text:
Glow stick

A glow stick, also known as a light stick, chem light, light wand, light rod, and rave light, is a self-contained, short-term light-source. It consists of a translucent plastic tube containing isolated substances that, when combined, make light through chemiluminescence reaction. The light cannot be turned off and can be used only once. The used tube is then thrown away. Glow sticks are often used for recreation, such as for events, camping, outdoor exploration, and concerts. Glow sticks are also used for light in military and emergency services applications. Industrial uses include marine, transportation, and mining.

History
Bis(2,4,5-trichlorophenyl-6-carbopentoxyphenyl)oxalate, trademarked "Cyalume", was invented in 1971 by Michael M. Rauhut, of American Cyanamid, based on work by Edwin A. Chandross and David Iba Sr. of Bell Labs.

Other early work on chemiluminescence was carried out at the same time, by researchers under Herbert Richter at China Lake Naval Weapons Center.

Several US patents for glow stick-type devices were issued in 1973–74. A later 1976 patent recommended a single glass ampoule that is suspended in a second substance, that when broken and mixed together, provide the chemiluminescent light. The design also included a stand for the signal device so it could be thrown from a moving vehicle and remain standing in an upright position on the road. The idea was this would replace traditional emergency roadside flares and would be superior, since it was not a fire hazard, would be easier and safer to deploy, and would not be made ineffective if struck by passing vehicles. This design, with its single glass ampoule inside a plastic tube filled with a second substance that when bent breaks the glass and then is shaken to mix the substances, most closely resembles the typical glow stick sold today.

In the early 1980s the majority of glow sticks were produced in Novato, California by Omniglow Corp. Omniglow completed a leveraged buyout of American Cyanamid's chemical light division in 1994 and became the leading supplier of glow sticks worldwide until going out of business in 2014. Most glow sticks seen today are now made in China.

Uses
Glow sticks are waterproof, do not use batteries, consume no oxygen, generate no or negligible heat, produce neither spark nor flame, can tolerate high pressures such as those found under water, are inexpensive, and are reasonably disposable. This makes them ideal as light sources and light markers by military forces, campers, spelunkers, and recreational divers.

Entertainment

Glowsticking is the use of glow sticks in dancing (such as in glow poi and wotagei). They are frequently used for entertainment at parties (in particular raves), concerts, and dance clubs. They are used by marching band conductors for evening performances; glow sticks are also used in festivals and celebrations around the world. Glow sticks also serve multiple functions as toys, readily visible night-time warnings to motorists, and luminous markings that enable parents to keep track of their children. Another use is for balloon-carried light effects. Glow sticks are also used to create special effects in low light photography and film.

The Guinness Book of Records recorded the world's largest glow stick was cracked at 150 m tall. It was created by the University of Wisconsin–Whitewater's Chemistry Department to celebrate the school's sesquicentennial, or 150th birthday in Whitewater, Wisconsin and cracked on 9 September 2018.

Recreation and survival
Glow sticks are used for outdoor recreation, often used at night for marking. Scuba divers use diving-rated glow sticks to mark themselves during night dives, and then can turn off bright diving lights. This is done to enable visibility of bioluminescent marine organisms, which cannot be seen while a bright dive light is illuminated. Glow sticks are used on backpacks, tent pegs, and on jackets during overnight camping expeditions. Often, glow sticks are recommended as an addition to survival kits.

Industry
There are specific industrial uses of glow sticks, which are often used as a light source in circumstances where electric lighting and LEDs are not best suited. For example, in the mining industry, glow sticks are required for emergency evacuation in the case of a gas leak. Use of an electric light source in this case may cause an unintended explosion. Chemiluminescence, the type of light used in glow sticks, is a "cold-light" and does not use electricity, and will not cause a gas leak to ignite.

Glow sticks are also used worldwide in the marine industry, often used as fishing lures in long-line, recreational, and commercial fishing, as well as for personnel safety.

Military
Glow sticks are used by militaries, and occasionally also police tactical units, as light sources during night operations or close-quarters combat in dark areas. They are also used to mark secured areas or objects of note. When worn, they can be used to identify friendly soldiers during nighttime operations.

Emergency services
Glow sticks are used by police, fire, and emergency medical services as light sources, similar to their military applications. Often, emergency rescue crews will hand out glow sticks in order to keep track of people at night, who may not have access to their own lighting. Glow sticks are sometimes attached to life vests and lifeboats on passenger and commercial vessels, to ensure night time visibility.

Glow sticks are often part of emergency kits to provide basic lighting and provide ease of identification in dark areas. They can be found in emergency lighting kits in buildings, public transportation vehicles, and subway stations.

Operation
Glow sticks emit light when two chemicals are mixed. The reaction between the two chemicals is catalyzed by a base, usually sodium salicylate. The sticks consist of a tiny, brittle container within a flexible outer container. Each container holds a different solution. When the outer container is flexed, the inner container breaks, allowing the solutions to combine, causing the necessary chemical reaction. After breaking, the tube is shaken to thoroughly mix the components.

