Saul Kripke

Saul Kripke was an American philosopher and logician (1940-2022). He is best known for his work on [[modal logic]], in particular the possible worlds semantics discussed at [[geometric model for modal logics]], and for related work in metaphysics, especially in _[[Naming and Necessity]]_.

* [Wikipedia entry](https://en.wikipedia.org/wiki/Saul_Kripke)

* [Saul Kripke Center](https://saulkripkecenter.org)

## Selected writings

On [[S4 modal logic|S4]]/[[S5-modal logic|S5]]-[[modal logic]] and its [[geometric models for modal logics|geometric models]] that came to be called [[Kripke frames]]/[[Kripke models]]/[[possible worlds semantics]]:

* {#Kripke59} [[Saul A. Kripke]], *A Completeness Theorem in Modal Logic*, The Journal of Symbolic Logic **24** 1 (1959) 1-14 [[doi:10.2307/2964568](https://doi.org/10.2307/2964568), [jstor:2964568](https://www.jstor.org/stable/2964568), [pdf](https://www.filosoficas.unam.mx/~morado/Cursos/17Modal/Kripke1959.pdf)]

* [[Saul A. Kripke]], *Semantical Considerations on Modal Logic*, Acta Philosophical Fennica **16** (1963) 83-94 [[pdf](http://saulkripkecenter.org/wp-content/uploads/2019/03/Semantical-Considerations-on-Modal-Logic-PUBLIC.pdf)]

* {#Kripke63} [[Saul A. Kripke]], *Semantical Analysis of Modal Logic I. Normal Modal Propositional Calculi*, Mathematical Logic Quaterly **9** 5-6 (1963) 67-96 [[doi:10.1002/malq.19630090502](https://doi.org/10.1002/malq.19630090502)]

* {#Kripke65} [[Saul A. Kripke]], *Semantical Analysis of Modal Logic II. Non-Normal Modal Propositional Calculi*, in *The Theory of Models* (Proceedings of the 1963 International Symposium at Berkeley) Studies in Logic and the Foundations of Mathematics (1965) 206-220 [[doi:10.1016/B978-0-7204-2233-7.50026-5](https://doi.org/10.1016/B978-0-7204-2233-7.50026-5)]

On the [[philosophy of language]]:

* [[Saul A. Kripke]], *[[Naming and Necessity]]*, Harvard University Press (1980) [[pdf](http://people.exeter.ac.uk/sp344/naming_and_necessityocr.pdf)]

## Related concepts

* [[Kripke frame]]

* [[Kripke semantics]]

category: people

[[!redirects Kripke]]
[[!redirects Kripke's]]
[[!redirects Saul Kripke]]
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[[!redirects Saul A. Kripke]]


Saul Permutter

* [Wikipedia entry](https://en.wikipedia.org/wiki/Saul_Perlmutter)

* [Institute page](https://physics.berkeley.edu/people/faculty/saul-perlmutter)

## Selected writings

On observation of [[supernovae]] interpreted as indicating a [[positive number|positive]] [[cosmological constant]] ([[dark energy]], the origin of the modern [[standard model of cosmology]]):

* {#PerlmutterEtAl1999} [[Saul Permutter]] et al. *Measurements of Omega and Lambda from 42 High-Redshift Supernovae*, Astrophys. J. **517** (1999) 565-586 [[arXiv:astro-ph/9812133](https://arxiv.org/abs/astro-ph/9812133), [doi:10.1086/307221](https://doi.org/10.1086/307221)]

category: people


Saul Ramos-Sanchez

* [InSpire page](https://inspirehep.net/authors/1027454)

* [ReserachGate page](https://www.researchgate.net/scientific-contributions/Saul-Ramos-Sanchez-12675168)

## Seleted writings

On [[heterotic string theory|heterotic]] [[string phenomenology]]:

* [[Hans-Peter Nilles]], [[Saul Ramos-Sanchez]], Patrick K.S. Vaudrevange, Akin Wingerter, _Exploring the $SO(32)$ Heterotic String_, JHEP 0604:050 (2006) \[doi:10.1088/1126-6708/2006/04/050[arXiv:hep-th/0603086](https://arxiv.org/abs/hep-th/0603086)\]

* [[Saul Ramos-Sanchez]], _Towards Low Energy Physics from the Heterotic String_,  Fortsch. Phys. **10** (2009) 907-1036 \[doi:10.1002/prop.200900073[arXiv:0812.3560](https://arxiv.org/abs/0812.3560)\]

  ([[heterotic string|heterotic]] [[SemiSpin(32)]] [[R-parity]] [[MSSM]] [[string theory vacua|vacua]])

* [[Saul Ramos-Sanchez]], [[Michael Ratz]], *Heterotic Orbifold Models*, in *[[Handbook of Quantum Gravity]]*, Springer (2024) [[arXiv:2401.03125](https://arxiv.org/abs/2401.03125), [doi:10.1007/978-981-19-3079-9](https://doi.org/10.1007/978-981-19-3079-9)]

  > (on [[orbifolds]])

category: people


Saunders Mac Lane

With [[S. Eilenberg|"Sammy" Eilenberg]], Saunders Mac Lane was one of the original pioneers of
[[category theory]]. He initially worked on it as a language to enable 'natural transformations' to be described in a 'natural' way, and also developed, again with Eilenberg many of the strong links with group theory and the cohomology of groups.  He was the author of one of the key books on homological algebra, see below.

With [[Henry Whitehead]] he gave the first algebraic description of the [[homotopy 2-type]] of a space.

* [English Wikipedia entry](https://secure.wikimedia.org/wikipedia/en/wiki/Saunders_Mac_Lane)

* [MacTutor biography](http://www-history.mcs.st-andrews.ac.uk/Biographies/MacLane.html)

* [[Colin McLarty]], _The Last Mathematician from Hilbert's Göttingen: Saunders Mac Lane as Philosopher of Mathematics_,Brit. J. Phil. Sci. 2007 ([pdf](http://www.cwru.edu/artsci/phil/BJPSMacLane.pdf))

## Selected writings
 {#Writings}

On [[group extensions]] and [[group homology]] via [[Ext]]/[[Tor]]-[[functors]]:

* {#EilenbergMacLane42} [[Samuel Eilenberg]], [[Saunders MacLane]], *Group Extensions and Homology*, Annals of Mathematics **43** 4 (1942) 757-831 [[doi:10.2307/1968966](https://doi.org/10.2307/1968966), [jstor:1968966](https://www.jstor.org/stable/1968966)]

Introducing [[category theory]]:

* {#EilenbergMacLane45} [[Samuel Eilenberg]], [[Saunders MacLane]], _[[General Theory of Natural Equivalences]]_,  Transactions of the American Mathematical Society **58** 2 (1945) 231-294 [[doi:10.1090/S0002-9947-1945-0013131-6](https://doi.org/10.1090/S0002-9947-1945-0013131-6), [jstor:1990284](http://www.jstor.org/stable/1990284)]

Introducing abstract [[group cohomology]]:

* [[Samuel Eilenberg]], [[Saunders MacLane]], *Cohomology Theory in Abstract Groups. I*, Annals of Mathematics, Second Series **48** 1 (1947) 51-78 [[doi:10.2307/1969215](https://doi.org/10.2307/1969215)]

Introducing [[Eilenberg-MacLane spaces]]:

* {#EilenbergMacLane53I} [[Samuel Eilenberg]], [[Saunders Mac Lane]], _On the Groups $H(\Pi,n)$, I_, Annals of Mathematics Second Series, Vol. 58, No. 1 (Jul., 1953), pp. 55-106 ([jstor:1969820](https://www.jstor.org/stable/1969820))

* {#EilenbergMacLane54II} [[Samuel Eilenberg]], [[Saunders Mac Lane]], _On the Groups $H(\Pi,n)$, II: Methods of Computation_, Annals of Mathematics Second Series, Vol. 60, No. 1 (Jul., 1954), pp. 49-139 ([jstor:1969702](https://www.jstor.org/stable/1969702))

* {#EilenbergMacLane54III} [[Samuel Eilenberg]], [[Saunders Mac Lane]], _On the Groups $H(\Pi,n)$, III: Operations and Obstructions_, Annals of Mathematics Second Series, Vol. 60, No. 3 (Nov., 1954), pp. 513-557 ([jstor:1969849](https://www.jstor.org/stable/1969849))

Introducing the [[simplicial classifying space]] construction $\overline{W}G$:

* {#MacLane54} [[Saunders MacLane]], *Constructions simpliciales acycliques*, Colloque Henri Poincaré 1954 ([[MacLaneConstructionsSimplicialesAcycliques.pdf:file]])

On [[homotopy 2-types]] (N.B.  their 3-type is the modern 2-type)

* {#WhiteheadMacLane50} (with [[J. H. C. Whitehead]]) _On the 3-type of a complex_, Proc. Nat. Acad. Sci.  U.S.A., **36** (1950) 41-48

On [[categorical algebra]]:

* [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

On [[classical mechanics]]:

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

On [[Grothendieck universes]] in the [[mathematical foundations]] of [[category theory]]:

* [[Saunders MacLane]], *One universe as a foundation for category theory*,  In: *Reports of the Midwest Category Seminar III*, Lecture Notes in Mathematics **106**, Springer (1969) 192-200 [[doi:10.1007/BFb0059147](https://doi.org/10.1007/BFb0059147)]

On [[geometric realization of simplicial topological spaces]] for constructing [[classifying spaces]], understood as simplicial [[coends]] in [[compactly generated topological spaces]]:

* {#MacLane70} [[Saunders MacLane]], _The Milgram bar construction as a tensor product of functors_,  In: F.P. Peterson  (eds.) *The Steenrod Algebra and Its Applications: A Conference to Celebrate N.E. Steenrod's Sixtieth Birthday*, Lecture Notes in Mathematics *168*,  Springer 1970 ([doi:10.1007/BFb0058523](https://doi.org/10.1007/BFb0058523), [pdf](https://link.springer.com/content/pdf/10.1007/BFb0058523.pdf))

On [[Euclidean geometry]]:

* {#MacLane} [[Saunders Mac Lane]], _Metric postulates for plane geometry_, American Mathematical Monthly, **66** 7 (1959) 543-555. ([doi:10.2307/2309851](https://doi.org/10.2307/2309851), [web](https://www.jstor.org/stable/2309851))

On [[category theory]]:

* [[Saunders MacLane]], *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5**, Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

On [[homological algebra]]:

* [[Saunders MacLane]], *Homology*, Grundlehren der Mathematischen Wissenschaften **114**, Springer (1963, 1974), reprinted as: Classics in Mathematics, Springer (1995)  [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/maclane_homology.pdf), [doi:10.1007/978-3-642-62029-4](https://doi.org/10.1007/978-3-642-62029-4)]

On [[philosophy of mathematics]]:

* _Mathematics form and function_, Springer 1986

On some history of mathematics:

* {#MacLane88} [[Saunders Mac Lane]]: *Concepts and Categories in Perspective*, in: P. Duren, *A century of mathematics in America* Part 1, AMS (1988) 323-365. [[pdf](http://www.ams.org/samplings/math-history/hmath1-maclane25.pdf), [ISBN:0-8218-0124-4](https://www.ams.org/publicoutreach/math-history/hmath1-index)]

On [[sheaf and topos theory]]:

* [[Saunders MacLane]], [[Ieke Moerdijk]], *[[Sheaves in Geometry and Logic]]*, Springer (1992) [[doi:10.1007/978-1-4612-0927-0](https://link.springer.com/book/10.1007/978-1-4612-0927-0)]

)

See also:

* _Selected Papers_. Edited by I. Kaplansky. Springer 1979

category: people

[[!redirects Sauders MacLane]]
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Saunders Mac Lane > history

< [[Saunders Mac Lane]]
[[!redirects Saunders Mac Lane -- history]]


Savas Dimopoulos

* [webpage](https://physics.stanford.edu/people/faculty/savas-dimopoulos)

## Selected writings

* [[Savas Dimopoulos]], [[Howard Georgi]], _Softly broken supersymmetry and $SU(5)$_, Nuclear Physics B Volume 193, Issue 1, 21 1981 (doi:10.1016/0550-3213(81)90522-8)

* [[Savas Dimopoulos]] [[Stuart Raby]], [[Frank Wilczek]], _Proton decay in supersymmetric models_, Physics Letters B Volume 112, Issue 2, 6 May 1982, Pages 133-136 (doi:10.1016/0370-2693(82)90313-6)

## Related $n$Lab entries

* [[split supersymmetry]]

* [[R-parity]]

category: people


Savdeep Sethi

* [webpage](http://theory.uchicago.edu/~sethi/)

## Selected writings

On [[C-field tadpole cancellation]]:

* [[Savdeep Sethi]], [[Cumrun Vafa]], [[Edward Witten]], _Constraints on Low-Dimensional String Compactifications_, Nucl.Phys.B480:213-224, 1996 ([arXiv:hep-th/9606122](https://arxiv.org/abs/hep-th/9606122))

* Keshav Dasgupta, Govindan Rajesh, [[Savdeep Sethi]], _M Theory, Orientifolds and G-Flux_, JHEP 9908 (1999) 023 ([arXiv:hep-th/9908088](https://xxx.lanl.gov/abs/hep-th/9908088))

* {#Sethi07} [[Savdeep Sethi]], around (2) in _A Note on Heterotic Dualities via M-theory_, Phys.Lett.B659:385-387, 2008 ([arXiv:0707.0295](https://arxiv.org/abs/0707.0295))

On [[string theory]] [[perturbative string theory vacuum|vacua]] [[KK-compactification|compactified]] on [[orbifolds]] and [[orientifolds]]:

