pdf](http://www.math.miami.edu/~galloway/papers/220_2006_19_OnlinePDF.pdf))
{#GallowaySchoen}

* [[Robert Wald]], _Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics_, University of Chicago Press 1994; _The back reaction effect in particle creation in curved spacetime_, Commun. Math. Phys. __54__, 1 (1977)

* [[Pietro Fré]], *Gravity, a Geometrical Course*, Volume 2: *Black Holes, Cosmology and Introduction to Supergravity*, Springer (2013) [[doi:10.1007/978-94-007-5443-0](https://doi.org/10.1007/978-94-007-5443-0)]

With focus on regular black holes:

* Chen Lan, Hao Yang, Yang Guo, Yan-Gang Miao, *Regular black holes: A short topic review* [[arXiv:2303.11696](https://arxiv.org/abs/2303.11696)]

### In supergravity

Black holes in [[supergravity]]:

* [[Riccardo D'Auria]], [[Pietro Fre]], _BPS black holes in supergravity_, ([hep-th/9812160](http://arxiv.org/abs/hep-th/9812160))

* Antonio Gallerati, _Constructing black hole solutions in supergravity theories_ ([arXiv:1905.04104](https://arxiv.org/abs/1905.04104))

### Holographic description

Discussion of black holes in the context of the [[holographic principle]] and the [[AdS-CFT correspondence]] is in

* [[Monica Guica]], [[Thomas Hartman]], Wei Song, [[Andrew Strominger]], _The Kerr/CFT Correspondence_, Phys. Rev. D **80** (2009) 124008 [[arXiv:0809.4266](http://arxiv.org/abs/0809.4266), [doi:10.1103/PhysRevD.80.124008](https://doi.org/10.1103/PhysRevD.80.124008)]

* Alejandra Castro, Alexander Maloney, [[Andrew Strominger]], _Hidden Conformal Symmetry of the Kerr Black Hole_ ([arXiv:1004.0996](http://arxiv.org/abs/1004.0996))

The nature of the [[event horizon]], specifically, is discussed in

* Kyriakos Papadodimas, Suvrat Raju, _An Infalling Observer in AdS/CFT_ ([arXiv:1211.6767](http://arxiv.org/abs/1211.6767))

### Cosmology inside black holes

* Razieh Pourhasan, Niayesh Afshordi, Robert B. Mann, _Out of the White Hole: A Holographic Origin for the Big Bang_ ([arxiv:1309.1487](https://arxiv.org/abs/1309.1487))

[[!redirects black holes]]

[[!redirects black hole spacetime]]
[[!redirects black hole spacetimes]]


black hole firewall > history

< [[black hole firewall]]


black hole firewall problem

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## Idea

In [AMPS 2012](#AMPS), it is proposed that infalling [[observers]] encounter a "firewall" at (or just outside) the [[event horizon]] of a [[black hole]].  This is part of an attempted resolution of the [[black hole information paradox]].

For a rough summary, see [Wikipedia](http://en.wikipedia.org/wiki/Firewall_(physics%29) for now.

## Related entries

* [[holographic entanglement entropy]]

## References

The suggestion that there is a black hole "firewall" phenomenon originates in

* {#AMPS} [[Ahmed Almheiri]], [[Donald Marolf]], [[Joseph Polchinski]], James Sully, _Black holes: complementarity or firewalls?_ ([arXiv:1207.3123](http://arxiv.org/abs/arXiv:1207.3123))

with a folowup in:

 * {#AMPSS} [[Ahmed Almheiri]], [[Don Marolf]], [[Joseph Polchinski]], [[Douglas Stanford]], [[James Sully]], _An Apologia for Firewalls_ 2013 ([arXiv:1304.6483](http://arxiv.org/abs/1304.6483))

A systematic analysis of the (non-)phenomenon using actual computations in [[AdS-CFT duality]] is in

* Kyriakos Papadodimas, Suvrat Raju, _An Infalling Observer in AdS/CFT_ ([arXiv:1211.6767](http://arxiv.org/abs/1211.6767))
  {#PR}

whose authors say in their introduction:

> These $[$BH firewall$]$ proposals originate not from direct calculations in [[quantum gravity]], $[$...$]$ While [[quantum gravity]] is a mysterious subject, the [[AdS-CFT correspondence]] provides us with a setting where we can examine these ideas within a perfectly well defined [[theory (physics)|theory]].