The glow stick contains two chemicals, a base catalyst, and a suitable dye (sensitizer, or fluorophor). This creates an exergonic reaction. The chemicals inside the plastic tube are a mixture of the dye, the base catalyst, and diphenyl oxalate. The chemical in the glass vial is hydrogen peroxide. By mixing the peroxide with the phenyl oxalate ester, a chemical reaction takes place, yielding two moles of phenol and one mole of peroxyacid ester (1,2-dioxetanedione). The peroxyacid decomposes spontaneously to carbon dioxide, releasing energy that excites the dye, which then relaxes by releasing a photon. The wavelength of the photon—the color of the emitted light—depends on the structure of the dye. The reaction releases energy mostly as light, with very little heat. The reason for this is that the reverse [2 + 2] photocycloadditions of 1,2-dioxetanedione is a forbidden transition (it violates Woodward–Hoffmann rules) and cannot proceed through a regular thermal mechanism.

By adjusting the concentrations of the two chemicals and the base, manufacturers can produce glow sticks that glow either brightly for a short amount of time or more dimly for an extended length of time. This also allows glow sticks to perform satisfactorily in hot or cold climates, by compensating for the temperature dependence of reaction. At maximum concentration (typically found only in laboratory settings), mixing the chemicals results in a furious reaction, producing large amounts of light for only a few seconds. The same effect can be achieved by adding copious amounts of sodium salicylate or other bases. Heating a glow stick also causes the reaction to proceed faster and the glow stick to glow more brightly for a brief period. Cooling a glow stick slows the reaction a small amount and causes it to last longer, but the light is dimmer. This can be demonstrated by refrigerating or freezing an active glow stick; when it warms up again, it will resume glowing. The dyes used in glow sticks usually exhibit fluorescence when exposed to ultraviolet radiation—even a spent glow stick may therefore shine under a black light.

The light intensity is high immediately after activation, then exponentially decays. Leveling of this initial high output is possible by refrigerating the glow stick before activation.

A combination of two fluorophores can be used, with one in the solution and another incorporated to the walls of the container. This is advantageous when the second fluorophore would degrade in solution or be attacked by the chemicals. The emission spectrum of the first fluorophore and the absorption spectrum of the second one have to largely overlap, and the first one has to emit at shorter wavelength than the second one. A downconversion from ultraviolet to visible is possible, as is conversion between visible wavelengths (e.g., green to orange) or visible to near-infrared. The shift can be as much as 200 nm, but usually the range is about 20–100 nm longer than the absorption spectrum. Glow sticks using this approach tend to have colored containers, due to the dye embedded in the plastic. Infrared glow sticks may appear dark-red to black, as the dyes absorb the visible light produced inside the container and reemit near-infrared. On the other hand, various colors can also be achieved by simply mixing several fluorophores within the solution to achieve the desired effect. These various colors can be achieved due to the principles of additive color. For example, a combination of red, yellow, and green fluorophores is used in orange light sticks, and a combination of several fluorescers is used in white light sticks.

Fluorophores used

 * 9,10-Diphenylanthracene (DPA) emits blue light
 * 9-(2-Phenylethenyl) anthracene emits teal light
 * 1-Chloro-9,10-diphenylanthracene (1-chloro(DPA)) and 2-chloro-9,10-diphenylanthracene (2-chloro(DPA)) emit blue-green light more efficiently than nonsubstituted DPA
 * 9,10-Bis(phenylethynyl)anthracene (BPEA) emits green light with maximum at 486 nm
 * 1-Chloro-9,10-bis(phenylethynyl)anthracene emits yellow-green light, used in 30-minute high-intensity Cyalume sticks
 * 2-Chloro-9,10-bis(phenylethynyl)anthracene emits green light, used in 12-hour low-intensity Cyalume sticks
 * 1,8-Dichloro-9,10-bis(phenylethynyl)anthracene emits yellow light, used in Cyalume sticks
 * Rubrene emits orange-yellow at 550 nm
 * 2,4-Di-tert-butylphenyl 1,4,5,8-tetracarboxynaphthalene diamide emits deep red light, together with DPA is used to produce white or hot-pink light, depending on their ratio
 * Rhodamine B emits red light. It is rarely used, as it breaks down in contact with CPPO, shortening the shelf life of the mixture.
 * 5,12-Bis(phenylethynyl)naphthacene emits orange light
 * Violanthrone emits orange light at 630 nm
 * 16,17-(1,2-Ethylenedioxy)violanthrone emits red at 680 nm
 * 16,17-Dihexyloxyviolanthrone emits infrared at 725 nm
 * 16,17-Butyloxyviolanthrone emits infrared
 * N,N′-Bis(2,5,-di-tert-butylphenyl)-3,4,9,10-perylenedicarboximide emits red
 * 1-(N,N-Dibutylamino)anthracene emits infrared
 * 6-Methylacridinium iodide emits infrared

Toxicity
In glow sticks, phenol is produced as a byproduct. It is advisable to keep the mixture away from skin and to prevent accidental ingestion if the glow stick case splits or breaks. If spilled on skin, the chemicals could cause slight skin irritation, swelling, or, in extreme circumstances, vomiting and nausea. Some of the chemicals used in older glow sticks were thought to be potential carcinogens. The sensitizers used are polynuclear aromatic hydrocarbons, a class of compounds known for their carcinogenic properties.