* {#BDHKMMS01} [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], section 3 of _Triples, Fluxes, and Strings_, Adv.Theor.Math.Phys. 4 (2002) 995-1186 ([arXiv:hep-th/0103170](https://arxiv.org/abs/hep-th/0103170))

On [[de Sitter spacetime]] (or not) among the [[landscape of string theory vacua]]:

* {#Sethi17} [[Savdeep Sethi]], _Supersymmetry Breaking by Fluxes_ ([arXiv:1709.03554](https://arxiv.org/abs/1709.03554))

  re-amplified in:

  [[Savdeep Sethi]], _Hope or No Hope for a String Landscape?_, talk at [StringPheno2019](https://indico.cern.ch/event/782251/) ([pdf](https://indico.cern.ch/event/782251/contributions/3441886/attachments/1870138/3076971/3_Sethi.pdf), [[SethiLandscape2019.pdf:file]])

On an analogue of the [[DBI action]] via [[TT deformation]] of 2d [[Yang-Mills theory]]:

* {#BrennanFerkoSethi20} [[T. Daniel Brennan]], [[Christian Ferko]], [[Savdeep Sethi]], *A Non-Abelian Analogue of DBI from $T \bar T$*, SciPost Phys. **8** 052 (2020) [[arXiv:1912.12389](https://arxiv.org/abs/1912.12389), [doi:10.21468/SciPostPhys.8.4.052](https://doi.org/10.21468/SciPostPhys.8.4.052)]

## Related entries

* [[string phenomenology]]

* [[flux compactification]]

* [[landscape of string theory vacua]]

* [[discrete torsion]]

* [[C-field tadpole cancellation]]

category: people


Sayantani Bhattacharyya

* [webpage](https://www.iitk.ac.in/new/dr-sayantani-bhattacharyya)

## Selected writings

Introducing the [[fluid/gravity correspondence]]:

* [[Sayantani Bhattacharyya]], [[Veronika Hubeny]], [[Shiraz Minwalla]], [[Mukund Rangamani]], _Nonlinear Fluid Dynamics from Gravity_, JHEP 0802:045, 2008 ([arXiv:0712.2456](https://arxiv.org/abs/0712.2456))

category: people


Scala

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## Idea

A [[functional programming language]].

## Related pages

* [[Haskell]]

## References

* [www.scala-lang.org](https://www.scala-lang.org)

* Wikipedia, *[Scala (programming language)](https://en.wikipedia.org/wiki/Scala_(programming_language))*

* {#Winitzki22} [[Sergei Winitzki]], Section 10 of: *The Science of Functional Programming -- A tutorial, with examples in Scala* (2022) [[leanpub:sofp](https://leanpub.com/sofp), [github:sofp](https://github.com/winitzki/sofp)]


scalar

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## Idea

Broadly, a _scalar quantity_ is a "basic form of quantity", in terms of which more sophisticated objects of [[algebra]] are defined; and/or a "plain form of quantity", not subject to non-trivial transformation laws.

Specifically:

* **in [[mathematics]]**, by a _scalar_ one typically means an [[element]] of a [[ground ring]] or [[ground field]] (see also: [[number]]);

  this usage appears in concepts such as _[[scalar product]]_, _[[extension of scalars]]_, _[[restriction of scalars]]_, ...

* **in [[physics]]**, by a _scalar_ one typically means an element of a [[trivial representation]] of a given [[symmetry group]];

  this usage appears in concepts such as _[[scalar field]]_, _[[scalar meson]]_, ..., and in partial negation in concepts such as _[[pseudoscalar]]_, ...

These two usages do overlap: The 1-[[dimension|dimensional]] [[trivial representation]] $\mathbf{1} \in Rep_k(G)$ of any [[symmetry group]] $G$ over any [[ground field]] $k$ has as [[underlying set]] that ground field itself: $\mathbf{1} \,\simeq_k\, k$.

For instance, the _[[scalar curvature]]_ in [[Riemannian geometry]] is a scalar(-valued [[function]]) in both senses of the word.

## Related concepts

* [[unit]]

## References

See also

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

Discussion of scalars  in the context of ([[dagger-compact closed categories|dagger-]], [[compact closed category|compact-]], [[closed monoidal category|closed]]) [[monoidal categories]], as forming the [[endomorphism ring]] of the [[tensor unit]] (cf. [[quantum information theory via dagger-compact categories]]):

* {#AbramskyCoecke04} [[Samson Abramsky]], [[Bob Coecke]], §6 in: *A categorical semantics of quantum protocols*, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004) $[$[arXiv:quant-ph/0402130](http://arxiv.org/abs/quant-ph/0402130), [doi:10.1109/LICS.2004.1319636](https://doi.org/10.1109/LICS.2004.1319636)$]$

* [[Jamie Vicary]], §3 in: *Completeness of dagger-categories and the complex numbers*,  J. Math. Phys. **52** (2011) 082104 [[arXiv:0807.2927](https://arxiv.org/abs/0807.2927), [doi:10.1063/1.3549117](https://doi.org/10.1063/1.3549117)]

* [[Jamie Vicary]], §4 in: *Categorical Properties of The Complex Numbers*, Electronic Notes in Theoretical Computer Science **270** 2 (2011) 163-189 [[doi:10.1016/j.entcs.2011.01.030](https://doi.org/10.1016/j.entcs.2011.01.030)]

* {#HeunenVicary19} [[Chris Heunen]], [[Jamie Vicary]], §2.1 in: *Categories for Quantum Theory*, Oxford University Press 2019 [[ISBN:9780198739616](https://global.oup.com/academic/product/categories-for-quantum-theory-9780198739616)]

  based on:

  {#HeunenVicary12} [[Chris Heunen]], [[Jamie Vicary]], §2.1 in: *Lectures on categorical quantum mechanics* (2012) [[pdf](https://www.cs.ox.ac.uk/files/4551/cqm-notes.pdf), [[HeunenVicary-QuantumLectures.pdf:file]]]

* {#HeunenKornell21} [[Chris Heunen]], [[Andre Kornell]], *Axioms for the category of Hilbert spaces*, PNAS **119** 9 (2022) e2117024119 [[arXiv:2109.07418](https://arxiv.org/abs/2109.07418), [doi:10.1073/pnas.2117024119](https://doi.org/10.1073/pnas.2117024119)]

* [[Chris Heunen]], [[Andre Kornell]], [[Nesta van der Schaaf]], *Axioms for the category of Hilbert spaces and linear contractions* [[arXiv:2211.02688](https://arxiv.org/abs/2211.02688)]

[[!redirects scalars]]


scalar curvature

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## Definition

For $(X,e)$ a ([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]] with [[smooth manifold]] $X$ and [[vielbein field]] $e$, its **scalar curvature** is the [[smooth function]]

$$
  R(e) \in C^\infty(X, \mathbb{R})
$$

defined to be the [[trace]] of the [[Ricci tensor]] of $e$

$$
  R(e) \coloneqq tr_e Ric(c)
  \,.
$$

## Related concepts

* [[scalar]]

* The product of the scalar curvature with the [[volume form]] is the [[Lagrangian]] of the [[theory (physics)]] of [[gravity]]. The corresponding [[action functional]] is the [[Einstein-Hilbert action]].

* [[Lichnerowicz formula]]

[[!include curvature in Riemannian geometry -- table]]

[[!redirects scalar curvatures]]

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scalar extension bialgebroid

#Contents#
* table of contents
{:toc}

## Motivation and historical appearance

Given a finite-dimensional Hopf algebra $H$ and a $D(H)$-module algebra $A$ (where $D(H)$ is the Drinfeld double of $H$), [Lu1996](#Lu1996) has introduced the structure of a left $A$-[[associative bialgebroid|bialgebroid]] on the [[smash product algebra]] $A\sharp H$ as a noncommutative generalization of the algebra of functions on an [[action groupoid]]. In this finite-dimensional case, the category of left-right Yetter-Drinfeld modules is equivalent to the category of [[Yetter-Drinfeld module|Yetter-Drinfeld]] $H$-modules. In this new context of Yetter-Drinfeld module algebras, the construction has been generalized
by the so called Brzeziński-Militaru construction or scalar extension bialgebroid in [BrzezinskiMilitaru2002](#BrzezinskiMilitaru). They also proved a converse, roughly that the formulas given for the scalar extension bialgebroid for an arbitrary smash product satisfy the axioms of a bialgebroid only if the module algebra which is also a comodule is in fact a braided commutative Yetter-Drinfeld algebra.

## Definition

If $(H,\Delta,\epsilon)$ (where $\Delta(h) = \sum h_{(1)}\otimes h_{(2)}$ is a bialgebra and $A$ a left-right Yetter-Drinfeld $H$-module algebra with Hopf action $\triangleright:H\otimes A\to A$ and a coaction $\rho: A\to A\otimes H$, then the smash product $A\sharp H$ inherits the left $A$-bialgebroid structure $(A\sharp H,\alpha,\beta,\Delta^{A\sharp H},\epsilon^{A\sharp H})$ where source $\alpha(a) = a\sharp 1$, target $\beta$ is $\rho$ followed by the identification of vector spaces $A\sharp H\cong A\otimes H$ and the coproduct $\Delta^{A\sharp H}$ is obtained by extending $\Delta$ along inclusion $H\to A\sharp H$, that is,
$$
\Delta^{A\sharp H} : A\sharp H\to (A\sharp H)\otimes_A (A\sharp H), \,\,\,\,a\sharp h\mapsto a\sharp h_{(1)}\otimes 1\sharp h_{(2)},
$$
which is an extension of $\Delta$ along the canonical embedding $H\to A\sharp H$. The counit is
$$
\epsilon^{A\sharp H}:A\sharp H\to A,\,\,\,\,
\epsilon^{A\sharp H}(a\sharp h) = a\epsilon(h).
$$

## Scalar extension Hopf algebroid

If $H$ is a Hopf algebra the Brzeziński-Militaru construction gives a Lu-Hopf bialgebroid and if the
antipode $S$ of $H$ is
invertible then it gives also a
Böhm-Szlachányi symmetric Hopf algebroid.
If the antipode $S$ is not invertible
then the data of a Yetter-Drinfeld module algebra has to be replaced by a compatible pair of a left-right Yetter-Drinfeld module algebra and a right-left Yetter-Drinfeld module algebra yielding again a symmetric Hopf algebroid [Stojic2023](#Stojic2023).

## Special case: Heisenberg double

Heisenberg double $H^*\sharp H$ of a finite-dimensional Hopf algebra is a specific example of a Hopf algebroid. There are many infinite-dimensional cases of Heisenberg double which can be cast into bialgebroids, but most often
complicated issues with completions are involved (e.g. in
[MSSncphasespace2017](#MSSncphasespace2017))

## Twists of scalar extension bialgebroids

## Literature

* {#Lu1996} [[Jiang-Hua Lu]], _Hopf algebroids and quantum groupoids_, Int. J. Math. __7__, 1 (1996) pp. 47-70, [q-alg/9505024](https://arxiv.org/abs/q-alg/9505024), [MR95e:16037](https://mathscinet.ams.org/mathscinet-getitem?mr=95e:16037), [doi](https://doi.org/10.1142/S0129167X96000050); _On the Drinfeld double and the Heisenberg double of a Hopf algebra_, Duke Math. J. __7__:3 (1994) 763-776, [MR1277953](https://www.ams.org/mathscinet-getitem?mr=1277953), [doi](https://doi.org/10.1215/S0012-7094-94-07428-0)

* {#BrzezinskiMilitaru} [[Tomasz Brzeziński]], [[Gigel Militaru]], *Bialgebroids, $\times_A$-bialgebras and duality*,  J. Algebra __251__ (2002) 279-294 [[math.QA/0012164](https://arxiv.org/abs/math.QA/0012164), [doi:10.1006/jabr.2001.9101](https://doi.org/10.1006/jabr.2001.9101)]

Standard reference is now

* [[G. Böhm]], _Hopf algebroids_, in Handbook of algebra, vol. 6, ed. M. Hazewinkel, arXiv:[math.RA/0805.3806](https://arxiv.org/abs/0805.3806)

Remaining issues about the antipode are settled in

* {#Stojic2023} M. Stojić, _Scalar extension Hopf algebroids_, Journal of algebra and applications 2023 [doi](https://doi.org/10.1142/S0219498824501147) [arXiv:2208.11696](https://arxiv.org/abs/2208.11696)

Every invertible counital 2-cocycle $F = \sum F^1\otimes F^2$  (Drinfeld twist) for a bialgebra $H$, with inverse $F^{-1} = G^1\otimes G^2$, induces a Drinfeld-Xu 2-cocycle $\mathcal{G}=\sum 1\sharp G^1\otimes_A 1\sharp G^2$ for the scalar extension bialgebroid $A\sharp H$. The scalar extension for the $F$-twisted data $A_F\sharp H^F$ is isomorphic as a $A_F$-bialgebroid to the $\mathcal{G}$-twist of $A\sharp H$.

