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| In [[mathematics]] and [[physics]], in particular in the theory of the [[orthogonal group]]s (such as the [[rotation group SO(3)|rotation]] or the [[Lorentz group]]s)<!--please keep this and similar remarks for readers like physics students -->, a '''spinor''' {{IPAc-en|'|s|p|ɪ|n|ə|r}} is an element of a complex [[vector space]]. Unlike [[spatial vector]]s, spinors only [[Coordinate_transformation#Transformations|transform]] "[[up to]] a sign" under the full [[orthogonal group]]. This means that a [[turn (geometry)|360 degree]] rotation transforms the numeric coordinates of a spinor into their negatives, and so it takes a rotation of 720 degrees to re-obtain the original values. Spinors are objects associated to a vector space with a [[quadratic form]] (like [[Euclidean space]] with the standard metric or [[Minkowski space]] with the [[Lorentz metric]]), and are realized as elements of representation spaces of [[Clifford algebras]]. For a given quadratic form, several different spaces of spinors with extra properties may exist. | |
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| Spinors are like vectors and [[tensors]] in that their definition includes their transformation properties, although unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors. Like vectors, spinors can be transformed under the [[Infinitesimal transformation|infinitesimal]] orthogonal transformations (like infinitesimal [[rotation]]s or infinitesimal [[Lorentz transformation]]s).
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| Spinors in general were discovered by [[Élie Cartan]] in 1913.<ref>{{Harvnb|Cartan|1913}}.</ref> Later, spinors were adopted by [[quantum mechanics]] in order to study the properties of the [[intrinsic angular momentum]] of the [[electron]] and other [[fermion]]s. Today spinors enjoy a wide range of physics applications. Classically, [[spinors in three dimensions]] are used to describe the spin of the non-relativistic electron and other [[spin-½]] particles. Via the [[Dirac equation]], [[Dirac spinor]]s are required in the mathematical description of the [[quantum state]] of the [[Theory of Relativity|relativistic]] electron. In [[quantum field theory]], spinors describe the state of relativistic many-particle systems. In mathematics, particularly in [[differential geometry]] and [[global analysis]], spinors have since found broad applications to [[algebraic topology|algebraic]] and [[differential topology]],<ref>{{Harvnb|Hitchin|1974}}, {{Harvnb|Lawson|Michelsohn|1989}}.</ref> [[symplectic geometry]], [[gauge theory]], [[complex variety|complex algebraic geometry]],<ref>{{Harvnb|Hitchin|1974}}, {{Harvnb|Penrose|Rindler|1988}}.</ref> [[index theory]],<ref>{{Harvnb|Gilkey|1984}}, {{Harvnb|Lawson|Michelsohn|1989}}.</ref> and [[holonomy|special holonomy]].<ref>{{Harvnb|Lawson|Michelsohn|1989}}, {{Harvnb|Harvey|1990}}. These two books also provide good mathematical introductions and fairly comprehensive bibliographies on the mathematical applications of spinors as of 1989–1990.</ref>
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| ==Overview==
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| In the classical geometry of space, a vector exhibits a certain behavior when it is acted upon by a rotation or reflected in a hyperplane. However, in a certain sense rotations and reflections contain finer geometrical information than can be expressed in terms of their actions on vectors. Spinors are objects constructed in order to encompass more fully this geometry. (See [[orientation entanglement]].)
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| There are essentially two frameworks for viewing the notion of a spinor.
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| One is [[representation theory|representation theoretic]]. In this point of view, one knows ''a priori'' that there are some representations of the [[Lie algebra]] of the [[orthogonal group]] that cannot be formed by the usual tensor constructions. These missing representations are then labeled the '''[[spin representation]]s''', and their constituents ''spinors''. In this view, a spinor must belong to a [[group representation|representation]] of the [[covering space|double cover]] of the [[special orthogonal group|rotation group]] {{nowrap|SO(''n'', '''R''')}}, or more generally of the [[generalized special orthogonal group]] {{nowrap|SO<sup>+</sup>(''p'', ''q'', '''R''')}} on spaces with [[metric signature]] {{nowrap|(''p'', ''q'')}}. These double-covers are [[Lie groups]], called the [[spin group]]s {{nowrap|Spin(''p'', ''q'')}}. All the properties of spinors, and their applications and derived objects, are manifested first in the spin group. Representations of the double covers of these groups yield [[projective representation]]s of the groups themselves, which do not meet the full definition of a representation.
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| The other point of view is geometrical. One can explicitly construct the spinors, and then examine how they behave under the action of the relevant Lie groups. This latter approach has the advantage of providing a concrete and elementary description of what a spinor is. However, such a description becomes unwieldy when complicated properties of spinors, such as [[Fierz identity|Fierz identities]], are needed.
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| ===Clifford algebras===
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| {{details|Clifford algebra}}
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| The language of [[Clifford algebra]]s<ref>Named after [[William Kingdon Clifford]],</ref> (also called [[geometric algebra]]s) provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via the [[classification of Clifford algebras]]. It largely removes the need for ''ad hoc'' constructions.
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| In detail, if ''V'' is a finite-dimensional complex vector space with nondegenerate bilinear form ''g'', the Clifford algebra {{nowrap|Cℓ(''V'', ''g'')}} is the algebra generated by ''V'' along with the anticommutation relation {{nowrap|1=''xy'' + ''yx'' = 2''g''(''x'', ''y'')}}. It is an abstract version of the algebra generated by the [[gamma matrices|gamma]] or [[Pauli matrices]]. Such a Clifford algebra may also be denoted Cℓ<sub>''n''</sub>('''C'''), where the '''C''' denotes that the vectors of V are built over the complex numbers, and {{nowrap|1=''n'' = dim(''V'')}}. The Clifford algebra Cℓ<sub>''n''</sub>('''C''') is algebraically isomorphic to the algebra {{nowrap|Mat(2<sup>''k''</sup>, '''C''')}} of {{nowrap|2<sup>''k''</sup> × 2<sup>''k''</sup>}} complex matrices, if {{nowrap|1=''n'' = dim(''V'') = 2''k''}} is even; or the algebra {{nowrap|Mat(2<sup>''k''</sup>, '''C''') ⊕ Mat(2<sup>''k''</sup>, '''C''')}} of two copies of the {{nowrap|2<sup>''k''</sup> × 2<sup>''k''</sup>}} matrices, if {{nowrap|1=''n'' = dim(''V'') = 2''k'' + 1}} is odd. It therefore has a unique irreducible representation (also called simple [[Clifford module]]), commonly denoted by Δ, whose dimension is 2<sup>''k''</sup>. The Lie algebra {{nowrap|'''so'''(''V'', ''g'')}} is embedded as a Lie subalgebra in {{nowrap|Cℓ(''V'', ''g'')}} equipped with the Clifford algebra [[commutator]] as Lie bracket. Therefore, the space Δ is also a Lie algebra representation of {{nowrap|'''so'''(''V'', ''g'')}} called a [[spin representation]]. If ''n'' is odd, this representation is irreducible. If ''n'' is even, it splits again into two irreducible representations {{nowrap|1=Δ = Δ<sub>+</sub> ⊕ Δ<sub>−</sub>}} called the ''half-spin representations''.
