# 重整化

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模板:Renormalization and regularization

**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**. 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.^{[1]}

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. 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.

重整化是量子场论、场的统计力学和自相似几何结构理论中的一系列技术，这些技术通过改变这些量的值来处理计算量中产生的无穷大，以补偿它们自相互作用的影响。但是，即使在量子场论的环路图中没有无穷大，也可以证明有必要对原拉格朗日函数中出现的质量和场进行重整化。

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，电子理论可以从假定电子具有初始质量和电荷开始。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-系统的行为好像它有一个不同的质量和电荷比最初的假设。重整化，在这个例子中，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。

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-质子的不期望碰撞与数据收集相互作用，以便同时进行附近期望的测量时，就会出现连续大型强子对撞机的概念。从物理上来说，一个问题所涉及的无限量级的贡献累积起来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”。重整化过程的基础是要求某些物理量(如电子的质量和电荷)等于观察到的(实验)值。也就是说，物理量的实验值产生实际应用，但由于它们的经验性质，所观察到的测量代表了量子场论的领域，需要从理论基础进行更深入的推导。

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.

重整化最早是在20世纪量子电动力学发展起来的，用来解释摄动理论中的无穷积分。重整化最初被认为是一个可疑的临时过程，甚至被一些发起者认为是可疑的，最终在物理学和数学的几个领域被认为是一个重要的和自洽的实际尺度物理机制。

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: 基于重整化群尼古拉·博戈柳博夫和 Kenneth Wilson 的突破性见解，重点在于连续尺度间物理量的变化，而远距离尺度通过有效的描述彼此相关。所有尺度都以广泛系统的方式联系在一起，与每个尺度相关的实际物理学用适合每个尺度的特定计算技术提取出来。威尔逊澄清了系统中哪些变量是至关重要的，哪些是冗余的。

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.

重整化不同于正则化，后者是另一种通过假设新的未知物理学存在于新的尺度来控制无穷大的技术。

## 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 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.

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。

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 r_{e}. 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

- [math]\displaystyle{ 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]\displaystyle{ 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 >

which becomes infinite as *r*_{e} → 0. This implies that the point particle would have infinite inertia, making it unable to be accelerated. Incidentally, the value of r_{e} that makes [math]\displaystyle{ m_\text{em} }[/math] equal to the electron mass is called the classical electron radius, which (setting [math]\displaystyle{ q = e }[/math] and restoring factors of c and [math]\displaystyle{ \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]\displaystyle{ m_\text{em} }[/math] equal to the electron mass is called the classical electron radius, which (setting [math]\displaystyle{ q = e }[/math] and restoring factors of and [math]\displaystyle{ \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

- [math]\displaystyle{ r_e = \frac{e^2}{4\pi\varepsilon_0 m_e c^2} = \alpha \frac{\hbar}{m_e c} \approx 2.8 \times 10^{-15}~\text{m}, }[/math]

[math]\displaystyle{ r_e = \frac{e^2}{4\pi\varepsilon_0 m_e c^2} = \alpha \frac{\hbar}{m_e c} \approx 2.8 \times 10^{-15}~\text{m}, }[/math]

4 pi varepsilon 0 m e c ^ 2} = alpha frac { hbar }{ m e c }大约2.8乘以10 ^ {-15} ~ text { m } ，</math >

where [math]\displaystyle{ \alpha \approx 1/137 }[/math] is the fine-structure constant, and [math]\displaystyle{ \hbar/(m_e c) }[/math] is the Compton wavelength of the electron.

where [math]\displaystyle{ \alpha \approx 1/137 }[/math] is the fine-structure constant, and [math]\displaystyle{ \hbar/(m_e c) }[/math] is the Compton wavelength of the electron.

其中 < math > alpha 大约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。

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.^{[citation needed]} 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。

(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。)

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。

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。

The trouble was worse in classical field theory than in quantum field theory, because in quantum field theory a charged particle experiences Zitterbewegung due to interference with virtual particle-antiparticle pairs, thus effectively smearing out the charge over a region comparable to the Compton wavelength. In quantum electrodynamics at small coupling, the electromagnetic mass only diverges as the logarithm of the radius of the particle.