* Andrzej Borowiec, Anna Pachoł, _Twisted bialgebroids versus bialgebroids from Drinfeld twist_, J. Phys. A50:5 (2017) [arXiv:1603.09280](https://arxiv.org/abs/1603.09280)

Some corrections and a slight generalization are given in

* [[Zoran Škoda]], Martina Stojić, _Comment on "Twisted bialgebroids versus bialgebroids from Drinfeld twists"_, [arXiv:2308.05083](https://arxiv.org/abs/2308.05083)

An infinite-dimensional case of a Heisenberg double of a universal enveloping algebra $U(g)$ of a finite dimensional Lie algebra over a field of characteristic zero is described as a version of a completed Hopf algebroid in

* {#MSSncphasespace2017} S. Meljanac, Z. Škoda, M. Stojić, _Lie algebra type noncommutative phase spaces are Hopf algebroids_, Lett. Math. Phys. 107:3, 475–503 (2017) [doi](https://doi.org/10.1007/s11005-016-0908-9)  [arXiv:1409.8188](https://arxiv.org/abs/1409.8188)

While all the statements in the above article are rigorous, the axioms are not the most natural. A natural version where the Heisenberg double of $U(g)$ has been described as an internal Hopf algebroid (of an internal scalar extension type) in a symmetric monoidal category of (countably cofinite) filtered-cofiltered vector spaces in

* Martina Stojić, _Completed Hopf algebroids_ (in Croatian: Upotpunjeni Hopfovi algebroidi), Ph. D. thesis, University of Zagreb, 2017, 306 pp. [pdf](https://web.math.pmf.unizg.hr/~stojic/Stojic-disertacija.pdf)

Other articles include

* X. Han, [[Shahn Majid]], _Bisections and cocycles on action Hopf algebroids_, [arXiv:2305.12465](https://arxiv.org/abs/2305.12465)

Scalar extension Hopf algebroids can be recast also in the form fitting the axioms of the Hopf algebroid with a balancing subalgebra, see Sec. 4 in

* [[Zoran Škoda]], Martina Stojić, _Hopf algebroids with balancing subalgebra_, J. Alg. __598__ (2022) 445--469 [arXiv:1610.03837](https://arxiv.org/abs/1610.03837) [doi](https://doi.org/10.1016/j.jalgebra.2022.01.027)

category: algebra
[[!redirects Brzeziński-Militaru construction]]
[[!redirects scalar extension Hopf algebroid]]
[[!redirects action Hopf algebroid]]


scalar field

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In [[physics]], a __scalar field__ is a [[field (physics)|field]] on [[spacetime]]/[[worldvolume]] which is simply a [[function]] with values in the [[field of scalars]], typically the [[real numbers]] $\mathbb{R}$ or [[complex numbers]] $\mathbb{C}$, sometimes the [[quaternions]] $\mathbb{H}$. Hence it is a [[field (physics)|field]] whose [[field bundle]] (of which the [[field (physics)|field]] is a [[section]]) is a [[trivial line bundle]].

One fundamental (complex, charged) scalar field is seen in [[experiment]], the _[[Higgs field]]_, which is one component of the [[standard model of particle physics]]. A widely hypothesized scalar field is the [[inflaton]] field in [[model (physics)|models]] of [[cosmic inflation]], which however remains speculative and might in any case be an effective compound of more fundamental fields.

But scalar fields also serve as a key toy example in theoretical studies of [[field theory]], such as in [[phi^4 theory]] or in the [[Ising model]]. The usefulness of the scalar field as a toy example of [[classical field theory]] and [[perturbative quantum field theory]] is due to
it already exhibiting much of the core structure of [[field theory]]. For instance the general formulas for [[propagators]]
and the [[S-matrix]] of general [[local field theories]] are structurally those of the
scalar field, just with some more fairly evident [[representation theory|representation theoretic]] structure thrown in.

## Definition

### Free scalar field

We discuss here the [[free field|free]] [[scalar field]] on general [[spacetimes]],
hence the scalar field subject to the [[force]] of a [[background field]] of [[gravity]], but
not [[interaction|interacting]] with itself. This means that its [[local Lagrangian density]]
is [[quadratic form|quadratic]] in the fields and its first derivatives (def. \ref{LocalLagrangianOfFreeScalarField})
and its [[equation of motion]] is the [[Klein-Gordon equation]] (hence the [[wave equation]] in the case of
vanishing [[mass]]) (prop. \ref{FreeScalarFieldEOM} below).

The [[Poisson bracket]] on the [[covariant phase space]] of this system (prop. \ref{FreeScalarFieldEOM} below) turns out
to have as [[integral kernel]] the [[causal propagator]] of the [[Klein-Gordon operator]]
(i.e. the [[Green function]] whose [[support of a distribution|support]] is inside the [[light cone]]).
Accordingly, the other associated [[Green functions]] of the [[Klein-Gordon operator]]
(the "[[propagators]]", such as the [[Feynman propagator]]) govern the
[[perturbative quantum field theory]] of the scalar field (see at _[[S-matrix]]_ for more).

#### Covariant phase space

Recall that a _[[classical field theory|classical]] [[local field theory]]_ is for some prescribed class of [[manifolds]] $\Sigma$ of given [[dimension]] $p+1 \in \mathbb{N}$ interpreted as [[spacetimes]]/[[worldvolumes]]:

1. a choice of [[fiber bundle]] $E \to\Sigma$, called the _[[field bundle]]_;

1. a choice $L \in \Omega^{p+1,0}(J^\infty(E))$ of [[horizontal differential form]] of degree $p+1$ on the [[jet bundle]] of the field bundle, called the _[[local Lagrangian density]]_.

Given a [[classical field theory|classical]] [[local field theory]] defined by a [[local Lagrangian density]] $L \in \Omega^{p+1,0}(J^\infty(E))$, then

1. the [[configuration space]] is the [[smooth space|smooth]] [[space of sections]]   $\Gamma_X(E)$ of the [[field bundle]];

1. the [[equations of motion]] is the [[partial differential equation]] on elements $\phi \in \Gamma_X(E)$ given by

   $$
     (\delta_{EL} L)(j^\infty \phi) = 0
     \,,
   $$

   where

   1. $\delta_{EL} \;\colon\; \Omega^{n,0}(J^\infty_X(E)) \to \mathcal{F}^1(J^\infty_X(E))$ denotes the _[[Euler-Lagrange operator]]_

   1. $j^\infty \;\colon\; \Gamma_X(E) \longrightarrow \Gamma_X(J^\infty_X(E))$ denotes [[jet prolongation]].

1. The [[covariant phase space]] $(Sol_{\delta_{EL}L = 0}, d\theta)$ is the subspace $Sol_{\delta_{EL} L = 0} \subset \Gamma_X(E)$ of solutions to the [[equations of motion]], equipped with the canonical [[presymplectic form]].

##### On Minkowski spacetime

We discuss the [[covariant phase space]] of the [[free field|free]] scalar field on [[Minkowski spacetime]].
For a more detailed exposition see at _[[geometry of physics -- A first idea of quantum field theory]]_.

+-- {: .num_defn #LagrangianForFreeScalarFieldOnMinkowskiSpacetime}
###### Definition
**([[local Lagrangian density]] for [[free field|free]] [[scalar field]] on [[Minkowski spacetime]])**

For $p \in \mathbb{N}$, let [[spacetime]] $\Sigma \coloneqq \mathbb{R}^{p,1} = (\mathbb{R}^d, \eta)$ be [[Minkowski spacetime]] of [[dimension]] $p + 1$, where $\eta$ denotes the Minkowski [[metric tensor]] of [[signature of a quadratic form|signature]] $(-,+,\cdots, +)$. We write $\mathrm{dvol}_\Sigma \in \Omega^{p+1}(\Sigma)$ for the corresponding [[volume form]] and $\{x^\mu \colon \Sigma \to \mathbb{R}\}_{\mu = 0}^p$ for the canonical [[coordinate functions]].

Let the [[field bundle]] $E \to \Sigma$ be the [[trivial vector bundle|trivial]] [[real line bundle]] over $\Sigma$.

Then its [[jet bundle]] $J^\infty E$ has canonical coordinates

$$
  \{ \{x^\mu\}, \phi, \{\phi_{,\mu}\}, \{\phi_{,\mu \nu}\}, \cdots \}
  \,.
$$

In these coordinates, the [[local Lagrangian density]]

$$
  L \in \Omega^{p+1,0}(\Sigma)
$$

defining the [[free field|free]] [[scalar field]] of [[mass]] $m \in [0,\infty)$ on $\Sigma$ is

$$
  L
    \coloneqq
  \tfrac{1}{2}
  \left(
    \eta^{\mu \nu} \phi_{,\mu} \phi_{,\nu}
    +
    m^2 \phi^2
  \right)
  \mathrm{dvol}_\Sigma
  \,.
$$

=--

+-- {: .num_prop #FreeScalarFieldEOM}
###### Proposition
**([[covariant phase space]] of [[free field|free]] [[scalar field]])**

In the situation of def. \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}
for $\Phi : \Sigma \to \mathbb{R}$ the $\phi$-component of a [[section]] of the [[field bundle]], its [[equation of motion]] is the [[Klein-Gordon equation]]

$$
  \left(\eta^{\mu \nu} \partial_\mu \partial_\nu + m^2 \right) \Phi = 0
  \,.
$$

Moreover, the induced [[pre-symplectic current]] $\omega \in \Omega^{p-1,2}(E)$ is, in local coordinates,

$$
  \omega
    =
  \left(\eta^{\mu \nu} d_V \phi_{,\mu} \wedge d_V \phi \right) \wedge \iota_{\partial_\nu} dvol_{\Sigma}
$$

and hence the induced [[symplectic form]] on the [[covariant phase space]] of the free scalar field
takes two [[smooth function]] $w_1,w_2 \in C^\infty(\Sigma)$, regarded as [[tangent vectors]] at zero to

$$
  \mathbf{\Omega}_{\Sigma_{p}}(w_1, w_2)
    \;=\;
    \int_{\Sigma_{p}}
    \left(
      (\partial_n w_1) w_2
      -
      w_1 \partial_n w_2
    \right)
    dvol_{\Sigma_{p}}
    \,,
$$

where $\Sigma_{p} \hookrightarrow \Sigma$ is any [[Cauchy surface]]
and where $n \in N \Sigma_{p}$ denotes its time-like normal vector field.

=--

+-- {: .proof}
###### Proof

We need to show that [[Euler-Lagrange operator]] $\delta_{EL} \colon \Omega^{p+1,0}(\Sigma) \to \Omega^{p+1,1}_S(\Sigma)$ takes the [[local Lagrangian density]] for the [[free field|free]] [[scalar field]] to

$$
  \delta_EL L
  \;=\;
  \left(
    \eta^{\mu \nu} \phi_{,\mu \nu}  + m^2 \phi
  \right)
  d_V \phi \wedge \mathrm{dvol}_\Sigma
  \,.
$$

First of all, the result of applying the [[variational bicomplex|vertical differential]] to the local Lagrangian density is

$$
  d_V L
    =
  \left(
    \eta^{\mu \nu} \phi_{,\mu} d_V \phi_{,\nu}
      +
    m^2 \phi d_V \phi
  \right)
  \wedge \mathrm{dvol}_\Sigma
  \,.
$$

By definition of the [[Euler-Lagrange operator]], in order to find $\mathrm{EL}$ and $\theta$, we need to exhibit this as the
sum of the form $(-) \wedge d_V \phi  - d_H \theta$.

The key to find $\theta$ is  to realize $d_V \phi_{,\nu}\wedge \mathrm{dvol}_\Sigma$ as a [[horizontal derivative]]. Since $d_H \phi = \phi_{,\mu} d x^\mu$ this is accomplished by

$$
  d_V \phi_{,\nu} \wedge \mathrm{dvol}_\Sigma
  =
  d_V d_H \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma
$$

Hence we may set

$$
  \theta
    \coloneqq
  \eta^{\mu \nu} \phi_{,\mu} d_V \phi \wedge \iota_{\partial_\nu}
  \mathrm{dvol}_\Sigma
  \,,
$$

because with this we have

$$
  d_H \theta
  =
  \eta^{\mu \nu}
  \left(
    \phi_{,\mu \nu} d_V \phi
    -
    \eta^{\mu \nu} \phi_{,\mu} d_V \phi_{,\nu}
  \right) \wedge \mathrm{dvol}_\Sigma
  \,.
$$

In conclusion this yields the decomposition of the vertical differential of the Lagrangian density

$$
  d_V L
  =
  \underset{
    = \delta_{EL} L
  }{
  \underbrace{
    \left(
      \eta^{\mu \nu} \phi_{,\mu \nu}  + m^2 \phi
    \right)
    d_V \phi \wedge \mathrm{dvol}_\Sigma
  }
  }
  -
  d_H \theta
  \,,
$$

which shows that $\delta_{EL} L$ is as claimed, and that $\theta$ is a presymplectic potential current.
Hence the presymplectic current itself is

$$
  \begin{aligned}
    \omega &\coloneqq d_V \theta
    \\
    & =
    d_V \left( \eta^{\mu \nu} \phi_{,\mu} d_V \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma \right)
    \\
    & =
    \left(\eta^{\mu \nu} d_V \phi_{,\mu} \wedge d_V \phi \right) \wedge \iota_{\partial_\nu} dvol_{\Sigma}
  \end{aligned}
  \,.
$$

For $\Sigma_p \hookrightarrow \Sigma$ a [[Cauchy surface]], the [[transgression of variational differential forms|transgression]] of
this presymplectic current to the [[infinitesimal neighbourhood]] of $\Sigma$ is

$$
  \begin{aligned}
    \omega_{\Sigma_{p}}(w_1, w_2)
    & =
    \left(
    \int_{\Sigma_p}
      \left(
        \eta^{\mu \nu} \mathbf{d} \partial_\mu \phi \wedge \mathbf{d} \phi
      \right)
      \iota_{\partial_\nu} dvol_\Sigma
    \right)
    (w_1, w_2)
    \\
    & =
    \int_{\Sigma_{p}}
    \left(
      (\partial_n w_1) w_2
      -
      w_1 \partial_n w_2
    \right)
    dvol_{\Sigma_{p}}
  \end{aligned}
$$