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| Irreducible representations over the reals in the case when ''V'' is a real vector space are much more intricate, and the reader is referred to the [[Clifford algebra#Spinors|Clifford algebra]] article for more details.
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| ===Terminology in physics===
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| The most typical type of spinor, the '''[[Dirac spinor]]''',<ref>Named after [[Paul Dirac]].</ref> is an element of the fundamental representation of {{nowrap|Cℓ<sub>''p''+''q''</sub>('''C''')}}, the complexification of the [[Clifford algebra]] {{nowrap|Cℓ<sub>''p'', ''q''</sub>('''R''')}}, into which the spin group {{nowrap|Spin(''p'', ''q'')}} may be embedded. On a 2''k''- or 2''k''+1-dimensional space a Dirac spinor may be represented as a vector of 2<sup>''k''</sup> [[complex number]]s. (See [[Special unitary group]].) In even dimensions, this representation is [[reducible representation|reducible]] when taken as a [[representations of Lie groups/algebras|representation]] of {{nowrap|Spin(''p'', ''q'')}} and may be decomposed into two: the left-handed and right-handed '''Weyl spinor'''<ref>Named after [[Hermann Weyl]].</ref> representations. In addition, sometimes the non-complexified version of {{nowrap|Cℓ<sub>''p'',''q''</sub>('''R''')}} has a smaller real representation, the '''Majorana spinor''' representation.<ref>Named after [[Ettore Majorana]].</ref> If this happens in an even dimension, the Majorana spinor representation will sometimes decompose into two '''Majorana–Weyl spinor''' representations.
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| Of all these, only the Dirac representation exists in all dimensions.{{Clarify|date=February 2011}}{{Citation needed|date=September 2012}} Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.
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| The Dirac, Lorentz, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra.<ref>Matthew R. Francis, Arthur Kosowsky: ''The Construction of Spinors in Geometric Algebra'', submitted 20 March 2004, version of 18 October 2004 [http://arxiv.org/abs/math-ph/0403040v2 arXiv:math-ph/0403040]</ref>
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| ===Spinors in representation theory===
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| {{main|Spin representation}}
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| One major mathematical application of the construction of spinors is to make possible the explicit construction of [[linear representation]]s of the [[Lie algebra]]s of the [[special orthogonal group]]s, and consequently spinor representations of the groups themselves. At a more profound level, spinors have been found to be at the heart of approaches to the [[Atiyah–Singer index theorem]], and to provide constructions in particular for [[discrete series]] representations of [[semisimple group]]s.
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| The spin representations of the special orthogonal Lie algebras are distinguished from the [[tensor]] representations given by [[Young symmetrizer|Weyl's construction]] by the [[weight (representation theory)|weights]]. Whereas the weights of the tensor representations are integer linear combinations of the roots of the Lie algebra, those of the spin representations are half-integer linear combinations thereof. Explicit details can be found in the [[spin representation]] article.
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| ===Attempts at intuitive understanding===
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| The spinor can be described, in simple terms, as “vectors of a space the transformations of which are related in a particular way to rotations in physical space”.<ref>Jean Hladik: ''Spinors in Physics'', translated by J. M. Cole, Springer 1999, ISBN 978-0-387-98647-0, p. 3</ref> Several ways of illustrating everyday analogies have been formulated in terms of the [[plate trick]], [[tangloids]] and other examples of [[orientation entanglement]].
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| Nonetheless, the concept is generally considered notoriously difficult to understand, as illustrated by [[Michael Atiyah]]'s statement that is recounted by Dirac's biographer Graham Farmelo:<ref>Graham Farmelo: ''The Strangest Man. The Hidden Life of Paul Dirac, Quantum Genius'', Faber & Faber, 2009, ISBN 978-0-571-22286-5, p. 430</ref>
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| :''No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the “square root” of geometry and, just as understanding the [[square root of −1]] took centuries, the same might be true of spinors.''
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| == History ==
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| 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>
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| 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.
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| Spinor spaces were represented as [[left ideal]]s of a matrix algebra in 1930, by G. Juvet<ref>G. Juvet: ''Opérateurs de Dirac et équations de Maxwell'', Commentarii Mathematici Helvelvetici, 2 (1930), pp. 225–235, {{doi|10.1007/BF01214461}} ([http://www.springerlink.com/content/m7p8j4436220040t/ abstract] in French language)</ref> and by [[Fritz Sauter]].<ref>F. Sauter: ''Lösung der Diracschen Gleichungen ohne Spezialisierung der Diracschen Operatoren'', Zeitschrift für Physik, Volume 63, Numbers 11–12, 803–814, {{doi|10.1007/BF01339277}} ([http://www.springerlink.com/content/x62t12x0484056t5/ abstract] in German language)</ref><ref name="lounesto-1995-p151">Pertti Lounesto: ''[[Albert Crumeyrolle|Crumeyrolle]]'s bivectors and spinors'', pp. 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, [http://books.google.de/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 {{nowrap|Mat(2, '''C''')}}.<ref>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 {{nowrap|Mat(''N'', '''C''')}} – 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''. See also: Pertti Lounesto: ''Clifford algebras and spinors'', London Mathematical Society Lecture Notes Series 286, Cambridge University Press, Second Edition 2001, DOI 978-0-521-00551-7, p. 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, DOI 978-0-521-00551-7, p. 148 f. and [http://books.google.de/books?id=DTecU6UpkSgC&pg=PA327 p. 327 f.]</ref>
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| 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>D. Hestenes: ''Real spinor fields'', J. Math. Phys. 8 (1967), pp. 798–808</ref> replaced spinor spaces by the [[even subalgebra]] Cℓ<sup>0</sup><sub>1,3</sub>('''R''') of the [[spacetime algebra]] Cℓ<sub>1,3</sub>('''R''').<ref name="lounesto-2001-p148f-p327f"/><ref name="lounesto-1995-p151"/> 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.
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| == Examples ==
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| Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra {{nowrap|Cℓ<sub>''p'', ''q''</sub>('''R''')}}. This is an algebra built up from an orthonormal basis of {{nowrap|1=''n'' = ''p'' + ''q''}} mutually orthogonal vectors under addition and multiplication, ''p'' of which have norm +1 and ''q'' of which have norm −1, with the product rule for the basis vectors
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| :<math>e_i e_j = \Bigg\{ \begin{matrix} +1 & i=j, \, i \in (1 \ldots p) \\
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| -1 & i=j, \, i \in (p+1 \ldots n) \\
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| - e_j e_i & i \not = j. \end{matrix}</math>
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| === Two dimensions ===
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| The Clifford algebra Cℓ<sub>2,0</sub>('''R''') is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, ''σ''<sub>1</sub> and ''σ''<sub>2</sub>, and one unit [[pseudoscalar]] {{nowrap|1=''i'' = ''σ''<sub>1</sub>''σ''<sub>2</sub>}}. From the definitions above, it is evident that {{nowrap|1=(''σ''<sub>1</sub>'')<sup>2</sup> = (''σ''<sub>2</sub>'')<sup>2</sup> = 1}}, and {{nowrap|1=(''σ''<sub>1</sub>''σ''<sub>2</sub>)(''σ''<sub>1</sub>''σ''<sub>2</sub>) = −''σ''<sub>1</sub>''σ''<sub>1</sub>''σ''<sub>2</sub>''σ''<sub>2</sub> = −1}}.