Dibutyl phthalate, a plasticizer sometimes used in glow sticks (and many plastics), has raised some health concerns. It was put on California's list of suspected teratogens in 2006. Glow stick liquid contains ingredients that can act as a plasticizer, softening plastics onto which it leaks. Diphenyl oxalate can sting and burn eyes, irritate and sting skin and can burn the mouth and throat if ingested.

Researchers in Brazil, concerned about waste from glowsticks used in fishing in their country, published a study in 2014 on this topic. It measured the secondary reactions that continue within used glow sticks, toxicity to cells in culture, and chemical reactions with DNA in vitro. The authors found "high toxicity" of light stick solutions, and evidence of reactivity with DNA. They concluded that light stick solutions "are hazardous and that the health risks associated with exposure have not yet been properly evaluated."

Single-use plastics
Glow sticks also contribute to the plastic waste problem, as glow sticks are single-use and made from plastic. Additionally, since the inner vial is often made from glass and the chemicals inside are dangerous if improperly handled, the plastic used for glow sticks is non-recoverable by recycling services, so glow sticks are categorized as non-recyclable waste.

Safety improvements
By the 2020s, work was being done to create safer glow sticks and alternatives. Canadian company LUX BIO, for example, developed glow stick alternatives such as the Light Wand which is biodegradable and glows with bioluminescence, rather than the chemiluminescence and LÜMI, which is a reusable and non-toxic alternative that glows with phosphorescence and is chemically and biologically inert.

title: Glow stick


text:
Anglo

Anglo is a prefix indicating a relation to, or descent from England, English culture, the English people or the English language, such as in the term Anglosphere. It is often used alone, somewhat loosely, to refer to people of British descent in Anglo-America, the Anglophone Caribbean, South Africa, Namibia, Australia, and New Zealand. It is used in Canada to differentiate between French-speaking Canadians (Francophones), located mainly in Quebec but found across Canada, and English-speaking Canadians (Anglophones), also located across Canada, including in Quebec. It is also used in the United States to distinguish the Latino population from the non-Latino white majority.

Anglo is a Late Latin prefix used to denote English- in conjunction with another toponym or demonym. The word is derived from Anglia, the Latin name for England and still used in the modern name for its eastern region, East Anglia. It most likely refers to the Angles, a Germanic people originating in the north German peninsula of Angeln, that is, the region of today's Lower Saxony that joins the Jutland Peninsula. (There are also various hypotheses for the origin of the name 'Angeln'.) The first recorded use of the word in Latin is in Tactitus's Germania,where he mentions the "Angles" as a Suebian tribe living near the Elbe. Bede writes that the Angles came from a place called Angulus "which lies between the province of the Jutes and the Saxons." Anglia and England both mean land of the English.

It is also often used to refer to British in historical and other contexts after the Acts of Union 1707, for example such as in the Anglo-Dutch Treaty of 1824, where in later years agreement was between the British government and the Dutch, not an English government. Typical examples of this use are also shown below, where non-English people from the British Isles are described as being Anglo.

Anglo is not an easily defined term. For traditionalists, there are linguistic problems with using the word as an adjective or noun on its own. For example, the purpose of the -o ending is to enable the formation of a compound term (for example Anglo-Saxon meaning of English and Saxon origin), so there is only an apparent parallelism between, for example, Latino and Anglo. However, a semantic change has taken place in many English-speaking regions so that in informal usage the meanings listed below are common. The definition is changed in each region which defines how it is identified.

Africa
The term Anglo-African has been used historically to self-identify by people of mixed British and African ancestry born in the United States and in Africa. The Anglo-African and The Weekly Anglo-African were the names of newspapers published by African American abolitionist Robert Hamilton (1819–1870) in New York during the American Civil War era. The Anglo-African was also the name of a newspaper published in Lagos (now part of Nigeria) from 1863 to 1865. It was founded and edited by Robert Campbell (1829–1884), a Jamaican born son of a Scottish father and Mulatto mother. The term has also been used historically to describe people living in the British Empire in Africa. The Anglo-African Who's Who and Biographical Sketch-Book published in London in 1905 includes details of prominent British and Afrikaner people in Africa at that time.

Australia
In Australia, Anglo is used as part of the terms Anglo-Australian and Anglo-Celtic, which refer to the majority of Australians, who are of English, Scottish, Welsh and Irish descent.

Canada
In Canada, and especially in Canadian French, Anglophone is widely used to designate someone whose mother tongue is English, as opposed to Francophone, which describes someone whose mother tongue is French, and to Allophone, which describes someone whose mother tongue is a language other than English or French. Anglo-Métis is also sometimes used to refer to an ethnic group.

Israel
Jewish immigrants making Aliyah to the State of Israel are sometimes referred to as Anglos.