=--

+-- {: .num_example #PoissonBracketsOverMinkowskiSpacetime}
###### Example
**([[Poisson brackets]] over [[Minkowski spacetime]])

Consider the covariant phase space over [[Minkowski spacetime]] of dimension $p+1$ as in def. \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime},
with [[pre-symplectic current]] according to prop. \ref{FreeScalarFieldEOM} given by

$$
  \omega
   =
  \eta^{\mu \nu} d_V \phi_{,\mu} \wedge d_V \phi \wedge \iota_{\partial_\mu} dvol_\Sigma
  \,
$$

The corresponding [[Poisson bracket Lie n-algebra|Poisson bracket Lie (p+1)-algebra]] has in degree 0
[[Hamiltonian forms]] such as

$$
  Q \coloneqq \phi \iota_{\partial_0} dvol_\Sigma \in \Omega^{p,0}(E)
$$

and

$$
  P \coloneqq \eta^{\mu \nu} \phi_{,\mu} \iota_{\partial_\nu} dvol_{\Sigma} \in \Omega^{p,0}(E)
  \,.
$$

The corresponding [[Hamiltonian vector fields]] are

$$
 v_P = \partial_{\phi_{,0}}
$$

and

$$
  v_Q = - \partial_{\phi}
  \,.
$$

Hence the corresponding bracket is

$$
  \{Q,P\} = \iota_{v_Q} \iota_{v_P} \omega = \iota_{\partial_0} dvol_\Sigma
  \,.
$$

More generally for $b_1, b_2 \in C^\infty_c(\Sigma)$ two [[bump functions]] then

$$
  \{ b_1 Q, b_2 P \} = \pm b_1 b_2 \iota_{\partial_0} dvol_\Sigma
  \,.
$$

Upon [[transgression of variational differential forms|transgression]] to the [[Cauchy surface]] $\Sigma^t_{p} \coloneqq \{x \in  \Sigma \vert x^0 = t \}$ this yields the [[Poisson bracket]]

$$
  \left\{
    \int_{\Sigma_p} b_1(\vec x) \phi(t,\vec x) \iota_{\partial_0} dvol_\Sigma(x)
    d^p \vec x
    \;,\;
    \int_{\Sigma_p} b_2(\vec x) \partial_0 \phi(t,\vec x) \iota_{\partial_0} dvol_\Sigma(\vec x)
  \right\}
  \;=\;
  \int_{\Sigma_p} b_1(\vec x) b_2(\vec x) \iota_{\partial_0} dvol_\Sigma(\vec x)
  d^p \vec x
  \,.
$$

where now

$$
  \phi(x), \partial_0 \phi(x) : PhaseSpace(\Sigma_p^t) \to \mathbb{R}
$$

are the point-evaluation functions ([[functionals]]), which act on a field configuration $\Phi \in \Gamma_\Sigma(E) = C^\infty(\Sigma)$ as

$$
 \phi(x)(\Phi) \coloneqq \Phi(x)
 \phantom{AAAAAAAA}
 \partial_0 \phi(x) (\Phi) \coloneqq \partial_0 \Phi(x)
 \,.
$$

Notice that these point-evaluation functions themselves do not arise
as the transgression of elements in $\Omega^{p,0}(E)$, only their smearings such as $\int_{\Sigma_p} b_1 \phi dvol_{\Sigma_p}$ do.
Nevertheless we may express the above Poisson bracket conveniently via the [[integral kernel]]

$$
  \label{PoissonBracketOfScalarFieldPointEvaluationOnMinkowskiSpacetime}
  \{\phi(t,\vec x), \phi(t,\vec y) \}
  =
  \delta(\vec x - \vec y)
  \,.
$$

=--

More generally one may express the integral kernel for the Poisson bracket of evaluation functions for  different values of $t$.
Notice that for each time interval $[t_1, t_2]$ we have a [[Lagrangian correspondence]]

$$
  \array{
    && Trajectories([t_1,t_2])
    \\
    & {}^{\mathllap{ \Phi(-) \mapsto ( \Phi(t_1,-), \partial_0 \Phi(t_1,-) ) }}\swarrow
      &&
    \searrow^{\mathrlap{ \Phi(-) \mapsto ( \Phi(t_2,-), \partial_0 \Phi(t_2,-) ) }}
    \\
    PhaseSpace(\Sigma^{t_1}_p)
     && &&
    PhaseSpace(\Sigma^{t_2}_p)
    \\
    & {}_{\mathllap{\Omega^{t_1}}}\searrow && \swarrow_{\mathrlap{\Omega^{t_2}}}
    \\
    && \mathbf{\Omega}_{cl}^2
  }
  \,,
$$

where $Trajectories([t_1,t_2])$ is the space of solutions to the [[Klein-Gordon equation]]
on $[t_1,t_2] \times \mathbb{R}^p \subset \mathbb{R}^{p,1}$.

There is a unique function on $PhaseSpace(\Sigma^{t_1}_p)$ whose pullback to $Trajectories([t_1,t_2])$
is the evaluation function $\phi(x)$ for any $x^0 \in [t_1,t_2]$. By convenient abuse of notation,
we also call that function $\phi(x)$.

+-- {: .num_prop #IntegralKernelForPoissonBracketOfFreeScalarFieldOnMinkowskiSpacetime}
###### Proposition
**([[integral kernel]] for [[Poisson bracket]] on [[Minkowski spacetime]] is the [[causal propagator]])

The integral kernel on $\mathbb{R}^{p,1}$ for the Poisson bracket of the scalar field
over [[Minkowski spacetime]] (example \ref{PoissonBracketsOverMinkowskiSpacetime})
is the _[[causal propagator]]_

$$
  \Delta
    \;\in\;
  \mathcal{D}'(\Sigma \times \Sigma)
$$

(also known as the _[[Pauli-Jordan distribution]]_ or _[[Peierls bracket]]_)
on Minkowski spacetime:

$$
  \begin{aligned}
    \{ \phi(x), \phi(y) \}
    & =
    \Delta(x,y)
    \\
    & \coloneqq
    -i (2\pi)^{-p} \int \tfrac{1}{2 E(\vec k)}\left(
    e^{- i E(\vec k) (x-y)^0 - \vec k \cdot (\vec x - \vec y)}
    -
    e^{+ i E(\vec k) (x-y)^0 + \vec k \cdot (\vec x - \vec y)}
  \right) d^p \vec k
    \\
    & =
   -i (2\pi)^{-p} \int \delta( k_\mu k^\mu + m^2 ) sgn( k_0 ) e^{ - i k_\mu (x-y)^\mu } d^{p+1} k
    \,.
  \end{aligned}
$$

=--

(e. g. [Scharf 01 (1.1.13)](#Scharf01))

+-- {: .proof}
###### Proof

By [[Fourier transform]] the general solution to the [[Klein-Gordon equation]] may be expressed in the form

$$
  \Phi(x)
    \;=\;
  (2\pi)^{-p} \int  Amp(k) e^{- i k_\mu x^\mu } \delta( k_\mu k^\mu + m^2 ) d^{p+1} k
  \,,
$$

where $Amp(k) \in \mathbb{C}$ is the _complex amplitude_ of the $k$th mode $(k \in \mathbb{R}^{p+1})$.

We may split this into the contributions with positive and those with negative [[energy]] $k_0$
by decomposing the integral over $k_0$ as

$$
  \begin{aligned}
    \Phi(x)
    & = \phantom{+}
    (2\pi)^{-p/2}
    \int_0^\infty  \int Amp(k_0, \vec k) e^{- i k_0 x^0 - i \vec k \cdot \vec x} \delta(- k_0^2 + \vec k + m^2) d^p \vec k \, d k_0
    \\
    & \phantom{=} +
    (2\pi)^{-p/2}
    \int_0^\infty  \int Amp(-k_0, \vec k) e^{+ i k_0 x^0 - i \vec k \cdot \vec x} \delta(-k_0^2 + \vec k^2 + m^2) d^p \vec k \, d k_0
    \,.
  \end{aligned}
$$

By [[changing integration variables]] via $k_0 = +\sqrt{ h }$ this yields

$$
  \begin{aligned}
    \Phi(x)
    & = \phantom{+}
    (2\pi)^{-p/2}
    \int_0^\infty  \int Amp(\sqrt{h}, \vec k) e^{- i \sqrt{h} x^0 - i \vec k \cdot \vec x} \delta(- h + \vec k + m^2) d^p \vec k \, \frac{d h}{2\sqrt{h}}
    \\
    & \phantom{=} +
    (2\pi)^{-p/2}
    \int_0^\infty  \int Amp(- \sqrt{h}, \vec k) e^{+ i \sqrt{h} x^0 - i \vec k \cdot \vec x} \delta(- h + \vec k^2 + m^2) d^p \vec k \, \frac{d h}{2 \sqrt{h}}
    \\
    & = \phantom{+}
    (2\pi)^{-p/2}
    \int Amp(E(\vec k), \vec k) e^{- i E(\vec k) x^0 - i \vec k \cdot \vec x} d^p \vec k
    \\
    & \phantom{=} +
    (2\pi)^{-p/2}
    \int Amp(-E(\vec k), \vec k) e^{+ i E(\vec k) x^0 - i \vec k \cdot \vec x} d^p \vec k
  \end{aligned}
$$

where we defined the _on-shell [[energy]]_

$$
  E(\vec k) \coloneqq + \sqrt{ \vec k^2 + m^2 }
  \,.
$$

It is convenient to also change variables $\vec k \mapsto - \vec k$ in the second integral. This yields

$$
  \begin{aligned}
    \Phi(x)
    &= \phantom{+}
      (2\pi)^{-p/2}
      \int Amp(E(\vec k), \vec k) e^{- i E(\vec k) x^0 - i \vec k \cdot \vec x} d^p \vec k
      \\
      & \phantom{=} +
      (-1)^p
      (2\pi)^{-p/2}
      \int Amp(-E(\vec k), \vec k) e^{+ i E(\vec k) x^0 + i \vec k \cdot \vec x} d^p \vec k
  \end{aligned}
    \,.
$$

Since $\Phi$ is real-valued, it follows that under [[complex conjugation]] $(-)^\ast$
the amplitudes are related by

$$
  Amp(E(\vec k), \vec k)^\ast = (-1)^p Amp(-E(\vec k), \vec k)
  \,.
$$

We abbreviate (cf. [Scharf 01 (1.1.18)](#Scharf01))

$$
  A(\vec k) \coloneqq E(\vec k) Amp(E(\vec k), \vec k)
  \,,
$$

where the prefactor just serves to make some of the following formulas come out conveniently.

With this the general solution to the Klein-Gordon equation is finally of the form

$$
  \label{FourierModeExpansionOfScalarFieldOnminkowskiSpacetime}
  \Phi(x)
  =
    (2\pi)^{-p/2}
    \int \frac{1}{\sqrt{2 E(\vec k)}}
      \left(
        A(\vec k) e^{- i E(\vec k) x^0 - i \vec k \cdot \vec x}
        +
        A(\vec k)^\ast e^{+ i E(\vec k) x^0 + i \vec k \cdot \vec x}
      \right)
    d^p \vec k
    \,.
$$

and hence its time derivative is

$$
  \partial_0 \Phi(x)
  =
    (2\pi)^{-p/2}
    \int -i \sqrt{E(k)/2}
      \left(
        A(\vec k) e^{- i E(\vec k) x^0 - i \vec k \cdot \vec x}
        -
        A(\vec k)^\ast e^{+ i E(\vec k) x^0 + i \vec k \cdot \vec x}
      \right)
    d^p \vec k
    \,.
$$

This allows to express the modes in terms of the value of the field and its time derivative at $t = 0$:

$$
  A(\vec k)
  =
   (2 \pi)^{-p/2}
   \int
     \left( \sqrt{E(k)/2}\Phi(0,\vec x) + i \frac{1}{2\sqrt{E(k)}} \partial_0 \Phi(0,\vec x)\right) \exp( i \vec k \cdot \vec x )
   d^p \vec x
   \,.
$$

As in example \ref{PoissonBracketsOverMinkowskiSpacetime} we denote the corresponding evaluation functional

$$
  a(\vec k) \;\colon\; C^\infty(\Sigma) \longrightarrow \mathbb{C}
$$

by the corresponding lower case symbol:

$$
  a(\vec k)
   \;\coloneqq\;
   (2 \pi)^{-p/2}
   \int
     \left( \sqrt{E(k)/2}\phi(\vec x) + i \frac{1}{\sqrt{2 E(k)}} \partial_0 \phi(\vec x)\right) \exp( i \vec k \cdot \vec x )
   d^p \vec x
   \,.
$$

With the Poisson bracket kernel $\{\phi(\vec x), \phi(\vec y)\} = \delta(\vec x - \vec y)$ from example \ref{PoissonBracketsOverMinkowskiSpacetime} (eq:PoissonBracketOfScalarFieldPointEvaluationOnMinkowskiSpacetime),
it follow that the  (integral kernel for the) Poisson bracket of these mode functionals is
that of the [[canonical commutation relations]]:

$$
  \begin{aligned}
    \label{CanonicalPoissonCommutationOfModesOfFreeScalarFieldOnMinkowskiSpacetime}
    \{ a(\vec k_1), a(\vec k_2)^\ast  \}
    & =
    -i
    (2\pi)^{-p}
    \int \delta(\vec x_1 - \vec x_2) e^{i  ( \vec k_1 \cdot \vec x_1 - \vec k_2 \cdot \vec x_2) } d \vec x_1 d\vec x_2
    \\
    & =
    -i
    (2\pi)^{-p}
    \int e^{i (\vec k_1 - \vec k_2) \cdot \vec x } d \vec x
    & =
    \\
    & =  -i \delta(\vec k_1 - \vec k_2)
    \,,
  \end{aligned}
$$

where in the last step we used the [[Fourier transform]] representation of the [[delta distribution]] ([this prop.](Dirac+distribution#FourierTransform)).

