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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.

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.

例如，一个电子理论会先假定电子具有初始质量和电荷。在量子场论中，一个由诸如光子、正电子等虚粒子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。重整化，在这个例子中，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的质量和电荷。数学和实验证明，正电子和质子等质量更大的粒子，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，一个'''<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。重整化，在这个例子中，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的质量和电荷。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。

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 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.

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这样的高能粒子加速器中，当质子-质子的不和需求的碰撞与同时临近的可取测量数据相互作用时，就会产生连环相撞（？pileup）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连续统时，某些统计的和量子力学的结构没有得到很好的定义。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这样的高能粒子加速器中，当质子-质子的不和需求的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。

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.

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 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.

今天，观点发生了转变: 基于尼古拉·博戈柳博夫和 Kenneth Wilson 对重整化群的突破性见解，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。

Renormalization is distinct from 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.

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

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

带电粒子的质量应包括其静电场(电磁质量)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

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

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

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惯性，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电子经典半径，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

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.

其中 <math>\alpha \approx 1/137</math> 是精细结构常数，<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。

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.

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.

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正则化和重整化的尝试。

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。

(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.)

(假设在小尺度上存在新的物理学，另见正则化(物理学)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。)

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.

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.

在计算带电粒子的电磁相互作用时，人们很容易忽略粒子自身的场对自己的反作用（？）。(类似于电路分析的反电动势)。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，因为场是平方反比的。

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。