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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。

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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. | 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。

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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>

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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 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。

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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。

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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 ===

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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>

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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。

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=== 历史 ===

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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。

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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。

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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

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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

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| title=Triviality Pursuit: Can Elementary Scalar Particles Exist?| journal=[[Physics Reports]]

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| 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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