In order to finally compute $\{\phi(x), \phi(y)\}$, it is convenient to break this up into two contributions:
Write

$$
  \phi^{(+)}(x)
  \;\coloneqq\;
    (2\pi)^{-p/2}
    \int \frac{1}{\sqrt{2 E(\vec k)}}
        a(\vec k)^\ast e^{+ i E(\vec k) x^0 + i \vec k \cdot \vec x}
    d^p \vec k
  \phantom{AAAA}
  \phi^{(-)}(x)
  \;\coloneqq\;
    (2\pi)^{-p/2}
    \int \frac{1}{\sqrt{2 E(\vec k)}}
        a(\vec k) e^{- i E(\vec k) x^0 - i \vec k \cdot \vec x}
    d^p \vec k
$$

for the positive and negative energy contributions from the Fourier expansion in (eq:FourierModeExpansionOfScalarFieldOnminkowskiSpacetime), so that

$$
  \phi(x) = \phi^{(-)}(x)  + \phi^{(+)}(x)
  \,.
$$

Using the [[canonical commutation relation]] of the mode functions (eq:CanonicalPoissonCommutationOfModesOfFreeScalarFieldOnMinkowskiSpacetime), we find

$$
  \label{2PointFunctionFreeScalarFieldOnMinkowski}
  \begin{aligned}
    -i \omega(x,y)
     & \coloneqq
     \{ \phi^{(-)}(x), \phi^{(+)}(y) \}
    \\
    &=
    -i (2\pi)^{-p} \int \tfrac{1}{2 E(\vec k)} e^{- i E(\vec k) (x-y)^0 - \vec k \cdot (\vec x - \vec y)} d^{p} \vec k
    \\
    & =
    -i (2\pi)^{-p} \int \delta( k_\mu k^\mu + m^2 ) \Theta( k_0 ) e^{ - i k_\mu (x-y)^\mu } d^{p+1} k
    \,,
  \end{aligned}
$$

where in the last line we again applied [[change of integration variables]].
This $\omega$ is known as the _[[2-point function]]_ or _[[Hadamard propagator]]_ on Minkowski spacetime (see def. \ref{2PointFunctionOfScalarFieldOnMinkowskiSpacetime} below).

Similarly

$$
  \begin{aligned}
  \{ \phi^{(+)}(x), \phi^{(-)}(y) \}
  & =
  + i (2\pi)^{-p} \int \tfrac{1}{2 E(\vec k)} e^{i E(\vec k) (x-y)^0 + \vec k \cdot (\vec x - \vec y)} d^{p} \vec k
  \\
  & =
  + i (2\pi)^{-p} \int \delta( k_\mu k^\mu + m^2 ) \Theta( -k_0 ) e^{ - i k_\mu (x-y)^\mu } d^{p+1} k
  \\
  \end{aligned}
  \,.
$$

In particular this says that

$$
  \{ \phi^{(+)}(x), \phi^{(-)}(y) \}
  =
  - \omega(y,x)
  \coloneqq
  -
  \{ \phi^{(-)}(y), \phi^{(+)}(x) \}
  \,.
$$

With this we finally obtain the expression for the [[causal propagator]] as the skew-symmetrization
of the [[2-point function]]:

$$
  \begin{aligned}
  \{
    \phi(x),
    \phi(y)
  \}
  & =
  \{ \phi^{(-)}(x), \phi^{(+)}(y) \}
  +
  \{\phi^{(+)}(x), \phi^{(-)}(y)\}
  \\
  & =
  \omega(x,y) - \omega(y,x)
  \\
  & =
  -i (2\pi)^{-p} \int \tfrac{1}{2 E(\vec k)}\left(
    e^{- i E(\vec k) (x-y)^0 - \vec k \cdot (\vec x - \vec y)}
    -
    e^{+ i E(\vec k) (x-y)^0 + \vec k \cdot (\vec x - \vec y)}
  \right) d^p \vec k
  \\
  & =
  -i (2\pi)^{-p} \int \delta( k_\mu k^\mu + m^2 ) sgn( k_0 ) e^{ - i k_\mu (x-y)^\mu } d^{p+1} k
 \,.
  \end{aligned}
$$

=--

We record the 2-point function that appeared in this computation:

+-- {: .num_defn #2PointFunctionOfScalarFieldOnMinkowskiSpacetime}
###### Definition
**([[2-point function]] of [[scalar field]] on [[Minkowski spacetime]])**

The _[[2-point function]]_ or _[[Hadamard propagator]]_ of the scalar field on [[Minkowski spacetime]]

$$
  \omega \;\in\; \mathcal{D}'(\Sigma \times \Sigma)
$$

is  (eq:2PointFunctionFreeScalarFieldOnMinkowski)

$$
  \omega(x,y)
    \;\coloneqq\;
  i \{\phi^{(-)}(x), \phi^{(+)}(y)\}
  \,.
$$

=--

##### On general spacetimes

We discuss the [[covariant phase space]] of the [[free field|free]] scalar field on general [[spacetimes]].

+-- {: .num_defn #LocalLagrangianOfFreeScalarField}
###### Definition
**([[local Lagrangian density]] for [[free field|free]] [[scalar field]] on general [[spacetime]])**

As a [[classical field theory|classical]] [[local field theory|local field]] the _relativistic [[free field|free]] scalar field_ in [[dimension]] $p+1 \in \mathbb{N}$ of [[mass]] $m \in [0,\infty)$ is

* for each [[globally hyperbolic spacetime|globally hyperbolic]] [[orientation|oriented]] and [[time orientation|time-oriented]] [[spacetime]] $(\Sigma,e)$ ($\Sigma$ a [[smooth manifold]], $e$ a [[pseudo-orthogonal structure]]/[[vielbein]]  inducing a [[pseudo-Riemannian metric]] $g$)

1. the [[field bundle]] given by the [[trivial line bundle]] $X \times k \overset{pr_1}{\to} X$ over $X$;

1. the [[local Lagrangian density]] $L \in \Omega^{p+1,0}(J^\infty_X(X \times k))$ (a [[horizontal differential form]] on the [[jet bundle]] of the [[trivial line bundle]] over $X$) given by

   $$
     L \;\coloneqq\; \left( \vert \nabla \phi\vert^2 + m^2 \phi^2\right) dvol_ \Sigma
   $$

   where

   1. $\vert \nabla \phi\vert^2$ denotes the [[norm]] square of the first order jets with respect to the given [[metric]] $g$, hence in a local [[coordinate chart]] $J^\infty(E\vert_U) \coloneqq \left\{ \{x^\mu\}, \phi, \{\phi_{,\mu} \cdots\} \right\}$ of $J^\infty(X \times k)$ the function

      $$
        \vert \nabla \phi\vert^2\vert_{J^{\infty}(E\vert_U)}
          \;=\;
        g^{\mu \nu} \phi_{,\mu} \phi_{,\nu}
      $$

   1. $dvol$ denotes the [[volume form]] of $(\Sigma,e)$, canonically regarded as a [[horizontal differential form]] on $J^\infty(\Sigma \times k)$.

=--

The analogue of prop. \ref{IntegralKernelForPoissonBracketOfFreeScalarFieldOnMinkowskiSpacetime}
holds true for general spacetimes:

+-- {: .num_prop #FreeScalarFieldGreenFunctions}
###### Proposition
**([[advanced propagator]] and [[retarded propagator]])**

On a [[time orientation|time oriented]] [[globally hyperbolic spacetime]] the [[Klein-Gordon operator]] admits unique [[advanced propagator]] and [[retarded propagator]].

=--

(e.g. [Bär-Ginoux-Pfäffle 07, corollary 3.4.3](BaerGinouxPfaeffle07))

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

The induced [[Poisson bracket]] on the [[covariant phase space]] of the free scalar field (def. \ref{LocalLagrangianOfFreeScalarField}) is given by the [[Peierls bracket]]. (...)

=--

By [this prop.](Peierls+bracket#PeierlsPoissonBracket) (e.g. [Khavkine 14](Peierls+bracket#Khavkine14), [Collini 16, lemma 21](#Collini16)). See also [Fredenhagen-Rejzner 15, 3.3 Example](locally+covariant+perturbative+quantum+field+theory#FredenhagenRejzner15)

### Interacting scalar field

We discuss [[perturbative quantum field theory]] of the free scalar field perturbed by an [[interaction]] term
via [[locally covariant perturbative algebraic quantum field theory]].

#### On Minkowski spacetime

We disscuss the interacting scalar field on [[Minkowski spacetime]] via [[causal perturbation theory]].

By the discussion at _[[S-matrix]]_ we need to determine the [[Feynman propagator]], which
is a linear combination of the [[2-point function]] with the [[advanced propagator]]:

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

[[advanced propagator]]

$$
  \Delta_A(x.y) \coloneqq \theta((x-y)^0) \Delta(x,y)
$$

[[retarded propagator]]

$$
  \Delta_R(x.y) \coloneqq \theta((y-x)^0) \Delta(x,y)
$$

=--

## Related concepts

* [[phi^n-theory|$\phi^n$-theory]]

* [[pseudo-scalar field]]

* [[wave equation]], [[Klein-Gordon equation]]

* [[causal propagator]], [[Feynman propagator]]

* [[Wick algebra]]

\linebreak

## References

For instance

* {#Scharf01} [[Günter Scharf]], section 1.1 of _[[Quantum Gauge Theories -- A True Ghost Story]], Wiley 2001

Most of the literatur on [[causal perturbation theory]] and [[perturbative AQFT]] focuses on the scalar field, for ease of exposition.
See the references there.

The standard [[perturbative quantum field theory]] (made rigorous via [[causal perturbation theory]]) of the [[interaction|interacting]] scalar field is shown to be [[Fedosov deformation quantization]] of the corresponding [[covariant phase space]] in

* {#Collini16} [[Giovanni Collini]], section 2.2 of _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

For references on the construction of [[perturbative quantum field theory|perturbative]] scalar field theory in [[causal perturbation theory]] see at _[[locally covariant perturbative quantum field theory]]_.

Discussion of scalar fields in [[cosmology]] includes

* J.W. van Holten, _On single scalar field cosmology_ ([arXiv:1301.1174](http://arxiv.org/abs/1301.1174))

Examples in [[AQFT]] of [[non-perturbative quantum field theory|non-perturbative]] [[interacting quantum field theory|interacting]] [[scalar field theory]] in _any_ [[spacetime]] [[dimension]] (in particular in $d \geq 4$) are claimed in

* [[Detlev Buchholz]], [[Klaus Fredenhagen]], _A $C*$-algebraic approach to interacting quantum field theories_ ([arXiv:1902.06062](https://arxiv.org/abs/1902.06062))

[[!redirects scalar field]]
[[!redirects scalar fields]]

[[!redirects real scalar field]]
[[!redirects reak scalar fields]]

[[!redirects complex scalar field]]
[[!redirects complex scalar fields]]

[[!redirects scalar field theory]]
[[!redirects scalar field theories]]

[[!redirects scalar particle]]
[[!redirects scalar particles]]


scalar meson

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

#Contents#
* table of contents
{:toc}

## Idea

A type of [[meson]] which (in [[Wigner classification]]) is a [[Lorentz group]] [[scalar]] or [[pseudoscalar]] is called a _(pseudo-)scalar meson_, for short. This is in contrast to (pseudo-)[[vector mesons]], which instead are ([[pseudovector representation|pseudo-]])[[vector representations]]

## Examples

Examples of [[pseudo-scalar mesons]]:

* [[pion]]

* [[eta-meson]]

* [[kaon]]

Examples of [[scalar mesons]]:

* [[sigma-meson]]

\begin{imagefromfile}
    "file_name": "LightAndChiralPartnerMesonFieldsPinII.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": 0,
        "right": 10,
        "bottom": 0,
        "left": 20
    }
\end{imagefromfile}

## Related concepts

* [[vector meson]], [[tensor meson]]

## References

(See also the references at _[[meson]]_.)

* Eef van Beveren, George Rupp, _Scalar mesons as simple quark+antiquark states_, AIP Conf. Proc. 508 (1999) 310-317, Hadron Physics, Sept. 1999, Coimbra, Portugal ([arXiv:hep-ph/9912332](https://arxiv.org/abs/hep-ph/9912332))

* L. Albino, I. M. Higuera-Angulo, K. Raya, A. Bashir, *Pseudo-scalar mesons: light front wave functions, GPDs and PDFs* [[arXiv:2207.06550](https://arxiv.org/abs/2207.06550)]

[[!redirects scalar mesons]]

[[!redirects pseudoscalar meson]]
[[!redirects pseudoscalar mesons]]

[[!redirects pseudo-scalar meson]]
[[!redirects pseudo-scalar mesons]]


scale

> This entry is about scales in [[algebra]] and [[linear logic]]. For scales in [[geometry]] and [[physics]], see [[length scale]].

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The idea of a scale comes from [[Peter Freyd]], in his attempt to give an algebraic (in the sense of universal algebra) description of the real [[interval]], which he believed to be more fundamental than the [[real numbers]] themselves in [[analysis]], and of concepts from analysis such as [[Lipschitz continuity]], [[limits]], [[differentiation]] and [[differential equations]]. Freyd also discovered that scales are models of multiplicative-additive [[linear logic]] with a midpoint operation.

