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There was a reference to 'pileup' at the LHC, this is not related to renormalization. Pileup at the LHC is a reference to multiple collisions at macroscopic distances, up to many centimetres, within a small enough time window to confuse detectors, (usually one bunch crossing). It is not relevant to renormalisation. The sentence was simply removed. |
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{{Short description|Method in physics used to deal with infinities}}
{{Use American English|date=January 2019}}
{{Use mdy dates|date=January 2019}}
{{Renormalization and regularization}}
{{Quantum field theory|cTopic=Tools}}
'''Renormalization''' is a collection of techniques in [[quantum field theory]],
For example, an [[electron]] theory may begin by postulating an electron with an initial mass and charge. In [[quantum field theory]] a cloud of [[virtual particle]]s, such as [[photon]]s, [[positron]]s, and others surrounds and interacts with the initial electron. Accounting for the interactions of the surrounding particles (e.g. collisions at different energies) shows that the electron-system behaves as if it had a different mass and charge than initially postulated. Renormalization, in this example, mathematically replaces the initially postulated mass and charge of an electron with the experimentally observed mass and charge. Mathematics and experiments prove that positrons and more massive particles like [[proton]]s
Renormalization specifies relationships between parameters in the theory when parameters describing large distance scales differ from parameters describing small distance scales. Physically, the pileup of contributions from an infinity of scales involved in a problem may then result in further infinities. When describing
Renormalization was first developed in [[quantum electrodynamics]] (QED) to make sense of [[infinity|infinite]] integrals in [[perturbation theory (quantum mechanics)|perturbation theory]]. Initially viewed as a suspect provisional procedure even by some of its originators, renormalization eventually was embraced as an important and [[self-consistent]] actual mechanism of scale physics in several fields of [[physics]] and [[mathematics]]. Despite his later skepticism, it was [[Paul Dirac]] who pioneered renormalization.<ref>{{Cite journal |last1=Sanyuk |first1=Valerii I. |last2=Sukhanov |first2=Alexander D. |date=2003-09-01 |title=Dirac in 20th century physics: a centenary assessment |url=https://ufn.ru/en/articles/2003/9/c/ |journal=Physics-Uspekhi |language=en |volume=46 |issue=9 |pages=937–956 |doi=10.1070/PU2003v046n09ABEH001165 |issn=1063-7869}}</ref><ref name=":62">{{Cite thesis |last=Kar |first=Arnab |title=Renormalization from Classical to Quantum Physics |date=2014 |publisher=University of Rochester |url=https://inspirehep.net/literature/1411969 |degree=}}</ref>
Today, the point of view has shifted: on the basis of the breakthrough [[renormalization group]] insights of [[Nikolay Bogolyubov]] and [[Kenneth G. Wilson|Kenneth Wilson]], the focus is on variation of physical quantities across contiguous scales, while distant scales are related to each other through "effective" descriptions. All scales are linked in a broadly systematic way, and the actual physics pertinent to each is extracted with the suitable specific computational techniques appropriate for each. Wilson clarified which variables of a system are crucial and which are redundant.
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The problem of infinities first arose in the [[classical electrodynamics]] of [[Elementary particle|point particles]] in the 19th and early 20th century.
The mass of a charged particle should include the
:<math>m_\text{em} = \int \frac{1}{2} E^2 \, dV = \int_{
which becomes infinite as {{math|''r''<sub>e</sub> → 0}}. This implies that the point particle would have infinite [[inertia]]
:<math>
where <math>\alpha \approx 1/137</math> is the [[fine-structure constant]], and <math>\hbar/(
Renormalization: The total effective mass of a spherical charged particle includes the actual bare mass of the spherical shell (in addition to the mass mentioned above associated with its electric field). If the shell's bare mass is allowed to be negative, it might be possible to take a consistent point limit.{{Citation needed|date=March 2015}} This was called ''renormalization'', and [[Hendrik Lorentz|Lorentz]] and [[Max Abraham|Abraham]] attempted to develop a classical theory of the electron this way. This early work was the inspiration for later attempts at [[regularization (physics)|regularization]] and renormalization in quantum field theory.
