Editing Renormalization (section)

=== Renormalization in QED ===
[[Image:Counterterm.png|thumb|upright=1.1|Figure 3. The vertex corresponding to the {{math|''Z''<sub>1</sub>}} counterterm cancels the divergence in Figure 2.]]

For example, in the [[quantum electrodynamics|Lagrangian of QED]]
<math display="block">\mathcal{L}=\bar\psi_B\left[i\gamma_\mu \left (\partial^\mu + ie_BA_B^\mu \right )-m_B\right]\psi_B -\frac{1}{4}F_{B\mu\nu}F_B^{\mu\nu}</math>
the fields and coupling constant are really ''bare'' quantities, hence the subscript {{mvar|B}} above. Conventionally the bare quantities are written so that the corresponding Lagrangian terms are multiples of the renormalized ones:
<math display="block">\left(\bar\psi m \psi\right)_B = Z_0 \bar\psi m \psi</math>
<math display="block">\left(\bar\psi\left(\partial^\mu + ieA^\mu \right )\psi\right)_B = Z_1 \bar\psi \left (\partial^\mu + ieA^\mu \right)\psi</math>
<math display="block">\left(F_{\mu\nu}F^{\mu\nu}\right)_B = Z_3\, F_{\mu\nu}F^{\mu\nu}.</math>

[[Gauge invariance]], via a [[Ward–Takahashi identity]], turns out to imply that we can renormalize the two terms of the [[covariant derivative]] piece
<math display="block">\bar \psi (\partial + ieA) \psi</math>
together (Pokorski 1987, p.&nbsp;115), which is what happened to {{math|''Z''<sub>2</sub>}}; it is the same as {{math|''Z''<sub>1</sub>}}.

A term in this Lagrangian, for example, the electron–photon interaction pictured in Figure 1, can then be written
<math display="block">\mathcal{L}_I = -e \bar\psi \gamma_\mu A^\mu \psi - (Z_1 - 1) e \bar\psi \gamma_\mu A^\mu \psi</math>

The physical constant {{mvar|e}}, the electron's charge, can then be defined in terms of some specific experiment:  we set the renormalization scale equal to the energy characteristic of this experiment, and the first term gives the interaction we see in the laboratory (up to small, finite corrections from loop diagrams, providing such exotica as the high-order corrections to the [[magnetic moment]]).  The rest is the counterterm.  If the theory is ''renormalizable'' (see below for more on this), as it is in QED, the ''divergent'' parts of loop diagrams can all be decomposed into pieces with three or fewer legs, with an algebraic form that can be canceled out by the second term (or by the similar counterterms that come from {{math|''Z''<sub>0</sub>}} and {{math|''Z''<sub>3</sub>}}).

The diagram with the {{math|''Z''<sub>1</sub>}} counterterm's interaction vertex placed as in Figure 3 cancels out the divergence from the loop in Figure 2.

Historically, the splitting of the "bare terms" into the original terms and counterterms came before the [[renormalization group]] insight due to [[Kenneth G. Wilson|Kenneth Wilson]].<ref name=Wilson1975>{{cite journal | last=Wilson | first=Kenneth G. |author-link=Kenneth G. Wilson| title=The renormalization group: Critical phenomena and the Kondo problem | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=47 | issue=4 | date=1975-10-01 | issn=0034-6861 | doi=10.1103/revmodphys.47.773 | pages=773–840| bibcode=1975RvMP...47..773W }}</ref> According to such [[renormalization group]] insights, detailed in the next section, this splitting is unnatural and actually unphysical, as all scales of the problem enter in continuous systematic ways.