Figure 1. Renormalization in QED: The simple electron-photon interaction that determines the electron's charge at one renormalization point is revealed to consist of more complicated interactions at another.

In quantum field theory (QFT) and the statistical mechanics of fields, renormalization refers to a collection of techniques used to construct mathematical relationships or approximate relationships between observable quantities, when the standard assumption that the parameters of the theory are finite breaks down (giving the result that many observables are infinite). Renormalization arose in quantum electrodynamics as a means of making sense of the infinite results of various calculations and extracting finite answers to properly posed physical questions. Initially viewed as a suspect, provisional procedure by most of its originators, renormalization eventually was embraced as an important tool in several fields of physics, as a result of work in effective field theory and the renormalization group. Image File history File links Download high resolution version (1271x894, 32 KB)The basic electron-photon interaction at one scale becomes a sum of more complex interactions at another. ... Image File history File links Download high resolution version (1271x894, 32 KB)The basic electron-photon interaction at one scale becomes a sum of more complex interactions at another. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... The infinity symbol ∞ in several typefaces The word infinity comes from the Latin infinitas or unboundedness. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... In physics, an effective field theory is an approximate theory (usually a quantum field theory) that contains the appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, but ignores the substructure and the degrees of freedom at shorter distances (or, equivalently, higher energies). ... In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ...

Prehistory: Self-interactions in classical mechanics

Some of the problems and phenomena eventually addressed by renormalization actually appeared earlier in the classical electrodynamics of point particles in the 19th and early 20th century. When calculating the electromagnetic interactions of charged particles, one often ignores the back-reaction of a particle's own field upon itself. It was realized early on that such a treatment is incomplete, and some theorists explored the intriguing idea that an electron's inertial mass could be entirely due to the back-reaction. However, if the electron is assumed to be a point, the calculated value of this back-reaction diverges, essentially because of the singularity at the origin in the inverse-square law. One potential solution was to assume that the electron had a nonzero size, comparable to the number known as the classical electron radius, about 2.8 x 10^-15m. Then, however, according to Henri Poincaré, the theory became inconsistent unless the electron possessed additional forces to hold it together internally against the repulsion of like charges. Today, the hypothesis of a classical electron radius might be seen as an early attempt at regularization. Attempts to deal with the back-reaction, such as the Abraham-Lorentz force, exhibited bizarre phenomena such as acausal "pre-acceleration", in which an electron would start moving shortly before a force was applied (Jackson 1998). This is because an extended rigid body, when struck, cannot maintain causal subluminal transmission of the impulse: the far end starts moving as soon as the near end is struck. If rigidity goes, the particle ceases to be fundamental since it has internal dynamics; but if subluminality or causality go, so does most of physics. Classical electromagnetism is a theory of electromagnetism that was developed over the course of the 19th century, most prominently by James Clerk Maxwell. ... In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not made up of smaller particles. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999... Electromagnetism is the force observed as static electricity, and causes the flow of electric charge (electric current) in electrical conductors. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... The electron is a fundamental subatomic particle that carries an electric charge. ... The principle of inertia is one of the fundamental laws of classical physics which are used to describe the motion of matter and how it is affected by applied forces. ... Unsolved problems in physics: What causes anything to have mass? Mass is a property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... This diagram shows how the law works. ... The electron is a fundamental subatomic particle that carries an electric charge. ... The classical electron radius, also known as the Compton radius or the Thomson scattering length is based on a classical (i. ... Jules Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [][1]), generally known as Henri Poincaré, was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... The mathematical term regularization has two main meanings, both associated with making a function more `regular or smooth. ... The Abraham-Lorentz force is the average force on an accelerating charged particle caused by the particle emitting electromagnetic radiation. ...

In classical field theory, therefore, the contribution of field interactions to a particle's physical properties was already problematic. Indeed, in some ways, the trouble was worse than in QFT, since the short-distance divergences involved were typically stronger than the ones encountered in quantum theories.

