March 15, 2018

New PDF release: An introduction to QED and QCD

By Forshaw J.

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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. The double covering of SO(3) by SU(2) underlies the behaviour of spin-1/2 and other half-odd-integer spin particles under rotations: they really transform under SU(2), and rotating them by 2π only gets you half way around SU(2), so you pick up a minus sign.

The Ti are called the generators and the fijk are called the structure constants. Find a set of 3 independent 2 × 2 matrices which are generators for SU(2) and a set of 8 independent 3 × 3 generators for SU(3). 3). The i Tadj define the adjoint representation. The W ’s of the weak interactions and the gluons of the strong interactions belong to the adjoint representations of SU(2) L , left-handed weak SU(2), and SU(3), the strong interaction colour algebra, respectively. The generators, and hence the algebra, were found by looking at group elements near the identity.

Experiment tells us that eˆ2 1 ≈ . 4π 137 We call eˆ the renormalised coupling constant of qed. We can calculate eˆ in terms of e in perturbation theory. 1) where a1 and b1 are constants obtained from the calculation. The e 3 term is divergent, so we have introduced a cutoff M to regulate it. This is called an ultraviolet divergence since it arises from the propagation of high momentum modes in the loops. The cutoff amounts to selecting only those modes where each component of momentum is less than M in magnitude.

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An introduction to QED and QCD by Forshaw J.


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