class object

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#### Universes
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\tableofcontents

## Idea

The [[internalization]] of the notion of [[class]] inside of any [[(2,1)-category]].

Might be better to define this inside of an [[(infinity,1)-category]] so that it admits models inside of [[homotopy type theory]] as [[embeddings]] $m:C \hookrightarrow \mathrm{Set}_U$, but I do not know how to do so.

## Definition

Let $\mathcal{C}$ be an [[(2,1)-category]] with a [[terminal object]], [[interval object]], finite [[2-pullback|(2,1)-pullbacks]], and a $\kappa$-small [[discrete object classifier]] $\mathrm{Set}_\kappa \in \mathrm{Ob}(\mathcal{C})$ for [[cardinal number]] $\kappa$.

An *internal set* $A$ in $\mathcal{C}$ is a [[morphism]] $A:\mathrm{Hom}(1, \mathrm{Set}_\kappa)$ from the [[terminal object]] $1 \in \mathrm{Ob}(\mathcal{C})$ to $\mathrm{Set}_\kappa$.

A **class object relative to $\mathrm{Set}_\kappa$** in $\mathcal{C}$ is an [[object]] $C \in \mathrm{Ob}(\mathcal{C})$ with a [[monomorphism]] $m:C \hookrightarrow \mathrm{Set}_\kappa$.

## Category of class objects

Given a [[(2,1)-category]] $\mathcal{C}$ with a [[terminal object]], [[interval object]], finite [[2-pullback|(2,1)-pullbacks]], and a $\kappa$-small [[discrete object classifier]] $\mathrm{Set}_\kappa \in \mathrm{Ob}(\mathcal{C})$ for [[cardinal number]] $\kappa$, one could define the **[[(2,1)-category]] of class objects** $\mathrm{Mono}(\mathrm{Set}_\kappa)$ as the category of [[monomorphisms]] into the $\kappa$-small discrete object classifier, whose morphisms preserve the monomorphism into $\mathrm{Set}_\kappa$: for every object $A \in \mathrm{Mono}(\mathrm{Set}_\kappa)$ with monomorphism $a:\mathrm{Hom}(A,\mathrm{Set}_\kappa)$ and $B \in \mathrm{Mono}(\mathrm{Set}_\kappa)$ with monomorphism $b:\mathrm{Hom}(B,\mathrm{Set}_\kappa)$ and morphism $f:\mathrm{Hom}(A, B)$, there is a [[natural isomorphism]] $i(a, b, f):\mathrm{id}_{\mathrm{Set}_\kappa} \circ a \cong b \circ f$.

$$\array{& A & \overset{a}\hookrightarrow & \mathrm{Set}_\kappa & \\
          f & \downarrow & \cong\swArrow &\downarrow & \mathrm{id}_{\mathrm{Set}_\kappa}\\
          &B & \underset{b}\hookrightarrow& \mathrm{Set}_\kappa & \\
}$$

## See also

* [[discrete object classifier]]

* [[class]]

* [[class theory]]

* [[category of classes]]

* [[type of classes]]

[[!redirects class objects]]

[[!redirects category of class objects]]
[[!redirects (2,1)-category of class objects]]
[[!redirects (n,1)-category of class objects]]
[[!redirects (infinity,1)-category of class objects]]


class of adapted objects

Let $A,B$ be [[abelian category|abelian categories]], $R\subset \mathrm{Ob}A$ a class of objects in $A$ and $F:A\to B$ an [[additive functor]].

If $F$ is left [[exact functor]] we say that $R$ is a **class of objects adapted to** $F$ if $F$ sends acyclic and bounded below [[cochain complex|cochain complexes]] of objects in $R$ to acyclic complexes (= having trivial cohomology), and every object in $A$ is a [[subobject]] of an object from $R$.

If $F$ is right [[exact functor]] we say that $R$ is a **class of objects adapted to** $F$ if $F$ sends acyclic and bounded above [[chain complex|chain complexes]] of objects in $R$ to acyclic complexes (= having trivial homology), and every object in $A$ is a [[quotient object]] of an object from $R$.

For example, if $R = I$ (resp. $R = P$) is the class of all [[injective object|injective]] (resp. [[projective object|projective]]) objects in an abelian category $A$ with sufficiently many injectives (resp. projectives), then $R$ is adapted to any left (resp. right) exact functor whose domain is $A$. If $F$ is left (resp. right) exact and a family $R$ of objects adapted to $F$ exists then the right [[derived functor]] $D^+(F):D^+(A)\to D^+(B)$ (resp. left derived functor $D^-(F):D^-(A)\to D^-(B)$) of $F$ exists by a standard construction using resolutions by objects in $R$. Flexibility in choosing an adapted class is often useful.

References:

* A. Grothendieck: [[Tohoku]]

* S. I. Gel'fand, Yu. I. Manin, Methods of homological algebra, Moskva 1988 (Russian); Springer 1996 (English): chapter 3

* (for generalizations in nonabelian setup) A. Rosenberg, Homological algebra of noncommutative 'spaces' I, [preprint MPIM2008-91](http://www.mpim-bonn.mpg.de/preprints/send?bid=3623)


class S field theory

#Contents#
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{:toc}

## Overview

Class $\mathcal{S}$ (for "six") originally denoted a specific class of $N=2$ 4d [[super Yang-Mills theory]] introduced in [Gaiotto, Moore & Neitzke 2009](#GMN2009). They originate from [[KK-compactification]] of [[M5-branes]] on punctured [[Riemann surfaces]] $C$ and are labelled by a [[simply laced Lie group]], a [[Riemann surface]] $C$, and a decoration of the punctures of $C$ by defect operators.

By further compactifications, one relates theories of this class with certain [[superconformal symmetry|superconformal]] field theories in  dimension $2$ (or, in other formalism, [[chiral algebras]]) which are now also said to be of class $S$ (see at [[AGT correspondence]]).

## References

Original articles

* {#GMN2009} [[Davide Gaiotto]], [[Greg Moore]], [[Andrew Neitzke]], _Wall-crossing, Hitchin systems, and the WKB approximation, Advances in Mathematics __234__ (2013) 239--403 [arXiv:0907.3987](https://arxiv.org/abs/0907.3987) [doi](https://doi.org/10.1016/j.aim.2012.09.027)

see also

* [Zafrir 2016](orientifold+plane#Zafrir2016).

In the [[AGT correspondence]]:

* Bruno Le Floch, _A slow review of the AGT correspondence_, J. Phys. A: Math. Theor. __55__ (2022) 353002 [doi](https://doi.org/10.1088/1751-8121/ac5945) [arXiv:2006.14025](https://arxiv.org/abs/2006.14025)

On [[vertex operator algebras]]/[[chiral algebras]] of class $\mathcal{S}$:

* [[Tomoyuki Arakawa]], _Chiral algebras of class S and Moore-Tachikawa symplectic varieties_, [arXiv:1811.01577](https://arxiv.org/abs/1811.01577)

  > We give a functorial construction of the genus zero chiral algebras of class $\mathcal{S}$, that is, the vertex algebras corresponding to the theory of class $\mathcal{S}$ associated with genus zero pointed Riemann surfaces via the 4d/2d duality discovered by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees in physics. We show that there is a unique family of vertex algebras satisfying the required conditions and show that they are all simple and conformal. In fact, our construction works for any complex semisimple group G that is not necessarily simply laced. Furthermore, we show that the associated varieties of these vertex algebras are exactly the genus zero Moore-Tachikawa symplectic varieties that have been recently constructed by Braverman, Finkelberg and Nakajima using the geometry of the affine Grassmannian for the Langlands dual group.

On [[superconformal symmetry|superconformal]] theories of class $\mathcal{S}$:

* [[Leonardo Rastelli]], S. S. Razamat, _The superconformal index of theories of class $\mathcal{S}$_, in: Teschner, J. (eds) New Dualities of Supersymmetric Gauge Theories. Mathematical Physics Studies. Springer 2015 [doi](https://doi.org/10.1007/978-3-319-18769-3)

* [[Christopher Beem]], Wolfger Peelaers, [[Leonardo Rastelli]], Balt C. van Rees, _Chiral algebras of class S_,   JHEP 05 (2015) 020 [doi](https://doi.org/10.1007/JHEP05%282015%29020) [arXiv:1408.6522](https://arxiv.org/abs/1408.6522)

On [[quantum Seiberg-Witten curves]] in relation to [[class S-theories]] and "[[M3]]"-[[defect branes]] inside [[M5-branes]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], *Elliptic Quantum Curves of Class $\mathcal{S}_k$*,  J. High Energ. Phys. **2021** 28 (2021) [[arXiv:2008.05155](https://arxiv.org/abs/2008.05155), doi:10.1007/JHEP03(2021)028]

category: physics

[[!redirects class S field theories]]

[[!redirects class S theory]]
[[!redirects class S theories]]

[[!redirects class S-theory]]
[[!redirects class S-theories]]

[[!redirects class S]]


class theory

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#### Foundations
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## Idea

In the [[foundations of mathematics]], a **class theory** or **class-set theory** is a [[theory]] of [[classes]] and [[sets]]. Class theories are usually formulated in the language of [[logic over type theory]]. Class theories are important because the allow for the formal definition of [[universes]], a [[universal class]] or a class of all sets, allowing for [[size issues]] in [[mathematics]] to be handled where needed, such as in [[category theory]] in defining [[large categories]] and [[locally small categories]].

Similar to the case with [[set theory]], there are [[material versus structural|material and structural versions]] of class theory as well:

Material approaches to class theories include *[[Morse-Kelley class theory]]* and *[[von Neumann–Bernays–Gödel class theory]]*, and are two-sorted theories with a sort of [[sets]] and a sort of [[classes]].

Structural approaches to class theories include *[[category with class structure]]* and the general field of *[[algebraic set theory]]*. Here, classes are defined as [[objects]] in a suitably presented [[well-pointed category]], and [[sets]] are defined as a class with some smallness structure.

In contrast to [[material set theories]] like [[ZFC]] and [[structural set theories]] like [[ETCS]] or [[SEAR]], the notion of [[set]] is not a primitive in the theory. It is instead a defined notion from the primitive notion of [[class]].

## Semantics and models

Let $U$ be a [[universe]], and let $\mathrm{Set}_U$ be the [[universe]] of [[sets]] in $U$, and let $\Omega$ be a collection of [[propositions]]. A [[model]] of [[set theory]] is $\mathrm{Set}_U$, while a model of [[class theory]] is the collection of [[predicates]] $\mathrm{Set}_U \to \Omega$.