A useful comment on that article is in

* [[Ted Jacobson]], _Boundary unitarity without firewalls_ ([arXiv:1212.6944](http://arxiv.org/abs/1212.6944))

In the course of this debate, the following article proposed that there is a close relation between [[wormholes]] ([[Einstein-Rosen bridges]]) and [[quantum entanglement]] ([[EPR paradox|Einstein-Podolsky-Rosen phenomenon]]), now known as "[[ER = EPR]]":

 * {#EREPR} [[Juan Maldacena]], [[Leonard Susskind]], _Cool horizons for entangled black holes_ 2013 ([arXiv:1306.0533](http://arxiv.org/abs/1306.0533))

Other contributions to the issue include the following

* [[Erik Verlinde]],  [[Herman Verlinde]], _Black Hole Information as Topological Qubits_ 2013 ([arXiv:1306.0516](http://arxiv.org/abs/1306.0516))

* [[Erik Verlinde]],  [[Herman Verlinde]], _Passing through the Firewall_ 2013 ([arXiv:1306.0515](http://arxiv.org/abs/1306.0515))

* [[Leonard Susskind]], _Black Hole Complementarity and the Harlow-Hayden Conjecture_  2013 ([arXiv:1301.4505](http://arxiv.org/abs/1301.4505))

* [[Daniel Harlow]], [[Patrick Hayden]], _Quantum Computation vs. Firewalls_ 2013 ([arXiv:1301.4504](http://arxiv.org/abs/1301.4504)) -- points out the significance of the so-called "overlap criterion," and suggests a way around the firewall paradox by arguing that the criterion should be relaxed.

* S. H. Shenker, D. Stanford, _Black holes and the butterfly effect_ 2013 ([arXiv:1306.0622](http://arxiv.org/abs/1306.0622))

* [[Don Page]], _Excluding Black Hole Firewalls with Extreme Cosmic Censorship_ 2013 ([arXiv:1306.0562](http://arxiv.org/abs/1306.0562))

* [[Raphael Bousso]], _Complementarity Is Not Enough_ 2012 ([arXiv:1207.5192](http://arxiv.org/abs/1207.5192))

* [[Tom Banks]], [[Willy Fischler]], _No Firewalls in Holographic Space-Time or Matrix Theory_ 2013 ([arXiv:1305.3923](http://arxiv.org/abs/1305.3923))

* [[Mark van Raamsdonk]], _Building up spacetime with quantum entanglement_, Gen. Rel. Grav.42:2323-2329,2010; Int.J.Mod.Phys.D19:2429-2435,2010 ([arXiv:1005.3035](http://arxiv.org/abs/1005.3035))

* [[Samuel L. Braunstein]]; S. Pirandola, K. Życzkowski,  *Better Late than Never: Information Retrieval from Black Holes* (2013) [[arXiv:0907.1190](http://arxiv.org/abs/0907.1190)]

* Samir D. Mathur, David Turton, Comments on black holes I: The possibility of complementarity 2012, [arXiv:1208.2005](http://arxiv.org/abs/1208.2005)

* Samir D. Mathur, The information paradox: A pedagogical introduction 2009, [arXiv:0909.1038](http://arxiv.org/abs/arXiv:0909.1038)

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black hole in anti-de Sitter spacetime

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## Idea

These are solutions to [[Einstein's equations]] which are [[black holes]] in [[spacetimes]] that far away from the black hole are asymptotic (not to [[Minkowski spacetime]] but) to [[anti de Sitter spacetime]].

## Properties

### In AdS/CFT

Under the [[AdS-CFT correspondence]] black holes in the anti-de Sitter spacetime translate to the [[conformal field theory]] on the [[asymptotic boundary]] being at [[thermal field theory|positive temperature]] (e.g. [Duff 99, section 6](#Duff99), [Natsuume 15, section 7](#Natsuume15)).

## Related concepts

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

## References

Original articles:

* Hawking, Stephen W., and Don N. Page. "Thermodynamics of black holes in anti-de Sitter space." Communications in Mathematical Physics 87.4 (1983): 577-588.

Review:

* {#Duff99} [[Mike Duff]], section 6 of _TASI Lectures on Branes, Black Holes and Anti-de Sitter Space_ ([arXiv:hep-th/9912164](https://arxiv.org/abs/hep-th/9912164))

* {#Natsuume15} [[Makoto Natsuume]], section 7of _AdS/CFT Duality User Guide_, Lecture Notes in Physics 903, Springer 2015 ([arXiv:1409.3575](https://arxiv.org/abs/1409.3575))

* {#Socolovsky17} M. Socolovsky, _Schwarzschild Black Hole in Anti-De Sitter Space_ ([arXiv:1711.02744](https://arxiv.org/abs/1711.02744))

* Peng Zhao, _Black Holes in Anti-de Sitter Spacetime_ ([pdf](https://pdfs.semanticscholar.org/9939/335e0d282e70201d6087988bf4f5aedfc8f6.pdf))

