Philosophy of Nature

> under construction

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this page is to eventually collect material on

* {#EPSBOI} [[Georg Hegel]],

  _Philosophie der Natur_

  in

  _Enzyklopädie der philosophischen Wissenschaften im Grundrisse_

  (1830)

  in W. Bonsiepen und H.-C. Lucas (eds.), _Gesammelte Werke_, Rheinisch-Westfälischen Akademie der Wissenschaften, and xx. Hamburg: Felix Meiner, 1992 ( Bonsiepen/Lucas 1992)

on, roughly, [[natural philosophy]]. For the moment see at

* _[Science of Logic -- Philosophy of Nature](Science+of+Logic#PhilosophyOfNature)_.

## Some material...

> {#202} §202 Other mathematical determinations, such as infinity and its relationships, the infinitesimal, factors, powers, and so on, have their true concepts in philosophy itself. It is awkward to want to take and derive these from mathematics, where they are employed in a nonconceptual, often meaningless way; rather, they must await their justification and significance from philosophy.

> §202 The truly philosophical science of mathematics as theory of magnitude would be the science of measures, but this already presupposes the real particularity of things, which is only at hand in concrete nature.

Compare to [Science of Logic §709](Sciene+of+Logic#709):

> SoL §709 The development of measure which has been attempted in the following chapters is extremely difficult. Starting from immediate, external measure it should, on the one hand, go on to develop the abstract determination of the quantitative aspects of natural objects (a mathematics of nature), and on the other hand, to indicate the connection between this determination of measure and the qualities of natural object

## References

Beware that the first version greatly differs from the third. English translations of the final version are in

* A. V. Miller, Oxford Clarendon Press 1970

* M. J. Petry, London 1970

for review of these see [jstor/3750018](http://www.jstor.org/stable/3750018)

category: reference


philosophy of physics

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## Idea

The [[philosophy]] of [[physics]].

## Sentiments

* [[universal exceptionalism]]

* [[empiricism]]

* [[idealism]]

## Related concepts

* [[coordination]], [[measurement problem]]

* [[natural philosophy]]

* [[metaphysics]]

* [[philosophy of science]]

  * [[philosophy of mathematics]]

## References

### General

* Wikipedia, _[Philosophy of physics](http://en.wikipedia.org/wiki/Philosophy_of_physics)_

* [[Georg Hegel]], _[[Philosophy of Nature]]_

* {#ButterfieldList} [[Jeremy Butterfield]], _[Philosophy of physics: a short reading list](http://www.hps.cam.ac.uk/research/pp.html)_

* [[Lawrence Sklar]], _Philosophy of Physics_, Oxford University Press (1992)

* [[Roland Omnès]], _[[Converging Realities -- Toward a common philosophy of physics and mathematics]]_, Princeton University Press, 2005

* {#Albert12} [[David Albert]], _[On the origin of Everything](http://www.nytimes.com/2012/03/25/books/review/a-universe-from-nothing-by-lawrence-m-krauss.html?_r=1&adxnnl=1&adxnnlx=1415727315-pCXR9NUmWeBxo+FViR+NlA)_, 2012

* [[Gabriel Catren]] et al. (org.) _Philosophy of Mechanics: Mathematical Foundations_, Workshop Feb 12-14, 2014 ([website](http://phil.physico-math.gie.im), [poster](http://phil.physico-math.gie.im/static/PhilosophyofMechanics25size.jpg))

### Interpretation of quantum mechanics

Reference specifically on the [[interpretation of quantum mechanics]] include

* [[Roland Omnès]], _[[The Interpretation of Quantum Mechanics]]

* Valia Allori (2017). *A New Argument for the Nomological Interpretation of the Wave Function: The Galilean Group and the Classical Limit of Nonrelativistic Quantum Mechanics.* ([philsci-archive:14023](https://philsci-archive.pitt.edu/14023/))


philosophy of science

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## Idea

[[philosophy]] of science

* [[philosophy of physics]]

* [[philosophy of mathematics]]

## Sentiments

* [[universal exceptionalism]]

* [[empiricism]]

* [[idealism]]

* [[structuralism]]

### Research programs in science

* [[Thomas Kuhn]]

* [[Imre Lakatos]]

* [[paradigm]]

* [[paradigm shift]]

This entire subject about past research programs, paradigms in science and paradigm shifts could be expanded on in the nLab. Examples include the shift from Newtonian spacetime to Minkowski spacetime in the early 20th century in physics, the shift from geocentrism to heliocentrism in the 17th century, and the shift to galactocentrism and finally to acentrism in the early 20th century in cosmology, et cetera.

## References

* {#Friedman01} [[Michael Friedman]], _The idea of scientific philosophy_, chapter 1 in  _Dynamics of Reason_, Chicago 2001 ([chapter pdf](http://web.stanford.edu/group/cslipublications/cslipublications/pdf/1575862921.pdf), [publisher webpage](http://press.uchicago.edu/ucp/books/book/distributed/D/bo3634648.html))

* [[David Corfield]], _[[davidcorfield:Friedman's Dynamics of Reason]]_

* {#Ladyman01} [[James Ladyman]], _Understanding Philosophy of Science_, 2001  ([pdf](http://srbiau.ac.ir/Files/Ladyman,%20Understanding%20Philosophy%20of%20Science.pdf), [review](http://www.jstor.org/stable/3093608))


philosophy resources

* [Stanford Encyclopedia of Philosophy](http://plato.stanford.edu)

## Related entries

* [[mathematics resources]] -- list for mathematics resources of $n$Lab

* [[physics resources]] -- list for physics resources of $n$Lab

* [[philosophy]], [[logic]]


phonon

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## Idea

By *phonons* one refers to the [[quanta]] of [[sound]] [[waves]] in [[condensed matter]], such as in a [[crystal]].

Compare to *[[photons]]*, which are the [[quanta]] of [[light]] [[waves]].

## Related concepts

* [[phononic crystal]]

## References

See also:

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

On [[AdS-CMT correspondence|holographic]] description of phonon gases in non-merallic [[crystals]]:

* Xiangqing Kong, Tao Wang, Liu Zhao, *High temperature AdS black holes are low temperature quantum phonon gases* [[arXiv:2209.12230](https://arxiv.org/abs/2209.12230)]

[[!redirects phonons]]


phononic crystal

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## Idea

A *phononic crystal* or *acoustic crystal* is a material with periodic elastic properties in which [[sound]] [[waves]] (ultimately: [[phonons]]) behave much like [[Bloch waves]] of [[electrons]] inside an actual [[crystal]], such as in that they form [[energy bands]] separated by band gaps ([KHDD93](#KHDD93)).

By suitably engineering phononic crystals they may emulate various phenomena seen elsewhere in nature or not seen anywhere else, whence one also speaks of "[[metamaterial|meta-materials]]".

In much the same way, there are *[[phononic crystals]]*.

For instance, various aspects of [[topological phases of matter]] (such as
[[Berry phases]], see [Liu, Chen & Xu 2020](#LiuChenXu20);
or
[[phononic semimetals]], see [Li et al. 2022](#LiEtAl21)) have analogs realizations in phononic crystals ("topological phononics", see the references [below](#ReferencesTopologicalPhononics)). Notably  [[braiding]] of [[band nodes]] has been demonstrated in [[phononic crystals]] ([Peng, Bouhon, Monserrat & Slager 2022](#PBMS22), [Peng, Bouhon, Slager & Monserrat 2022](#PBSM22)) and similarly in
[[photonic semimetals]] ([Park, Gao, Zhang & Oh 2022](#ParkGaoZhangOh22)).

## References

### General

Early articles on acoustic [[energy band structure]] in phononic crystals:

* {#KHDD93} M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, *Acoustic band structure of periodic elastic composites*, Phys. Rev. Lett. **71** 2022 (1993) [[doi:10.1103/PhysRevLett.71.2022](https://doi.org/10.1103/PhysRevLett.71.2022)]

* M. M. Sigalas, E. N. Economou, *Elastic and acoustic wave band structure*, Journal of Sound and Vibration **158** 2 (1992) 377-382 [doi:10.1016/0022-460X(92)90059-7]

Review of phononic crystals:

* Ming-Hui Lu, Liang Feng, Yan-Feng Chen, *Phononic crystals and acoustic metamaterials*, Materials Today **12** 12 (2009) 34-42 [doi:10.1016/S1369-7021(09)70315-3]

* {#LiuChenXu20} Yizhou Liu, Xiaobin Chen, Yong Xu, *Topological Phononics: From Fundamental Models to Real Materials*, Advanced Functional Materials **30** 8 (2020) [[doi:10.1002/adfm.201904784](https://doi.org/10.1002/adfm.201904784)]

Further resources:

* *[Phononics 2017](https://phononics2017.phononics.org/conf/index.php/phononics/2017/index.html)* -- The 4th International Conference on Phononic Crystals/Metamaterials, Phonon Transport/Coupling and Topological Phononics (2017)

* *[Phononics 2019](https://phononics2019.phononics.org/conf/index.php/phononics/2019/index.html)* -- 5th International Conference on Phononic Crystals/Metamaterials, Phonon Transport, Topological Phononics (2019)

* [META 2022](https://metaconferences.org/META/index.php/META2022) -- 12th International Conference on Metamaterials, Photonic Crystals and Plasmonics

* [Iwamoto Laboratory -- Topological Photonics and Topological Phononics](http://www.iwamoto.iis.u-tokyo.ac.jp/en/detail3.html)

### Topological phononics

On [[topological phononics]]:

* S. Huber, *Topological mechanics*, Nature Phys **12*** 621–623 (2016) [[doi:10.1038/nphys3801](https://doi.org/10.1038/nphys3801)]

* {#LiEtAl21} J. Li, J. Liu, S.A. Baronett, *Computation and data driven discovery of topological phononic materials*, Nat Commun **12** 1204 (2021) [[doi:10.1038/s41467-021-21293-2](https://doi.org/10.1038/s41467-021-21293-2)]

* {#LiuChenXu20} Yan Du, Weiguo Wu, Wei Chen, Yongshui Lin, Qingjia Chi: *Control the structure to optimize the performance of sound absorption of acoustic metamaterial: A review*, AIP Advances **11** 060701 (2021) [[doi:10.1063/5.0042834](https://doi.org/10.1063/5.0042834)]

The phononic ananlog of [[topological insulators]]/[[quantum Hall effect]]:

* Roman Süsstrunk, Sebastian D. Huber, *Observation of phononic helical edge states in a mechanical 'topological insulator'*, Science **349** 47 (2015) [[arXiv:1503.06808](https://arxiv.org/abs/1503.06808)]

[[!include anyonic braiding in momentum space -- references]]

[[!redirects phononic crystals]]

[[!redirects phononic semimetal]]
[[!redirects phononic semimetals]]

[[!redirects topological phononics]]


photon

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## Idea

The _photon_ is the [[gauge boson]] of the [[electromagnetic field]], hence of _[[electromagnetic radiation]]_ and of _[[light]]_.

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

Compare to *[[phonons]]* which are the [[quanta]] of [[sound]] [[waves]] in [[condensed matter]].

## Related concepts

* [[photon propagator]]

* [[laser]], [[quantum optics]]

## Refereces

### General

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

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

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

[[!redirects photons]]


photon propagator

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## Idea

Considering the [[Lagrangian field theory]] of the [[free field theory]] [[electromagnetic field]], the corresponding [[Feynman propagator]] $\Delta_F(x,y)$ describes the [[time-ordered product]] of linear observables of the [[vector potential]] $A$ and may be thought of as the [[probability amplitude]] for a [[photon]] to travel from $x$ to $y$. Therefore this is often called the _photon propagator_.

For details see at _[[A first idea of quantum field theory]]_ [this prop.](A+first+idea+of+quantum+field+theory#PhotonPropagatorInGaussianAveragedLorenzGauge).

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

## Related concepts

* [[electron propagator]]

## References

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

[[!redirects photon propagator]]


photonic crystal

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## Idea

A *photonic crystal* ([Benisty et a. 1999](#BenistyEtAl99), going back to [Yablonovitch 1987](#Yablonovitch87), [John 1987](#John87)) (often: a *waveguide array*, see [Trompeter et al. 2003](#TrompeterEtAl03)) is a material with periodic dielectric structure/refractive index in which [[light waves]] behave like [[Bloch waves]] of [[electrons]] in an actual [[crystal]], such as in that they form [[energy bands]] separated by band gaps ([Yablonovitch 1987](#Yablonovitch87)). (Of course, the underlying material may itself consist of actual [[crystal|crystalline]] structures, such as [[graphene]], see e.g. [BBKKL10](#BBKKL10)).