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| The even subalgebra Cℓ<sup>0</sup><sub>2,0</sub>('''R'''), spanned by ''even-graded'' basis elements of Cℓ<sub>2,0</sub>('''R'''), determines the space of spinors via its representations. It is made up of real linear combinations of 1 and ''σ''<sub>1</sub>''σ''<sub>2</sub>. As a real algebra, Cℓ<sup>0</sup><sub>2,0</sub>('''R''') is isomorphic to field of [[complex numbers]] '''C'''. As a result, it admits a conjugation operation (analogous to [[complex conjugate|complex conjugation]]), sometimes called the ''reverse'' of a Clifford element, defined by
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| :<math>(a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1\,</math>.
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| which, by the Clifford relations, can be written
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| :<math>(a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1 = a-b\sigma_1\sigma_2\,</math>.
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| The action of an even Clifford element {{nowrap|''γ'' ∈ Cℓ<sup>0</sup><sub>2,0</sub>('''R''')}} on vectors, regarded as 1-graded elements of Cℓ<sub>2,0</sub>('''R'''), is determined by mapping a general vector {{nowrap|1=''u'' = ''a''<sub>1</sub>''σ''<sub>1</sub> + ''a''<sub>2</sub>''σ''<sub>2</sub>}} to the vector
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| :<math>\gamma(u) = \gamma u \gamma^*\,</math>,
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| where ''γ''<sup>∗</sup> is the conjugate of ''γ'', and the product is Clifford multiplication. In this situation, a '''spinor'''<ref>These are the right-handed Weyl spinors in two dimensions. For the left-handed Weyl spinors, the representation is via {{nowrap|1=''γ''(''ϕ'') = ''{{overline|γ}}ϕ''}}. The Majorana spinors are the common underlying real representation for the Weyl representations.</ref> is an ordinary complex number. The action of ''γ'' on a spinor ''φ'' is given by ordinary complex multiplication:
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| :<math> \gamma(\phi) = \gamma\phi</math>.
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| An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the even-graded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors:
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| :<math> \gamma(u) = \gamma u \gamma^* = \gamma^2 u\,</math>.
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| On the other hand, comparing with the action on spinors {{nowrap|1=''γ''(''φ'') = ''γφ''}}, ''γ'' on ordinary vectors acts as the ''square'' of its action on spinors.
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| Consider, for example, the implication this has for plane rotations. Rotating a vector through an angle of ''θ'' corresponds to {{nowrap|1=''γ''<sup>2</sup> = exp(''θ σ''<sub>1</sub>''σ''<sub>2</sub>)}}, so that the corresponding action on spinors is via {{nowrap|1=''γ'' = ± exp(''θ σ''<sub>1</sub>''σ''<sub>2</sub>/2)}}. In general, because of [[branch cut|logarithmic branching]], it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is two-valued.
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| In applications of spinors in two dimensions, it is common to exploit the fact that the algebra of even-graded elements (that is just the ring of complex numbers) is identical to the space of spinors. So, by [[abuse of notation|abuse of language]], the two are often conflated. One may then talk about "the action of a spinor on a vector." In a general setting, such statements are meaningless. But in dimensions 2 and 3 (as applied, for example, to [[computer graphics]]) they make sense.
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| ;Examples
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| * The even-graded element
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| ::<math>\gamma = \tfrac{1}{\sqrt{2}} (1 - \sigma_1 \sigma_2) \,</math>
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| :corresponds to a vector rotation of 90° from ''σ''<sub>1</sub> around towards ''σ''<sub>2</sub>, which can be checked by confirming that
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| ::<math>\tfrac{1}{2} (1 - \sigma_1 \sigma_2) \, \{a_1\sigma_1+a_2\sigma_2\} \, (1 - \sigma_2 \sigma_1) = a_1\sigma_2 - a_2\sigma_1 \,</math>
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| :It corresponds to a spinor rotation of only 45°, however:
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| ::<math>\tfrac{1}{\sqrt{2}} (1 - \sigma_1 \sigma_2) \, \{a_1+a_2\sigma_1\sigma_2\}=
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| \frac{a_1+a_2}{\sqrt{2}} + \frac{-a_1+a_2}{\sqrt{2}}\sigma_1\sigma_2</math>
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| * Similarly the even-graded element {{nowrap|1=''γ'' = −''σ''<sub>1</sub>''σ''<sub>2</sub>}} corresponds to a vector rotation of 180°:
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| ::<math> (- \sigma_1 \sigma_2) \, \{a_1\sigma_1 + a_2\sigma_2\} \, (- \sigma_2 \sigma_1) = - a_1\sigma_1 -a_2\sigma_2 \,</math>
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| : but a spinor rotation of only 90°:
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| ::<math>(- \sigma_1 \sigma_2) \, \{a_1 + a_2\sigma_1\sigma_2\}
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| =a_2 - a_1\sigma_1\sigma_2</math>
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| * Continuing on further, the even-graded element {{nowrap|1=''γ'' = −1}} corresponds to a vector rotation of 360°:
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| ::<math> (-1) \, \{a_1\sigma_1+a_2\sigma_2\} \, (-1) = a_1\sigma_1+a_2\sigma_2 \,</math>
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| : but a spinor rotation of 180°.
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| === Three dimensions ===
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| :''Main articles [[Spinors in three dimensions]], [[Quaternions and spatial rotation]]''
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| The Clifford algebra Cℓ<sub>3,0</sub>('''R''') is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, ''σ''<sub>1</sub>, ''σ''<sub>2</sub> and ''σ''<sub>3</sub>, the three unit bivectors ''σ''<sub>1</sub>''σ''<sub>2</sub>, ''σ''<sub>2</sub>''σ''<sub>3</sub>, ''σ''<sub>3</sub>''σ''<sub>1</sub> and the [[pseudoscalar]] {{nowrap|1=''i'' = ''σ''<sub>1</sub>''σ''<sub>2</sub>''σ''<sub>3</sub>}}. It is straightforward to show that {{nowrap|1=(''σ''<sub>1</sub>)<sup>2</sup> = (''σ''<sub>2</sub>)<sup>2</sup> = (''σ''<sub>3</sub>)<sup>2</sup> = 1}}, and {{nowrap|1=(''σ''<sub>1</sub>''σ''<sub>2</sub>)<sup>2</sup> = (''σ''<sub>2</sub>''σ''<sub>3</sub>)<sup>2</sup> = (''σ''<sub>3</sub>''σ''<sub>1</sub>)<sup>2</sup> = (''σ''<sub>1</sub>''σ''<sub>2</sub>''σ''<sub>3</sub>)<sup>2</sup> = −1}}.
