Spinors

Abstract

Starting from an abstract complex 2-dimensional vector space with a fixed alternating tensor, there is constructed what is called a spinor space. This spinor space, it turns out, is intimately connected to what is known as a Lorentz vector space – a 4-dimensional vector space endowed with a metric of Lorentz signature. Finally, this connection between these two kinds of spaces is exploited to introduce, on virtually any spacetime, spinor fields.

Spinor fields, in many ways, merely reflect tensor fields. Every tensor field can be expressed in terms of one or more spinor fields; and the derivative operator on tensor fields extends uniquely to one on spinor fields. Thus, every algebraic calculation and every differential equation involving tensor fields has a direct spinor analog. It turns out, however, that for a number of topics – though by no means for all – the spinor version is simpler and more transparent than the tensor version. Examples include the classification of the Weyl tensor, the structure of the Maxwell field and the properties of null geodetic congruences. In addition, there are some topics for which spinors seem to be essential. These include the spin-s fields (for s a half-integer) and the Witten proof of positivity of gravitational mass. Acting on the spinor space is a certain group, the spinor group. The spinors generate the representations of that group, and in addition show how this group and its representations are related to those of the Lorentz group.

There are two common variants of spinors: 4-component spinors (which are used extensively in particle physics); and Euclidean spinors (which are used in, among other things, the Witten proof). There is also a number of subtleties involved in using spinors. For example: Some spacetimes admit no spinors at all, and for those that do admit a spinor structure it is not in general unique; there is in general not available any notion of the Lie derivative of a spinor field; and variational calculations involving spinors must be done with some care. Finally, there exist entire computational schemes – the spin-coefficient formalisms – based on spinors.