By suitably engineering photonic crystals they may emulate various phenomena seen elsewhere in nature (for instance transport properties analogous to those in [[semiconductors]]) or not seen anywhere else, whence one also speaks of "[[metamaterial|meta-materials]]".

In much the same way, there are *[[phononic crystals]]*.

For instance, many aspects of [[topological phases of matter]] ([[topological insulators]], [[semimetals]], ...) have analogs realizations in photonic crystals ("[[topological photonics]]", see the references [below](#ReferencesTopologicalPhotonics)). Notably [[nodal lines]] have been demonstrated in [[photonic semimetals]] ([Park, Wong, Zhang& Oh 2021](#ParkWongZhangOh21), [Park, Gao, Zhang & Oh 2022](#ParkGaoZhangOh22)) and similarly for [[phonons]] ([Peng, Bouhon, Monserrat & Slager 2022](#PBMS22)).

## Related concepts

* [[crystal]]

* [[phononic crystal]]

## References

### General

Original articles:

* {#Yablonovitch87} Eli Yablonovitch, *Inhibited Spontaneous Emission in Solid-State Physics and Electronics*, Phys. Rev. Lett. **58** 2059 (1987) [[doi:10.1103/PhysRevLett.58.2059](https://doi.org/10.1103/PhysRevLett.58.2059)]

* {#John87} Sajeev John, *Strong localization of photons in certain disordered dielectric superlattices* Phys. Rev. Lett. **58** 2486 (1987) [[doi:10.1103/PhysRevLett.58.2486](https://doi.org/10.1103/PhysRevLett.58.2486)]

Further articles:

* Thomas F. Krauss, Richard M. De La Rue,  Stuart Brand, *Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths*, Nature **383** (1996) 699–702 [[doi:10.1038/383699a0](https://doi.org/10.1038/383699a0)]

* {#BenistyEtAl99} H. Benisty et al., *Optical and confinement properties of two-dimensional photonic crystals*, Journal of Lightwave Technology **17** 11 (1999) 2063-2077 [[doi:10.1109/50.802996](https://doi.org/10.1109/50.802996)]

* D. N. Chigrin & C. M. Sotomayor Torres, *Periodic thin-film interference filters as one-dimensional photonic crystals*, Optics and Spectroscopy **91** (2001) 484–489 [[doi:10.1134/1.1405232](https://doi.org/10.1134/1.1405232)]

Early history:

* David Lindley, *Landmarks -- The Birth of Photonic Crystals*,  Physics **6** 94 (2013) [[physics:v6/94](https://physics.aps.org/articles/v6/94)]

Review of experimental realizations:

* Steven G. Johnson, *Photonic Crystals: Periodic Surprises in Electromagnetism*, lecture notes (2003) [[webpage](http://ab-initio.mit.edu/photons/tutorial/), [pdf](http://ab-initio.mit.edu/photons/tutorial/L3-fab.pdf)]

and for waveguide arrays:

* {#TrompeterEtAl03} Henrike Trompeter et al., *Tailoring guided modes in waveguide arrays*, **11** 25 (2003) 3404-3411 [[doi:10.1364/OE.11.003404](https://doi.org/10.1364/OE.11.003404)]

Further review:

* Ivan L.Garanovicha, Stefano Longhib, Andrey A.Sukhorukova, Yuri S. Kivshar, *Light propagation and localization in modulated photonic lattices and waveguides*, Physics Reports **518** 1–2 (2012) 1-79 [[doi:10.1016/j.physrep.2012.03.005](https://doi.org/10.1016/j.physrep.2012.03.005)]

On the effective appearance of the [[Schrödinger equation]] for [[electromagnetic waves]] in photonic crystals:

* Max Lein, *Taking Inspiration from Quantum-Wave Analogies --- Recent Results for Photonic Crystals*, in Springer Proceedings in Mathematics & Statistics **270** (2018) [[arXiv:1804.01234](https://arxiv.org/abs/1804.01234), [doi:10.1007/978-3-030-01602-9_10](https://link.springer.com/chapter/10.1007/978-3-030-01602-9_10)]

Textbook account:

* John D. Joannopoulos, Steven G. Johnson, Joshua N. Winn, and Robert D. Meade, *Photonic Crystals: Molding the Flow of Light* [[webpage](http://ab-initio.mit.edu/book/), [ISBN:9780691124568](https://press.princeton.edu/books/hardcover/9780691124568/photonic-crystals)]

Description of photonic crystals by a [[non-linear Schrödinger equation]]:

* A. Pankov, *Periodic Nonlinear Schrödinger Equation with Application to Photonic Crystals*,  Milan J. Math. **73** (2005) 259–287 [[doi:10.1007/s00032-005-0047-8](https://doi.org/10.1007/s00032-005-0047-8)]

* Anatoli Babin, Alexander Figotin, *Nonlinear Photonic Crystals: IV. Nonlinear Schrödinger Equation Regime* **15** (2005) 145-228  [[arXiv:math-ph/0409079](https://arxiv.org/abs/math-ph/0409079), [doi:10.1080/17455030500196929](https://doi.org/10.1080/17455030500196929)]

* Waleed Alrefai, *Schrödinger equation for propagation in photonic crystal fibers* Eureka Physics & Engineering **1** (2016) [[doi:10.21303/2461-4262.2016.00021](https://doi.org/10.21303/2461-4262.2016.00021)]

Realization with [[graphene]]:

* {#BBKKL10} Oleg L. Berman, Vladimir S. Boyko, Roman Ya. Kezerashvili, Anton A. Kolesnikov, Yurii E. Lozovik, *Graphene-based photonic crystal*, Physics Letters A **374** (2010) 4784-4786 [[arXiv:1012.4143](https://arxiv.org/abs/1012.4143), [doi:10.1016/j.physleta.2010.09.064](https://doi.org/10.1016/j.physleta.2010.09.064)]

Photonic analog *of* [[graphene]]:

* *Observation of unconventional edge states in ‘photonic graphene’*, Nature Materials **13** (2014) 57–62 [[doi:10.1038/nmat3783](https://doi.org/10.1038/nmat3783)]

See also:

* Wikipedia, *[Photonic crystal](https://en.wikipedia.org/wiki/Photonic_crystal)*

Claim of photonic [[hyperbolic tesselations]]:

* Alicia J. Kollár, Mattias Fitzpatrick, Andrew A. Houck, *Hyperbolic Lattices in Circuit Quantum Electrodynamics*, Nature **571**  (2019) 45–50 [[arXiv:1802.09549](https://arxiv.org/abs/1802.09549), [doi:10.1038/s41586-019-1348-3](https://doi.org/10.1038/s41586-019-1348-3)]

### Topological photonics
 {#ReferencesTopologicalPhotonics}

Original idea of [[topological photonics]]:

* [[Duncan Haldane]], S. Raghu, *Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry*, Phys. Rev. Lett. **100** (2008) 013904 [[doi:10.1103/PhysRevLett.100.013904](https://doi.org/10.1103/PhysRevLett.100.013904)]

#### Solitons
 {#ReferencesSolitons}

On [[soliton|solitonic]] excitations in waveguide arrays:

* Nikolaos K. Efremidis  et al., *Spatial photonics in nonlinear waveguide arrays*, Optics Express **13** 6 (2005) 1780-1796 [[doi:10.1364/OPEX.13.001780](https://doi.org/10.1364/OPEX.13.001780)]

* Thawatchai Mayteevarunyoo, Boris A. Malomed, *Solitons in one-dimensional photonic crystals*, Journal of the Optical Society of America B **25** 11 (2008) 1854-1863 [[doi:10.1364/JOSAB.25.001854](https://doi.org/10.1364/JOSAB.25.001854)]

* [[Amaria Javed]], Alaa Shaheen U. Al Khawaja, *Amplifying optical signals with discrete solitons in waveguide arrays*, Physics Letters A **384** 26 (2020) 126654 [[doi:10.1016/j.physleta.2020.126654](https://doi.org/10.1016/j.physleta.2020.126654)]

* Alaa Shaheen, [[Amaria Javed]], U. Al Khawaja, *Adding binary numbers with discrete solitons in waveguide arrays*, Phys. Scr. **95** (2020) 085107 [[arXiv:2108.01406](https://arxiv.org/abs/2108.01406), [doi:10.1088/1402-4896/aba2b2](https://doi.org/10.1088/1402-4896/aba2b2)]

#### Photonic topological phases

On photonic analogs of the [[quantum Hall effect]]:

* Zheng Wang, Yidong Chong, J. D. Joannopoulos, Marin Soljačić, *Observation of unidirectional backscattering-immune topological electromagnetic states*, Nature **461** (2009) 772–775 [[doi:10.1038/nature08293](https://doi.org/10.1038/nature08293)]

On photonic analogs of [[topological insulators]] ([[photonic topological insulator|photonic topological insulators]]):

* *Photonic topological insulators*, Nature Materials **12** (2013) 233-239 [[doi:10.1038/nmat3520](https://doi.org/10.1038/nmat3520)]

On photonic analogs of [[topological semimetals]] ([[photonic topological semimetal|photonic topological semimetals]]):

* Shuqi sehn et al., *A Review of Topological Semimetal Phases in Photonic Artificial Microstructures*, Front. Phys., **16** [[doi:10.3389/fphy.2021.771481](https://doi.org/10.3389/fphy.2021.771481)]

* Ruey-Lin Chern and You-Zhong Yu, *Photonic topological semimetals in bigyrotropic metamaterials*,  **30** 14 (2022) 25162-25176 [[doi:10.1364/OE.459097](https://doi.org/10.1364/OE.459097)]

Photonic analog of [[nodal lines]]:

* Shuang Zhang et al., *Experimental observation of photonic nodal line degeneracies in metacrystals*, Nature Communications **9** 950 (2018) [[doi:10.1038/s41467-018-03407-5](https://doi.org/10.1038/s41467-018-03407-5)]

Movement of [[nodal points]] in photonic crystals:

* Yong-Heng Lu et al., *Observing movement of Dirac cones from single-photon dynamics*, Phys. Rev. B **103** 064304 (2021) [[doi:10.1103/PhysRevB.103.064304](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.064304)]

* María Blanco de Paz, Alejandro González-Tudela, Paloma Arroyo Huidobro, *Manipulating Generalized Dirac Cones In Quantum Metasurfaces* [[arXiv:2203.11195](https://arxiv.org/abs/2203.11195)]

On something like [[anyon]] [[braiding]] in photonic crystals:

* {#ZhangEtAl22} Xu-Lin Zhang et al., *Non-Abelian braiding on photonic chips*, Nature Photonics **16** (2022) 390–395 [[arXiv:2112.01776](https://arxiv.org/abs/2112.01776), [doi:10.1038/s41566-022-00976-2](https://doi.org/10.1038/s41566-022-00976-2)]

See also:

* Hui Liu et al., *Topological phases and non-Hermitian topology in photonic artificial microstructures*, Nanophotonic (Feb. 2023) [[arXiv:10.1515/nanoph-2022-0778](https://doi.org/10.1515/nanoph-2022-0778)]

#### Topological lasers

On [[topological lasers]]:

* Babak Bahari, Abdoulaye Ndao, Felipe Vallini, Abdelkrim El Amili, Yeshaiahu Fainman, Boubacar Kanté, *Nonreciprocal lasing in topological cavities of arbitrary geometries*, Science
 **358** 6363 (2017) 636-640 [[doi:10.1126/science.aao4551](https://www.science.org/doi/10.1126/science.aao4551), [pdf](https://triontech.com/wp-content/uploads/2017/12/grandcentr_5a33e9da82b1d2045611513355242.pdf)]

* Gal Harari, Miguel A. Bandres, Yaakov Lumer, Mikael C. Rechtsman, Y. D. Chong, Mercedeh Khajavikhan, Demetrios N. Christodoulides, Mordechai Segev, *Topological insulator laser: Theory*, Science **359** 6 (2018) [[doi:10.1126/science.aar4003](https://doi.org/10.1126/science.aar4003), [pdf](https://dr.ntu.edu.sg/bitstream/10356/90137/1/Topological%20insulator%20laser-Theory.pdf)]

* Natsuko Ishida*, Yasutomo Ota*, Wenbo Lin, [[Tim Byrnes]], Yasuhiko Arakawa and Satoshi Iwamoto, *A large-scale single-mode array laser based on a topological edge mode*, Nanophotonics **11** 9 (2022) 2169–2181 [[doi:10.1515/nanoph-2021-0608](https://doi.org/10.1515/nanoph-2021-0608)]

[[!include anyonic braiding in momentum space -- references]]

[[!redirects photonic crystals]]

[[!redirects metamaterial]]
[[!redirects metamaterials]]

[[!redirects waveguide array]]
[[!redirects waveguide arrays]]

[[!redirects photonic semimetal]]
[[!redirects photonic semimetals]]

[[!redirects photonic topological insulator]]
[[!redirects photonic topological insulators]]

[[!redirects photonic topological semimetal]]
[[!redirects photonic topological semimetals]]

[[!redirects topological laser]]
[[!redirects topological lasers]]


physical field > history

* [[field (physics)]]


physical law

[[!redirects physical laws]]

## References

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


Physical Mathematics and the Future

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#### Physics
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#### String theory
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=--
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This page collects links related to

* {#Moore14} [[Gregory Moore]]

  _Physical Mathematics and the Future_

  talk at [Strings 2014](http://physics.princeton.edu/strings2014/)

  ([talk slides](http://physics.princeton.edu/strings2014/slides/Moore.pdf), [companion text pdf](http://www.physics.rutgers.edu/~gmoore/PhysicalMathematicsAndFuture.pdf), [[MooreVisionTalk2014.pdf:file]])

on the [[mathematical physics]] of [[string theory]].