## See also

* [[set theory]]

* [[class]]

* [[category of classes]]

## References

* [[Henrik Forssell]], *Categorical Models of Intuitionistic Theories of Sets and Classes*. Master's thesis, Carnegie Mellon University, 2004. ([PDF](https://www.phil.cmu.edu/projects/ast/Papers/forssell_thesis.pdf))

[[!redirects class theory]]
[[!redirects class theories]]

[[!redirects class-set theory]]
[[!redirects class-set theories]]

[[!redirects material class theory]]
[[!redirects material class theories]]

[[!redirects material class-set theory]]
[[!redirects material class-set theories]]

[[!redirects structural class theory]]
[[!redirects structural class theories]]

[[!redirects structural class-set theory]]
[[!redirects structural class-set theories]]


class-locally presentable category

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#### Category theory
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#### Limits and colimits
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#Contents#
* table of contents
{:toc}

## Idea

The notion of a **class-locally presentable category** is a generalisation of that of a [[locally presentable category]]: In a class-locally presentable category one may need a [[proper class]] of [[objects]] (instead of just a [[small set]]) to build all the others by suitable [[colimits]].

## Definition

Let $\lambda$ be a [[regular cardinal]], and let the [[category]] $C$ be [[complete category|complete]] and [[cocomplete category|cocomplete]]. Then $C$ is called **class $\lambda$-presentable** if there is a [[class]] $A \subset C$ of $\lambda$-presentable objects such that every object of $C$ is a $\lambda$-[[filtered colimit]] of objects in $A$. A category is class-locally presentable if it is class $\lambda$-presentable for some $\lambda$.

If $C$ merely has $\lambda$-filtered colimits, rather than all colimits (and limits), then it is called **class $\lambda$-accessible**, and **class-accessible** if it is class $\lambda$-accessible for some $\lambda$.

## Related concepts

* [[macrotopos]]

## References

* [[Boris Chorny]], [[Jiří Rosický]], _Class-locally presentable and class-accessible categories_, Journal of Pure and Applied Algebra
**216**, Issue 10 (2012) pp 2113–2125 ([doi:10.1016/j.jpaa.2012.01.015](https://doi.org/10.1016/j.jpaa.2012.01.015))

[[!redirects class-locally presentable categories]]

[[!redirects class-accessible category]]

[[!redirects class-accessible categories]]


classical Adams spectral sequence

> This entry is about the classical Adams spectral sequence only. For more general discussion see at _[[Adams spectral sequence]]_.

> under construction

#Contents#
* table of contents
{:toc}

## Idea

The _classical Adams spectral sequence_ ([Adams 58](#Adams58)) is a type of [[spectral sequences]] used for computations  in [[stable homotopy theory]]. It computes the [[homotopy groups of spheres]] at prime 2 from   [[homology]]/[[cohomology]], as [[modules]]/[[comodules]] over its [[Steenrod operations]]. The Adams spectral sequence may be seen as a variant of the [[Serre spectral sequence]] obtained by replacing a single fibration by an "[[Adams resolution]]".

The original _clasical Adams spectral sequenc_ for [[ordinary cohomology]] is further refined by the _[[Adams-Novikov spectral sequence]]_ ([Novikov 67](#Novikov67)) by replacing [[ordinary cohomology]] modulo $p$ by [[complex cobordism cohomology theory]] or [[Brown-Peterson theory]] or the like.

Generally, for $E$ a suitable [[E-infinity algebra]] there is a corresponding _$E$-[[Adams spectral sequence]]_ whose second page is given by $E$-[[generalized cohomology]] and which arises as the [[spectral sequence of a simplicial stable homotopy type]] of the [[cosimplicial object|cosimplicial]] object which is the [[Cech nerve]]/[[Sweedler coring]]/[[Amitsur complex]] of $E$. As such the Adams spectral sequence is an analog in [[stable homotopy theory]] of the [[Bousfield-Kan spectral sequence|Bousfield-Kan]] [[homotopy spectral sequence]] in unstable [[homotopy theory]].

### Via iterated Hurewicz theorem and Serre spectral sequence
 {#MotivationFromHurewiczTheoremAndSerreSpectralSequence}

The classical Adams spectral sequence may be motivated from the strategy to
compute [[homotopy groups]] from [[cohomology groups]] by subsequently applying the [[Hurewicz theorem]] to compute the lowest-degree non-trivial homotopy group from the corresponding [[cohomology group]], then co-killing that by forming its [[homotopy fiber]], finally applying the [[Serre spectral sequence]] to  identify the next lowest non-trivial cohomology group of that fiber, and then iterating this process. The Adams spectral sequence arises when in this kind of strategy instead of co-killing only the lowest lying [[cohomology group]], one at a time, one co-kills _all_ nontrivial cohomology groups, then forms the corresponding [[homotopy fiber]] and so on.

This was apparently historically the way that [[John Adams]] indeed proceeded from [[Jean-Pierre Serre]]'s approach and this is still a good motivation for the whole construction. A nice exposition is in  ([Wilson 13, 1.1](#Wilson13)).

We now say this again in more detail.

Given $n \in \mathbb{N}$, consider the probem of computing the [[homotopy groups of spheres|homotopy groups]] $\pi_k(S^n) \;mod \;2$ of the [[n-sphere]] $S^n$. For $k \leq n$ this is clear: first for $k \lt n$ they all vanish, and second for $k = n$ we have, by the very nature of [[Eilenberg-MacLane spaces]] $K(\mathbb{Z}_2, n)$, that the [[ordinary cohomology]] is

$$
  H^n(S^n, \mathbb{Z}_2) \simeq [S^n, K(\mathbb{Z}_2,n)] \simeq \pi_n(K(\mathbb{Z}_2,n)) \simeq \mathbb{Z}_2
$$

so that by the [[Hurewicz theorem]] it follows that also

$$
  \pi_n(S^n) \;mod\;2 \;\simeq \mathbb{Z}_2
  \,.
$$

The [[Hurewicz theorem]] does not say anything beyond the first non-vanishing cohomology group, but so to apply it again we can move up one step in the [[Whitehead tower]] of $S^n$ and hence consider the [[homotopy fiber]]

$$
  \array{
     F_1
     \\
     \downarrow
     \\
     S^n &\stackrel{c_1}{\longrightarrow}& K(\mathbb{Z}_2,n)
  }
$$

of the generator $[c_1] = 1 \in \pi_n(S^n) \simeq \mathbb{Z}_2$.

To apply the [[Hurewicz theorem]] to that fiber we need to know its lowest non-trivial [[cohomology group]] again, and this is computed via the [[Serre spectral sequence]] applied to this [[fiber sequence]].

From here on the process repeats, and one moves higher through the [[Whitehead tower]] of $S^n$

$$
  \array{
     \vdots
     \\
     \downarrow
     \\
     F_1 &\stackrel{c_2}{\longrightarrow}& K(\mathbb{Z}_2, n+1)
     \\
     \downarrow
     \\
     S^n &\stackrel{c_1}{\longrightarrow}& K(\mathbb{Z}_2,n)
  }
  \,.
$$

The Adams spectral sequence arises from this strategy by co-killing not just the first non-trivial [[cohomology group]] at each stage, but _all_ nontrivial cohomology groups at a given stage.

This is done in [[stable homotopy theory]], so let now $X$ be a [[spectrum]] (for instance the [[sphere spectrum]] $X = \mathbb{S}$ if we still with the computation of the [[stable homotopy groups of spheres]]). Write $H \mathbb{F}_2$ for the [[Eilenberg-MacLane spectrum]] for [[ordinary cohomology]] with [[coefficients]] in $\mathbb{Z}_2$, so that an element in [[cohomology]]

$$
  [c] \in H^n(X)
$$

is represented by the [[homotopy class]] of a [[homomorphism]] of [[spectra]] of the form

$$
  c \;\colon\;  X \longrightarrow \Sigma^n H\mathbb{F}_2
$$

(a [[cocycle]]), where "$\Sigma$" denotes [[suspension]], as usual.

If $X$ is a [[spectrum]] of [[finite type]] then there is a [[finite]] $I$ of non-trivial cohomology classes like this, and a choice of [[cocycles]] $c_i$ for each of them gives a single map

$$
  f_0
    \coloneqq
   (c_i)_I \;\colon\;
   X \longrightarrow K_0 \coloneqq \bigvee_{i \in I} \Sigma^{n_i}H \mathbb{F}_2
$$

into a [[generalized Eilenberg-MacLane spectrum]]. As before, this map classifies its [[homotopy fiber]]

$$
  \array{
    F_1
    \\
    \downarrow
    \\
     X &\stackrel{f_0}{\longrightarrow}& K_0
  }
$$

which may be thought of as encoding all information about $X$ beyond its [[cohomology groups]]. Iterating this process gives the corresponding analog of the [[Whitehead tower]], called the _[[Adams resolution]]_ of $X$:

$$
  \array{
    \vdots
    \\
    \downarrow
    \\
    F_2 &\stackrel{f_2}{\longrightarrow}& K_2
    \\
    \downarrow
    \\
    F_1 &\stackrel{f_1}{\longrightarrow}& K_1
    \\
    \downarrow
    \\
     X &\stackrel{f_0}{\longrightarrow}& K_0
  }
  \,.
$$

The _Adams spectral sequence_ is that induced by the [[exact couple]] obtained by applying $\pi_\bullet$ to this [[Adams resolution]].

We now say this more in detail.

The [[long exact sequences of homotopy groups]] for all the [[homotopy fibers]] in this diagram arrange into a diagram of the form

$$
  \array{
    \vdots
    \\
    \downarrow & \nwarrow
    \\
    \pi_\bullet(F_2) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow}& \pi_\bullet(K_2)
    \\
    \downarrow & \nwarrow^{\mathrlap{\partial_2}}
    \\
    \pi_\bullet(F_1) &\stackrel{\pi_\bullet(f_1)}{\longrightarrow}& \pi_\bullet(K_1)
    \\
    \downarrow & \nwarrow^{\mathrlap{\partial_1}}
    \\
    \pi_\bullet(X) &\stackrel{\pi_\bullet(f_0)}{\longrightarrow}& \pi_\bullet(K_0)
  }
  \,,
$$

where the diagonal maps are the [[connecting homomorphisms]] and hence decrease degree in $\pi_\bullet$ by one.
The idea now is to compute the [[homotopy groups]] of $X$ from the decomposed information in this diagram as follows.

First, by construction the homotopy groups $\pi_\bullet(K_s)$ are known, therefore we can identify elements

$$
  \sigma \in \pi_\bullet(X)
$$

if they come from elements

$$
  \sigma_s \in \pi_\bullet(X_s)
$$

whose image

$$
  \pi_\bullet(f_s)(\sigma_s) \in \pi_\bullet(K_s)
$$

we understand. So the task is to understand the image of $\pi_\bullet(f_s)$ in $\pi_\bullet(K_s)$, for each $s$.