* Jakob Gath, _The role of black holes in the AdS/CFT correspondence_ ([pdf](https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/dissertations/2009/Jakob-Gath-Dissertation-2009.pdf))

See also

* Wikipedia, _[AdS black hole](https://en.wikipedia.org/wiki/AdS_black_hole)_

Discussion of [[black hole entropy]]:

* Dao-Quan Sun, Ping Li, Xian-Ru Hu, Jian-Bo Deng, _Insight into the Microscopic Structure of an AdS Black Hole from the quantization_ ([arXiv:1811.09952](https://arxiv.org/abs/1811.09952))

Discussion via [[AdS/CFT correspondence]]:

* Sinya Aoki, Tetsuya Onogi, Shuichi Yokoyama, _What does a quantum black hole look like?_ ([arXiv:2004.03779](https://arxiv.org/abs/2004.03779))

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black hole information paradox

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## Idea

(...)

## Related concepts

* [[Bekenstein-Hawking entropy]]

* [[paradox]]

## References

See also:

* Wikipedia, _[Black hole information paradox](http://en.wikipedia.org/wiki/Black_hole_information_paradox)_

[[!include entanglement island proposal for black hole paradox -- references]]


black hole radiation

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## Related concepts

* [[black hole entropy]]

* [[weak gravity conjecture]]

## References

In [[AQFT]]:

* [[Klaus Fredenhagen]], [[Rudolf Haag]], _On The Derivation Of Hawking Radiation Associated With The Formation Of A Black Hole_, Commun. Math. Phys. 127 (1990) 273.

For more recent developments see at _[[locally covariant perturbative AQFT]]_.

See also

* Wikipedia, _[Hawking radiation](https://en.wikipedia.org/wiki/Hawking_radiation)_

[[!redirects Hawking raditation]]


black holes in string theory

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## Black hole entropy

In ([[perturbative string theory|perturbative]]) [[string theory]] the [[Bekenstein-Hawking entropy]] associated to a macroscopic [[black hole]]  finds an explanation as follows:

The [[black hole]] [[spacetime]] is regarded as a [[non-perturbative QFT|strongly coupled]] [[condensate]] of string states ([[closed strings]], carrying [[gravity]]). One looks for a corresponding weakly-coupled background for closed strings in flat [[Minkowski space]] that is known to turn into the black hole spacetime as the [[string coupling]] is turned on. Such turn out to be certain [[D-brane]] configurations in Minkowski space: at weak coupling the D-brane does not back-react on the spacetime (since that back-reaction is mediated by closed string quanta) and so it just sits there, whereas at strong coupling it curves spacetime and may collapse to a configuration that looks like the prescribed black hole spacetime.

> graphics grabbed from [Ibanez-Uranga 12](#IbanezUranga12)

Now, if the configuration has a sufficient amount of [[supersymmetry]] preserved ([[BPS state]]), then one argues ([Witten 95, section 2.3](#Witten95)) that certain observables are actually independent of the coupling constant ("[[protection from quantum corrections]]"), and accordingly these observables are the same for black holes as for the corresponding D-brane configurations in flat space, where they may be computed in string perturbation theory.

The best studied such configuration is that of [[D1-D5 brane bound states]]. For these configurations one may compute the number of [[BPS states]] (which are "protected"), hence the  [[entropy]], via the [[Witten genus]] ([Strominger-Vafa 96](#StromingerVafa96)), see at _[Witten genus -- Relation to BPS state counting](Witten+genus#RelationToBPSStateCounting)_. By the above reasoning this may then be compared to the [[Bekenstein-Hawking entropy]] of the corresponding (supersymmetric) black hole. And indeed the results match the semiclassical BH-entropy to leading order and in addition provide their higher order quantum corrections. See the _[References](#References)_ below.

The [[T-duality|T-dual]] version of the D1/D5-bound states are D0/D6-bound states, coming from the [[Kaluza-Klein monopole]] in [[11-dimensional supergravity]] (e.g. [Nelson 93](#Nelson93)).

## Related concepts

* [[extremal black hole]]

* [[black ring]]

* [[black brane]]

  * [[black hole]]

  * [[black string]]

  * [[black membrane]]

  * [[black fivebrane]]

* [[AdS/CFT correspondence]]

  * [[Sachdev-Ye-Kitaev model]]

* gravitational entropy

  * [[Bekenstein-Hawking entropy]]

  * [[generalized second law of thermodynamics]]

  * [[black holes in string theory]]

  * [[holographic entanglement entropy]]

[[!include table of branes]]

* [[5d supergravity]]

## References
 {#References}

A careful non-technical exposition is in

* Sebastian De Haro, Jeroen van Dongen, Manus Visser, [[Jeremy Butterfield]], _Conceptual Analysis of Black Hole Entropy in String Theory_ ([arXiv:1904.03232](https://arxiv.org/abs/1904.03232))