Scotland
In Scotland, and in related cultures, the term Anglo-Scot, sometimes shortened to Anglo or Anglos, is used to refer to people with some permutation of mixed Scottish-English ancestry, association and/or birth; such as English people of Scottish descent, Scottish people of English descent, or heavily Anglicised members of the Scottish nobility who are indistinguishable from English members of the British upper class and speak with a Received Pronunciation, or other elite Southron accent.

A great number of Anglo-Scots have made their mark in the fields of sport, politics, law, diplomacy, the Military history of the United Kingdom, medicine, engineering, technical invention, maritime history, geographical exploration, journalism and on the stage and screen. The London-born writer Ian Fleming being one such example of this mixed ancestry and his James Bond character being the preeminent fictional example of the Anglo-Scot.

At the same time, however, John Lorne Campbell, whose decades long work as a collector alongside his wife, American ethnomusicologist Margaret Fay Shaw, preserved countless works of Canadian Gaelic and Scottish Gaelic literature, Hebridean mythology and folklore, and Scottish traditional music that would otherwise have been lost, was also an Anglo-Scot. Campbell was raised to speak only Received Pronunciation English as an Argyllshire landlord at the height of the British Empire, but his decision as a young adult to reject the traditionally pro-English and pro-Empire politics of his family in favor of Scottish nationalism, decolonisation, and fighting for the survival of his threatened ancestral heritage language of Scottish Gaelic, may well be said to have changed the course of modern Scottish history. The modern Gaelic literary and language revivals, as well as the growing use of immersion schools in both Scotland and Nova Scotia are his legacy.

The term Anglo-Scot is often used to describe Scottish sports players who are based in England or playing for English teams, or vice versa. This is especially so in football, and notably so in Rugby union, where the Anglo Scots were a Scottish non-native select provincial District side that competed in the Scottish Inter-District Championship.

United States
In many parts of the United States, especially those with high Latino populations, the term "Anglo" is applied to white Americans who are not of Latino origin. In the Southwest United States, "Anglo", short for "Anglo American", is used as a synonym for non-Latino whites; that is European Americans, most of whom speak the English language, even those who are not necessarily of English or British descent. Some non-Latino whites in the United States who speak English but are not of English or British ancestry do not identify with the term "Anglo" and find the term offensive. For instance, some Cajuns in southern Louisiana use the term to refer to white people who do not have Francophone backgrounds. Irish Americans, the second largest self-identified ethnic group in the United States following German-Americans, also sometimes take umbrage at being called "Anglo".

Countries with significant populations
Although conceptions of "Anglo" identity vary from country to country, the below table provides estimates of native English-speaking "white" populations by country.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Anglo


text:
Robert Carr (disambiguation)

Robert Carr (1916–2012) was a British Conservative politician.

Robert Carr may also refer to:

Politicians

 * Robert Carr (MP for Boston) (c. 1511–1590), MP for Boston 1559
 * Robert Kerr (MP) or Carr (1578–1654), English politician
 * Robert Carr, 1st Earl of Somerset (c. 1587–1645), Scottish politician
 * Sir Robert Carr, 3rd Baronet, British politician, Privy Counsellor of England, MP for Lincolnshire 1665–1685
 * Robert S. Carr (1845–1925), American politician, president of the West Virginia Senate from 1889 to 1891

Others

 * Robert Carr, 1st Earl of Ancram (c. 1578–1654), Scottish nobleman and writer
 * Sir Robert Carr, Kt., English officer who secured the surrender of Fort Cassimir, New Netherland in 1664
 * Robert Carr (bishop) (1774–1841), English churchman, bishop of Chichester, 1824, and bishop of Worcester, 1831
 * Robert Carr (baritone) (1881–1948), English baritone singer and recording artist
 * Robert Frederick Carr (1943–2007), American serial killer
 * Robert K. Carr (1908–1979), American scholar in the field of government and political science
 * Robert Spencer Carr (1909–1994), American writer
 * Milton Robert Carr (born 1943), commonly known as Bob Carr, American politician, U.S. Representative from Michigan
 * Robert Carr (programmer) (born 1956), American computer programmer
 * Robert Carr (activist) (1963–2011), Trinidadian scholar and human rights activist

title: Robert Carr (disambiguation)


text: Talk:Camorra

title: Talk:Camorra


text:
User:Steinsky/todo

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title: User:Steinsky/todo


text:
Renormalization

Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian.

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 particles, such as photons, positrons, 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 protons exhibit precisely the same observed charge as the electron – even in the presence of much stronger interactions and more intense clouds of virtual particles.

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 spacetime as a continuum, certain statistical and quantum mechanical constructions are not well-defined. To define them, or make them unambiguous, a continuum limit must carefully remove "construction scaffolding" of lattices at various scales. Renormalization procedures are based on the requirement that certain physical quantities (such as the mass and charge of an electron) equal observed (experimental) values. That is, the experimental value of the physical quantity yields practical applications, but due to their empirical nature the observed measurement represents areas of quantum field theory that require deeper derivation from theoretical bases.

Renormalization was first developed in quantum electrodynamics (QED) to make sense of infinite integrals in 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.

