Editing Spinor (section)

== History ==
The most general mathematical form of spinors was discovered by [[Élie Cartan]] in 1913.<ref>{{Harvnb|Cartan|1913}}</ref> The word "spinor" was coined by [[Paul Ehrenfest]] in his work on [[quantum physics]].<ref>{{harvnb|Tomonaga|1998|p=129}}</ref>

Spinors were first applied to [[mathematical physics]] by [[Wolfgang Pauli]] in 1927, when he introduced his [[Pauli matrices|spin matrices]].<ref>{{Harvnb|Pauli|1927}}.</ref> The following year, [[Paul Dirac]] discovered the fully [[special relativity|relativistic]] theory of [[electron]] [[Spin (physics)|spin]] by showing the connection between spinors and the [[Lorentz group]].<ref>{{Harvnb|Dirac|1928}}.</ref> By the 1930s, Dirac, [[Piet Hein (Denmark)|Piet Hein]] and others at the [[Niels Bohr Institute]] (then known as the Institute for Theoretical Physics of the University of Copenhagen) created toys such as [[Tangloids]] to teach and model the calculus of spinors.

Spinor spaces were represented as [[left ideal]]s of a matrix algebra in 1930, by [[Gustave Juvett]]<ref>{{cite journal |last1=Juvet |first1=G. |year=1930 |title=Opérateurs de Dirac et équations de Maxwell |language=fr |journal=[[Commentarii Mathematici Helvetici]] |volume=2 |pages=225–235 |doi=10.1007/BF01214461|s2cid=121226923 }}</ref> and by [[Fritz Sauter]].<ref>{{cite journal |first=F. |last=Sauter |title=Lösung der Diracschen Gleichungen ohne Spezialisierung der Diracschen Operatoren |journal=Zeitschrift für Physik |volume=63 |issue=11–12 |pages=803–814 |doi=10.1007/BF01339277 |bibcode=1930ZPhy...63..803S |year=1930|s2cid=122940202 }}</ref><ref name="lounesto-1995-p151">Pertti Lounesto: ''[[Albert Crumeyrolle|Crumeyrolle]]'s bivectors and spinors'', pp.&nbsp;137–166, In: Rafał Abłamowicz, Pertti Lounesto (eds.): ''Clifford algebras and spinor structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992)'', {{isbn|0-7923-3366-7}}, 1995, [https://books.google.com/books?id=DnyUDg483kEC&pg=PA151 p. 151]</ref> More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero. In this manner the spinor space became a [[minimal ideal|minimal left ideal]] in {{math|Mat(2, <math>\Complex</math>)}}.{{efn|The matrices of dimension ''N'' × ''N'' in which only the elements of the left column are non-zero form a ''left ideal'' in the ''N'' × ''N'' matrix algebra {{math|Mat(''N'', <math>\Complex</math>)}} – multiplying such a matrix ''M'' from the left with any ''N'' × ''N'' matrix ''A'' gives the result ''AM'' that is again an ''N'' × ''N'' matrix in which only the elements of the left column are non-zero. Moreover, it can be shown that it is a ''minimal left ideal''.<ref>See also: Pertti Lounesto: ''Clifford algebras and spinors'', London Mathematical Society Lecture Notes Series 286, Cambridge University Press, Second Edition 2001, {{ISBN|978-0-521-00551-7}}, p.&nbsp;52</ref>}}<ref name="lounesto-2001-p148f-p327f">Pertti Lounesto: ''Clifford algebras and spinors'', London Mathematical Society Lecture Notes Series 286, Cambridge University Press, Second Edition 2001, {{ISBN|978-0-521-00551-7}}, p.&nbsp;148&nbsp;f. and [https://books.google.com/books?id=DTecU6UpkSgC&pg=PA327 p. 327 f.]</ref>

In 1947 [[Marcel Riesz]] constructed spinor spaces as elements of a minimal left ideal of [[Clifford algebra]]s. In 1966/1967, [[David Hestenes]]<ref>D. Hestenes: ''Space–Time Algebra'', Gordon and Breach, New York, 1966, 1987, 1992</ref><ref>{{cite journal |first=D. |last=Hestenes |title=Real spinor fields |journal=[[Journal of Mathematical Physics|J. Math. Phys.]] |volume=8 |year=1967 |issue=4 |pages=798–808 |doi=10.1063/1.1705279 |bibcode=1967JMP.....8..798H |s2cid=13371668 |doi-access=free |url=https://davidhestenes.net/geocalc/pdf/RealSpinorFields.pdf }}</ref> replaced spinor spaces by the [[even subalgebra]] Cℓ<sup>0</sup><sub>1,3</sub>(<math>\Reals</math>) of the [[spacetime algebra]] Cℓ<sub>1,3</sub>(<math>\Reals</math>).<ref name="lounesto-1995-p151" /><ref name="lounesto-2001-p148f-p327f" /> As of the 1980s, the theoretical physics group at [[Birkbeck College]] around [[David Bohm]] and [[Basil Hiley]] has been developing [[Basil Hiley#Implicate orders, pre-space and algebraic structures|algebraic approaches to quantum theory]] that build on Sauter and Riesz' identification of spinors with minimal left ideals.