4884:. 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 quantityâthat is, if the theory is only an effective description of physics up to some maximum energy or minimum distance scaleâthen 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.
1308:, 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 such as
135:
2203:. 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
2892:
1375:, 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.
2054:
1391:
7329:
2493:
3209:. 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.
2544:
3252:
4933:
4063:, 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
2070:
4732:. 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.
1620:
2062:
3169:
2305:, 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
4602:, 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
1880:
4759:
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
4647:
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
4668:
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
3715:
1328:
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
4565:
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
1974:
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
4822:
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
4133:
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
4771:
Field-theoretic infinities â first encountered in
Lorentz's computation of electron self-mass â 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
1315:
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
2313:
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
4128:
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
4787:
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
4841:
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
2887:{\displaystyle -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 {-ig_{\mu \nu }}{q^{2}+i\epsilon }}.}
4836:
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
4471:
2042:
due to interference with virtual particleâantiparticle 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.
4794:
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
2019:
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
4054:
4772:
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 â and finite values for the renormalization constants computed â is considered irrational. Compare
4827:
has put it, "In the
Quantum Theory, these 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."
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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.
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its equations, ignoring them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves disregarding a quantity when it is small â not neglecting it just because it is infinitely great and you do not want it!
4630:
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.
4164:. 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.
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1742:
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Figure 1. Renormalization in quantum electrodynamics: The simple electron/photon interaction that determines the electron's charge at one renormalization point is revealed to consist of more complicated interactions at
2500:
The diagram in Figure 2 shows one of the several one-loop contributions to electronâelectron scattering in QED. The electron on the left side of the diagram, represented by the solid line, starts out with 4-momentum
4334:
5010:
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.
5428:
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
4149:
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
2322:
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
4566:
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.
6796:
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
3921:
2390:
6167:
Koba, Z.; Tomonaga, S.-i. (September 1, 1948). "On
Radiation Reactions in Collision Processes. I: Application of the "Self-Consistent" Subtraction Method to the Elastic Scattering of an Electron".
4075:
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
1329:
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.
110:
4704:, and despite the fact that Dirac and various othersâall of whom belonged to the older generationânever withdrew their criticisms, attitudes began to change, especially among younger theorists.
7140:. 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.
3889:
1615:{\displaystyle 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}}}},}
5403:
5219:
4156:
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
3164:{\displaystyle 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}}}.}
4673:
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
2479:
1965:
1917:
4589:, 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.
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7155:. A masterpiece on applications of renormalization methods to the calculation of critical exponents in statistical mechanics, following Wilson's ideas (Kenneth Wilson was
1673:
2197:
4819:. Instead of being only a worrisome problem, renormalization has become an important theoretical tool for studying the behavior of field theories in different regimes.
2936:
5143:
5110:
4558:, 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
4887:
Nonrenormalizable interactions in effective field theories rapidly become weaker as the energy scale becomes much smaller than the cutoff. The classic example is the
4569:
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
103:
6720:
5584:
The
Conceptual Completion and Extensions of Quantum Mechanics 1932â1941. Epilogue: Aspects of the Further Development of Quantum Theory 1942â1999: Volumes 6, Part 2
5282:
4160:
becomes small at large energy scales, the theory behaves more like a free theory as the energy exchanged in an interaction becomes large â a phenomenon known as
1703:
1241:
1875:{\displaystyle 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}},}
5302:
5077:
3721:
2938:
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
2273:(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
4120:
insights, detailed in the next section, this splitting is unnatural and actually unphysical, as all scales of the problem enter in continuous systematic ways.
1312:
exhibit precisely the same observed charge as the electron â even in the presence of much stronger interactions and more intense clouds of virtual particles.
3710:{\displaystyle \left({\bar {\psi }}\left(\partial ^{\mu }+ieA^{\mu }\right)\psi \right)_{B}=Z_{1}{\bar {\psi }}\left(\partial ^{\mu }+ieA^{\mu }\right)\psi }
6545:
4617:, where ultraviolet divergences are avoided from the start in calculations by performing well-defined mathematical operations only within the framework of
304:
96:
6857:
5024:
A deeper understanding of the physical meaning and generalization of the renormalization process, which goes beyond the dilatation group of conventional
4637:
argued that these infinities are of a basic nature and cannot be eliminated by any formal mathematical procedures, such as the renormalization method.
6079:"On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. [IV]: Case of Interacting Electromagnetic and Meson Fields"
5146:
2523:
to transfer energy and momentum to the other electron. But in this diagram, before that happens, it emits another virtual photon carrying 4-momentum
344:
3481:
2327:. From a mathematical point of view, the IR divergences can be regularized by assuming fractional differentiation w.r.t. a parameter, for example:
1344:. Initially viewed as a suspect provisional procedure even by some of its originators, renormalization eventually was embraced as an important and
4870:
2105:
1341:
4466:{\displaystyle I=\ln a-\ln b-\ln {\varepsilon _{a}}+\ln {\varepsilon _{b}}=\ln {\tfrac {a}{b}}-\ln {\tfrac {\varepsilon _{a}}{\varepsilon _{b}}}}
7186:
6030:"On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. III: Case of Interacting Electromagnetic and Electron Fields"
5981:"On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. II: Case of Interacting Electromagnetic and Electron Fields"
4543:
is ill-defined, in order to make this notion of canceling divergences precise, the divergences first have to be tamed mathematically using the
7120:
2038:
The trouble was worse in classical field theory than in quantum field theory, because in quantum field theory a charged particle experiences
4903:
almost all of the particle interactions we see are describable by renormalizable theories. It may be that any others that may exist at the
6626:
6120:"On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields V: Case of Interacting Electromagnetic and Meson Fields"
1099:
3475:
above. Conventionally the bare quantities are written so that the corresponding
Lagrangian terms are multiples of the renormalized ones:
4986:
In actual calculations, the counterterms introduced to cancel the divergences in
Feynman diagram calculations beyond tree level must be
2199:
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
4669:
still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.
2529:, and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the 4-momentum
1234:
4696:
The general unease was almost universal in texts up to the 1970s and 1980s. Beginning in the 1970s, however, inspired by work on the
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5043:. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the
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tend to decrease, and the weak hypercharge coupling of the electroweak force tends to increase. At the colossal energy scale of 10
7274:, June 15â26, 1998, Les Houches, France, Kluwer Academic Publishers, NATO ASI Series C 530, 375â388 (1999). Full text available in
5566:. The latter did not appear in print until the Proceedings of the Solvay Conference, published in 1950 (see Laurie M. Brown (ed.),
4585:
3235:). The parts of the Lagrangian left over, involving the remaining portions of the bare quantities, could then be reinterpreted as
1281:
in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original
869:
64:
5039:
This approach covered the conceptual point and was given full computational substance in the extensive important contributions of
4562:; taking this cutoff to infinity (or, equivalently, the corresponding length/time scale to zero) recovers the original integrals.
4049:{\displaystyle {\mathcal {L}}_{I}=-e{\bar {\psi }}\gamma _{\mu }A^{\mu }\psi -(Z_{1}-1)e{\bar {\psi }}\gamma _{\mu }A^{\mu }\psi }
5437:
7101:), ĐĐ°ŃĐ”ĐŒĐ°ŃĐžŃĐ”ŃĐșОД ĐŸŃĐœĐŸĐČŃ ĐșĐČĐ°ĐœŃĐŸĐČĐŸĐč ŃĐ”ĐŸŃОО ĐżĐŸĐ»Ń, (Mathematical aspects of quantum field theory), Atomizdat, Moscow, 1975. 368 pp.
6866:
Cao, Tian Yu; Schweber, Silvan S. (1993). "The conceptual foundations and the philosophical aspects of renormalization theory".
5047:, in 1974, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and
4862:
4815:), they all become approximately the same size (Grotz and Klapdor 1990, p. 254), a major motivation for speculations about
2251:
Shown in the pictures at the right margin, there are exactly three one-loop divergent loop diagrams in quantum electrodynamics:
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40:
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1987:
attempted to develop a classical theory of the electron this way. This early work was the inspiration for later attempts at
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Makogon, D.; Morais Smith, C. (2022). "Median-point approximation and its application for the study of fermionic systems".
7359:
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7268:
Renormalization and renormalization group: From the discovery of UV divergences to the concept of effective field theories
6567:
Isham, C. J.; Salam, Abdus; Strathdee, J. (May 15, 1972). "Infinity
Suppression in Gravity-Modified Electrodynamics. II".
3185:
The solution was to realize that the quantities initially appearing in the theory's formulae (such as the formula for the
1227:
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suggested to them. They think hope is irrational and that, in sitting down to lazy despair, they are merely facing facts.
4583:, which tames the integrals by carrying them into a space with a fictitious fractional number of dimensions. Another is
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2244:
very short proper-time between particle emission and absorption, if the loop is thought of as a sum over particle paths.
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1998:
for an alternative way to remove infinities from this classical problem, assuming new physics exists at small scales.)
417:
7074:
6862:, (2002). Summary of a lecture; has more information about specific regularization and divergence-subtraction schemes.
354:
5036:
is a way to define the components of the theory at large distances as aggregates of components at shorter distances.
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84:
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A term in this Lagrangian, for example, the electronâphoton interaction pictured in Figure 1, can then be written
2028:
407:
6521:
P.A.M. Dirac, "The Evolution of the Physicist's Picture of Nature", in Scientific American, May 1963, p. 53.
5807:
5761:
Schwinger, J. (1949). "III. The electromagnetic properties of the electron radiative corrections to scattering".
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4693:, does not have a Landau pole. Feynman, along with most others, accepted that QCD was a fully consistent theory.
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7287:, Poincaré Seminar (Paris, Oct. 12, 2002), published in : Duplantier, Bertrand; Rivasseau, Vincent (Eds.);
7168:, Poincaré Seminar (Paris, Oct. 12, 2002), published in : Duplantier, Bertrand; Rivasseau, Vincent (Eds.);
7023:, Poincaré Seminar (Paris, Oct. 12, 2002), published in : Duplantier, Bertrand; Rivasseau, Vincent (Eds.);
2496:
Figure 2. A diagram contributing to electronâelectron scattering in QED. The loop has an ultraviolet divergence.