Today, the point of view has shifted: on the basis of the breakthrough renormalization group insights of Nikolay Bogolyubov and 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.

Renormalization is distinct from regularization, another technique to control infinities by assuming the existence of new unknown physics at new scales.

Self-interactions in classical physics
The problem of infinities first arose in the classical electrodynamics of point particles in the 19th and early 20th century.

The mass of a charged particle should include the mass–energy in its electrostatic field (electromagnetic mass). Assume that the particle is a charged spherical shell of radius $r_{e}$. The mass–energy in the field is

 * $$m_\text{em} = \int \frac{1}{2} E^2 \, dV = \int_{r_\text{e}}^\infty \frac{1}{2} \left( \frac{q}{4\pi r^2} \right)^2 4\pi r^2 \, dr = \frac{q^2}{8\pi r_\text{e}},$$

which becomes infinite as $r_{e} → 0$. This implies that the point particle would have infinite inertia and thus cannot be accelerated. Incidentally, the value of $r_{e}$ that makes $$m_\text{em}$$ equal to the electron mass is called the classical electron radius, which (setting $$q = e$$ and restoring factors of $c$ and $$\varepsilon_0$$) turns out to be

 * $$r_\text{e} = \frac{e^2}{4\pi\varepsilon_0 m_\text{e} c^2} = \alpha \frac{\hbar}{m_\text{e} c} \approx 2.8 \times 10^{-15}~\text{m},$$

where $$\alpha \approx 1/137$$ is the fine-structure constant, and $$\hbar/(m_\text{e} c)$$ is the reduced Compton wavelength of the electron.

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. This was called renormalization, and Lorentz and Abraham attempted to develop a classical theory of the electron this way. This early work was the inspiration for later attempts at regularization and renormalization in quantum field theory.

(See also regularization (physics) for an alternative way to remove infinities from this classical problem, assuming new physics exists at small scales.)

When calculating the electromagnetic interactions of 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.

The Abraham–Lorentz theory had a noncausal "pre-acceleration". Sometimes an electron would start moving before the force is applied. This is a sign that the point limit is inconsistent.

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 particle–antiparticle pairs, thus effectively smearing out the charge over a region comparable to the Compton wavelength. In quantum electrodynamics at small coupling, the electromagnetic mass only diverges as the logarithm of the radius of the particle.

Divergences in quantum electrodynamics

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 corrections' divergences was discovered in 1947–49 by Hans Kramers, Hans Bethe, Julian Schwinger,   Richard Feynman,   and Shin'ichiro Tomonaga,       and systematized by Freeman Dyson in 1949. The divergences appear in radiative corrections involving Feynman diagrams with closed loops of virtual particles 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, $$E^2 - p^2$$ 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 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 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" (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 wavelengths and high-frequencies fluctuations of the fields, in the 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:
 * (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.
 * (c) An electron emits a photon, emits a second photon, and reabsorbs the first. This process is shown in the section below in figure 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:
 * 1) The field normalization Z.
 * 2) The mass of the electron.
 * 3) 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:

 * $$ \left( p^2 - a^2 \right)^{\frac{1}{2}} $$

is well defined at $p = a$ but is UV divergent; if we take the $3/2$-th fractional derivative with respect to $−a^{2}$, we obtain the IR divergence

 * $$ \frac{1}{p^2 - a^2},$$

so we can cure IR divergences by turning them into UV divergences.

A loop 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 $p^{μ}$ and ends up with 4-momentum $r^{μ}$. It emits a virtual photon carrying $r^{μ} − p^{μ}$ to transfer energy and momentum to the other electron. But in this diagram, before that happens, it emits another virtual photon carrying 4-momentum $q^{μ}$, and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the 4-momentum $q^{μ}$ 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

 * $$-ie^3 \int \frac{d^4 q}{(2\pi)^4} \gamma^\mu \frac{i (\gamma^\alpha (r - q)_\alpha + m)}{(r - q)^2 - m^2 + i \epsilon} \gamma^\rho \frac{i (\gamma^\beta (p - q)_\beta + m)}{(p - q)^2 - m^2 + i \epsilon} \gamma^\nu \frac{-i g_{\mu\nu}}{q^2 + i\epsilon}.$$

The various $γ^{μ}$ 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 $e$ are the electric coupling constant, while the $$i\epsilon$$ 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 $q^{μ}$ of the three big factors in the integrand, which are from the propagators of the two electron lines and the photon line in the loop.

This has a piece with two powers of $q^{μ}$ on top that dominates at large values of $q^{μ}$ (Pokorski 1987, p. 122):

 * $$e^3 \gamma^\mu \gamma^\alpha \gamma^\rho \gamma^\beta \gamma_\mu \int \frac{d^4 q}{(2\pi)^4} \frac{q_\alpha q_\beta}{(r - q)^2 (p - q)^2 q^2}.$$

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.

