Index Concordia

Relativity – Week 03

Posted in Relativity by Index Guy on September 25, 2007

During the third week we discussed spinor notation.

The basic idea is to write traditional 3-vectors as 2×2 matrices instead of the usual ordered triplet. We can accomplish this by defining some basis (usually the Pauli matrices). Then the rule that takes us from vector V (in 2×2 matrices) to vector V^{a} (in traditional Gibbs notation) reads as

V = V^{a}\sigma_{a}

We will require this vectors to be hermitian. Since a hermitian matrix has 4 independent (complex) components, we fixed one by requiring also vectors to be traceless. Using some general identities, this traceless-ness leads us to all define the norm of a vector in terms of the determinant.

Rotations of vectors now are given by

V^{\prime} = UVU^{-1},

with the transformation matrices taken as unitary matrices in order to preserve hermiticity and traceless-ness of vectors.

When one considers four-vectors, then the traceless-ness requirement can be relaxed. This is not a trivial generalization since now it is convenient to distinguish left and right indices. Since most of the lectures was verbatim from Fields (which is perfectly acceptable since the professor wrote that monster), I will just mentioned the sections that we covered:

  • Part 1, chapter II, section A with the first 5 subsections, mention of what a Dirac spinor was and the projection operators and the last subsection on chirality and duality.

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