Divergences in quantum electrodynamics

Figure 2. A diagram contributing to electron-electron scattering in QED. The loop has an ultraviolet divergence.

Vacuum polarization aka charge screening. this has a quadratic divergence

When developing quantum electrodynamics in the 1940s, Shin'ichiro Tomonaga, Julian Schwinger, Richard Feynman, and Freeman Dyson discovered that, in perturbative calculations, problems with divergent integrals abounded. One way of describing the divergences in QED is that they appear as the consequence of calculations involving Feynman diagrams with closed loops of virtual particles in them. These diagrams appear in the perturbative approximation of quantum field theory. Each looped diagram represents a perturbation, or small correction, to a diagram without loops. Intuitively, diagrams with more and more loops should give smaller and smaller corrections to the values of diagrams which do not contain any loops. However, when the contributions of these loop diagrams are naively calculated, they become infinitely large. One type of loop would be a situation in which a virtual electron-positron pair appear out of the vacuum, interact with various photons, and then annihilate. Another would be an electron-photon interaction as in Figure 1. Image File history File links Download high resolution version (958x929, 55 KB)A loop diagram contributing to electron-electron scattering in QED. Drawn by Matt McIrvin. ... Image File history File links Download high resolution version (958x929, 55 KB)A loop diagram contributing to electron-electron scattering in QED. Drawn by Matt McIrvin. ... 1-loop vacuum polarization diagram This is the one loop contribution to the photon propagator Π due to vacuum polarization in QED. It causes a wave function renormalization for the photon, leading to a charge screening. ... // Events and trends World War II was a truly global conflict with many facets: immense human suffering, fierce indoctrination, and the use of new, extremely devastating weapons such as the atomic bomb. ... Sin-Itiro Tomonaga or Shinichirō Tomonaga (朝永 振一郎 Tomonaga Shinichirō, March 31, 1906–July 8, 1979) was a Japanese physicist, influential in the development of quantum electrodynamics, work for which he was jointly awarded the Nobel Prize in Physics in 1965 along with... Julian Seymour Schwinger (February 12, 1918 -- July 16, 1994) was an American theoretical physicist. ... Richard Phillips Feynman (May 11, 1918 in Queens, New York – February 15, 1988 in Los Angeles, California) (surname pronounced FINE-man; in IPA) was an influential American physicist known for expanding greatly on the theory of quantum electrodynamics, particle theory, and the physics of the superfluidity of supercooled liquid helium. ... // Freeman Dyson in San Francisco in 2005 (Photo: Jacob Appelbaum) Freeman John Dyson (born December 15, 1923) is an English-born American physicist and mathematician, famous for his work in quantum mechanics, nuclear weapons design and policy, and for his serious theorizing in futurism and science fiction concepts, including the... In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ... In physics, a virtual particle is a particle-like abstraction used in some models of quantum field theory. ... The first detection of the positron in 1932 by Carl D. Anderson The positron is the antiparticle or the antimatter counterpart of the electron. ... Look up Vacuum in Wiktionary, the free dictionary. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves, radio waves, and visible light are all forms of light. ...

While virtual particles obey conservation of energy and momentum, they can possess combinations of energies and momenta not allowed by the classical laws of motion; physicists say that they are not on shell. Furthermore, whenever a loop appears, the particles involved in the loop are not individually constrained by the energies and momenta of incoming and outgoing particles, since a variation in, say, the energy of one particle in the loop can be balanced by an equal and opposite variation in the energy of another particle in the loop. Therefore, in order to calculate the contribution to a probability amplitude, one must integrate over all possible combinations of energy and momentum in the loop—and these integrals are often divergent, that is, they give infinite answers. The most theoretically troublesome divergences are the "ultraviolet" (UV) ones associated with large energies and momenta of the virtual particles in the loop, or, equivalently, very short wavelengths and high frequencies of the fields for which these particles are the quanta. These divergences are, therefore, fundamentally short-distance, short-time phenomena. (There are infrared divergences as well, but they are not as hard to interpret and are beyond the scope of this article.) Conservation of energy states that the total amount of energy (often expressed as the sum of kinetic energy and potential energy) in an isolated system remains constant. ... In classical mechanics, momentum (pl. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ... In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ... In calculus, the integral of a function is a generalization of area, mass, volume and total. ... In physics, an ultraviolet divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very high energy approaching infinity, or, equivalently, because of physical phenomena at very short distances. ... The wavelength is the distance between repeating units of a wave pattern. ... Sine waves of various frequencies; the lower waves have higher frequencies than those above. ... In physics, an infrared divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very small energy approaching zero, or, equivalently, because of physical phenomena at very long distances. ...