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| The sub-algebra of even-graded elements is made up of scalar dilations,
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| :<math>u^{\prime} = \rho^{(1/2)} u \rho^{(1/2)} = \rho u, </math>
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| and vector rotations
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| :<math>u^{\prime} = \gamma \, u \, \gamma^*, </math>
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| where
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| :<math>\left.\begin{matrix} \gamma & = & \cos(\theta/2) - \{a_1 \sigma_2\sigma_3 + a_2 \sigma_3\sigma_1 + a_3 \sigma_1\sigma_2\} \sin(\theta/2) \\
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| & = & \cos(\theta/2) - i \{a_1 \sigma_1 + a_2 \sigma_2 + a_3 \sigma_3\} \sin(\theta/2) \\
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| & = & \cos(\theta/2) - i v \sin(\theta/2) \end{matrix}\right\}</math> (1)
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| corresponds to a vector rotation through an angle ''θ'' about an axis defined by a unit vector {{nowrap|1=''v'' = ''a''<sub>1</sub>''σ''<sub>1</sub> + ''a''<sub>2</sub>''σ''<sub>2</sub> + ''a''<sub>3</sub>''σ''<sub>3</sub>}}.
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| As a special case, it is easy to see that, if {{nowrap|1=''v'' = ''σ''<sub>3</sub>}}, this reproduces the ''σ''<sub>1</sub>''σ''<sub>2</sub> rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in the ''σ''<sub>3</sub> direction invariant, since
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| :<math>(\cos(\theta/2) - i \sigma_3 \sin(\theta/2)) \, \sigma_3 \, (\cos(\theta/2) + i \sigma_3 \sin(\theta/2))
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| = (\cos^2(\theta/2) + \sin^2(\theta/2)) \, \sigma_3 = \sigma_3.</math>
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| The bivectors ''σ''<sub>2</sub>''σ''<sub>3</sub>, ''σ''<sub>3</sub>''σ''<sub>1</sub> and ''σ''<sub>1</sub>''σ''<sub>2</sub> are in fact [[William Rowan Hamilton|Hamilton's]] [[quaternion]]s '''i''', '''j''' and '''k''', discovered in 1843:
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| :<math>\begin{matrix}\mathbf{i} = -\sigma_2 \sigma_3 = -i \sigma_1 \\
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| \mathbf{j} = -\sigma_3 \sigma_1 = -i \sigma_2 \\
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| \mathbf{k} = -\sigma_1 \sigma_2 = -i \sigma_3. \end{matrix}</math>
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| With the identification of the even-graded elements with the algebra '''H''' of quaternions, as in the case of two dimensions the only representation of the algebra of even-graded elements is on itself.<ref>Since, for a [[skew field]], the kernel of the representation must be trivial. So inequivalent representations can only arise via an [[automorphism]] of the skew-field. In this case, there are a pair of equivalent representations: {{nowrap|1=''γ''(''ϕ'') = ''γϕ''}}, and its quaternionic conjugate {{nowrap|1=''γ''(''ϕ'') = ''ϕ{{overline|γ}}''}}.</ref> Thus the (real<ref>The complex spinors are obtained as the representations of the [[tensor product]] {{nowrap|1='''H''' ⊗<sub>'''R'''</sub> '''C''' = Mat<sub>2</sub>('''C''')}}. These are considered in more detail in [[spinors in three dimensions]].</ref>) spinors in three-dimensions are quaternions, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication.
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| Note that the expression (1) for a vector rotation through an angle ''θ'', the angle appearing in ''γ'' was halved. Thus the spinor rotation {{nowrap|1=''γ''(''ψ'') = ''γψ''}} (ordinary quaternionic multiplication) will rotate the spinor ''ψ'' through an angle one-half the measure of the angle of the corresponding vector rotation. Once again, the problem of lifting a vector rotation to a spinor rotation is two-valued: the expression (1) with {{nowrap|(180° + ''θ''/2)}} in place of ''θ''/2 will produce the same vector rotation, but the negative of the spinor rotation.
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| The spinor/quaternion representation of rotations in 3D is becoming increasingly prevalent in computer geometry and other applications, because of the notable brevity of the corresponding spin matrix, and the simplicity with which they can be multiplied together to calculate the combined effect of successive rotations about different axes.
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| == Explicit constructions ==
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| | |
| A space of spinors can be constructed explicitly with concrete and abstract constructions. The
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| equivalence of these constructions are a consequence of the uniqueness of the spinor representation of the complex Clifford algebra. For a complete example in dimension 3, see [[spinors in three dimensions]].
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| ===Component spinors===
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| Given a vector space ''V'' and a quadratic form ''g'' an explicit matrix representation of the Clifford algebra {{nowrap|Cℓ(''V'', ''g'')}} can be defined as follows. Choose an orthonormal basis {{nowrap|''e''<sup>1</sup> … ''e''<sup>n</sup>}} for ''V'' i.e. {{nowrap|1=''g''(''e''<sup>''μ''</sup>''e''<sup>''ν''</sup>) = ''η''<sup>''μν''</sup>}} where {{nowrap|1=''η''<sup>''μμ''</sup> = ±1}} and {{nowrap|1=''η''<sup>''μν''</sup> = 0}} for {{nowrap|''μ'' ≠ ''ν''}}. Let {{nowrap|1=''k'' = ⌊ ''n''/2 ⌋}}. Fix a set of {{nowrap|2<sup>''k''</sup> × 2<sup>''k''</sup>}} matrices {{nowrap|''γ''<sup>1</sup> … ''γ''<sup>''n''</sup>}} such that {{nowrap|1=''γ''<sup>''μ''</sup>''γ''<sup>''ν''</sup> + ''γ''<sup>''ν''</sup>''γ''<sup>''μ''</sup> = ''η''<sup>''μν''</sup>1}} (i.e. fix a convention for the [[gamma matrices]]). Then the assignment {{nowrap|''e''<sup>''μ''</sup> → ''γ''<sup>''μ''</sup>}} extends uniquely to an algebra homomorphism {{nowrap|Cℓ(''V'', ''g'') → Mat(2<sup>''k''</sup>, '''C''')}} by sending the monomial {{nowrap|''e''<sup>''μ''<sub>1</sub></sup> … ''e''<sup>''μ''<sub>''k''</sub></sup>}} in the Clifford algebra to the product {{nowrap|''γ''<sup>''μ''<sub>1</sub></sup> … ''γ''<sup>''μ''<sub>''k''</sub></sup>}} of matrices and extending linearly. The space {{nowrap|1=Δ = '''C'''<sup>2<sup>''k''</sup></sup>}} on which the gamma matrices act is a now a space of spinors. One needs to construct such matrices explicitly, however. In dimension 3, defining the gamma matrices to be the [[Pauli matrices|Pauli sigma matrices]] gives rise to the familiar two component spinors used in non relativistic [[quantum mechanics]]. Likewise using the {{nowrap|4 × 4}} Dirac gamma matrices gives rise to the 4 component Dirac spinors used in 3+1 dimensional relativistic [[quantum field theory]]. In general, in order to define gamma matrices of the required kind, one can use the [[Weyl–Brauer matrices]].
