From Special Relativity to Feynman Diagrams.pdf

125 amplitudes in the momentum representation 481

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considerations of this subsection to the Möller scattering.
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12.5 Amplitudes in the Momentum Representation 481 particle with m 2 = k 2 . While for a real photon only the two transverse polariza- tions are physical, when a virtual photon is exchanged, all four polarization vectors ε (λ) μ ( k ), λ = 0 , 1 , 2 , 3 contribute to the amplitude. It is then interesting to see what is the role of the time-like and longitudinal photons ε ( 0 ) μ ( k ), ε ( 3 ) μ ( k ) in the interpretation of the process. Let us refer for concreteness to the second diagram of the Möller scattering whose lowest order amplitude is given by the second term of ( 12.163 ). We observe that the sum over the indices μ of the gamma-matrices is due to the η μν factor of the photon propagator ˜ D F μν = − i η μν ( k 2 + i ) 1 , which in turn comes from the completeness relation ( 12.166 ). Therefore the amplitude corresponding to the first term of ( 12.163 ) could have been alternatively written as ¯ u ( q 2 , s 2 μ u ( p 2 , r 2 ) λ = 3 λ = 0 ε (λ) μ ( k (λ) ν ( k ) i ( p 1 q 1 ) 2 ¯ u ( q 1 , s 1 ν u ( p 1 , r 1 ). (12.164) For a virtual photon we must take as polarization vectors a set which for k 2 0 reduces to the set used for a real photon in (11.236) and (11.237). As seen in Sect. 11.7, such set is obtained by simply replacing the longitudinal vector ε ( 3 ) μ ( k ) of (11.237) with ε ( 3 ) μ ( k ) = k μ η μ ( k · η) ( k · η) 2 k 2 . (12.165) Let us now decompose the sum appearing in the completeness relation 3 λ = 0 ε (λ) μ ( k (λ)ν ( k ) = η μν , (12.166) into the sum over λ = 0 , 3, corresponding to the exchange of timelike and longitu- dinal photons and the sum over the transverse polarizations λ = 1 , 2 . In particular, using ( 12.165 ) for ε ( 3 ) μ ( k ) and the value ε ( 0 ) μ ( k ) = η μ of (11.239), we have ε ( 0 ) μ ( k ( 0 ) ν ( k ) ε ( 3 ) μ ( k ( 3 ) ν ( k ) = η μ η ν [ k μ η μ ( k · η) ][ k ν η ν ( k · η) ] ( k · η) 2 k 2 . Since we are interested in the contribution to the amplitude of the λ = 0 , 3 polariza- tions, we substitute the right hand side of this expression into ( 12.164 ) with the sum restricted to the values λ = 0 , λ = 3 . Since, as already seen in the previous section, the terms proportional to k μ do not contribute by virtue of gauge invariance 17 we 17 In this special case this can be also seen directly. Indeed k μ ¯ u ( q 2 , s 2 μ u ( p 2 , r 2 ) = ¯ u ( q 2 , s 2 )( p 2 q 2 ) μ γ μ u ( p 2 , r 2 ) = − m ¯ u ( q 2 , s 2 ) u ( p 2 , r 2 ) + m ¯ u ( q 2 , s 2 ) u ( p 2 , r 2 ) = 0 , ( 12 . 167 ) and similarly for the other factor of ( 12.168 ).
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482 12 Fields in Interaction obtain i k 2 ¯ u ( q 2 , s 2 μ u ( p 2 , r 2 ) ¯ u ( q 1 , s 1 ν u ( p 1 , r 1 ) × η μ η ν 1 ( k · η) 2 ( k · η) 2 k 2 = − i ¯ u ( q 2 , s 2 0 ( p 2 , r 2 ) 1 k 2 ( k 0 ) 2 ¯ u ( q 1 , s 1 0 u ( p 1 , r 1 ) = iu ( q 2 , s 2 ) u ( p 2 , r 2 ) 1 | k | 2 u ( q 1 , s 1 ) u ( p 1 , r 1 ), (12.168) where in the second step we have used the explicit value η = ( 1 , 0 , 0 , 0 ) valid in the Lorentz frame where ε ( 1 , 2 ) μ ( k ) are transverse (see Sect.11.7 of Chap.11 ). We now observe that u ( q 2 , s 2 ) u ( p 2 , r 2 ) and u ( q 1 , s 1 ) u ( p 1 , r 1 ) are the Fourier transform in the momentum space of the charge densities, while 1 | k | 2 is the Fourier transform of 1 /( 4 π r ). It follows that ( 12.168 ) represents an “istantaneous” Coulomb interaction between the two electrons. Adding the sum over the two transverse polarizations
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