The trouble was worse in classical field theory than in quantum field theory, because in quantum field theory a charged particle experiences Zitterbewegung due to interference with virtual particle-antiparticle pairs, thus effectively smearing out the charge over a region comparable to the Compton wavelength. In quantum electrodynamics at small coupling, the electromagnetic mass only diverges as the logarithm of the radius of the particle.

这个问题在经典场论中比在量子场论中更严重，因为在量子场论中，由于虚粒子-反粒子对的干涉，带电粒子经历了 Zitterbewegung，从而有效地抹去了一个可以与康普顿波长相比的区域上的电荷。在小耦合的量子电动力学中，电磁质量只随着粒子半径的对数发散。

## Divergences in quantum electrodynamics

(a) Vacuum polarization, a.k.a. charge screening. This loop has a logarithmic ultraviolet divergence.

(a) Vacuum polarization, a.k.a.电荷屏蔽。这个环有一个对数的紫外辐散。

(b) Self-energy diagram in QED

(b)量子电动力学中的自能图

(c) Example of a “penguin” diagram

(c)「企鹅」图示例子

When developing quantum electrodynamics in the 1930s, Max Born, Werner Heisenberg, Pascual Jordan, and Paul Dirac discovered that in perturbative corrections many integrals were divergent (see The problem of infinities).

When developing quantum electrodynamics in the 1930s, Max Born, Werner Heisenberg, Pascual Jordan, and Paul Dirac discovered that in perturbative corrections many integrals were divergent (see The problem of infinities).

在20世纪30年代开发量子电动力学时，Max Born，维尔纳·海森堡，Pascual Jordan 和 Paul Dirac 发现许多积分在微扰修正中是发散的(见无穷大问题)。

One way of describing the perturbation theory corrections' divergences was discovered in 1947–49 by Hans Kramers,^{[2]} Hans Bethe,^{[3]}

One way of describing the perturbation theory corrections' divergences was discovered in 1947–49 by Hans Kramers, Hans Bethe,

1947年至1949年间，汉斯 · 克莱默斯按照时间顺序发现了一种描述摄动理论修正差异的方法——1947年6月,

Julian Schwinger,^{[4]}^{[5]}^{[6]}^{[7]} Richard Feynman,^{[8]}^{[9]}^{[10]} and Shin'ichiro Tomonaga,^{[11]}^{[12]}^{[13]}^{[14]}^{[15]}^{[16]}^{[17]} and systematized by Freeman Dyson in 1949.^{[18]} The divergences appear in radiative corrections involving Feynman diagrams with closed *loops* of virtual particles in them.

Julian Schwinger, Richard Feynman, and Shin'ichiro Tomonaga, and systematized by Freeman Dyson in 1949. The divergences appear in radiative corrections involving Feynman diagrams with closed loops of virtual particles in them.

根据 s. s. schweber，qed，1994，p. 269，koba-Tomonaga 包含了至关重要的计算公式，并在1949年由 Freeman Dyson 系统化。发散出现在费曼图的辐射修正中，其中包含虚粒子的闭环。

While virtual particles obey conservation of energy and momentum, they can have any energy and momentum, even one that is not allowed by the relativistic energy–momentum relation for the observed mass of that particle (that is, [math]\displaystyle{ E^2 - p^2 }[/math] is not necessarily the squared mass of the particle in that process, e.g. for a photon it could be nonzero). Such a particle is called off-shell. When there is a loop, the momentum of the particles involved in the loop is not uniquely determined by the energies and momenta of incoming and outgoing particles. A variation in the energy of one particle in the loop can be balanced by an equal and opposite change in the energy of another particle in the loop, without affecting the incoming and outgoing particles. Thus many variations are possible. So to find the amplitude for the loop process, one must integrate over *all* possible combinations of energy and momentum that could travel around the loop.

