From Special Relativity to Feynman Diagrams.pdf

# In our problem it

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In our problem it representstheelectromagneticfieldgeneratedbyaparticlewhosestateisunperturbed

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12.4 Quantum Electrodynamics and Feynman Rules 475 Fig.12.3 Interaction with an external fields by the interaction process and will be referred to as an external field . The amplitude ( 12.143 ) can the be recast in the following first order form: ψ out | S ( 2 ) | ψ in = out e | ie d 4 x : ψ( x μ ψ( x ) : A ext μ ( x ) | in e = out e | i d 4 x H ext I ( x ) | in e , (12.146) where H ext I ( x ) = − e : ψ( x μ ψ( x ) : A ext μ ( x ). (12.147) We have shown that, in all interaction processes in which particle q is just a “spectator”, its effect on the electron can be accounted for by means of the external field A ext μ it generates. This is done by adding to the QED Hamiltonian describing just the electron and the electromagnetic field, the corresponding interaction term, generalizing thus the definition of the interaction Hamiltonian H I ( x ) H I ( x ) + H ext I ( x ) = − e : ψ( x μ ψ( x )( A μ ( x ) + A ext μ ( x )) : . This amounts in turn to redefining the electromagnetic potential in the QED Lagrangian ( 12.129 ) as the sum A μ ( x ) + A ext μ ( x ) of the field operator A μ ( x ) and the external field A ext μ ( x ). Let us stress that A ext μ ( x ) is a classical field and not an operator, namely it is a number and thus acts as the identity on the Fock space of free photons. Therefore the interaction term H ext I ( x ) contains just two field operators, ψ, ψ. Graphically it will be represented by a 2-line vertex, with the external field being represented by a cross, as in Fig. 12.3 . 12.5 Amplitudes in the Momentum Representation 12.5.1 Möller Scattering Let us start considering a specific process describing the scattering between two electrons (labeled by 1 , 2 respectively): e + e −→ e + e . (12.148)
476 12 Fields in Interaction The initial state describes the incoming electrons with momenta p 1 , p 2 and polariza- tions r 1 , r 2 , respectively. The final momenta and polarizations of the two electrons are q 1 , q 2 and s 1 , s 2 respectively: | ψ in = | p 1 , r 1 | p 2 , r 2 , | ψ out = | q 1 , s 1 | q 2 , s 2 . (12.149) We shall compute the amplitude of the process to lowest order, namely the matrix element of S ( 2 ) between the initial and final states. The only term contributing to the amplitude is the one described by the diagram (2) in Fig. 12.2 , so that: ψ out | S ( 2 ) | ψ in = ( ie ) 2 2 ! d 4 xd 4 y × q 1 , s 1 | q 2 , s 2 | : ψ( x μ ψ( x ) ψ( y ν ψ( y ) : | p 1 , r 1 | p 2 , r 2 × D F μν ( x y ) . (12.150) We can convince ourselves that the only term in the normal product which contributes to the matrix element is the one of the form c c cc , since we need to destroy the two incoming electrons and to create the two outgoing ones. Let us explicitly compute the corresponding matrix element, bearing in mind that the two c operators come from the ψ fields, while the two c operator originate from the ψ fields. We write the initial and final states in terms of creation operators acting on the vacuum: | p 1 , r 1 | p 2 , r 2 = c ( p 1 , r 1 ) c ( p 2 , r 2 ) | 0 , | q 1 , s 1 | q 2 , s 2 = c ( q 1 , s 1 ) c ( q 2 , s 2 ) | 0 . (12.151)

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• Fall '17
• Chris Odonovan

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