Spinors, B2 SYMMETRY AND RELATIVITY, physical, Oxford University

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Spinors, B2 SYMMETRY AND RELATIVITY, physical, Oxford University Introducing spinors Spinors are mathematical entities somewhat like tensors, that allow a more general treatment of the notion of in... variance under rotation and Lorentz boosts. To every tensor of rank k there corresponds a spinor of rank 2k, and some kinds of tensor can be associated with a spinor of the same rank. For example, a general 4-vector would correspond to a Hermitian spinor of rank 2, which can be represented by a 2 × 2 Hermitian matrix of complex numbers. A null 4-vector can also be associated with a spinor of rank 1, which can be represented by a complex vector with two components. We shall see why in the following. Spinors can be used without reference to relativity, but they arise naturally in discussions of the Lorentz group. One could say that a spinor is the most basic sort of mathematical object that can be Lorentz-transformed. The main facts about spinors are given in the box. This summary is placed here rather than at the end of the chapter in order to help the reader follow the main thread of the argument. It appears that Klein originally designed the spinor to simplify the treatment of the classical spinning top in 1897. The more thorough understanding of spinors as mathematical objects is credited to Elie Cartan in 1913. They are closely related to Hamilton’s quaternions (about ´ 1845). Spinors began to find a more extensive role in physics when it was discovered that electrons and other particles have an intrinsic form of angular momentum now called ‘spin’, and the behaviour of this type of angular momentum is correctly captured by the mathematics discovered by Cartan. Pauli formalized this connection in a non-relativistic (i.e. low velocity) context, modelling the electron spin using a two-component complex vector, and introducing the Paul [Show More]

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