重整化 - 版本历史

Swarma：/* 态度以及解读 */

2021-09-04T01:53:14Z

态度以及解读

Swarma：/* 单圈发散 */

2021-09-04T01:50:21Z

单圈发散

2021年9月1日 (三) 03:10 唐糖糖

2021-09-01T03:10:54Z

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2021-09-01T02:59:46Z

唐糖糖：/* 运转联轴器 */

2021-08-31T14:40:53Z

运转联轴器

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2021-08-31T14:36:44Z

运转联轴器

2021年8月31日 (二) 08:32 唐糖糖

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2021年8月31日 (二) 08:30 唐糖糖

2021-08-31T08:30:31Z

2021年8月31日 (二) 08:27 唐糖糖

2021-08-31T08:27:56Z

@@ 第292行： / 第292行： @@
 狄拉克的批评是最持久的。<ref>P.A.M. Dirac, "The Evolution of the Physicist's Picture of Nature," in Scientific American, May 1963, p. 53.</ref>直到1975年，他仍表示:<ref>Kragh, Helge; ''Dirac: A scientific biography'', CUP 1990, p. 184</ref>
-:大多数物理学家对这种情况非常满意。他们说: ‘量子电动力学理论是一个很好的理论，我们再也不用担心它了’。但我必须说，我对这种情况非常不满意，因为这种所谓的“好理论”确实有涉及忽视方程式中出现的无穷大，且以一种武断的方式忽视它们。这根本不是合理的数学。合理的数学涉及对微小量的忽略，而不是仅仅因为一个量无限大，你不想要它就可以忽略它！
+:''<small>“大多数物理学家对这种情况非常满意。他们说: ‘量子电动力学理论是一个很好的理论，我们再也不用担心它了’。但我必须说，我对这种情况非常不满意，因为这种所谓的“好理论”确实有涉及忽视方程式中出现的无穷大，且以一种武断的方式忽视它们。这根本不是合理的数学。合理的数学涉及对微小量的忽略，而不是仅仅因为一个量无限大，你不想要它就可以忽略它！”</small>''
 另一位重要的评论家是费曼。尽管他在量子电动力学的发展中扮演了关键角色，他在1985年写道:<ref>Feynman, Richard P.; ''QED: The Strange Theory of Light and Matter'', Penguin 1990, p. 128</ref>
-:我们玩的这个骗局在技术上叫做“重整化”。但是不管这个词多么聪明，它仍然是我所说的一个含糊的过程！不得不求助于这样的骗术阻碍了我们证明量子电动力学理论在数学上是自洽的脚步。令人惊讶的是，到目前为止，这个理论仍然没有以这样或那样的方式被证明是自洽的; 我怀疑重整化在数学上是不正当的。
+:<small>''“我们玩的这个骗局在技术上叫做“重整化”。但是不管这个词多么聪明，它仍然是我所说的一个含糊的过程！不得不求助于这样的骗术阻碍了我们证明量子电动力学理论在数学上是自洽的脚步。令人惊讶的是，到目前为止，这个理论仍然没有以这样或那样的方式被证明是自洽的; 我怀疑重整化在数学上是不正当的。”''</small>
@@ 第309行： / 第309行： @@
-尽管如此，萨拉姆在1972年的言论似乎仍然有意义
+尽管如此，萨拉姆在1972年的言论似乎仍然有意义：
-:场论的无穷大首次出现在洛伦兹对电子自质量的计算过程中，它在经典电动力学中已经存在了七十年，在量子电动力学中也已存在了35年。这么多年的挫折使得研究这个课题的人对无穷大产生了一种奇怪的感情，并且热切地相信它们是自然界不可避免的一部分; 以至于即使他们有可能避开了正解，有限的重整化常数可被计算--这样的希望都被认为是不合理的。将罗素的附言与他的自传《最后的岁月，1944-1969》(乔治 · 艾伦和安文出版社，伦敦，1969年)的第三卷相比较，于第221页:
+:场论的无穷大首次出现在洛伦兹对电子自质量的计算过程中，它在经典电动力学中已经存在了七十年，在量子电动力学中也已存在了35年。这么多年的挫折使得研究这个课题的人对无穷大产生了一种奇怪的感情，并且热切地相信它们是自然界不可避免的一部分; 以至于即使他们有可能避开了正解，有限的重整化常数可被计算--这样的希望都被认为是不合理的。将罗素的附言与他的自传《最后的岁月，1944-1969》(乔治 · 艾伦和安文出版社，伦敦，1969年)的第三卷相比较，在第221页:
-::在现代社会，如果团体不幸福，那往往是因为他们有那些比幸福甚至生命更重要的无知、习惯、信仰和激情。我发现在我们这个危险的时代，有许多人似乎爱上了痛苦和死亡，以至于当面对希望时，他们会生气起来。他们认为希望是非理性的，坐下来懒洋洋地绝望只是在面对事实而已。
+::''<small>“在现代社会，如果团体不幸福，那往往是因为他们有那些比幸福甚至生命更重要的无知、习惯、信仰和激情。我发现在我们这个危险的时代，有许多人似乎爱上了痛苦和死亡，以至于当面对希望时，他们会生气起来。他们认为希望是非理性的，坐下来懒洋洋地绝望只是在面对事实而已。”</small>''

