Editing Renormalization (section)

== Divergences in quantum electrodynamics ==
{{anchor|renormalization_loop_divergence}}
[[Image:vacuum polarization.svg|thumb|(a) Vacuum polarization, a.k.a. charge screening. This loop has a logarithmic ultraviolet divergence.]]
[[Image:selfE.svg|thumb|(b) Self-energy diagram in QED]]
[[Image:Penguin diagram.JPG|thumb|(c) Example of a “penguin” diagram]]

When developing [[quantum electrodynamics]] in the 1930s, [[Max Born]], [[Werner Heisenberg]], [[Pascual Jordan]], and [[Paul Dirac]] discovered that in perturbative  corrections  many integrals were divergent (see [[The problem of infinities]]).

One way of describing the [[perturbation theory (quantum mechanics)|perturbation theory]] corrections' divergences was discovered in 1947–49 by<!--in chronological order--> [[Hans Kramers]]<!--June 1947-->,<ref>Kramers presented his work at the 1947 [[Shelter Island Conference]], repeated in 1948 at the [[Solvay Conference]]. The latter did not appear in print until the Proceedings of the Solvay Conference, published in 1950 (see Laurie M. Brown (ed.), ''Renormalization: From Lorentz to Landau (and Beyond)'', Springer, 2012, p.&nbsp;53). Kramers' approach was [[nonrelativistic]] (see [[Jagdish Mehra]], [[Helmut Rechenberg]], ''The Conceptual Completion and Extensions of Quantum Mechanics 1932–1941. Epilogue: Aspects of the Further Development of Quantum Theory 1942–1999: Volumes 6, Part 2'', Springer, 2001, p.&nbsp;1050).</ref> [[Hans Bethe]]<!--August 1947-->,<ref>{{cite journal |author=H. Bethe |author-link=Hans Bethe |year=1947 |title=The Electromagnetic Shift of Energy Levels |journal=[[Physical Review]] |volume=72 |pages=339–341 |doi=10.1103/PhysRev.72.339 |bibcode=1947PhRv...72..339B |issue=4|s2cid=120434909 }}</ref> 
[[Julian Schwinger]]<!--February 1948-->,<ref>{{cite journal |author=Schwinger, J. |title=On quantum-electrodynamics and the magnetic moment of the electron |journal=[[Physical Review]] |volume=73 |issue=4 |pages=416–417 |year=1948|doi=10.1103/PhysRev.73.416 |bibcode=1948PhRv...73..416S |doi-access=free }}</ref><ref>{{cite journal |author=Schwinger, J. |series=Quantum Electrodynamics |title=I. A covariant formulation |journal=[[Physical Review]] |volume=74 |issue=10 |pages=1439–1461 |year=1948|doi=10.1103/PhysRev.74.1439 |bibcode=1948PhRv...74.1439S }}</ref><ref>{{cite journal |author=Schwinger, J. |series=Quantum Electrodynamics |title=II. Vacuum polarization and self-energy |journal=[[Physical Review]] |volume=75 |issue=4 |pages=651–679 |year=1949|doi=10.1103/PhysRev.75.651 |bibcode=1949PhRv...75..651S }}</ref><ref>{{cite journal |author=Schwinger, J. |series=Quantum Electrodynamics |title=III. The electromagnetic properties of the electron radiative corrections to scattering |journal=[[Physical Review]] |volume=76 |issue=6 |pages=790–817 |year=1949|doi=10.1103/PhysRev.76.790 |bibcode=1949PhRv...76..790S }}</ref> [[Richard Feynman]]<!--April 1948-->,<ref>{{cite journal |first=Richard P. |last=Feynman |title=Space-time approach to non-relativistic quantum mechanics |journal=[[Reviews of Modern Physics]] |volume=20 |pages=367–387 |year=1948 |doi=10.1103/RevModPhys.20.367 |bibcode=1948RvMP...20..367F |issue=2|url=https://authors.library.caltech.edu/47756/1/FEYrmp48.pdf }}</ref><ref>{{cite journal |last=Feynman  |first= Richard P. |title=A relativistic cut-off for classical electrodynamics   |journal=[[Physical Review]] |volume=74  |issue=8 |pages= 939–946 |year=1948  |doi=10.1103/PhysRev.74.939 |bibcode=1948PhRv...74..939F|url= https://authors.library.caltech.edu/3516/1/FEYpr48a.pdf }}</ref><ref>{{cite journal |first=Richard P. |last=Feynman |title=A relativistic cut-off for quantum electrodynamics |journal=[[Physical Review]] |volume=74 |pages=1430–1438 |year=1948 |doi=10.1103/PhysRev.74.1430 |bibcode=1948PhRv...74.1430F |issue=10|url=https://authors.library.caltech.edu/3517/1/FEYpr48b.pdf }}</ref> and [[Shin'ichiro Tomonaga]]<!--July 1948 (Koba–Tomonaga); according to S. S. Schweber, ''QED'', 1994, p.&nbsp;269, Koba–Tomonaga contains the crucial calculation-->,<ref>{{cite journal | last=Tomonaga | first=S. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=1 | issue=2 | date=1946-08-01 | issn=1347-4081 | doi=10.1143/ptp.1.27 | pages=27–42|doi-access=free| bibcode=1946PThPh...1...27T }}</ref><ref>{{cite journal | last1=Koba | first1=Z. | last2=Tati | first2=T. | last3=Tomonaga | first3=S.-i. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. II: Case of Interacting Electromagnetic and Electron Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=2 | issue=3 | date=1947-10-01 | issn=0033-068X | doi=10.1143/ptp/2.3.101 | pages=101–116|doi-access=free| bibcode=1947PThPh...2..101K }}</ref><ref>{{cite journal | last1=Koba | first1=Z. | last2=Tati | first2=T. | last3=Tomonaga | first3=S.-i. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. III: Case of Interacting Electromagnetic and Electron Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=2 | issue=4 | date=1947-12-01 | issn=0033-068X | doi=10.1143/ptp/2.4.198 | pages=198–208|doi-access=free| bibcode=1947PThPh...2..198K }}</ref><ref>{{cite journal | last1=Kanesawa | first1=S. | last2=Tomonaga | first2=S.-i. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. [IV]: Case of Interacting Electromagnetic and Meson Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=3 | issue=1 | date=1948-03-01 | issn=0033-068X | doi=10.1143/ptp/3.1.1 | pages=1–13|doi-access=free}}</ref><ref>{{cite journal | last1=Kanesawa | first1=S. | last2=Tomonaga | first2=S.-i. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields V: Case of Interacting Electromagnetic and Meson Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=3 | issue=2 | date=1948-06-01 | issn=0033-068X | doi=10.1143/ptp/3.2.101 | pages=101–113|doi-access=free| bibcode=1948PThPh...3..101K }}</ref><ref>{{cite journal | last1=Koba | first1=Z. | last2=Tomonaga | first2=S.-i. | title=On Radiation Reactions in Collision Processes. I: Application of the "Self-Consistent" Subtraction Method to the Elastic Scattering of an Electron | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=3 | issue=3 | date=1948-09-01 | issn=0033-068X | doi=10.1143/ptp/3.3.290 | pages=290–303|doi-access=| bibcode=1948PThPh...3..290K }}</ref><ref>{{cite journal | last1=Tomonaga | first1=Sin-Itiro | last2=Oppenheimer | first2=J. R. |author-link2=J. Robert Oppenheimer| title=On Infinite Field Reactions in Quantum Field Theory | journal=Physical Review | publisher=American Physical Society (APS) | volume=74 | issue=2 | date=1948-07-15 | issn=0031-899X | doi=10.1103/physrev.74.224 | pages=224–225| bibcode=1948PhRv...74..224T }}</ref> and systematized by [[Freeman Dyson]] in 1949.<ref>{{cite journal |author=Dyson, F. J. |title=The radiation theories of Tomonaga, Schwinger, and Feynman |journal=Phys. Rev. |volume=75 |pages=486–502 |year=1949|doi=10.1103/PhysRev.75.486 |issue=3 |bibcode=1949PhRv...75..486D |doi-access=free }}</ref> The divergences appear in radiative corrections  involving [[Feynman diagram]]s with closed ''loops'' of [[virtual particle]]s in them.

