In quantum field theory, and especially in quantum electrodynamics, the interacting theory leads to infinite quantities that have to be absorbed in a renormalization procedure, in order to be able to predict measurable quantities. The renormalization scheme can depend on the type of particles that are being considered. For particles that can travel asymptotically large distances, or for low energy processes, the on-shell scheme, also known as the physical scheme, is appropriate. If these conditions are not fulfilled, one can turn to other schemes, like the minimal subtraction scheme (MS scheme).
Fermion propagator in the interacting theory Knowing the different propagators is the basis for being able to calculate Feynman diagrams which are useful tools to predict, for example, the result of scattering experiments. In a theory where the only field is the Dirac field, the Feynman propagator reads
⟨ 0 | T ( ψ ( x ) ψ ¯ ( 0 ) ) | 0 ⟩ = i S F ( x ) = ∫ d 4 p ( 2 π ) 4 i e − i p ⋅ x p / − m + i ϵ {\displaystyle \langle 0|T(\psi (x){\bar {\psi }}(0))|0\rangle =iS_{F}(x)=\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {ie^{-ip\cdot x}}{p\!\!\!/-m+i\epsilon }}}
where T {\displaystyle T} is the time-ordering operator, | 0 ⟩ {\displaystyle |0\rangle } the vacuum in the non interacting theory, ψ ( x ) {\displaystyle \psi (x)} and ψ ¯ ( x ) {\displaystyle {\bar {\psi }}(x)} the Dirac field and its Dirac adjoint, and where the left-hand side of the equation is the two-point correlation function of the Dirac field. In a new theory, the Dirac field can interact with another field, for example with the electromagnetic field in quantum electrodynamics, and the strength of the interaction is measured by a parameter, in the case of QED it is the bare electron charge, e {\displaystyle e} . The general form of the propagator should remain unchanged, meaning that if | Ω ⟩ {\displaystyle |\Omega \rangle } now represents the vacuum in the interacting theory, the two-point correlation function would now read
⟨ Ω | T ( ψ ( x ) ψ ¯ ( 0 ) ) | Ω ⟩ = ∫ d 4 p ( 2 π ) 4 i Z 2 e − i p ⋅ x p / − m r + i ϵ {\displaystyle \langle \Omega |T(\psi (x){\bar {\psi }}(0))|\Omega \rangle =\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {iZ_{2}e^{-ip\cdot x}}{p\!\!\!/-m_{r}+i\epsilon }}}
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