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| | |
| In this construction the representation of the Clifford algebra {{nowrap|Cℓ(''V'', ''g'')}}, the Lie algebra {{nowrap|'''so'''(''V'', ''g'')}}, and the Spin group {{nowrap|Spin(''V'', ''g'')}}, all depend on the choice of the orthonormal basis and the choice of the gamma matrices. This can cause confusion over conventions, but invariants like traces are independent of choices. In particular, all physically observable quantities must be independent of such choices. In this construction a spinor can be represented as a vector of 2<sup>''k''</sup> complex numbers and is denoted with spinor indices (usually ''α'', ''β'', ''γ''). In the physics literature, [[abstract indices|abstract spinor indices]] are often used to denote spinors even when an abstract spinor construction is used.
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| ===Abstract spinors===
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| There are at least two different, but essentially equivalent, ways to define spinors abstractly. One approach seeks to identify the minimal ideals for the left action of {{nowrap|Cℓ(''V'', ''g'')}} on itself. These are subspaces of the Clifford algebra of the form {{nowrap|Cℓ(''V'', ''g'')''ω''}}, admitting the evident action of {{nowrap|Cℓ(''V'', ''g'')}} by left-multiplication: {{nowrap|''c'' : ''xω'' → ''cxω''}}. There are two variations on this theme: one can either find a primitive element ''ω'' that is a [[nilpotent]] element of the Clifford algebra, or one that is an [[idempotent]]. The construction via nilpotent elements is more fundamental in the sense that an idempotent may then be produced from it.<ref>This construction is due to Cartan. The treatment here is based on {{Harvtxt|Chevalley|1954}}.</ref> In this way, the spinor representations are identified with certain subspaces of the Clifford algebra itself. The second approach is to construct a vector space using a distinguished subspace of ''V'', and then specify the action of the Clifford algebra ''externally'' to that vector space.
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| | |
| In either approach, the fundamental notion is that of an [[isotropic subspace]] ''W''. Each construction depends on an initial freedom in choosing this subspace. In physical terms, this corresponds to the fact that there is no measurement protocol that can specify a basis of the spin space, even if a preferred basis of ''V'' is given.
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| As above, we let {{nowrap|(''V'', ''g'')}} be an ''n''-dimensional complex vector space equipped with a nondegenerate bilinear form. If ''V'' is a real vector space, then we replace ''V'' by its [[complexification]] {{nowrap|''V'' ⊗<sub>'''R'''</sub> '''C'''}} and let ''g'' denote the induced bilinear form on {{nowrap|''V'' ⊗<sub>'''R'''</sub> '''C'''}}. Let ''W'' be a maximal isotropic subspace, i.e. a maximal subspace of ''V'' such that {{nowrap|1=''g''{{!}}<sub>''W''</sub> = 0}}. If {{nowrap|1=''n'' = 2''k''}} is even, then let ''W''′ be an isotropic subspace complementary to ''W''. If {{nowrap|1=''n'' = 2''k'' + 1}} is odd let ''W''′ be a maximal isotropic subspace with {{nowrap|1=''W'' ∩ ''W''′ = 0}}, and let ''U'' be the orthogonal complement of {{nowrap|''W'' ⊕ ''W''′}}. In both the even and odd dimensional cases ''W'' and ''W''′ have dimension ''k''. In the odd dimensional case, ''U'' is one dimensional, spanned by a unit vector ''u''.
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| | |
| === Minimal ideals ===
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| Since ''W''′ is isotropic, multiplication of elements of ''W''′ inside {{nowrap|Cℓ(''V'', ''g'')}} is [[alternative algebra|skew]]. Hence vectors in ''W''′ anti-commute, and {{nowrap|1=Cℓ(''W''′, ''g''{{!}}<sub>''W''′</sub>) = Cℓ(''W''′, 0)}} is just the [[exterior algebra]] Λ<sup>∗</sup>''W''′. Consequently, the ''k''-fold product of ''W''′ with itself, ''W''′<sup>''k''</sup>, is one-dimensional. Let ''ω'' be a generator of ''W''′<sup>''k''</sup>. In terms of a basis {{nowrap|''w''′<sub>1</sub>,..., ''w''′<sub>k</sub>}} of in ''W''′, one possibility is to set
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| :<math>\omega=w'_1w'_2\cdots w'_k.</math>
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| Note that {{nowrap|1=''ω''<sup>2</sup> = 0}} (i.e., ''ω'' is nilpotent of order 2), and moreover, {{nowrap|1=''w''′''ω'' = 0}} for all {{nowrap|''w''′ ∈ ''W''′}}. The following facts can be proven easily:
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| # If {{nowrap|1=''n'' = 2''k''}}, then the left ideal {{nowrap|1=Δ = Cℓ(''V'', ''g'')''ω''}} is a minimal left ideal. Furthermore, this splits into the two spin spaces {{nowrap|1=Δ<sub>+</sub> = Cℓ<sup>even</sup>''ω''}} and {{nowrap|1=Δ<sub>−</sub> = Cℓ<sup>odd</sup>''ω''}} on restriction to the action of the even Clifford algebra.
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| # If {{nowrap|1=''n'' = 2''k'' + 1}}, then the action of the unit vector ''u'' on the left ideal {{nowrap|Cℓ(''V'', ''g'')''ω''}} decomposes the space into a pair of isomorphic irreducible eigenspaces (both denoted by Δ), corresponding to the respective eigenvalues +1 and −1.
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| In detail, suppose for instance that ''n'' is even. Suppose that ''I'' is a non-zero left ideal contained in {{nowrap|Cℓ(''V'', ''g'')''ω''}}. We shall show that ''I'' must be equal to {{nowrap|Cℓ(''V'', ''g'')''ω''}} by proving that it contains a nonzero scalar multiple of ''ω''.
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| | |
| Fix a basis ''w''<sub>i</sub> of ''W'' and a complementary basis ''w''<sub>i</sub>′ of ''W''′ so that
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| :''w''<sub>i</sub>''w''<sub>j</sub>′ +''w''<sub>j</sub>′ ''w''<sub>i</sub> = δ<sub>ij</sub>, and
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| :(''w''<sub>i</sub>)<sup>2</sup> = 0, (''w''<sub>i</sub>′)<sup>2</sup> = 0.
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| | |
| Note that any element of ''I'' must have the form ''αω'', by virtue of our assumption that {{nowrap|''I'' ⊂ Cℓ(''V'', ''g'') ''ω''}}. Let {{nowrap|''αω'' ∈ ''I''}} be any such element. Using the chosen basis, we may write
| |
| | |
| :<math>\alpha = \sum_{i_1<i_2<\cdots<i_p} a_{i_1\dots i_p}w_{i_1}\cdots w_{i_p} + \sum_j B_jw'_j</math>
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| | |
| where the ''a''<sub>i<sub>1</sub>…i<sub>p</sub></sub> are scalars, and the ''B''<sub>j</sub> are auxiliary elements of the Clifford algebra. Observe now that the product
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| :<math>\alpha\omega = \sum_{i_1<i_2<\cdots<i_p} a_{i_1\dots i_p}w_{i_1}\cdots w_{i_p} \omega.</math>
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| Pick any nonzero monomial ''a'' in the expansion of ''α'' with maximal homogeneous degree in the elements ''w''<sub>i</sub>:
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| :<math>a = a_{i_1\dots i_{max}}w_{i_1}\dots w_{i_{max}}</math> (no summation implied),
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| then
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| :<math>w_{i_{max}}\cdots w_{i_1}\alpha\omega = a_{i_1\dots i_{max}}\omega</math>
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| is a nonzero scalar multiple of ''ω'', as required.
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| Note that for ''n'' even, this computation also shows that
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| :<math>\Delta = \mathrm{C}\ell(W)\omega = (\Lambda^* W)\omega</math>.
| |
| as a vector space. In the last equality we again used that ''W'' is isotropic. In physics terms, this shows that Δ is built up like a [[Fock space]] by [[creation and annihilation|creating]] spinors using anti-commuting creation operators in ''W'' acting on a vacuum ''ω''.
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| | |
| === Exterior algebra construction ===
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| The computations with the minimal ideal construction suggest that a spinor representation can
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| also be defined directly using the [[exterior algebra]] {{nowrap|1=Λ<sup>∗</sup> ''W'' = ⊕<sub>''j''</sub> Λ<sup>''j''</sup> ''W''}} of the isotropic subspace ''W''.
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| Let {{nowrap|1=Δ = Λ<sup>∗</sup> ''W''}} denote the exterior algebra of ''W'' considered as vector space only. This will be the spin representation, and its elements will be referred to as spinors.<ref>One source for this subsection is {{Harvtxt|Fulton|Harris|1991}}.</ref>
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| The action of the Clifford algebra on Δ is defined first by giving the action of an element of ''V'' on Δ, and then showing that this action respects the Clifford relation and so extends to a [[homomorphism]] of the full Clifford algebra into the [[endomorphism ring]] End(Δ) by the [[Clifford algebra#Universal property and construction|universal property of Clifford algebras]]. The details differ slightly according to whether the dimension of ''V'' is even or odd.
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| When dim(''V'') is even, {{nowrap|1=''V'' = ''W'' ⊕ ''W''′}} where ''W''′ is the chosen isotropic complement. Hence any {{nowrap|''v'' ∈ ''V''}} decomposes uniquely as {{nowrap|1=''v'' = ''w'' + ''w''′}} with {{nowrap|''w'' ∈ ''W''}} and {{nowrap|''w''′ ∈ ''W''′}}. The action of ''v'' on a spinor is given by
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| :<math> c(v) w_1 \wedge\cdots\wedge w_n = (\epsilon(w) + i(w'))\left(w_1 \wedge\cdots\wedge w_n\right)</math>
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| where ''i''(''w''′) is [[interior product]] with ''w''′ using the non degenerate quadratic form to identify ''V'' with ''V''<sup>∗</sup>, and ε(w) denotes the [[exterior product]]. It may be verified that
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| :''c''(''u'')''c''(''v'') + ''c''(''v'')''c''(''u'') = 2 ''g''(''u'',''v''),
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| and so ''c'' respects the Clifford relations and extends to a homomorphism from the Clifford algebra to End(Δ).
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| The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group<ref>Via the even-graded Clifford algebra.</ref> (the half-spin representations, or Weyl spinors) via
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| :<math>\Delta_+ = \Lambda^{even} W,\, \Delta_- = \Lambda^{odd} W</math>.
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| | |
| When dim(''V'') is odd, {{nowrap|1=''V'' = ''W'' ⊕ ''U'' ⊕ ''W''′}}, where ''U'' is spanned by a unit vector ''u'' orthogonal to ''W''. The Clifford action ''c'' is defined as before on {{nowrap|''W'' ⊕ ''W''′}}, while the Clifford action of (multiples of) ''u'' is defined by
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| :<math> c(u) \alpha = \left\{\begin{matrix}
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| \alpha&\hbox{if } \alpha\in \Lambda^{even} W\\
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| -\alpha&\hbox{if } \alpha\in \Lambda^{odd} W
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| \end{matrix}\right.</math>
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| As before, one verifies that ''c'' respects the Clifford relations, and so induces a homomorphism.
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| | |
| === Hermitian vector spaces and spinors ===
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| If the vector space ''V'' has extra structure that provides a decomposition of its complexification into two maximal isotropic subspaces, then the definition of spinors (by either method) becomes natural.
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| The main example is the case that the real vector space ''V'' is a [[hermitian form|hermitian vector space]] {{nowrap|(''V'', ''h'')}}, i.e., ''V'' is equipped with a [[Linear complex structure|complex structure]] ''J'' that is an [[orthogonal transformation]] with respect to the inner product ''g'' on ''V''. Then {{nowrap|''V'' ⊗<sub>'''R'''</sub> '''C'''}} splits in the ±''i'' eigenspaces of ''J''. These eigenspaces are isotropic for the complexification of ''g'' and can be identified with the complex vector space {{nowrap|(''V'', ''J'')}} and its complex conjugate {{nowrap|(''V'', −''J'')}}. Therefore for a hermitian vector space {{nowrap|(''V'', ''h'')}} the vector space Λ{{su|p=⋅|b='''C'''}}{{overline|''V''}} (as well as its complex conjugate Λ{{su|p=⋅|b='''C'''}}''V'') is a spinor space for the underlying real euclidean vector space.
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| With the Clifford action as above but with contraction using the hermitian form, this construction gives a spinor space at every point of an [[almost Hermitian manifold]] and is the reason why every [[almost complex manifold]] (in particular every [[symplectic manifold]]) has a [[Spin-c structure|Spin<sup>c</sup> structure]]. Likewise, every complex vector bundle on a manifold carries a Spin<sup>c</sup> structure.<ref>{{Harvnb|Lawson|Michelsohn|1989|loc=Appendix D}}.</ref>
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| | |
| == Clebsch–Gordan decomposition ==
| |
| A number of [[Clebsch–Gordan coefficients|Clebsch–Gordan decompositions]] are possible on the [[tensor product]] of one spin representation with another.<ref>{{Harvnb|Brauer|Weyl|1935}}.</ref> These decompositions express the tensor product in terms of the alternating representations of the orthogonal group.
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| | |
| For the real or complex case, the alternating representations are
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| * {{nowrap|1=Γ<sub>''r''</sub> = Λ<sup>''r''</sup>''V''}}, the representation of the orthogonal group on skew tensors of rank ''r''.
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| | |
| In addition, for the real orthogonal groups, there are three [[character theory|characters]] (one-dimensional representations)
| |
| * ''σ''<sub>+</sub> : O(''p'', ''q'') → {−1, +1} given by {{nowrap|1=''σ''<sub>+</sub>(R) = −1}}, if ''R'' reverses the spatial orientation of ''V'', +1, if ''R'' preserves the spatial orientation of ''V''. (''The spatial character''.)