By [[exact sequence|exactness]] an element $\kappa_s \in \pi_\bullet(K_s)$ is in this image if its image

$$
  \rho_{s+1} \coloneqq \partial(\kappa_s) \in \pi_{\bullet-1}(X_{s+1})
$$

vanishes. Now, by construction of the resolution, "evidence" for this is that $f_{s+1}(\partial(\kappa_s)) \in \pi_{\bullet-1}(K_{s+1})$ vanishes, which in turn by [[exact sequence|exactness]] means equivalently that $\partial(\kappa_s)$ is the image of an element  $\rho_{s+2} \in  \pi_{\bullet-1}(X_{s+2}) \to \pi_{\bullet-1}(X_{s+1})$. Now again "evidence" for $\rho_{s+2}$ to vanish is that its image $f_{s+2}(\rho(s+2))$ vanishes, which again means that it comes from an element $\rho_{s+3} \in  \pi_{\bullet-1}(X_{s+3}) \to \pi_{\bullet-1}(X_{s+2})$.

Proceeding by [[induction]] this way, we find that accumulated "evidence" in homotopy groups of $K_\bullet$ for an element $\kappa_s$ to represent an element in $\pi_\bullet(X)$ is that its differential $\partial \kappa_s$ factors through all the $\pi_{\bullet-1}(X_{s+k}) \to \pi_{\bullet-1}(X_s)$. This in turn means that it factors through the [[inverse limit]] $\underset{\leftarrow}{\lim}_s \pi_{\bullet-1}(X_s)$.
Such an element $\kappa_s$ with

$$
  \partial \kappa_s \in \underset{\leftarrow}{\lim}_s \pi_{\bullet-1}(X_s) \to \pi_{\bullet-1}(X_{s+1})
$$

is called a _permanent cycle_.

In good cases, the [[Adams resolution]] is indeed a [[resolution]] which means that  the [[inverse limit]] $\underset{\leftarrow}{\lim}_s X_s $ is in fact [[contractible]]. This means that all the "evidence" accumulated in a permanent cycle is indeed sufficient evidence to prove the existence of an element $\sigma_s \in \pi_\bullet(X_s)$ and hence of an element $\sigma \in \pi_\bullet(X)$.

A trivial way for this to be the case is that the original $\sigma_s$ is itself in the image under $\partial$ of some element, in which case $\kappa_s = 0$ already all by itself. These elements are called _eventual boundaries_. Therefore if the Adams resolution is indeed a resolution, then the quotient group

$$
  \frac{permanent\;cycles}{eventual\;boundaries}
$$

gives elements in $\pi_\bullet(S)$, and this quotient is what the Adams spectral sequence computes.

## Properties

### Generators in low range

For the moment see at _[[May spectral sequence]]_.

### Further

$h_j$ is a permanent cycle in the Adams spectral sequence if $\mathbb{R}^{(2^j)}$ admits the structure of a real [[division algebra]]

Adams: ...

$$
  d_2(h_{j+1}) = h_0 h_j^2
$$

Mahowald-Tangora: $h_4^2$ is a permanent cycle

Barratt-Jones-Mahowald: $h_5^2$ is a permanent cycle

Hill-Hopkins-Ravenel: for $j \gt 7$ then $h_j^2$ is not a permanent cycle.

## Related concepts

* [[May spectral sequence]], [[Curtis algorithm]]

* [[Adams-Novikov spectral sequence]]

## References

The original article is

* {#Adams58} [[Frank Adams]], _On the structure and applications of the Steenrod algebra_, Comm. Math. Helv. 32 (1958), 180–214.

Textbook accounts proceeding in the [[coalgebra]] picture include

* [[Doug Ravenel]], chapter 3 of _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986/2003

* {#Kochman96} [[Stanley Kochman]], chapter 5 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

Further review includes

* {#MosherTangora68} [[Robert Mosher]], [[Martin Tangora]], section 18 of _Cohomology Operations and Application in Homotopy Theory_, Harper and Row (1968) ([pdf](www.maths.ed.ac.uk/~aar/papers/moshtang.pdf))

* [[John McCleary]], chapter 9 of _A User's Guide to Spectral Sequences_, Cambridge University Press (1985, 2001)

* {#Bruner09} [[Robert Bruner]], _An Adams spectral sequence primer_, 2009 ([pdf](http://www.math.wayne.edu/~rrb/papers/adams.pdf))

* {#Rognes12} [[John Rognes]], _The Adams spectral sequence_ (following [Bruner 09](#Bruner09)), 2012 ([pdf](http://folk.uio.no/rognes/papers/notes.050612.pdf))

and

* [[Alexander Kupers]], _An introduction to the Adams spectral sequence_ (following [Rognes 12](#Rognes12)) ([pdf](http://math.stanford.edu/~kupers/adamsss.pdf))

* Paolo Masulli, _Stable homotopy and the Adams spectral sequence_ ([pdf](http://www.math.ku.dk/~jg/students/masulli.msproject.2011.pdf))

* [pdf](http://www.math.harvard.edu/~sia/notes/classical_topology_adams.pdf)


classical anomaly

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#Contents#
* table of contents
{:toc}

## Idea

In the context of [[classical mechanics]], a _classical anomaly_ is a  [[central extension]] of a [[Noether current]] algebra (e.g. [Toppan 01](#Toppan01)).

## Definition

In terms of [[symplectic geometry]] this means the following ([Arnold 78, appendix 5.A](#Arnold78)). Given a [[Lie group]] $G$ acting by [[symplectomorphisms]] on a [[phase space]] [[symplectic manifold]] $(X,\omega)$, this [[symmetry]] has a _classical anomaly_ if it does not lift to a genuine $G$-[[Hamiltonian action]], but only to a [[projective representation|projective]] $G$-[[Hamiltonian action]], hence to a [[Hamiltonian action]] of a [[central extension]] $\widehat G$ of $G$. Specifically, the _classical anomaly_ of the original symplectic $G$-action is the 2-[[cocycle]] which classifies this extension.

If the original $G$-action is by [[flows]] of  [[Hamiltonian vector fields]] (just not with explicitly chosen [[Hamiltonians]]), then there is a  universal classical anomaly given by the [[pullback]] of the [[quantomorphism group]] [[extension]]

$$
  \array{
     \widehat G &\longrightarrow& QuantMorph(X,\omega)
     \\
     \downarrow &(pb)& \downarrow
     \\
    G &\stackrel{}{\longrightarrow}& HamSympl(X,\omega)
  }
  \,.
$$

This $\widehat G$ is the [[Heisenberg group]] of the given $G$-action (See also [Fiorenza-Rogers-Schreiber 13](#FiorenzaRogersSchreiber13) for discussion in [[higher prequantum geometry]]).

Notice that on the infinitesimal level of [[Lie algebras]], using that the Lie algebra of the [[quantomorphism group]] is the [[Poisson Lie algebra]] $\mathfrak{pois}(X,\omega)$, this means that an infinitesimal action of a Lie algebra $\mathfrak{g}$ via [[Hamiltonian vector fields]] on $X$ has a classical anomaly if it lifts to an action with consistently chosen [[Hamiltonians]] -- also called a [[moment map]] -- only after passing to a central [[Lie algebra extension]]

$$
  \array{
     \widehat \mathfrak{g} &\longrightarrow& \mathfrak{pois}(X,\omega)
     \\
     \downarrow && \downarrow
     \\
     \mathfrak{g} &\stackrel{}{\longrightarrow}& \Gamma(T X)_{\omega}
  }
  \,.
$$

## Examples

### Mass-anomaly of Galilean action on non-relativistic mechanics
 {#GalileoAnomaly}

The canonical [[Galileo group]]-[[action]] on the [[phase space]] of non-relativistic [[classical mechanics]] has a classical anomaly, given by a [[group cohomology|group 2-cocycle]] proportional to the [[mass]] of the system, the _[[Galileo 2-cocycle]]_ (e.g. [Chen-Shaw-Yen 85](#ChenShawYen85), [Azcárraga-Izquierdo 95](#AzcarragaIzquierdo95) [Marle 14](#Marle14)).

## Related concepts

* [[quantum anomaly]]

* [[fiber bundles in physics]]

[[!include geometric quantization extensions - table]]

## References

A textbook discussion of the concept is (without the terminology yet) is in appendix 5.A of

* [[Vladimir Arnol'd]], _[[Mathematical methods of classical mechanics]]_, Graduate texts in Mathematics 60 (1978)
 {#Arnold78}

A discussion under the term "classical [[central charge]]" is in

* J. D. Brown, [[Marc Henneaux]], _Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity_, Commun. Math. Phys. 104 (1986) 207. ([web](http://dx.doi.org/10.1007/BF01211590))

For a list of some examples and further pointers to the (historical) literature, see

* {#Toppan01} Francesco Toppan, _On anomalies in classical mechanical systems_, Journal of Nonlinear Mathematical Physics Volume 8, Number 3 (2001), 518–533  ([[ToppanClassicalAnomaly.pdf:file]])

See also

* [[Glenn Barnich]], Cédric Troessaert, _Comments on holographic current algebras and asymptotically flat four dimensional spacetimes at null infinity_ ([arXiv:1309.0794](http://arxiv.org/abs/1309.0794))

Discussion in terms of [[Heisenberg group]] [[group extensions|extensions]] and generalization to [[higher symplectic geometry]] is in

* {#FiorenzaRogersSchreiber13} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantum theory]]_ ([arXiv:1304.0236](http://arxiv.org/abs/1304.0236))

Discussion in the context of formalization of [[classical field theory]] in [[cohesive homotopy theory]] is in

* [[Urs Schreiber]], section 2.14 of _[[schreiber:Classical field theory via Cohesive homotopy types]]_ ([pdf](https://dl.dropboxusercontent.com/u/12630719/classicalinhigher.pdf))

The example of the [[Galileo 2-cocycle]] is discussed for instance in

* {#ChenShawYen85} Chen, Shaw, Yen, _An example of a 2-cocycle_, [pdf](http://psroc.phys.ntu.edu.tw/cjp/v23/4.pdf)

* {#Marle14} Charles-Michel Marle, _The manifold of Motions and the total mass of a mechanical system_, 2014 ([pdf](http://charles-michel.marle.pagesperso-orange.fr/diaporamas/ManifoldMotionsMass.pdf))

and in the broader context of [[WZW model]] terms in

* {#AzcarragaIzquierdo95} [[José de Azcárraga]], José M. Izquierdo, section 8.3 of _[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]_ , Cambridge monographs of mathematical physics, (1995)

following

* [[José de Azcárraga]], _Wess-Zumino terms, extended algebras and anomalies in classical physics_, Contemp.Math. 132 (1992) 75-98 ([spire](https://inspirehep.net/record/320052?ln=de))

[[!redirects classical anomalies]]

[[!redirects Galileo 2-cocycle]]


classical context

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum mechanics]], given an [[algebra of quantum observables]], a _classical context_ is a [[poset of commutative subalgebras|commutative subalgebra]].

The [[quantum observables]] in such a commutative subalgebra may all be [[measurement|measured]] simultaneously without the [[uncertainty principle]] of [[quantum mechanics]] interfering, hence they may be measured as [[observables]] in [[classical mechanics]].

