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{{Short description|Renormalization scheme in quantum field theory}}
{{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 [[
▲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]].
Knowing the different [[propagator
▲== Fermion propagator in the interacting theory ==
▲Knowing the different [[propagator (Quantum Theory)|propagators]] is the basis for being able to calculate [[Feynman diagram|Feynman diagrams]] which are useful tools to predict, for example, the result of scattering experiments. In a theory where the only field is the [[Fermionic field|Dirac field]], the Feynman propagator reads
:<math> \langle 0 | T(\psi(x)\bar{\psi}(0))| 0 \rangle =iS_F(x) = \int \frac{d^4p}{(2\pi)^4}\frac{ie^{-ip\cdot x}}{p\!\!\!/-m+i\epsilon} </math>
where <math>T</math> is the [[Time ordered#Time ordering|time-ordering operator]], <math>|0\rangle</math> the vacuum in the non interacting theory, <math>\psi(x)</math> and <math>\bar{\psi}(x)</math> the Dirac field and its Dirac adjoint, and where the left-hand
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, <math>e</math>. The general form of the propagator should remain unchanged, meaning that if <math>|\Omega\rangle</math> now represents the vacuum in the interacting theory, the two-point correlation function would now read
:<math> \langle \Omega | T(\psi(x)\bar{\psi}(0))| \Omega \rangle = \int \frac{d^
Two new quantities have been introduced. First the renormalized mass <math>m_r</math> has been defined as the pole in the Fourier transform of the Feynman propagator. This is the main prescription of the on-shell renormalization scheme (there is then no need to introduce other mass scales like in the minimal
This means that <math>m_r</math> and <math>Z_2</math> can be defined as a
:<math>Z_2=1+\delta_2</math>
:<math>
On the other hand, the modification to the propagator can be calculated up to a certain order in <math>e</math> using Feynman diagrams. These modifications are summed up in the fermion [[self energy]] <math>\Sigma(p)</math>
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:<math> \langle \Omega | T(\psi(x)\bar{\psi}(0))| \Omega \rangle = \int \frac{d^4p}{(2\pi)^4}\frac{ie^{-i p\cdot x}}{p\!\!\!/-m - \Sigma(p) +i\epsilon} </math>
These corrections are often divergent because they
By identifying the two expressions of the correlation function up to a certain order in <math>e</math>, the counterterms can be defined, and they are going to absorb the divergent contributions of the corrections to the fermion propagator. Thus, the renormalized quantities, such as <math>m_r</math>, will remain finite, and will be the quantities measured in experiments.
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:<math> \langle \Omega | T(A^{\mu}(x)A^{\nu}(0))| \Omega \rangle = \int \frac{d^4q}{(2\pi)^4}\frac{-i\eta^{\mu\nu}e^{-i p\cdot x}}{q^2(1 - \Pi(q^2)) +i\epsilon} = \int \frac{d^4q}{(2\pi)^4}\frac{-iZ_3 \eta^{\mu\nu}e^{-i p\cdot x}}{q^2 +i\epsilon} </math>
The behaviour of the counterterm <math>\delta_3=Z_3-1</math> is independent of the momentum of the incoming photon <math>q</math>. To fix it, the behaviour of QED at large distances (which should help recover [[classical electrodynamics]]),
:<math>\frac{-i\eta^{\mu\nu}e^{-i p\cdot x}}{q^2(1 - \Pi(q^2)) +i\epsilon}\sim\frac{-i\eta^{\mu\nu}e^{-i p\cdot x}}{q^2}</math>
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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 [[
==
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
:<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>
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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>
The <math>\delta_i</math> are called counterterms (some other definitions of them are possible). They are supposed to be small in the parameter <math>e</math>. 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).▼
▲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 <math>A</math> when they do not interact (which corresponds to removing the term <math>e \bar{\psi} \gamma^\mu \psi A_{\mu} </math> in the Lagrangian).
==References==
* {{Cite book|author=M. Peskin
[[Category:Quantum field theory]]
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