## Definition

A __scale__ is a [[minor scale]] $M$ satisfying the scale identities:

* for all $a$ and $b$ in $M$, $a \otimes b = (a^\vee \otimes b^\wedge) \vert (a^\wedge \otimes b^\vee)$

* for all $a$ and $b$ in $M$, $a \oplus b = (a^\wedge \oplus b^\vee) \vert (a^\vee \oplus b^\wedge)$

## Properties

Every scale with $\bot = \top$ is [[trivial object|trivial]].

As a scale is a [[closed midpoint algebra]], a scale has a [[partial order]] $\leq$. For all $a$ and $b$ in $M$, $a \leq b$ if and only if $a \multimap b = \top$.

A subset $\mathcal{I}$ of a scale $M$ is an __[[ideal]]__ if $\bot \in \mathcal{I}$, and $x \vee y \in \mathcal{I}$ if and only if $x \in \mathcal{I}$ and $y \in \mathcal{I}$. An ideal is a __zoom-invariant ideal__ if it is closed under $\bot$-zooming.

A subset $\mathcal{I}$ of a scale $M$ is a __$\bot$-face__ if $\bot \in \mathcal{I}$, and $x \vert y \in \mathcal{I}$ if and only if $x \in \mathcal{I}$ and $y \in \mathcal{I}$.

Every $\bot$-face is a zoom-invariant ideal.

Given an element $a$ in scale $M$, a __principal $\bot$-face__ $((a))$ is the subset of all $b$ in $M$ such that $(\bot\vert)^n b \leq a$ for all large $n$ in $\mathbb{N}$.

A __[[Jacobson radical]]__ of a scale $M$ is the set $J(M)$ of all $a$ in $M$ such that for all $n$ in $\mathbb{N}$, $x \leq (\bot\vert)^n \top$. $M$ is __semi-simple__ if $J(M)$ is [[trivial]].

+-- {: .standout}
The proof of the Linear Representation Theorem in section 8 of _Algebraic Real Analysis_ by Peter Freyd requires the use of [[excluded middle]] through its implicit definition of the [[quasiorder]] $\lt$ from the algebraically defined [[partial order]] $\leq$. In particular, that every scale is a [[*-autonomous category]] and thus a model for [[linear logic]] and that every equational axiom added to the theory of minor scales is either a consequence of the scale identity for scales or is inconsistent with the theory of minor scales are classical results, as certain lemmas used in the proofs have only been derived from the scale identities through the Linear Representation Theorem. The same is true of the definition of simple scales in section 10, of the algebraic construction of the standard interval $I$ from simple scales in section 11, and various results involving absolute retracts in section 25. Since quasiorders can be constructed from partial orders in any [[inequality space]], these results hold if the scales have a [[tight apartness relation]], but it is unknown if these results still hold for general scales in [[constructive mathematics]].
=--

## Examples

The [[unit interval]] with $a \vert b \coloneqq \frac{a + b}{2}$, $\odot = \frac{1}{2}$, $a^\bullet = 1 - a$, $\bot = 0$, $\top = 1$, $a^\wedge = max(2a-1,0)$, and $a^\vee = min(2a,1)$ is an example of a scale.

The set of truth values in Girard's [[linear logic]] is a scale.

## Related concepts

* [[minor scale]]

* [[Heyting scale]]

* [[linear logic]]

## References

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

[[!redirects scales]]


scale transformation

## Related concepts

* [[length scale]]

* [[conformal transformation]]

* [[scaling degree of a distribution]]

* [[renormalization]]

## References

* {#Duetsch18} [[Michael Dütsch]],
  def 3.19 of _[[From classical field theory to perturbative quantum field theory]]_, 2018

See also

* Wikipedia, _Scaling (geometry)_

[[!redirects scale transformations]]

[[!redirects scaling transformation]]
[[!redirects scaling transformations]]

[[!redirects rescaling]]
[[!redirects rescalings]]


scaling degree of a distribution

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _scaling degree_ or _degree of divergence_ ([Steinmann 71](#Steinmann71)) or more generally the _degree_ ([Weinstein 78](#Weinstein78)) of a [[distribution]] on [[Cartesian space]] $\mathbb{R}^n$  is a measure for how it behaves at the origin $0 \in \mathbb{R}^n$ under [[rescaling]] $x \mapsto \lambda x$ of the canonical [[coordinates]].

The concept controls the problem of [[extension of distributions]] from the [[complement]] $\mathbb{R}^n \setminus \{0\}$ of the origin to all of $\mathbb{R}^n$. Such extensions are important notably in the construction of [[perturbative quantum field theories]] via [[causal perturbation theory]], where the freedom in the choice of such extensions models the [[renormalization|("re"-)normalization]] freedom ("counter-terms") in the construction.

## Definition

+-- {: .num_defn #RescaledDistribution}
###### Definition
**(rescaled distribution)**

Let $n \in \mathbb{N}$. For $\lambda \in (0,\infty) \subset \mathbb{R}$ a [[positive number|positive]] [[real number]] write

$$
  \array{
    \mathbb{R}^n
      &\overset{s_\lambda}{\longrightarrow}&
    \mathbb{R}^n
    \\
    x &\mapsto& \lambda x
  }
$$

for the [[diffeomorphism]] given by multiplication with $\lambda$, using the canonical [[real vector space]]-structure of $\mathbb{R}^n$.

Then for $u \in \mathcal{D}'(\mathbb{R}^n)$ a [[distribution]] on the [[Cartesian space]] $\mathbb{R}^n$ the _rescaled distribution_ is the [[pullback of a distribution|pullback]] of $u$ along $m_\lambda$

$$
  u_\lambda
  \coloneqq
   s_\lambda^\ast u
  \;\in\;
  \mathcal{D}'(\mathbb{R}^n)
  \,.
$$

Explicitly, this is given by

$$
  \array{
     \mathcal{D}(\mathbb{R}^n)
       &\overset{ \langle  u_\lambda, - \rangle}{\longrightarrow}&
     \mathbb{R}
     \\
     b &\mapsto& \lambda^{-n} \langle u , b(\lambda^{-1}\cdot (-))\rangle
  }
  \,.
$$

Similarly for $X \subset \mathbb{R}^n$ an [[open subset]] which is invariant under $s_\lambda$, the rescaling of a distribution $u \in \mathcal{D}'(X)$ is is $u_\lambda \coloneqq s_\lambda^\ast u$.

=--

+-- {: .num_defn #ScalingDegree}
###### Definition
**([[scaling degree of a distribution]])**

Let $n \in \mathbb{N}$ and let $X \subset \mathbb{R}^n$ be an [[open subset]] of [[Cartesian space]] which is invariant under [[rescaling]] $s_\lambda$ (def. \ref{RescaledDistribution}) for all $\lambda \in (0,\infty)$, and let $u \in \mathcal{D}'(X)$ be a [[distribution]] on this subset. Then

1. The _[[scaling degree of a distribution|scaling degree]]_ of $u$ is the [[infimum]]

   $$
     sd(u)
     \;\coloneqq\;
     inf
     \left\{
       \omega \in \mathbb{R}
        \;\vert\;
       \underset{\lambda \to 0}{\lim}
        \lambda^\omega u_\lambda
        = 0
     \right\}
   $$

   of the set of [[real numbers]] $\omega$ such that the [[limit of a sequence|limit]] of the rescaled distribution $\lambda^\omega u_\lambda$ (def. \ref{RescaledDistribution}) vanishes. If there is no such $\omega$ one sets $sd(u) \coloneqq \infty$.

1. The _[[degree of divergence of a distribution|degree of divergence]]_ of $u$ is the difference of the scaling degree by the [[dimension]] of the underlying space:

  $$
    deg(u) \coloneqq sd(u) - n
     \,.
  $$

=--

## Examples

+-- {: .num_example #NonSingularDistributionsScalingDegree}
###### Example
**([[scaling degree of distributions|scaling degree]] of [[non-singular distributions]])**

If $u = u_f$ is a [[non-singular distribution]] given by [[bump function]] $f \in C^\infty(X) \subset \mathcal{D}'(X)$, then its [[scaling degree of a distribution|scaling degree]] (def. \ref{ScalingDegree}) is non-[[positive number|positive]]

$$
  sd(u_f) \leq 0
  \,.
$$

Specifically if the first non-vanishing [[partial derivative]] $\partial_\alpha f(0)$ of $f$ at 0 occurs at order ${\vert \alpha\vert} \in \mathbb{N}$, then the scaling degree of $u_f$ is $-{\vert \alpha\vert}$.

=--

+-- {: .proof}
###### Proof

By definition we have for $b \in C^\infty_{cp}(\mathbb{R}^n)$ any [[bump function]] that

$$
  \begin{aligned}
    \left\langle
      \lambda^{\omega}
      (u_f)_\lambda,
      n
    \right\rangle
    & =
    \lambda^{\omega-n}
    \underset{\mathbb{R}^n}{\int}
       f(x) g(\lambda^{-1} x)
    d^n x
    \\
    & =
    \lambda^{\omega}
    \underset{\mathbb{R}^n}{\int}
       f(\lambda x) g(x)
    d^n x
  \end{aligned}
  \,,
$$

where in last line we applied [[change of integration variables]].

The limit of this expression is clearly zero for all  $\omega \gt 0$, which shows the first claim.

If moreover the first non-vanishing [[partial derivative]] of $f$ occurs at order ${\vert \alpha \vert} = k$, then [[Hadamard's lemma]] says that $f$ is of the form

$$
  f(x)
    \;=\;
  \left( \underset{i}{\prod} \alpha_i ! \right)^{-1}
  (\partial_\alpha f(0))
  \underset{i}{\prod} (x^i)^{\alpha_i}
  +
  \underset{ {\beta \in \mathbb{N}^n} \atop { {\vert \beta\vert} = {\vert \alpha \vert} + 1 }  }{\sum}
  \underset{i}{\prod} (x^i)^{\beta_i}
  h_{\beta}(x)
$$

where the $h_{\beta}$ are [[smooth functions]]. Hence in this case

$$
  \begin{aligned}
    \left\langle
      \lambda^{\omega}
      (u_f)_\lambda,
      n
    \right\rangle
    & =
    \lambda^{\omega + {\vert \alpha\vert }}
    \underset{\mathbb{R}^n}{\int}
      \left( \underset{i}{\prod} \alpha_i ! \right)^{-1}
     (\partial_\alpha f(0))
     \underset{i}{\prod} (x^i)^{\alpha_i}
     b(x)
    d^n x
    \\
    & \phantom{=}
    +
    \lambda^{\omega + {\vert \alpha\vert} + 1}
    \underset{\mathbb{R}^n}{\int}
      \underset{i}{\prod} (x^i)^{\beta_i}
      h_{\beta}(x)
      b(x)
    d^n x
  \end{aligned}
  \,.
$$

This makes manifest that the expression goes to zero with $\lambda \to 0$ precisely for $\omega \gt - {\vert \alpha \vert}$, which means that

$$
  sd(u_f) = -{\vert \alpha \vert}
$$

in this case.

=--

+-- {: .num_example #DerivativesOfDeltaDistributionScalingDegree}
###### Example
**([[scaling degree of a distribution|scaling degree]] of [[derivative of a distribution|derivatives]] of [[delta-distributions]])**

Let $\alpha \in \mathbb{N}^n$ be a multi-index and $\partial_\alpha \delta \in \mathcal{D}'(X)$ the corresponding [[partial derivative|partial]] [[derivative of distributions|derivatives]] of the [[delta distribution]] $\delta_0 \in \mathcal{D}'(\mathbb{R}^n)$ [[support of a distribution|supported]] at $0$. Then the [[degree of divergence of a distribution|degree of divergence]] (def. \ref{ScalingDegree}) of $\partial_\alpha \delta_0$ is the total order the derivatives

$$
  deg\left(
    {\, \atop \,} \partial_\alpha\delta_0{\, \atop \,}
  \right)
  \;=\;
  {\vert \alpha \vert}
$$

where ${\vert \alpha\vert} \coloneqq \underset{i}{\sum} \alpha_i$.

=--

+-- {: .proof}
###### Proof

By definition we have for $b \in C^\infty_{cp}(\mathbb{R}^n)$ any [[bump function]] that

$$
  \begin{aligned}
    \left\langle
      \lambda^\omega (\partial_\alpha \delta_0)_\lambda,
      b
    \right\rangle
    & =
    (-1)^{{\vert \alpha \vert}}
    \lambda^{\omega-n}
    \left(
      \frac{
        \partial^{{\vert \alpha \vert}}
      }{
       \partial^{\alpha_1} x^1  \cdots \partial^{\alpha_n}x^n
      }
      b(\lambda^{-1}x)
    \right)_{\vert x = 0}
    \\
    & =
    (-1)^{{\vert \alpha \vert}}
    \lambda^{\omega - n - {\vert \alpha\vert}}
    \frac{
      \partial^{{\vert \alpha \vert}}
    }{
     \partial^{\alpha_1} x^1  \cdots \partial^{\alpha_n}x^n
    }
    b(0)
  \end{aligned}
  \,,
$$

where in the last step we used the [[chain rule]] of [[differentiation]]. It is clear that this goes to zero with $\lambda$ as long as $\omega \gt n + {\vert \alpha\vert}$. Hence $sd(\partial_{\alpha} \delta_0) = n + {\vert \alpha \vert}$.