* Jeroen van Dongen, Sebastian De Haro, Manus Visser, [[Jeremy Butterfield]], _Emergence and Correspondence for String Theory Black Holes_ ([arXiv:1904.03234](https://arxiv.org/abs/1904.03234))

### Introductions and Review

* [[Juan Maldacena]], _Black Holes in String Theory_, PhD thesis 1996 ([arXiv:hep-th/9607235](http://arxiv.org/abs/hep-th/9607235))

* [[Juan Maldacena]], _Black Holes and D-branes_, Nucl. Phys. Proc. Suppl. 61A (1998) 111-123; Nucl. Phys. Proc. Suppl. 62 (1998) 428-442 ([arXiv:hep-th/9705078](http://arxiv.org/abs/hep-th/9705078))

* [[Mike Duff]], chapter 5 of _[[The World in Eleven Dimensions]]: Supgergravity, Supermembranes and M-theory_, IoP 1999 ([publisher](https://www.crcpress.com/The-World-in-Eleven-Dimensions-Supergravity-supermembranes-and-M-theory/Duff/9780750306720))

* [[Kostas Skenderis]], _Black holes and branes in string theory_,  	Lect.Notes Phys. 541 (2000) 325-364 ([arXiv:hep-th/9901050](https://arxiv.org/abs/hep-th/9901050))

* [[Thomas Mohaupt]], _Black Hole Entropy, Special Geometry and Strings_, Fortsch.Phys. 49 (2001) 3-161 ([arXiv:hep-th/0007195](http://arxiv.org/abs/hep-th/0007195))

* Sumit R. Das, [[Samir Mathur]], _The Quantum Physics of Black Holes: Results from String Theory_ ([arXiv:gr-qc/0105063](http://arxiv.org/abs/gr-qc/0105063))

* Justin R. David, Gautam Mandal, Spenta R. Wadia, _Microscopic Formulation of Black Holes in String Theory_ ([arXiv:hep-th/0203048](http://arxiv.org/abs/hep-th/0203048))

* [[Thibault Damour]], _The entropy of black holes: a primer_ ([arXiv:hep-th/0401160](http://arxiv.org/abs/hep-th/0401160))

* [[Barton Zwiebach]], chapter 22 of _A first course in string theory_

* [[Per Kraus]], _Lectures on black holes and the $AdS_3/CFT_2$ correspondence_,  Lect. Notes Phys.755:193-247, 2008 ([arXiv:hep-th/0609074](https://arxiv.org/abs/hep-th/0609074))

  _Stringy black holes in five dimensions_, 2007 ([pdf slides](http://hep.ps.uci.edu/~arajaram/Irvine.07.pdf))

* [[Xi Yin]], _Black Holes, Anti de Sitter space, and Topological Strings_, 2006 ([pdf](http://www.people.fas.harvard.edu/~xiyin/Site/Research_files/allthesis.pdf))

* {#Sen07} [[Ashoke Sen]], _Black Hole Entropy Function, Attractors and Precision Counting of Microstates_, Gen. Rel. Grav. 40: 2249-2431, 2008 ([arXiv:0708.1270](http://arxiv.org/abs/0708.1270))

* Joao Gomes, _Quantum entropy of supersymmetric black holes_ ([arXiv:1111.2025](http://arxiv.org/abs/1111.2025))

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 6.2 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

* [[Dieter Lüst]], Ward Vleeshouwers, _Black Hole Information and Thermodynamics_ ([arXiv:1809.01403](https://arxiv.org/abs/1809.01403))

* {#Zaffaroni19} [[Alberto Zaffaroni]], _Lectures on AdS Black Holes, Holography and Localization_ ([arXiv:1902.07176](https://arxiv.org/abs/1902.07176))

* Lorenzo Pieri, _Black Holes in String Theory_ ([arXiv:1911.11670](https://arxiv.org/abs/1911.11670))

Lectures on AdS Black Holes, Holography and Localization
Alberto Zaffaroniee also

* [String Theory Wiki](http://www.stringwiki.org), _[Black hole entropy](http://www.stringwiki.org/w/index.php?title=Black_Hole_Entropy)_

### Original articles

The argument that properties of [[BPS states]] are preserved as the coupling increases beyond [[perturbation theory]] and are not destroyed by [[non-perturbative effects]] originates in

* {#Witten95} [[Edward Witten]], section 2.3 of _String Theory Dynamics In Various Dimensions_, Nucl.Phys.B443:85-126,1995 ([arXiv:hep-th/9503124](http://arxiv.org/abs/hep-th/9503124))

The original computations of stringy black hole entropy using this argument are due to

