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第36行： − − <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> − − <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> − 第49行：
第44行： − <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> − − − − 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. This was called renormalization, and Lorentz and Abraham attempted to develop a classical theory of the electron this way. This early work was the inspiration for later attempts at regularization and renormalization in quantum field theory. − − (See also [[regularization (physics)]] for an alternative way to remove infinities from this classical problem, assuming new physics exists at small scales.) − − (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 electromagnetic interactions of charged particles, it is tempting to ignore the back-reaction of a particle's own field on itself. (Analogous to the back-EMF of circuit analysis.) But this back-reaction is necessary to explain the friction on charged particles when they emit radiation. If the electron is assumed to be a point, the value of the back-reaction diverges, for the same reason that the mass diverges, because the field is inverse-square. − − − The [[Abraham–Lorentz 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 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. − − == Divergences in quantum electrodynamics == − + − − (a) Vacuum polarization, a.k.a. charge screening. This loop has a logarithmic ultraviolet divergence. − − (a) Vacuum polarization, a.k.a.电荷屏蔽。这个环有一个对数的紫外辐散。 − − [[Image:selfE.svg|thumb|(b) Self-energy diagram in QED|链接=Special:FilePath/SelfE.svg]] − − (b) Self-energy diagram in QED − − (b)量子电动力学中的自能图 − − [[Image:Penguin diagram.JPG|thumb|(c) Example of a “penguin” diagram|链接=Special:FilePath/Penguin_diagram.JPG]] − − (c) Example of a “penguin” diagram − (c)「企鹅」图示例子 + − 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). 第136行：
第84行： − 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 corrections' divergences was discovered in 1947–49 by<!--in chronological order--> Hans Kramers<!--June 1947-->, Hans Bethe<!--August 1947-->, − − [[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-->, Richard Feynman<!--April 1948-->, and Shin'ichiro Tomonaga<!--July 1948 (Koba–Tomonaga); according to S. S. Schweber, QED, 1994, p. 269, Koba–Tomonaga contains the crucial calculation-->, and systematized by Freeman Dyson in 1949. The divergences appear in radiative corrections involving Feynman diagrams with closed loops of virtual particles in them. − − 一种描述微扰理论修正发散的方法是由Hans Kramers，[2] Hans Bethe，[3] Julian Schwinger，[4][5][6][7] Richard Feynman，[8][9][10]和Shin'ichiro Tomonaga，[11][12][13][14][15][16][17]在1947-49年发现的，并在1949年被Freeman Dyson系统化。发散出现在含虚粒子闭环的费曼图的辐射校正中。 − − − 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 off-shell. When there is a loop, the momentum of the particles involved in the loop is not uniquely determined by the energies and momenta of incoming and outgoing particles. A variation in the energy of one particle in the loop can be balanced by an equal and opposite change in the energy of another particle in the loop, without affecting the incoming and outgoing particles. Thus many variations are possible. So to find the amplitude for the loop process, one must integrate over all possible combinations of energy and momentum that could travel around the loop. − − − These integrals are often ''divergent'', that is, they give infinite answers. The divergences that are significant are the "[[ultraviolet 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" (UV) ones. An ultraviolet divergence can be described as one that comes from − * the region in the integral where all particles in the loop have large energies and momenta, − + + + − * very short proper-time between particle emission and absorption, if the loop is thought of as a sum over particle paths. − − 所有圈中粒子具有很大的能量和动量的积分区域； − 在场的路径积分中，场具有非常短的波长和高频涨落； − 如果这个圈是粒子路径的和，粒子发射和吸收之间的固有时间很短。 − − So these divergences are short-distance, short-time phenomena. − − So these divergences are short-distance, short-time phenomena. 第180行：
第102行： − 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 10}}</ref>+ − − Shown in the pictures at the right margin, there are exactly three one-loop divergent loop diagrams in quantum electrodynamics: − − 如右图所示。量子电动力学中有三个单圈发散圈图： − − :(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. − − (a)一个光子产生一个虚拟电子-正电子对，然后这个电子-正电子对湮灭。这是真空极化图。 − − :(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. − (b)电子迅速发射并重新吸收虚光子，称为自能。 − + − + − + − − − 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. − − 1.场归一化因子Z − − # The mass of the electron. − − The mass of the electron. − − 2.电子的质量 − − # The charge of the electron. − The charge of the electron.+ − 3.电子的电荷+ − − − 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: 第241行：
第127行： − − <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''<sup>2</sup>}}, we obtain the IR divergence − − is well defined at a}} but is UV divergent; if we take the -th fractional derivative with respect to , 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. − − === A loop divergence === − + + − Figure 2. A diagram contributing to electron–electron scattering in QED. The loop has an ultraviolet divergence. − 图2。量子电动力学中电子-电子散射的图解。这个环有一个紫外辐射。 − + + + + +
无编辑摘要
:<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>
:<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>
:<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>
其中 <math>\alpha \approx 1/137</math> 是'''<font color="#32cd32"> 精细结构常数 Fine-structure Constant </font>'''精细结构常数，<math>\hbar/(m_e c)</math> 是电子的康普顿波长。
其中 <math>\alpha \approx 1/137</math> 是'''<font color="#32cd32"> 精细结构常数 Fine-structure Constant </font>'''精细结构常数，<math>\hbar/(m_e c)</math> 是电子的康普顿波长。
重整化: 球形带电粒子的总有效质量包括球壳的实际裸质量(在上述与其电场相关的质量之上)。如果允许壳体的裸质量允许为负值，则可能取一个一致的点极限。这就是所谓的重整化，洛伦兹和亚伯拉罕试图用这种方式发展出电子的经典理论。这项早期的工作启发了后来在量子场论中'''<font color="#ff8000"> 正则化 </font>'''和重整化的尝试。
重整化: 球形带电粒子的总有效质量包括球壳的实际裸质量(在上述与其电场相关的质量之上)。如果允许壳体的裸质量允许为负值，则可能取一个一致的点极限。这就是所谓的重整化，洛伦兹和亚伯拉罕试图用这种方式发展出电子的经典理论。这项早期的工作启发了后来在量子场论中'''<font color="#ff8000"> 正则化 </font>'''和重整化的尝试。
(假设在小尺度上存在新的物理学，另见'''<font color="#ff8000"> 正则化 </font>'''从这个经典问题中去除无穷大的替代方法。)
(假设在小尺度上存在新的物理学，另见'''<font color="#ff8000"> 正则化 </font>'''从这个经典问题中去除无穷大的替代方法。)
在计算'''<font color="#32cd32"> 带电 Electric Charged </font>'''粒子的'''<font color="#ff8000"> 电磁 electromagnetic </font>'''相互作用时，人们很容易忽略粒子自身的场对自己的'''<font color="#32cd32"> 反作用 Back-reaction </font>'''。(类似于电路分析的'''<font color="#32cd32"> 反电动势 Back-EMF </font>''')。但是这种反作用对于解释带电粒子发射辐射时的摩擦是必要的。如果假设电子是一个点，反作用的值就会发散，这和质量发散的原因是一样的，因为场是呈'''<font color="#32cd32"> 平方反比 Inverse-square </font>'''的。
在计算'''<font color="#32cd32"> 带电 Electric Charged </font>'''粒子的'''<font color="#ff8000"> 电磁 electromagnetic </font>'''相互作用时，人们很容易忽略粒子自身的场对自己的'''<font color="#32cd32"> 反作用 Back-reaction </font>'''。(类似于电路分析的'''<font color="#32cd32"> 反电动势 Back-EMF </font>''')。但是这种反作用对于解释带电粒子发射辐射时的摩擦是必要的。如果假设电子是一个点，反作用的值就会发散，这和质量发散的原因是一样的，因为场是呈'''<font color="#32cd32"> 平方反比 Inverse-square </font>'''的。
亚伯拉罕-洛伦兹理论有一个非因果的“预加速度”。有时，电子在施加力之前就开始移动了。这是点极限不一致的标志。
亚伯拉罕-洛伦兹理论有一个非因果的“预加速度”。有时，电子在施加力之前就开始移动了。这是点极限不一致的标志。
这个问题在经典场论中比在量子场论中更严重，因为在量子场论中，由于虚粒子-反粒子对的干涉，带电粒子经历了 Zitterbewegung，从而有效地抹去了一个可以与康普顿波长相比的区域上的电荷。在小耦合的量子电动力学中，电磁质量只随着粒子半径的对数发散。
这个问题在经典场论中比在量子场论中更严重，因为在量子场论中，由于虚粒子-反粒子对的干涉，带电粒子经历了 Zitterbewegung，从而有效地抹去了一个可以与康普顿波长相比的区域上的电荷。在小耦合的量子电动力学中，电磁质量只随着粒子半径的对数发散。
== 量子电动力学中的发散 ==
== 量子电动力学中的发散 ==
{{anchor|renormalization_loop_divergence}}
{{anchor|renormalization_loop_divergence}}
[[Image:vacuum polarization.svg|thumb|(a) Vacuum polarization, a.k.a. charge screening. This loop has a logarithmic ultraviolet divergence.|链接=Special:FilePath/Vacuum_polarization.svg]]
[[Image:vacuum polarization.svg|thumb|(a) Vacuum polarization, a.k.a.电荷屏蔽。这个环有一个对数的紫外辐散。|链接=Special:FilePath/Vacuum_polarization.svg]]
[[Image:selfE.svg|thumb|(b)量子电动力学中的自能图|链接=Special:FilePath/SelfE.svg]]
[[Image:Penguin diagram.JPG|thumb|(c)「企鹅」图示例子|链接=Special:FilePath/Penguin_diagram.JPG]]
在20世纪30年代发展量子电动力学时，马克斯·伯恩、维尔纳·海森堡、帕斯夸尔·乔丹和保罗·狄拉克发现，在微扰修正中，许多积分是发散的(见无穷大问题)。
在20世纪30年代发展量子电动力学时，马克斯·伯恩、维尔纳·海森堡、帕斯夸尔·乔丹和保罗·狄拉克发现，在微扰修正中，许多积分是发散的(见无穷大问题)。
一种描述微扰理论修正发散的方法是由Hans Kramers<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<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> ，[3] Julian Schwinger，<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<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>和Shin'ichiro Tomonaga，<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>在1947-49年发现的，并在1949年被Freeman Dyson系统化。发散出现在含虚粒子闭环的费曼图的辐射校正中。
虚粒子遵循能量和动量守恒，它们可以有任何能量和动量，甚至是观测到的粒子质量的相对能量和动量关系所不允许的能量和动量(即，{\displaystyle E^{2}-p^{2}}不一定是这个过程中粒子质量的平方，例如，对于光子它可能是非零的)。这样的粒子叫做离壳粒子。当有一个圈时，参与圈的粒子的动量不是唯一由入射和输出粒子的能量和动量决定的。圈中一个粒子能量的变化可以被圈中另一个粒子能量相等而相反的变化所平衡，而不影响进入和流出的粒子。因此，有许多变化是可能的。因此，为了找到圈过程的振幅，必须对所有可能的能量和动量的组合进行积分，这些组合可能在圈中传播。
虚粒子遵循能量和动量守恒，它们可以有任何能量和动量，甚至是观测到的粒子质量的相对能量和动量关系所不允许的能量和动量(即，{\displaystyle E^{2}-p^{2}}不一定是这个过程中粒子质量的平方，例如，对于光子它可能是非零的)。这样的粒子叫做离壳粒子。当有一个圈时，参与圈的粒子的动量不是唯一由入射和输出粒子的能量和动量决定的。圈中一个粒子能量的变化可以被圈中另一个粒子能量相等而相反的变化所平衡，而不影响进入和流出的粒子。因此，有许多变化是可能的。因此，为了找到圈过程的振幅，必须对所有可能的能量和动量的组合进行积分，这些组合可能在圈中传播。
这些积分通常是发散的，也就是说，它们给出无限的结果。其中“紫外”发散较为显著。紫外发散来自于以下几种情形：
这些积分通常是发散的，也就是说，它们给出无限的结果。其中“紫外”发散较为显著。紫外发散来自于以下几种情形：
* very short [[wavelength]]s and high-[[frequency|frequencies]] fluctuations of the fields, in the [[Path integral formulation|path integral]] for the field,
*所有圈中粒子具有很大的能量和动量的积分区域；
*在场的路径积分中，场具有非常短的波长和高频涨落；
*如果这个圈是粒子路径的和，粒子发射和吸收之间的固有时间很短。
所以这些发散是短距离，短时间的现象。
所以这些发散是短距离，短时间的现象。
如右图所示。量子电动力学中有三个单圈发散圈图：<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 10}}</ref>
:(c) An electron emits a photon, emits a second photon, and reabsorbs the first. This process is shown in the section below in figure 2, and it is called a ''[[vertex renormalization]]''. The Feynman diagram for this is also called a “[[penguin diagram]]” due to its shape remotely resembling a penguin.
:(a)一个光子产生一个虚拟电子-正电子对，然后这个电子-正电子对湮灭。这是真空极化图。
(c) An electron emits a photon, emits a second photon, and reabsorbs the first. This process is shown in the section below in figure 2, and it is called a vertex renormalization. The Feynman diagram for this is also called a “penguin diagram” due to its shape remotely resembling a penguin.
:(b)电子迅速发射并重新吸收虚光子，称为自能。
(c)电子发射一个光子，发射第二个光子，并重新吸收第一个光子。这个过程如下面的图2所示，它被称为顶点重正化。费曼图也被称为“企鹅图”，因为它的形状很像企鹅。
:(c)电子发射一个光子，发射第二个光子，并重新吸收第一个光子。这个过程如下面的图2所示，它被称为顶点重正化。费曼图也被称为“企鹅图”，因为它的形状很像企鹅。
这三中发散对应于所考虑理论中的三个参数:
这三中发散对应于所考虑理论中的三个参数:
# The field normalization Z.
# 场归一化因子Z
# 电子的质量
# 电子的电荷
第二类发散称为红外发散，由无质量粒子造成的，比如光子。每一个涉及带电粒子的过程都会发射出无限多个波长无限的相干光子，而发射任意有限数量光子的振幅为零。对于光子来说，这些发散过程研究透彻，理解清晰。例如在单圈阶处，顶点函数既有紫外散度也有红外散度。与紫外发散相反，红外发散在理论中不需要参数的重整化。顶点图的红外散度通过包含一个类似于顶点图的图来消除，该图具有以下重要的特征：连接电子（两条腿？）的光子被切断并被两个波长趋向于无穷大的在壳(实)光子所取代；该图图相当于轫致辐射过程。该图被包含在内是必要的，因为没有物理方法来区分在顶点图中流过圈的零能量光子和通过轫致辐射发射的零能量光子。从数学的角度来看，红外发散可以通过假设对参数进行分数阶微分来正则化，例如:
第二类发散称为红外发散，由无质量粒子造成的，比如光子。每一个涉及带电粒子的过程都会发射出无限多个波长无限的相干光子，而发射任意有限数量光子的振幅为零。对于光子来说，这些发散过程研究透彻，理解清晰。例如在单圈阶处，顶点函数既有紫外散度也有红外散度。与紫外发散相反，红外发散在理论中不需要参数的重整化。顶点图的红外散度通过包含一个类似于顶点图的图来消除，该图具有以下重要的特征：连接电子（两条腿？）的光子被切断并被两个波长趋向于无穷大的在壳(实)光子所取代；该图图相当于轫致辐射过程。该图被包含在内是必要的，因为没有物理方法来区分在顶点图中流过圈的零能量光子和通过轫致辐射发射的零能量光子。从数学的角度来看，红外发散可以通过假设对参数进行分数阶微分来正则化，例如:
:<math> \left( p^2 - a^2 \right)^{\frac{1}{2}} </math>
:<math> \left( p^2 - a^2 \right)^{\frac{1}{2}} </math>
式1在{{math|''p'' {{=}} ''a''}}处定义良好，不过却是紫外散度；如果我们对{{math|−''a''<sup>2</sup>}}2求{{frac|3|2}}分数阶导数，就可以得到红外散度：
式1在{{math|''p'' {{=}} ''a''}}处定义良好，不过却是紫外散度；如果我们对{{math|−''a''<sup>2</sup>}}2求{{frac|3|2}}分数阶导数，就可以得到红外散度：
:<math> \frac{1}{p^2 - a^2},</math>
:<math> \frac{1}{p^2 - a^2},</math>
因此我们可以通过将红外发散转化为紫外发散对其进行修正。
因此我们可以通过将红外发散转化为紫外发散对其进行修正。
=== 单圈发散 ===
=== 单圈发散 ===
[[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.|链接=Special:FilePath/Loop-diagram.png]]
[[Image:Loop-diagram.png|thumb|upright=1.1|图2。量子电动力学中电子-电子散射的图解。这个环有一个紫外辐射。
|链接=Special:FilePath/Loop-diagram.png]]
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<sup>μ</sup>''}} and ends up with 4-momentum {{math|''r<sup>μ</sup>''}}. It emits a virtual photon carrying {{math|''r<sup>μ</sup>'' − ''p<sup>μ</sup>''}} 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<sup>μ</sup>''}}, and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the 4-momentum {{math|''q<sup>μ</sup>''}} uniquely, so all possibilities contribute equally and we must integrate.
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<sup>μ</sup>''}} and ends up with 4-momentum {{math|''r<sup>μ</sup>''}}.
It emits a virtual photon carrying {{math|''r<sup>μ</sup>'' − ''p<sup>μ</sup>''}} 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<sup>μ</sup>''}}, and it reabsorbs this one after emitting the other virtual photon.
Energy and momentum conservation do not determine the 4-momentum {{math|''q<sup>μ</sup>''}} uniquely, so all possibilities contribute equally and we must integrate.
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 and ends up with 4-momentum . It emits a virtual photon carrying to transfer energy and momentum to the other electron. But in this diagram, before that happens, it emits another virtual photon carrying 4-momentum , and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the 4-momentum uniquely, so all possibilities contribute equally and we must integrate.
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 and ends up with 4-momentum . It emits a virtual photon carrying to transfer energy and momentum to the other electron. But in this diagram, before that happens, it emits another virtual photon carrying 4-momentum , and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the 4-momentum uniquely, so all possibilities contribute equally and we must integrate.