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By Forshaw J.

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However, we have seen that both SU(2) and the rotation group SO(3) have the same, angular momentum, algebra. What is going on? It must be that SO(3) is not simply connected. In fact, there is a mapping, called a covering, from SU(2) to SO(3) which preserves the group property: that is if U ∈ SU(2) is mapped to f (U) ∈ SO(3), then f (UV ) = f (U)f (V ). In the SU(2) → SO(3) case, two elements of SU(2) are mapped on to every element of SO(3). Whenever a group G has the same Lie algebra as a simply connected group S there must be such a covering S → G.

A second 2π rotation gets you back to where you started.

However, this is too restrictive. We need only ensure that the gauge condition holds for matrix elements of ∂ µ Aµ and now we can impose non-zero commutation relations. 30) with all other commutators vanishing. 31) λ=0 where ǫλµ are a set of four linearly independent basis vectors (λ = 0, 1, 2, 3). For example we might choose ǫ0 = (1, 0, 0, 0), ǫ1 = (0, 1, 0, 0), ǫ2 = (0, 0, 1, 0) and ǫ3 = (0, 0, 0, 1). If k is along the z-axis then ǫ1 and ǫ2 are polarisation vectors for transverse polarisations whilst ǫ0 is referred to as the timelike polarisation vector and ǫ 3 is referred to as the longitudinal polarisation vector.

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An introduction to QED and QCD by Forshaw J.


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