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When calculating the [[electromagnetism|electromagnetic]] interactions of [[electric charge|charged]] particles, it is tempting to ignore the ''[[back-reaction]]'' of a particle's own field on itself. (Analogous to the [[back-EMF]] of circuit analysis.) But this back-reaction is necessary to explain the friction on charged particles when they emit radiation. If the electron is assumed to be a point, the value of the back-reaction diverges, for the same reason that the mass diverges, because the field is [[inverse-square law|inverse-square]].
The [[Abraham–Lorentz force|Abraham–Lorentz theory]] had a noncausal "pre-acceleration
The trouble was worse in classical field theory than in quantum field theory, because in quantum field theory a charged particle experiences [[Zitterbewegung]] due to interference with virtual
== Divergences in quantum electrodynamics ==
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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.
[[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.
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.
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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 10}}</ref>
:(a) A photon creates a virtual electron–[[positron]] pair, which then annihilates. This is a [[vacuum polarization]] diagram.
:(b) An electron quickly emits and reabsorbs a virtual photon, called a [[self-energy]].
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together (Pokorski 1987, p. 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
:<math>\mathcal{L}_I = -e \bar\psi \gamma_\mu A^\mu \psi - (Z_1 - 1) e \bar\psi \gamma_\mu A^\mu \psi</math>
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With the regulator in place, and a finite value for the cutoff, divergent terms in the integrals then turn into finite but cutoff-dependent terms. After canceling out these terms with the contributions from cutoff-dependent counterterms, the cutoff is taken to infinity and finite physical results recovered. If physics on scales we can measure is independent of what happens at the very shortest distance and time scales, then it should be possible to get cutoff-independent results for calculations.
Many different types of regulator are used in quantum field theory calculations, each with its advantages and disadvantages. One of the most popular in modern use is ''[[dimensional regularization]]'', invented by [[Gerardus 't Hooft]] and [[Martinus J. G. Veltman]],<ref>{{Cite journal | last1 = 't Hooft | first1 = G. | last2 = Veltman | first2 = M. | doi = 10.1016/0550-3213(72)90279-9 | title = Regularization and renormalization of gauge fields | journal = Nuclear Physics B | volume = 44 | issue = 1 | pages = 189–213 | year = 1972 |bibcode = 1972NuPhB..44..189T | hdl = 1874/4845 | url = https://repositorio.unal.edu.co/handle/unal/81144 | hdl-access = free }}</ref> which tames the integrals by carrying them into a space with a fictitious fractional number of dimensions. Another is ''[[Pauli–Villars regularization]]'', which adds fictitious particles to the theory with very large masses, such that loop integrands involving the massive particles cancel out the existing loops at large momenta.
Yet another regularization scheme is the ''[[Lattice field theory|lattice regularization]]'', introduced by [[Kenneth G. Wilson|Kenneth Wilson]], which pretends that hyper-cubical lattice constructs our
A rigorous mathematical approach to renormalization theory is the so-called [[causal perturbation theory]], where ultraviolet divergences are avoided from the start in calculations by performing well-defined mathematical operations only within the framework of [[Distribution (mathematics)|distribution]] theory. In this approach, divergences are replaced by ambiguity: corresponding to a divergent diagram is a term which now has a finite, but undetermined, coefficient. Other principles, such as gauge symmetry, must then be used to reduce or eliminate the ambiguity.