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1988:
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counterterm's interaction vertex placed as in Figure 3 cancels out the divergence from the loop in Figure 2.
3216:
quantities. The charge of the electron, say, would be defined in terms of a quantity measured at a specific
7073:, Encyclopaedia of Mathematics, Kluwer Academic Publisher (1996). Full text available in PostScript and pdf
6976:(3 volumes) Cambridge University Press (1995). A monumental treatise on Q.F.T. written by a leading expert,
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3457:{\displaystyle {\mathcal {L}}={\bar {\psi }}_{B}\left\psi _{B}-{\frac {1}{4}}F_{B\mu \nu }F_{B}^{\mu \nu }}
1273:
arising in calculated quantities by altering values of these quantities to compensate for effects of their
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4656:. Despite his crucial role in the development of quantum electrodynamics, he wrote the following in 1985:
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1382:, another technique to control infinities by assuming the existence of new unknown physics at new scales.
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4301:{\displaystyle I=\int _{0}^{a}{\frac {1}{z}}\,dz-\int _{0}^{b}{\frac {1}{z}}\,dz=\ln a-\ln b-\ln 0+\ln 0}
4108:
Historically, the splitting of the "bare terms" into the original terms and counterterms came before the
3174:
This integral is divergent and infinite, unless we cut it off at finite energy and momentum in some way.
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The possible macroscopic states of the system, at a large scale, are given by this set of fixed points.
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7309:) of renormalization, and its link to the Riemann-Hilbert problem. Full text (in French) available at
6954:
Highly readable textbook, certainly the best introduction to relativistic Q.F.T. for particle physics.
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2057:(a) Vacuum polarization, a.k.a. charge screening. This loop has a logarithmic ultraviolet divergence.
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To make contact with reality, then, the formulae would have to be rewritten in terms of measurable,
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Dyson, F. J. (February 15, 1952). "Divergence of Perturbation Theory in Quantum Electrodynamics".
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or Planck scale simply become too weak to detect in the realm we can observe, with one exception:
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actually correspond to the physical constants measured in the laboratory. As written, they were
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5451:, but the nature of the quantum field theories associated with these remains an open question.
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quantities that did not take into account the contribution of virtual-particle loop effects to
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or something different), then there may be no real problem with short-distance divergences in
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The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics
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In more technical terms, let us assume that we have a theory described by a certain function
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5618:
5516:
5150:
4911:, whose exceedingly weak interaction is magnified by the presence of the enormous masses of
4866:
4773:
4752:
4717:
4559:
3824:
2287:
The three divergences correspond to the three parameters in the theory under consideration:
2141:
2117:
2002:
1297:
1199:
1114:
1094:
1064:
1014:
839:
834:
829:
714:
699:
629:
314:
241:
231:
166:
6698:
5799:
5260:
2226:
the region in the integral where all particles in the loop have large energies and momenta,
7190:
6744:
6336:(October 1, 1975). "The renormalization group: Critical phenomena and the Kondo problem".
5895:
5887:
5851:
5843:
5763:
5726:
5689:
5650:
5601:
5571:
5029:
4858:
4653:
4064:
3814:{\displaystyle \left(F_{\mu \nu }F^{\mu \nu }\right)_{B}=Z_{3}\,F_{\mu \nu }F^{\mu \nu }.}
3190:
2318:(i.e. real) photons whose wavelengths tend to infinity; this diagram is equivalent to the
2314:
difference: the photon connecting the two legs of the electron is cut and replaced by two
2306:
2280:
2155:, they can have any energy and momentum, even one that is not allowed by the relativistic
2133:
2121:
2006:
1980:
1410:
The mass of a charged particle should include the massâenergy in its electrostatic field (
1345:
1325:
1266:
1144:
1139:
1109:
1104:
1074:
1034:
1004:
984:
889:
784:
779:
744:
729:
704:
684:
664:
594:
584:
505:
467:
329:
161:
139:
134:
4799:
of particle physics vary in different ways with increasing energy scale: the coupling of
1682:
7198:
The Critical Point: A Historical Introduction to the Modern Theory of Critical Phenomena
6824:
6757:
6663:
6580:
6484:
6441:
6396:
6349:
6274:
6227:
6180:
6135:
6045:
5996:
5947:
5908:
5864:
5820:
5776:
5739:
5702:
5663:
5614:
4837:
calculation. If the Lagrangian contains combinations of field operators of high enough
7251:
7098:
6928:
6894:
6303:
5287:
5062:
4850:
4796:
4682:
2908:
2904:
2324:
2319:
2090:
2039:
1169:
1119:
1039:
1024:
1009:
989:
959:
954:
949:
934:
924:
919:
884:
874:
804:
739:
709:
644:
609:
599:
574:
445:
402:
266:
2218:, that is, they give infinite answers. The divergences that are significant are the "
7343:
6848:
6765:
6679:
6404:
5630:
5575:
5443:
If these fixed points correspond to free field theory, the theory is said to exhibit
4895:, a nonrenormalizable effective theory whose cutoff is comparable to the mass of the
4744:
4740:
4686:
4634:
4607:
4129:
physical predictions of the theory, calculated to all orders, should in principle be
2538:
This diagram's amplitude ends up with, among other things, a factor from the loop of
2129:
2011:
1174:
1164:
1154:
1149:
1069:
1054:
854:
849:
679:
674:
634:
604:
569:
559:
490:
485:
6887:
6471:
Stern, A. W. (November 7, 1952). "Space, Field, and Ether in Contemporary Physics".
6380:
5520:
4853:
of particle physics contains only renormalizable operators, but the interactions of
7306:
6840:
6692:
6554:
6549:. Princeton: Princeton University Press, 1985, p. 128. The quoted passage is
5044:
3232:
2109:
2053:
1390:
1321:
1209:
1134:
1019:
994:
979:
914:
794:
789:
669:
659:
654:
619:
614:
589:
544:
412:
374:
196:
176:
6902:
5932:"On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields"
5051:
in 1971. He was awarded the Nobel prize for these decisive contributions in 1982.
4857:
become nonrenormalizable operators if one attempts to construct a field theory of
6611:
6492:
2097:
discovered that in perturbative corrections many integrals were divergent (see
1320:
as a continuum, certain statistical and quantum mechanical constructions are not
6671:
4824:
4764:
4678:
4677:, made it plausible that quantum field theories were all inconsistent. In 1974,
4674:
4603:
3236:
2267:
2211:
possible combinations of energy and momentum that could travel around the loop.
1984:
1353:
1292:
theory may begin by postulating an electron with an initial mass and charge. In
1189:
1124:
1084:
1044:
999:
864:
769:
764:
749:
734:
719:
3554:{\displaystyle \left({\bar {\psi }}m\psi \right)_{B}=Z_{0}{\bar {\psi }}m\psi }
2222:" (UV) ones. An ultraviolet divergence can be described as one that comes from
7226:
7063:
6357:
5828:
5596:
4896:
4640:
3217:
2945:
2230:
2113:
2094:
2069:
1357:
844:
759:
624:
564:
236:
171:
6596:
6500:
6479:(3019). American Association for the Advancement of Science (AAAS): 493â496.
6457:
6365:
6243:
6196:
6153:
6104:
6063:
6014:
5965:
5528:
4606:
to grid size 0, or our natural universe. This presupposes the existence of a
2031:
had a noncausal "pre-acceleration". Sometimes an electron would start moving
7328:
7305:(Fields medallist 1982) describe the mathematical underlying structure (the
7270:, in: de Witt-Morette C., Zuber J.-B. (eds), Proceedings of the NATO ASI on
6588:
5916:
5710:
5543:
4613:
A rigorous mathematical approach to renormalization theory is the so-called
2234:
1317:
7275:
6508:
6449:
6283:
6258:
6235:
5872:
5784:
5747:
5672:
5645:
5622:
5157:, etc. It must contain the whole description of the physics of the system.
2035:
the force is applied. This is a sign that the point limit is inconsistent.
6188:
6144:
6119:
6054:
6029:
6005:
5980:
4142:, and the general theory of this kind of scale-dependence is known as the
2492:
5436:
The ability of several theories to flow to the same fixed point leads to
5160:
Now we consider a certain blocking transformation of the state variables
4881:
2535:
uniquely, so all possibilities contribute equally and we must integrate.
2315:
2256:
2204:
2200:
2152:
2082:
2016:
1337:
1305:
1289:
1270:
809:
6214:(July 15, 1948). "On Infinite Field Reactions in Quantum Field Theory".
6095:
6078:
4314:
To eliminate the divergence, simply change lower limit of integral into
3251:
7259:
7036:
7011:
6879:
6815:
5504:
4908:
2266:(b) An electron quickly emits and reabsorbs a virtual photon, called a
1636:
1363:
Today, the point of view has shifted: on the basis of the breakthrough
1349:
6832:
6742:(1988). "Triviality Pursuit: Can Elementary Scalar Particles Exist?".
5032:'s paper in 1966 proposed the "block-spin" renormalization group. The
4643:'s criticism was the most persistent. As late as 1975, he was saying:
3197:, as well as the normalizations of the quantum fields themselves, did
7314:
7166:
Phase Transitions & Renormalization Group: from Theory to Numbers
6413:
5956:
5931:
4916:
4725:
1309:
1301:
5646:"On quantum-electrodynamics and the magnetic moment of the electron"
4708:
and others demonstrated that the renormalization group is useful in
2132:
in 1949. The divergences appear in radiative corrections involving
6654:
2484:
so we can cure IR divergences by turning them into UV divergences.
1414:). Assume that the particle is a charged spherical shell of radius
3831:, turns out to imply that we can renormalize the two terms of the
3250:
2491:
2068:
2060:
2052:
1389:
6612:
The Autobiography of Bertrand Russell: The Final Years, 1944-1969
6557:(2014 electronic version of 2006 reprint of 1985 first printing).
5724:
Schwinger, J. (1949). "II. Vacuum polarization and self-energy".
4554:
An essentially arbitrary modification to the loop integrands, or
3894:
together (Pokorski 1987, p. 115), which is what happened to
7002:
The Glorious Days of Physics â Renormalization of Gauge theories
6901:, C.E.R.N. Courrier 41(7) (2001). Full text available at :
5503:
Sanyuk, Valerii I.; Sukhanov, Alexander D. (September 1, 2003).
