From Special Relativity to Feynman Diagrams.pdf

We know however that the photon is a massless

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We know, however, that the photon is a massless particle and, as such, there exists no RF in which its linear momentum vanishes: p = 0 . This implies that there is no RF in which the total angular momentum J M + S = x × p + S , where M is the orbital part and S is the spin (see Chap.9 ), coincides with S and thus acts on the internal degrees of freedom only. The only component of J which acts only on the internal degrees of freedom of the photon and which thus can be taken as a definition of its spin, is its component along p , called the “helicity” and denoted by : J · p | p | = ( x × p ) · p | p | + S · p | p | = S · n k . (6.76) The helicity generates rotations about the direction n k of p : R (θ) = e i θ . (6.77) On the internal components (polarization) of the photon, which are components of a four-vector ( μ ( k )) = ( 0 , k ) (transverse components of A μ ), this transformation acts as a particular Lorentz transformation. Let us choose a RF in which p is aligned to the X direction, p = ( p , 0 , 0 ) = k . The infinitesimal generator of rotations about the X axis is represented, on the four-vector k μ , by the matrix J 1 :
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6.3 Spin of the Photon 179 = J 1 = − i 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 . (6.78) Since k is transverse to the direction X of motion, we have: ( μ ( k )) = ( 0 , 0 , 2 , 3 ) , we easily find that has two eigenvalues i ( i ) = ± with eigenvectors: ( + ) μ ( k ) = 0 0 1 i and ( ) μ ( k ) = 0 0 1 i . (6.79) We define the spin of a massless particle as the number s such that its states are eigenstates of to the eigenvalues ± s . It then follows that the photon has spin s = 1. Note that the transformation R (θ) precisely coincides with the transformation ( 0 ) given in ( 5.116 ) so that the definition of spin of a photon given here corresponds to the definition of spin of a plane wave given in Sect.5.6.1 . Reference For further reading see Refs. [8] (Vol. 4). [9].
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Chapter 7 Group Representations and Lie Algebras 7.1 Lie Groups As already mentioned in Chap.4 several properties of the rotation group SO ( 3 ) and of the Lorentz group SO ( 1 , 3 ) are actually valid for any Lie group G and do not depend of the particular representation of their elements in terms of matrices. Such representation independent features are encoded in the notion of an abstract group . In this chapter we give the definition of an abstract group, restricting to Lie groups only. Without any pretension to rigour or completeness, we define the general concept of representation and that of a Lie algebra . This will be essential to showing the deep relation, existing in classical and quantum field theories, between symmetry and/or invariance properties of a system, to be described in group theoretical language, and conservation laws of physical quantities. These interrelations will be discussed in the next chapters. Let us first give the general axioms defining an abstract group.
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