* [[Ashoke Sen]], _Extremal black holes and elementary string states_,  Mod. Phys. Lett. A10: 2081-2094, 1995 ([arXiv:hep-th/9504147](http://arxiv.org/abs/hep-th/9504147))

* {#StromingerVafa96} [[Andrew Strominger]], [[Cumrun Vafa]], _Microscopic Origin of the Bekenstein-Hawking Entropy_, Phys. Lett. B379: 99-104, 1996 ([arXiv:hep-th/9601029](http://arxiv.org/abs/hep-th/9601029))

* [[Curtis Callan]], [[Juan Maldacena]], _D-brane Approach to Black Hole Quantum Mechanics_, Nucl. Phys. B472:591-610, 1996 ([arXiv:hep-th/9602043](http://arxiv.org/abs/hep-th/9602043))

* [[Juan Maldacena]], [[Gregory Moore]], [[Andrew Strominger]], _Counting BPS Blackholes in Toroidal Type II String Theory_ ([arXiv:hep-th/9903163](http://arxiv.org/abs/hep-th/9903163))

* [[Davide Gaiotto]], [[Andrew Strominger]], X. Yin, _New Connections Between 4D and 5D Black Holes_, JHEP 0602:024,2006 ([arXiv:hep-th/0503217](http://arxiv.org/abs/hep-th/0503217))

See also

* [[Emil Martinec]], V. Sahakian, _Black Holes and Five-brane Thermodynamics_, Phys.Rev. D60 (1999) 064002 ([arXiv:hep-th/9901135](http://arxiv.org/abs/hep-th/9901135))

Discussion for [[M-theory]]/[[11-dimensional supergravity]] includes

* [[Juan Maldacena]], [[Andrew Strominger]], [[Edward Witten]], _Black Hole Entropy in M-Theory_, JHEP 9712:002,1997 ([arXiv:hep-th/9711053](http://arxiv.org/abs/hep-th/9711053))

* [[Gary Horowitz]] , M. Roberts, _Counting the microstates of a Kerr black hole in M theory_, Phys Rev Lett. 2007 Nov 30;99(22):221601. Epub 2007 Nov 30. ([web](https://www.ncbi.nlm.nih.gov/pubmed/18233277)), ([arXiv:0708.1346](https://arxiv.org/abs/0708.1346))

* [[Davide Gaiotto]], [[Andrew Strominger]], Xi Yin, _The M5-Brane Elliptic Genus: Modularity and BPS States_, JHEP 0708:070,2007 ([arXiv:hep-th/0607010](http://arxiv.org/abs/hep-th/0607010))

Discussion relating to the [[Kaluza-Klein monopole]] includes

* {#Nelson93} William Nelson, _Kaluza-Klein Black Holes in String Theory_, Phys.Rev.D49:5302-5306,1994 ([arXiv:hep-th/9312058](http://arxiv.org/abs/hep-th/9312058))

More recent developments include

* Alejandra Castro, Joshua L. Davis, [[Per Kraus]], Finn Larsen, _String Theory Effects on Five-Dimensional Black Hole Physics_ ([arXiv:0801.1863](http://arxiv.org/abs/0801.1863))

* Oleg Lunin, [[Samir Mathur]], _A toy black hole S-matrix in the D1-D5 CFT_ ([arXiv:1211.5830](http://arxiv.org/abs/1211.5830))

* Alejandra Castro, Joshua M. Lapan, Alexander Maloney, Maria J. Rodriguez, _Black Hole Monodromy and Conformal Field Theory_ ([arXiv:1303.0759](http://arxiv.org/abs/1303.0759))

* H. L. Dao, Parinya Karndumri, _Supersymmetric $AdS_5$ black holes and strings from 5D $N=4$ gauged supergravity_ ([arXiv:1812.10122](https://arxiv.org/abs/1812.10122))

* Behnam Pourhassan, Mubasher Jamil, Mir Faizal, _Black Remnants from T-Duality_ ([arXiv:1912.09235](https://arxiv.org/abs/1912.09235))

See also:

* Nejc Čeplak, Roberto Emparan, Andrea Puhm, Marija Tomašević, *The correspondence between rotating black holes and fundamental strings*, JHEP 11 (2023) 226 [[arXiv:2307.03573](https://arxiv.org/abs/2307.03573), doi:10.1007/JHEP11(2023)226]

###  BPS black holes in $AdS_5/CFT_4$ duality

(...)