== Attitudes and interpretation ==
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[[Freeman Dyson]] argued that these infinities are of a basic nature and cannot be eliminated by any formal mathematical procedures, such as the renormalization method.<ref>{{cite journal | last=Dyson | first=F. J. | title=Divergence of Perturbation Theory in Quantum Electrodynamics | journal=Physical Review | publisher=American Physical Society (APS) | volume=85 | issue=4 | date=1952-02-15 | issn=0031-899X | doi=10.1103/physrev.85.631 | pages=631–632| bibcode=1952PhRv...85..631D }}</ref><ref>{{cite journal | last=Stern | first=A. W. | title=Space, Field, and Ether in Contemporary Physics | journal=Science | publisher=American Association for the Advancement of Science (AAAS) | volume=116 | issue=3019 | date=1952-11-07 | issn=0036-8075 | doi=10.1126/science.116.3019.493 | pages=493–496| pmid=17801299 | bibcode=1952Sci...116..493S }}</ref>
[[Paul Dirac|Dirac]]'s criticism was the most persistent.<ref>P.A.M. Dirac, "The Evolution of the Physicist's Picture of Nature
: Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics is a good theory and we do not have to worry about it any more.' I must say that I am very dissatisfied with the situation because this so-called 'good theory' does involve neglecting infinities which appear in its equations, ignoring them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves disregarding a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!
Another important critic was [[Richard Feynman|Feynman]]. Despite his crucial role in the development of quantum electrodynamics, he wrote the following in 1985:<ref>Feynman, Richard P.
: The shell game that we play to find ''n'' and ''j'' is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.
Feynman was concerned that all field theories known in the 1960s had the property that the interactions become infinitely strong at short enough distance scales. This property called a [[Landau pole]], made it plausible that quantum field theories were all inconsistent. In 1974, [[David Gross|Gross]], [[David Politzer|Politzer]] and [[Frank Wilczek|Wilczek]] showed that another quantum field theory, [[quantum chromodynamics]], does not have a Landau pole. Feynman, along with most others, accepted that QCD was a fully consistent theory.{{Citation needed|date=December 2009}}
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Be that as it may, [[Abdus Salam|Salam]]'s remark<ref>{{cite journal | last1=Isham | first1=C. J. | last2=Salam | first2=Abdus | last3=Strathdee | first3=J. | title=Infinity Suppression in Gravity-Modified Electrodynamics. II | journal=Physical Review D | publisher=American Physical Society (APS) | volume=5 | issue=10 | date=1972-05-15 | issn=0556-2821 | doi=10.1103/physrevd.5.2548 | pages=2548–2565| bibcode=1972PhRvD...5.2548I }}</ref> in 1972 seems still relevant
: Field-theoretic infinities
:: In the modern world, if communities are unhappy, it is often because they have ignorances, habits, beliefs, and passions, which are dearer to them than happiness or even life. I find many men in our dangerous age who seem to be in love with misery and death, and who grow angry when hopes are suggested to them. They think hope is irrational and that, in sitting down to lazy despair, they are merely facing facts.
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In QFT, the value of a physical constant, in general, depends on the scale that one chooses as the renormalization point, and it becomes very interesting to examine the renormalization group running of physical constants under changes in the energy scale. The coupling constants in the [[Standard Model]] of particle physics vary in different ways with increasing energy scale: the coupling of [[quantum chromodynamics]] and the weak isospin coupling of the [[electroweak force]] tend to decrease, and the weak hypercharge coupling of the electroweak force tends to increase. At the colossal energy scale of 10<sup>15</sup> [[GeV]] (far beyond the reach of our current [[particle accelerator]]s), they all become approximately the same size (Grotz and Klapdor 1990, p. 254), a major motivation for speculations about [[grand unified theory]]. Instead of being only a worrisome problem, renormalization has become an important theoretical tool for studying the behavior of field theories in different regimes.
