旋量

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在數學與物理中，特別是在正交群(見群論)裡面，旋量是一種輔助性的數學物件，用以擴充向量（矢量）的表示。因為在一給定維度下，需要它們才能完整地描述旋轉，如此引入了額外數量的維度。

其乃自旋群的表象，類似於空間中的向量但差異在於其每次旋轉了2π弳度會發生變號。

[编辑] 概論

一種特定的旋量是旋轉群(李群SO(n,R) )的投影表象中的元素，或更廣義地說，是SO(p,q,R)群的投影表象中的元素，其中p + q = n for spinors in a space of nontrivial signature. This is equivalent to an ordinary (non-projective) representation of the double cover of SO(p,q,R), which is a real Lie group called the spinor group Spin(p,q).

旋量常被描述成「向量的平方根」，因為向量表象會出現在兩個相同旋量表象的張量積。

旋量中最典型的是狄拉克旋量，is an element of the fundamental representation of the complexified Clifford algebra C(p,q), into which Spin(p,q) may be embedded. In even dimensions, this representation is reducible when taken as a representation of Spin(p,q) and may be decomposed into two: the left-handed and right-handed Weyl spinor representations. In addition, sometimes the non-complexified version of C(p,q) has a smaller real representation, the Majorana spinor representation. If this happens in an even dimension, the Majorana spinor representation will sometimes decompose into two Majorana-Weyl spinor representations. Of all these, only the Dirac representation exists in all dimensions. Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.

A 2n- or 2n+1-dimensional Dirac spinor may be represented as a vector of 2ⁿ complex numbers. (See Special unitary group.)

There are also more complicated spinors like the Rarita-Schwinger spinor, which will not be covered here.