While virtual particles obey conservation of energy and momentum, they can have any energy and momentum, even one that is not allowed by the relativistic energy–momentum relation for the observed mass of that particle (that is, [math]\displaystyle{ E^2 - p^2 }[/math] is not necessarily the squared mass of the particle in that process, e.g. for a photon it could be nonzero). Such a particle is called off-shell. When there is a loop, the momentum of the particles involved in the loop is not uniquely determined by the energies and momenta of incoming and outgoing particles. A variation in the energy of one particle in the loop can be balanced by an equal and opposite change in the energy of another particle in the loop, without affecting the incoming and outgoing particles. Thus many variations are possible. So to find the amplitude for the loop process, one must integrate over all possible combinations of energy and momentum that could travel around the loop.

虽然虚粒子遵从能量和动量守恒，但是它们可以有任何能量和动量，甚至是那个粒子的观测质量的相对论能量-动量关系不允许的能量和动量(即 e ^ 2-p ^ 2 </math > 不一定是那个过程中粒子的质量的平方，例如:。对于一个光子来说，它可能是非零的)。这种粒子叫做脱壳粒子。当环路存在时，环路中的粒子的动量不是由入射粒子和出射粒子的能量和动量决定的。环中一个粒子能量的变化可以通过环中另一个粒子能量的相等和相反的变化来平衡，而不影响入射和出射粒子。因此有许多变化是可能的。所以为了找到这个循环过程的振幅，我们必须积分所有可能的能量和动量的组合，这些能量和动量可以在循环中传播。

These integrals are often *divergent*, that is, they give infinite answers. The divergences that are significant are the "ultraviolet" (UV) ones. An ultraviolet divergence can be described as one that comes from

These integrals are often divergent, that is, they give infinite answers. The divergences that are significant are the "ultraviolet" (UV) ones. An ultraviolet divergence can be described as one that comes from

这些积分常常是发散的，也就是说，它们给出无穷的答案。最显著的分歧是“紫外线”(UV)分歧。紫外线辐散可以描述为一个来自

- the region in the integral where all particles in the loop have large energies and momenta,

- very short wavelengths and high-frequencies fluctuations of the fields, in the path integral for the field,

- very short proper-time between particle emission and absorption, if the loop is thought of as a sum over particle paths.

So these divergences are short-distance, short-time phenomena.

So these divergences are short-distance, short-time phenomena.

所以这些分歧是短距离、短时间的现象。

Shown in the pictures at the right margin, there are exactly three one-loop divergent loop diagrams in quantum electrodynamics:^{[19]}

Shown in the pictures at the right margin, there are exactly three one-loop divergent loop diagrams in quantum electrodynamics:

在右边的图片中，我们可以看到在量子电动力学中有三个单圈的发散性循环图表:

- (a) A photon creates a virtual electron–positron pair, which then annihilates. This is a vacuum polarization diagram.

(a) A photon creates a virtual electron–positron pair, which then annihilates. This is a vacuum polarization diagram.

(a)光子产生一个虚电子-正电子对，这个对随后湮灭。这是一个真空极化图表。

- (b) An electron quickly emits and reabsorbs a virtual photon, called a self-energy.

(b) An electron quickly emits and reabsorbs a virtual photon, called a self-energy.

一个电子很快地放射并重新吸收一个虚拟的光子，称为自能。

- (c) An electron emits a photon, emits a second photon, and reabsorbs the first. This process is shown in the section below in figure 2, and it is called a
*vertex renormalization*. The Feynman diagram for this is also called a “penguin diagram” due to its shape remotely resembling a penguin.

(c) An electron emits a photon, emits a second photon, and reabsorbs the first. This process is shown in the section below in figure 2, and it is called a vertex renormalization. The Feynman diagram for this is also called a “penguin diagram” due to its shape remotely resembling a penguin.

(c)电子发射一个光子，第二个光子，并重新吸收第一个光子。这个过程如下面图2中的部分所示，它被称为顶点重整化。这幅费曼图也被称为“企鹅图” ，因为它的形状远远类似于企鹅。

The three divergences correspond to the three parameters in the theory under consideration:

The three divergences correspond to the three parameters in the theory under consideration:

这三种不同对应于所审议的理论中的三个参数:

- The field normalization Z.