If a theory featuring renormalization (e.g. QED) can only be sensibly interpreted as an effective field theory, i.e. as an approximation reflecting human ignorance about the workings of nature, then the problem remains of discovering a more accurate theory that does not have these renormalization problems. As [[Lewis Ryder]] has put it, "In the Quantum Theory, these [classical] divergences do not disappear; on the contrary, they appear to get worse. And despite the comparative success of renormalisation theory, the feeling remains that there ought to be a more satisfactory way of doing things."<ref>Ryder, Lewis. ''[https://books.google.com/books?id=L9YhYS7gcXAC
== Renormalizability ==
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== Renormalization schemes ==
{{Confusing|section|date=January 2022}}
In actual calculations, the counterterms introduced to cancel the divergences in Feynman diagram calculations beyond tree level must be ''fixed'' using a set of '' renormalisation conditions''. The common renormalization schemes in use include:
* [[Minimal subtraction scheme|Minimal subtraction (MS) scheme]] and the related modified minimal subtraction (MS-bar) scheme
* [[On shell renormalization scheme|On-shell scheme]]
Besides, there exists a "natural" definition of the renormalized coupling (combined with the photon propagator) as a propagator of dual free bosons, which does not explicitly require introducing the counterterms.<ref>{{Cite journal | last1 = Makogon | first1 = D. | last2 = Morais Smith | first2 = C. | doi = 10.1103/PhysRevB.105.174505 | title = Median-point approximation and its application for the study of fermionic systems | journal = Phys. Rev. B | volume = 105 | pages = 174505 | year = 2022| issue = 17 | arxiv = 1909.12553 | bibcode = 2022PhRvB.105q4505M | s2cid = 203591796 }}</ref>
==
===History===
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\{\tilde J_k\}</math>, then the theory is said to be
'''renormalizable'''.
The possible
macroscopic states of the system, at a large scale, are given by this
set of fixed points.
===Renormalization group fixed points===
The most important information in the RG flow is its '''fixed points'''. A fixed point is defined by the vanishing of the [[beta function (physics)|beta function]] associated to the flow. Then, fixed points of the renormalization group are by definition scale invariant. In many cases of physical interest scale invariance enlarges to conformal invariance. One then has a [[conformal field theory]] at the fixed point.
The ability of several theories to flow to the same fixed point leads to [[Universality (dynamical systems)|universality]].
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* [[Quantum triviality]]
* [[Zeno's paradoxes]]
* [[Nonoblique correction]]
== References ==
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* {{cite journal |doi=10.1119/1.1624112 |arxiv=hep-th/0212049|title=A hint of renormalization|journal=American Journal of Physics|volume=72|issue=2|pages=170–184|year=2004|last1=Delamotte|first1=Bertrand|bibcode=2004AmJPh..72..170D|s2cid=2506712}}
* Baez, John; [http://math.ucr.edu/home/baez/renormalization.html ''Renormalization Made Easy''], (2005). A qualitative introduction to the subject.
* Blechman, Andrew E.; [https://web.archive.org/web/20160801034555/http://www.pha.jhu.edu/~blechman/papers/renormalization/renormalization.pdf ''Renormalization: Our Greatly Misunderstood Friend''], (2002). Summary of a lecture; has more information about specific regularization and divergence-subtraction schemes.
* {{cite journal |doi=10.1007/BF01255832|title=The conceptual foundations and the philosophical aspects of renormalization theory|journal=Synthese|volume=97|pages=33–108|year=1993|last1=Cao|first1=Tian Yu|last2=Schweber|first2=Silvan S.|s2cid=46968305}}
* [[Dmitry Shirkov|Shirkov, Dmitry]]; ''Fifty Years of the Renormalization Group'', C.E.R.N. Courrier 41(7) (2001). Full text available at : [http://www.cerncourier.com/main/article/41/7/14 ''I.O.P Magazines''].
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* 't Hooft, Gerard; ''The Glorious Days of Physics – Renormalization of Gauge theories'', lecture given at Erice (August/September 1998) by the [http://nobelprize.org/physics/laureates/1999/thooft-autobio.html ''Nobel laureate 1999''] . Full text available at: [https://arxiv.org/abs/hep-th/9812203 ''hep-th/9812203''].
* Rivasseau, Vincent; ''An introduction to renormalization'', Poincaré Seminar (Paris, Oct. 12, 2002), published in : Duplantier, Bertrand; Rivasseau, Vincent (Eds.); ''Poincaré Seminar 2002'', Progress in Mathematical Physics 30, Birkhäuser (2003) {{ISBN|3-7643-0579-7}}. Full text available in [http://www.bourbaphy.fr/Rivasseau.ps ''PostScript''].