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−	这个表达式中的各种{{math\|''γ<sup>μ</sup>''}}因子是和狄拉克方程的协变公式一样的伽马矩阵; 它们与电子的自旋有关。{{mvar\|e}}的因子为电耦合常数，{\displaystyle i\epsilon}提供了动量空间中绕极点积分轮廓的启发式定义。对于我们的目的来说，重要的部分是被积函数中三个主要因子对{{math\|''q<sup>μ</sup>''}}的依赖，这三个因子来自圈中的两条电子线和光子线的传播子。	+	这个表达式中的各种{{math\|''γ<sup>μ</sup>''}}因子是和狄拉克方程的协变公式一样的伽马矩阵; 它们与电子的自旋有关。{{mvar\|e}}的因子为电耦合常数，提供了动量空间中绕极点积分轮廓的启发式定义。对于我们的目的来说，重要的部分是被积函数中三个主要因子对{{math\|''q<sup>μ</sup>''}}的依赖，这三个因子来自圈中的两条电子线和光子线的传播子。

@@ 第339行： / 第339行： @@
 在实际计算中，为了抵消费曼图计算中超出树图的散度而引入的反项必须使用一组重整化条件来解决。常用的重整化方案包括：
-* 最小减法(MS)方案和相关的改进最小减法(MS-bar)方案；
+* 最小减法（MS）方案和相关的改进最小减法（MS-bar）方案；
 * 在壳方案

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 ----
-本中文词条由[[用户:白开水|白开水]][[用户:Moonscar|Moonscar]]参与编译和审校，[[用户：唐糖糖|糖糖]]编辑，如有问题，欢迎在讨论页面留言。
+本中文词条由[[用户:白开水|白开水]]、[[用户:Moonscar|Moonscar]]参与编译和审校，[[用户：唐糖糖|糖糖]]编辑，如有问题，欢迎在讨论页面留言。
 '''本词条内容源自wikipedia及公开资料，遵守 CC3.0协议。'''