The field normalization Z.

场的归一化 z。

- The mass of the electron.

The mass of the electron.

电子的质量。

- The charge of the electron.

The charge of the electron.

电荷电子的电荷。

The second class of divergence called an infrared divergence, is due to massless particles, like the photon. Every process involving charged particles emits infinitely many coherent photons of infinite wavelength, and the amplitude for emitting any finite number of photons is zero. For photons, these divergences are well understood. For example, at the 1-loop order, the vertex function has both ultraviolet and *infrared* divergences. In contrast to the ultraviolet divergence, the infrared divergence does not require the renormalization of a parameter in the theory involved. The infrared divergence of the vertex diagram is removed by including a diagram similar to the vertex diagram with the following important difference: the photon connecting the two legs of the electron is cut and replaced by two on-shell (i.e. real) photons whose wavelengths tend to infinity; this diagram is equivalent to the bremsstrahlung process. This additional diagram must be included because there is no physical way to distinguish a zero-energy photon flowing through a loop as in the vertex diagram and zero-energy photons emitted through bremsstrahlung. From a mathematical point of view, the IR divergences can be regularized by assuming fractional differentiation w.r.t. a parameter, for example:

The second class of divergence called an infrared divergence, is due to massless particles, like the photon. Every process involving charged particles emits infinitely many coherent photons of infinite wavelength, and the amplitude for emitting any finite number of photons is zero. For photons, these divergences are well understood. For example, at the 1-loop order, the vertex function has both ultraviolet and infrared divergences. In contrast to the ultraviolet divergence, the infrared divergence does not require the renormalization of a parameter in the theory involved. The infrared divergence of the vertex diagram is removed by including a diagram similar to the vertex diagram with the following important difference: the photon connecting the two legs of the electron is cut and replaced by two on-shell (i.e. real) photons whose wavelengths tend to infinity; this diagram is equivalent to the bremsstrahlung process. This additional diagram must be included because there is no physical way to distinguish a zero-energy photon flowing through a loop as in the vertex diagram and zero-energy photons emitted through bremsstrahlung. From a mathematical point of view, the IR divergences can be regularized by assuming fractional differentiation w.r.t. a parameter, for example:

第二类发散叫做红外发散，是由于无质量的粒子，比如光子。每一个涉及带电粒子的过程都会发射无限多个波长相干的光子，而发射任意有限数量光子的振幅为零。对于光子来说，这些发散是可以理解的。例如，在1环阶段，顶点函数有紫外和红外发散。与紫外发散相比，红外发散不需要理论中参数的重整化。通过在顶点图中加入一个类似于顶点图的图来消除顶点图的红外发散，这个图有以下重要的区别: 连接电子两个支柱的光子被切割并被两个在壳层上的(即:。波长趋于无穷大的光子; 这个图表等价于韧致辐射过程。这个附加图必须包括在内，因为没有物理方法可以区分零能光子流过一个环，就像顶点图和通过韧致辐射发射的零能光子一样。从数学的角度来看，通过假设分数阶微分，可以正则化红外发散。一个参数，例如:

- [math]\displaystyle{ \left( p^2 - a^2 \right)^{\frac{1}{2}} }[/math]

[math]\displaystyle{ \left( p^2 - a^2 \right)^{\frac{1}{2}} }[/math]

左(p ^ 2-a ^ 2 right) ^ { frac {1}{2}} </math >

is well defined at *p* = *a* but is UV divergent; if we take the 模板:Frac-th fractional derivative with respect to −*a*^{2}, we obtain the IR divergence

is well defined at a}} but is UV divergent; if we take the -th fractional derivative with respect to , we obtain the IR divergence

在 a }处定义良好，但是是紫外发散的，如果我们对 a }取第-次分数导数，我们就得到了红外发散

- [math]\displaystyle{ \frac{1}{p^2 - a^2}, }[/math]

[math]\displaystyle{ \frac{1}{p^2 - a^2}, }[/math]

1}{ p ^ 2-a ^ 2} ,

so we can cure IR divergences by turning them into UV divergences.模板:Clarify

so we can cure IR divergences by turning them into UV divergences.