4912:
4729:
3194:
3177:
Similar loop divergences occur in other quantum field theories.
2911:; they have to do with the spin of the electron. The factors of
2061:
5344:. If this is achievable by a certain change in the parameters,
4861:
in the most straightforward manner (treating the metric in the
3227:(which will generally have a characteristic energy, called the
2248:
So these divergences are short-distance, short-time phenomena.
7258:, JINR Communication E2-96-15 (1996). Full text available at:
4926:
4808:
2944:
of the three big factors in the integrand, which are from the
7300:
6259:"The radiation theories of Tomonaga, Schwinger, and Feynman"
5001:
and the related modified minimal subtraction (MS-bar) scheme
4716:, where it provides important insights into the behavior of
3928:
3283:
7130:
Lectures on Phase Transitions and the Renormalization Group
5800:"Space-time approach to non-relativistic quantum mechanics"
2948:
of the two electron lines and the photon line in the loop.
1639:
and thus cannot be accelerated. Incidentally, the value of
1635:. This implies that the point particle would have infinite
4134:
exploit this fact to calculate the effective variation of
7291:, Progress in Mathematical Physics 30, BirkhÀuser (2003)
7172:, Progress in Mathematical Physics 30, BirkhÀuser (2003)
7027:, Progress in Mathematical Physics 30, BirkhÀuser (2003)
6801:
Delamotte, Bertrand (2004). "A hint of renormalization".
4899:. This fact may also provide a possible explanation for
4780:(George Allen and Unwin, Ltd., London 1969), p. 221:
7004:, lecture given at Erice (August/September 1998) by the
6028:
Koba, Z.; Tati, T.; Tomonaga, S.-i. (December 1, 1947).
4776:'s postscript to the third volume of his autobiography
7132:, Frontiers in Physics 85, Westview Press (June, 1992)
5979:
Koba, Z.; Tati, T.; Tomonaga, S.-i. (October 1, 1947).
5505:"Dirac in 20th century physics: a centenary assessment"
4873:
is not satisfactory in application to quantum gravity.
2385:{\displaystyle \left(p^{2}-a^{2}\right)^{\frac {1}{2}}}
2015:
of a particle's own field on itself. (Analogous to the
1348:
actual mechanism of scale physics in several fields of
5844:"A relativistic cut-off for classical electrodynamics"
5599:(1947). "The Electromagnetic Shift of Energy Levels".
4994:. The common renormalization schemes in use include:
4438:
4417:
2108:
corrections' divergences was discovered in 1947â49 by
6935:. New York, Interscience. The first text-book on the
6701:
5570:, Springer, 2012, p. 53). Kramers' approach was
5420:
The most important information in the RG flow is its
5350:
5314:
5290:
5263:
5227:
5166:
5118:
5085:
5065:
4337:
4176:
3924:
3844:
3724:
3568:
3484:
3280:
2972:
2921:
2547:
2435:
2336:
2165:
1929:
1891:
1745:
1715:
1685:
1654:
1432:
7213:
Renormalization: From Lorentz to Landau (and Beyond)
7086:, Springer Verlag Berlin Heidelberg New York (1995)
6853:, (2005). A qualitative introduction to the subject.
6381:"Regularization and renormalization of gauge fields"
5888:"A relativistic cut-off for quantum electrodynamics"
5687:
Schwinger, J. (1948). "I. A covariant formulation".
5568:
Renormalization: From Lorentz to Landau (and Beyond)
4138:
with changes in scale. This variation is encoded by
7084:
Finite quantum electrodynamics: The causal approach
5424:. A fixed point is defined by the vanishing of the
4071:(see below for more on this), as it is in QED, the
4067:). The rest is the counterterm. If the theory is
7231:Scaling and Renormalization in Statistical Physics
6714:
6575:(10). American Physical Society (APS): 2548â2565.
6302:
5397:
5336:
5296:
5276:
5249:
5213:
5137:
5104:
5071:
4465:
4300:
4048:
3884:{\displaystyle {\bar {\psi }}(\partial +ieA)\psi }
3883:
3813:
3709:
3553:
3456:
3163:
2930:
2886:
2473:
2384:
2191:
1959:
1911:
1874:
1728:
1697:
1667:
1614:
7272:Quantum Field Theory: Perspective and Prospective
7046:From perturbative to constructive renormalization
6859:Renormalization: Our Greatly Misunderstood Friend
5545:Renormalization from Classical to Quantum Physics
5398:{\displaystyle \{J_{k}\}\to \{{\tilde {J}}_{k}\}}
5214:{\displaystyle \{s_{i}\}\to \{{\tilde {s}}_{i}\}}
2283:â due to its shape remotely resembling a penguin.
2279:. The Feynman diagram for this is also called a â
2159:for the observed mass of that particle (that is,
6912:Zeta regularization techniques with Applications
5447:. Numerous fixed points appear in the study of
3243:the troublesome divergences for other diagrams.
7147:, Oxford University Press (4th edition â 2002)
6436:(4). American Physical Society (APS): 631â632.
6344:(4). American Physical Society (APS): 773â840.
6222:(2). American Physical Society (APS): 224â225.
6077:Kanesawa, S.; Tomonaga, S.-i. (March 1, 1948).
5028:theories, came from condensed matter physics.
4880:, "renormalizability" is, strictly speaking, a
3264:counterterm cancels the divergence in Figure 2.
6118:Kanesawa, S.; Tomonaga, S.-i. (June 1, 1948).
3189:), representing such things as the electron's
2915:are the electric coupling constant, while the
1324:. To define them, or make them unambiguous, a
16:Method in physics used to deal with infinities
6630:, page 390 (Cambridge University Press 1996).
6328:
6326:
6175:(3). Oxford University Press (OUP): 290â303.
6130:(2). Oxford University Press (OUP): 101â113.
6040:(4). Oxford University Press (OUP): 198â208.
5991:(3). Oxford University Press (OUP): 101â116.
1991:and renormalization in quantum field theory.
1399:The problem of infinities first arose in the
1269:geometric structures, that are used to treat
1235:
104:
8:
7285:Symétries Galoisiennes & Renormalisation
6695:(1966): "Scaling laws for Ising models near
6309:. Reading: Addison-Wesley. Chapter 10.
5392:
5370:
5364:
5351:
5208:
5186:
5180:
5167:
5132:
5119:
5099:
5086:
4954:. There might be a discussion about this on
3467:the fields and coupling constant are really
7145:Quantum Field Theory and Critical Phenomena
6546:QED: The Strange Theory of Light and Matter
5942:(2). Oxford University Press (OUP): 27â42.
6089:(1). Oxford University Press (OUP): 1â13.
4689:showed that another quantum field theory,
3255:Figure 3. The vertex corresponding to the
1242:
1228:
133:
122:
111:
97:
20:
6814:
6706:
6700:
6653:
6412:
6282:
6143:
6094:
6053:
6004:
5955:
5671:
5386:
5375:
5374:
5358:
5349:
5328:
5317:
5316:
5313:
5289:
5268:
5262:
5241:
5230:
5229:
5226:
5202:
5191:
5190:
5174:
5165:
5126:
5117:
5093:
5084:
5064:
4974:Learn how and when to remove this message
4592:Yet another regularization scheme is the
4454:
4444:
4437:
4416:
4400:
4395:
4379:
4374:
4336:
4243:
4233:
4227:
4222:
4208:
4198:
4192:
4187:
4175:
4037:
4027:
4012:
4011:
3993:
3974:
3964:
3949:
3948:
3933:
3927:
3926:
3923:
3846:
3845:
3843:
3799:
3786:
3781:
3775:
3762:
3748:
3735:
3723:
3693:
3674:
3654:
3653:
3647:
3634:
3615:
3596:
3576:
3575:
3567:
3534:
3533:
3527:
3514:
3492:
3491:
3483:
3445:
3440:
3424:
3410:
3401:
3386:
3368:
3363:
3353:
3337:
3322:
3304:
3293:
3292:
3282:
3281:
3279:
3239:, involved in divergent diagrams exactly
3149:
3139:
3117:
3093:
3083:
3076:
3067:
3043:
3036:
3027:
3017:
3007:
2997:
2987:
2977:
2971:
2957:on top that dominates at large values of
2920:
2863:
2848:
2835:
2829:
2807:
2794:
2761:
2739:
2726:
2720:
2698:
2685:
2652:
2630:
2617:
2611:
2598:
2574:
2567:
2558:
2546:
2474:{\displaystyle {\frac {1}{p^{2}-a^{2}}},}
2459:
2446:
2436:
2434:
2371:
2360:
2347:
2335:
2301:The second class of divergence called an
2183:
2170:
2164:
1945:
1933:
1928:
1901:
1890:
1864:
1852:
1827:
1817:
1802:
1792:
1782:
1765:
1759:
1750:
1744:
1720:
1714:
1684:
1675:equal to the electron mass is called the
1659:
1653:
1600:
1583:
1577:
1567:
1561:
1545:
1532:
1516:
1501:
1495:
1488:
1483:
1469:
1463:
1449:
1437:
1431:
6899:Fifty Years of the Renormalization Group
5112:and a certain set of coupling constants
4728:ceases to be continuous on the scale of
2259:pair, which then annihilates. This is a
2255:(a) A photon creates a virtual electronâ
2009:particles, it is tempting to ignore the
6305:An Introduction to Quantum Field Theory
5558:Kramers presented his work at the 1947
5486:
4767:'s remark in 1972 seems still relevant
2907:as in the covariant formulation of the
1930:
1819:
1356:. Despite his later skepticism, it was
1332:Renormalization was first developed in
125:
23:
7069:Iagolnitzer, Daniel & Magnen, J.;
6969:. Another excellent textbook on Q.F.T.
2047:Divergences in quantum electrodynamics
1960:{\displaystyle \hbar /(m_{\text{e}}c)}
1386:Self-interactions in classical physics
7115:(Routledge Chapman & Hall 2004);
5493:See e.g., Weinberg vol I, chapter 10.
4811:(far beyond the reach of our current
4803:and the weak isospin coupling of the
2513:. It emits a virtual photon carrying
1277:. But even if no infinities arose in
7:
7256:The Bogoliubov Renormalization Group
7233:, Cambridge University Press (1996)
7048:, Princeton University Press (1991)
6989:, Cambridge University Press (1987)
6961:, Princeton University Press (2003)
6946:(Cambridge University Press, 1985),
4660:The shell game that we play to find
2951:This has a piece with two powers of
2405:but is UV divergent; if we take the
1912:{\displaystyle \alpha \approx 1/137}
1407:in the 19th and early 20th century.
2237:fluctuations of the fields, in the
7215:, Springer-Verlag (New York-1993)
7021:An introduction to renormalization
6959:Quantum Field Theory in a Nutshell
6379:'t Hooft, G.; Veltman, M. (1972).
5548:(Thesis). University of Rochester.
5416:Renormalization group fixed points
3860:
3671:
3593:
3334:
2073:(c) Example of a âpenguinâ diagram
1496:
1423:. The massâenergy in the field is
24:Renormalization and regularization
14:
5257:must be lower than the number of
4724:short-distance regulator exists:
4720:. In condensed matter physics, a
3207:the physical constants themselves
1378:Renormalization is distinct from
1257:is a collection of techniques in
7327:
6724:Physics (Long Island City, N.Y.)
5586:, Springer, 2001, p. 1050).
5405:, then the theory is said to be
5337:{\displaystyle {\tilde {s}}_{i}}
5284:. Now let us try to rewrite the
5250:{\displaystyle {\tilde {s}}_{i}}
4931:
4739:holds all the way down past the
3471:quantities, hence the subscript
3181:Renormalized and bare quantities
1729:{\displaystyle \varepsilon _{0}}
6789:Introduction to Renormalization
6301:; Schroeder, Daniel V. (1995).
6169:Progress of Theoretical Physics
6124:Progress of Theoretical Physics
6083:Progress of Theoretical Physics
6034:Progress of Theoretical Physics
5985:Progress of Theoretical Physics
5936:Progress of Theoretical Physics
5930:Tomonaga, S. (August 1, 1946).
5521:10.1070/PU2003v046n09ABEH001165
5461:History of quantum field theory
4999:Minimal subtraction (MS) scheme
2903:factors in this expression are
1360:who pioneered renormalization.
7071:Renormalization group analysis
6933:The Theory of Quantized Fields
5380:
5367:
5322:
5235:
5196:
5183:
4017:
4005:
3986:
3954:
3875:
3857:
3851:
3659:
3581:
3539:
3497:
3298:
3136:
3123:
3114:
3101:
3064:
3054:
2963:(Pokorski 1987, p. 122):
2791:
2778:
2773:
2758:
2745:
2732:
2682:
2669:
2664:
2649:
2636:
2623:
2595:
2585:
2426:, we obtain the IR divergence
2065:(b) Self-energy diagram in QED
1954:
1938:
433:Renormalization group equation
85:Point-splitting regularization
1:
6532:Dirac: A scientific biography
4652:Another important critic was
2147:While virtual particles obey
1975:point limit. This was called
1668:{\displaystyle m_{\text{em}}}
262:Spontaneous symmetry breaking
222:Symmetry in quantum mechanics
6974:The Quantum Theory of Fields
6919:Mainly: quantum field theory
6766:10.1016/0370-1573(88)90008-7
6493:10.1126/science.116.3019.493
6405:10.1016/0550-3213(72)90279-9
5886:Feynman, Richard P. (1948).
5842:Feynman, Richard P. (1948).
5798:Feynman, Richard P. (1948).
4865:as a perturbation about the
4626:Attitudes and interpretation
4586:PauliâVillars regularization
2507:and ends up with 4-momentum
1316:infinities. When describing
73:Zeta function regularization
65:PauliâVillars regularization
7106:Mainly: statistical physics
6803:American Journal of Physics
6672:10.1103/PhysRevB.105.174505
5767:. Quantum Electrodynamics.
5730:. Quantum Electrodynamics.
5693:. Quantum Electrodynamics.
4863:EinsteinâHilbert Lagrangian
2297:The charge of the electron.
2192:{\displaystyle E^{2}-p^{2}}
7376:
7200:, CRC Press (March, 1996)
7010:. Full text available at:
5562:, repeated in 1948 at the
4992:renormalisation conditions
4778:The Final Years, 1944â1969
4615:causal perturbation theory
4572:dimensional regularization
2931:{\displaystyle i\epsilon }
2291:The field normalization Z.
2214:These integrals are often
2104:One way of describing the
1625:which becomes infinite as
257:Explicit symmetry breaking
77:Causal perturbation theory
61:Dimensional regularization
45:Minimal subtraction scheme
7189:October 15, 2005, at the
7180:. Full text available in
7056:. Full text available in
7035:. Full text available in
6850:Renormalization Made Easy
6358:10.1103/revmodphys.47.773
6338:Reviews of Modern Physics
5829:10.1103/RevModPhys.20.367
5808:Reviews of Modern Physics
5560:Shelter Island Conference
5145:. This function may be a
5138:{\displaystyle \{J_{k}\}}
5105:{\displaystyle \{s_{i}\}}
4743:(where it might yield to
2309:has both ultraviolet and
2294:The mass of the electron.
2099:The problem of infinities
1705:and restoring factors of
1677:classical electron radius
1401:classical electrodynamics
423:BargmannâWigner equations
350:Path Integral Formulation
7211:Brown, Laurie M. (Ed.);
4714:condensed matter physics
4712:field theory applied to
4547:, in a process known as
2157:energyâmomentum relation
1996:regularization (physics)
1263:statistical field theory
428:Schwinger-Dyson equation
7299:. French mathematician
6589:10.1103/physrevd.5.2548
6534:, CUP 1990, p. 184
5917:10.1103/PhysRev.74.1430
5711:10.1103/PhysRev.74.1439
5079:of the state variables
4923:Renormalization schemes
3829:WardâTakahashi identity
2079:quantum electrodynamics
1921:fine-structure constant
1336:(QED) to make sense of
1334:quantum electrodynamics
458:Electroweak interaction
453:Quantum electrodynamics
418:WheelerâDeWitt equation
295:Background field method
81:Hadamard regularization
7332:Quotations related to
6716:
6450:10.1103/physrev.85.631
6284:10.1103/PhysRev.75.486
6236:10.1103/physrev.74.224
5873:10.1103/PhysRev.74.939
5785:10.1103/PhysRev.76.790
5748:10.1103/PhysRev.75.651
5673:10.1103/PhysRev.73.416
5644:Schwinger, J. (1948).
5623:10.1103/PhysRev.72.339
5449:lattice Higgs theories
5431:conformal field theory
5399:
5338:
5298:
5278:
5251:
5215:
5139:
5106:
5073:
5015:In statistical physics
4878:effective field theory
4801:quantum chromodynamics
4702:effective field theory
4691:quantum chromodynamics
4595:lattice regularization
4581:Martinus J. G. Veltman
4467:
4302:
4158:quantum chromodynamics
4059:The physical constant
4050:
3885:
3815:
3711:
3555:
3458:
3265:
3247:Renormalization in QED
3165:
2932:
2888:
2497:
2475:
2386:
2276:vertex renormalization
2193:
2149:conservation of energy
2128:, and systematized by
2074:
2066:
2058:
2029:AbrahamâLorentz theory
1961:
1913:
1876:
1730:
1699:
1669:
1616:
1396:
463:Quantum chromodynamics
320:Effective field theory
69:Lattice regularization
7355:Renormalization group
7289:Poincaré Seminar 2002
7170:Poincaré Seminar 2002
7025:Poincaré Seminar 2002
6944:Quantum Field Theory
6937:renormalization group
6856:Blechman, Andrew E.;
6717:
6715:{\displaystyle T_{c}}
6257:Dyson, F. J. (1949).
6210:Tomonaga, Sin-Itiro;
5476:Nonoblique correction
5400:
5339:
5299:
5279:
5277:{\displaystyle s_{i}}
5252:
5216:
5140:
5107:
5074:
4813:particle accelerators
4698:renormalization group
4468:
4303:
4144:renormalization group
4118:renormalization group
4110:renormalization group
4096:The diagram with the
4051:
3886:
3816:
3712:
3556:
3459:
3254:
3229:renormalization scale
3221:renormalization point
3166:
2933:
2889:
2495:
2476:
2417:fractional derivative
2387:
2194:
2072:
2064:
2056:
2001:When calculating the
1962:
1914:
1877:
1731:
1700:
1670:
1617:
1393:
1365:renormalization group
408:KleinâGordon equation
340:LSZ reduction formula
37:Renormalization group
7360:Mathematical physics
7350:Quantum field theory
7044:Rivasseau, Vincent;
7019:Rivasseau, Vincent;
6987:Gauge Field Theories
6781:General introduction
6699:
6627:Quantum Field Theory
6615:(Bantam Books, 1970)
6543:Feynman, Richard P.
5433:at the fixed point.
5348:
5312:
5288:
5261:
5225:
5164:
5116:
5083:
5063:
4944:confusing or unclear
4817:grand unified theory
4335:
4174:
4116:. According to such
3922:
3903:; it is the same as
3842:
3833:covariant derivative
3722:
3566:
3482:
3278:
3268:For example, in the
2970:
2919:
2545:
2433:
2334:
2163:
2126:Shin'ichiro Tomonaga
1927:
1889:
1743:
1713:
1683:
1652:
1430:
1412:electromagnetic mass
1294:quantum field theory
1265:, and the theory of
1259:quantum field theory
501:Theory of everything
335:Lattice field theory
305:Correlation function
127:Quantum field theory
7266:Zinn-Justin, Jean;
7164:Zinn-Justin, Jean;
7158:Nobel laureate 1982
7143:Zinn-Justin, Jean;
7064:PDF (draft version)
7007:Nobel laureate 1999
6979:Nobel laureate 1979
6825:2004AmJPh..72..170D
6758:1988PhR...167..241C
6664:2022PhRvB.105q4505M
6609:Russell, Bertrand.
6581:1972PhRvD...5.2548I
6485:1952Sci...116..493S
6442:1952PhRv...85..631D
6397:1972NuPhB..44..189T
6350:1975RvMP...47..773W
6275:1949PhRv...75..486D
6228:1948PhRv...74..224T
6189:10.1143/ptp/3.3.290
6181:1948PThPh...3..290K
6145:10.1143/ptp/3.2.101
6136:1948PThPh...3..101K
6055:10.1143/ptp/2.4.198
6046:1947PThPh...2..198K
6006:10.1143/ptp/2.3.101
5997:1947PThPh...2..101K
5948:1946PThPh...1...27T
5909:1948PhRv...74.1430F
5865:1948PhRv...74..939F
5821:1948RvMP...20..367F
5777:1949PhRv...76..790S
5740:1949PhRv...75..651S
5703:1948PhRv...74.1439S
5664:1948PhRv...73..416S
5615:1947PhRv...72..339B
5542:Kar, Arnab (2014).
4952:clarify the section
4871:perturbation theory
4869:), suggesting that
4763:Be that as it may,
4539:Since the quantity
4232:
4197:
3453:
3373:
2395:is well defined at
2303:infrared divergence
2261:vacuum polarization
2106:perturbation theory
1698:{\displaystyle q=e}
1500:
1342:perturbation theory
480:Incomplete theories
7000:'t Hooft, Gerard;
6985:Pokorski, Stefan;
6972:Weinberg, Steven;
6880:10.1007/BF01255832
6794:Santa Fe Institute
6712:
6334:Wilson, Kenneth G.
6299:Peskin, Michael E.
6212:Oppenheimer, J. R.
5466:Quantum triviality
5445:quantum triviality
5395:
5334:
5294:
5274:
5247:
5211:
5147:partition function
5135:
5102:
5069:
5049:critical phenomena
4893:weak nuclear force
4855:general relativity
4551:(Weinberg, 1995).
4463:
4461:
4426:
4298:
4218:
4183:
4162:asymptotic freedom
4136:physical constants
4046:
3881:
3811:
3707:
3551:
3454:
3436:
3359:
3266:
3161:
2928:
2884:
2498:
2471:
2382:
2189:
2075:
2067:
2059:
1969:Compton wavelength
1957:
1909:
1872:
1736:) turns out to be
1726:
1695:
1665:
1612:
1479:
1397:
1369:Nikolay Bogolyubov
345:Partition function
272:Topological charge
192:General relativity
187:Special relativity
7121:978-0-415-31002-4
6942:Ryder, Lewis H.;
6833:10.1119/1.1624112
6740:D. J. E. Callaway
6569:Physical Review D
6385:Nuclear Physics B
6096:10.1143/ptp/3.1.1
5903:(10): 1430â1438.
5697:(10): 1439â1461.
5580:Helmut Rechenberg
5564:Solvay Conference
5383:
5325:
5297:{\displaystyle Z}
5238:
5199:
5072:{\displaystyle Z}
4984:
4983:
4976:
4844:nonrenormalizable
4832:Renormalizability
4805:electroweak force
4749:causal set theory
4718:phase transitions
4706:Kenneth G. Wilson
4577:Gerardus 't Hooft
4460:
4425:
4241:
4206:
4124:Running couplings
4020:
3957:
3854:
3662:
3584:
3542:
3500:
3418:
3301:
3270:Lagrangian of QED
3225:subtraction point
3156:
3074:
2879:
2823:
2714:
2605:
2488:A loop divergence
2466:
2379:
2142:virtual particles
2087:Werner Heisenberg
1971:of the electron.
1948:
1867:
1863:
1837:
1830:
1809:
1795:
1753:
1679:, which (setting
1662:
1607:
1603:
1539:
1509:
1491:
1457:
1440:
1298:virtual particles
1275:self-interactions
1252:
1251:
325:Expectation value
300:BRST quantization
247:Poincaré symmetry
202:YangâMills theory
182:Quantum mechanics
121:
120:
7367:
7331:
7126:Nigel Goldenfeld
7111:A. N. Vasil'ev;
7082:Scharf, GĂŒnter;
6925:N. N. Bogoliubov
6891:
6844:
6818:
6770:
6769:
6736:
6730:
6721:
6719:
6718:
6713:
6711:
6710:
6690:
6684:
6683:
6657:
6637:
6631:
6622:
6616:
6607:
6601:
6600:
6564:
6558:
6541:
6535:
6528:
6522:
6519:
6513:
6512:
6468:
6462:
6461:
6425:
6419:
6418:
6416:
6376:
6370:
6369:
6330:
6321:
6320:
6308:
6295:
6289:
6288:
6286:
6254:
6248:
6247:
6207:
6201:
6200:
6164:
6158:
6157:
6147:
6115:
6109:
6108:
6098:
6074:
6068:
6067:
6057:
6025:
6019:
6018:
6008:
5976:
5970:
5969:
5959:
5957:10.1143/ptp.1.27
5927:
5921:
5920:
5892:
5883:
5877:
5876:
5848:
5839:
5833:
5832:
5804:
5795:
5789:
5788:
5758:
5752:
5751:
5721:
5715:
5714:
5684:
5678:
5677:
5675:
5641:
5635:
5634:
5593:
5587:
5556:
5550:
5549:
5539:
5533:
5532:
5500:
5494:
5491:
5471:Zeno's paradoxes
5404:
5402:
5401:
5396:
5391:
5390:
5385:
5384:
5376:
5363:
5362:
5343:
5341:
5340:
5335:
5333:
5332:
5327:
5326:
5318:
5308:in terms of the
5303:
5301:
5300:
5295:
5283:
5281:
5280:
5275:
5273:
5272:
5256:
5254:
5253:
5248:
5246:
5245:
5240:
5239:
5231:
5221:, the number of
5220:
5218:
5217:
5212:
5207:
5206:
5201:
5200:
5192:
5179:
5178:
5144:
5142:
5141:
5136:
5131:
5130:
5111:
5109:
5108:
5103:
5098:
5097:
5078:
5076:
5075:
5070:
4979:
4972:
4968:
4965:
4959:
4935:
4934:
4927:
4867:Minkowski metric
4753:particle physics
4598:, introduced by
4545:theory of limits
4542:
4531:
4529:
4527:
4526:
4521:
4518:
4504:
4502:
4500:
4499:
4491:
4488:
4472:
4470:
4469:
4464:
4462:
4459:
4458:
4449:
4448:
4439:
4427:
4418:
4406:
4405:
4404:
4385:
4384:
4383:
4327:
4320:
4311:is ill-defined.
4307:
4305:
4304:
4299:
4242:
4234:
4231:
4226:
4207:
4199:
4196:
4191:
4104:
4092:
4083:
4062:
4055:
4053:
4052:
4047:
4042:
4041:
4032:
4031:
4022:
4021:
4013:
3998:
3997:
3979:
3978:
3969:
3968:
3959:
3958:
3950:
3938:
3937:
3932:
3931:
3911:
3902:
3890:
3888:
3887:
3882:
3856:
3855:
3847:
3825:Gauge invariance
3820:
3818:
3817:
3812:
3807:
3806:
3794:
3793:
3780:
3779:
3767:
3766:
3761:
3757:
3756:
3755:
3743:
3742:
3716:
3714:
3713:
3708:
3703:
3699:
3698:
3697:
3679:
3678:
3664:
3663:
3655:
3652:
3651:
3639:
3638:
3633:
3629:
3625:
3621:
3620:
3619:
3601:
3600:
3586:
3585:
3577:
3560:
3558:
3557:
3552:
3544:
3543:
3535:
3532:
3531:
3519:
3518:
3513:
3509:
3502:
3501:
3493:
3474:
3463:
3461:
3460:
3455:
3452:
3444:
3435:
3434:
3419:
3411:
3406:
3405:
3396:
3392:
3391:
3390:
3378:
3374:
3372:
3367:
3358:
3357:
3342:
3341:
3327:
3326:
3309:
3308:
3303:
3302:
3294:
3287:
3286:
3263:
3170:
3168:
3167:
3162:
3157:
3155:
3154:
3153:
3144:
3143:
3122:
3121:
3099:
3098:
3097:
3088:
3087:
3077:
3075:
3073:
3072:
3071:
3052:
3048:
3047:
3037:
3032:
3031:
3022:
3021:
3012:
3011:
3002:
3001:
2992:
2991:
2982:
2981:
2962:
2956:
2943:
2937:
2935:
2934:
2929:
2914:
2902:
2893:
2891:
2890:
2885:
2880:
2878:
2868:
2867:
2857:
2856:
2855:
2836:
2834:
2833:
2824:
2822:
2812:
2811:
2799:
2798:
2776:
2766:
2765:
2744:
2743:
2727:
2725:
2724:
2715:
2713:
2703:
2702:
2690:
2689:
2667:
2657:
2656:
2635:
2634:
2618:
2616:
2615:
2606:
2604:
2603:
2602:
2583:
2579:
2578:
2568:
2563:
2562:
2534:
2528:
2522:
2512:
2506:
2480:
2478:
2477:
2472:
2467:
2465:
2464:
2463:
2451:
2450:
2437:
2425:
2419:with respect to
2414:
2413:
2409:
2404:
2391:
2389:
2388:
2383:
2381:
2380:
2372:
2370:
2366:
2365:
2364:
2352:
2351:
2198:
2196:
2195:
2190:
2188:
2187:
2175:
2174:
2134:Feynman diagrams
2118:Julian Schwinger
2077:When developing
2005:interactions of
1966:
1964:
1963:
1958:
1950:
1949:
1946:
1937:
1918:
1916:
1915:
1910:
1905:
1881:
1879:
1878:
1873:
1868:
1865:
1861:
1860:
1859:
1838:
1836:
1832:
1831:
1828:
1818:
1810:
1808:
1807:
1806:
1797:
1796:
1793:
1787:
1786:
1770:
1769:
1760:
1755:
1754:
1751:
1735:
1733:
1732:
1727:
1725:
1724:
1708:
1704:
1702:
1701:
1696:
1674:
1672:
1671:
1666:
1664:
1663:
1660:
1647:
1634:
1621:
1619:
1618:
1613:
1608:
1606:
1605:
1604:
1601:
1588:
1587:
1578:
1566:
1565:
1550:
1549:
1544:
1540:
1538:
1537:
1536:
1517:
1510:
1502:
1499:
1494:
1493:
1492:
1489:
1468:
1467:
1458:
1450:
1442:
1441:
1438:
1422:
1288:For example, an
1244:
1237:
1230:
315:Effective action
242:Lorentz symmetry
167:Electromagnetism
137:
123:
113:
106:
99:
21:
7375:
7374:
7370:
7369:
7368:
7366:
7365:
7364:
7340:
7339:
7334:Renormalization
7324:
7283:Connes, Alain;
7252:Shirkov, Dmitry
7248:
7191:Wayback Machine
7108:
6921:
6904:I.O.P Magazines
6895:Shirkov, Dmitry
6865:
6800:
6783:
6778:
6776:Further reading
6773:
6745:Physics Reports
6738:
6737:
6733:
6702:
6697:
6696:
6691:
6687:
6639:
6638:
6634:
6623:
6619:
6608:
6604:
6566:
6565:
6561:
6542:
6538:
6529:
6525:
6520:
6516:
6470:
6469:
6465:
6430:Physical Review
6427:
6426:
6422:
6378:
6377:
6373:
6332:
6331:
6324:
6317:
6297:
6296:
6292:
6256:
6255:
6251:
6216:Physical Review
6209:
6208:
6204:
6166:
6165:
6161:
6117:
6116:
6112:
6076:
6075:
6071:
6027:
6026:
6022:
5978:
5977:
5973:
5929:
5928:
5924:
5896:Physical Review
5890:
5885:
5884:
5880:
5852:Physical Review
5846:
5841:
5840:
5836:
5802:
5797:
5796:
5792:
5764:Physical Review
5760:
5759:
5755:
5727:Physical Review
5723:
5722:
5718:
5690:Physical Review
5686:
5685:
5681:
5651:Physical Review
5643:
5642:
5638:
5602:Physical Review
5595:
5594:
5590:
5572:nonrelativistic
5557:
5553:
5541:
5540:
5536:
5509:Physics-Uspekhi
5502:
5501:
5497:
5492:
5488:
5484:
5457:
5418:
5373:
5354:
5346:
5345:
5315:
5310:
5309:
5286:
5285:
5264:
5259:
5258:
5228:
5223:
5222:
5189:
5170:
5162:
5161:
5122:
5114:
5113:
5089:
5081:
5080:
5061:
5060:
5057:
5030:Leo P. Kadanoff
5022:
5017:
5005:On-shell scheme
4990:using a set of
4980:
4969:
4963:
4960:
4949:
4936:
4932:
4925:
4876:However, in an
4859:quantum gravity
4834:
4628:
4540:
4537:
4522:
4519:
4514:
4513:
4511:
4506:
4497:
4492:
4489:
4486:
4481:
4480:
4478:
4477:
4450:
4440:
4396:
4375:
4333:
4332:
4326:
4322:
4319:
4315:
4172:
4171:
4126:
4112:insight due to
4103:
4097:
4091:
4085:
4082:
4076:
4065:magnetic moment
4060:
4033:
4023:
3989:
3970:
3960:
3925:
3920:
3919:
3910:
3904:
3901:
3895:
3840:
3839:
3795:
3782:
3771:
3744:
3731:
3730:
3726:
3725:
3720:
3719:
3689:
3670:
3669:
3665:
3643:
3611:
3592:
3591:
3587:
3574:
3570:
3569:
3564:
3563:
3523:
3490:
3486:
3485:
3480:
3479:
3472:
3420:
3397:
3382:
3349:
3333:
3332:
3328:
3318:
3314:
3310:
3291:
3276:
3275:
3262:
3256:
3249:
3191:electric charge
3183:
3145:
3135:
3113:
3100:
3089:
3079:
3078:
3063:
3053:
3039:
3038:
3023:
3013:
3003:
2993:
2983:
2973:
2968:
2967:
2958:
2952:
2939:
2917:
2916:
2912:
2898:
2859:
2858:
2844:
2837:
2825:
2803:
2790:
2777:
2757:
2735:
2728:
2716:
2694:
2681:
2668:
2648:
2626:
2619:
2607:
2594:
2584:
2570:
2569:
2554:
2543:
2542:
2530:
2524:
2514:
2508:
2502:
2490:
2455:
2442:
2441:
2431:
2430:
2420:
2411:
2407:
2406:
2396:
2356:
2343:
2342:
2338:
2337:
2332:
2331:
2307:vertex function
2281:penguin diagram
2179:
2166:
2161:
2160:
2122:Richard Feynman
2049:
2003:electromagnetic
1977:renormalization
1967:is the reduced
1941:
1925:
1924:
1887:
1886:
1848:
1823:
1822:
1798:
1788:
1778:
1771:
1761:
1746:
1741:
1740:
1716:
1711:
1710:
1706:
1681:
1680:
1655:
1650:
1649:
1646:
1640:
1632:
1626:
1596:
1589:
1579:
1557:
1528:
1521:
1512:
1511:
1484:
1459:
1433:
1428:
1427:
1421:
1415:
1405:point particles
1388:
1346:self-consistent
1326:continuum limit
1255:Renormalization
1248:
1219:
1218:
1217:
1215:
519:
511:
510:
506:Quantum gravity
481:
473:
472:
468:Higgs mechanism
448:
438:
437:
413:Proca equations
398:
390:
389:
385:Wightman Axioms
370:Renormalization
330:Feynman diagram
285:
277:
276:
217:
207:
206:
157:
142:
140:Feynman diagram
117:
88:
87:
83:
79:
75:
71:
67:
63:
58:
48:
47:
43:
41:On-shell scheme
39:
34:
32:Renormalization
17:
12:
11:
5:
7373:
7371:
7363:
7362:
7357:
7352:
7342:
7341:
7338:
7337:
7323:
7322:External links
7320:
7319:
7318:
7281:
7264:
7261:hep-th/9602024
7247:
7244:
7243:
7242:
7224:
7209:
7194:
7162:
7141:
7123:
7107:
7104:
7103:
7102:
7099:Albert Schwarz
7095:
7080:
7067:
7042:
7017:
7013:hep-th/9812203
6998:
6983:
6970:
6957:Zee, Anthony;
6955:
6940:
6920:
6917:
6916:
6915:
6908:
6892:
6863:
6854:
6845:
6816:hep-th/0212049
6809:(2): 170â184.
6798:
6786:DeDeo, Simon;
6782:
6779:
6777:
6774:
6772:
6771:
6752:(5): 241â320.
6731:
6709:
6705:
6685:
6648:(17): 174505.
6632:
6624:Ryder, Lewis.
6617:
6602:
6559:
6551:available here
6536:
6530:Kragh, Helge;
6523:
6514:
6463:
6420:
6391:(1): 189â213.
6371:
6322:
6315:
6290:
6269:(3): 486â502.
6249:
6202:
6159:
6110:
6069:
6020:
5971:
5922:
5878:
5859:(8): 939â946.
5834:
5815:(2): 367â387.
5790:
5771:(6): 790â817.
5753:
5734:(4): 651â679.
5716:
5679:
5658:(4): 416â417.
5636:
5609:(4): 339â341.
5588:
5551:
5534:
5515:(9): 937â956.
5495:
5485:
5483:
5480:
5479:
5478:
5473:
5468:
5463:
5456:
5453:
5417:
5414:
5407:renormalizable
5394:
5389:
5382:
5379:
5372:
5369:
5366:
5361:
5357:
5353:
5331:
5324:
5321:
5293:
5271:
5267:
5244:
5237:
5234:
5210:
5205:
5198:
5195:
5188:
5185:
5182:
5177:
5173:
5169:
5134:
5129:
5125:
5121:
5101:
5096:
5092:
5088:
5068:
5056:
5053:
5041:Kenneth Wilson
5026:renormalizable
5021:
5018:
5016:
5013:
5008:
5007:
5002:
4982:
4981:
4939:
4937:
4930:
4924:
4921:
4851:Standard Model
4833:
4830:
4823:problems. As
4797:Standard Model
4792:
4791:
4790:
4789:
4782:
4781:
4671:
4670:
4650:
4649:
4627:
4624:
4600:Kenneth Wilson
4575:, invented by
4549:regularization
4536:
4535:Regularization
4533:
4495:
4484:
4474:
4473:
4457:
4453:
4447:
4443:
4436:
4433:
4430:
4424:
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4415:
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4409:
4403:
4399:
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4364:
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4340:
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4317:
4309:
4308:
4297:
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4285:
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4279:
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4270:
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4264:
4261:
4258:
4255:
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4249:
4246:
4240:
4237:
4230:
4225:
4221:
4217:
4214:
4211:
4205:
4202:
4195:
4190:
4186:
4182:
4179:
4140:beta-functions
4125:
4122:
4114:Kenneth Wilson
4101:
4089:
4080:
4069:renormalizable
4057:
4056:
4045:
4040:
4036:
4030:
4026:
4019:
4016:
4010:
4007:
4004:
4001:
3996:
3992:
3988:
3985:
3982:
3977:
3973:
3967:
3963:
3956:
3953:
3947:
3944:
3941:
3936:
3930:
3908:
3899:
3892:
3891:
3880:
3877:
3874:
3871:
3868:
3865:
3862:
3859:
3853:
3850:
3822:
3821:
3810:
3805:
3802:
3798:
3792:
3789:
3785:
3778:
3774:
3770:
3765:
3760:
3754:
3751:
3747:
3741:
3738:
3734:
3729:
3717:
3706:
3702:
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3692:
3688:
3685:
3682:
3677:
3673:
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3642:
3637:
3632:
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3610:
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3599:
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3580:
3573:
3561:
3550:
3547:
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3526:
3522:
3517:
3512:
3508:
3505:
3499:
3496:
3489:
3465:
3464:
3451:
3448:
3443:
3439:
3433:
3430:
3427:
3423:
3417:
3414:
3409:
3404:
3400:
3395:
3389:
3385:
3381:
3377:
3371:
3366:
3362:
3356:
3352:
3348:
3345:
3340:
3336:
3331:
3325:
3321:
3317:
3313:
3307:
3300:
3297:
3290:
3285:
3260:
3248:
3245:
3231:or simply the
3182:
3179:
3172:
3171:
3160:
3152:
3148:
3142:
3138:
3134:
3131:
3128:
3125:
3120:
3116:
3112:
3109:
3106:
3103:
3096:
3092:
3086:
3082:
3070:
3066:
3062:
3059:
3056:
3051:
3046:
3042:
3035:
3030:
3026:
3020:
3016:
3010:
3006:
3000:
2996:
2990:
2986:
2980:
2976:
2927:
2924:
2909:Dirac equation
2905:gamma matrices
2895:
2894:
2883:
2877:
2874:
2871:
2866:
2862:
2854:
2851:
2847:
2843:
2840:
2832:
2828:
2821:
2818:
2815:
2810:
2806:
2802:
2797:
2793:
2789:
2786:
2783:
2780:
2775:
2772:
2769:
2764:
2760:
2756:
2753:
2750:
2747:
2742:
2738:
2734:
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2723:
2719:
2712:
2709:
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2701:
2697:
2693:
2688:
2684:
2680:
2677:
2674:
2671:
2666:
2663:
2660:
2655:
2651:
2647:
2644:
2641:
2638:
2633:
2629:
2625:
2622:
2614:
2610:
2601:
2597:
2593:
2590:
2587:
2582:
2577:
2573:
2566:
2561:
2557:
2553:
2550:
2489:
2486:
2482:
2481:
2470:
2462:
2458:
2454:
2449:
2445:
2440:
2393:
2392:
2378:
2375:
2369:
2363:
2359:
2355:
2350:
2346:
2341:
2325:bremsstrahlung
2320:bremsstrahlung
2299:
2298:
2295:
2292:
2285:
2284:
2271:
2264:
2246:
2245:
2242:
2241:for the field,
2227:
2186:
2182:
2178:
2173:
2169:
2091:Pascual Jordan
2081:in the 1930s,
2048:
2045:
2040:Zitterbewegung
2022:inverse-square
1989:regularization
1956:
1953:
1944:
1940:
1936:
1932:
1908:
1904:
1900:
1897:
1894:
1883:
1882:
1871:
1858:
1855:
1851:
1847:
1844:
1841:
1835:
1826:
1821:
1816:
1813:
1805:
1801:
1791:
1785:
1781:
1777:
1774:
1768:
1764:
1758:
1749:
1723:
1719:
1694:
1691:
1688:
1658:
1644:
1630:
1623:
1622:
1611:
1599:
1595:
1592:
1586:
1582:
1576:
1573:
1570:
1564:
1560:
1556:
1553:
1548:
1543:
1535:
1531:
1527:
1524:
1520:
1515:
1508:
1505:
1498:
1487:
1482:
1478:
1475:
1472:
1466:
1462:
1456:
1453:
1448:
1445:
1436:
1419:
1387:
1384:
1380:regularization
1373:Kenneth Wilson
1250:
1249:
1247:
1246:
1239:
1232:
1224:
1221:
1220:
1213:
1212:
1207:
1202:
1197:
1192:
1187:
1182:
1177:
1172:
1167:
1162:
1157:
1152:
1147:
1142:
1137:
1132:
1127:
1122:
1117:
1112:
1107:
1102:
1097:
1092:
1087:
1082:
1077:
1072:
1067:
1062:
1057:
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1047:
1042:
1037:
1032:
1027:
1022:
1017:
1012:
1007:
1002:
997:
992:
987:
982:
977:
972:
967:
962:
957:
952:
947:
942:
937:
932:
927:
922:
917:
912:
907:
902:
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892:
887:
882:
877:
872:
867:
862:
857:
852:
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
617:
612:
607:
602:
597:
592:
587:
582:
577:
572:
567:
562:
557:
552:
547:
542:
537:
532:
527:
521:
520:
517:
516:
513:
512:
509:
508:
503:
498:
493:
488:
482:
479:
478:
475:
474:
471:
470:
465:
460:
455:
449:
446:Standard Model
444:
443:
440:
439:
436:
435:
430:
425:
420:
415:
410:
405:
403:Dirac equation
399:
396:
395:
392:
391:
388:
387:
382:
380:Wick's theorem
377:
372:
367:
365:Regularization
362:
357:
352:
347:
342:
337:
332:
327:
322:
317:
312:
307:
302:
297:
292:
286:
283:
282:
279:
278:
275:
274:
269:
267:Noether charge
264:
259:
254:
252:Gauge symmetry
249:
244:
239:
234:
229:
224:
218:
213:
212:
209:
208:
205:
204:
199:
194:
189:
184:
179:
174:
169:
164:
158:
155:
154:
151:
150:
144:
143:
138:
130:
129:
119:
118:
116:
115:
108:
101:
93:
90:
89:
59:
56:Regularization
54:
53:
50:
49:
35:
30:
29:
26:
25:
15:
13:
10:
9:
6:
4:
3:
2:
7372:
7361:
7358:
7356:
7353:
7351:
7348:
7347:
7345:
7335:
7330:
7326:
7325:
7321:
7316:
7312:
7308:
7304:
7303:
7298:
7297:3-7643-0579-7
7294:
7290:
7286:
7282:
7279:
7278:
7273:
7269:
7265:
7263:
7262:
7257:
7253:
7250:
7249:
7246:Miscellaneous
7245:
7240:
7239:0-521-49959-3
7236:
7232:
7228:
7225:
7222:
7221:0-387-97933-6
7218:
7214:
7210:
7207:
7206:0-7484-0435-X
7203:
7199:
7196:Domb, Cyril;
7195:
7192:
7188:
7185:
7184:
7179:
7178:3-7643-0579-7
7175:
7171:
7167:
7163:
7160:
7159:
7154:
7153:0-19-850923-5
7150:
7146:
7142:
7139:
7138:0-201-55409-7
7135:
7131:
7127:
7124:
7122:
7118:
7114:
7110:
7109:
7105:
7100:
7097:A. S. Ć varc (
7096:
7093:
7092:3-540-60142-2
7089:
7085:
7081:
7078:
7077:
7072:
7068:
7065:
7061:
7060:
7055:
7054:0-691-08530-7
7051:
7047:
7043:
7040:
7039:
7034:
7033:3-7643-0579-7
7030:
7026:
7022:
7018:
7015:
7014:
7009:
7008:
7003:
6999:
6996:
6995:0-521-47816-2
6992:
6988:
6984:
6981:
6980:
6975:
6971:
6968:
6967:0-691-01019-6
6964:
6960:
6956:
6953:
6952:0-521-33859-X
6949:
6945:
6941:
6938:
6934:
6930:
6929:D. V. Shirkov
6926:
6923:
6922:
6918:
6913:
6910:E. Elizalde;
6909:
6906:
6905:
6900:
6896:
6893:
6889:
6885:
6881:
6877:
6873:
6869:
6864:
6861:
6860:
6855:
6852:
6851:
6846:
6842:
6838:
6834:
6830:
6826:
6822:
6817:
6812:
6808:
6804:
6799:
6795:
6791:
6790:
6785:
6784:
6780:
6775:
6767:
6763:
6759:
6755:
6751:
6747:
6746:
6741:
6735:
6732:
6728:
6725:
6707:
6703:
6694:
6693:L.P. Kadanoff
6689:
6686:
6681:
6677:
6673:
6669:
6665:
6661:
6656:
6651:
6647:
6643:
6636:
6633:
6629:
6628:
6621:
6618:
6614:
6613:
6606:
6603:
6598:
6594:
6590:
6586:
6582:
6578:
6574:
6570:
6563:
6560:
6556:
6552:
6548:
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6533:
6527:
6524:
6518:
6515:
6510:
6506:
6502:
6498:
6494:
6490:
6486:
6482:
6478:
6474:
6467:
6464:
6459:
6455:
6451:
6447:
6443:
6439:
6435:
6431:
6424:
6421:
6415:
6410:
6406:
6402:
6398:
6394:
6390:
6386:
6382:
6375:
6372:
6367:
6363:
6359:
6355:
6351:
6347:
6343:
6339:
6335:
6329:
6327:
6323:
6318:
6316:9780201503975
6312:
6307:
6306:
6300:
6294:
6291:
6285:
6280:
6276:
6272:
6268:
6264:
6260:
6253:
6250:
6245:
6241:
6237:
6233:
6229:
6225:
6221:
6217:
6213:
6206:
6203:
6198:
6194:
6190:
6186:
6182:
6178:
6174:
6170:
6163:
6160:
6155:
6151:
6146:
6141:
6137:
6133:
6129:
6125:
6121:
6114:
6111:
6106:
6102:
6097:
6092:
6088:
6084:
6080:
6073:
6070:
6065:
6061:
6056:
6051:
6047:
6043:
6039:
6035:
6031:
6024:
6021:
6016:
6012:
6007:
6002:
5998:
5994:
5990:
5986:
5982:
5975:
5972:
5967:
5963:
5958:
5953:
5949:
5945:
5941:
5937:
5933:
5926:
5923:
5918:
5914:
5910:
5906:
5902:
5898:
5897:
5889:
5882:
5879:
5874:
5870:
5866:
5862:
5858:
5854:
5853:
5845:
5838:
5835:
5830:
5826:
5822:
5818:
5814:
5810:
5809:
5801:
5794:
5791:
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5576:Jagdish Mehra
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5426:beta function
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5387:
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5045:Kondo problem
5042:
5037:
5035:
5034:blocking idea
5031:
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5019:
5014:
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5006:
5003:
5000:
4997:
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4989:
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4957:
4956:the talk page
4953:
4947:
4945:
4940:This section
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4745:string theory
4742:
4741:Planck length
4738:
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4727:
4723:
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4707:
4703:
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4635:Freeman Dyson
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4608:scaling limit
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3241:canceling out
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2239:path integral
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2130:Freeman Dyson
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2012:back-reaction
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1340:integrals in
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1319:
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1279:loop diagrams
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531:
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492:
491:Supersymmetry
489:
487:
486:String theory
484:
483:
477:
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454:
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66:
62:
57:
52:
51:
46:
42:
38:
33:
28:
27:
22:
19:
7336:at Wikiquote
7315:math/0211199
7307:Hopf algebra
7302:Alain Connes
7301:
7288:
7284:
7276:
7271:
7267:
7260:
7255:
7230:
7212:
7197:
7182:
7169:
7165:
7157:
7144:
7129:
7112:
7083:
7075:
7070:
7058:
7045:
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7024:
7020:
7012:
7006:
7001:
6986:
6978:
6973:
6958:
6943:
6932:
6911:
6903:
6898:
6871:
6867:
6858:
6849:
6847:Baez, John;
6806:
6802:
6788:
6749:
6743:
6734:
6726:
6723:
6688:
6645:
6642:Phys. Rev. B
6641:
6635:
6625:
6620:
6610:
6605:
6572:
6568:
6562:
6555:Google Books
6544:
6539:
6531:
6526:
6517:
6476:
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6388:
6384:
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6252:
6219:
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6205:
6172:
6168:
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6123:
6113:
6086:
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6072:
6037:
6033:
6023:
5988:
5984:
5974:
5939:
5935:
5925:
5900:
5894:
5881:
5856:
5850:
5837:
5812:
5806:
5793:
5768:
5762:
5756:
5731:
5725:
5719:
5694:
5688:
5682:
5655:
5649:
5639:
5606:
5600:
5591:
5583:
5567:
5554:
5544:
5537:
5512:
5508:
5498:
5489:
5442:
5438:universality
5435:
5422:fixed points
5421:
5419:
5411:
5406:
5305:
5159:
5058:
5038:
5033:
5025:
5023:
5009:
4991:
4987:
4985:
4970:
4964:January 2022
4961:
4950:Please help
4941:
4900:
4889:Fermi theory
4886:
4875:
4848:
4843:
4835:
4821:
4793:
4777:
4762:
4756:
4734:
4721:
4695:
4672:
4665:
4661:
4651:
4639:
4633:
4629:
4619:distribution
4612:
4604:extrapolated
4593:
4591:
4584:
4570:
4568:
4564:
4555:
4553:
4538:
4523:
4515:
4507:
4493:
4482:
4476:Making sure
4475:
4313:
4310:
4166:
4152:
4148:
4130:
4127:
4107:
4098:
4095:
4086:
4077:
4072:
4068:
4058:
3914:
3905:
3896:
3893:
3823:
3468:
3466:
3267:
3257:
3240:
3237:counterterms
3233:energy scale
3228:
3224:
3220:
3214:renormalized
3213:
3211:
3206:
3202:
3198:
3184:
3176:
3173:
2959:
2953:
2950:
2940:
2899:
2897:The various
2896:
2537:
2531:
2525:
2519:
2515:
2509:
2503:
2499:
2483:
2422:
2401:
2397:
2394:
2310:
2300:
2286:
2274:
2250:
2247:
2215:
2213:
2208:
2146:
2137:
2136:with closed
2110:Hans Kramers
2103:
2076:
2050:
2037:
2032:
2026:
2010:
2000:
1993:
1976:
1973:
1884:
1641:
1627:
1624:
1416:
1409:
1398:
1377:
1367:insights of
1362:
1331:
1322:well-defined
1314:
1287:
1274:
1267:self-similar
1254:
1253:
1214:
1060:Stueckelberg
800:Jona-Lasinio
375:Vacuum state
369:
360:Quantization
197:Gauge theory
177:Strong force
162:Field theory
31:
18:
7227:Cardy, John
5155:Hamiltonian
4825:Lewis Ryder
4710:statistical
4675:Landau pole
4131:independent
2946:propagators
2268:self-energy
2235:frequencies
2231:wavelengths
2229:very short
2220:ultraviolet
1648:that makes
1354:mathematics
1296:a cloud of
1200:Zinn-Justin
1050:Sommerfield
975:Pomeranchuk
945:Osterwalder
940:Oppenheimer
870:ĆopuszaĆski
695:Fredenhagen
496:Technicolor
7344:Categories
7277:PostScript
7183:PostScript
7059:PostScript
7038:PostScript
6874:: 33â108.
6655:1909.12553
5482:References
5055:Principles
4946:to readers
4897:W particle
3187:Lagrangian
2114:Hans Bethe
2095:Paul Dirac
1994:(See also
1358:Paul Dirac
1300:, such as
1283:Lagrangian
1271:infinities
1195:Zimmermann
1090:Vainshtein
835:Kontsevich
780:Iliopoulos
755:Heisenberg
580:Bogoliubov
518:Scientists
355:Propagator
237:T-symmetry
232:P-symmetry
227:C-symmetry
215:Symmetries
172:Weak force
156:Background
6680:203591796
6597:0556-2821
6501:0036-8075
6458:0031-899X
6414:1874/4845
6366:0034-6861
6263:Phys. Rev
6244:0031-899X
6197:0033-068X
6154:0033-068X
6105:0033-068X
6064:0033-068X
6015:0033-068X
5966:1347-4081
5631:120434909
5529:1063-7869
5381:~
5368:→
5323:~
5304:function
5236:~
5197:~
5184:→
4839:dimension
4556:regulator
4452:ε
4442:ε
4435:
4429:−
4414:
4398:ε
4393:
4377:ε
4372:
4366:−
4360:
4354:−
4348:
4293:
4281:
4275:−
4269:
4263:−
4257:
4220:∫
4216:−
4185:∫
4073:divergent
4044:ψ
4039:μ
4029:μ
4025:γ
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4015:ψ
4000:−
3984:−
3981:ψ
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3966:μ
3962:γ
3955:¯
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3943:−
3879:ψ
3861:∂
3852:¯
3849:ψ
3804:ν
3801:μ
3791:ν
3788:μ
3753:ν
3750:μ
3740:ν
3737:μ
3705:ψ
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3672:∂
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3296:ψ
3218:kinematic
3130:−
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3009:ρ
3005:γ
2999:α
2995:γ
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2926:ϵ
2876:ϵ
2853:ν
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2827:γ
2820:ϵ
2801:−
2785:−
2763:β
2752:−
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2737:γ
2722:ρ
2718:γ
2711:ϵ
2692:−
2676:−
2654:α
2643:−
2632:α
2628:γ
2613:μ
2609:γ
2592:π
2565:∫
2549:−
2453:−
2354:−
2233:and high-
2216:divergent
2205:integrate
2201:off-shell
2177:−
2144:in them.
1931:ℏ
1896:≈
1893:α
1854:−
1846:×
1840:≈
1820:ℏ
1815:α
1780:ε
1776:π
1718:ε
1594:π
1555:π
1526:π
1497:∞
1481:∫
1447:∫
1318:spacetime
1306:positrons
1130:Wetterich
1115:Weisskopf
1065:Sudarshan
1015:Schwinger
930:Nishijima
895:Maldacena
860:Leutwyler
825:Kinoshita
725:Goldstone
715:Gell-Mann
630:Doplicher
397:Equations
7187:Archived
6931:(1959):
6888:46968305
6868:Synthese
6792:(2017).
6553:through
6509:17801299
5597:H. Bethe
5455:See also
4882:misnomer
4760:useful.
4755:either;
4722:physical
4683:Politzer
3827:, via a
2316:on-shell
2311:infrared
2263:diagram.
2257:positron
2153:momentum
2083:Max Born
2017:back-EMF
1395:another.
1338:infinite
1290:electron
1145:Wightman
1110:Weinberg
1100:Virasoro
1080:Tomonaga
1075:Thirring
1070:Symanzik
1030:Semenoff
1005:Schrader
970:Polyakov
890:Majorana
830:Klebanov
785:Ivanenko
775:'t Hooft
745:Guralnik
690:Fröhlich
685:Fritzsch
680:Frampton
595:Buchholz
540:Bargmann
530:Anderson
310:Crossing
7062:and in
6939:theory.
6841:2506712
6821:Bibcode
6797:theory.
6754:Bibcode
6660:Bibcode
6577:Bibcode
6481:Bibcode
6473:Science
6438:Bibcode
6393:Bibcode
6346:Bibcode
6271:Bibcode
6224:Bibcode
6177:Bibcode
6132:Bibcode
6042:Bibcode
5993:Bibcode
5944:Bibcode
5905:Bibcode
5861:Bibcode
5817:Bibcode
5773:Bibcode
5736:Bibcode
5699:Bibcode
5660:Bibcode
5611:Bibcode
5020:History
4942:may be
4917:planets
4909:gravity
4891:of the
4774:Russell
4687:Wilczek
4654:Feynman
4528:
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4505:, then
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4153:running
2410:⁄
2007:charged
1985:Abraham
1981:Lorentz
1919:is the
1637:inertia
1350:physics
1310:protons
1302:photons
1155:Wilczek
1120:Wentzel
1095:Veltman
1040:Shirkov
1035:Shifman
1025:Seiberg
1010:Schwarz
990:Rubakov
915:Naimark
865:Lipatov
855:Lehmann
820:Kendall
710:Gelfand
705:Glashow
665:Feynman
645:Faddeev
640:Englert
610:Coleman
600:Cachazo
585:Brodsky
570:Bjorken
560:Berezin
550:Belavin
290:Anomaly
148:History
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2093:, and
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1979:, and
1923:, and
1885:where
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1210:Zumino
1175:Yukawa
1165:Witten
1160:Wilson
1150:Wigner
1085:Tyutin
1045:Skyrme
995:Ruelle
965:Plefka
960:Peskin
950:Parisi
910:MĂžller
900:Migdal
885:Maiani
880:LĂŒders
845:Landau
840:Kuraev
815:KÀllén
805:Jordan
790:Jackiw
730:Gribov
620:DeWitt
615:Dashen
605:Callan
575:Bleuer
545:Becchi
535:Anselm
7311:arXiv
6884:S2CID
6837:S2CID
6811:arXiv
6676:S2CID
6650:arXiv
5891:(PDF)
5847:(PDF)
5803:(PDF)
5627:S2CID
5574:(see
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4541:â â â
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1000:Salam
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980:Popov
955:Pauli
935:Oehme
925:Neveu
920:Nambu
905:Mills
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770:Hagen
765:Higgs
740:Gupta
735:Gross
720:Glimm
700:Furry
670:Fierz
660:Fermi
655:Fayet
650:Fadin
635:Dyson
625:Dirac
590:Brout
565:Bethe
525:Adler
284:Tools
7293:ISBN
7235:ISBN
7217:ISBN
7202:ISBN
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6963:ISBN
6948:ISBN
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2415:-th
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2027:The
1983:and
1709:and
1371:and
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1170:Yang
1140:Wick
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1125:Wess
1105:Ward
810:Jost
760:Hepp
750:Haag
675:Fock
555:Bell
6876:doi
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