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−	量子场论计算中使用了许多不同类型的调节器，它们各有优缺点。在现代应用中最流行的是由Gerardus 't Hooft和Martinus J. G. Veltman<ref>{{Cite journal \| last1 = 't Hooft \| first1 = G. \| last2 = Veltman \| first2 = M. \| doi = 10.1016/0550-3213(72)90279-9 \| title = Regularization and renormalization of gauge fields \| journal = Nuclear Physics B \| volume = 44 \| issue = 1 \| pages = 189–213 \| year = 1972 \| pmid = \| pmc = \|bibcode = 1972NuPhB..44..189T \| hdl = 1874/4845 \| hdl-access = free }}</ref>~~发明的量纲正则化[21]，它通过将积分带入一个虚构的分数维数的空间来驯服积分。另一种是保利~~-维拉斯正则化，它以非常大的质量将虚构的粒子添加到理论中，这样涉及大质量粒子的圈积分在大动量中抵消了现有的圈。	+	量子场论计算中使用了许多不同类型的调节器，它们各有优缺点。在现代应用中最流行的是由Gerardus 't Hooft和Martinus J. G. Veltman<ref>{{Cite journal \| last1 = 't Hooft \| first1 = G. \| last2 = Veltman \| first2 = M. \| doi = 10.1016/0550-3213(72)90279-9 \| title = Regularization and renormalization of gauge fields \| journal = Nuclear Physics B \| volume = 44 \| issue = 1 \| pages = 189–213 \| year = 1972 \| pmid = \| pmc = \|bibcode = 1972NuPhB..44..189T \| hdl = 1874/4845 \| hdl-access = free }}</ref>发明的量纲正则化，它通过将积分带入一个虚构的分数维数的空间来驯服积分。另一种是保利-维拉斯正则化，它以非常大的质量将虚构的粒子添加到理论中，这样涉及大质量粒子的圈积分在大动量中抵消了现有的圈。

@@ 第206行： / 第206行： @@
 :<math>I=\ln a-\ln b-\ln{\varepsilon_a}+\ln{\varepsilon_b} = \ln \tfrac{a}{b} - \ln \tfrac{\varepsilon_a}{\varepsilon_b}</math>
-确保{{math|{{sfrac|''ε<sub>b</sub>''|''ε<sub>a</sub>''}} → 1}}，然后 {{math|''I'' {{=}} ln {{sfrac|''a''|''b''}}.}}。
 == 正则化 ==

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	:<math>I=\ln a-\ln b-\ln{\varepsilon_a}+\ln{\varepsilon_b} = \ln \tfrac{a}{b} - \ln \tfrac{\varepsilon_a}{\varepsilon_b}</math>		:<math>I=\ln a-\ln b-\ln{\varepsilon_a}+\ln{\varepsilon_b} = \ln \tfrac{a}{b} - \ln \tfrac{\varepsilon_a}{\varepsilon_b}</math>

		+	<math>\frac{\varepsilon_{b}}{\varepsilon_{a}}</math>

	确保{{math\|{{sfrac\|''ε<sub>b</sub>''\|''ε<sub>a</sub>''}} → 1}}，然后 {{math\|''I'' {{=}} ln {{sfrac\|''a''\|''b''}}.}}。		确保{{math\|{{sfrac\|''ε<sub>b</sub>''\|''ε<sub>a</sub>''}} → 1}}，然后 {{math\|''I'' {{=}} ln {{sfrac\|''a''\|''b''}}.}}。

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 {{#seo:
-|keywords=埃尔德什，菲尔兹奖章
+|keywords=量子场论，统计力学，可重整性，量子电动力学
-|description=埃尔德什数根据数学论文的著作权来来对数学家保罗·埃尔德什与其他作者之间的“协作距离”进行描述。同样的原则也应用于很多当特定某个人与众多同行之间保持合作关系的其他领域。
+|description=重整化 Renormalization是应用于量子场论、场的统计力学和自相似几何结构理论中的一类方法。通过重整化，可以改变计算量的值以抵消其自相互作用，进而消除计算量中产生的无穷大。但是，即使在量子场论的圈图中没有出现无穷大，对原拉格朗日场理论中出现的质量和场进行重整化也是必要的。
 }}