=--

+-- {: .num_example #FeynmanPropagatorOnMinkowskiScalingDegree}
###### Example
**([[scaling degree of a distribution|scaling degree]] of [[Feynman propagator]] on [[Minkowski spacetime]])**

Let

$$
  \Delta_F(x)
  \;=\;
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu x^\mu}
  }{
    - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon
  }
  \, d k_0 \, d^p \vec k
$$

be the [[Feynman propagator]] for the massive [[free field|free]] [[real scalar field]] on $n = p+1$-dimensional [[Minkowski spacetime]] ([this prop.](Feynman+propagator#FeynmanPropagatorAsACauchyPrincipalvalue)). Its [[scaling degree of a distribution|scaling degree]] is

$$
  \begin{aligned}
    sd(\Delta_{F})
    & = n - 2
    \\
    & = p -1
  \end{aligned}
  \,.
$$

=--

([Brunetti-Fredenhagen 00, example 3 on p. 22](#BrunettiFredenhagen00))

+-- {: .proof}
###### Proof

Regarding $\Delta_F$ as a [[generalized function]] via the given [[Fourier transform of distributions|Fourier-transform]] expression, we find by [[change of integration variables]] in the Fourier integral that in the scaling limit the Feynman propagator becomes that for vannishing [[mass]], which scales homogeneously:

$$
  \begin{aligned}
    \underset{\lambda \to 0}{\lim}
    \left(
    \lambda^\omega
    \;
    \Delta_F(\lambda x)
    \right)
    & =
    \underset{\lambda \to 0}{\lim}
    \left(
    \lambda^{\omega}
    \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
    \frac{+i}{(2\pi)^{p+1}}
    \int
    \int_{-\infty}^\infty
    \frac{
       e^{i k_\mu \lambda x^\mu}
    }{
      - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon
    }
    \, d k_0 \, d^p \vec k
    \right)
    \\
    & =
    \underset{\lambda \to 0}{\lim}
    \left(
    \lambda^{\omega-n}
    \;
    \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
    \frac{+i}{(2\pi)^{p+1}}
    \int
    \int_{-\infty}^\infty
    \frac{
       e^{i k_\mu \lambda x^\mu}
    }{
      - (\lambda^{-2}) k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon
    }
    \, d k_0 \, d^p \vec k
    \right)
    \\
    & =
    \underset{\lambda \to 0}{\lim}
    \left(
    \lambda^{\omega-n + 2 }
    \;
    \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
    \frac{+i}{(2\pi)^{p+1}}
    \int
    \int_{-\infty}^\infty
    \frac{
       e^{i k_\mu \lambda x^\mu}
    }{
      - k_\mu k^\mu + i \epsilon
    }
    \, d k_0 \, d^p \vec k
    \right)
    \,.
  \end{aligned}
$$

=--

## Properties

+-- {: .num_prop #ScalingDegreeOfDistributionsBasicProperties}
###### Proposition
**(basic properties of [[scaling degree of distributions]])**

Let $X \subset \mathbb{R}^n$ and $u \in \mathcal{D}'(X)$ be a [[distribution]] as in def. \ref{RescaledDistribution}, such that its [[scaling degree of a distribution|scaling degree]] is finite: $sd(u) \lt \infty$ (def. \ref{ScalingDegree}). Then

1. For $\alpha \in \mathbb{N}^n$, the [[partial derivative|partial]] [[derivative of distributions]] $\partial_\alpha$ increases scaling degree at most by ${\vert \alpha\vert }$:

   $$
     deg(\partial_\alpha u) \;\leq\; deg(u) + {\vert \alpha\vert}
   $$

1. For $\alpha \in \mathbb{N}^n$, the [[product of distributions]] with the smooth coordinate functions $x^\alpha$ decreases scaling degree at least by ${\vert \alpha\vert }$:

   $$
     deg(x^\alpha u) \;\leq\; deg(u) - {\vert \alpha\vert}
   $$

1. Under [[tensor product of distributions]] their scaling degrees add:

   $$
     sd(u \otimes v) \leq sd(u) + sd(v)
   $$

   for $v \in \mathcal{D}'(Y)$ another distribution on $Y \subset \mathbb{R}^{n'}$;

1. $deg(f u) \leq deg(u) - k$ for $f \in C^\infty(X)$ and $f^{(\alpha)}(0) = 0$ for ${\vert \alpha\vert} \leq k-1$;

=--

([Brunetti-Fredenhagen 00, lemma 5.1](#BrunettiFredenhagen00), [Dütsch 18, exercise 3.34](#Duetsch18))

+-- {: .proof}
###### Proof

The first three statements follow with manipulations as in example \ref{NonSingularDistributionsScalingDegree} and example \ref{DerivativesOfDeltaDistributionScalingDegree}.

For the fourth...

=--

+-- {: .num_prop #ScalingDegreeOfProductDistribution}
###### Proposition
**([[scaling degree of distributions|scaling degree]] of [[product of distributions|product distribution]])**

Let $u,v \in \mathcal{D}'(\mathbb{R}^n)$ be two [[distributions]] such that

1. both have finite [[degree of divergence of a distribution|degree of divergence]] (def. \ref{ScalingDegree})

   $$
     deg(u), deg(v) \lt \infty
   $$

1. their [[product of distributions]] is well-defined

   $$
     u v \in \mathcal{D}'(\mathbb{R}^n)
   $$

   (in that their [[wave front sets]] satisfy [[Hörmander's criterion]])

then the product distribution has [[degree of divergence of a distribution|degree of divergence]] bounded by the sum of the separate degrees:

$$
  deg(u v)
  \;\leq\;
  deg(u) + deg(v)
  \,.
$$

=--

([Brunetti-Fredenhagen 00, special case of lemma 6.6](#BrunettiFredenhagen00))

## Related concepts

* [[support of a distribution]]

* [[order of a distribution]]

## References

The concept of scaling degree is due to

* {#Steinmann71} O. Steinmann, _Perturbation Expansions in Axiomatic Field Theory_, volume 11 of Lecture Notes in Physics, Springer, Berlin Springer Verlag, 1971.

and the more general concept of _degree_ due to

* {#Weinstein78} [[Alan Weinstein]], _The order and symbol of a distribution_, Trans. Amer. Math. Soc. 241, 1–54 (1978).

Review and further developments in the context of [[renormalization|("re"-)normalization]] in [[causal perturbation theory]]/[[pAQFT]] is in

* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], section 5.1 of _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661, 2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028))

* Dorothea Bahns, Michał Wrochna, _On-shell extension of distributions_ ([arXiv:1210.5448](https://arxiv.org/abs/1210.5448))

* {#Duetsch18} [[Michael Dütsch]], def. 3.32 of _[[From classical field theory to perturbative quantum field theory]]_, 2018

[[!redirects scaling degree of distributions]]

[[!redirects degree of divergence of a distribution]]
[[!redirects degree of divergence of distributions]]

[[!redirects degree of a distribution]]
[[!redirects degree of distributions]]


scallop decomposition

## Definition

([[Quasi-Coherent Sheaves and Tannaka Duality Theorems|LurieQC, def. 2.5.5]])

## Related concepts

* [[derived Deligne-Mumford stack]]

[[!redirects scallop decompositions]]


scattered space

+-- {: .rightHandSide}
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###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

[[!redirects scattered set]]
[[!redirects dense-in-itself subset]]
[[!redirects scattered spaces]]

#Contents#
* automatic table of contents goes here
{:toc}

## Idea
A **scattered space** is a [[topological space]] that on the scale of 'connectivity' is at the extreme opposite of [[perfect space|perfect spaces]].

Scattered spaces are used to provide topological models of [[provability logic]].

## Definition

A subset $A$ of a topological space is called _dense in itself_ if  each point of $A$ is a [[limit point]] of $A$.

A topological space $X$ is called _scattered_ if $X$ doesn't contain nonempty dense-in-itself subsets.

## Example

* The [[Sierpinski space]] $X=\{a,b\}$ with topology $\{\emptyset ,\{a\}, X\}$ is scattered: whereas $b$ is a limit point of $X$, $a$ isn't, whence $X$ fails to be dense in itself. $\{a\}$ fails to be dense in itself since, whereas $b$ is a limit point, $a$ isn't. Finally, $\{b\}$ has no limit points at all.

* Discrete spaces are scattered.

## Properties

* Every topological space $X$ is the disjoint union of a scattered subset and a [[perfect space|perfect]] subset.

## Remark

Given a space $X$, using the derivative $\partial : 2^X\to 2^X$ that maps a subset to the set of its limit points, the property of being dense-in-itself amounts to $A\subseteq\partial A$. Sierpiński ([1927](#Sierpinski27)) studies the abstract properties of dense-in-itself and corresponding scattered subsets relative to arbitrary monotone maps $2^X\to 2^X$.

## Related entries

* [[scattered topos]]

* [[perfect space]]

* [[Magari algebra]]

* [[provability logic]]

## Reference

* L. Beklemishev, D. Gabelaia, _Topological interpretations of provability logic_ , arXiv:1210.7317 (2012).  ([abstract](http://arxiv.org/abs/1210.7317))

* {#Sierpinski27} W. Sierpiński, _La notion de derivée comme base d'une théorie des ensembles abstraits_ , Math. Ann. **97** (1927) pp.321-337. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN235181684_0097%7CLOG_0017))

* S. Willard, _General Topology_ , Addison-Wesley Reading 1970. (Dover reprint 2004, p.219)


scattered topos

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

# Contents
* automatic table of contents goes here
{:toc}

##Idea
A **scattered topos** is the topos-theoretic analogue of a [[scattered space|scattered topological space]] and therefore provides a natural environment for an internal interpretation of [[provability logic]].

##Definition
A topos $\mathcal{E}$ is called $\bot\negthinspace$-scattered if the subtopos $Sh_{\neg\neg}(\mathcal{E})$ of [[double negation|double negation sheaves]] is an [[open subtopos]].

A topos $\mathcal{E}$ is called _scattered_ if every [[closed subtopos]] of $\mathcal{E}$ is $\bot$-scattered.

##Examples

Every [[Boolean topos]] $\mathcal{E}$ is $\bot$-scattered since it obviously coincides with $Sh_{\neg\neg}(\mathcal{E})$ and is [[open subtopos|open]] in itself (and [[closed subtopos|closed]] as well). Accordingly, $\mathcal{E}$ is scattered since subtoposes of Boolean toposes are Boolean.

A simple example of a non-Boolean scattered topos is the [[Sierpinski topos]] $Set^\rightarrow$ that consists of two copies of $Set$ [[Artin gluing|glued]] together such that one copy corresponding to $Sh_{\neg\neg}(Set^\rightarrow)$ is open and the other one is closed. Since $Set^\rightarrow$ and the closed copy of $Set$ are both $\bot$-scattered the claim follows. (For another simple example see at [[hypergraph]].)

##Properties

* That the Sierpinski-topos is $\bot$-scattered is an instance of the more general fact that _the topos $Sh(X)$ of sheaves on a $T_0$-space $X$ is $\bot$-scattered_ iff _open points are dense in $X$_ (cf. [Esakia-Jibladze-Pataraia 2000](#EJP00), p.101).

* A topos is scattered iff the (internal) Heyting algebra of [[Lawvere-Tierney topologies]] is Boolean (cf. [Esakia-Jibladze-Pataraia 2000](#EJP00), p.103). Since Boolean algebras are precisely the Heyting algebras where every element is complemented this says that a topos where all subtoposes are complemented is scattered.

* A spatial topos $Sh(X)$ is scattered iff every non degenerate subtopos has a [[point of a topos|point]] (cf. [Esakia-Jibladze-Pataraia 2000](#EJP00), p.103).

##Related entries

* [[scattered space]]

* [[provability logic]]

* [[double negation]]

* [[De Morgan topos]]

* [[locally closed subtopos]]

##References

The concept was introduced in

* {#EJP00} Leo Esakia, [[Mamuka Jibladze]], Dito Pataraia, _Scattered Toposes_ , APAL **103** (2000) pp.97-107.

For the wider context see also

* Leo Esakia, _Quantification with a provability smack_ , Bull. Sect. Log. **27**  (1998) pp.26-28. ([pdf](http://www.filozof.uni.lodz.pl/bulletin/pdf/27_12_6.pdf))


scattering

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###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Outline

The concept of _scattering_ refers to [[physics|physical]] process in which some matter or wave which has stable, say linear trajectory (or wave distribution) in distant past, comes into the area where it interacts with some localized perturbation (or other such waves), which results in different stable state (or distribution of states) in far future.  Examples include light scattering, scattering of matter beams, scatterring of solitons and multisolitons (which preserve their identity after a long while),
and scattering in quantum mechanics, including variants like QFT and superstring theory. The condition that the perturbation (interaction) is localized is someties relaxed (for example, in the case of the scattering for Schroedinger operator corresponding to the Coulomb potential, the waves are substantially perturbed even
asymptotically at infinity, because the Coulomb potential does not fall sufficiently fast far away from the source).

The basic concepts are time evolution operator, adiabatic switching, [[interaction picture]] and the ingoing and outgoing states.

### Direct and inverse problems of scattering

Scattering theory wants to predict the so called __scattering data__ which give the distribution of scattered waves (particles), i.e. of outgoing states in terms of ingoing states and knowledge of the __scattering interaction/potential__ on which the waves are scattered. This is the direct problem of scattering. Conversely, one can try to find out the potential/interaction by observing the results of scattering experiments, what boils to the knowledge of scattering data; hence finding the physics from knowledge of scattering data. Symbolically, fundamental physics experiments
in fact seek for the interaction which will correspond to scattering data. However,
when we try to do this literally, calculating the interaction potential from the scattering data, we call it the inverse scattering method. While in quantum mechanics, the inverse scattering method applies to a linear problem a variant applies to the study of nonlinear equations like nonlinear Schroedinger equation;
some cases lead to the integrability at classical or at quantized level. Thus
we have the [[classical inverse scattering method|classical]] and the [[quantum inverse scattering method]] within the subject of [[integrable systems]].