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 four-momentum p^μ and ends up with four-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 four-momentum q^μ, and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the four-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

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 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. The gamma matrices, also known as Dirac matrices, were developed by P.A.M. Dirac in order to serve as coefficients of the Dirac equation. ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by Paul Dirac in 1928 and provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ... In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. ...

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

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 study in the first place; so finite measured quantities would in general imply divergent bare quantities. A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... Unsolved problems in physics: What causes anything to have mass? Mass is a property of a physical object that quantifies the amount of matter and energy it is equivalent to. ...

In order 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. This article or section may be confusing for some readers, and should be edited to be clearer or more simplified. ... In physics, energy scale is a particular value of energy determined with the precision of one order (or a few orders) of magnitude. ...

Renormalization in QED

Figure 3. The vertex corresponding to the Z₁ counterterm cancels the divergence in Figure 2.

For example, in the Lagrangian of QED Image File history File links Download high resolution version (958x929, 48 KB)A counterterm cancels a loop divergence in QED. Drawn by Matt McIrvin. ... Image File history File links Download high resolution version (958x929, 48 KB)A counterterm cancels a loop divergence in QED. Drawn by Matt McIrvin. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ...

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:

.

(Gauge invariance, via a Ward-Takahashi identity, turns out to imply that we can renormalize the two terms of the covariant derivative piece together (Pokorski 1987, p. 115), which is what happened to Z₂; it is the same as Z₁.) Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In quantum field theory a Ward-Takahashi identity is nowadays used to designate an identity between correlation functions that follows from symmetries, either global or gauged, of the theory, and which remains valid after renormalization. ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...

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

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 we are lucky, 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 Z₀ and Z₃). In QED, we are lucky: the theory is renormalizable (see below for more on this). In physics, the magnetic moment or magnetic dipole moment is a measure of the strength of a magnetic source. ...

The diagram with the Z₁ counterterm's interaction vertex placed as in Figure 3 cancels out the divergence from the loop in Figure 2.

The splitting of the "bare terms" into the original terms and counterterms came before the renormalization group insights due to Kenneth Wilson. According to the renormalization group insights, this splitting is unnatural and unphysical. In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ... Kenneth Geddes Wilson (born June 8, 1936) is an American physicist. ... In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ...

Running constants

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 actually 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 leftover 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. In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. ... In mathematics, the beta function (occasionally written as Beta function), also called the Euler integral of the first kind, is a special function defined by for Re(x), Re(y) > 0. ... In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ...

Colloquially, particle physicists often speak of certain physical constants as varying with the energy of an 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 constant 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. In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. ... Quantum chromodynamics (QCD) is the theory of the strong interaction, a fundamental force describing the interactions of the quarks and gluons found in nucleons (such as the proton and neutron). ... In physics, asymptotic freedom is the property of some gauge theories in which the interaction between the particles, such as quarks, becomes arbitrarily weak at ever shorter distances, i. ...

Regularization

Since the quantity 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. In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ... In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator (for example, the minimal distance in space which is useful if the divergences arise from short-distance physical effects). ...

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. In theoretical physics, cutoff usually represents a particular energy scale or length scale. ...