Related:

* G. Moore _[[The Impact of D-Branes on Mathematics]]_, 2014

***

> The most spectacular successes and interactions with math have been in the areas of [[algebra]], [[geometry]], and [[topology]]. A hallmark of the subject is the astonishing converse of [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences|Wigner's title]]: [[The Unreasonable Effectiveness of Physics in the Mathematical Sciences|The unreasonable effectiveness of physics in the mathematical sciences]]. The subject is dauntingly vast. I will limit my comments to just a few possible future paths.

> ([Moore 14, p. 10](#Moore14))

> {#WeStillHaveNo} We still have no fundamental formulation of "[[M-theory]]" - the hypothetical theory of which [[11-dimensional supergravity]] and the five [[string theories]] are all special limiting cases. Work on formulating the fundamental principles underlying M-theory has noticeably waned.

> {#AGoodStartWasGivenByTheMatrixTheory} A good start was given by the [[BFSS matrix model|Matrix theory]] approach of [[Tom Banks|Banks]], [[Willy Fischler|Fischler]], [[Stephen Shenker|Shenker]] and [[Leonard Susskind|Susskind]]. We have every reason to expect that this theory produces the correct [[scattering amplitudes]] of modes in the [[11-dimensional supergravity]] multiplet in 11-dimensional [[Minkowski space]] - even at energies sufficiently large that [[black holes]] should be created. (This latter phenomenon has never been explicitly demonstrated). But [[BFSS matrix model|Matrix theory]] is only a beginning and does not give us the whole picture of [[M-theory]]. The program ran into increasing technical difficulties when more complicated [[KK-compactification|compactifications]] were investigated. (For example, compactification on a six-dimensional torus is not very well understood at all. $[...]$). Moreover, to my mind, as it has thus far been practiced it has an important flaw: It has not led to much significant new mathematics.

> If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of [[M-theory]] is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside - temporarily. But, ultimately, Physical Mathematics must return to this grand issue.

> ([Moore 14, section 12, p. 43-44](#Moore14))

> Perhaps we need to understand the nature of time itself better. $[...]$ One natural way to approach that question would be to understand in what sense time itself is an emergent concept, and one natural way to make sense of such a notion is to understand how pseudo-Riemannian geometry can emerge from more fundamental
and abstract notions such as categories of branes.

> ([Moore 14, section 9](#Moore14))

category: reference


physical neural network

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# Contents
* table of contents
{: toc}

## Idea

A [[neural network]] which relies on a physical medium, such as a [[nonlinear system|nonlinear]] [[optics|optical system]], a mechanical plate oscillator, or an [[electronics|electronic]] [[circuit]], to emulate neurons, rather than [[software]] [[code]].

## See also

* [[neural network]]

* [[machine learning]]

## References

* Logan G. Wright, Tatsuhiro Onodera, Martin M. Stein, Tianyu Wang, Darren T. Schachter, Zoey Hu & Peter L. McMahon, *Deep physical neural networks trained with backpropagation*, Nature 601, 549–555 (2022). doi:10.1038/s41586-021-04223-6. ([web](https://www.nature.com/articles/s41586-021-04223-6))

* Victor Lopez-Pastor, Florian Marquardt, *Self-learning Machines based on Hamiltonian Echo Backpropagation*, ([arXiv:2103.04992](https://arxiv.org/abs/2103.04992))

* Sam Dillavou, Menachem Stern, Andrea J. Liu, Douglas J. Durian, *Demonstration of Decentralized, Physics-Driven Learning*, ([arXiv:2108.00275](https://arxiv.org/abs/2108.00275))

[[!redirects physical neural networks]]


physical unit

> under construction

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#Contents#
* table of contents
{:toc}

## Idea

When formulating a [[theory of physics]] in terms of [[mathematics]] one typically models the range of certain physical quantities by [[torsors]] over some [[group]] of transformations.

For instance a [[wavelength]] would be identified as an element in the [[positive real numbers]], $\mathbb{R}_{\gt0}$ , being a torsor over the [[multiplicative group]] $\mathbb{R}_{\gt0}^\times$ of [[positive reals]].

In order for the "[[coordination]]" of the mathematical theory with physical [[experiment]] to take place, one needs to choose an identification of this abstract torsor with the (idealized) one that it is supposed to model in nature. Such a choice is equivalent to a choice of [[unit]] (in the mathematical sense), hence a choice of element of the torsor. In this context this is then a _physical unit_.

For instance picking an element in $\mathbb{R}_{\gt0}$ and declaring this to be the length of the path travelled by light in a vacuum in
1/299 792 458 second means defining a _physical unit of length_ (in this example: of the [[meter]]).

Notice that _choice of unit_ is also called _choice of gauge_. This is indeed the same "[[gauge]]" as in "[[gauge theory]]", as it is how ([Weyl 23](gauge#Weyl23)) introduced the concept of gauge theory: as a theory in which the choice of unit of length may change along paths in space.

## Relation to physical constants

Physical units are often called _physical constants_. But by definition physical units are arbitrary choices made in the desciption of a physical system. Of course once made, one wants to keep these choices constant, such as to be useful.

The actual _constants of nature_, when they are "dimensionful", are instead elements of these abstract torsors. For example, the [[Planck length]] belongs to the torsor of length, a torsor over $\mathbb{R}_{\gt0}^\times$. We can describe it by its ratio to another element of the same torsor, such as the meter — the Planck length is approximately $1.616 \cdot 10^{-35} \mathrm{m}$ — but the number appearing in this statement is completely driven by our arbitrary choice of the meter.

An actual constant of nature corresponds to a specific, non-arbitrary real number only when it is a _dimensionless_ quotient of physical quantities. For instance the [[fine structure constant]] is the [[quotient]]

$$
  \alpha \coloneqq \frac{e^2}{ (4 \pi \epsilon_0) \hbar c} \in \mathbb{R}
$$

where $e$ is the [[electric charge]] of the [[electron]], $\hbar$ is [[Planck's constant]], etc. These quantities are such that their dimensionalities cancel out, meaning their quotient is a member not of an abstract torsor but of $\mathbb{R}$ itself, and hence a [[real number]] characterizing nature independently of any conventions about how to parameterize it.

For computing such a quotient concretely, one expresses each of the constituent quantities as some real multiple of an appropriate physical unit: e.g. $e$ as a multiple of the [[coulomb]], $c$ as a multiple of the meter per second, etc. If the units are chosen such that their quotient is unity, then the quotient of the numbers is the actual physical constant.

## Units of length in Lagrangian field theory
 {#UnitsOfLengthInLagrangianFieldTheory}

> under construction

Let $\Sigma \simeq \mathbb{R}^{p,1}$ be [[Minkowski spacetime]] and
let $E \overset{fb}{\to} \Sigma$ be a [[fiber bundle]] thought of as a [[field]] bundle. Write $\{\phi^a\}$ for local [[coordinates]] on the typical fiber of this bundle.

The total space of the corresponding [[jet bundle]] $J^\infty_\Sigma(E) \overset{jb^\infty}{\to} \Sigma$ carries an [[action]]

$$
  sc \;\colon\; \mathbb{R}^\times \times J^\infty_\Sigma(E) \longrightarrow J^\infty_\Sigma(E)
$$

of the multiplicative [[group of units]] $\mathbb{R}^\times$ of the [[real numbers]], given on the induced jet coordinates by

$$
  \begin{aligned}
    x^\mu & \mapsto r x^\mu
    \\
    \phi^a & \mapsto \phi^a
    \\
    \phi^a_{,\mu_1, \cdots \mu_k} & \mapsto r^{-k} \phi^a_{,\mu_1 \cdots \mu_k}
  \end{aligned}
  \,.
$$

Let then

$$
  \mathbf{L}_{(-)}
   \;\colon\;
  \mathbb{R}^n \longrightarrow \mathbf{\Omega}^{p+1}(J^\infty_\Sigma(E))
$$

be a smoothly $n$-parameterized collection of [[Lagrangian densities]], equipped with an $R^\times$-action

$$
  scp \;\colon\; \mathbb{R}^\times \times \mathbb{R}^n \longrightarrow \mathbb{R}^n
$$

on $\mathbb{R}^n$.

Observe that the [[Euler-Lagrange equations]] induced by a Lagrangian density $\mathbf{L}$ equal those induced by the rescaled Lagrangian $r \mathbf{L}$, and that the [[presymplectic current]] $\Omega_{BFV}$ induced by $\mathbf{L}$ scales linearly with $r$ itself. Upon [[quantization]], this rescaling of $\Omega_{BFV}$ may be absorbed in [[Planck's constant]]. In conclusion, as long as Lagrangian densities scale _homogeneously_ the rescaled Lagrangian induces the same physics.

Hence we require that the combined scaling action of $\mathbb{R}^\times$ on $J^\infty_\Sigma(E)$ via $sc$ and on the parameters in $\mathbb{R}^n$ via $scp$ is homogeneous on $\mathbf{L}$ in that there exists $dim \in \mathbb{Z}$ such that  for every $r \in \mathbb{R}^\times$ we have

$$
  sc_r^\ast \mathbf{L}_{( scp(-))} = r^{dim} \mathbf{L}_{(-)}
  \,.
$$

Then a parameter $a \colon \mathbb{R}^n \to \mathbb{R}$ such that there exists $w \in \mathbb{Z}$ with

$$
  scp_r^\ast a = r^{w} a
$$

is said to have _dimension_ $[length]^{w}$.

For example the Lagrangian density for the [[free field theory|free]] [[scalar field]]

$$
  \mathbf{L}_{(-)}
     \;\colon\;
   \mathbb{R}^1
     \longrightarrow
   \mathbf{\Omega}^{p+1}(J^\infty_\Sigma(\Sigma \times \mathbb{R})
$$

given by

$$
  \mathbf{L}_{m}
  \;\coloneqq\;
  \tfrac{1}{2}
  \left(
    \eta^{\mu \nu}\phi_{,\mu} \phi_{,\nu}
    -
    m^2 \phi^2
  \right)
  dvol
$$

is parameterized by the [[mass]] $m$. For the Lagrangian to scale homogenously with $r^{p-1}$ the mass parameter has to have dimension $[length]^{-1}$. To indicate this action one writes the mass in the combination $m c / \hbar$, called the inverse _[[Compton wavelength]]_, so that the homogenously scaling collection of Lagrangians is

$$
  \mathbf{L}_{m c / \hbar}
  \;\coloneqq\;
  \tfrac{1}{2}
  \left(
    \eta^{\mu \nu}\phi_{,\mu} \phi_{,\nu}
    -
    \left(
      \tfrac{m c}{\hbar}
    \right)
   \phi^2
  \right)
  dvol
$$

(...)

## Examples

* [[meter]], [[femtometer]]

* [[electronvolt]], [[MeV]], [[GeV]], [[TeV]]

* [[Boltzmann constant]]

[[!include fundamental scales -- contents]]

## Related concepts

* [[comoving time]]
* [[natural units]]

## References

For a mathematical description of physical units and the associated “physical dimensions”,
including a discussion on how [[densities]] can be used to define real and complex powers of physical quantities, see

* [[Dmitri Pavlov]], Answer to Question 402515 on [[MathOverflow]] (How do we give mathematical meaning to 'physical dimensions'?), <https://mathoverflow.net/questions/402497/how-do-we-give-mathematical-meaning-to-physical-dimensions/402515#402515>.