因此，我们可以通过将红外辐射分歧转化为紫外辐射分歧来治疗红外辐射分歧。

### A loop divergence

Figure 2. A diagram contributing to electron–electron scattering in QED. The loop has an ultraviolet divergence.

图2。量子电动力学中电子-电子散射的图解。这个环有一个紫外辐射。

The diagram in Figure 2 shows one of the several one-loop contributions to electron–electron scattering in QED. The electron on the left side of the diagram, represented by the solid line, starts out with 4-momentum *p ^{μ}* and ends up with 4-momentum

*r*. It emits a virtual photon carrying

^{μ}*r*−

^{μ}*p*to transfer energy and momentum to the other electron. But in this diagram, before that happens, it emits another virtual photon carrying 4-momentum

^{μ}*q*, and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the 4-momentum

^{μ}*q*uniquely, so all possibilities contribute equally and we must integrate.

^{μ}The diagram in Figure 2 shows one of the several one-loop contributions to electron–electron scattering in QED. The electron on the left side of the diagram, represented by the solid line, starts out with 4-momentum and ends up with 4-momentum . It emits a virtual photon carrying to transfer energy and momentum to the other electron. But in this diagram, before that happens, it emits another virtual photon carrying 4-momentum , and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the 4-momentum uniquely, so all possibilities contribute equally and we must integrate.

图2中的图表显示了 QED 中电子-电子散射的几个单环贡献之一。图中左边的电子，用实线表示，以4- 动量开始，以4- 动量结束。它放射出一个虚光子，携带能量和动量传递给另一个电子。但是在这个图表中，在它发生之前，它会放出另一个虚光子，携带4个动量，在放出另一个虚光子之后，它会重新吸收这个。能量守恒和动量守恒不能独特地决定4- 动量，因此所有的可能性都平等地作出贡献，我们必须加以整合。

This diagram's amplitude ends up with, among other things, a factor from the loop of

This diagram's amplitude ends up with, among other things, a factor from the loop of

这个图表的振幅，除了其他因素之外，还有一个来自

- [math]\displaystyle{ -ie^3 \int \frac{d^4 q}{(2\pi)^4} \gamma^\mu \frac{i (\gamma^\alpha (r - q)_\alpha + m)}{(r - q)^2 - m^2 + i \epsilon} \gamma^\rho \frac{i (\gamma^\beta (p - q)_\beta + m)}{(p - q)^2 - m^2 + i \epsilon} \gamma^\nu \frac{-i g_{\mu\nu}}{q^2 + i\epsilon}. }[/math]

[math]\displaystyle{ -ie^3 \int \frac{d^4 q}{(2\pi)^4} \gamma^\mu \frac{i (\gamma^\alpha (r - q)_\alpha + m)}{(r - q)^2 - m^2 + i \epsilon} \gamma^\rho \frac{i (\gamma^\beta (p - q)_\beta + m)}{(p - q)^2 - m^2 + i \epsilon} \gamma^\nu \frac{-i g_{\mu\nu}}{q^2 + i\epsilon}. }[/math]

(2 pi) ^ 4} gamma ^ mu frac { i (gamma ^ alpha (r-q) _ alpha + m)}{(r-q) ^ 2-m ^ 2 + i epsilon } gamma ^ rho frac { i (gamma ^ beta (p-q) _ beta + m)}{(p-q) ^ 2-m ^ 2 + i epsilon } nu frac { i { nu ^ 2 + i epsilon }.数学

The various *γ ^{μ}* factors in this expression are gamma matrices as in the covariant formulation of the Dirac equation; they have to do with the spin of the electron. The factors of e are the electric coupling constant, while the [math]\displaystyle{ i\epsilon }[/math] provide a heuristic definition of the contour of integration around the poles in the space of momenta. The important part for our purposes is the dependency on

*q*of the three big factors in the integrand, which are from the propagators of the two electron lines and the photon line in the loop.

