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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球壳。场中的质量-能量是

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

<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>

1}{2} e ^ 2，dV = int _ { r _ e } ^ infty frac {1}{2}左(frac { q }{4 pi r ^ 2}右) ^ 24 pi r ^ 2，dr = frac { q ^ 2}{8 pi r _ e } ，</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},

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

变得无穷无尽。这意味着点粒子具有无穷大的惯性，使它无法被加速。顺便说一句，这个值使得 < math > m text { em } </math > 等于电子质量，这个值被称为电子经典半径，它(设置 < math > q = e </math > 和 < math > varepssilon 0 </math > 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

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(除了上述与其电场相关的质量)。如果允许壳体的裸质量为负值，则可以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。这项早期的工作对于后来在量子场论中正则化和重整化的尝试是一种启发。

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。

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场对自身的反作用。(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，反向反应的值就会发散，这和质量发散的原因是一样的，因为场是平方反比。

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。

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 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.

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。这是点限制不一致的标志。

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。