From Special Relativity to Feynman Diagrams.pdf

In our problem it

Info icon This preview shows pages 455–458. Sign up to view the full content.

View Full Document Right Arrow Icon
In our problem it representstheelectromagneticfieldgeneratedbyaparticlewhosestateisunperturbed
Image of page 455

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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)
Image of page 456
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)
Image of page 457

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 458
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern