更改

{{Use American English|date=January 2019}}

{{Use American English|date=January 2019}}

'''Renormalization''' is a collection of techniques in [[quantum field theory]], the [[statistical mechanics]] of fields, and the theory of [[self-similarity|self-similar]] geometric structures, that are used to treat [[infinity|infinities]] arising in calculated quantities by altering values of these quantities to compensate for effects of their '''self-interactions'''<!--boldface per WP:R#PLA; 'Self-interaction' and 'Self-interactions' redirect here-->. But even if no infinities arose in [[One-loop Feynman diagram|loop diagrams]] in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original [[Lagrangian (field theory)|Lagrangian]].<ref>See e.g., Weinberg vol I, chapter 10.</ref>

'''Renormalization''' is a collection of techniques in [[quantum field theory]], the [[statistical mechanics]] of fields, and the theory of [[self-similarity|self-similar]] geometric structures, that are used to treat [[infinity|infinities]] arising in calculated quantities by altering values of these quantities to compensate for effects of their '''self-interactions'''<!--boldface per WP:R#PLA; 'Self-interaction' and 'Self-interactions' redirect here-->. But even if no infinities arose in [[One-loop Feynman diagram|loop diagrams]] in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original [[Lagrangian (field theory)|Lagrangian]].<ref>See e.g., Weinberg vol I, chapter 10.</ref>

Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions<!--boldface per WP:R#PLA; 'Self-interaction' and 'Self-interactions' redirect here-->. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian. in the extensive important contributions of Kenneth Wilson. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the Kondo problem, in 1974,  as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and critical phenomena in 1971. He was awarded the Nobel prize for these decisive contributions in 1982.

Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions<!--boldface per WP:R#PLA; 'Self-interaction' and 'Self-interactions' redirect here-->. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian.

For example, an [[electron]] theory may begin by postulating an electron with an initial mass and charge. In [[quantum field theory]] a cloud of [[virtual particle]]s, such as [[photon]]s, [[positron]]s, and others surrounds and interacts with the initial electron. Accounting for the interactions of the surrounding particles (e.g. collisions at different energies) shows that the electron-system behaves as if it had a different mass and charge than initially postulated. Renormalization, in this example, mathematically replaces the initially postulated mass and charge of an electron with the experimentally observed mass and charge. Mathematics and experiments prove that positrons and more massive particles like [[proton]]s, exhibit ''precisely the same'' observed charge as the electron - even in the presence of much stronger interactions and more intense clouds of virtual particles.

For example, an [[electron]] theory may begin by postulating an electron with an initial mass and charge. In [[quantum field theory]] a cloud of [[virtual particle]]s, such as [[photon]]s, [[positron]]s, and others surrounds and interacts with the initial electron. Accounting for the interactions of the surrounding particles (e.g. collisions at different energies) shows that the electron-system behaves as if it had a different mass and charge than initially postulated. Renormalization, in this example, mathematically replaces the initially postulated mass and charge of an electron with the experimentally observed mass and charge. Mathematics and experiments prove that positrons and more massive particles like [[proton]]s, exhibit ''precisely the same'' observed charge as the electron - even in the presence of much stronger interactions and more intense clouds of virtual particles.

Renormalization specifies relationships between parameters in the theory when parameters describing large distance scales differ from parameters describing small distance scales. In high-energy particle accelerators like the [[Large Hadron Collider|CERN Large Hadron Collider]] the concept named [[Pileup (disambiguation)|pileup]] occurs when undesirable proton-proton collisions interact with data collection for simultaneous, nearby desirable measurements. Physically, the pileup of contributions from an infinity of scales involved in a problem may then result in further infinities. When describing space-time as a [[Space-time continuum|continuum]], certain statistical and quantum mechanical constructions are not [[well-defined]]. To define them, or make them unambiguous, a continuum limit must carefully remove "construction scaffolding" of lattices at various scales. Renormalization procedures are based on the requirement that certain physical quantities (such as the mass and charge of an electron) equal observed (experimental) values. That is, the experimental value of the physical quantity yields practical applications, but due to their empirical nature the observed measurement represents areas of quantum field theory that require deeper derivation from theoretical bases.

Renormalization specifies relationships between parameters in the theory when parameters describing large distance scales differ from parameters describing small distance scales. In high-energy particle accelerators like the [[Large Hadron Collider|CERN Large Hadron Collider]] the concept named [[Pileup (disambiguation)|pileup]] occurs when undesirable proton-proton collisions interact with data collection for simultaneous, nearby desirable measurements. Physically, the pileup of contributions from an infinity of scales involved in a problem may then result in further infinities. When describing space-time as a [[Space-time continuum|continuum]], certain statistical and quantum mechanical constructions are not [[well-defined]]. To define them, or make them unambiguous, a continuum limit must carefully remove "construction scaffolding" of lattices at various scales. Renormalization procedures are based on the requirement that certain physical quantities (such as the mass and charge of an electron) equal observed (experimental) values. That is, the experimental value of the physical quantity yields practical applications, but due to their empirical nature the observed measurement represents areas of quantum field theory that require deeper derivation from theoretical bases.

by a certain function <math>Z</math> of the state variables

Renormalization specifies relationships between parameters in the theory when parameters describing large distance scales differ from parameters describing small distance scales. In high-energy particle accelerators like the CERN Large Hadron Collider the concept named pileup occurs when undesirable proton-proton collisions interact with data collection for simultaneous, nearby desirable measurements. Physically, the pileup of contributions from an infinity of scales involved in a problem may then result in further infinities. When describing space-time as a continuum, certain statistical and quantum mechanical constructions are not well-defined. To define them, or make them unambiguous, a continuum limit must carefully remove "construction scaffolding" of lattices at various scales. Renormalization procedures are based on the requirement that certain physical quantities (such as the mass and charge of an electron) equal observed (experimental) values. That is, the experimental value of the physical quantity yields practical applications, but due to their empirical nature the observed measurement represents areas of quantum field theory that require deeper derivation from theoretical bases.

Renormalization was first developed in [[quantum electrodynamics]] (QED) to make sense of [[infinity|infinite]] integrals in [[perturbation theory (quantum mechanics)|perturbation theory]]. Initially viewed as a suspect provisional procedure even by some of its originators, renormalization eventually was embraced as an important and [[self-consistent]] actual mechanism of scale physics in several fields of [[physics]] and [[mathematics]].

Renormalization was first developed in [[quantum electrodynamics]] (QED) to make sense of [[infinity|infinite]] integrals in [[perturbation theory (quantum mechanics)|perturbation theory]]. Initially viewed as a suspect provisional procedure even by some of its originators, renormalization eventually was embraced as an important and [[self-consistent]] actual mechanism of scale physics in several fields of [[physics]] and [[mathematics]].

<math>\{J_k\}</math>. This function may be a partition function,

Renormalization was first developed in quantum electrodynamics (QED) to make sense of infinite integrals in perturbation theory. Initially viewed as a suspect provisional procedure even by some of its originators, renormalization eventually was embraced as an important and self-consistent actual mechanism of scale physics in several fields of physics and mathematics.

Today, the point of view has shifted: on the basis of the breakthrough [[renormalization group]] insights of [[Nikolay Bogolyubov]] and [[Kenneth G. Wilson|Kenneth Wilson]], the focus is on variation of physical quantities across contiguous scales, while distant scales are related to each other through "effective" descriptions. ''All scales'' are linked in a broadly systematic way, and the actual physics pertinent to each is extracted with the suitable specific computational techniques appropriate for each. Wilson clarified which variables of a system are crucial and which are redundant.

Today, the point of view has shifted: on the basis of the breakthrough [[renormalization group]] insights of [[Nikolay Bogolyubov]] and [[Kenneth G. Wilson|Kenneth Wilson]], the focus is on variation of physical quantities across contiguous scales, while distant scales are related to each other through "effective" descriptions. ''All scales'' are linked in a broadly systematic way, and the actual physics pertinent to each is extracted with the suitable specific computational techniques appropriate for each. Wilson clarified which variables of a system are crucial and which are redundant.

whole description of the physics of the system.

Today, the point of view has shifted: on the basis of the breakthrough renormalization group insights of Nikolay Bogolyubov and Kenneth Wilson, the focus is on variation of physical quantities across contiguous scales, while distant scales are related to each other through "effective" descriptions. All scales are linked in a broadly systematic way, and the actual physics pertinent to each is extracted with the suitable specific computational techniques appropriate for each. Wilson clarified which variables of a system are crucial and which are redundant.

Renormalization is distinct from [[Regularization (physics)|regularization]], another technique to control infinities by assuming the existence of new unknown physics at new scales.

Renormalization is distinct from [[Regularization (physics)|regularization]], another technique to control infinities by assuming the existence of new unknown physics at new scales.

Now we consider a certain blocking transformation of the state

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

 

== Self-interactions in classical physics ==

== Self-interactions in classical physics ==

[[Image:Renormalized-vertex.png|thumbnail|upright=1.3|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 another.]]

[[Image:Renormalized-vertex.png|thumbnail|upright=1.3|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 another.]]

<math>s_i</math>. Now let us try to rewrite the <math>Z</math>

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 another.

“ math”“ math”。现在让我们尝试重写 < math > z </math >

图1。量子电动力学中的重整化: 确定一个重整化点上电子电荷的简单电子/光子相互作用被揭示为由另一个重整化点上更复杂的相互作用组成。

The problem of infinities first arose in the [[classical electrodynamics]] of [[Elementary particle|point particles]] in the 19th and early 20th century.

The problem of infinities first arose in the [[classical electrodynamics]] of [[Elementary particle|point particles]] in the 19th and early 20th century.

function only in terms of the <math>\tilde s_i</math>. If this is achievable by a

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

 

 

The mass of a charged particle should include the mass-energy in its electrostatic field ([[electromagnetic mass]]). Assume that the particle is a charged spherical shell of radius {{mvar|r_e}}. The mass–energy in the field is

The mass of a charged particle should include the mass-energy in its electrostatic field ([[electromagnetic mass]]). Assume that the particle is a charged spherical shell of radius {{mvar|r_e}}. The mass–energy in the field is

\{\tilde J_k\}</math>, then the theory is said to be

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

:<math>m_\text{em} = \int \frac{1}{2} E^2 \, dV = \int_{r_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_e},</math>

1}{2} e ^ 2，dV = int _ { r _ e } ^ infty frac {1}{2}左(frac { q }{4 pi r ^ 2}右) ^ 24 pi r ^ 2，dr = frac { q ^ 2}{8 pi r _ e } ，</math >  

which becomes infinite as {{math|''r''_e → 0}}. This implies that the point particle would have infinite [[inertia]], making it unable to be accelerated. Incidentally, the value of {{mvar|r_e}} that makes <math>m_\text{em}</math> equal to the electron mass is called the [[classical electron radius]], which (setting <math>q = e</math> and restoring factors of {{mvar|c}} and <math>\varepsilon_0</math>) turns out to be

which becomes infinite as {{math|''r''_e → 0}}. This implies that the point particle would have infinite [[inertia]], making it unable to be accelerated. Incidentally, the value of {{mvar|r_e}} that makes <math>m_\text{em}</math> equal to the electron mass is called the [[classical electron radius]], which (setting <math>q = e</math> and restoring factors of {{mvar|c}} and <math>\varepsilon_0</math>) turns out to be

:<math>r_e = \frac{e^2}{4\pi\varepsilon_0 m_e c^2} = \alpha \frac{\hbar}{m_e c} \approx 2.8 \times 10^{-15}~\text{m},</math>

:<math>r_e = \frac{e^2}{4\pi\varepsilon_0 m_e c^2} = \alpha \frac{\hbar}{m_e c} \approx 2.8 \times 10^{-15}~\text{m},</math>

set of fixed points.

<math>r_e = \frac{e^2}{4\pi\varepsilon_0 m_e c^2} = \alpha \frac{\hbar}{m_e c} \approx 2.8 \times 10^{-15}~\text{m},</math>

where <math>\alpha \approx 1/137</math> is the [[fine-structure constant]], and <math>\hbar/(m_e c)</math> is the [[Compton wavelength]] of the electron.

where <math>\alpha \approx 1/137</math> is the [[fine-structure constant]], and <math>\hbar/(m_e c)</math> is the [[Compton wavelength]] of the electron.

Renormalization: The total effective mass of a spherical charged particle includes the actual bare mass of the spherical shell (in addition to the mass mentioned above associated with its electric field). If the shell's bare mass is allowed to be negative, it might be possible to take a consistent point limit.{{Citation needed|date=March 2015}} This was called ''renormalization'', and [[Hendrik Lorentz|Lorentz]] and [[Max Abraham|Abraham]] attempted to develop a classical theory of the electron this way. This early work was the inspiration for later attempts at [[regularization (physics)|regularization]] and renormalization in quantum field theory.

Renormalization: The total effective mass of a spherical charged particle includes the actual bare mass of the spherical shell (in addition to the mass mentioned above associated with its electric field). If the shell's bare mass is allowed to be negative, it might be possible to take a consistent point limit.{{Citation needed|date=March 2015}} This was called ''renormalization'', and [[Hendrik Lorentz|Lorentz]] and [[Max Abraham|Abraham]] attempted to develop a classical theory of the electron this way. This early work was the inspiration for later attempts at [[regularization (physics)|regularization]] and renormalization in quantum field theory.

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

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

When calculating the [[electromagnetism|electromagnetic]] interactions of [[electric charge|charged]] particles, it is tempting to ignore the ''[[back-reaction]]'' of a particle's own field on itself. (Analogous to the [[back-EMF]] of circuit analysis.)  But this back-reaction is necessary to explain the friction on charged particles when they emit radiation. If the electron is assumed to be a point, the value of the back-reaction diverges, for the same reason that the mass diverges, because the field is [[inverse-square law|inverse-square]].

When calculating the [[electromagnetism|electromagnetic]] interactions of [[electric charge|charged]] particles, it is tempting to ignore the ''[[back-reaction]]'' of a particle's own field on itself. (Analogous to the [[back-EMF]] of circuit analysis.)  But this back-reaction is necessary to explain the friction on charged particles when they emit radiation. If the electron is assumed to be a point, the value of the back-reaction diverges, for the same reason that the mass diverges, because the field is [[inverse-square law|inverse-square]].

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

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

The trouble was worse in classical field theory than in quantum field theory, because in quantum field theory a charged particle experiences [[Zitterbewegung]] due to interference with virtual particle-antiparticle pairs, thus effectively smearing out the charge over a region comparable to the Compton wavelength. In quantum electrodynamics at small coupling, the electromagnetic mass only diverges as the logarithm of the radius of the particle.

The trouble was worse in classical field theory than in quantum field theory, because in quantum field theory a charged particle experiences [[Zitterbewegung]] due to interference with virtual particle-antiparticle pairs, thus effectively smearing out the charge over a region comparable to the Compton wavelength. In quantum electrodynamics at small coupling, the electromagnetic mass only diverges as the logarithm of the radius of the particle.

[[Image:vacuum polarization.svg|thumb|(a) Vacuum polarization, a.k.a. charge screening. This loop has a logarithmic ultraviolet divergence.]]

[[Image:vacuum polarization.svg|thumb|(a) Vacuum polarization, a.k.a. charge screening. This loop has a logarithmic ultraviolet divergence.]]

[[Image:selfE.svg|thumb|(b) Self-energy diagram in QED]]

[[Image:selfE.svg|thumb|(b) Self-energy diagram in QED]]

[[Image:Penguin diagram.JPG|thumb|(c) Example of a “penguin” diagram]]

[[Image:Penguin diagram.JPG|thumb|(c) Example of a “penguin” diagram]]

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

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

One way of describing the [[perturbation theory (quantum mechanics)|perturbation theory]] corrections' divergences was discovered in 1947–49 by<!--in chronological order--> [[Hans Kramers]]<!--June 1947-->,<ref>Kramers presented his work at the 1947 [[Shelter Island Conference]], repeated in 1948 at the [[Solvay Conference]]. The latter did not appear in print until the Proceedings of the Solvay Conference, published in 1950 (see Laurie M. Brown (ed.), ''Renormalization: From Lorentz to Landau (and Beyond)'', Springer, 2012, p. 53). Kramers' approach was [[nonrelativistic]] (see [[Jagdish Mehra]], Helmut Rechenberg, ''The Conceptual Completion and Extensions of Quantum Mechanics 1932-1941. Epilogue: Aspects of the Further Development of Quantum Theory 1942-1999: Volumes 6, Part 2'', Springer, 2001, p. 1050).</ref> [[Hans Bethe]]<!--August 1947-->,<ref>{{cite journal |author=H. Bethe |authorlink=Hans Bethe |year=1947 |title=The Electromagnetic Shift of Energy Levels |journal=[[Physical Review]] |volume=72 |pages=339–341 |doi=10.1103/PhysRev.72.339 |bibcode=1947PhRv...72..339B |issue=4}}</ref>  

One way of describing the [[perturbation theory (quantum mechanics)|perturbation theory]] corrections' divergences was discovered in 1947–49 by<!--in chronological order--> [[Hans Kramers]]<!--June 1947-->,<ref>Kramers presented his work at the 1947 [[Shelter Island Conference]], repeated in 1948 at the [[Solvay Conference]]. The latter did not appear in print until the Proceedings of the Solvay Conference, published in 1950 (see Laurie M. Brown (ed.), ''Renormalization: From Lorentz to Landau (and Beyond)'', Springer, 2012, p. 53). Kramers' approach was [[nonrelativistic]] (see [[Jagdish Mehra]], Helmut Rechenberg, ''The Conceptual Completion and Extensions of Quantum Mechanics 1932-1941. Epilogue: Aspects of the Further Development of Quantum Theory 1942-1999: Volumes 6, Part 2'', Springer, 2001, p. 1050).</ref> [[Hans Bethe]]<!--August 1947-->,<ref>{{cite journal |author=H. Bethe |authorlink=Hans Bethe |year=1947 |title=The Electromagnetic Shift of Energy Levels |journal=[[Physical Review]] |volume=72 |pages=339–341 |doi=10.1103/PhysRev.72.339 |bibcode=1947PhRv...72..339B |issue=4}}</ref>  

[[Julian Schwinger]]<!--February 1948-->,<ref>{{cite journal |author=Schwinger, J. |title=On quantum-electrodynamics and the magnetic moment of the electron |journal=[[Physical Review]] |volume=73 |issue=4 |pages=416–417 |year=1948|doi=10.1103/PhysRev.73.416 |bibcode=1948PhRv...73..416S |doi-access=free }}</ref><ref>{{cite journal |author=Schwinger, J. |series=Quantum Electrodynamics |title=I. A covariant formulation |journal=[[Physical Review]] |volume=74 |issue=10 |pages=1439–1461 |year=1948|doi=10.1103/PhysRev.74.1439 |bibcode=1948PhRv...74.1439S }}</ref><ref>{{cite journal |author=Schwinger, J. |series=Quantum Electrodynamics |title=II. Vacuum polarization and self-energy |journal=[[Physical Review]] |volume=75 |issue=4 |pages=651–679 |year=1949|doi=10.1103/PhysRev.75.651 |bibcode=1949PhRv...75..651S }}</ref><ref>{{cite journal |author=Schwinger, J. |series=Quantum Electrodynamics |title=III. The electromagnetic properties of the electron radiative corrections to scattering |journal=[[Physical Review]] |volume=76 |issue=6 |pages=790–817 |year=1949|doi=10.1103/PhysRev.76.790 |bibcode=1949PhRv...76..790S }}</ref> [[Richard Feynman]]<!--April 1948-->,<ref>{{cite journal |first=Richard P. |last=Feynman |title=Space-time approach to non-relativistic quantum mechanics |journal=[[Reviews of Modern Physics]] |volume=20 |pages=367–387 |year=1948 |doi=10.1103/RevModPhys.20.367 |bibcode=1948RvMP...20..367F |issue=2|url=https://authors.library.caltech.edu/47756/1/FEYrmp48.pdf }}</ref><ref>{{cite journal |last=Feynman  |first= Richard P. |title=A relativistic cut-off for classical electrodynamics  |journal=[[Physical Review]] |volume=74  |issue=8 |pages= 939–946 |year=1948  |doi=10.1103/PhysRev.74.939 |bibcode=1948PhRv...74..939F|url= https://authors.library.caltech.edu/3516/1/FEYpr48a.pdf }}</ref><ref>{{cite journal |first=Richard P. |last=Feynman |title=A relativistic cut-off for quantum electrodynamics |journal=[[Physical Review]] |volume=74 |pages=1430–1438 |year=1948 |doi=10.1103/PhysRev.74.1430 |bibcode=1948PhRv...74.1430F |issue=10|url=https://authors.library.caltech.edu/3517/1/FEYpr48b.pdf }}</ref> and [[Shin'ichiro Tomonaga]]<!--July 1948 (Koba–Tomonaga); according to S. S. Schweber, ''QED'', 1994, p. 269, Koba–Tomonaga contains the crucial calculation-->,<ref>{{cite journal | last=Tomonaga | first=S. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=1 | issue=2 | date=1946-08-01 | issn=1347-4081 | doi=10.1143/ptp.1.27 | pages=27–42|doi-access=free| bibcode=1946PThPh...1...27T }}</ref><ref>{{cite journal | last1=Koba | first1=Z. | last2=Tati | first2=T. | last3=Tomonaga | first3=S.-i. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. II: Case of Interacting Electromagnetic and Electron Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=2 | issue=3 | date=1947-10-01 | issn=0033-068X | doi=10.1143/ptp/2.3.101 | pages=101–116|doi-access=free| bibcode=1947PThPh...2..101K }}</ref><ref>{{cite journal | last1=Koba | first1=Z. | last2=Tati | first2=T. | last3=Tomonaga | first3=S.-i. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. III: Case of Interacting Electromagnetic and Electron Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=2 | issue=4 | date=1947-12-01 | issn=0033-068X | doi=10.1143/ptp/2.4.198 | pages=198–208|doi-access=free| bibcode=1947PThPh...2..198K }}</ref><ref>{{cite journal | last1=Kanesawa | first1=S. | last2=Tomonaga | first2=S.-i. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. [IV]: Case of Interacting Electromagnetic and Meson Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=3 | issue=1 | date=1948-03-01 | issn=0033-068X | doi=10.1143/ptp/3.1.1 | pages=1–13|doi-access=free}}</ref><ref>{{cite journal | last1=Kanesawa | first1=S. | last2=Tomonaga | first2=S.-i. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields V: Case of Interacting Electromagnetic and Meson Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=3 | issue=2 | date=1948-06-01 | issn=0033-068X | doi=10.1143/ptp/3.2.101 | pages=101–113|doi-access=free| bibcode=1948PThPh...3..101K }}</ref><ref>{{cite journal | last1=Koba | first1=Z. | last2=Tomonaga | first2=S.-i. | title=On Radiation Reactions in Collision Processes. I: Application of the "Self-Consistent" Subtraction Method to the Elastic Scattering of an Electron | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=3 | issue=3 | date=1948-09-01 | issn=0033-068X | doi=10.1143/ptp/3.3.290 | pages=290–303|doi-access=free| bibcode=1948PThPh...3..290K }}</ref><ref>{{cite journal | last1=Tomonaga | first1=Sin-Itiro | last2=Oppenheimer | first2=J. R. |author-link2=J. Robert Oppenheimer| title=On Infinite Field Reactions in Quantum Field Theory | journal=Physical Review | publisher=American Physical Society (APS) | volume=74 | issue=2 | date=1948-07-15 | issn=0031-899X | doi=10.1103/physrev.74.224 | pages=224–225| bibcode=1948PhRv...74..224T }}</ref> and systematized by [[Freeman Dyson]] in 1949.<ref>{{cite journal |author=Dyson, F. J. |title=The radiation theories of Tomonaga, Schwinger, and Feynman |journal=Phys. Rev. |volume=75 |pages=486–502 |year=1949|doi=10.1103/PhysRev.75.486 |issue=3 |bibcode=1949PhRv...75..486D |doi-access=free }}</ref> The divergences appear in radiative corrections  involving [[Feynman diagram]]s with closed ''loops'' of [[virtual particle]]s in them.

[[Julian Schwinger]]<!--February 1948-->,<ref>{{cite journal |author=Schwinger, J. |title=On quantum-electrodynamics and the magnetic moment of the electron |journal=[[Physical Review]] |volume=73 |issue=4 |pages=416–417 |year=1948|doi=10.1103/PhysRev.73.416 |bibcode=1948PhRv...73..416S |doi-access=free }}</ref><ref>{{cite journal |author=Schwinger, J. |series=Quantum Electrodynamics |title=I. A covariant formulation |journal=[[Physical Review]] |volume=74 |issue=10 |pages=1439–1461 |year=1948|doi=10.1103/PhysRev.74.1439 |bibcode=1948PhRv...74.1439S }}</ref><ref>{{cite journal |author=Schwinger, J. |series=Quantum Electrodynamics |title=II. Vacuum polarization and self-energy |journal=[[Physical Review]] |volume=75 |issue=4 |pages=651–679 |year=1949|doi=10.1103/PhysRev.75.651 |bibcode=1949PhRv...75..651S }}</ref><ref>{{cite journal |author=Schwinger, J. |series=Quantum Electrodynamics |title=III. The electromagnetic properties of the electron radiative corrections to scattering |journal=[[Physical Review]] |volume=76 |issue=6 |pages=790–817 |year=1949|doi=10.1103/PhysRev.76.790 |bibcode=1949PhRv...76..790S }}</ref> [[Richard Feynman]]<!--April 1948-->,<ref>{{cite journal |first=Richard P. |last=Feynman |title=Space-time approach to non-relativistic quantum mechanics |journal=[[Reviews of Modern Physics]] |volume=20 |pages=367–387 |year=1948 |doi=10.1103/RevModPhys.20.367 |bibcode=1948RvMP...20..367F |issue=2|url=https://authors.library.caltech.edu/47756/1/FEYrmp48.pdf }}</ref><ref>{{cite journal |last=Feynman  |first= Richard P. |title=A relativistic cut-off for classical electrodynamics  |journal=[[Physical Review]] |volume=74  |issue=8 |pages= 939–946 |year=1948  |doi=10.1103/PhysRev.74.939 |bibcode=1948PhRv...74..939F|url= https://authors.library.caltech.edu/3516/1/FEYpr48a.pdf }}</ref><ref>{{cite journal |first=Richard P. |last=Feynman |title=A relativistic cut-off for quantum electrodynamics |journal=[[Physical Review]] |volume=74 |pages=1430–1438 |year=1948 |doi=10.1103/PhysRev.74.1430 |bibcode=1948PhRv...74.1430F |issue=10|url=https://authors.library.caltech.edu/3517/1/FEYpr48b.pdf }}</ref> and [[Shin'ichiro Tomonaga]]<!--July 1948 (Koba–Tomonaga); according to S. S. Schweber, ''QED'', 1994, p. 269, Koba–Tomonaga contains the crucial calculation-->,<ref>{{cite journal | last=Tomonaga | first=S. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=1 | issue=2 | date=1946-08-01 | issn=1347-4081 | doi=10.1143/ptp.1.27 | pages=27–42|doi-access=free| bibcode=1946PThPh...1...27T }}</ref><ref>{{cite journal | last1=Koba | first1=Z. | last2=Tati | first2=T. | last3=Tomonaga | first3=S.-i. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. II: Case of Interacting Electromagnetic and Electron Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=2 | issue=3 | date=1947-10-01 | issn=0033-068X | doi=10.1143/ptp/2.3.101 | pages=101–116|doi-access=free| bibcode=1947PThPh...2..101K }}</ref><ref>{{cite journal | last1=Koba | first1=Z. | last2=Tati | first2=T. | last3=Tomonaga | first3=S.-i. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. III: Case of Interacting Electromagnetic and Electron Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=2 | issue=4 | date=1947-12-01 | issn=0033-068X | doi=10.1143/ptp/2.4.198 | pages=198–208|doi-access=free| bibcode=1947PThPh...2..198K }}</ref><ref>{{cite journal | last1=Kanesawa | first1=S. | last2=Tomonaga | first2=S.-i. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. [IV]: Case of Interacting Electromagnetic and Meson Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=3 | issue=1 | date=1948-03-01 | issn=0033-068X | doi=10.1143/ptp/3.1.1 | pages=1–13|doi-access=free}}</ref><ref>{{cite journal | last1=Kanesawa | first1=S. | last2=Tomonaga | first2=S.-i. | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields V: Case of Interacting Electromagnetic and Meson Fields | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=3 | issue=2 | date=1948-06-01 | issn=0033-068X | doi=10.1143/ptp/3.2.101 | pages=101–113|doi-access=free| bibcode=1948PThPh...3..101K }}</ref><ref>{{cite journal | last1=Koba | first1=Z. | last2=Tomonaga | first2=S.-i. | title=On Radiation Reactions in Collision Processes. I: Application of the "Self-Consistent" Subtraction Method to the Elastic Scattering of an Electron | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=3 | issue=3 | date=1948-09-01 | issn=0033-068X | doi=10.1143/ptp/3.3.290 | pages=290–303|doi-access=free| bibcode=1948PThPh...3..290K }}</ref><ref>{{cite journal | last1=Tomonaga | first1=Sin-Itiro | last2=Oppenheimer | first2=J. R. |author-link2=J. Robert Oppenheimer| title=On Infinite Field Reactions in Quantum Field Theory | journal=Physical Review | publisher=American Physical Society (APS) | volume=74 | issue=2 | date=1948-07-15 | issn=0031-899X | doi=10.1103/physrev.74.224 | pages=224–225| bibcode=1948PhRv...74..224T }}</ref> and systematized by [[Freeman Dyson]] in 1949.<ref>{{cite journal |author=Dyson, F. J. |title=The radiation theories of Tomonaga, Schwinger, and Feynman |journal=Phys. Rev. |volume=75 |pages=486–502 |year=1949|doi=10.1103/PhysRev.75.486 |issue=3 |bibcode=1949PhRv...75..486D |doi-access=free }}</ref> The divergences appear in radiative corrections  involving [[Feynman diagram]]s with closed ''loops'' of [[virtual particle]]s in them.

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

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

These integrals are often ''divergent'', that is, they give infinite answers.  The divergences that are significant are the "[[ultraviolet divergence|ultraviolet]]" (UV) ones. An ultraviolet divergence can be described as one that comes from

These integrals are often ''divergent'', that is, they give infinite answers.  The divergences that are significant are the "[[ultraviolet divergence|ultraviolet]]" (UV) ones. An ultraviolet divergence can be described as one that comes from

* the region in the integral where all particles in the loop have large energies and momenta,

* the region in the integral where all particles in the loop have large energies and momenta,

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

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

Shown in the pictures at the right margin, there are exactly three one-loop divergent loop diagrams in quantum electrodynamics:<ref>{{cite book |author1-link=Michael E. Peskin |first1=Michael E. |last1=Peskin |first2=Daniel V. |last2=Schroeder |title=An Introduction to Quantum Field Theory |url=https://archive.org/details/introductiontoqu0000pesk |url-access=registration |publisher=Addison-Wesley |location=Reading |year=1995 |at=Chapter&nbsp;10}}</ref>

Shown in the pictures at the right margin, there are exactly three one-loop divergent loop diagrams in quantum electrodynamics:<ref>{{cite book |author1-link=Michael E. Peskin |first1=Michael E. |last1=Peskin |first2=Daniel V. |last2=Schroeder |title=An Introduction to Quantum Field Theory |url=https://archive.org/details/introductiontoqu0000pesk |url-access=registration |publisher=Addison-Wesley |location=Reading |year=1995 |at=Chapter&nbsp;10}}</ref>

:(a) A photon creates a virtual electron–[[positron]] pair, which then annihilates. This is a [[vacuum polarization]] diagram.

:(a) A photon creates a virtual electron–[[positron]] pair, which then annihilates. This is a [[vacuum polarization]] diagram.

:(b) An electron quickly emits and reabsorbs a virtual photon, called a [[self-energy]].

:(b) An electron quickly emits and reabsorbs a virtual photon, called a [[self-energy]].

:(c) An electron emits a photon, emits a second photon, and reabsorbs the first. This process is shown in the section below in figure&nbsp;2, and it is called a ''[[vertex renormalization]]''. The Feynman diagram for this is also called a “[[penguin diagram]]” due to its shape remotely resembling a penguin.

:(c) An electron emits a photon, emits a second photon, and reabsorbs the first. This process is shown in the section below in figure&nbsp;2, and it is called a ''[[vertex renormalization]]''. The Feynman diagram for this is also called a “[[penguin diagram]]” due to its shape remotely resembling a penguin.

The three divergences correspond to the three parameters in the theory under consideration:

The three divergences correspond to the three parameters in the theory under consideration:

# The field normalization Z.

# The field normalization Z.

# The mass of the electron.

# The mass of the electron.

# The charge of the electron.

# The charge of the electron.

The second class of divergence called an [[infrared divergence]], is due to massless particles, like the photon. Every process involving charged particles emits infinitely many coherent photons of infinite wavelength, and the amplitude for emitting any finite number of photons is zero. For photons, these divergences are well understood. For example, at the 1-loop order, the [[vertex function]] has both ultraviolet and ''infrared'' divergences. In contrast to the ultraviolet divergence, the infrared divergence does not require the renormalization of a parameter in the theory involved. The infrared divergence of the vertex diagram is removed by including a diagram similar to the vertex diagram with the following important difference: the photon connecting the two legs of the electron is cut and replaced by two [[on-shell]] (i.e. real) photons whose wavelengths tend to infinity; this diagram is equivalent to the [[bremsstrahlung]] process. This additional diagram must be included because there is no physical way to distinguish a zero-energy photon flowing through a loop as in the vertex diagram and zero-energy photons emitted through [[bremsstrahlung]]. From a mathematical point of view, the IR divergences can be regularized by assuming fractional differentiation w.r.t. a parameter, for example:

The second class of divergence called an [[infrared divergence]], is due to massless particles, like the photon. Every process involving charged particles emits infinitely many coherent photons of infinite wavelength, and the amplitude for emitting any finite number of photons is zero. For photons, these divergences are well understood. For example, at the 1-loop order, the [[vertex function]] has both ultraviolet and ''infrared'' divergences. In contrast to the ultraviolet divergence, the infrared divergence does not require the renormalization of a parameter in the theory involved. The infrared divergence of the vertex diagram is removed by including a diagram similar to the vertex diagram with the following important difference: the photon connecting the two legs of the electron is cut and replaced by two [[on-shell]] (i.e. real) photons whose wavelengths tend to infinity; this diagram is equivalent to the [[bremsstrahlung]] process. This additional diagram must be included because there is no physical way to distinguish a zero-energy photon flowing through a loop as in the vertex diagram and zero-energy photons emitted through [[bremsstrahlung]]. From a mathematical point of view, the IR divergences can be regularized by assuming fractional differentiation w.r.t. a parameter, for example:

:<math> \left( p^2 - a^2 \right)^{\frac{1}{2}} </math>

:<math> \left( p^2 - a^2 \right)^{\frac{1}{2}} </math>

is well defined at {{math|''p'' {{=}} ''a''}} but is UV divergent; if we take the {{frac|3|2}}-th [[fractional derivative]] with respect to {{math|−''a''²}}, we obtain the IR divergence

is well defined at {{math|''p'' {{=}} ''a''}} but is UV divergent; if we take the {{frac|3|2}}-th [[fractional derivative]] with respect to {{math|−''a''²}}, we obtain the IR divergence

:<math> \frac{1}{p^2 - a^2},</math>

:<math> \frac{1}{p^2 - a^2},</math>

so we can cure IR divergences by turning them into UV divergences.{{clarify|date=May 2012}}

so we can cure IR divergences by turning them into UV divergences.{{clarify|date=May 2012}}

[[Image:Loop-diagram.png|thumb|upright=1.1|Figure 2. A diagram contributing to electron–electron scattering in QED. The loop has an ultraviolet divergence.]]

[[Image:Loop-diagram.png|thumb|upright=1.1|Figure 2. A diagram contributing to electron–electron scattering in QED. The loop has an ultraviolet divergence.]]

Category:Concepts in physics

Category:Concepts in physics

分类: 物理概念

分类: 物理概念

The diagram in Figure 2 shows one of the several one-loop contributions to electron–electron scattering in QED. The electron on the left side of the diagram, represented by the solid line, starts out with 4-momentum {{math|''p^μ''}} and ends up with 4-momentum {{math|''r^μ''}}. It emits a virtual photon carrying {{math|''r^μ'' − ''p^μ''}} to transfer energy and momentum to the other electron.  But in this diagram, before that happens, it emits another virtual photon carrying 4-momentum {{math|''q^μ''}}, and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the 4-momentum {{math|''q^μ''}} uniquely, so all possibilities contribute equally and we must integrate.

* Blechman, Andrew E.; [http://www.pha.jhu.edu/~blechman/papers/renormalization/ ''Renormalization: Our Greatly Misunderstood Friend''], (2002). Summary of a lecture; has more information about specific regularization and divergence-subtraction schemes.

Category:Particle physics

Category:Particle physics

类别: 粒子物理学

类别: 粒子物理学

 

Category:Quantum field theory

Category:Quantum field theory

范畴: 量子场论

范畴: 量子场论

This diagram's amplitude ends up with, among other things, a factor from the loop of

* [[Dmitry Shirkov|Shirkov, Dmitry]]; ''Fifty Years of the Renormalization Group'', C.E.R.N. Courrier 41(7) (2001). Full text available at : [http://www.cerncourier.com/main/article/41/7/14 ''I.O.P Magazines''].

Category:Renormalization group

Category:Renormalization group

类别: 重整化群

类别: 重整化群

 

Category:Mathematical physics

Category:Mathematical physics