## Related concepts

* [[beable]]

* [[order-theoretic structure in quantum mechanics]]

  * [[Gleason's theorem]]

  * [[Kochen-Specker theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

  * [[Bohr topos]]

[[!redirects classical contexts]]


classical double copy

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### Context
#### Gravity
+-- {: .hide}
[[!include gravity contents]]
=--
#### Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
=--
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _classical double copy_-method is the counterpart in [[classical field theory]] of the [[double copy]]-phenomenon for [[scattering amplitudes]] in [[perturbative quantum field theory]]. It relates classical solutions of the [[equation of motion|field equations]] of a [[Yang-Mills theory|Yang-Mills]] [[gauge theory]] with solutions of the [[Einstein equation]] in [[general relativity]].

## Double copy and classical field theory

### Double copy and Kerr-Schild metric

#### Definitions

* A [[Kerr-Schild spacetime|Kerr-Schild metric]] is a [[perturbation]] of a flat [[Minkowski metric]] $\eta_{\mu\nu}$ of the form

  $$ g_{\mu\nu} = \eta_{\mu\nu} + \kappa \phi k_{\mu}k_{\nu} $$

  where $\kappa=\sqrt{32\pi G_{\mathrm{N}}}$ is a constant with $G_{\mathrm{N}}$ [[Newton's constant]], $\phi$ is a [[scalar field]] and $k_\mu$ is a [[lightlike|null]] [[covector]] satisfying the [[geodesic]] property, i.e.

  $$ \eta_{\mu\nu}k^\mu k^\nu = g_{\mu\nu}k^\mu k^\nu =0, \quad (k\cdot\partial)k^\mu =0.$$

* The _single copy_ gauge field ([MOW 15](#MOW15)) of this gravitational field is defined for any [[gauge group]] $G$ by

  $$A_{\mu} = (c^a \mathbf{T}_a) \phi k_\mu$$

  where $c^a\mathbf{T}_a \in \mathfrak{g}$ is an arbitrary constant [[color charge]], specified by a [[vector]] $c^a$ in the [[linear basis|basis]] $\{\mathbf{T}_a\}$ of the [[Lie algebra]] $\mathfrak{g}$.

* Conversely, if we start from a gauge field of the form $A_{\mu} = (c^a \mathbf{T}_a) \phi k_\mu$ for any constant [[color charge]] $c^a\mathbf{T}_a \in \mathfrak{g}$ and [[lightlike|null]] [[covector]] $k_\mu$ satisfying the [[geodesic]] property, we can define its _double copy_ gravitational field by the [[Kerr-Schild spacetime|Kerr-Schild metric]] $g_{\mu\nu}=\eta_{\mu\nu} + \kappa \phi k_{\mu}k_{\nu}$.

* Otherwise, if we repeat the procedure of replacing a covector $k_\mu$ with any fixed [[color charge]] $(\tilde{c}^b \tilde{\mathbf{T}}_b)\in\tilde{\mathfrak{g}}$ we can get a _zeroth copy_ [[scalar field]], defined by

  $$\Phi = (c^a \mathbf{T}_a)\otimes(\tilde{c}^b \tilde{\mathbf{T}}_b)\phi $$

  where the new [[gauge group]] $\tilde{G}$ can be chosen different from the previous $G$.

#### Field equations

By following ([MOW 15](#MOW15)) we have a comparison of the field equations. Assume without loss of generality that $k^0=1$. We get the following:

* The vacuum [[Einstein equations]] for the metric $g_{\mu\nu}$ are $R=0$ (where $R$ is the [[Ricci curvature]]), which reduce to
$$\begin{aligned}
R^0_{\;0} &= \frac{1}{2}\nabla^2\phi && = 0 \\
R^i_{\;0} &= \frac{1}{2}\partial_j \left(\partial^j(\phi k^i)-\partial^i(\phi k^j)\right) && =0 \\
R^i_{\;j} &= \frac{1}{2}\partial_l \left(\partial^i(\phi k^l k^j)+\partial_j(\phi k^l k^i)-\partial^l(\phi k^i k_j)\right) && =0
\end{aligned}$$

* The [[Maxwell equations]] for the gauge field $A$ are $\mathrm{d} F = 0$, which reduce to
$$\begin{aligned}
(\mathrm{d}F)^0 &= \nabla^2\phi && = 0 \\
(\mathrm{d}F)^i &= \partial_j \left(\partial^j(\phi k^i)-\partial^i(\phi k^j)\right) && = 0
\end{aligned}$$

* The [[Klein-Gordon equation]] for the scalar field $\Phi$ are
$$\nabla^2\Phi = 0$$

#### Outlook

Summarizing, we have the following table:

| zeroth copy | single copy | double copy |
|--|--|--|
| $\;\Phi = (c^a \mathbf{T}_a)\otimes(\tilde{c}^b \tilde{\mathbf{T}}_b)\phi\;$ | $\;A_{\mu} = (c^a \mathbf{T}_a) \phi k_\mu\;$ | $\;g_{\mu\nu} = \eta_{\mu\nu} + \kappa \phi k_{\mu}k_{\nu}\;$ |

\begin{imagefromfile}
"file_name": "doublecopy.png",
"width": 700,
"unit": "px",
"caption": "Picture grabbed from ([BSW 20](#BSW20))."
\end{imagefromfile}

## Examples

Examples of classical double copy of gauge fields:

| [[gauge theory]] solution | [[gravity]] solution | ref. |
|--|--|--|
| electric monopole | [[Schwarzschild spacetime]] | ([MOW 15](#MOW15)) |
| magnetic monopole | massless [[Taub-NUT spacetime]] | ([LMOW 15](#LMOW15)) |
| planar wave | [[pp-wave]] | ([MOW 15](#MOW15)) |
|planar shockwave | Aichelburg-Sexl shockwave | ([BSW 20](#BSW20)) |

## Double copy and topology

From ([LMOW 15](#LMOW15)) we know that in terms of [[charges]] we have the following correspondence:

| [[gauge theory]] solution | [[gravity]] solution |
|--|--|
| [[electric charge]] | [[mass]] |
| [[magnetic charge]] | [[NUT charge]] |

The topological consequences were explored by ([AWW 20](#AWW20)):

* A [[magnetic monopole]] is geometrically a [[principal bundle]] of the form
$$\underset{ \text{worldline} }\underbrace{\mathbb{R}^{1}} \times \underset{ \text{transverse space} }\underbrace{ (\mathbb{R}^3-\{0\}) } \;\,\simeq_{\mathrm{diff}}\;\,  \underset{ \text{worldline} }\underbrace{\mathbb{R}^{1}} \times \underset{ \text{radial dir.} }\underbrace{\mathbb{R}^+} \times \underset{ \text{angular dir.} }\underbrace{S^2} \longrightarrow B U(1) $$
which is trivial only on the worldline $\mathbb{R}^{1}$ of the monopole. Therefore, since we have the homotopy $\mathbb{R}^3-\{0\} \simeq S^2$, the [[first Chern class]] of the bundle will be an element $[F] \in H^2(S^2,\mathbb{Z})\cong \mathbb{Z}$. In other words we have
$$ [F] = \tilde{g} [\mathrm{Vol}_{S^2}] $$
where $\mathrm{Vol}_{S^2}$ is the volume form of $S^2$ and $\tilde{g}\in\mathbb{Z}$ is the quantized [[magnetic charge]].

* The massless [[Taub-NUT spacetime]] with [[NUT charge]] $N$ is a [[circle bundle]] too. In fact it is diffeomorphic to the manifold $\mathbb{R}^+\times L(1,N)$, where $\times L(1,N)$ is the $3$-dimensional Lens space with quantized [[first Chern class]] $N\in\mathbb{Z}\cong H^2(S^2,\mathbb{Z})$. In this case the $S^1$ fiber has the interpretation of [[time]] direction, which is periodic and non-trivially fibrated on the sphere $S^2$ of the angular directions.

Therefore the double copy procedure exchange the [[first Chern class]] of the [[magnetic monopole]] with the one of [[Taub-NUT spacetime]], i.e.
$$ \tilde{g} \mapsto N. $$

## Double copy and Wilson lines

The classical double copy of [[Wilson lines]] was introduced by ([AWW 20](#AWW20)). We can use as gravitational [[Wilson lines]] on spacetime $M$ the [[action functional]] $e^{i S_{\mathrm{kin}}}: [S^1, M]\rightarrow U(1)$ of a test particle. For any loop $\gamma\in[S^1, M]$ we can then write

$$ W_{\mathrm{grav}}(\gamma) = e^{i S_{\mathrm{kin}}}(\gamma) = \exp \left(i m \int_\gamma \mathrm{d}s \left(g_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}s}\frac{\mathrm{d}x^\nu}{\mathrm{d}s}\right)^{\frac{1}{2}} \right) \;\in\, U(1) $$

If we assume that the metric is of the form $g_{\mu\nu} = \eta_{\mu\nu} + \kappa h_{\mu\nu}$, we can expand $W_{\mathrm{grav}}(\gamma)$ at first order in $\kappa$ and obtain

$$ W_{\mathrm{grav}}(\gamma) = \exp \left(\frac{i \kappa}{2} \int_\gamma \mathrm{d}s\, h_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}s}\frac{\mathrm{d}x^\nu}{\mathrm{d}s} \right) \;\in\, U(1) $$

where the mass $m$ is absorbed into the new parameter $s$. If now we write the [[holonomy]] of the _single copy_ gauge field along the same path $\gamma$ we get

$$ W_{\mathrm{gauge}}(\gamma) = \mathcal{P}\exp \left(i g \int_\gamma \mathrm{d}s\, A_{\mu}^a \frac{\mathrm{d}x^\mu}{\mathrm{d}s}\mathbf{T}_a \right) \;\in\, G$$

Thus we immediately see that the double copy rules for a [[Wilson line]] are the following:

$$ \mathbf{T}_a \mapsto  \frac{\mathrm{d}x^\nu}{\mathrm{d}s}, \quad\; g \mapsto  \frac{\kappa}{2}$$

Notice that they precisely mirror the BCJ prescription of [[double copy]] for [[scattering amplitudes]] by exchanging color data with kinematic data and gauge coupling constant with its gravitational analogue.

This suggests that this formulation can be a bridge to formally connect [[classical double copy]] with [[double copy]] for [[scattering amplitudes]].