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. In theoretical physics, dimensional regularization is a particular way to get rid of infinities that occur when one evaluates Feynman diagrams in quantum field theory. ... Gerard t Hooft at Harvard University Gerardus (Gerard) t Hooft [ut-hooft] (The prefix ’t is pronounced as ‘ut’ and stands for ‘het’) (born July 5, 1946) is a professor in theoretical physics at Utrecht University, The Netherlands. ... Martinus J.G. Veltman (Tini for short) (born June 27, 1931) is a 1999 Nobel Prize in Physics laureate for elucidating the quantum structure of electroweak interactions in physics, work done at Utrecht University, The Netherlands. ... In theoretical physics, the Pauli-Villars regularization is a particular procedure to get rid of infinities in divergent integrals that correspond to Feynman diagrams. ...

Yet another regularization scheme is the Lattice regularization, introduced by Kenneth Wilson, which pretends that our space-time is constructed by hyper-cubical lattice 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 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. It has been suggested that this article or section be merged with Lattice gauge theory. ... Kenneth Geddes Wilson (born June 8, 1936) is an American physicist. ... In mathematics, extrapolation is a type of interpolation. ... If we wish to have a lattice model which approximates a continuum quantum field theory in the limit as the lattice spacing goes to zero, then this corresponds to finding a second order phase transition of the model. ...

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. The disadvantage of the method is the fact that the approach is quite technical and requires a high level of mathematical knowledge. Causal perturbation theory is a mathematically rigorous approach to renormalization theory, which makes it possible to put the theoretical setup of perturbative quantum field theory on a sound mathematical basis. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ...

Zeta function regularization

Julian Schwinger discovered a relationship between zeta function regularization and renormalization, using the asymptotic relation: Julian Seymour Schwinger (February 12, 1918 -- July 16, 1994) was an American theoretical physicist. ... In mathematics and theoretical physics, zeta function regularization is a summability method assign finite values to superficially divergent sums. ...

as the regulator . Based on this, he considered using the values of ζ( − n) to get finite results. Although he reached inconsistent results, an improved formula by Hartle, J. Garcia, M:A Valle and E. Elizalde includes Jim Hartle at Harvard University James Hartle is an American physicist. ...

,

where the B's are the Bernoulli numbers and In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ...

.

So every I(m,Λ) can be written as a linear combination of ζ( − 1),ζ( − 3),ζ( − 5),......ζ( − m)

Or simply using Able-Plana formula we have for every divergent integral:

valid when m>0, Here the Zeta function is Hurwitz zeta function and Beta is a positive real number. In mathematics, the Hurwitz zeta function is one of the many zeta functions. ...

The "Geommetric" analogy is given by, (if we use rectangle method) to evaluate the integral so: In mathematics, the rectangle method of integral calculus uses an approximation to a definite integral, made by finding the area of a series of rectangles. ...

Using Hurwitz zeta regularization plus rectangle method with step h (not to mess up with Planck's constant)

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.

Dirac's criticism was the most persistent. As late as 1975, he was saying: Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ...

"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, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small - not neglecting it just because it is infinitely great and you do not want it!" ^[1]

Another important critic was Feynman. Despite his crucial role in the development of quantum electrodynamics, he wrote: Richard Phillips Feynman (May 11, 1918 in Queens, New York – February 15, 1988 in Los Angeles, California) (surname pronounced FINE-man; in IPA) was an influential American physicist known for expanding greatly on the theory of quantum electrodynamics, particle theory, and the physics of the superfluidity of supercooled liquid helium. ...

"The shell game that we play ... 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." ^[2]

The general unease was almost universal in texts up to the 1970s and 1980s. The 1970s decade refers to the years from 1970 to 1979, inclusive. ... The examples and perspective in this article or section may not represent a worldwide view. ...