While virtual particles obey [[conservation of energy]] and [[momentum]], they can have any energy and momentum, even one that is not allowed by the relativistic [[energy–momentum relation]] for the observed mass of that particle (that is, <math>E^2 - p^2</math> is not necessarily the squared mass of the particle in that process, e.g. for a photon it could be nonzero). Such a particle is called [[on shell|off-shell]]. When there is a loop, the momentum of the particles involved in the loop is not uniquely determined by the energies and momenta of incoming and outgoing particles. A variation in the energy of one particle in the loop can be balanced by an equal and opposite change in the energy of another particle in the loop, without affecting the incoming and outgoing particles. Thus many variations are possible. So to find the amplitude for the loop process, one must [[integral|integrate]] over ''all'' possible combinations of energy and momentum that could travel around the loop.

These integrals are often ''divergent'', that is, they give infinite answers.  The divergences that are significant are the "[[ultraviolet divergence|ultraviolet]]" (UV) ones. An ultraviolet divergence can be described as one that comes from
* the region in the integral where all particles in the loop have large energies and momenta,
* very short [[wavelength]]s and high-[[frequency|frequencies]] fluctuations of the fields, in the [[Path integral formulation|path integral]] for the field,
* very short proper-time between particle emission and absorption, if the loop is thought of as a sum over particle paths.

So these divergences are short-distance, short-time phenomena.