## Related concepts

* [[scattering amplitude]]

* [[scattering cross section]]

* [[scattering matrix]]

* [[Møller operator]]

* [[n-point function]], [[correlator]]

* [[Feynman diagram]]

* [[perturbation theory]]

* [[abstract scattering theory]]

## References

### General

Textbook account:

* [[Michael Reed]] and [[Barry Simon]]: *[[Methods of Modern Mathematical Physics]]*, Academic Press (1978), Volume III: *Scattering Theory* $[$[ISBN:9780080925387](https://www.elsevier.com/books/iii-scattering-theory/reed/978-0-08-092538-7)$]$

For more see the references at

* *[[S-matrix]]*

See also:

* Wikipedia: [scattering](http://en.wikipedia.org/wiki/Scattering), [scattering theory](http://en.wikipedia.org/wiki/Scattering_theory), [inverse scattering](http://en.wikipedia.org/wiki/Inverse_scattering), [Raman scattering](http://en.wikipedia.org/wiki/Raman_scattering), [Rayleigh scattering](http://en.wikipedia.org/wiki/Rayleigh_scattering),

[[!include classification of long-range forces -- references]]

category: physics

[[!redirects scattering theory]]

[[!redirects scattering process]]
[[!redirects scattering processes]]


scattering amplitude

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

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]] scattering amplitudes are the  [[probability amplitudes]] for processes of [[scattering]] of [[fundamental particles]] (or [[fundamental strings]] etc.) off each other. The collection of scattering amplitudes forms the _[[S-matrix]]_. In [[perturbative quantum field theory]] its contributions may be labeled by [[Feynman diagrams]], whence it is also called the _[[Feynman perturbation series]]_.

Of particular interest are [[vacuum amplitudes]] which are scattering amplitudes "where nothing external scatters" hence for no incoming and no outgoing states. The [[1-loop]] vacuum amplitudes are [[regularization (physics)|regularized]] [[traces]] over [[Feynman propagators]]. These are the incarnations of [[zeta functions]], [[L-functions]] and [[eta functions]] in [[physics]].

## Examples

### In Chern-Simons theory

The Feynman amplitudes of [[higher Chern-Simons theory]], such as [[AKSZ sigma-models]], regarded in their incarnation as [[Feynman amplitudes on compactified configuration spaces of points]], serve to exhibit a [[graph complex]]-model for the [[de Rham complex]] of [[Fulton-MacPherson compactifications]] of [[configuration spaces of points]] by the construction recalled [there](Fulton-MacPherson+operad#RelationToGraphComplexes). See the pointers at _[[Chern-Simons theory]]_ [here](Chern-Simons+theory#FeynmanPerturbationSeries).

### Of monopoles

See at _[[moduli space of monopoles]]_ the section _[Scattering amplitudes of monopoles](moduli+space+of+monopoles#ScatteringAmplitudesofMonopoles)_.

## Related concepts

* [[scattering]]

* [[scattering matrix]]

* [[scattering cross section]]

* [[leptonic decay]] (of [[mesons]]), [[Dalitz decay]]

* [[perturbative quantum field theory]]

* [[n-point function]], [[correlator]]

* [[Feynman diagram]]

* [[abstract scattering theory]]

* [[cluster decomposition]]

* [[renormalization]]

* [[on-shell recursion]], [[planar limit]]

* [[form factor (QFT)]]

* [[KLT relations]]

* [[amplituhedron]]

* [[celestial amplitude]]

* [[string theory results applied elsewhere]]

* [[string scattering amplitude]]

* [[motives in physics]]

* [[graph complex]], [[formality of the little n-disk operad]]

## References

### General

Introduction and review:

* [[Henriette Elvang]], [[Yu-tin Huang]], *Scattering Amplitudes*, Cambridge University Press (2015) [[arXiv:1308.1697](http://arxiv.org/abs/1308.1697), [doi:10.1017/CBO9781107706620]( https://doi.org/10.1017/CBO9781107706620)]

  > "The purpose of this review is to bridge the gap between a standard course in quantum field theory and recent fascinating developments in the studies of on-shell scattering amplitudes."

* Tomasz R. Taylor, _A Course in Amplitudes_ [[arXiv:1703.05670](https://arxiv.org/abs/1703.05670)]

* {#SummersBuchholz} Stephen J. Summers, [[Detlev Buchholz]], _Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools_ [[arXiv:math-ph/0509047] (https://arxiv.org/abs/math-ph/0509047)]

* Clifford Cheung, _TASI Lectures on Scattering Amplitudes_ [[arXiv:1708.03872](https://arxiv.org/abs/1708.03872)]

* Simon Badger, Johannes Henn, [[Jan Plefka]], Simone Zoia, *Scattering Amplitudes in Quantum Field Theory* [[arXiv:2306.05976](https://arxiv.org/abs/2306.05976)]

  > "These lecture notes bridge a gap between introductory quantum field theory (QFT) courses and state-of-the-art research in scattering amplitudes"

For related reference on the *[[S-matrix]]* see there, such as

* [[Sebastian Mizera]], *Physics of the Analytic S-Matrix* [[arXiv:2306.05395](https://arxiv.org/abs/2306.05395)]

A historical overview of the development of on-shell methods/analytic methods:

* {#Dixon13} [[Lance Dixon]], *Calculating Amplitudes*, (December 2013) [[web](http://www.preposterousuniverse.com/blog/2013/10/03/guest-post-lance-dixon-on-calculating-amplitudes/)]

Annual conference series:

* *[Amplitudes 2019](https://indico.cern.ch/event/750565/timetable/#20190701.detailed)*

* *[Aplitudes 2020](https://indico.cern.ch/event/908370/)*

* *[Amplitudes 2021](https://indico.nbi.ku.dk/event/1321/)*

* *[Amplitudes 2022](https://indico.cern.ch/event/1101193/)*

* *[Amplitudes 2023](https://indico.cern.ch/event/1228963/)*

### Analytic methods

* Ruth Britto, _Loop amplitudes in gauge theories: modern analytic approaches_ ([arXiv:1012.4493](http://arxiv.org/abs/1012.4493))

* [[Zvi Bern]], [[Lance Dixon]], [[David Kosower]], _On-Shell Methods in Perturbative QCD_ ([arXiv:0704.2798](http://arxiv.org/abs/0704.2798))

* [[Joseph Polchinski]], [[Matthew Strassler]], _Hard Scattering and Gauge/String Duality_, Phys.Rev.Lett.88:031601,2002, ([arXiv:hep-th/0109174](http://lanl.arxiv.org/abs/hep-th/0109174))

See also at _[[string theory results applied elsewhere]]_ and at _[[motivic multiple zeta values]]_.

### In super Yang-Mills theory

In [[super Yang-Mills theory]] (and there in particular in the [[planar limit]] of [[N=4 D=4 super Yang-Mills theory]]) scattering amplitudes enjoy special symmetry properties, some of which can be used to extract information about scattering amplitudes in non-supersymmetric theories (see also at _[[amplituhedron]]_):

* [[Nima Arkani-Hamed]], Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, [[Alexander Postnikov]], Jaroslav Trnka, _Scattering Amplitudes and the Positive Grassmannian_ ([arXiv:1212.5605](http://arxiv.org/abs/1212.5605))

* [[Nima Arkani-Hamed]], Jaroslav Trnka, _The Amplituhedron_, [arxiv/1312.2007](http://arxiv.org/abs/1312.2007)

### Classification of massless scattering

Classification of possible long-range [[forces]], hence of scattering processes of [[mass|massless]] [[field (physics)|fields]], by classification of suitably factorizing and decaying [[Poincaré group|Poincaré]]-[[invariant]] [[S-matrices]] depending on particle spin, leading to uniqueness statements about [[Maxwell theory|Maxwell]]/[[photon]]-, [[Yang-Mills theory|Yang-Mills]]/[[gluon]]-, [[gravity]]/[[graviton]]- and [[supergravity]]/[[gravitino]]-[[interactions]]:

* [[Steven Weinberg]], _Feynman Rules for Any Spin. 2. Massless Particles_, Phys. Rev. 134 (1964) B882 ([doi:10.1103/PhysRev.134.B882](https://doi.org/10.1103/PhysRev.134.B882))

* [[Steven Weinberg]], _Photons and Gravitons in $S$-Matrix Theory: Derivation of Charge Conservationand Equality of Gravitational and Inertial Mass_, Phys. Rev. 135 (1964) B1049 ([doi:10.1103/PhysRev.135.B1049](https://doi.org/10.1103/PhysRev.135.B1049))

* [[Steven Weinberg]], _Photons and Gravitons in Perturbation Theory: Derivation of Maxwell’s and Einstein’s Equations,” Phys. Rev. 138 (1965) B988 ([doi:10.1103/PhysRev.138.B988](https://doi.org/10.1103/PhysRev.138.B988))

* Paolo Benincasa, Freddy Cachazo, _Consistency Conditions on the S-Matrix of Massless Particles_ ([arXiv:0705.4305](https://arxiv.org/abs/0705.4305))

* David A. McGady, Laurentiu Rodina, _Higher-spin massless S-matrices in four-dimensions_, Phys. Rev. D 90, 084048 (2014) ([arXiv:1311.2938](https://arxiv.org/abs/1311.2938), [doi:10.1103/PhysRevD.90.084048](https://doi.org/10.1103/PhysRevD.90.084048))

Quick review:

* [[Daniel Baumann]], _What long-range forces are allowed?_, 2019 ([[BaumannLongRange19.pdf:file]])

### In string theory and higher supergravity

* Wieland Staessens, Bert Vercnocke, _Lectures on Scattering Amplitudes in String Theory_ ([arXiv:1011.0456](http://arxiv.org/abs/1011.0456))

* [[Michael Green]], _Properties of low energy graviton scattering amplitudes_, June 2010 ([pdf](http://string.lpthe.jussieu.fr/cargese/2010/talks/green_cargese2010.pdf))

For more see at _[[string scattering amplitude]]_.

### Motivic structure

Motivic structures in scattering amplitudes (see at _[[motives in physics]]_) are discussed for instance in

* John Golden, Alexander B. Goncharov, Marcus Spradlin, Cristian Vergu, Anastasia Volovich, _Motivic Amplitudes and Cluster Coordinates_ ([arXiv:1305.1617](http://arxiv.org/abs/1305.1617))

See also the references at _[[period]]_.

[[!redirects scattering amplitudes]]

[[!redirects Feynman amplitude]]
[[!redirects Feynman amplitudes]]

[[!redirects decay]]
[[!redirects decays]]

[[!redirects decay mode]]
[[!redirects decay modes]]


scattering cross section

+-- {: .rightHandSide}
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

## Related concepts

* [[scattering amplitude]]

* [[scattering matrix]]

## References

* Wikipedia, _Cross section (physics)_

[[!redirects scattering cross sections]]


SCFT

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#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[CFT]] with [[supersymmetry]], i.e. with [[superconformal geometry|superconformal symmetry]]

[[!include superconformal symmetry -- table]]

## Examples

### $d = 2$

* [[2d SCFT]]

  * [[NSR string]]

  * [[heterotic string]], [[2d (2,0)-superconformal QFT]]

### $d = 3$

* [[3d superconformal gauge field theory]]

### $d = 4$

* [[N=4 D=4 super Yang-Mills theory]]

### $d = 6$

* [[D=6 N=(2,0) SCFT]]

* [[D=6 N=(1,0) SCFT]]

## Applications

* [[mirror symmetry]]

## Related concepts

* [[superconformal geometry]]

* [[supersymmetric sigma-model]]

* [[AdS/CFT correspondence]]

## References

### General

Survey of [[SCFTs]] in higher dimensions ([[D=4 SYM]], [[D=5 SYM]], [[D=6 N=(1,0) SCFT|D=6 N=1]], [[D=6 N=(2,0) SCFT|D=6, N=2]]) their [[moduli]] ([[Higgs branches]]/[[Coulomb branches]]) and their [[duality in string theory|dualities]] ([[KK-compactification]], [[AGT correspondence]], [[AdS/CFT duality]], ...):

* [[Shlomo Razamat]], *News from the higher $\mathcal{D}$ SQFT frontier*, talk at *[[Strings 2022]]* [[indico:4940856](https://indico.cern.ch/event/1085701/contributions/4940856), [slides](https://indico.cern.ch/event/1085701/contributions/4940856/attachments/2482369/4261653/Shlomo%20Razamat%20Tuesday-Review-Razamat.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=ad1f1a6b-5018-4b69-9e38-e499daabda1c)]

### Classification
 {#Classification}

* {#Nahm78} [[Werner Nahm]], _[[Supersymmetries and their Representations]]_, Nucl.Phys. B135 (1978) 149 ([spire](https://inspirehep.net/record/120988/), [pdf](http://cds.cern.ch/record/132743/files/197709213.pdf))

* {#Shnider88} [[Steven Shnider]], _The superconformal algebra in higher dimensions_, Letters in Mathematical Physics November 1988, Volume 16, Issue 4, pp 377-383

* [[Shiraz Minwalla]], _Restrictions imposed by superconformal invariance on quantum field theories_ Adv. Theor. Math. Phys. 2, 781 (1998)
([arXiv:hep-th/9712074](http://arxiv.org/abs/hep-th/9712074)).

* [[Clay Cordova]], [[Thomas Dumitrescu]], [[Kenneth Intriligator]], _Deformations of Superconformal Theories_, ([arXiv:1602.01217](https://arxiv.org/abs/1602.01217))

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