Beginning in the 1970s, however, inspired by work on the renormalization group and effective field theory, 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 real 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 actually always finite, and it makes perfect sense that the bare quantities are cutoff-dependent. In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ... In physics, an effective field theory is an approximate theory (usually a quantum field theory) that contains the appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, but ignores the substructure and the degrees of freedom at shorter distances (or, equivalently, higher energies). ... Kenneth Geddes Wilson (born June 8, 1936) is an American physicist. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... In physics, a phase transition, (or phase change) is the transformation of a thermodynamic system from one phase to another. ... In physics, matter is commonly defined as the substance of which physical objects are composed, not counting the contribution of various energy or force-fields, which are not usually considered to be matter per se (though they may contribute to the mass of objects). ... Properties For other articles with similar names, see Atom (disambiguation). ...

If QFT holds all the way down past the Planck length (where it might yield to "string 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. The Planck length, denoted by , is the unit of length in the system of units known as Planck units. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles... Particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ...

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 civilization's 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 a worrisome problem, renormalization has become an important theoretical tool for studying the behavior of field theories in different regimes. The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... Quantum chromodynamics (QCD) is the theory of the strong interaction, a fundamental force describing the interactions of the quarks and gluons found in nucleons (such as the proton and neutron). ... In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ... A GEV (or Ground Effect Vehicle) is vehicle that takes advantage of the aerodynamic principle of ground effect (or Wing-in-ground). ... A 1960s single stage 2 MeV linear Van de Graaff accelerator, here opened for maintenance A particle accelerator is a device that uses electric fields to propel electrically charged particles to high speeds and magnetic fields to contain them. ... Grand unification, grand unified theory, or GUT is a theory in physics that unifies the strong interaction and electroweak interaction. ...

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 excessively high 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. In physics, the adjective renormalizable refers to a theory (usually a quantum field theory) in which all ultraviolet divergences, infinities and other seemingly meaningless results can be cured by the process of renormalization. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ...

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, suggesting that perturbation theory is useless in application to quantum gravity. The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... Quantum gravity is the field of theoretical physics attempting to unify the theory of quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ...

However, in an effective field theory, "renormalizability" is, strictly speaking, a misnomer. In a 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—if, that is, the theory is only an effective description of physics up to some maximum energy or minimum distance scale—then these extra 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. An editor has expressed a concern that the topic of this article may be unencyclopedic. ...

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. In physics, Fermis interaction is an old explanation of the weak force, proposed by Enrico Fermi. ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... In physics, the W and Z bosons are the elementary particles that mediate the weak nuclear force. ... Gravity is a force of attraction that acts between bodies that have mass. ... The Pleiades, an open cluster of stars in the constellation of Taurus. ... The eight planets and three dwarf planets of the Solar System. ...

Further reading

General Introduction

• Delamotte, Bertrand ; A hint of renormalization, American Journal of Physics 72 (2004) pp. 170-184. Beautiful elementary introduction to the ideas, no prior knowledge of field theory being necessary. Full text available at: hep-th/0212049
• 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.
• Cao, Tian Yu & Schweber, Silvian S. ; The Conceptual Foundations and Philosophical Aspects of Renormalization Theory, Synthese, 97(1) (1993), 33-108.
• 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

• 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.
• Iagolnitzer, Daniel & Magnen, J. ; Renormalization group analysis, Encyclopaedia of Mathematics, Kluwer Academic Publisher (1996) [ISBN ]. 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].
• Introduction to renormalization using zeta regularization http://arxiv.org/pdf/math.GM/0402259

Mainly: Statistical Physics

• 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].

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) [ISBN ]. 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 (Hopf algebra) of renormalization, and its link to the Riemann-Hilbert problem. Full text (in french) available in PostScript.
• http://demoprints.eprints.org/5120/01/Renormalization.pdf A new approach to renormalization M.A Valle and J. Garcia.

References

1. ^ Kragh, Helge ; Dirac: A scientific biography, CUP 1990, p. 184
2. ^ Feynman, Richard P. ; QED, The Strange Theory of Light and Matter, Penguin 1990, p. 128