## Double copy and S-duality

In ([ABSP 19](#ABSP19)) it was proved that an [[electric-magnetic duality]] (i.e. [[S-duality]]) transformation on the single copy gauge fields corresponds to an [[Ehlers transformation]] on the double copy gravitational field. In other words the following ideal diagram commutes:

$$\array{{electric \; monopole} & \overset{{\;\; double \; copy \;\;}}{\to} & {Schwarzschild \; black \; hole}\\
& \\
  ^{{S-duality}}\downarrow && \downarrow^{{Ehlers \; transformation}}\\
& \\
  {magnetic \; monopole}& \underset{{\;\; double \; copy \;\;}}{\to} & {NUT-charged \; spacetime}}$$

## Related concepts

* [[KLT relations]]

* [[Schwarzschild spacetime]], [[Taub-NUT space]]

* [[duality in physics]]

* [[string theory results applied elsewhere]], [[open/closed string duality]]

* [[magic pyramid]], [[magic supergravity]]

* [[Kaluza-Klein mechanism]], [[Kaluza-Klein monopole]]

* [[effective QFT]] incarnations of [[open/closed string duality]],

  relating ([[supergravity|super]]-)[[gravity]] to ([[super Yang-Mills theory|super]]-)[[Yang-Mills theory]]:

  * [[KLT relations]]

  * [[BCJ relations]]

  * [[classical double copy]]

## References

Fundamental bibliography:

* {#MOW15} Ricardo Monteiro, Donal O'Connell, Chris D. White, _Black holes and the double copy_ ([arXiv:1410.0239](https://arxiv.org/abs/1410.0239))

* {#LMOW15} Andrés Luna, Ricardo Monteiro, Donal O'Connell, Chris D. White, _The classical double copy for Taub-NUT spacetime_ ([arXiv:1507.01869](https://arxiv.org/abs/1507.01869))

* Chris D. White, _The double copy: gravity from gluons_ ([arXiv:1708.07056](https://arxiv.org/abs/1708.07056))

* [[David Berman]], Erick Chacón, Andrés Luna, Chris D. White, _The self-dual classical double copy, and the Eguchi-Hanson instanton_ ([arXiv:1809.04063](https://arxiv.org/abs/1809.04063))

* Kwangeon Kim, Kanghoon Lee, Ricardo Monteiro, Isobel Nicholson, David Peinador Veiga, _The Classical Double Copy of a Point Charge_ ([arXiv:1912.02177](https://arxiv.org/abs/1912.02177))

* {#BSW20} Nadia Bahjat-Abbas, Ricardo Stark-Muchão, Chris D. White, _Monopoles, shockwaves and the classical double copy_ ([arXiv:2001.09918](https://arxiv.org/abs/2001.09918))

Foundational issues:

* Chris D. White, _A Twistorial Foundation for the Classical Double Copy_ ([arXiv:2012.02479](https://arxiv.org/abs/2012.02479))

Some global aspects of the [[classical double copy]] were explored in the following paper:

* {#AWW20} [[Luigi Alfonsi]], Chris D. White, Sam Wikeley, _Topology and Wilson lines: global aspects of the double copy_ ([arXiv:2004.07181](https://arxiv.org/abs/2004.07181))

In the following paper it is shown that a [[S-duality]] on a gauge field corresponds to an [[Ehlers transformation]] on its double copy:

* {#ABSP19} Rashid Alawadhi, [[David Berman]], Bill Spence, David Peinador Veiga, _S-duality and the Double Copy_ ([arXiv:1911.06797](https://arxiv.org/abs/1911.06797))

The following paper is a proposal of extension of [[classical double copy]] to [[double field theory]]:

* Kanghoon Lee, _Kerr-Schild Double Field Theory and Classical Double Copy_ ([arXiv:1807.08443](https://arxiv.org/abs/1807.08443))

See also:

* Andres Luna, Silvia Nagy, Chris White, _The convolutional double copy: a case study with a point_ ([arXiv:2004.11254](https://arxiv.org/abs/2004.11254))

Discussion for [[D=11 supergravity]]:

* [[David Berman]], Kwangeon Kim, Kanghoon Lee, _The Classical Double Copy for M-theory from a Kerr-Schild Ansatz for Exceptional Field Theory_ ([arXiv:2010.08255](https://arxiv.org/abs/2010.08255))

Description via [[L-infinity algebras|$L_\infty$-algebras]]:

* [[Leron Borsten]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], _Double copy from homotopy algebras_, Fortschritte der Physik, **69** 8-9 (2021) 2100075 $[$[arXiv:2102.11390](https://arxiv.org/abs/2102.11390), [doi:10.1002/prop.202100075](https://doi.org/10.1002/prop.202100075)$]$

For curved spacetimes:

* Gokhan Alkac, Mehmet Kemal Gumus, Mustafa Tek, _The Classical Double Copy in Curved Spacetime_ ([arXiv:2103.06986](https://arxiv.org/abs/2103.06986))


classical electrodynamics

# Contents
* the following line creates the automatic table of contents
{: toc}

## Idea ##
**Classical electrodynamics** is about the field theory of the electric and magnetic fields and its interactions with point charges from the viewpoint of [[classical physics]].

## References ##

Relativistic electrodynamics in the vacuum is a theory that can be very elegantly formulated with tools from differential geometry, see e.g.

* Stephen Parrott: _Relativistic electrodynamics and differential geometry_ ([ZMATH](http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0609.53045))


classical field theory

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## Idea

Ordinary [[classical mechanics]] of point particles may be regarded as the theory of [[action functional]]s on [[mapping space]]s of maps from the real line to some space.

In _classical [[field theory]]_ one instead studies functionals on [[mapping space]]s on higher dimensional domains.

### Classical gauge theory ##

Of particular interest are classical field theories that are [[gauge theory|gauge theories]]. A powerful formalism for handling these is provided by [[BV theory]], which effectively realizes spaces of classical fields as [[Lie infinity-algebroid|∞-Lie algebroid]]s. BV-formalism can be understood as a means to capture a classical gauge field theory in such a way that it lends itself to [[quantization]]. (See below)

## Examples

Important examples of classical field theories are

* [[electromagnetism]]

* [[gravity]], [[supergravity]]

## Quantization of classical field theory

When it was realized that fundamental physics is governed by [[quantum field theory]] it became clear that classical field theory of fundamental fields can only be an approximation to the corresponding [[quantum field theory]]. If we think of quantum field theory in terms of [[FQFT|functorial quantum field theory]], then the domains of the mapping spaces mentioned above are the [[cobordism]]s that this [[FQFT]] is a functor on. The [[quantization]] of classical field theories to [[quantum field theories]] is a major issue in theoretical and mathematical [[physics]] (see also [[renormalization]] and [[geometric quantization]]).

## References

Textbook accounts:

* [[Clifford Truesdell]], [[Richard A. Toupin]], *The Classical Field Theories*, in: *Principles of Classical Mechanics and Field Theory*, Encyclopedia of Physics, volume III/1, Springer (1960) 226-858 [[doi:10.1007/978-3-642-45943-6_2](https://doi.org/10.1007/978-3-642-45943-6_2)]

* {#DeligneFreed99} [[Pierre Deligne]], [[Daniel Freed]], _Classical field theory_ (1999) ([pdf](https://publications.ias.edu/sites/default/files/79_ClassicalFieldTheory.pdf))

  this is a chapter in

  [[Pierre Deligne|P. Deligne]], [[Pavel Etingof|P. Etingof]], [[Dan Freed|D.S. Freed]], L. Jeffrey, [[David Kazhdan|D. Kazhdan]], J. Morgan, D.R. Morrison, [[Edward Witten|E. Witten]] (eds.)  _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

* S. Flügge (ed.), _Encyclopedia of Physics_ Volume III/I, _Principles of Classical Mechanics and Field Theory_, Springer 1960

For more see the references at _[[multisymplectic geometry]]_.

Discussion of recursive solutions to classical field equations and their relation to the quantum [[perturbation theory]]:

* [[Robert Helling]], _Solving classical field equations_ ([pdf](http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf))

On [[smooth sets]] as a [[convenient category of spaces|convenient category]] for [[variational calculus]] of [[Lagrangian quantum field theory|Lagrangian]] [[classical field theory]]:

* [[Grigorios Giotopoulos]], [[Hisham Sati]], *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]

[[!redirects classical field theories]]


classical homotopy category > history

> see at _[[Ho(Top)]]


classical inverse scattering method

Given a partial differential equation containing a parametrized part called a potential and describing time evolution and having suitable notion of asymptotic solution at infinity, the operator may be suitable for describing
a variant of the phenomenon of [[scattering]], hence one can establish the appropriate notions of scattering data and scattering potential. __Inverse problem of scattering__ is to determine the scattering potental from the knowledge of scattering data.

Classical inverse scattering method is to apply the methods of solving the invrese scattering problem to finding the exact solutions of nonlinear PDE-s.
Its analogue at (second) quantized level is the [[quantum inverse scattering method]].

category: physics
[[!redirects inverse scattering method]]
[[!redirects inverse scattering problem]]
[[!redirects inverse problem of scattering]]


classical Lie group

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{:toc}

## Idea

The [[classification of simple Lie groups]] contains four infinite series of "classical" simple Lie groups:

* $A_n = SU(n+1)$, the [[special unitary group]]

* $B_n = SO(2n+1)$, the [[special orthogonal group]]

* $C_n = Sp(n)$, the [[symplectic group]]

* $D_n = SO(2n)$, the [[special orthogonal group]]

Apart from this there are five [[exceptional Lie groups]].

> graphics grabbed from [Schwichtenberg](http://jakobschwichtenberg.com/classification-of-simple-lie-groups/)

More generally, other [[matrix Lie groups]], such as the [[general linear group]] $GL(n)$, may be called "classical".

## Related concepts

* [[exceptional isomorphism]]

## References

* {#Bredon72} [[Glen Bredon]], Section 0.2 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

* [[Howard Georgi]], §19  in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

  > with an eye towards application to (the [[standard model of particle physics|standard model]] of) [[particle physics]] and [[grand unified theory]]

[[!redirects classical Lie group]]
[[!redirects classical Lie groups]]

[[!redirects classical Lie algebra]]
[[!redirects classical Lie algebras]]

[[!redirects classical simple Lie group]]
[[!redirects classical simple Lie groups]]


classical limit

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## Idea

To a large extent [[quantum physics]] is a [[deformation theory|deformation]] of [[classical physics]] parameterized by [[Planck's constant]] $\hbar$. For a given [[quantum system]], the limit $\hbar \to 0$ (in as far as that can be made sense of), is called its _classical limit_.

One imagines that from a [[quantum field theory]] one can derive a [[classical field theory]] of which the original quantum field theory may be regarded as a [[quantization]].  Whereas quantization may be haphazard, since quantum physics seems to be a more fundamental description of reality that need not be derivable from anything classical, we expect taking the classical limit to be more systematic, since the appearance of a classical world in some regimes should be explicable by fundamental quantum theories.  However, it may not be as systematic as some people would like, as in this [blog discussion](http://golem.ph.utexas.edu/category/2009/10/this_weeks_finds_in_mathematic_43.html#c028689).

