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第3行： − '''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. − + − − 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 particles, such as photons, positrons, 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 protons, 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 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 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 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, another technique to control infinities by assuming the existence of new unknown physics at new scales. 第47行：
第24行： − == Self-interactions in classical physics == − + − 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. − − 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 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<sub>e</sub>}}. 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 . The mass–energy in the field is 第79行：
第42行： − − which becomes infinite as {{math|''r''<sub>e</sub> → 0}}. This implies that the point particle would have infinite [[inertia]], making it unable to be accelerated. Incidentally, the value of {{mvar|r<sub>e</sub>}} 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 . This implies that the point particle would have infinite inertia, making it unable to be accelerated. Incidentally, the value of 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 and <math>\varepsilon_0</math>) turns out to be
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|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=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。
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'''<font color="#ff8000">重整化 Renormalization </font>'''是应用于 '''<font color="#ff8000">量子场论 Quantum Field Theory </font>'''、场的'''<font color="#ff8000">统计力学 Statistical Mechanics </font>'''和'''<font color="#32cd32">自相似 Self-similar</font>'''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'''<font color="#32cd32"> 自相互作用 Self-interaction </font>'''，进而消除计算量中产生的'''<font color="#ff8000"> 无穷大 infinities</font>'''。但是，即使在量子场论的'''<font color="#32d32"> 圈图 loop diagrams </font>'''中没有出现无穷大，对原'''<font color="#32d32"> 拉格朗日场理论 Lagrangian (Field Theory) </font>'''中出现的质量和场进行重整化也是必要的。
'''<font color="#ff8000">重整化 Renormalization </font>'''是应用于 '''<font color="#ff8000">量子场论 Quantum Field Theory </font>'''、场的'''<font color="#ff8000">统计力学 Statistical Mechanics </font>'''和'''<font color="#32cd32">自相似 Self-similar</font>'''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'''<font color="#32cd32"> 自相互作用 Self-interaction </font>'''，进而消除计算量中产生的'''<font color="#ff8000"> 无穷大 infinities</font>'''。但是，即使在量子场论的'''<font color="#32d32"> 圈图 loop diagrams </font>'''中没有出现无穷大，对原'''<font color="#32d32"> 拉格朗日场理论 Lagrangian (Field Theory) </font>'''中出现的质量和场进行重整化也是必要的。<ref>See e.g., Weinberg vol I, chapter 10.</ref>
例如，'''<font color="#ff8000"> 电子 Electron </font>'''理论会先假定电子具有初始质量和电荷。在'''<font color="#ff8000"> 量子场论 </font>'''中，一个由诸如'''<font color="#ff8000"> 光子 Photon</font>'''、'''<font color="#ff8000">正电子 Positron </font>'''等'''<font color="#ff8000"> 虚粒子 Virtual Particle </font>'''组成的云团围绕着初始电子并与之相互作用。考虑到周围粒子的相互作用（例如: 不同能量的碰撞）表明电子-系统的行为宛如它有不同于最初假设的质量和电荷。在这个例子中，重整化在数学上用实验观察到的质量和电荷代替了最初假设的电子质量和电荷。数学和实验证明，正电子和'''<font color="#ff8000"> 质子 Proton </font>'''等质量更大的粒子，即使存在更强烈的相互作用和更密集的虚粒子云，其电荷也与电子完全相同。
例如，'''<font color="#ff8000"> 电子 Electron </font>'''理论会先假定电子具有初始质量和电荷。在'''<font color="#ff8000"> 量子场论 </font>'''中，一个由诸如'''<font color="#ff8000"> 光子 Photon</font>'''、'''<font color="#ff8000">正电子 Positron </font>'''等'''<font color="#ff8000"> 虚粒子 Virtual Particle </font>'''组成的云团围绕着初始电子并与之相互作用。考虑到周围粒子的相互作用（例如: 不同能量的碰撞）表明电子-系统的行为宛如它有不同于最初假设的质量和电荷。在这个例子中，重整化在数学上用实验观察到的质量和电荷代替了最初假设的电子质量和电荷。数学和实验证明，正电子和'''<font color="#ff8000"> 质子 Proton </font>'''等质量更大的粒子，即使存在更强烈的相互作用和更密集的虚粒子云，其电荷也与电子完全相同。
当描述大距离尺度的参数不同于描述小距离尺度的参数时，重整化指定了理论中参数之间的关系。在像欧洲核子研究中心的高能粒子加速器中，当不理想的质子-质子碰撞与同时临近的可取测量数据相互作用时，就会产生'''<font color="#32cd32"> 连环相撞 Pileup </font>'''的概念。从物理上来说，涉及某一问题的无限量级在累积后可能会导致进一步的无限量。当把时空描述为一个'''<font color="#32cd32"> 时空连续统 Space-time Continuum</font>'''时，某些统计的和量子力学的结构没有得到'''<font color="#32cd32"> 明确定义 Well-defined </font>'''。为了定义它们，或者使它们毫不含糊，连续统的限制必须能够小心地移除不同尺度的晶格的“结构脚手架（？）”。重整化过程的基础要求某些物理量(如电子的质量和电荷)等于观察到的(实验)值。也就是说，物理量的实验值虽能产生实际应用，但由于它们的经验性本质，所观察到的测量代表了量子场论中那些需要从理论基础进行更深入的推导的领域。
当描述大距离尺度的参数不同于描述小距离尺度的参数时，重整化指定了理论中参数之间的关系。在像欧洲核子研究中心的高能粒子加速器中，当不理想的质子-质子碰撞与同时临近的可取测量数据相互作用时，就会产生'''<font color="#32cd32"> 连环相撞 Pileup </font>'''的概念。从物理上来说，涉及某一问题的无限量级在累积后可能会导致进一步的无限量。当把时空描述为一个'''<font color="#32cd32"> 时空连续统 Space-time Continuum</font>'''时，某些统计的和量子力学的结构没有得到'''<font color="#32cd32"> 明确定义 Well-defined </font>'''。为了定义它们，或者使它们毫不含糊，连续统的限制必须能够小心地移除不同尺度的晶格的“结构脚手架（？）”。重整化过程的基础要求某些物理量(如电子的质量和电荷)等于观察到的(实验)值。也就是说，物理量的实验值虽能产生实际应用，但由于它们的经验性本质，所观察到的测量代表了量子场论中那些需要从理论基础进行更深入的推导的领域。
重整化最早发展于'''<font color="#ff8000"> 量子电动力学 Quantum Electrodynamics </font>'''，以解释'''<font color="#ff8000"> 微扰理论 Perturbation Theory </font>'''中的无穷积分。重整化最初被人认为是一个存疑的临时程序，甚至包括它的一些发明者。即便如此，重整化最终作为一个重要的且'''<font color="#ff8000"> 自洽 Self-consistent </font>'''的实际尺度物理机制被'''<font color="#ff8000"> 物理学 Physics </font>'''和'''<font color="#ff8000"> 数学 Mathematics </font>'''的几个领域所接受。
重整化最早发展于'''<font color="#ff8000"> 量子电动力学 Quantum Electrodynamics </font>'''，以解释'''<font color="#ff8000"> 微扰理论 Perturbation Theory </font>'''中的无穷积分。重整化最初被人认为是一个存疑的临时程序，甚至包括它的一些发明者。即便如此，重整化最终作为一个重要的且'''<font color="#ff8000"> 自洽 Self-consistent </font>'''的实际尺度物理机制被'''<font color="#ff8000"> 物理学 Physics </font>'''和'''<font color="#ff8000"> 数学 Mathematics </font>'''的几个领域所接受。
今天，观点发生了转变: 基于尼古拉·博戈柳博夫和 Kenneth Wilson 对'''<font color="#ff8000"> 重整化群 Renormalization Group </font>'''的突破性见解，关注点成为连续尺度间物理量的变化，而相隔较远的尺度通过“有效的“（？）描述彼此相关。广泛来说，所有尺度都以系统的方式联系在一起。同时，与每个尺度相关的实际物理学被适合于每个尺度的特定计算技术提取出来。威尔逊阐明了系统中哪些变量是至关重要的，而哪些又是冗余的。
今天，观点发生了转变: 基于尼古拉·博戈柳博夫和 Kenneth Wilson 对'''<font color="#ff8000"> 重整化群 Renormalization Group </font>'''的突破性见解，关注点成为连续尺度间物理量的变化，而相隔较远的尺度通过“有效的“（？）描述彼此相关。广泛来说，所有尺度都以系统的方式联系在一起。同时，与每个尺度相关的实际物理学被适合于每个尺度的特定计算技术提取出来。威尔逊阐明了系统中哪些变量是至关重要的，而哪些又是冗余的。
重整化不同于'''<font color="#ff8000"> 正则化 Regularization </font>'''，后者是另一种通过假设新尺度中存在新的未知的物理学以控制无穷大的技术。
重整化不同于'''<font color="#ff8000"> 正则化 Regularization </font>'''，后者是另一种通过假设新尺度中存在新的未知的物理学以控制无穷大的技术。
== 经典物理中的自相互作用 ==
== 经典物理中的自相互作用 ==
[[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.|链接=Special:FilePath/Renormalized-vertex.png]]
[[Image:Renormalized-vertex.png|thumbnail|upright=1.3|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。|链接=Special:FilePath/Renormalized-vertex.png]]
无穷大问题最早出现在19世纪和20世纪初的点粒子经典电动力学中。
无穷大问题最早出现在19世纪和20世纪初的点粒子经典电动力学中。
带电粒子的质量应包括其静电场('''<font color="#32cd32"> 电磁质量 </font>''')中的质能。假设这个粒子是一个带电的半径为r_e的球壳。场中的质能是：
带电粒子的质量应包括其静电场('''<font color="#32cd32"> 电磁质量 </font>''')中的质能。假设这个粒子是一个带电的半径为r_e的球壳。场中的质能是：
当r_e趋于0时，它会变得无穷大。这意味着点粒子具有无穷大的'''<font color="#ff8000"> 惯性 Inertia </font>'''，使它无法被加速。顺带一提，使得 < math > m text { em } <nowiki></math ></nowiki> 等于电子质量的这个值被称为'''<font color="#32cd32"> 电子经典半径 Classical Electron Radius </font>'''，它(设置 < math > q = e <nowiki></math ></nowiki> 和 < math > varepssilon 0 <nowiki></math ></nowiki> 的还原因子)被证明是
当r_e趋于0时，它会变得无穷大。这意味着点粒子具有无穷大的'''<font color="#ff8000"> 惯性 Inertia </font>'''，使它无法被加速。顺带一提，使得 < math > m text { em } <nowiki></math ></nowiki> 等于电子质量的这个值被称为'''<font color="#32cd32"> 电子经典半径 Classical Electron Radius </font>'''，它(设置 < math > q = e <nowiki></math ></nowiki> 和 < math > varepssilon 0 <nowiki></math ></nowiki> 的还原因子)被证明是