# Spinor

An element of a spinor representation space. For example, if $Q$ is a non-degenerate quadratic form on an $n$-dimensional space $V$ over a field $k$ with maximal Witt index $m=[n/2]$ (the latter condition always holds when $k$ is algebraically closed), then as the spinor space corresponding to $Q$ one can take the exterior algebra over the maximal ($m$-dimensional) totally-isotropic subspace of $V$.

Spinors were first studied in 1913 by E. Cartan in his investigations of the theory of representations of topological groups, and were taken up again in 1929 by B.L. van der Waerden in his research on quantum mechanics. (Thus, it was discovered that the occurrence of spin in an electron and in other elementary particles is characterized by physical variables of till then unknown type (such as tensors and pseudo-tensors); for example, they are only defined up to sign, and by rotating the coordinate system through $2\pi$ about some axis, all the components of these variables change sign.)

Spinor calculus currently finds wide application in many branches of mathematics, and has made it possible to solve a series of difficult problems in algebraic and differential topology (for example, the problem of the number of non-zero vector fields on a $k$-dimensional sphere, the problem of the index of an elliptic operator, and problems in $K$-theory).

#### References

[1] | B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, "Modern geometry - methods and applications" , 1–2 , Springer (1985) (Translated from Russian) |

[2] | V.A. Zhelnorovich, "The theory of spinors and its applications in physics and mechanics" , Moscow (1982) (In Russian) |

[3] | E. Cartan, "Leçons sur la théorie des spineurs" , 1–2 , Hermann (1938) |

[4] | M. Karoubi, "$K$-theory: an introduction" , Springer (1978) (Translated from French) |

#### Comments

#### References

[a1] | D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) |

[a2] | R. Penrose, W. Rindler, "Spinors and space-time" , Cambridge Univ. Press (1984) |

**How to Cite This Entry:**

Spinor.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Spinor&oldid=33748