What precisely this means and to which degree precisely it makes sense depends on which precise formulation of these concepts one considers. For instance when regarding [[deformation quantization]] as definition the quantum theory then taking the classical limit is an obvious straightforward operation. But in more complete contexts, such as when the [[quantum field theory]] is realized as an [[FQFT]] or [[AQFT]], the passage to the classical limit may be subtle or not make sense at all.

Most prescriptions for taking classical limits are formulated in algebraic settings. To fit for instance a 1-dimensional [[FQFT]] into the discussion one is likely to find in the literature, one will have to encode it equivalently in its [[spectral triple]] (as described there) and then proceed with that.

## Formalizations

In various formalizations of (aspects of) quantum physics, the classical limit has a corresponding formalization. The following list discusses a few.

### Relations to coherent states

One method for producing [[classical mechanics]] from a quantum theory is by looking at [[coherent states]] of the quantum theory. The standard (Glauber) coherent states
have a localized probability distribution in classical [[phase space]] whose center follows the classical equations of motion when the Hamiltonian is quadratic in positions and momenta.

(For nonquadratic Hamiltonians, this only holds approximately over short times. For example,
for the 2-body problem with a $1/r^2$ interaction,
Glauber coherent states are not preserved by the dynamics.
In this particular case, there are, however, alternative $SO(2,4)$-based coherent states that are preserved by the dynamics, smeared over Kepler-like orbits. The reason is that the Kepler 2-body problem -- and its quantum version, the hydrogen atom -- are superintegrable systems with the large dynamical symmetry group $SO(2,4)$.)

In general, roughly, coherent states form a nice orbit of unit vectors of a Hilbert space $H$ under a dynamical symmetry group $G$ with a triangular decomposition, such that the linear combinations of coherent states are dense in $H$, and the inner product $\phi^*\psi$ of coherent states $\phi$ and $\psi$ can be calculated explicitly in terms of the highest weight representation theory of $G$. The diagonal of the $N$-th tensor power of $H$ has coherent states $\psi_N$ (labelled by the same classical phase space as the original coherent states, and corresponding to the $N$-fold highest weight) with inner product

$$
\phi_N^* \psi_N = (\phi^* \psi)^N
$$

and for $N\to \infty$, one gets a good classical limit.
For the Heisenberg group, $\phi^*\psi$ is a $1/\hbar$-th power,
and the $N$-th power corresponds to replacing $\hbar$ by $\hbar/N$. Thus one gets the standard classical limit.

### Formulation in geometric quantization

In the context of [[geometric quantization]] the classical limit corresponds to taking high [[tensor powers]] of the [[prequantum line bundle]] with itself. See at _[[Planck's constant]]_ the section [Planck's constant - In geometric quantization](Planck%27s%20constant#InGeometricQuantization).

## References

Basic literature on relations between [[coherent state]]s and the classical limit, based on irreducible unitary representations of Lie groups includes the book

* A. M. Perelomov,
Generalized Coherent States and Their Applications,
Springer-Verlag, Berlin, 1986.

and the paper

* L. Yaffe, _Large $N$ limits as classical mechanics_,
Rev. Mod. Phys. 54, 407--435 (1982)

Both references assume that the Lie group is finite-dimensional and semisimple. This excludes the Heisenberg group, in terms of which the standard (Glauber) coherent states are usually defined. However, the Heisenberg group has a triangular decomposition, and this suffices to apply
Perelomov's theory in spirit.

* [[Roland Omnès]], §6 of: *[[The Interpretation of Quantum Mechanics]]*, Princeton University Press (1994) [[ISBN:9780691036694](http://press.princeton.edu/titles/5525.html)]

The online book

* [[Arnold Neumaier]], [[Dennis Westra]], _Classical and Quantum Mechanics via Lie algebras_ ([arXiv](http://lanl.arxiv.org/abs/0810.1019))

contains a general discussion of the relations between classical mechanics and quantum mechanics, and discusses
in Chapter 16 the concept of a triangular decomposition of Lie algebras and a summary of the associated representation theory (though in its present version not the general relation to coherent states).

For other relevant approaches to a rigorous classical limit, see the online sources

* <http://www.projecteuclid.org/Dienst/Repository/1.0/Disseminate/euclid.cmp/1103859040/body/pdf>

* <http://www.univie.ac.at/nuhag-php/bibtex/open_files/si80_SIMON!!!.pdf>

* <http://arxiv.org/abs/quant-ph/9504016>

* <http://arxiv.org/pdf/math-ph/9807027>


classical logic

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## Idea
 {#Idea}

There are many systems of _[[formal logic]]_. By "classical logic" one broadly refers to those such systems which reflect the kind of logic as understood, quite literally, by the classics, say starting with [Aristotle, Metaphysics 1011b24](excluded+middle#AristotleMetaphysics). If you have never heard of any alternative system of logic, then classical logic is just the kind of logic that you have heard about.

There is some variance in what exactly counts as classical and as non-classical in [[logic]], but one main characteristic of classical logic is its use of the _[[principle of excluded middle]]_. Informally this says that every [[proposition]] is either [[true]] or [[false]]. (Slightly more carefully it says that for any proposition $P$, either it or its [[negation]] is true, where the 'or' is to be read in an 'internal' sense, i.e., as a formal [[disjunction]] $P \vee \neg P$.)

It is the apparent self-evidence of this principle of excluded middle that made people take it for granted for a long period of human civilization, so that it became "classical".

One consequence of the [[principle of excluded middle]] in classical logic is the possibility to obtain [[proof]] of a [[proposition]] by showing that its [[negation]] is [[false]] ([[proof by contradiction]]). Evident as this may superficially seem, it has the noteworthy consequence that one may for instance prove [[existential quantifier|existence]] of mathematical objects without having any way to actually construct an example.

This "non-constructive" nature of classical logic eventually led people to consider alternative and hence _non-classical_ systems of logic, such as [[constructive logic]]/[[intuitionistic logic]], where the [[principle of excluded middle]], and possibly other principle of classical logic, are not considered by default.  There is still a close relation between classical and [[constructive logic]]. For example a [[proposition]] is provable in classical logic precisely if its [[double negation]] is provable in [[constructive logic]] ("[[double negation translation]]").

Other considerations such as the [[internal logic]] of [[categories]] also led to constructive/intuitionistic logics.  More recently, other "nonclassical" logics such as [[linear logic]], [[paraconsistent logic]], [[relevance logic]], [[quantum logic]], and so on have also gained prominence.

There are other principles that are often associated with classical logic, which still seemed self-evident at a time, but maybe less so than the principle of excluded middle. One such is the [[set theory|set-theoretic]] _[[axiom of choice]]_ or one of its equivalent incarnations, such as [[Zorn's lemma]]. The presently most popular formal [[foundation of mathematics]] via the [[set theory]], called _[[ZFC]]_, includes both the [[principle of excluded middle]] as well as the [[axiom of choice]]. In this form classical logic serves as the foundation for _[[classical mathematics]]_.

## Details

Classical logic is the form of [[logic]] usually accepted and taught as standard among working mathematicians, and traced back (at least) to Aristotle.  Some particular features that distinguish classical logic are (perhaps not a complete list):

* a [[distributive lattice]] of logical operations ($\wedge$ and $\vee$);
* the [[structural rule]]s of weakening, contraction, and (where meaningful) exchange;
* an involutory [[negation]].

In contrast, [[minimal logic|minimal]], [[intuitionistic logic|intuitionistic]], and (some forms of) [[paraconsistent logic|paraconsistent]] logics have the distributive lattice and the structural rules but no involutory negation.  On the other hand, [[linear logic]] (at least, "classical" linear logic) and (other forms of) paraconsistent logic have logical operations with at least some distributive properties and an involutory negation, but lack some structural rules.  Then again, [[quantum logic]] and (yet other forms of) paraconsistent logic have the structural rules and the involutory negation, but lack any sort of distributivity.

In [[category theory]] (and in the [[foundations]] of mathematics generally), it is intuitionistic logic that is most often contrasted to classical logic; the difference is given by the law of [[excluded middle]], which holds classically but not intuitionistically.

## Classical propositional logic

Before describing what makes a [[propositional logic]] 'classical', let us quickly recall what propositional logic is in the first place.

In general, a (deductive) [[logic]] specifies rules of [[syntax]] for forming well-formed [[formulas]] which can be considered as assertions, and rules of inference for deriving one formula (called a 'conclusion') from other formulas (called 'premises'), in such a way that truth of the conclusion follows on the assumption of truth of the premises. In short, a [[formal logic]] specifies a mode of reasoning that is considered valid for a particular mathematical context.

In [[propositional logic]], one forms a formula by applying basic logical operations such as $\wedge$ ('[[logical conjunction|and]]'), $\vee$ ('[[logical disjunction|or]]'), and $\Rightarrow$ ('[[implication|implies]]')  to simpler formulas. The simplest formulas are called atomic formulas, and usually take the form of an [[equation]] $s = t$ between two [[terms]] (and underneath there are rules for term formation as well), or something like $R(a_1, \ldots, a_n)$ where $R$ stands for some basic relation (between terms $a_1, \ldots, a_n$) belonging to whatever language one is working with. A standard list of *propositional* operations would be

* $\wedge, \vee, \Rightarrow$ (binary operations), $\neg$ ('not', an unary operator), and $\top, \bot$ ('true' and 'false', basic logical constants or nullary operators).

Missing from propositional logic *per se* are [[quantifiers]] such as $\exists$ ([[existential quantifier]]) and $\forall$ ([[universal quantifier]]), which properly speaking belong to [[predicate logic]]. (One way of thinking about propositional logic is that it treats formulas as assertions "all of the same type", whereas quantifiers allow one to shift between classes of assertions of one type to assertions of another type.)

As we said, there are also rules of inference. There are various styles in which these are presented, such as [[natural deduction]] or [[sequent calculus]]. But a simple example might look like this:

$$\frac{P \wedge Q}{P}$$

("from $P \wedge Q$ we may infer $P$"). This might also be written $P \wedge Q \vdash P$ ("pronounced $P \wedge Q$ entails $P$"), thought of as asserting a relation between propositions called entailment. Two of the universal rules of propositional logic (whether classical or intuitionistic or quantum or whatever) is that the entailment relation should be [[reflexive relation|reflexive]] and [[transitive relation|transitive]]: the rules of inference allow an entailment $P \vdash P$ for any formula $P$, and if they allow entailments $A \vdash B$ and $B \vdash C$, then they allow $A \vdash C$.

Thus, in a propositional logic, formulas together with the entailment relation that is provided for by the rules of inference form a [[preorder]]. Sometimes one goes a little further: in situations where one can derive $A \vdash B$ and $B \vdash A$, one says that $A$ and $B$ are provably equivalent (often abbreviated as $A \equiv B$), and one may pass to the [[poset]] of provable equivalence classes (the usual way in which a poset is associated with a preorder).