Additional references:

* [[Robert Loren Jaffe]], *Natural Units and the Scales of Fundamental Physics*, Course Notes 2007 ([pdf](https://stuff.mit.edu/afs/athena/course/8/8.06/spring08/handouts/units.pdf), [[Jaffe_NaturalUnits.pdf:file]])

Discussion in the context of [[philosophy of science]] includes

* [[Georg Hegel]], [Book I, third section, first chapter](Science+of+Logic#TheMeasureFirstChapter)  of _[[Science of Logic]]_

[[!redirects physical unit]]
[[!redirects physical units]]

[[!redirects unit of measurement]]
[[!redirects units of measurement]]


physics

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# Physics
* table of contents
{: toc}

## Idea

_Physics_ studies the constituting mechanisms of the [[observable universe]].

Starting with suggestions by [[Galileo]] around 1600 and then clearly so since [[Newton]] in 1687 laid the foundations of physics proper with what is since called _[[Newton's laws of physics]]_, distinguishing it from the earlier _[[natural philosophy]]_, physics is specifically concerned with modelling [[experiment|experimentally]] observed [[phenomenology]] by "[[theory (physics)|physical laws]]" formulated in the language of [[mathematics]].

> [[natural philosophy|Philosophy]] $[$i.e. [[physics]] $]$ is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of [[mathematics]], and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth. ([[Galileo Galilei]] in [Galilei, Il Saggiatore, 1623](http://en.wikipedia.org/wiki/The_Assayer)).

The relation to mathematics is typically the stronger the more one turns to _fundamental physics_ (e.g. [[fundamental particles]]) and away from [[complex systems]] or else the more one coarse-grains irregular complex behaviour and studies [[statistical physics|statistical]] aspects in [[statistical physics]].

The present most fundamental understanding of the physics of the observable universe is summarized in [[models (in theoretical physics)]] called

1. the [[standard model of particle physics]]

1. the [[standard model of cosmology]];

which are formulated in the two fundamental [[theories]] called

1. [[Yang-Mills theory|Yang-Mills]]-[[gauge theory]]

1. [[Einstein]]-[[gravity]]

which in turn are both instances of the general framework of [[classical field theory|classical]]/[[quantum field theory]].

Theoretical physics also studies lots of approximate, hypothetical "toy"- and test-case [[models (in theoretical physics)]], exploring the "space of physical theories", and parts of [[mathematical physics]] overlaps with pure [[mathematics]], the relation to physics only serving as a motivation that may or may not eventually lead back to statements about [[phenomenology]].

## Related entries

* [[topological physics -- contents]]

* [[fields and quanta - table]]

* [[geometry of physics]]

## References

* [[physics resources]]

* _[The Feynman Lectures on Physics](http://www.feynmanlectures.caltech.edu)_

[[!redirects physics]]

[[!redirects fundamental physics]]


Physics > history

< [[Physics]]
[[!redirects Physics -- history]]


physics resources

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This page lists (or will some day) some (online) resources on [[physics]] in general and probably [[theoretical physics]] and [[mathematical physics]], in particular.

#Contents#
* table of contents
{:toc}

## Lecture notes

* [[Raymond Brock]], *Quarks, Spacetime, and the Big Bang* lecture notes [full text:[pdf](https://web.pa.msu.edu/people/brock/file_sharing/QSandBB/QS&BB_manuscript/QS&BB.pdf)]

## Textbooks

* [[Lev Landau]], [[Evgeny Lifshitz]], *[[Course of Theoretical Physics]]* (1950s)

## Online question-and-answers

* [Physics.Stack exchange](http://physics.stackexchange.com/)

  basic physics

* [Theoretical Physics.Stack Exchange](http://theoreticalphysics.stackexchange.com/)

  research-level theoretical physics

* [Bartleby](https://www.bartleby.com/)

  basic and research-level Physics, and other STEM subjects

## Related entries

* [[mathematics resources]]

* [[philosophy resources]]


physicscontents

**[[physics]]**, [[mathematical physics]], [[philosophy of physics]]

## Surveys, textbooks and lecture notes

* _[[higher category theory and physics|(higher) category theory and physics]]_

* _[[geometry of physics]]_

* [[books and reviews in mathematical physics|books and reviews]], [[physics resources]]

***

[[theory (physics)]], [[model (physics)]]

[[experiment]], [[measurement]], [[computable physics]]

* **[[mechanics]]**

  * [[mass]], [[charge]], [[momentum]], [[angular momentum]], [[moment of inertia]]

  * [[dynamics on Lie groups]]

    * [[rigid body dynamics]]

* [[field (physics)]]

  * [[Lagrangian mechanics]]

    * [[configuration space]], [[state]]

    * [[action functional]], [[Lagrangian]]

    * [[covariant phase space]], [[Euler-Lagrange equations]]

  * [[Hamiltonian mechanics]]

    * [[phase space]]

    * [[symplectic geometry]]

      * [[Poisson manifold]]

      * [[symplectic manifold]]

      * [[symplectic groupoid]]

    * [[multisymplectic geometry]]

      * [[n-symplectic manifold]]

  * [[spacetime]]

    * [[smooth Lorentzian manifold]]

    * [[special relativity]]

    * [[general relativity]]

    * [[gravity]]

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

      * [[black hole]]

* **[[classical field theory|Classical field theory]]**

  * [[classical physics]]

    * [[classical mechanics]]
    * [[waves]] and [[optics]]
    * [[thermodynamics]]

* **[[quantum mechanics|Quantum Mechanics]]**

  * [[quantum mechanics in terms of dagger-compact categories|in terms of ∞-compact categories]]

  * [[quantum information]]

  * [[Hamiltonian operator]]

  * [[density matrix]]

  * [[Kochen-Specker theorem]]

  * [[Bell's theorem]]

  * [[Gleason's theorem]]

* **[[quantization|Quantization]]**

  * [[geometric quantization]]

  * [[deformation quantization]]

  * [[path integral|path integral quantization]]

  * [[semiclassical approximation]]

* **[[quantum field theory|Quantum Field Theory]]**

  * Axiomatizations

    * [[AQFT|algebraic QFT]]

      * [[Wightman axioms]]

      * [[Haag-Kastler axioms]]

        * [[operator algebra]]

        * [[local net]]

          * [[conformal net]]

      * [[Reeh-Schlieder theorem]]

      * [[Osterwalder-Schrader theorem]]

      * [[PCT theorem]]

      * [[Bisognano-Wichmann theorem]]

        * [[modular theory]]

      * [[spin-statistics theorem]]

        * [[boson]], [[fermion]]

    * [[FQFT|functorial QFT]]

      * [[cobordism]]

      * [[(∞,n)-category of cobordisms]]

      * [[cobordism hypothesis]]-theorem

      * [[extended topological quantum field theory]]

  * Tools

    * [[perturbative quantum field theory]], [[vacuum]]

    * [[effective quantum field theory]]

    * [[renormalization]]

    * [[BV-BRST formalism]]

    * [[geometric ∞-function theory]]

  * [[particle physics]]

    * [[phenomenology]]

    * [[model (in particle phyiscs)|models]]

      * [[standard model of particle physics]]

      * [[fields and quanta - table|fields and quanta]]

      * [[GUT|Grand Unified Theories]], [[MSSM]]

    * [[scattering amplitude]]

      * [[on-shell recursion]], [[KLT relations]]

  * Structural phenomena

    * [[universality class]]

    * [[quantum anomaly]]

      * [[Green-Schwarz mechanism]]

    * [[instanton]]

    * [[spontaneously broken symmetry]]

    * [[Kaluza-Klein mechanism]]

    * [[integrable systems]]

    * [[holonomic quantum fields]]

  * Types of quantum field thories

    * [[TQFT]]

      * [[2d TQFT]]

      * [[Dijkgraaf-Witten theory]]

      * [[Chern-Simons theory]]

    * [[TCFT]]

      * [[A-model]], [[B-model]]

      * [[homological mirror symmetry]]

    * [[QFT with defects]]

    * [[conformal field theory]]

    * [[(1,1)-dimensional Euclidean field theories and K-theory]]

    * [[(2,1)-dimensional Euclidean field theory|(2,1)-dimensional Euclidean field theory and elliptic cohomology]]

    * [[CFT]]

      * [[WZW model]]

      * [[6d (2,0)-supersymmetric QFT]]

    * [[gauge theory]]
      * [[field strength]]
      * [[gauge group]], [[gauge transformation]], [[gauge fixing]]
      * examples

        * [[electromagnetic field]], [[QED]]
          * [[electric charge]]
          * [[magnetic charge]]

        * [[Yang-Mills field]], [[QCD]]

          * [[Yang-Mills theory]]
          * [[spinors in Yang-Mills theory]]
          * [[topological Yang-Mills theory]]

        * [[Kalb-Ramond field]]
        * [[supergravity C-field]]
        * [[RR field]]
        * [[first-order formulation of gravity]]
          * [[general covariance]]
          * [[supergravity]]
            * [[D'Auria-Fre formulation of supergravity]]
          * [[gravity as a BF-theory]]

    * [[sigma-model]]

      * [[particle]], [[relativistic particle]], [[fundamental particle]], [[spinning particle]], [[superparticle]]

      * [[string]], [[spinning string]], [[superstring]]

      * [[membrane]]

      * [[AKSZ theory]]

* [[string theory|String Theory]]

  * [[string theory results applied elsewhere]]

* [[number theory and physics]]

  * [[Riemann hypothesis and physics]]

[Edit this sidebar](/nlab/edit/physicscontents)

[[!redirects physics contents]]


pi

# Contents
* table of contents
{: toc}

## Idea

The [[real number]] $\pi \in \mathbb{R}$ is the [[length]] of one [[hemisphere|semi-circle]] of the [[unit circle]]. Hence $2 \pi$ is the circumference of the full [[unit circle]].

This number plays a central role in [[trigonometry]], but -- especially via its relation to [[complex analysis]] via [[Euler's formula]] -- it also appears in many other situations.

## Definition via real analysis

The [[real number]] $\pi \in \mathbb{R}$ (occasionally called _Ludolph's number_ ---make that _very_ occasionally, nowadays--- or _Archimedes' constant_) is the minimal [[positive real number|positive]] [[root]] of the [[sine function]] $sin_\mathbb{R} : \mathbb{R}\to\mathbb{R}$, which itself is the unique solution to the [[differential equation]]/[[initial value problem]]
$$ sin_\mathbb{R}'' = -sin_\mathbb{R} $$
$$ sin_\mathbb{R}(0) = 0 $$
$$ sin_\mathbb{R}'(0) = 1 .$$

Alternatively, $\pi$ may be defined as $\frac{1}{2} \tau $, where $\tau$ is the [[period]] of the same function [[sin]].

An alternative description of $sin$ is as the unique function $\mathbb{R}\to\mathbb{R}$ continuous at $0$ and otherwise satisfying
$$ 1 - x^2 \leq \frac{sin(x)}{x} \leq 1 $$
and
$$ sin 3 x = 3 sin x - 4 sin^3 x $$
although this makes practical calculations rather difficult. (If you really care to know why this characterization works, you can look [here](http://ncatlab.org/toddtrimble/published/Characterization+of+sine). Or at [this nForum discussion](https://nforum.ncatlab.org/discussion/5773/a-short-note-for-todd-trimble/#Item_4).)

### Tangential Remarks

Either of the definitions of $sin$ above mentions a thing, and claims some properties for it, in order to indirectly describe some other thing (in this case, $\pi$).  To properly make sense of either definition, it is necessary to ensure that one description or the other actually does define some thing, that there is in fact exactly one such thing so defined, and that it has the needed properties for the definition of $\pi$ to make sense — particularly, that the roots of $sin$ are isolated, and that there is a positive root.  These might make a decent [[exercise]] in a first course in classical analysis.

## Definition via complex analysis

A related but perhaps more conceptual description is via complex analysis. Recall that the standard [[exponential function]] is defined by the formula

$$\exp(z) = \sum_{n \geq 0} \frac{z^n}{n!} = 1 + z + \frac{z^2}{2} + \frac{z^3}{6} + \ldots$$

for all [[complex numbers]] $z$ (as an absolutely [[Cauchy sequence|convergent series]]), and satisfies the equations

* $\exp(z + w) = \exp(z) \cdot \exp(w)$,

* $\exp(\widebar{z}) = \widebar{\exp(z)}$.

It follows that for $z \in i \mathbb{R}$, where $z + \widebar{z} = 0$, we have ${|\exp(i t)|} = 1$.

