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We have considered some proportionality factors (like the <math>Z_i</math>) that have been defined from the form of the propagator. However they can also be defined from the QED lagrangian, which will be done in this section, and these definitions are equivalent. The Lagrangian that describes the physics of [[quantum electrodynamics]] is
:<math> \mathcal L = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \bar{\psi}(i \partial\!\!\!/ - m )\psi + e \bar{\psi} \gamma^\mu \psi A_{\mu} </math>
where <math>F_{\mu \nu}</math> is the [[Electromagnetic tensor|field strength tensor]], <math>\psi</math> is the Dirac spinor (the relativistic equivalent of the [[wavefunction]]), and A the [[electromagnetic four-potential]]. The parameters of the theory are <math>\psi,\; A,\;m</math> and <math>e</math>. These quantities happen to be infinite due to [[Renormalization#A_loop_divergence|loop corrections]] (see below). One can define the renormalized quantities (which will be finite and observable) :
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The <math>\delta_i</math> are called counterterms (some other definitions of them are possible). They are supposed to be small in the parameter e. The Lagrangian now reads in terms of renormalized quantities (to first order in the counterterms) :
:<math> \mathcal L = -\frac{1}{4} Z_3 F_{\mu \nu,r} F^{\mu \nu}_r + Z_2 \bar{\psi}_r(i \partial\!\!\!/ - m_r )\psi_r - \bar{\psi}_r\delta m \psi_r + Z_1 e_r \bar{\psi}_r \gamma^\mu \psi_r A_{\mu,r} </math>
A renormalization prescription is a set of rules that describes what part of the divergences should be in the renormalized quantities and what parts should be in the counterterms. The prescription is often based on the theory of free fields, that is of the behaviour of <math>\psi</math> and A when they do not interact (which corresponds to removing the term <math>e \bar{\psi} \gamma^\mu \psi A_{\mu} </math> in the Lagrangian).
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