One knows that one is in the realm of *classical* propositional logic if the poset of provable equivalence classes is a [[Boolean algebra]], in which $\wedge$ induces the meet operation and $\vee$ the join operation. Here the rules of inference allow bi-entailments such as

$$A \Rightarrow B \equiv \neg A \vee B$$

and in fact the different types of propositional logic are usually conveniently described by specifying what type of algebraic structure we get on the poset of provable equivalence classes. For example, in [[intuitionistic logic]] this poset forms a [[Heyting algebra]].

## Related concepts

* [[logic]]
* [[classical mathematics]]
* [[constructive logic]] / [[intuitionistic logic]]
* [[material set theory]]

[[!include categories and logic - table]]

[[!redirects classical logics]]


classical mathematics

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## Idea

**Classical mathematics** is [[mathematics]] as it is "normally" practised (or, sometimes, as it used to be practiced), and particularly using commonly accepted [[foundations]].

This is a vague term, but some of the things that it might mean are:

*  use of [[classical logic]] and the [[axiom of choice]], in contrast to [[constructive mathematics]];

*  free use of [[power sets]] and [[infinite sets]], in contrast to [[predicative mathematics]] and [[finite mathematics]];

*  working in [[Set]], in contrast to working [[internalization|internally]] to some other [[topos]] or more general [[category]];

*  violating the [[principle of equivalence]] or other normative perspectives of [[higher category theory]];

*  use of non-[[structural set theory|structural]] aspects of [[material set theory]].

Despite the name, classical mathematics is only about 100 years old; anything much before that can usually interpreted just as well nonclassically as classically.  In fact, it was developments in mathematics around the beginning of the 20th century which sparked the famous "foundational crisis," leading to the development of intuitionistic, constructive, and predicative mathematics as alternatives, while the approaches accepted by the mainstream were called "classical" in contrast.

Note that the term 'classical' also has meanings within many specific fields of mathematics that may have nothing in particular to do with 'classical mathematics' as a whole.  Compare also [[ordinary mathematics]].

## Related pages

* [[intuitionistic mathematics]]

* [[constructive mathematics]]

* [[structural set theory]], [[ETCS]]

* [[type theory]], [[homotopy type theory]]


classical mechanics

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## Idea

Classical mechanics is that part of [[classical physics]] dealing with the deterministic physics of point [[particle|particles]] and rigid bodies; often the systems with the infinitely many degrees of freedom are also included (like infinite arrays of particles and their continuous limits like classical mechanics of strings, membranes, elastic media and of classical fields). For the continuous systems, the equations of motion can often be explained by the partial differential equations, describing classical [[physical field|physical fields]] of quantities (typically smooth possibly vector valued functions on manifolds), including background fields like metric; the latter (sub)area is the [[classical field theory]], but it is often studied separately from the classical mechanics of the finite systems of particles; especially if non-classical features or interpretations are involved (e.g. supersymmetry, or unusual case of non-variational  equations of motion etc.). In Hamiltonian reduction, due to conservation laws, many systems with infinitely many degrees of freedom, reduce to the finite ones.

Nondissipative systems with finitely many degrees of freedom may be described geometrically using [[symplectic manifolds]], or more generally [[Poisson manifolds]]; the later may also sometimes appear as reductions of the systems with infinitely many degrees of freedom.

Classical mechanics of a system of point particles and rigid bodies is usually divided into statics, kinematics and dynamics. __Statics__ studies the balance of forces in a system which does not move, or in a stationary flow. __Kinematics__ studies the relation between position, velocity and acceleration of bodies in a mechanical system, without reference to the causes of motion. __Dynamics__ studies motion with reference to the causes of motion and interaction between bodies and its manifestation via (quantified) [[force|forces]], energy and mass assigned to bodies in motion and interaction.

For a theoretical classical mechanics one often starts with a concrete system of bodies with pulleys, strings, spins, external and internal forces, and dissipative sinks and sources (e.g. friction forces), which are then analysed to get the configuration or phase space of the system, the equations of motion and possibly to determine some special [[observable|observables]] of interest. Once abstracted that way, the rest of the study is a rather special case of the theory of [[dynamical system|dynamical systems]], which itself studies general (either deterministic or stochastic) spatially-parametrized systems in a (discrete or continuous) time evolution.

A terminological and scope discussion is archived [here](https://nforum.ncatlab.org/comments.php?DiscussionID=2807&Focus=23543#Comment_23543).

## Traditional formulation

The fundamental distinction is between open and closed mechanical systems. Open systems have exchange of energy with the rest of universe, that is with the energy sources or sinks which are not described by the mechanical system: for example the energy can be absorbed through forces from external bodies not belonging to the system but accounted in terms of such forces, or the energy can be lost by heating (described by friction force and alike). Closed systems are conservative in energetic sense. The terminology open/closed is wider than conservative/nonconservative, as it pertains also to statistical and quantum systems.

The standard formalisms for classical mechanics are Newtonian mechanics, Lagrangean formalism, and the Hamiltonian formalism which can be studied in the generality of symplectic manifolds and more general (allowing degeneracies) formalism of Poisson manifolds; Poisson manifolds can generalize to he study of somewhat nonclassically to (the opposite of) more general Poisson algebras.

### Newtonian formalism

...

### Lagrangean formalism

...

### Hamiltonian formalism and symplectic manifolds

(...) [[symplectic manifold]], [[Hamiltonian]]  (...)

###  Mechanical systems based on Poisson algebras

We set up some basic notions of classical mechanics.

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

A _real [[Poisson algebra]]_ is a unital (commutative) [[associative algebra]] $(A, \dot)$ over the [[real number|real numbers]] that is equipped with with an additional bilinear operation

$$
  [-,-] : A \otimes_{\mathbb{R}} A \to A
$$

that makes $A$ into a [[Lie algebra]] such that for each element $a \in A$ the operation $[a,-] : A \to A$ is a [[derivation]] of the product $\cdot : A \otimes A \to A$.

This definition readily generalized to symmetric monoidal categories. For example, a real _super Poisson algebra_ is a graded-commutative [[superalgebra]] equipped with the compatible structure of a [[super Lie algebra]].

A _[[homomorphism]]_ $(A, \dot, [-,-]) \to (B, \cdot, [-,-])$ of (super) Poisson algebras is a [[linear function]] $A \to B$ that respects both the associative product and the Lie bracket.

Write [[Poiss]] for the resulting category of
(super) Poisson algebras.

=--

The standard setup of conservative classical mechanical system is a Poisson manifold.

+-- {: .num_defn #PoissonManifoldExample}
###### Definition - principal example

A [[Poisson manifold]] is a smooth real manifold $M$ equipped with a Poisson structure (that is the Poisson bracket) on the commutative algebra $C^\infty(M)$ of smooth real valued functions on $M$.
=--

Recall that every [[symplectic manifold]] provides an example of a Poisson manifold. Possibly infinite-dimensional generalization of this example is called a [[phase space]]. Hence commutative Poisson algebras should be viewed as formal opposites to phase spaces in a generalized sense.

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

The [[opposite category]] of that of commutative real (super) Poisson algebras we call the category of **classical mechanical systems**

$$
  ClassMechSys \coloneqq CPoiss^{op}
  \,.
$$

=--

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

Poisson superalgebras describe systems with [[fermion|fermions]]. Systems without fermions may be described by plain Poisson algebras.

=--

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

This definition captures most notions of "mechanical systems". Exceptions contain
[[open system|open systems]] for which there is no conservation laws (examples: externally-driven
and [[dissipative system|dissipative systems]]). In real worlds, physicists believe that such systems may be realized only as parts of larger systems (eventually, the universe) which are conservative; hence either describable by a Poisson algebra or it entails energy types which can not be described using classical mechanics.

=--

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

For $(A, \cdot, \{-,-\})$ a Poisson algebra, $A$ together with its [[module]] $\Omega^1(A)$ of [[Kähler differential|Kähler differentials]] naturally form a [[Lie-Rinehart pair]], with bracket given by

$$
  [d a, d b ] \coloneqq d \{a,b\}
  \,.
$$

If the Poisson algebra comes from a [[Poisson manifold]] $X$, then this Lie-Rinehart pair is the [[Chevalley-Eilenberg algebra]] of the given [[Poisson Lie algebroid]] over $X$. We can therefore identify classical mechanical systems over a phase space manifold also with Poisson Lie algebroids.

=--

### Observables and states

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

Given $S \coloneqq (A,\cdot, \{-,-\})$ a classical mechanical system, we say

* an **[[observable]]** of $S$ is an element $a \in A$, hence we call $A$ the **algebra of observables**;

* a **[[classical state]]** of $S$ is

  * a [[linear function]] $\rho : A \to \mathbb{R}$;

    * which is **positive** in that for all $a \in A$ we have that

      $\rho(a \cdot a) \geq 0$;

    * and which is **normalized** in that $\rho(1) = 1$.

* a **[[pure state]]** of $S$ is a state that is not only a linear map, but even an [[associative algebra]] [[homomorphism]] $\rho : A \to \mathbb{R}$.

Write $States((A, \cdot))$ for the set of states of $A$.

For $\rho \in States$ and $a \in A$ we say that $\rho(a)$ is the **value of the observable $a$ on the system in state $\rho$**.

=--

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

If the classical mechanical system comes from a [[Poisson manifold]] $(X, \{-,-\})$ by
example \ref{PoissonManifoldExample}, then the pure states correspond precisely to the points of the manifold $X$. So each point of $X$ is one _specific_ (= "pure") state that the mechanical system defined by $(X, \{-,-\})$ can be in, whereas a general state $\rho : A \to \mathbb{R}$ is a distribution of such specific states.

=--

### Flows and evolution

Let $M$ be a [[Poisson manifold]], possibly infinite-dimensional with $A = C^\infty(M)$. More generally, $A$ can be a Poisson algebra with a good notion of weak topology and corresponding notion of a derivative.

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

For any observable $a \in A$ we say that a **1-parameter flow induced by the observable** is, if it exists, an [[action]] of the [[real number|real numbers]] on the observables

$$
  F_a : \mathbb{R} \times A \to A
$$

which is [[differentiation|differentiable]] in $\mathbb{R}$ and satisfies for all $b \in A$ the [[differential equation]]

$$
  \frac{d}{d t} F_a(b) (t) = \{a,b\}
  \,.
$$

with initial value $F_a(b)(0) = b$. The differentiation is understood pointwise on $M$ (pointwise we get to differentiate a function $\mathbb{R}\to\mathbb{R}$).

=--

In non-relativistic classical mechanics every system comes up with a choice of one element $H \in A$ and declare that the corresponding flow is the _time evolution_ of observables of the system. One calls this $H$ the **[[Hamiltonian]]** or *energy observable* of the system. The physical meaning, however, does not change if $H$ is changed by an overall constant.

### Quantization

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

Write $StarAlg_{\mathbb{C}}$ for the category of [[star-algebra|star-algebras]] over the [[complex number|complex numbers]].