* Alejandro Cabo-Bizet, Davide Cassani, Dario Martelli, Sameer Murthy, _Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric $AdS_5$ black holes_ ([arXiv:1810.11442](https://arxiv.org/abs/1810.11442))

* {#CKJN18} Sunjin Choi, [[Joonho Kim]], [[Seok Kim]], June Nahmgoong, _Large AdS black holes from QFT_ ([arxiv:1810.12067](https://arxiv.org/abs/1810.12067))

* Francesco Benini, Paolo Milan, _Black holes in 4d $\mathcal{N} = 4$ Super-Yang-Mills_ ([arXiv:1812.09613](https://arxiv.org/abs/1812.09613))

(...)

### The D$p$-brane bound states

* [[Konstadinos Sfetsos]], [[Kostas Skenderis]], _Microscopic derivation of the Bekenstein-Hawking entropy formula for non-extremal black holes_, Nucl. Phys. B517 (1998) 179-204 ([arXiv:hep-th/9711138](https://arxiv.org/abs/hep-th/9711138))

* Mohammad R. Garousi, Ahmad Ghodsi, _Entropy Function for Non-extremal D1D5 and D2D6NS5-branes_, JHEP 0710:036, 2007 ([arXiv:0705.2149](https://arxiv.org/abs/0705.2149))

#### D1-D5 brane systems

Discussion of [[black hole entropy]] via [[D1-D5 brane bound states]] as [[black holes in string theory]]:

(...) many references go here (...)

* [[Emil Martinec]], *The branes behind black holes*, talk at *[Strings 2022](https://indico.cern.ch/event/1085701/)* [[indico:4940814](https://indico.cern.ch/event/1085701/contributions/4940814/)]

#### D2-D6-NS5 brane systems

Discussion of [[black hole entropy]] of [[D2-D6 brane bound states]] as [[black holes in string theory]]:

* [[Juan Maldacena]], [[Andrew Strominger]], _Statistical Entropy of Four-Dimensional Extremal Black Holes_, Phys. Rev. Lett. 77:428-429, 1996 ([arXiv:hep-th/9603060](https://arxiv.org/abs/hep-th/9603060))

### Black holes in supergravity

Black holes in [[supergravity]]:

* [[Riccardo D'Auria]], [[Pietro Fre]], _BPS black holes in supergravity_, ([hep-th/9812160](http://arxiv.org/abs/hep-th/9812160))

* Antonio Gallerati, _Constructing black hole solutions in supergravity theories_ ([arXiv:1905.04104](https://arxiv.org/abs/1905.04104))

Resolution of [[scalar field]]-dressed [[Schwarzschild black holes]] in [[D=11 supergravity]]:

* [[Ibrahima Bah]], Pierre Heidmann, *Geometric Resolution of Schwarzschild Horizon* [[arXiv:2303.10186](https://arxiv.org/abs/2303.10186)]

### Fat black holes

Discussion of black holes whose Schwarzschild radius is larger than the [[KK-compactification]]-scale:

* [[Juan Maldacena]], [[Leonard Susskind]], *D-branes and Fat Black Holes*, Nucl. Phys.B475:679-690 (1996) ([arXiv:hep-th/9604042](https://arxiv.org/abs/hep-th/9604042))

* E. Halyo, [[Arvind Rajaraman]], [[Leonard Susskind]], *Braneless Black Holes*, Phys. Lett.B392:319-322 (1997) ([arXiv:hep-th/9605112](https://arxiv.org/abs/hep-th/9605112))

* A.A. Bytsenko, S.D. Odintsov, p. 8 of: *Statistical entropy of near-extremal and fundamental black p-branes*, Prog. Theor. Phys. 98:987-1003 (1997) ([arXiv:hep-th/9611151](https://arxiv.org/abs/hep-th/9611151))

### Discussion in BFSS matrix theory

Discussion of black holes via the [[BFSS matrix model]] includes the following:

* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]],  _Schwarzschild Black Holes from Matrix Theory_, Phys.Rev.Lett.80:226-229,1998 ([arXiv:hep-th/9709091](https://arxiv.org/abs/hep-th/9709091))

* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzchild Black Holes in Matrix Theory II_, JHEP 9801:008,1998 ([arXiv:hep-th/9711005](https://arxiv.org/abs/hep-th/9711005))

* [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzschild Black Holes in Various Dimensions from Matrix Theory_, Phys.Lett.B416:62-66,1998 ([arXiv:hep-th/9709108](https://arxiv.org/abs/hep-th/9709108))

* Edi Halyo, _Six Dimensional Schwarzschild Black Holes in M(atrix) Theory_ ([arXiv:hep-th/9709225](https://arxiv.org/abs/hep-th/9709225))

* [[Gary Horowitz]], [[Emil Martinec]], _Comments on Black Holes in Matrix Theory_, Phys. Rev. D 57, 4935 (1998) ([arXiv:hep-th/9710217](https://arxiv.org/abs/hep-th/9710217))

* {#KabatTaylor97} Daniel Kabat, [[Washington Taylor]], _Spherical membranes in Matrix theory_, Adv.Theor.Math.Phys.2:181-206,1998 ([arXiv:hep-th/9711078](https://arxiv.org/abs/hep-th/9711078))

* Yoshifumi Hyakutake, _Black Hole and Fuzzy Objects in BFSS Matrix Model_
([arXiv:1801.07869](https://arxiv.org/abs/1801.07869))

### Relation to topological string amplitudes

Discussion of black hole microstates via the [[topological string]]'s [[Gopakumar-Vafa invariants]]:

* [[Rajesh Gopakumar]], [[Cumrun Vafa]], _M-Theory and Topological Strings--I_ ([arXiv:hep-th/9809187](https://arxiv.org/abs/hep-th/9809187))

* [[Rajesh Gopakumar]], [[Cumrun Vafa]], _M-Theory and Topological Strings--II_ ([arXiv:hep-th/9812127](https://arxiv.org/abs/hep-th/9812127))

[[!redirects black hole in string theory]]


black ring

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## Idea

The [[Einstein equations]] for [[gravity]] in [[dimension]] $\geq 5$ admit [[black hole]] solutions whose [[event horizon]] is not [[homeomorphism|homeomorphic]] to an [[n-sphere]], but has different topology. In particular in 5d-gravity and [[5d supergravity]] there are solutions with topology $S^1 \times S^2$. These are called _black rings_.

Their [[KK-reduction]] down to 4 dimensions yields again ordinary black holes. This plays a central role in the discussion of [[black holes in string theory]].

## Related concepts

* [[black string]]

* [[D=5 Einstein-Maxwell theory]]

## References

The first black ring solution was found, in [[D=5 gravity]], in

* [[Roberto Emparan]], [[Harvey Reall]], _A rotating black ring in five dimensions_, Phys. Rev. Lett.88:101101, 2002 ([arXiv:hep-th/0110260](https://arxiv.org/abs/hep-th/0110260))

The first black ring solution was found, in [[D=5 supergravity]], in:

* [[Henriette Elvang]], [[Roberto Emparan]], [[David Mateos]], [[Harvey Reall]], _Supersymmetric 4D Rotating Black Holes from 5D Black Rings_, JHEP0508:042,2005 ([arXiv:hep-th/0504125](http://arxiv.org/abs/hep-th/0504125))

Review:

* {#EmparanReall06} [[Roberto Emparan]], [[Harvey Reall]], _Black Rings_, Class. Quant. Grav. 23:R169, 2006 ([arXiv:hep-th/0608012](https://arxiv.org/abs/hep-th/0608012))

[[KK-compactification|Dimensional reduction]] to [[black holes]] in [[D=4 gravity]]:

* [[Davide Gaiotto]], [[Andrew Strominger]], [[Xi Yin]], _5D Black Rings and 4D Black Holes_, JHEP 0602:023, 2006 ([arXiv:hep-th/0504126](https://arxiv.org/abs/hep-th/0504126))

On non-uniqueness:

* [[Roberto Emparan]], _Rotating Circular Strings, and Infinite Non-Uniqueness of Black Rings_, JHEP0403:064, 2004 ([arXiv:hep-th/0402149](https://arxiv.org/abs/hep-th/0402149))

* [[Iosif Bena]], [[Simon Ross]], [[Nicholas Warner]], _Coiffured Black Rings_, Class.Quant.Grav. 31 (2014) 165015 ([arXiv:1405.5217](https://arxiv.org/abs/1405.5217))

As [[intersecting brane|intersecting]] [[black brane|black]] [[M5-branes]] [[wrapped brane|wrapped]] on a [[4-manifold]] [[product manifold|times]] the black ring's [[circle]]:

* [Emparan-Reall 06, Sec. 4.1, 4.2, 6.1](#EmparanReall06)

* [Castro-David-Kraus-Larsen 08, Sec 2.2](#CastroDavidKrausLarsen08)

following

* {#MaldacenaStromingerWitten97} [[Juan Maldacena]], [[Andrew Strominger]], [[Edward Witten]], _Black Hole Entropy in M-Theory_, 	JHEP 9712:002, 1997 ([arXiv:hep-th/9711053](https://arxiv.org/abs/hep-th/9711053))

More in the context of [[D1-D5-brane bound states]]

* [[Iosif Bena]], [[Per Kraus]], _Microscopic description of black rings in AdS/CFT_ JHEP 12 (2004) 070 ([hep-th/0408186](http://arxiv.org/abs/hep-th/0408186))

* [[Iosif Bena]], [[Per Kraus]], _Microstates of the D1-D5-KK system_ Phys. Rev. D72 (2005) 025007 ([hep-th/0503053](http://arxiv.org/abs/hep-th/0503053))

* {#CastroDavidKrausLarsen08}  Alejandra Castro, Joshua L. Davis, [[Per Kraus]], [[Finn Larsen]], _String Theory Effects on Five-Dimensional Black Hole Physics_ ([arXiv:0801.1863](http://arxiv.org/abs/0801.1863))

[[!redirects black rings]]


black string

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## Idea

A [[black brane]] of one spatial [[dimension]].