Renormalized and bare quantities
The solution was to realize that the quantities initially appearing in the theory's formulae (such as the formula for the Lagrangian), representing such things as the electron's electric charge and mass, as well as the normalizations of the quantum fields themselves, did not actually correspond to the physical constants measured in the laboratory. As written, they were bare quantities that did not take into account the contribution of virtual-particle loop effects to the physical constants themselves. Among other things, these effects would include the quantum counterpart of the electromagnetic back-reaction that so vexed classical theorists of electromagnetism. In general, these effects would be just as divergent as the amplitudes under consideration in the first place; so finite measured quantities would, in general, imply divergent bare quantities.

To make contact with reality, then, the formulae would have to be rewritten in terms of measurable, renormalized quantities. The charge of the electron, say, would be defined in terms of a quantity measured at a specific kinematic renormalization point or subtraction point (which will generally have a characteristic energy, called the renormalization scale or simply the energy scale). The parts of the Lagrangian left over, involving the remaining portions of the bare quantities, could then be reinterpreted as counterterms, involved in divergent diagrams exactly canceling out the troublesome divergences for other diagrams.

Renormalization in QED

For example, in the Lagrangian of QED

 * $$\mathcal{L}=\bar\psi_B\left[i\gamma_\mu \left (\partial^\mu + ie_BA_B^\mu \right )-m_B\right]\psi_B -\frac{1}{4}F_{B\mu\nu}F_B^{\mu\nu}$$

the fields and coupling constant are really bare quantities, hence the subscript $B$ above. Conventionally the bare quantities are written so that the corresponding Lagrangian terms are multiples of the renormalized ones:

 * $$\left(\bar\psi m \psi\right)_B = Z_0 \bar\psi m \psi$$
 * $$\left(\bar\psi\left(\partial^\mu + ieA^\mu \right )\psi\right)_B = Z_1 \bar\psi \left (\partial^\mu + ieA^\mu \right)\psi$$
 * $$\left(F_{\mu\nu}F^{\mu\nu}\right)_B = Z_3\, F_{\mu\nu}F^{\mu\nu}.$$

Gauge invariance, via a Ward–Takahashi identity, turns out to imply that we can renormalize the two terms of the covariant derivative piece

 * $$\bar \psi (\partial + ieA) \psi$$

together (Pokorski 1987, p. 115), which is what happened to $Z_{1}$; it is the same as $Z_{2}$.

A term in this Lagrangian, for example, the electron–photon interaction pictured in Figure 1, can then be written

 * $$\mathcal{L}_I = -e \bar\psi \gamma_\mu A^\mu \psi - (Z_1 - 1) e \bar\psi \gamma_\mu A^\mu \psi$$

The physical constant $e$, 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 $Z_{1}$ and $Z_{0}$).

The diagram with the $Z_{3}$ 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 Wilson. 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.

Running couplings
To minimize the contribution of loop diagrams to a given calculation (and therefore make it easier to extract results), one chooses a renormalization point close to the energies and momenta exchanged in the interaction. However, the renormalization point is not itself a physical quantity: the physical predictions of the theory, calculated to all orders, should in principle be independent of the choice of renormalization point, as long as it is within the domain of application of the theory. Changes in renormalization scale will simply affect how much of a result comes from Feynman diagrams without loops, and how much comes from the remaining finite parts of loop diagrams. One can exploit this fact to calculate the effective variation of physical constants with changes in scale. This variation is encoded by beta-functions, and the general theory of this kind of scale-dependence is known as the renormalization group.

Colloquially, particle physicists often speak of certain physical "constants" as varying with the energy of interaction, though in fact, it is the renormalization scale that is the independent quantity. This running does, however, provide a convenient means of describing changes in the behavior of a field theory under changes in the energies involved in an interaction. For example, since the coupling in quantum chromodynamics becomes small at large energy scales, the theory behaves more like a free theory as the energy exchanged in an interaction becomes large – a phenomenon known as asymptotic freedom. Choosing an increasing energy scale and using the renormalization group makes this clear from simple Feynman diagrams; were this not done, the prediction would be the same, but would arise from complicated high-order cancellations.

For example,

 * $$I=\int_0^a \frac{1}{z}\,dz-\int_0^b \frac{1}{z}\,dz=\ln a-\ln b-\ln 0 +\ln 0$$

is ill-defined.

To eliminate the divergence, simply change lower limit of integral into $ε_{a}$ and $ε_{b}$:

 * $$I=\ln a-\ln b-\ln{\varepsilon_a}+\ln{\varepsilon_b} = \ln \tfrac{a}{b} - \ln \tfrac{\varepsilon_a}{\varepsilon_b}$$

Making sure $Z_{1}$, then $ε_{b}⁄ε_{a} → 1$

Regularization
Since the quantity $I = ln a⁄b.$ is ill-defined, in order to make this notion of canceling divergences precise, the divergences first have to be tamed mathematically using the theory of limits, in a process known as regularization (Weinberg, 1995).

An essentially arbitrary modification to the loop integrands, or regulator, can make them drop off faster at high energies and momenta, in such a manner that the integrals converge. A regulator has a characteristic energy scale known as the cutoff; taking this cutoff to infinity (or, equivalently, the corresponding length/time scale to zero) recovers the original integrals.