The [[opposite category]]

$$
  QuantMechSys \coloneqq StarAlg^{op}
$$

we call the category of **[[quantum mechanical systems]]**.

=--

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

For $(A, \cdot, \{-,-\})$ a classical mechanical system,
def. \ref{ClassMechSys}, a **[[quantization]]** of it is -- if it exists --

* a 1-parameter [[field of star-algebras]] $\{A_\hbar\}_{\hbar \in [0,\infty)}$;

* such that in the limit $\hbar \to 0$ we have

  1. $A_\hbar \to A_0 \coloneqq A \otimes_{\mathbb{R}}\mathbb{C}$;

  1. for all $a,b \in A_{\hbar}$: $\frac{1}{i \hbar } [a,b] = \{a,b\}$.

Conversely, given a quantum mechanical system $(A, \ast)$ and a field of star-algebras such that $A = A_1$, then we call the clasical system $A_0$ its (or rather: a)**[[classical limit]]**.

=--

## Formulation in higher geometry by local prequantum field theory

Traditional classical mechanics ([[Hamiltonian mechanics]], [[Lagrangian mechanics]], [[Hamilton-Jacobi theory]]) is naturally understood as a special case of -- and in fact as deriving from -- [[local prequantum field theory]] formulated in the [[higher differential geometry]] over $\mathbf{B}U(1)_{conn}$. This is discussed in some detail at

* _[[prequantized Lagrangian correspondence]]_.

## Particular classical systems

* [[Hamiltonian dynamics on Lie groups]]

  * [[rigid body dynamics]]

## Related concepts

* [[mechanics]], [[Poisson manifold]], [[symplectic manifold]], [[contact manifold]]

* [[continuum mechanics]]

* [[semiclassical approximation]]

* [[classical anomaly]]

* [[quantization]]: [[deformation quantization]], [[geometric quantization]]

* [[quantum mechanics]], [[supersymmetric quantum mechanics]]

* [[quantum field theory]]: [[FQFT]], [[AQFT]]

* [[generating function in classical mechanics]],

## References

The origin of [[Newton's laws of motion]]:

* {#Newton1687} [[Isaac Newton]], *[[Philosophiæ naturalis principia mathematica]]* (1687)

Classical textbooks:

* [[Tom W. B. Kibble]], [[Frank Berkshire]], *Classical mechanics*, McGraw-Hill (1966, 1973, 2004) [[pdf](https://physicaeducator.files.wordpress.com/2018/02/classical-mechanics-by-kibble-and-berkshire.pdf), Wikipedia entry]

* [[Vladimir Arnol'd]], _[[Mathematical methods of classical mechanics]]_, Graduate Texts in Mathematics **60**, Springer (1978) [[doi:10.1007/978-1-4757-1693-1](https://doi.org/10.1007/978-1-4757-1693-1)]

* [[Ralph Abraham]], [[Jerrold Marsden]], _[[Foundations of Mechanics]]_ (1978)

* [[Lev Landau]], [[Evgeny Lifshitz]], _Classical mechanics_, vol. I of the *[[Course of Theoretical Physics]]*

* [[Michael Spivak]], _Elementary mechanics from a mathematician's viewpoint_ ([pdf](http://alpha.math.uga.edu/~shifrin/Spivak_physics.pdf))

Detailed expository textbook with emphasis on the [[principle of extremal action]]:

* [[Douglas Cline]], *Variational Principles in Classical Mechanics*, University of Rochester (2021) [[pdf](http://classicalmechanics.lib.rochester.edu/pdf/Variational_Principles_in_Classical_Mechanics_3e_final_fbcover.pdf), online version]

Discussion with an eye also towards [[quantum mechanics]] is in

* [[Klaas Landsman]], _[[Mathematical Topics Between Classical and Quantum Mechanics]]_

See also

* {#MacLane68} [[Saunders MacLane]], _Geometrical Mechanics_, Lectures 1968 ([web](https://harrydole.com/wp/2017/09/10/saunders-mac-lane-geometrical-mechanics/))

* [[Valter Moretti]], *Analytical Mechanics* -- *Classical, Lagrangian and Hamiltonian Mechanics, Stability Theory, Special Relativity*, Springer (2023) [[doi:10.1007/978-3-031-27612-5](https://doi.org/10.1007/978-3-031-27612-5)]

[[!redirects classical mechanical system]]
[[!redirects classical mechanical systems]]
[[!redirects system of classical mechanics]]

[[!redirects Newtonian mechanics]]


classical modality

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## Idea

In as far as a [[bireflective subcategory]] inclusion
$$
  \mathcal{C}
    \xhookrightarrow{\phantom{-} \iota \phantom{-}}
  \mathcal{B}
    \xrightarrow{\phantom{-} \beta \phantom{-}}  \mathcal{C}
  \,,
  \;\;\;\;\;\;\;
  \beta \dashv \iota \dashv \beta
$$

is understood as an inclusion (via forming of [[zero object]]-bundles) of a [[category]] $\mathcal{C}$ of "classical" [[objects]] into a category $\mathcal{B}$ of [[bundles]] of [[infinitesimal objects|infinitesimal]] [[fibers]] over these  (as in [[infinitesimal cohesion]]), it makes sense to address the corresponding [[bireflective subcategory|bireflecting]] [[idempotent monad|idempotent]] [[Frobenius monad]]

$$
  \natural
    \,\coloneqq\,
  \iota \circ \beta
    \;\colon\;
  \mathcal{B} \longrightarrow \mathcal{B}
$$

as the [[modal operator]] which projects out the "[[modal objects|mode of being]] classical", hence the *classical modality*.

For example, interpreting a [[tangent (infinity,1)-topos|tangent $\infty$-topos]] $T \mathbf{H}$ (generally an [[(infinity,1)-topos|$\infty$-topos]] of [[parameterized spectrum|parameterized]] [[module spectra]]) as the home of a kind of higher [[quantum information theory]] (as discussed at *[[motivic quantization]]* and *[[quantum circuits via dependent linear types]]*) then its [[infinitesimal cohesion]] $\iota \,\colon\, \mathbf{H} \hookrightarrow T \mathbf{H}$ characterizes exactly the "classical objects" in the standard sense of [[classical mechanics|classical]] vs [[quantum mechanics]].

A [[modal homotopy type theory|modal homotopy type theoretic]] formulation of the classical modality $\natural \,\colon\, T \mathbf{H} \to T \mathbf{H}$ in the [[internal language]] of such [[tangent (infinity,1)-topos|tangent $\infty$-toposes]] is proposed in [Riley, Finster & Licata (2021)](#RileyFinsterLicata21), its combination with a [[linear type theory|linear]] [[multiplicative conjunction]] $\otimes$ (and the required [[bunched logic]]) has been developed by [Riley (2022)](#Riley22Thesis) and the interpretation as a "classical modality" in the sense of [[classical mechanics|classical]]/[[quantum mechanics]] is discussed in [Myers et al. (2023)](#MyersEtAl).

## Related concepts

[[!include cohesion - table]]

## References

A system of [[inference rules]] for a classical modality added to [[homotopy type theory]] has been proposed in

* {#RileyFinsterLicata21} [[Mitchell Riley]], [[Eric Finster]], [[Daniel R. Licata]], *Synthetic Spectra via a Monadic and Comonadic Modality* [[arXiv:2102.04099](https://arxiv.org/abs/2102.04099)]

and combined with a [[linear type theory|linear]] [[multiplicative conjunction]] and the necessary [[bunched logic]] in

* {#Riley22Thesis} [[Mitchell Riley]], *A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory*, PhD Thesis (2022) [[doi:10.14418/wes01.3.139](https://doi.org/10.14418/wes01.3.139), [ir:3269](https://digitalcollections.wesleyan.edu/object/ir%3A3269),  [pdf](https://mvr.hosting.nyu.edu/pubs/thesis.pdf)]

The role as a "classical modality" (and introducing this terminology) making the resulting [[linear homotopy type theory]] a [[quantum programming language|formal quantum language]] is indicated in

* {#MyersEtAl} [[David Jaz Myers]], [[Mitchell Riley]], [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Quantum Certification via Linear Homotopy Types]]*

with further exposition in:

* [[Urs Schreiber]]: *Quantum Data Types via Linear Homotopy Type Theory*, talk at *[Workshop on Quantum Software](https://quasar.unina.it/qtml2022/workshop.php)* @ *[QTML 2022](https://quasar.unina.it/qtml2022.html)*, Naples (Nov 2022) [slides:[pdf](/schreiber/files/QuantumDataInLHoTT-221117.pdf)]

[[!redirects classical modalities]]


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#### Model category theory
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[[!include model category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _classical [[model category]] structure on [[pointed topological spaces]]_ $Top^{\ast/}_{Quillen}$ is the [[model structure on pointed objects]] of the [[classical model structure on topological spaces]] $Top_{Quillen}$ under the point (a [[pointed model category]]).

Equipped with the [[smash product]] this is a [[monoidal model category]].

## Properties

### Cofibrant generation

Recall that the generatic cofibrations of the [[classical model structure on topological spaces]] are

$$
 I_{Top}
  \coloneqq
 \left\{
    S^{n-1} \overset{\iota_n}{\longrightarrow} D^n
 \right\}_{n \in \mathbb{N}}
$$

and the generating acylic cofibrations are

$$
 J_{Top}
  \coloneqq
 \left\{
    D^n \overset{(id,\delta_0)}{\longrightarrow} D^n \times I
 \right\}_{n \in \mathbb{N}}
  \,.
$$

Write

$$
  (-)_+
  \;\colon\;
  Top \longrightarrow Top^{\ast/}
$$

for the operation of freely adjoining a basepoint.

+-- {: .num_prop}
###### Proposition

The [[coslice model structure]] $(Top_{Quillen})^{\ast/}$ is itself [[cofibrantly generated model category|cofibrantly generated]], with generating cofibrations

$$
  I_{Top^{\ast/}}
  =
 \left\{
    S^{n-1}_+ \overset{(\iota_n)_+}{\longrightarrow} D^n_+
 \right\}
$$

and generating acyclic cofibrations

$$
  J_{Top^{\ast/}}
  =
 \left\{
    D^n_+ \overset{(id, \delta_0)_+}{\longrightarrow} (D^n \times I)_+
 \right\}
  \,.
$$

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This is a special case of a general statement about cofibrant generation of [[coslice model structures]], see [this proposition](model+structure+on+an+over+category#ModelStructureInheritsGoodProperties).

## Related concepts

* [[model structure on topological spaces]]

  * [[classical model structure on topological spaces]]

    * [[classical model structure on pointed topological spaces]]

  * [[Strøm model structure]]

  * [[model structure on Delta-generated topological spaces]]

## References

Textbook accounts:

* {#Hovey99} [[Mark Hovey]], Cor. 2.4.20 in: _[[Model Categories]]_, Mathematical Surveys and Monographs, Volume 63, AMS (1999) ([ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s), [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover))