Editing Spinor (section)

== Summary in low dimensions ==
* In 1 dimension (a trivial example), the single spinor representation is formally Majorana, a [[real representation|real]] 1-dimensional representation that does not transform.
* 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''.
* 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)|groups]] {{math|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]].
* In 4 Euclidean dimensions, the corresponding isomorphism is {{math|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.
* In 5 Euclidean dimensions, the relevant isomorphism is {{math|Spin(5) ≅ USp(4) ≅ Sp(2)}} that implies that the single spinor representation is 4-dimensional and quaternionic.
* In 6 Euclidean dimensions, the isomorphism {{math|Spin(6) ≅ SU(4)}} guarantees that there are two 4-dimensional complex Weyl representations that are complex conjugates of one another.
* 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.
* 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]].
* In {{math|''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]].
* 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 {{math|(''p'' + ''q'')}}-dimensional Euclidean space, but the reality projections mimic the structure in {{math|{{abs|''p'' − ''q''}}}} Euclidean dimensions. For example, in {{math|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 {{math|SL(2, <math>\Complex</math>) ≅ Spin(3,1)}}.

{| class="wikitable" style="margin:1em auto; text-align:center;"
! rowspan=2 | [[Metric signature]]
! colspan=2 | Weyl, complex
! rowspan=2 | Conjugacy
! rowspan=2 | Dirac, <br />complex
! colspan=2 | Majorana–Weyl, real
! rowspan=2 | Majorana, <br />real
|-
! Left-handed
! Right-handed
! Left-handed
! Right-handed
|-
|(2,0)||1||1||Mutual||2||–||–||2
|-
|(1,1)||1||1||Self||2||1||1||2
|-
|(3,0)||–||–||–||2||–||–||–
|-
|(2,1)||–||–||–||2||–||–||2
|-
|(4,0)||2||2||Self||4||–||–||–
|-
|(3,1)||2||2||Mutual||4||–||–||4
|-
|(5,0)||–||–||–||4||–||–||–
|-
|(4,1)||–||–||–||4||–||–||–
|-
|(6,0)||4||4||Mutual||8||–||–||8
|-
|(5,1)||4||4||Self||8||–||–||–
|-
|(7,0)||–||–||–||8||–||–||8
|-
|(6,1)||–||–||–||8||–||–||–
|-
|(8,0)||8||8||Self||16||8||8||16
|-
|(7,1)||8||8||Mutual||16||–||–||16
|-
|(9,0)||–||–||–||16||–||–||16
|-
|(8,1)||–||–||–||16||–||–||16
|}