On-shell renormalization scheme: Difference between revisions

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{{Short description|Renormalization scheme in quantum field theory}}

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]].

{{Renormalization and regularization}}

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 [[Minimalminimal subtraction scheme]] (MS scheme).

==Fermion propagator in the interacting theory==

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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 accountaccounts 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>

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==References==

* {{Cite book|author=M. Peskin and |author2=D. Schroeder, ''|title=An Introduction to Quantum Field Theory'' |publisher=Addison-Weasley, |___location=Reading, |year=1995}}

* M. Srednicki, ''[http://www.physics.ucsb.edu/~mark/qft.html http://www.physics.ucsb.edu/~mark/qft.html Quantum Field Theory]''

* T. Gehrmann, ''[https://www.mitschriften.ethz.ch/main.php?page=3&details=161 https://www.mitschriften.ethz.ch/main.php?page=3&details=161 Quantum Field Theory 1]''

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[[Category:Renormalization group]]