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, 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 regularization, introduced by Kenneth Wilson, which pretends that hyper-cubical lattice constructs our spacetime with fixed grid size. This size is a natural cutoff for the maximal momentum that a particle could possess when propagating on the lattice. And after doing a calculation on several lattices with different grid size, the physical result is extrapolated to grid size 0, or our natural universe. This presupposes the existence of a scaling limit.

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 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
The early formulators of QED and other quantum field theories were, as a rule, dissatisfied with this state of affairs. It seemed illegitimate to do something tantamount to subtracting infinities from infinities to get finite answers.

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.

Dirac's criticism was the most persistent. As late as 1975, he was saying:

 * 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 Feynman. Despite his crucial role in the development of quantum electrodynamics, he wrote the following in 1985:

 * 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, Gross, Politzer and 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.

The general unease was almost universal in texts up to the 1970s and 1980s. Beginning in the 1970s, however, inspired by work on the renormalization group and effective field theory, and despite the fact that Dirac and various others—all of whom belonged to the older generation—never withdrew their criticisms, attitudes began to change, especially among younger theorists. Kenneth G. Wilson and others demonstrated that the renormalization group is useful in statistical field theory applied to condensed matter physics, where it provides important insights into the behavior of phase transitions. In condensed matter physics, a physical short-distance regulator exists: matter ceases to be continuous on the scale of atoms. Short-distance divergences in condensed matter physics do not present a philosophical problem since the field theory is only an effective, smoothed-out representation of the behavior of matter anyway; there are no infinities since the cutoff is always finite, and it makes perfect sense that the bare quantities are cutoff-dependent.

If QFT holds all the way down past the Planck length (where it might yield to string theory, causal set theory or something different), then there may be no real problem with short-distance divergences in particle physics either; all field theories could simply be effective field theories. In a sense, this approach echoes the older attitude that the divergences in QFT speak of human ignorance about the workings of nature, but also acknowledges that this ignorance can be quantified and that the resulting effective theories remain useful.

Be that as it may, Salam's remark in 1972 seems still relevant

 * Field-theoretic infinities – first encountered in Lorentz's computation of electron self-mass – have persisted in classical electrodynamics for seventy and in quantum electrodynamics for some thirty-five years. These long years of frustration have left in the subject a curious affection for the infinities and a passionate belief that they are an inevitable part of nature; so much so that even the suggestion of a hope that they may, after all, be circumvented — and finite values for the renormalization constants computed – is considered irrational. Compare Russell's postscript to the third volume of his autobiography The Final Years, 1944–1969 (George Allen and Unwin, Ltd., London 1969), p. 221:

 * 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.

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 1015 GeV (far beyond the reach of our current particle accelerators), 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."

Renormalizability
From this philosophical reassessment, a new concept follows naturally: the notion of renormalizability. Not all theories lend themselves to renormalization in the manner described above, with a finite supply of counterterms and all quantities becoming cutoff-independent at the end of the calculation. If the Lagrangian contains combinations of field operators of high enough dimension in energy units, the counterterms required to cancel all divergences proliferate to infinite number, and, at first glance, the theory would seem to gain an infinite number of free parameters and therefore lose all predictive power, becoming scientifically worthless. Such theories are called nonrenormalizable.

The Standard Model of particle physics contains only renormalizable operators, but the interactions of general relativity become nonrenormalizable operators if one attempts to construct a field theory of quantum gravity in the most straightforward manner (treating the metric in the Einstein–Hilbert Lagrangian as a perturbation about the Minkowski metric), suggesting that perturbation theory is not satisfactory in application to quantum gravity.

However, in an effective field theory, "renormalizability" is, strictly speaking, a misnomer. In nonrenormalizable effective field theory, terms in the Lagrangian do multiply to infinity, but have coefficients suppressed by ever-more-extreme inverse powers of the energy cutoff. If the cutoff is a real, physical quantity—that is, if the theory is only an effective description of physics up to some maximum energy or minimum distance scale—then these additional terms could represent real physical interactions. Assuming that the dimensionless constants in the theory do not get too large, one can group calculations by inverse powers of the cutoff, and extract approximate predictions to finite order in the cutoff that still have a finite number of free parameters. It can even be useful to renormalize these "nonrenormalizable" interactions.

Nonrenormalizable interactions in effective field theories rapidly become weaker as the energy scale becomes much smaller than the cutoff. The classic example is the Fermi theory of the weak nuclear force, a nonrenormalizable effective theory whose cutoff is comparable to the mass of the W particle. This fact may also provide a possible explanation for why almost all of the particle interactions we see are describable by renormalizable theories. It may be that any others that may exist at the GUT or Planck scale simply become too weak to detect in the realm we can observe, with one exception: gravity, whose exceedingly weak interaction is magnified by the presence of the enormous masses of stars and planets.

Renormalization schemes
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: 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.
 * Minimal subtraction (MS) scheme and the related modified minimal subtraction (MS-bar) scheme
 * On-shell scheme

History
A deeper understanding of the physical meaning and generalization of the renormalization process, which goes beyond the dilatation group of conventional renormalizable theories, came from condensed matter physics. Leo P. Kadanoff's paper in 1966 proposed the "block-spin" renormalization group. The blocking idea is a way to define the components of the theory at large distances as aggregates of components at shorter distances.