Hence $t \mapsto \exp(i t)$ defines a [[Lie group]] homomorphism $\phi$ from the additive group $\mathbb{R}$ to the multiplicative group $S^1$ of unit complex numbers. It may be checked that this is a local diffeomorphism: a local inverse to $z \mapsto \exp(z) - 1$ in a neighborhood of $0$ is given by

$$z \mapsto \log (1 + z) = z - \frac{z^2}{2} + \frac{z^3}{3} - \ldots$$

Thus $\phi: \mathbb{R} \to S^1$ maps a small neighborhood $U$ of $0$ onto a small neighborhood $V$ of $1$ in $S^1$. As $S^1$ is [[connected space|connected]], we have $S^1 = \bigcup_{n \geq 1} V^n$ where $V^n$ consists of $n$-fold products of elements in $V$, and in fact a finite union $\bigcup_{n=1}^m V^n$ suffices since $S^1$ is [[compact space|compact]]. It follows easily that the homomorphism $\phi: \mathbb{R} \to S^1$ is surjective.

In particular, there exists some element $t \in \mathbb{R}$ such that $\phi(t) = -1$. We may define $\pi$ to be the least positive $t$ such that $\phi(t) = -1$ (cf. Euler's formula $e^{\pi i} + 1 = 0$). Indeed, it is slightly nicer just to observe that $\ker(\phi)$ is nontrivial (since $S^1$ has [[torsion group|torsion elements]] whereas $\mathbb{R}$ does not) and must be a discrete subgroup, generated by a unique positive $\tau$, and one defines $\pi = \tau/2$.

These considerations suggest why mathematicians sometimes consider $\tau = 2\pi$ to be "more fundamental" than $\pi$, or even that $2\pi i$ is the truly fundamental quantity, being (up to sign) the unique generator of the kernel of $\exp: \mathbb{C} \to \mathbb{C}^\times$. This is also reflected in the familiar contour integral formula

$$\int_{{|z|} = 1} \frac{d z}{z} = 2\pi i$$

that on some level pervades complex analysis.

## "Archimedean" definition

A sequence that approximates $\pi$ that was available to [[Archimedes]] (using his "method of exhaustion") involves calculating perimeters of inscribed $2^n$-gons (squares, octagons, etc.) in a unit circle, using half-angle formulas. This brief description is, for that matter, available to high-school students, so we can just summarize the result.

Define a sequence by the recursion $x_1 = 0$, $x_{n+1} = \sqrt{2 + x_n}$. A typical term has nested square roots, like $x_4 = \sqrt{2 + \sqrt{2 + \sqrt{2}}}$. This sequence rapidly approaches $2$ in the limit; put

$$y_n = 2^n \sqrt{2 - x_n}.$$

The statement is that $y_n$ is half the perimeter of the inscribed $2^{n+1}$-gon for $n \geq 1$, so that $y_n$ is an increasing convergent sequence which tends to $\pi$ in the limit. Incidentally, simple high-school algebra shows

$$y_n = 2 \prod_{k=2}^n \frac{2}{x_k}$$

and thus we may deduce Vieta's formula (see below).

Archimedes used a similar idea, in effect viewing $2\pi$ as squeezed between the perimeters of inscribed and circumscribed polygons with $3 \cdot 2^n$ sides each (starting with a hexagon). By considering $n = 5$, he was eventually led to his famous estimate $3 + \frac{10}{71} \lt \pi \lt 3 + \frac1{7}$.

## Other formulas

* Leibniz formula: $\frac{\pi}{4} = 1 - \frac1{3} + \frac1{5} - \ldots$. Based on the power series for the [[arctangent]]
$$\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \ldots,$$
this series for $\pi/4$ obviously converges slowly, but more sophisticated expressions in terms of the arctangent have better convergence rates, such as the following first observed by Machin:
$$\frac{\pi}{4} = 4\arctan(\frac1{5}) - \arctan(\frac1{239})$$

* Integral formulas (based ultimately on the [[Riemann integral]], the [[Henstock integral]], the [[Lebesgue integral]], or the like):

  * Also based on the [[arctangent]]:
    $$ \pi = 4 \int_{x=0}^1 \frac{\mathrm{d}x}{x^2 + 1} .$$

  * Or the [[arcsine]]:
    $$ \pi = 2 \int_{x=0}^1 \frac{\mathrm{d}x}{\sqrt{1 - x^2}} $$
    (but this is an [[improper integral]] if one is using the [[Riemann integral]]).

  Of course, you can come up with many modifications of these (including proper Riemann integrals based on the arcsine).

* Product formulas:

  * Vieta's formula:
$$\pi= 2\times\frac{2}{\sqrt{2}}\times
\frac{2}{\sqrt{2+\sqrt{2}}}\times
\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\times\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}\times\cdots$$
This was perhaps the first [[infinite product]] in the history of mathematics.

  * Euler's [[product formula for the sine function|product formula]]:
$$\sin(x) = x \prod_{n \geq 1} (1 - \frac{x^2}{\pi^2 n^2})$$
One point is that putting $x = \frac{\pi}{2}$, one can derive another famous formula, namely

  * Wallis's formula:
$$\frac{\pi}{2} = \lim_{n \to \infty} \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \ldots \cdot \frac{2 n}{2 n - 1} \cdot \frac{2 n}{2 n + 1}$$
Alternatively, this may be derived from [[Stirling's approximation]] for $n!$, or rather starting from an asymptotic formula which follows from Stirling's approximation:
$$\binom{2 n}{n} \sim \frac{2^{2 n}}{\sqrt{\pi n}}.$$

* Euler series: let $\zeta(s)$ be the Riemann [[zeta function]]. By applying the logarithmic derivative to Euler's product formula above, one may derive the following partial fraction expansion for the cotangent:
$$\cot(x) = \frac1{x} - 2\sum_{n \geq 1} \frac{x}{\pi^2 n^2 - x^2}.$$
Expanding each of the summands in a geometric series and rearranging terms, one may derive
$$x \cot(x) = 1 - 2\sum_{m \geq 1} \zeta(2 m)\frac{x^{2m}}{\pi^{2 m}}.$$
On the other hand, one has
$$x\cot(x) = i x + \frac{2 i x}{e^{2 i x}-1} = 1 + \sum_{n \geq 2} B_n \frac{(2 i x)^n}{n!}$$
where the $B_n$ are the [[Bernoulli numbers]]. By comparing coefficients, one arrives at
$$-\frac{2}{\pi^{2 m}} \zeta(2 m) = (-1)^m \frac{2^{2 m}}{(2 m)!}B_{2 m}$$
or, after rearranging,
$$\pi^{2 k} = (-1)^{k-1} \frac{(2 k)!}{2^{2 k - 1}B_{2 k}}\zeta(2 k)$$
which has as special cases the formulas $\pi^2 = 6\zeta(2)$, $\pi^4 = 90\zeta(4)$, $\pi^6 = 945\zeta(6)$.

* Continued fractions: the regular [[continued fraction]] expansion of $\pi$ is highly irregular, but several generalized continued fraction expansions listed in [Wikipedia](http://en.wikipedia.org/wiki/Pi#Continued_fractions) include
$$\pi=\textstyle \frac{4}{1+\textstyle \frac{1^2}{2+\textstyle \frac{3^2}{2+\textstyle \frac{5^2}{2+\textstyle \frac{7^2}{2+\textstyle \frac{9^2}{2+\ddots}}}}}}
$$

## Irrationality and transcendence

Famously, $\pi$ is an [[irrational number]], although proving this fact is no triviality. The first substantive idea of proof was given by Lambert, who discovered the following [[continued fraction]] expression for the [[tangent]]:

$$\tan(z) = \textstyle \frac{z}{1 - \textstyle \frac{z^2}{3 - \textstyle \frac{z^2}{5 - \textstyle \frac{z^2}{7 - \ldots}}}}.$$

The basic inference was that $\tan(z)$ must therefore be irrational for rational values of $z$, and since $\tan(\pi/4) = 1$, it follows that $\pi/4$ is irrational. This was not a completely rigorous proof since Lambert had failed to prove the irrationality of certain infinite continued fractions, but Legendre provided the missing lemma in 1806. Actually [Legendre](#Leg) proves more: applying his lemma to the continued fraction for $\sqrt{q}\; \tan(\sqrt{q})$ (derived from the above), he shows this expression is irrational for rational $q$, and therefore not even $\pi^2$ can be rational. He goes on to speculate that $\pi$ is in fact transcendental.

The transcendence of $\pi$ was finally established by Lindemann in 1882, by adapting methods of [[Charles Hermite]] who had established the transcendence of [[e]] in 1873. This definitively proved the impossibility of solving the problem of "squaring the circle" by means of ruler-and-compass constructions. A far more general theorem was then established by Weierstrass:

+-- {: .num_theorem #LindWeier}
###### Theorem
If $\alpha_1, \ldots, \alpha_n$ are algebraic numbers linearly independent over $\mathbb{Q}$, then $\exp(\alpha_1), \ldots, \exp(\alpha_n)$ are _algebraically_ independent over $\mathbb{Q}$ (which is to say that the [[field]] $\mathbb{Q}(\exp(\alpha_1), \ldots, \exp(\alpha_n))$ has [[transcendence degree]] $n$ over $\mathbb{Q}$).
=--

A proof is given in [Wikipedia](http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_transcendental#Proof). It follows that if $\pi$ and therefore $i\pi$ were algebraic, then $\exp(i\pi) = -1$ would be transcendental, which is absurd.

## Related entries

* [[circumference of a circle]]

* [[circle constant]]

* [[radius]]

* [[e]]

* [[golden ratio]]

## References

A whole chapter on $\pi$ written for the mathematically literate is given in

* Ebbinghaus et al., _Numbers_, GTM (Readings in Mathematics) 123, Springer (1991).

The material on the Euler series for $\pi^{2 n}$ in terms of the zeta function was extracted from

* Kenneth Ireland and Michael Rosen, _A Classical Introduction to Modern Number Theory_, Graduate Texts in Mathematics 84, Springer-Verlag (1982).

The proof of irrationality of $\pi$ given by Legendre is said to be given in the following treatise:

* {#Leg}  A.-M. Legendre, _Eléments de géométrie_ (11th edition), Note IV. F. Didot, Paris (1817). Translation by Charles Davies: ([web](http://quod.lib.umich.edu/cgi/t/text/text-idx?c=moa;idno=ABN7066.0001.001))

{#BBPFormulaReferences} On efficient representations in [[exact real computer arithmetic]] of [[pi]] and related [[irrational numbers]] ([[Bailey-Borwein-Plouffe algorithm]])

* [[David Bailey]], [[Peter Borwein]], [[Simon Plouffe]]: *On the Rapid Computation of Various Polylogarithmic Constants*, Math. Comp. **66** (1997) 903-913 [[doi:1997-66-218/S0025-5718-97-00856-9](https://www.ams.org/journals/mcom/1997-66-218/S0025-5718-97-00856-9), [pdf](https://www.davidhbailey.com/dhbpapers/digits.pdf)]

further discussed in:

* [[Jerzy Karczmarczuk]], *Infinite precision real fractions, and lazy carry propagation* or: *The Most Unreliable Technique in the World to Compute $\pi$*, A Braga School (1998) [[pdf](https://karczmarczuk.users.greyc.fr/arpap/lazypi.pdf), [[Karczmarczuk-Pi.pdf:file]]]

* [[Simon Plouffe]], *On the computation of the $n^{th}$ decimal digit of various transcendental numbers* [[arXiv:0912.0303](https://arxiv.org/abs/0912.0303)]

* MathWorld, *[BBP-Type formula](https://mathworld.wolfram.com/BBP-TypeFormula.html)*

[[!redirects pi]]
[[!redirects π]]
[[!redirects Ludolph number]]
[[!redirects Ludolph's number]]
[[!redirects Ludolph's number]]
[[!redirects Ludolph\'s number]]
[[!redirects Ludolphian number]]
[[!redirects Ludolphian]]
[[!redirects Archimedes constant]]
[[!redirects Archimedes' constant]]
[[!redirects Archimedes' constant]]
[[!redirects Archimedes\' constant]]
[[!redirects Archimedes's constant]]
[[!redirects Archimedes's constant]]
[[!redirects Archimedes\'s constant]]


Pi-algebra

[[!redirects Pi-algebras]]
[[!redirects Pi-algebras]]

+-- {: .rightHandSide}
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###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

##Idea

A _$\Pi$-algebra_ is an algebraic model for the [[homotopy group]]s $\pi_*X$ of a pointed [[topological space]], $X$, together with the [[action]] of the [[primary homotopy operation]]s on them, in the same sense that algebras over the [[Steenrod algebra]] are models for the [[cohomology]] of a space.

Constructions of this type exist in many pointed model categories. It suffices to have a collection of [[spherical object]]s.

