On-shell renormalization scheme: Difference between revisions

Revision as of 00:20, 14 January 2019

Revision as of 22:24, 24 November 2019

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==Vertex function==

A similar reasoning using the [[vertex function]] leads to the renormalization of the electric charge <math>e_r</math>. This renormalization, and the fixing of renormalization terms is done using what is known from classical electrodynamics at large space scales. This leads to the value of the counterterm <math>\delta_1</math>, which is, in fact, equal to <math>\delta_2</math> because of the [[Ward-TakahashiWard–Takahashi identity]]. It is this calculation that account for the [[anomalous magnetic dipole moment]] of fermions.

==Rescaling of the QED Lagrangian==

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 lagrangianLagrangian, 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 <math>A</math> the [[electromagnetic four-potential]]. The parameters of the theory are <math>\psi</math>,\; <math>A</math>,\; <math>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):

:<math>

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m = m_r + \delta m \;\;\;\;\;

e = \frac{Z_1}{Z_2 \sqrt{Z_3}} e_r \;\;\;\;\;

\text{with} \;\;\;\;\; Z_i = 1 + \delta_i

</math>