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-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 |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，<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 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。
+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 |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，<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. | 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 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。
@@ 第85行： / 第85行： @@
-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&nbsp;10}}</ref>
+如右图所示。量子电动力学中有三个单圈发散圈图：<ref>{{cite book|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)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。
@@ 第185行： / 第185行： @@
-从历史上看，将“裸项”分解为原始项和反项的做法，早于肯尼思 · 威尔逊对重整化群的洞察。<ref name=Wilson1975>{{cite journal | last=Wilson | first=Kenneth G. |author-link=Kenneth G. Wilson| title=The renormalization group: Critical phenomena and the Kondo problem | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=47 | issue=4 | date=1975-10-01 | issn=0034-6861 | doi=10.1103/revmodphys.47.773 | pages=773–840| bibcode=1975RvMP...47..773W }}</ref>根据这些重整化群的洞察，在更细节的部分里这种分裂是非自然的也是非物理的，因为问题的所有尺度都是以连续的系统方式进入的。
+从历史上看，将“裸项”分解为原始项和反项的做法，早于肯尼思 · 威尔逊对重整化群的洞察。<ref name=Wilson1975>{{cite journal | last=Wilson | first=Kenneth G. | title=The renormalization group: Critical phenomena and the Kondo problem | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=47 | issue=4 | date=1975-10-01 | issn=0034-6861 | doi=10.1103/revmodphys.47.773 | pages=773–840| bibcode=1975RvMP...47..773W }}</ref>根据这些重整化群的洞察，在更细节的部分里这种分裂是非自然的也是非物理的，因为问题的所有尺度都是以连续的系统方式进入的。
 === 运转联轴器 ===
@@ 第294行： / 第294行： @@
-另一位重要的评论家是费曼。尽管他在量子电动力学的发展中扮演了关键角色，他在1985年写道:<ref>Feynman, Richard P.; ''[[QED: The Strange Theory of Light and Matter]]'', Penguin 1990, p. 128</ref>
+另一位重要的评论家是费曼。尽管他在量子电动力学的发展中扮演了关键角色，他在1985年写道:<ref>Feynman, Richard P.; ''QED: The Strange Theory of Light and Matter'', Penguin 1990, p. 128</ref>
 :我们玩的这个骗局在技术上叫做“重整化”。但是不管这个词多么聪明，它仍然是我所说的一个含糊的过程！不得不求助于这样的骗术阻碍了我们证明量子电动力学理论在数学上是自洽的脚步。令人惊讶的是，到目前为止，这个理论仍然没有以这样或那样的方式被证明是自洽的; 我怀疑重整化在数学上是不正当的。
@@ 第347行： / 第347行： @@
 === 历史 ===
-凝聚态物理学对重整化过程的物理意义和推广提供了更深入的理解，它超越了传统重整化理论的膨胀群。Leo P. Kadanoff在1966年的论文中提出了“块区自旋”重整群。<ref>[[Leo Kadanoff|L.P. Kadanoff]] (1966): "Scaling laws for Ising models near <math>T_c</math>", ''Physics (Long Island City, N.Y.)'' '''2''', 263.</ref>分块思想是一种将理论中远距离的分量定义为较短距离分量的集合的方法。
+凝聚态物理学对重整化过程的物理意义和推广提供了更深入的理解，它超越了传统重整化理论的膨胀群。Leo P. Kadanoff在1966年的论文中提出了“块区自旋”重整群。<ref>Leo Kadanoff|L.P. Kadanoff (1966): "Scaling laws for Ising models near <math>T_c</math>", ''Physics (Long Island City, N.Y.)'' '''2''', 263.</ref>分块思想是一种将理论中远距离的分量定义为较短距离分量的集合的方法。
@@ 第371行： / 第371行： @@
-几种理论都可以流动到同一固定点的性质产生了普遍性。如果这些固定点与自由场论相对应，那么这个理论就表现出了量子的平凡性。在格子希格斯理论的研究中出现了许多固定点，但与之相关的量子场论的本质仍然是一个悬而未决的问题。<ref name="TrivPurs">{{cite journal| author=D. J. E. Callaway | year=1988
+几种理论都可以流动到同一固定点的性质产生了普遍性。如果这些固定点与自由场论相对应，那么这个理论就表现出了量子的平凡性。在格子希格斯理论的研究中出现了许多固定点，但与之相关的量子场论的本质仍然是一个悬而未决的问题。<ref name="TrivPurs">{{cite journal | year=1988
-| title=Triviality Pursuit: Can Elementary Scalar Particles Exist?| journal=[[Physics Reports]]
+| title=Triviality Pursuit: Can Elementary Scalar Particles Exist?| journal=Physics Reports
 |volume=167| issue=5 | pages=241–320| doi=10.1016/0370-1573(88)90008-7
 |bibcode = 1988PhR...167..241C | author-link=David J E Callaway

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−	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> ，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 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。	+	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 \|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，<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 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。