## The category $\Pi$ of homotopy operations

The [[category]] $\Pi$ of homotopy operations has

* as [[object]]s - pointed [[CW-complex]]es with the [[homotopy type]] of a finite [[wedge product]] of [[sphere]]s of [[dimension]]s $\geq 1$;

* as [[morphism]]s - homotopy classes of (pointed) [[continuous function]]s between them.

### Properties

*  $\Pi$ is a pointed category and has finite coproducts (given by the finite wedges), but not products.

 *  There is a [[functor]], _smash product_ $i : \Pi\times \Pi \to \Pi$, which sends an object $(U,V)$ to $U\wedge V = (U\times V)/((U\times *)\vee(*\times V))$, which preserves [[coproduct]]s in each variable.

This category $\Pi^{op}$ is a finite product theory, in the sense of [[algebraic theories]] whose models are:

##$\Pi$-algebras

Let $Set_*$ denote the category of [[pointed sets]].

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

A _$\Pi$-algebra_ is a [[functor]] $A: \Pi^{op}\to Set_*$, which sends [[coproduct]]s to [[products]].

A morphism of $\Pi$-algebras is a natural transformation between the corresponding functors.

=--

### Properties

* A $\Pi$-algebra $A$ satisfies $A* = *$.

* The values of a $\Pi$-algebra $A$ are determined by the values $A_n = A(S^n)$, that it takes on the spheres, $S^n$, $n\geq 1$.

*  A $\Pi$-algebra can be considered to be a graded [[group]] $\{A_n\}_{n=1}^\infty$ with $A_n$ abelian for $n\gt 1$, together with

   * '[[Whitehead product]]' homomorphisms :

$$[-,-] : A_p\otimes A_q \to A_{p+q-1}$$

for $p,q \geq 1$ (the case where they are equal to 1 needs special mention, see below.)

   * '[[composition operation]]s', $-\cdot \alpha : A_p\to A_r$ for $\alpha \in \pi_r(S^p)$, $1\lt p\lt r$,

which satisfy the identities that hold for the Whitehead products and composition operations on the higher homotopy groups of a pointed space, and

   * a left action of $A_1$ on the $A_n$, $n\gt 1$, which commutes with these operations.

The [[Whitehead products]] include

   * $[\xi,a] = {}^\xi a - a$, where ${}^\xi a$ is the result of the $A_1$-action of $\xi \in A_1$ on $a\in A_r$, $r\gt 1$; similarly for a right action;

   * the commutators $[a,b] = a b a^{-1} b^{-1}$, for $a,b \in A_1$.

## The homotopy $\Pi$-algebra of a pointed topological space.

For a pointed space $X$, and $U \in \Pi$, define a $\Pi$-algebra $\pi_* X$ by $\pi_* X(U) = [U,X]_*$, the set of pointed homotopy classes of pointed maps from $U$ to $X$.

This is a $\Pi$-algebra called the **homotopy $\Pi$-algebra** of $X$.

##The realisability problem

Suppose $A: \Pi \to sets_*$ is an abstract $\Pi$-algebra, the **realisability problem** for $A$ is to construct, if possible, a pointed space $X$, such that $A\simeq \pi_* X$.  The space $X$ is called a _realisation_ of $A$.

Things can be complicated!

1.  The homotopy type of $X$ is not always determined by $A$ (hence 'a' rather than 'the" realisation) , so that raises the additional problem of classifying the realisations.

1. Not all $\Pi$-algebras can be realised, in fact

####Theorem (Blanc 1995)

Given a $\Pi$-algebra, $A$, there is a sequence of [[higher homotopy operation]]s depending only on maps between wedges of spheres, such that $A$ is realisable if and only if the operations vanish coherently.

####Example (Blanc 1995)

For $p\neq 2$, a prime and $r\geq 4(p-1)$, $\pi_*S^r \otimes \mathbb{Z}/p$ cannot be realised (and if $p = 2$, one uses $r\geq 6$).

(There is a problem, discussed in Blanc's 1995 paper, that the composition operations need not be
homomorphisms, so tensoring with $\mathbb{Z}/p$ has to be interpreted carefully.)

##Simply connected $\Pi$-algebras

A $\Pi$-algebra, $A$, is said to be **simply connected** if $A_1= 0$.

In this case the universal identities for the [[primary homotopy operations]] can be described more easily (see Blanc 1993).  These include
the structural information that the [[Whitehead products]] make $A$ into a graded [[Lie algebra|Lie ring]] (with a shift of indices).

##Truncated $\Pi$-algebras

The beginnings of a classification theory for $n$-truncated $\Pi$-algebras can be found in Frankland's thesis (link given below).

## References

* C.R. Stover, _A Van Kampen spectral sequence for higher homotopy groups_, Topology 29 (1990) 9 - 26.

[[David Blanc]] has written a lot on the theory of these objects. An example is

* [[David Blanc]], _Loop spaces and homotopy operations_, Fund. Math. 154 (1997) 75 - 95.

The realisability problem is discussed in

* [[David Blanc]], _Higher homotopy operations and the realizability of homotopy groups_, Proc.  London Math. Soc. (3) 70 (1995) 214 -240,

and further in

*  [[David Blanc]], _Algebraic invariants for homotopy types_, Math. Proc. Camb. Phil. Soc. 127 (3)(1999) 497 - 523. (preprint version on the [ArXiv](http://arxiv.org/abs/math/9812035).)

There are more recent results on the realisability problem in [[Martin Frankland]]'s [thesis](http://www.math.uiuc.edu/~franklan/Frankland_Thesis_20100513.pdf).


pi-calculus

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#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

\tableofcontents

## Idea

The *$\pi$-calculus* is a [[computational]] model for concurrent processes that communicate messages through named channels. It subsumes [[λ-calculus]].

In the monadic (here meaning: *[[unary]]*) $\pi$-calculus, outputs and inputs only send and receive single names, while the polyadic (here meaning: *[[n-ary|$n$-ary]]*) $\pi$-calculus allows outputs of tuples of names and corresponding inputs.

This calculus permits non-[[confluent]] reductions by design, as opposed to other process calculi such as the Interactive Lambda Calculus (ILC).

## Syntax

A process is either:

* an inactive process $0$;
* the parallel composition of two processes $P\mid Q$;
* the replication of a process $!P$;
* a restriction $(\nu n)\,P$ (that is the generation of a fresh name only known to $P$);
* prefixed by an input $n(m).P$;
* prefixed by an output $\overline{n}\langle m\rangle.P$.

Sometimes an [[nondeterministic computation|indeterministic]] choice operator $+$ is added to the syntax.

## Example

* a server that increments the value it receives: $S=!\mathrm{incr}(a,x).\overline{a}\langle x+1\rangle$;
* a client that calls to the $\mathrm{incr}$ server: $C=(\nu a)\bigl(\overline{\mathrm{incr}}\langle a,4\rangle\mid a(y)\bigr)$;
* the final process by composing the server and the client: $S\mid C$.

## Reduction-based operational semantics

The central reduction rule of the $\pi$-calculus is the following:
$$\overline{n}\langle a\rangle.P\mid n(b).Q\longrightarrow P\mid Q_{a\mapsto b}\,.$$

## Typed $\pi$-calculus

Several attempts have been made to give the $\pi$-calculus [[typing systems]]. It is crucial for $\mathrm{HO}\pi$, a [[higher-order  logic|higher-order]] $\pi$-calculus in which messages can be processes themselves.

### Simple types

Simple types are defined by the following grammar:
$$T\longrightarrow\sharp\widetilde{T}$$
where $\widetilde{T}$ ranges over (possibly empty) tuples of simple types.

[[typing context|Typing contexts]] are lists of hypotheses of the form $n:T$ where $n$ is a channel name and $T$ is a simple type.

Simply typed $\pi$-calculus corresponds to a [[distributed logic]]: using a property proven at some place supposes therefore a call to that property under a name. Theorems can be used many times, but lemmas are limited to one use.

Typing | Intuitive signification
-------|------------------------
$a(x,y).P:p,q\to r$ | **Lemma 1.** Under hypotheses $p$, $q$, then $r$.
$\overline{a}\langle M,N\rangle:r$ | Use Lemma 1, knowing that $p$ and $q$, to prove $r$.

### Session types

[[session type|Session types]] were introduced by Kohei Honda. A session links exactly two subprocesses that interact with each other, and multiple sessions can run simultaneously.

\begin{example}
The process
$x(n).\overline{x}\langle n+1\rangle.0$
may be given type
$x:?\mathsf{int}.!\mathsf{int}.\mathsf{end}$.
\end{example}

[Caires and Pfenning 2010](#Caires2010) established a correspondence between [[dual intuitionistic linear logic]] (DILL) and session typed $\pi$-calculus.
Later on, [Wadler 2012](#Walder2012) established a correspondence between [[classical linear logic]] (CLL) and the same calculus.

## References

* [[Robin Milner]], Joachim Parrow and David Walker, _A Calculus of Mobile Processes_, Information and Computation, **100** (1992) 1-40.
* Peter Sewell, _Applied $\pi$ – A Brief Tutorial_, (2000) ([pdf](https://www.cl.cam.ac.uk/~pes20/apppi.pdf)).
* Jeannette M. Wing, _FAQ on $\pi$-Calculus_, (2002) ([pdf](https://www.cs.cmu.edu/~wing/publications/Wing02a.pdf)).
* Daniel Hirschkoff, _A brief survey of the theory of the $\pi$-calculus_, (2003) ([pdf](https://hal-lara.archives-ouvertes.fr/hal-02101985/file/RR2003-13.pdf)).

On types:

* Kohei Honda, _Types for dyadic interaction_, 4th International Conference on Concurrency Theory (1993) 509–523.
* Noël Bernard, _Investigating the Logical Aspects of the Pi-calculus_, Schedae Informaticae, **12** (2003) 57-66 ([hal-00386109](https://hal.science/hal-00386109)).
* {#Caires10} Luís Caires and [[Frank Pfenning]], _Session Types as Intuitionistic Linear Propositions_, 21st International Conference on Concurrency Theory (2010) 222-236 ([pdf](https://www.cs.cmu.edu/~fp/papers/concur10.pdf)).
* {#Walder2012} Philip Wadler, _Propositions as Sessions_ ([pdf](https://homepages.inf.ed.ac.uk/wadler/papers/propositions-as-sessions/propositions-as-sessions.pdf), [doi:10.1145/2398856.2364568](https://dl.acm.org/doi/10.1145/2398856.2364568)).

Other process calculi:

* Kevin Liao, Matthew A. Hammer and Andrew Miller, _ILC: A Calculus for Composable, Computational Cryptography_, Proceedings of the 40th ACM SIGPLAN Conference on Programming Language Design and Implementation, (2019) 640-654 ([doi:10.1145/3314221.3314607](https://dl.acm.org/doi/10.1145/3314221.3314607), [pdf](https://eprint.iacr.org/2019/402.pdf)).

[[!redirects π-calculus]]


pi-finite type

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#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Combinatorics
+-- {: .hide}
[[!include combinatorics - contents]]
=--
=--
=--

\tableofcontents

## Idea

In [[dependent type theory]] ([[homotopy type theory]]), by *$\pi$-finite types* one means a kind of [[finite types]] which are not necessarily [[h-sets]] but have [[finite group|finite]] [[homotopy groups]] in each degree (what in [[homotopy theory]] are called a *[[pi-finite homotopy types]]* or similar).

## Definition
 {#Definition}

For given [[natural number]] $n$, a [[type]] is called *$\pi_n$-finite* if

1. it has a [[finite type]] of [[connected components]] ([[0-truncation]]),

1. all its [[homotopy groups]] up to [[h-level]] $n$ at all base points are [[finite group|finite]].

A type is **$\pi$-finite type** if it is $\pi_n$-finite for all $n \colon \mathbb{N}$.

## Properties

Given two $\pi$-finite types $A$ and $B$, the types $A \times B$ and $A + B$ are both $\pi$-finite types. Similarly, given a family of $\pi$-finite types $B(x)$ indexed by a $\pi$-finite type $x:A$, the dependent sum type $\sum_{x:A} B(x)$ is a $\pi$-finite type.

## Examples

Given a [[univalent universe]] $U$ and a natural number $n:\mathbb{N}$,

* the type of all $U$-small types with $n$ elements is a $\pi$-finite type.

* The type of all $U$-small [[finite groups]] of [[order]] $n$ is a $\pi$-finite type.

## See also

* [[finite type]]

* [[locally finite type]]

* [[pi-finite homotopy type]]

* [[combinatorics]]

## References

* [[UniMath project]], *π-finite types* ([web](https://unimath.github.io/agda-unimath/univalent-combinatorics.pi-finite-types.html))

* [[UniMath project]], *Univalent combinatorics* ([web](https://unimath.github.io/agda-unimath/univalent-combinatorics.html))

[[!redirects π-finite type]]
[[!redirects π-finite types]]

[[!redirects pi-finite type]]
[[!redirects pi-finite types]]


pi-r-squared team

This is a research group formed within [[PPS]], working on [[rewriting]] and [[Coq]].