## Related concepts

* [[black ring]]

* [[black holes in string theory]]

## References

### General

See also

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

### From wrapped M5-branes

On [[black strings]] in [[D=5 gravity]]/[[D=5 supergravity]] from [[M5-branes]] [[wrapped brane|wrapped]] on [[4-manifolds]]:

* [[David Kastor]], Section 3 of: _From wrapped M-branes to Calabi-Yau black holes and strings_, JHEP 0307 (2003) 040 ([arXiv:hep-th/0305261](https://arxiv.org/abs/hep-th/0305261))

[[!redirects black strings]]


Blake Stacey

I'm a [Research Assistant Professor](https://www.umb.edu/academics/csm/physics/faculty_staff) at the University of Massachusetts Boston. My current research is in [quantum information theory](https://arxiv.org/abs/1703.07901); my background is in [statistical physics, stochastic processes and evolutionary dynamics](https://arxiv.org/abs/1509.02958). I'm also [Blake Stacey](https://scirate.com/blake-stacey) at SciRate.

Evidence of my lack of education:

* [...about combinatorial games](http://golem.ph.utexas.edu/category/2009/11/combinatorialgame_categories.html#c029299)
* [...about generating functions and stuff types](http://golem.ph.utexas.edu/category/2010/06/this_weeks_finds_in_mathematic_60.html#c033601)
* [...about thermodynamics, chaos theory and probability](http://golem.ph.utexas.edu/category/2011/05/categorytheoretic_characteriza_1.html#c037874)

On the advice of [[John Baez]], I'm going to try writing a bit about [[universality]].  And having started a page on [[universality class|universality classes]], I now realise that I should get a few other topics going:

* [[c-theorem]]
* [[Reggeon field theory]]
* [[RSOS model]]
* [[ADE classification]]

### Scratchpad ###

**Restricted Solid-on-Solid** or **RSOS models** are field theories defined on a lattice, in which the field values at adjacent lattice sites are energetically constrained in a way specified by a graph structure.  If the graph is one of the special $A$, $D$ or $E$ Dynkin diagrams, then the critical point of the model is governed by a conformal field theory whose central charge is fixed by the Coxeter number of that Dynkin diagram.  RSOS models therefore exemplify the possible universality classes of 2D systems in thermal equilibrium.

$$ Z = \sum_{heights} \prod_{faces} W \left(\begin{array}{cc} d & c \\ a & b\end{array}\right). $$

The Boltzmann weights, as defined by Pasquier, depend on the _crossing parameter_ $\lambda = \pi/\hat{h}$, where $\hat{h}$ is the Coxeter number of the Dynkin diagram, and the _spectral parameter_ $u \in (0, \lambda)$.

$$ W \left(\begin{array}{cc} d & c \\ a & b\end{array}\mid u\right) = \frac{\sin\lambda - u}{\sin\lambda} \delta_{a, c} G_{a, b} G_{a, d} + \frac{\sin u}{\sin\lambda} \left(\frac{S_a S_c}{S_b S_d}\right)^{1/2} \delta_{b, d} G_{a, b} G_{b, c}. $$

$G_{a, b}$ is the adjacency matrix of the graph.

Michael E. Fisher (1968), _Renormalization of Critical Exponents by Hidden Variables_ Physical Review **176**: 257--72. ([web](http://link.aps.org/doi/10.1103/PhysRev.176.257))

:The mathematical content of this article consists chiefly in the analysis of some thermodynamic manipulations appropriate to the imposition of "constraints" on certain previously free thermodynamic variables and is quite elementary.  To motivate our analysis and reveal its theoretical significance we ask the following fundamental question: "To what extent are the observed values of the critical-point exponents universal?" ... the fairly close concurrence of all the exponent values for fluids and magnets does suggest the existence of a universal set of exponents.

## Selected writings

On [[quantum Bayesianism]]/[[QBism]]

* {#FuchsSchlosshauerStacey14} [[Christopher A. Fuchs]], [[Maximilian Schlosshauer]] (foreword), [[Blake C. Stacey]] (ed.) *My Struggles with the Block Universe* [[arXiv:1405.2390](https://arxiv.org/abs/1405.2390)]

category: people

[[!redirects Blake C. Stacey]]