* Rivasseau, Vincent; ''From perturbative to constructive renormalization'', Princeton University Press (1991) {{ISBN|0-691-08530-7}}. Full text available in [http://cpth.polytechnique.fr/cpth/rivass/articles/book.ps ''PostScript'']{{Dead link|date=October 2022 |bot=InternetArchiveBot |fix-attempted=yes }} and in [http://www.rivasseau.com/resources/book.pdf PDF (draft version)]. <!-- PDF link from author's homepage: http://www.rivasseau.com/3.html -->
* Iagolnitzer, Daniel & Magnen, J.; ''Renormalization group analysis'', Encyclopaedia of Mathematics, Kluwer Academic Publisher (1996). Full text available in PostScript and pdf [https://web.archive.org/web/20060630015233/http://www-spht.cea.fr/articles/t96/037/ ''here''].
* Scharf, Günter; ''Finite quantum electrodynamics: The causal approach'', Springer Verlag Berlin Heidelberg New York (1995) {{ISBN|3-540-60142-2}}.
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* [[Nigel Goldenfeld]]; ''Lectures on Phase Transitions and the Renormalization Group'', Frontiers in Physics 85, Westview Press (June, 1992) {{ISBN|0-201-55409-7}}. Covering the elementary aspects of the physics of phases transitions and the renormalization group, this popular book emphasizes understanding and clarity rather than technical manipulations.
* Zinn-Justin, Jean; ''Quantum Field Theory and Critical Phenomena'', Oxford University Press (4th edition – 2002) {{ISBN|0-19-850923-5}}. A masterpiece on applications of renormalization methods to the calculation of critical exponents in statistical mechanics, following Wilson's ideas (Kenneth Wilson was [http://nobelprize.org/physics/laureates/1982/wilson-autobio.html ''Nobel laureate 1982'']).
* Zinn-Justin, Jean; ''Phase Transitions & Renormalization Group: from Theory to Numbers'', Poincaré Seminar (Paris, Oct. 12, 2002), published in : Duplantier, Bertrand; Rivasseau, Vincent (Eds.); ''Poincaré Seminar 2002'', Progress in Mathematical Physics 30, Birkhäuser (2003) {{ISBN|3-7643-0579-7}}. Full text available in [http://parthe.lpthe.jussieu.fr/poincare/textes/octobre2002/Zinn.ps ''PostScript''] {{Webarchive|url=https://web.archive.org/web/20051015150706/http://parthe.lpthe.jussieu.fr/poincare/textes/octobre2002/Zinn.ps |date=October 15, 2005 }}.
* Domb, Cyril; ''The Critical Point: A Historical Introduction to the Modern Theory of Critical Phenomena'', CRC Press (March, 1996) {{ISBN|0-7484-0435-X}}.
* Brown, Laurie M. (Ed.); ''Renormalization: From Lorentz to Landau (and Beyond)'', Springer-Verlag (New York-1993) {{ISBN|0-387-97933-6}}.
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* Connes, Alain; ''Symétries Galoisiennes & Renormalisation'', Poincaré Seminar (Paris, Oct. 12, 2002), published in : Duplantier, Bertrand; Rivasseau, Vincent (Eds.); ''Poincaré Seminar 2002'', Progress in Mathematical Physics 30, Birkhäuser (2003) {{ISBN|3-7643-0579-7}}. French mathematician [http://www.alainconnes.org ''Alain Connes''] (Fields medallist 1982) describe the mathematical underlying structure (the [[Hopf algebra]]) of renormalization, and its link to the Riemann-Hilbert problem. Full text (in French) available at {{ArXiv|math/0211199}}.
== External links ==
* {{wikiquote-inline}}
[[Category:Quantum field theory]]
[[Category:Renormalization group]]
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