The team $\pi r^2$ covers three research topics:

*  Fundamental research concerning the correspondence between proofs and programs

*  Theoretical research on the formalism that underlies the Coq proof assistant

*  An implementation research field with the development of Coq, especially in the view of Coq as dependently-typed programming language

##References

* [homepage](http://www.pps.univ-paris-diderot.fr/pi.r2/)


Picard 2-group

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#### Group Theory
+-- {: .hide}
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=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The _[[Picard groupoid]]_ $PIC(\mathcal{C}, \otimes)$ of a [[monoidal category]] $(\mathcal{C}, \otimes)$ is the [[core]] of its [[full subcategory]] on those [[objects]] that are [[invertible objects]] under the [[tensor product]]. This inherits the monoidal structure from $(\mathcal{C}, \otimes)$ and hence becomes a [[2-group]]. This is the **Picard 2-group** of $(\mathcal{C}, \otimes)$.

In geometric contexts this is also called the _[[Picard stack]]_.

## Properties

### Relation to Picard group

The [[decategorification]] of the Picard 2-group, hence the [[group]] of connected components, is the ordinary [[Picard group]] $Pic(\mathcal{C}, \otimes)$.

$$
  Pic(\mathcal{C}, \otimes) \simeq \pi_0 PIC(\mathcal{C}, \otimes)
  \,.
$$

## Related concepts

* [[Picard 3-group]]

* [[Picard ∞-group]]

## References

* [[Alexei Davydov]], [[Dmitri Nikshych]], _Braided Picard groups and graded extensions of braided tensor categories_, ([arXiv:2006.08022](https://arxiv.org/abs/2006.08022))

Early discussion of [[2-groups]] as Picard 2-groups:

* {#Solian72} [[Alexandru Solian]], *Groupe dans une catégorie*, C. R. Acad. Sc. Paris **275** (1972) [[ark:/12148/bpt6k57310477/f7](https://gallica.bnf.fr/ark:/12148/bpt6k57310477/f7.item)]

* {#Solian80} [[Alexandru Solian]], *Coherence in categorical groups*,  Communications in Algebra **9** 10 (1980) 1039-1057 [[doi:10.1080/00927878108822631](https://doi.org/10.1080/00927878108822631)]

[[!redirects Picard 2-groups]]
[[!redirects Picard groupoid of a monoidal category]]


Picard 3-group

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## Definition

For $(\mathcal{C}, \otimes)$ a [[monoidal 2-category]], its **Picard 3-group** or **Picard-Brauer 3-group** is the [[3-group]] [[structure]] induced on the [[core]] of the full sub-2-groupoid $PIC(\mathcal{C}, \otimes)$ on those [[objects]] that are invertible under the [[tensor product]].

* If $(\mathcal{C}, \otimes)$ is a [[braided monoidal 2-category]], then $PIC(\mathcal{C}, \otimes)$ is a [[braided 3-group]].

* If $(\mathcal{C}, \otimes)$ is a [[sylleptic monoidal 2-category]], then $PIC(\mathcal{C}, \otimes)$ is a [[sylleptic 3-group]].

* If $(\mathcal{C}, \otimes)$ is a [[symmetric monoidal 2-category]], then $PIC(\mathcal{C}, \otimes)$ is a [[abelian 3-group]].

  In this case this is the [[n-truncated object in an (infinity,1)-category|3-truncation]] of the [[Picard ∞-group]].

## Examples

* For $R$ a [[commutative ring]] and $2Mod_R$ the 2-category of [[2-modules]] over $R$, hence of [[associative algebras]] over $R$ and [[bimodules]] between them and [[intertwiners]] between those, the corresponding Picard 3-group may be thought of as that of [[line 2-bundles]] over $Spec R$. It has as [[homotopy groups]] the [[Brauer group]], the [[Picard group]] and the [[group of units]] of $R$.

## Related concepts

* [[Picard group]], [[Brauer group]]

* [[Picard 2-group]]

* [[Picard ∞-group]]

## References
 {#References}

The Picard 3-group, or rather the [[monoidal 2-category]] that it sits in, was maybe first made explicit in the last part of

* R. Gordon, A.J. Power, [[Ross Street]], _Coherence for tricategories_, Memoirs of the American Math. Society 117 (1995) Number 558.

The corresponding [[Kan complex]] is discussed in

* [[John Duskin]], _The Azumaya complex of a commutative ring_, Categorical algebra and its applications (Louvain-La-Neuve, 1987), 107-117, Lecture Notes in Math., 1348, Springer, Berlin, 1988.

A summary of these considerations is in section 12 of

* [[Ross Street]], _Categorical and combinatorial aspects of descent theory_ ([arXiv:math.CT/0303175](http://arxiv.org/abs/math.CT/0303175))

A refinement to [[stable homotopy theory]] is discussed in

* {#Sagave11} [[Steffen Sagave]], _Spectra of units for periodic ring spectra_ ([arXiv:1111.6731](http://arxiv.org/abs/1111.6731))

See also the discussion of higher [[Brauer groups]] in stable homotopy theory (which in turn are a "non-connective delooping"of $Pic(-)$) in

* {#Szymik11} [[Markus Szymik]], _Brauer spaces for commutative rings and structured ring spectra_ ([arXiv:1110.2956](http://arxiv.org/abs/1110.2956))

* {#BakerRichterSzymik12} [[Andrew Baker]], [[Birgit Richter]], [[Markus Szymik]], _Brauer groups for commutative $\mathbb{S}$-algebras_, J. Pure Appl. Algebra 216 (2012) 2361–2376 ([arXiv:1005.5370](http://arxiv.org/abs/1005.5370))

See also:

* Urs Schreiber, [Picard and Brauer 2-groups](http://golem.ph.utexas.edu/string/archives/000786.html), String Theory Coffee Table, 2006.

* John Baez, [The Brauer 3-group](https://golem.ph.utexas.edu/category/2020/05/the_brauer_3group.html), $n$-Category Café, 2020.

[[!redirects Picard 3-groups]]

[[!redirects Picard-Brauer 3-group]]
[[!redirects Picard-Brauer 3-groups]]

[[!redirects Brauer-Picard 3-group]]
[[!redirects Brauer-Picard 3-groups]]

[[!redirects Picard 2-groupoid]]
[[!redirects Picard 2-groupoids]]


Picard group

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## Idea

Fully generally, a _Picard group_ is an [[abelian group]] defined for a [[symmetric monoidal category]] as the group of [[isomorphism classes]] of [[objects]] which are [[invertible object|invertible]] with respect to the [[tensor product]].

Traditionally though one speaks in the context of [[geometry]] of the _Picard group_ $Pic(X)$ of some kind of [[space]] and by default means the invertible objects in some monoidal category of something like [[vector bundles]] over $X$.
Specifically for $X$ a [[ringed topos]] (in particular a [[ringed space]]), then the [[monoidal category]] to be understood is that of locally free [[module sheaves]] over the [[structure sheaf]] and hence the Picard group in this case is that of locally free sheaves of $\mathcal{O}_X$-modules of [[rank]] $1$ (i.e. the [[line bundles]]).

Specifically in [[complex geometry]] these objects on a [[complex manifold]] $X$ are [[holomorphic vector bundles]] and hence in this case the Picard group of a $X$ is that of isomorphism classes of [[holomorphic line bundles]]. This case has an obvious genralization to [[schemes]] in [[algebraic geometry]], and in much of the literature a _Picard group_ is meant to be a Picard group of $\mathbb{G}_m$-[[torsors]] over a given [[scheme]]. In this (and other) geometric situations, the Picard group naturally inherits geometric structure itself and equipped with that it is then called the _[[Picard scheme]]_ (with [[formal geometry|formal]] completion the _[[formal Picard group]]_), see there for more.

Not [[decategorification|decategorifying]] by passing to isomorphism classes instead yields the concept of [[Picard 2-group]] and geometrically that of [[Picard stack]], see there for more.

## Definition

### General abstract

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

Given a ([[symmetric monoidal category]]) [[monoidal category]] $(C, \otimes)$, the _Picard group of $(C,\otimes)$ is the group of [[isomorphism classes]] of [[invertible objects]], those that have an [[inverse]] under the [[tensor product]] -- the [[line objects]].
Equivalently, this is the [[decategorification]] or [[0-truncated|0-truncation]] of the [[Picard 2-group]], the maximal [[2-group]] inside a monoidal category.

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

The Picard group is indeed a [[group]]: First, if $\mathcal{L}$ and $\mathcal{M}$ are elements of $Pic(X)$, then $\mathcal{L}\otimes \mathcal{M}$ is still locally free of rank $1$ as can be seen by taking intersections of the trivializing covers. So $Pic(X)$ is closed under tensor product.

There is an identity element, since $\mathcal{O}_X\otimes \mathcal{L}\simeq \mathcal{L}$. The tensor product is associative.

Lastly, given any invertible sheaf $\mathcal{L}$ we check that $\mathcal{L}^\wedge=\mathcal{Hom}(\mathcal{L}, \mathcal{O}_X)$ is its inverse. Consider $\mathcal{L}^\wedge \otimes \mathcal{L}\simeq \mathcal{Hom}(\mathcal{L}, \mathcal{L})\simeq \mathcal{O}_X$.

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### In algebraic geometry

Suppose that $X$ is an integral [[scheme]] over a [[field]]. The correspondence between [[Cartier divisors]] and [[invertible sheaves]] is given by $D\mapsto \mathcal{O}_X(D)$. If $D$ is represented by $\{(U_i, f_i)\}$, then $\mathcal{O}_X(D)$ is $\mathcal{O}_X$-submodule of $\mathcal{K}$, the sheaf of quotients, generated by $f_i^{-1}$ on $U_i$. Under our assumptions, this map is an isomorphism between the Cartier class divisor group and Picard group, but for a general scheme it is only injective. Under the additional assumptions that $X$ is [[separated scheme|separated]] and locally factorial, we get an isomorphism between the class divisor group and $Pic(X)$.

Another form the Picard group takes is from the isomorphism $Pic(X)\simeq H^1(X, \mathcal{O}_X^*)$. The isomorphism is most easily seen by looking at the transition functions for a trivializing cover of $\mathcal{L}$. Suppose $(\phi_i)$ trivialize $\mathcal{L}$ over the cover $(U_i)$. Then $\phi_i^{-1}\circ \phi_j$ is an automorphism of $\mathcal{O}_{U_i\cap U_j}$, i.e. a section of $\mathcal{O}_X^*(U_i\cap U_j)$. One can check this defines a [[?ech cohomology|?ech cocycle]] $\check{H}^1(\mathcal{U}, \mathcal{O}_X^*)$ which is isomorphic to the abelian [[sheaf cohomology]] $H^1(X, \mathcal{O}_X^*)$.

## Examples

* over a [[Riemann surface]]: see at _[Riemann surface -- Picard group of holomorphic line bundles](Riemann+surface#PicardGroupOfHolomorphicLineBundles)_.

## Properties

* Let $R$ be the [[ring of integers]] of an algebraic number field. Then $R$ is a [[unique factorization domain]] if and only if the Picard group of $R$ is [[trivial group|trivial]].

## Related concepts

* [[group of units]], **Picard group**, [[Picard stack]]

  * [[Brauer group]], [[Azumaya algebra]]

* [[Picard 2-group]], [[Picard ∞-group]], [[Picard ∞-stack]]

* [[Picard scheme]]

* [[Tate?Shafarevich group]]

[[!include moduli of higher lines -- table]]

## References

### General

* [[John Baez]], *Hoàng Xuân Sính’s thesis: categorifying group theory* ([pdf](https://math.ucr.edu/home/baez/sinh.pdf))

### In algebraic geometry

* Robin Hartshorne, Algebraic Geometry

[[!redirects Picard groups]]


Picard groupoid

> This page considers Picard groupoids in themselves. For the concept of [[Picard groupoid of a monoidal category]], see there.

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##Introduction

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###### Definition

A _Picard groupoid_ is a [[symmetric monoidal|strict symmetric monoidal]] [[category]] $(\mathcal{A}, \otimes, 1)$ in which every [[object]] and every [[morphism]] is strictly [[inverse|invertible]] with respect to $\otimes$; that is to say: there is, for every object $a$ of $\mathcal{A}$, an object $a^{-1}$ of $\mathcal{A}$ such that $a \otimes a^{-1} = 1$, and similarly for morphisms.