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| * ''σ''<sub>−</sub> : O(''p'', ''q'') → {−1, +1} given by {{nowrap|1=''σ''<sub>−</sub>(R) = −1}}, if ''R'' reverses the temporal orientation of ''V'', +1, if ''R'' preserves the temporal orientation of ''V''. (''The temporal character''.)
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| * ''σ'' = ''σ''<sub>+</sub>''σ''<sub>−</sub> . (''The orientation character''.)
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| | |
| The Clebsch–Gordan decomposition allows one to define, among other things:
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| * An action of spinors on vectors.
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| * A [[Hermitian metric]] on the complex representations of the real spin groups.
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| * A [[Dirac operator]] on each spin representation.
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| | |
| === Even dimensions ===
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| If {{nowrap|1=''n'' = 2''k''}} is even, then the tensor product of Δ with the [[contragredient representation]] decomposes as
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| : <math>\Delta\otimes\Delta^* \cong \bigoplus_{p=0}^n \Gamma_p \cong \bigoplus_{p=0}^{k-1} \left(\Gamma_p\oplus\sigma\Gamma_p\right)\, \oplus \Gamma_k</math>
| |
| which can be seen explicitly by considering (in the Explicit construction) the action of the Clifford algebra on decomposable elements {{nowrap|''αω'' ⊗ ''βω''′}}. The rightmost formulation follows from the transformation properties of the [[Hodge star operator]]. Note that on restriction to the even Clifford algebra, the paired summands {{nowrap|Γ<sub>''p''</sub> ⊕ ''σ''Γ<sub>''p''</sub>}} are isomorphic, but under the full Clifford algebra they are not.
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| | |
| There is a natural identification of Δ with its contragredient representation via the conjugation in the Clifford algebra:
| |
| :<math>(\alpha\omega)^*=\omega(\alpha^*).</math>
| |
| So {{nowrap|Δ ⊗ Δ}} also decomposes in the above manner. Furthermore, under the even Clifford algebra, the half-spin representations decompose
| |
| : <math>\begin{matrix}
| |
| \Delta_+\otimes\Delta^*_+ \cong \Delta_-\otimes\Delta^*_- &\cong& \bigoplus_{p=0}^k \Gamma_{2p}\\
| |
| \Delta_+\otimes\Delta^*_- \cong \Delta_-\otimes\Delta^*_+ &\cong& \bigoplus_{p=0}^{k-1} \Gamma_{2p+1}
| |
| \end{matrix}
| |
| </math>
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| | |
| For the complex representations of the real Clifford algebras, the associated [[reality structure]] on the complex Clifford algebra descends to the space of spinors (via the explicit construction in terms of minimal ideals, for instance). In this way, we obtain the complex conjugate {{overline|Δ}} of the representation Δ, and the following isomorphism is seen to hold:
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| :<math>\bar{\Delta} \cong \sigma_-\Delta^* </math>
| |
| | |
| In particular, note that the representation Δ of the orthochronous spin group is a [[unitary representation]]. In general, there are Clebsch–Gordan decompositions
| |
| :<math>\Delta\otimes\bar{\Delta} \cong \bigoplus_{p=0}^k\left(\sigma_-\Gamma_p\oplus \sigma_+\Gamma_p\right).</math>
| |
| | |
| In metric signature {{nowap|(''p'', ''q'')}}, the following isomorphisms hold for the conjugate half-spin representations
| |
| * If ''q'' is even, then <math>\bar{\Delta}_+ \cong \sigma_-\otimes \Delta_+^*</math> and <math>\bar{\Delta}_- \cong \sigma_-\otimes \Delta_-^*.</math>
| |
| * If ''q'' is odd, then <math>\bar{\Delta}_+ \cong \sigma_-\otimes \Delta_-^*</math> and <math>\bar{\Delta}_- \cong \sigma_-\otimes \Delta_+^*.</math>
| |
| Using these isomorphisms, one can deduce analogous decompositions for the tensor products of the half-spin representations {{nowrap|Δ<sub>±</sub> ⊗ {{overline|Δ}}<sub>±</sub>}}.
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| | |
| === Odd dimensions ===
| |
| If {{nowrap|1=''n'' = 2''k'' + 1}} is odd, then
| |
| : <math>\Delta\otimes\Delta^* \cong \bigoplus_{p=0}^k \Gamma_{2p}.</math>
| |
| In the real case, once again the isomorphism holds
| |
| : <math>\bar{\Delta} \cong \sigma_-\Delta^*.</math>
| |
| Hence there is a Clebsch–Gordan decomposition (again using the Hodge star to dualize) given by
| |
| : <math>\Delta\otimes\bar{\Delta} \cong \sigma_-\Gamma_0\oplus\sigma_+\Gamma_1\oplus\dots\oplus\sigma_\pm\Gamma_k</math>
| |
| | |
| === Consequences ===
| |
| There are many far-reaching consequences of the Clebsch–Gordan decompositions of the spinor spaces. The most fundamental of these pertain to Dirac's theory of the electron, among whose basic requirements are
| |
| * A manner of regarding the product of two spinors ''{{overline|ϕ}}ψ'' as a scalar. In physical terms, a spinor should determine a [[probability amplitude]] for the [[quantum state]].
| |
| * A manner of regarding the product ''ψ{{overline|ϕ}}'' as a vector. This is an essential feature of Dirac's theory, which ties the spinor formalism to the geometry of physical space.
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| * A manner of regarding a spinor as acting upon a vector, by an expression such as ''ψv{{overline|ψ}}''. In physical terms, this represents an [[electrical current]] of Maxwell's [[electromagnetic theory]], or more generally a [[probability current]].
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| == Summary in low dimensions ==
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| * In 1 dimension (a trivial example), the single spinor representation is formally Majorana, a [[real representation|real]] 1-dimensional representation that does not transform.
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| * In 2 Euclidean dimensions, the left-handed and the right-handed Weyl spinor are 1-component [[complex representation]]s, i.e. complex numbers that get multiplied by ''e''<sup>±''iφ''/2</sup> under a rotation by angle ''φ''.
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| * In 3 Euclidean dimensions, the single spinor representation is 2-dimensional and [[quaternionic representation|quaternionic]]. The existence of spinors in 3 dimensions follows from the isomorphism of the [[group (mathematics)|group]]s {{nowrap|SU(2) ≅ Spin(3)}} that allows us to define the action of Spin(3) on a complex 2-component column (a spinor); the generators of SU(2) can be written as [[Pauli matrices]].
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| * In 4 Euclidean dimensions, the corresponding isomorphism is {{nowrap|Spin(4) ≅ SU(2) × SU(2)}}. There are two inequivalent [[quaternionic representation|quaternionic]] 2-component Weyl spinors and each of them transforms under one of the SU(2) factors only.
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| * In 5 Euclidean dimensions, the relevant isomorphism is {{nowrap|Spin(5) ≅ USp(4) ≅ Sp(2)}} that implies that the single spinor representation is 4-dimensional and quaternionic.
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| * In 6 Euclidean dimensions, the isomorphism {{nowrap|Spin(6) ≅ SU(4)}} guarantees that there are two 4-dimensional complex Weyl representations that are complex conjugates of one another.