This approach covered the conceptual point and was given full computational substance in the extensive important contributions of Kenneth Wilson. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the Kondo problem, in 1974, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and critical phenomena in 1971. He was awarded the Nobel prize for these decisive contributions in 1982.

Principles
In more technical terms, let us assume that we have a theory described by a certain function $$Z$$ of the state variables $$\{s_i\}$$ and a certain set of coupling constants $$\{J_k\}$$. This function may be a partition function, an action, a Hamiltonian, etc. It must contain the whole description of the physics of the system.

Now we consider a certain blocking transformation of the state variables $$\{s_i\}\to \{\tilde s_i\}$$, the number of $$\tilde s_i$$ must be lower than the number of $$s_i$$. Now let us try to rewrite the $$Z$$ function only in terms of the $$\tilde s_i$$. If this is achievable by a certain change in the parameters, $$\{J_k\}\to \{\tilde J_k\}$$, 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 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.

If these fixed points correspond to free field theory, the theory is said to exhibit quantum triviality. Numerous fixed points appear in the study of lattice Higgs theories, but the nature of the quantum field theories associated with these remains an open question.

General introduction

 * DeDeo, Simon; Introduction to Renormalization (2017). Santa Fe Institute Complexity Explorer MOOC. Renormalization from a complex systems point of view, including Markov Chains, Cellular Automata, the real space Ising model, the Krohn-Rhodes Theorem, QED, and rate distortion theory.
 * Baez, John; Renormalization Made Easy, (2005). A qualitative introduction to the subject.
 * Blechman, Andrew E.; Renormalization: Our Greatly Misunderstood Friend, (2002). Summary of a lecture; has more information about specific regularization and divergence-subtraction schemes.
 * Shirkov, Dmitry; Fifty Years of the Renormalization Group, C.E.R.N. Courrier 41(7) (2001). Full text available at : I.O.P Magazines.
 * E. Elizalde; Zeta regularization techniques with Applications.
 * Shirkov, Dmitry; Fifty Years of the Renormalization Group, C.E.R.N. Courrier 41(7) (2001). Full text available at : I.O.P Magazines.
 * E. Elizalde; Zeta regularization techniques with Applications.

Mainly: quantum field theory

 * N. N. Bogoliubov, D. V. Shirkov (1959): The Theory of Quantized Fields. New York, Interscience. The first text-book on the renormalization group theory.
 * Ryder, Lewis H.; Quantum Field Theory  (Cambridge University Press, 1985), ISBN 0-521-33859-X Highly readable textbook, certainly the best introduction to relativistic Q.F.T. for particle physics.
 * Zee, Anthony; Quantum Field Theory in a Nutshell, Princeton University Press (2003) ISBN 0-691-01019-6. Another excellent textbook on Q.F.T.
 * Weinberg, Steven; The Quantum Theory of Fields (3 volumes) Cambridge University Press (1995). A monumental treatise on Q.F.T. written by a leading expert, Nobel laureate 1979.
 * Pokorski, Stefan; Gauge Field Theories, Cambridge University Press (1987) ISBN 0-521-47816-2.
 * 't Hooft, Gerard; The Glorious Days of Physics – Renormalization of Gauge theories, lecture given at Erice (August/September 1998) by the Nobel laureate 1999 . Full text available at: 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 PostScript.
 * Rivasseau, Vincent; From perturbative to constructive renormalization, Princeton University Press (1991) ISBN 0-691-08530-7. Full text available in PostScript and in PDF (draft version).
 * Iagolnitzer, Daniel & Magnen, J.; Renormalization group analysis, Encyclopaedia of Mathematics, Kluwer Academic Publisher (1996). Full text available in PostScript and pdf here.
 * Scharf, Günter; Finite quantum electrodynamics: The causal approach, Springer Verlag Berlin Heidelberg New York (1995) ISBN 3-540-60142-2.
 * A. S. Švarc (Albert Schwarz), Математические основы квантовой теории поля, (Mathematical aspects of quantum field theory), Atomizdat, Moscow, 1975. 368 pp.

Mainly: statistical physics

 * A. N. Vasil'ev; The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics (Routledge Chapman & Hall 2004); ISBN 978-0-415-31002-4
 * 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 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 PostScript.
 * 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.
 * Cardy, John; Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996) ISBN 0-521-49959-3.

Miscellaneous

 * Shirkov, Dmitry; The Bogoliubov Renormalization Group, JINR Communication E2-96-15 (1996). Full text available at: hep-th/9602024
 * Zinn-Justin, Jean; Renormalization and renormalization group: From the discovery of UV divergences to the concept of effective field theories, in: de Witt-Morette C., Zuber J.-B. (eds), Proceedings of the NATO ASI on Quantum Field Theory: Perspective and Prospective, June 15–26, 1998, Les Houches, France, Kluwer Academic Publishers, NATO ASI Series C 530, 375–388 (1999). Full text available in PostScript.
 * 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 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.

title: Renormalization