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###### Remark

The notion of a Picard groupoid [[categorification|categorifies]] that of an [[abelian group]]. Note in particular that $\oplus$ and $0$ equip the set of objects of $\mathcal{A}$ with the structure of an abelian group.

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###### Remark

The notion of a Picard groupoid can be weakened in two (or three) directions: the monoidal structure can be weak instead of strict (or just the symmetry part), and the invertibility criterion can be asked to hold only up to isomorphism. On this page, we shall, however, work with the fully strict notion.

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##2-category of Picard groupoids

Picard groupoids assemble into a [[strict 2-category]]. The objects are Picard groupoids, the 1-arrows are strict [[monoidal functor|monoidal functors]] (these necessarily preserve both the symmetry and the object inverses), and the 2-arrows are [[monoidal natural transformation|monoidal natural transformations]].

This strict 2-category admits a [[closed monoidal category|closed]] [[symmetric monoidal 2-category|(fully) strict symmetric monoidal structure]], which categorifies the usual closed monoidal structure of the category of abelian groups.

##Additivity of the homotopy category of Picard groupoids

The _homotopy category_ of the category of Picard groupoids consists, roughly speaking, of Picard groupoids up to equivalence. Formally, we can obtain it by [[decategorifying]] the 2-category of Picard groupoids, namely identifying all 1-arrows which are 2-isomorphic, and throwing away the 2-arrows.

The closed monoidal structure of the 2-category of Picard groupoids, together with the fact that the objects of a Picard groupoid define an abelian group under $\oplus$ and $0$, provides immediately an [[enriched category|enrichment]] of the homotopy category of Picard groupoids over abelian groups. Moreover, it is obvious that this category has all coproducts. Thus it is an additive category.

##Model for stable homotopy 1-types

Picard groupoids are well-known to model stable homotopy 1-types, at least if one adopts the weak version of the invertibility condition. This is a stable version of the 1-truncated [[homotopy hypothesis]].

## Related concepts

* [[monoidal groupoid]]

* [[2-group]]

[[!redirects Picard groupoids]]

[[!redirects Picard category]]
[[!redirects Picard categories]]


Picard infinity-group

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## Definition

For $(\mathcal{C}, \otimes)$ a [[monoidal (∞,1)-category]], its **Picard $\infty$-group** is the [[∞-group]] induced on the [[core]] of the [[full sub-(∞,1)-category|full sub-]][[∞-groupoid]] $PIC(\mathcal{C}, \otimes)$ on those [[objects]] that are invertible under the [[tensor product]].

## Properties

### Relation to Brauer $\infty$-group and $\infty$-group of units

For $E$ an [[E-∞ ring]] and $Mod(E)$ its [[(∞,1)-category of ∞-modules]], then the Picard $\infty$-group is a "non-connected [[delooping]]" of the [[∞-group of units]] in that

$$
  \Omega Pic(E)\simeq GL_1(E)
  \,.
$$

Conversely $Pic(-)$ itself has a further non-connected delooping by the [[Brauer ∞-group]] $Br(-)$ in that

$$
  \Omega Br(E)\simeq Pic(E)
$$

([Szymik 11, theorem 5.7](#Szymik11)).

## Related concepts

* [[Picard group]], [[Picard stack]]

  * [[Brauer group]]

* [[Picard 2-group]]

* [[Picard 3-group]]

* **Picard $\infty$-group**, [[Picard ∞-stack]]

## References

* {#Sagave11} [[Steffen Sagave]], _Spectra of units for periodic ring spectra_ ([arXiv:1111.6731](http://arxiv.org/abs/1111.6731))

See also the discussion of higher [[Brauer groups]] in stable homotopy theory (which in turn are a "non-connective delooping"of $Pic(-)$) in

* {#Szymik11} [[Markus Szymik]], _Brauer spaces for commutative rings and structured ring spectra_ ([arXiv:1110.2956](http://arxiv.org/abs/1110.2956))

* {#BakerRichterSzymik12} [[Andrew Baker]], [[Birgit Richter]], [[Markus Szymik]], _Brauer groups for commutative $\mathbb{S}$-algebras_, J. Pure Appl. Algebra 216 (2012) 2361–2376 ([arXiv:1005.5370](http://arxiv.org/abs/1005.5370))

[[!redirects Picard infinity-group]]
[[!redirects Picard infinity-groups]]
[[!redirects Picard ∞-groups]]

[[!redirects Picard ∞-group]]

[[!redirects Picard ∞-groupoid]]
[[!redirects Picard infinity-groupoid]]
[[!redirects Picard ∞-groupoids]]
[[!redirects Picard infinity-groupoids]]


Picard scheme

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## Idea

For a [[ringed space]] $(X, \mathcal{O}_X)$ there is its [[Picard group]] of [[invertible objects]] in the category of $\mathcal{O}_X$-[[modules]]. When $X$ is a [[projective morphism|projective]] [[integral scheme]] over $k$ the [[Picard group]] underlies a $k$-scheme, this is the _Picard scheme_ $Pic_X$.
This scheme varies in a family as $X$ varies in a family. From this starting point one can naturally generalize to more general relative situations.

Often one considers just the connected component $Pic_X^0$ of the neutral element in $Pic_X$, and often (such as in the discussion below, beware) it is that connected component (only) which is referred to by "Picard scheme". The difference between the two is measured by the [[quotient]] $Pic_X/Pic_X^0$, which is called the _[[Néron-Severi group]]_ of $X$. Though at least for $X$ an [[algebraic curve]], $Pic_X^0$ goes by a separate name: it is the _[[Jacobian variety]]_ of $X$.

The [[completion]] of the Picard scheme at its neutral element (hence either of $Pic_X$ or $Pic_X^0$) is the _[[formal Picard group]]_.

## Definition

The __Picard variety__ of a complete [[smooth scheme|smooth]] algebraic [[variety]] $X$ over an [[algebraically closed field]] parametrizes the [[Picard group]] of $X$, more precisely the set of classes of isomorphic invertible [[quasicoherent sheaves]] with vanishing [[first Chern class]].

The __Picard scheme__ is a scheme [[representable functor|representing]] the relative Picard functor $Pic_{X/S}: (Sch/S)^{op}\to Set$ by $T\mapsto Pic(X_T)/f^*Pic(T)$. In this generality the Picard functor has been introduced by [[Grothendieck]] in [[FGA]], along with the proof of representability. An alternate form of this functor (with respect to the Zariski topology) in terms of the [[derived functor]] of $f_*$ is $Pic_{X/S}(T)=H^0(T, R^1f_{T*}\mathcal{O}_{X_T}^*)$.

Note we must work with the relative functor because the global Picard functor $Pic_X(T)=Pic(X_T)$ has no hope of being representable as it is not even a  [[sheaf]]. Consider any non-trivial invertible sheaf $\mathcal{L}$ in $Pic(T)$. Then $f^*\mathcal{L}$ becomes trivial on some cover $\{T_i\to T\}$, so $Pic(X_T)\to \prod Pic(X_{T_i})$ is not injective.

## Representability

For this section suppose $f:X\to S$ is s [[separated morphism of schemes|separated]] map, [[finite type]] map of schemes. Many general forms of [[representable functor|representability]] have been proven several of which are given in _[[FGA explained]]_. Here we list several of the common forms:

* Suppose $\mathcal{O}_S\to f_*\mathcal{O}_X$ is universally an isomorphism (stays an [[isomorphism]] after any [[base change]]), then we have a comparison of relative Picard functors $Pic_{X/S}\hookrightarrow Pic_{X/S, zar}\hookrightarrow Pic_{X/S, et}\hookrightarrow Pic_{X/S, fppf}$. They are all isomorphisms if $f$ has a section.

* If $Pic_{X/S}$ is representable by a scheme, then by [[descent theory]] for sheaves it is representable by the same scheme in all the topologies listed above. In general, representability gives representability in a finer topology (of the ones listed).

* If $Pic_{X/S}$ is representable then a universal sheaf $\mathcal{P}$ on $X\times Pic_{X/S}$ is called a [[Poincaré sheaf]]. It is universal in the following sense: if $T\to S$ and $\mathcal{L}$ is invertible on $X_T$, then there is a unique $h:T\to Pic_{X/S}$ such that for some $\mathcal{N}$ invertible on $T$ we get $\mathcal{L}\simeq (1\times h)^*\mathcal{P}\otimes f_T^*\mathcal{N}$.

* If $f$ is (Zariski) [[projective morphism|projective]], [[flat morphism|flat]] with [[integral scheme|integral]] geometric fibers then $Pic_{X/S, et}$ is representable by a [[separated scheme|separated]] and [[morphism of finite type|locally of finite type]] scheme over $S$.

* Grothendieck's Generic Representability: If $f$ is [[proper morphism|proper]] and $S$ is [[integral scheme|integral]], then there is a nonempty open $V\subset S$ such that $Pic_{X_V/V, fppf}$ is representable and is a disjoint union of open [[quasi-projective scheme| quasi-projective]] subschemes.

* If $f$ is a flat, cohomologically flat in dimension 0, proper, [[finitely presented]] map of of [[algebraic space]]s, then $Pic_{X/S}$ is representable by an algebraic space locally of finite presentation over $S$.

## Picard Stack
 {#PicardStack}

The *[[Picard stack]]* $\mathcal{Pic}_{X/S}$ is the [[stack]] of invertible sheaves on $X/S$, i.e. the [[fiber category]] over $T\to X$ is the [[groupoid]] of [[line bundles]] on $X_T$ (not just their [[isomorphism classes]]). (Hence it is the [[Picard groupoid of a monoidal category|Picard groupoid]] equipped with geometric structure).

If $X$ is proper and flat, then $\mathcal{Pic}_{X/S}$ is an [[Artin stack]] since $\mathcal{Pic}_{X/S}=\mathcal{Hom}(X, B\mathbb{G}_m)$ is the [[Hom stack]] which is Artin.

Note the following "failure" of the relative Picard scheme: points on $Pic_{X/S}$ do not parametrize line bundles. The low degree terms of the Leray [[spectral sequence]] give the following exact sequence $H^1(X_T, \mathbb{G}_m)\to H^0(T, R^1f_*\mathbb{G}_m)\to H^2(T, \mathbb{G}_m)\to H^2(X_T, \mathbb{G}_m)$, but as noted above $Pic_{X/S}(T)=H^0(T, R^1f_*\mathbb{G}_m)$, so we see exactly when a $T$-point comes from a line bundle it is when that point maps to $0$ in this sequence.

This gives us an obstruction theory lying in $H^2(T, \mathbb{G}_m)$ for a point corresponding to a line bundle. If $Pic_{X/S}$ is representable we could take $T=Pic_{X/S}$ to find a universal obstruction. Intuitively this is because the Picard stack is the right object to look at for the moduli problem of line bundles over $X$. The Picard scheme is the $\mathbb{G}_m$-[[rigidification]] of the Picard stack.

The natural map $\mathcal{Pic}_{X/S}\to Pic_{X/S}$ is a $\mathbb{G}_m$-[[gerbe]]. But isomorphism classes of $\mathbb{G}_m$-gerbes over $T$ are in bijective correspondence with $H^2(T, \mathbb{G}_m)$ and so the above map could be thought of as a geometric realization of the universal obstruction class.

## Related concepts

* [[dual abelian group scheme]]

* [[Poincaré line bundle]]

* [[Albanese variety]]

* [[Prym variety]]

[[!include moduli of higher lines -- table]]

## References

* Springer eom: [Picard variety](http://eom.springer.de/p/p072690.htm), [Picard scheme](http://eom.springer.de/p/p072670.htm)
* wikipedia [Picard group](http://en.wikipedia.org/wiki/Picard_group)
* [[Steven Kleiman|Steven L. Kleiman]], _The Picard scheme_, pp. 235--321 in [[FGA explained]], MR2223410 (draft [pdf](http://cdsagenda5.ictp.it//askArchive.php?categ=a0255&id=a0255s6t3&ifd=15022&down=1&type=lecture_notes)), [arxiv](http://arxiv.org/abs/math/0504020)

* [[Akhil Mathew]], _[The Picard Scheme I](http://amathew.wordpress.com/2013/03/19/the-picard-scheme-i/)_, _[The Picard Scheme II: deformation theory](http://amathew.wordpress.com/2013/03/19/the-picard-scheme-ii-deformation-theory/)_

Specifically on the [[Picard stack]]:

* [[The Stacks Project]], _[The Picard stack](http://stacks.math.columbia.edu/tag/0372)_

category: algebraic geometry

[[!redirects Picard schemes]]

[[!redirects Picard variety]]
[[!redirects Picard varieties]]