Shown in the pictures at the right margin, there are exactly three one-loop divergent loop diagrams in quantum electrodynamics:<ref>{{cite book |author1-link=Michael E. Peskin |first1=Michael E. |last1=Peskin |first2=Daniel V. |last2=Schroeder |title=An Introduction to Quantum Field Theory |url=https://archive.org/details/introductiontoqu0000pesk |url-access=registration |publisher=Addison-Wesley |location=Reading |year=1995 |isbn=9780201503975 |at=Chapter&nbsp;10}}</ref>
{{olist|type=lower-alpha
| 1=A photon creates a virtual electron–[[positron]] pair, which then annihilates. This is a [[vacuum polarization]] diagram.
| 2=An electron quickly emits and reabsorbs a virtual photon, called a [[self-energy]].
| 3=An electron emits a photon, emits a second photon, and reabsorbs the first. This process is shown in the section below in figure&nbsp;2, and it is called a ''[[vertex renormalization]]''. The Feynman diagram for this is also called a “[[penguin diagram]]” due to its shape remotely resembling a penguin.}}

The three divergences correspond to the three parameters in the theory under consideration:
# The field normalization Z.
# The mass of the electron.
# The charge of the electron.

The second class of divergence called an [[infrared divergence]], is due to massless particles, like the photon. Every process involving charged particles emits infinitely many coherent photons of infinite wavelength, and the amplitude for emitting any finite number of photons is zero. For photons, these divergences are well understood. For example, at the 1-loop order, the [[vertex function]] has both ultraviolet and ''infrared'' divergences. In contrast to the ultraviolet divergence, the infrared divergence does not require the renormalization of a parameter in the theory involved. The infrared divergence of the vertex diagram is removed by including a diagram similar to the vertex diagram with the following important difference: the photon connecting the two legs of the electron is cut and replaced by two [[on-shell]] (i.e. real) photons whose wavelengths tend to infinity; this diagram is equivalent to the [[bremsstrahlung]] process. This additional diagram must be included because there is no physical way to distinguish a zero-energy photon flowing through a loop as in the vertex diagram and zero-energy photons emitted through [[bremsstrahlung]]. From a mathematical point of view, the IR divergences can be regularized by assuming fractional differentiation w.r.t. a parameter, for example:
<math display="block"> \left( p^2 - a^2 \right)^{\frac{1}{2}} </math>
is well defined at {{math|''p'' {{=}} ''a''}} but is UV divergent; if we take the {{frac|3|2}}-th [[fractional derivative]] with respect to {{math|−''a''<sup>2</sup>}}, we obtain the IR divergence
<math display="block"> \frac{1}{p^2 - a^2},</math>
so we can cure IR divergences by turning them into UV divergences.{{clarify|date=May 2012}}

=== A loop divergence ===
[[Image:Loop-diagram.png|thumb|upright=1.1|Figure 2. A diagram contributing to electron–electron scattering in QED. The loop has an ultraviolet divergence.]]
The diagram in Figure 2 shows one of the several one-loop contributions to electron–electron scattering in QED. The electron on the left side of the diagram, represented by the solid line, starts out with 4-momentum {{math|''p<sup>μ</sup>''}} and ends up with 4-momentum {{math|''r<sup>μ</sup>''}}. It emits a virtual photon carrying {{math|''r<sup>μ</sup>'' − ''p<sup>μ</sup>''}} to transfer energy and momentum to the other electron.  But in this diagram, before that happens, it emits another virtual photon carrying 4-momentum {{math|''q<sup>μ</sup>''}}, and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the 4-momentum {{math|''q<sup>μ</sup>''}} uniquely, so all possibilities contribute equally and we must integrate.

This diagram's amplitude ends up with, among other things, a factor from the loop of
<math display="block">-ie^3 \int \frac{d^4 q}{(2\pi)^4} \gamma^\mu \frac{i (\gamma^\alpha (r - q)_\alpha + m)}{(r - q)^2 - m^2 + i \epsilon} \gamma^\rho \frac{i (\gamma^\beta (p - q)_\beta + m)}{(p - q)^2 - m^2 + i \epsilon} \gamma^\nu \frac{-i g_{\mu\nu}}{q^2 + i\epsilon}.</math>

The various {{math|''γ<sup>μ</sup>''}} factors in this expression are [[gamma matrices]] as in the covariant formulation of the [[Dirac equation]]; they have to do with the spin of the electron. The factors of {{mvar|e}} are the electric coupling constant, while the <math>i\epsilon</math> provide a heuristic definition of the contour of integration around the poles in the space of momenta. The important part for our purposes is the dependency on {{math|''q<sup>μ</sup>''}} of the three big factors in the integrand, which are from the [[propagator]]s of the two electron lines and the photon line in the loop.

This has a piece with two powers of {{math|''q<sup>μ</sup>''}} on top that dominates at large values of {{math|''q<sup>μ</sup>''}} (Pokorski 1987, p.&nbsp;122):
<math display="block">e^3 \gamma^\mu \gamma^\alpha \gamma^\rho \gamma^\beta \gamma_\mu \int \frac{d^4 q}{(2\pi)^4} \frac{q_\alpha q_\beta}{(r - q)^2 (p - q)^2 q^2}.</math>

This integral is divergent and infinite, unless we cut it off at finite energy and momentum in some way.

Similar loop divergences occur in other quantum field theories.