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| * In 7 Euclidean dimensions, the single spinor representation is 8-dimensional and real; no isomorphisms to a Lie algebra from another series (A or C) exist from this dimension on.
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| * In 8 Euclidean dimensions, there are two Weyl–Majorana real 8-dimensional representations that are related to the 8-dimensional real vector representation by a special property of [[Spin(8)]] called [[triality]].
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| * In {{nowrap|''d'' + 8}} dimensions, the number of distinct irreducible spinor representations and their reality (whether they are real, pseudoreal, or complex) mimics the structure in ''d'' dimensions, but their dimensions are 16 times larger; this allows one to understand all remaining cases. See [[Bott periodicity]].
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| * In spacetimes with ''p'' spatial and ''q'' time-like directions, the dimensions viewed as dimensions over the complex numbers coincide with the case of the {{nowrap|(''p'' + ''q'')}}-dimensional Euclidean space, but the reality projections mimic the structure in {{nowrap|{{abs|''p'' − ''q''}}}} Euclidean dimensions. For example, in {{nowrap|3 + 1}} dimensions there are two non-equivalent Weyl complex (like in 2 dimensions) 2-component (like in 4 dimensions) spinors, which follows from the isomorphism {{nowrap|SL(2, '''C''') ≅ Spin(3,1)}}.
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| <center>
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| {| class="wikitable" style="text-align:center"
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| ![[Metric signature]]!!left-handed Weyl!!right-handed Weyl!!conjugacy!!Dirac!!left-handed Majorana–Weyl!!right-handed Majorana–Weyl!!Majorana
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| | ||complex||complex|| ||complex||real||real||real
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| |(2,0)||1||1||mutual||2||–||–||2
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| |(1,1)||1||1||self||2||1||1||2
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| |(3,0)||–||–||–||2||–||–||–
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| |(2,1)||–||–||–||2||–||–||2
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| |(4,0)||2||2||self||4||–||–||–
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| |(3,1)||2||2||mutual||4||–||–||4
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| |(5,0)||–||–||–||4||–||–||–
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| |(4,1)||–||–||–||4||–||–||–
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| |(6,0)||4||4||mutual||8||–||–||8
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| |(5,1)||4||4||self||8||–||–||–
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| |(7,0)||–||–||–||8||–||–||8
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| |(6,1)||–||–||–||8||–||–||–
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| |(8,0)||8||8||self||16||8||8||16
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| |(7,1)||8||8||mutual||16||–||–||16
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| |(9,0)||–||–||–||16||–||–||16
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| |(8,1)||–||–||–||16||–||–||16
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| |}</center>
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| == References ==
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| {{reflist|2}}
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| == See also ==
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| * [[Anyon]]
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| * [[Dirac equation in the algebra of physical space]]
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| * [[Einstein–Cartan theory]]
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| * [[Pure spinor]]
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| * [[Spin-½]]
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| * [[Spinor bundle]]
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| * [[Supercharge]]
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| * [[Twistor]]
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| == Further reading ==
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| * {{citation|last1=Brauer|first1=Richard|author1-link=Richard Brauer|last2=Weyl|first2= Hermann|author2-link=Hermann Weyl|title=Spinors in n dimensions|journal= American Journal of Mathematics|volume=57|issue= 2|year=1935|pages= 425–449|doi=10.2307/2371218|publisher=The Johns Hopkins University Press|jstor=2371218}}.
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| * {{citation|last=Cartan|first=Élie|authorlink=Élie Cartan|url=http://archive.numdam.org/article/BSMF_1913__41__53_1.pdf|title=Les groupes projectifs qui ne laissent invariante aucune multiplicité plane|journal= Bul. Soc. Math. France|volume= 41|year=1913|pages= 53–96}}.
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| * {{citation|last=Cartan|first=Élie|authorlink=Élie Cartan|year=1966|title=The theory of spinors|publisher = Paris, Hermann (reprinted 1981, Dover Publications)| isbn= 978-0-486-64070-9}}
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| * {{citation|last= Chevalley|first= Claude|authorlink=Claude Chevalley|title=The algebraic theory of spinors and Clifford algebras|publisher=Columbia University Press (reprinted 1996, Springer)|year=1954|isbn=978-3-540-57063-9}}.
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| * {{citation|last=Dirac|first= Paul M.|authorlink=Paul Dirac|title=The quantum theory of the electron|journal=Proceedings of the Royal Society of London|volume=A117|year=1928|pages=610–624 |jstor=94981}}.
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| * {{citation| last1=Fulton | first1=William| author1-link=William Fulton (mathematician) | last2=Harris| first2=Joe | author2-link=Joe Harris (mathematician) | title=Representation theory. A first course| publisher=[[Springer-Verlag]]| location=New York| series=[[Graduate Texts in Mathematics]], Readings in Mathematics| isbn=0-387-97495-4| mr=1153249 | year=1991| volume=129}}.
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| * {{citation|last=Gilkey|first= Peter B.|authorlink=Peter B. Gilkey|url=http://www.emis.de/monographs/gilkey/index.html|title= Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem|publisher= Publish or Perish|year= 1984|isbn= 0-914098-20-9}}.
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| * {{citation|title=Spinors and Calibrations|last=Harvey|first= F. Reese|authorlink=F. Reese Harvey|publisher=Academic Press|year=1990|isbn=978-0-12-329650-4}}.
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| * {{springer|title=Spinor|id=p/s086750}}
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| * {{citation|last=Hitchin|first= Nigel J.|authorlink=Nigel Hitchin|title=Harmonic spinors|journal= Advances in Mathematics|volume=14|year=1974|pages=1–55|mr=358873 |doi=10.1016/0001-8708(74)90021-8}}.
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| * {{citation|last1=Lawson|first1= H. Blaine|author1-link=H. Blaine Lawson|last2=Michelsohn|first2=Marie-Louise|author2-link=Marie-Louise Michelsohn|title=Spin Geometry|publisher= Princeton University Press|year=1989|isbn= 0-691-08542-0}}.
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| * {{citation|last=Pauli|first= Wolfgang|authorlink=Wolfgang Pauli|title=Zur Quantenmechanik des magnetischen Elektrons|journal=Zeitschrift für Physik|volume=43|year=1927|pages= 601–632|doi=10.1007/BF01397326|bibcode = 1927ZPhy...43..601P|issue=9–10 }}.
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| * {{citation|last1=Penrose|first1= Roger|author1-link=Roger Penrose|last2=Rindler|first2=W.|title=Spinors and Space–Time: Volume 2, Spinor and Twistor Methods in Space–Time Geometry|publisher= Cambridge University Press|year=1988|isbn=0-521-34786-6}}.
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| * {{citation|first=Sin-Itiro|last=Tomonaga|title=The story of spin|chapter=Lecture 7: The Quantity Which Is Neither Vector nor Tensor|page=129|isbn=0-226-80794-0|publisher=University of Chicago Press|year=1998}}
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| {{tensors}}
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| [[Category:Spinors| ]]
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| [[Category:Rotation in three dimensions]]
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| [[Category:Quantum mechanics]]
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| [[Category:Quantum field theory]]
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