^{μ}The various factors in this expression are gamma matrices as in the covariant formulation of the Dirac equation; they have to do with the spin of the electron. The factors of are the electric coupling constant, while the [math]\displaystyle{ i\epsilon }[/math] provide a heuristic definition of the contour of integration around the poles in the space of momenta. The important part for our purposes is the dependency on of the three big factors in the integrand, which are from the propagators of the two electron lines and the photon line in the loop.

这个表达式中的各种因素是类似于狄拉克方程的协变公式中的伽马矩阵，它们与电子的自旋有关。其因子是耦合常数，而动量空间中极点周围的积分等高线提供了一个启发式的定义。本文研究的重点是被积函数中三个重要因子的依赖性，它们分别来自于环路中的两电子线和光子线的传播子。

This has a piece with two powers of *q ^{μ}* on top that dominates at large values of

*q*(Pokorski 1987, p. 122):

^{μ}This has a piece with two powers of on top that dominates at large values of (Pokorski 1987, p. 122):

这里有一个顶上有两个幂的函数，它在大值上占主导地位(Pokorski 1987，第122页) :

- [math]\displaystyle{ e^3 \gamma^\mu \gamma^\alpha \gamma^\rho \gamma^\beta \gamma_\mu \int \frac{d^4 q}{(2\pi)^4} \frac{q_\alpha q_\beta}{(r - q)^2 (p - q)^2 q^2}. }[/math]

[math]\displaystyle{ e^3 \gamma^\mu \gamma^\alpha \gamma^\rho \gamma^\beta \gamma_\mu \int \frac{d^4 q}{(2\pi)^4} \frac{q_\alpha q_\beta}{(r - q)^2 (p - q)^2 q^2}. }[/math]

< math > e ^ 3 gamma ^ mu gamma ^ alpha gamma ^ rho ^ rho gamma ^ beta gamma mu int frac { d ^ 4 q }{(2 pi) ^ 4} frac { q _ alpha q _ beta }{(r-q) ^ 2(p-q) ^ 2 q ^ 2} . </math >

This integral is divergent and infinite, unless we cut it off at finite energy and momentum in some way.

This integral is divergent and infinite, unless we cut it off at finite energy and momentum in some way.

这个积分是发散的，也是无穷的，除非我们以某种方式把它截断，在有限的能量和动量下。

Similar loop divergences occur in other quantum field theories.

Similar loop divergences occur in other quantum field theories.

类似的圈发散现象在其他量子场理论中也有发生。

## Renormalized and bare quantities

The solution was to realize that the quantities initially appearing in the theory's formulae (such as the formula for the Lagrangian), representing such things as the electron's electric charge and mass, as well as the normalizations of the quantum fields themselves, did *not* actually correspond to the physical constants measured in the laboratory. As written, they were *bare* quantities that did not take into account the contribution of virtual-particle loop effects to *the physical constants themselves*. Among other things, these effects would include the quantum counterpart of the electromagnetic back-reaction that so vexed classical theorists of electromagnetism. In general, these effects would be just as divergent as the amplitudes under consideration in the first place; so finite measured quantities would, in general, imply divergent bare quantities.

The solution was to realize that the quantities initially appearing in the theory's formulae (such as the formula for the Lagrangian), representing such things as the electron's electric charge and mass, as well as the normalizations of the quantum fields themselves, did not actually correspond to the physical constants measured in the laboratory. As written, they were bare quantities that did not take into account the contribution of virtual-particle loop effects to the physical constants themselves. Among other things, these effects would include the quantum counterpart of the electromagnetic back-reaction that so vexed classical theorists of electromagnetism. In general, these effects would be just as divergent as the amplitudes under consideration in the first place; so finite measured quantities would, in general, imply divergent bare quantities.

解决方案是认识到最初出现在理论公式中的量(比如拉格朗日公式) ，代表电子的电荷和质量，以及量子场本身的归一化，实际上并不符合在实验室测量的物理常数。如上所述，它们是裸量，没有考虑虚粒子环效应对物理常数本身的贡献。除此之外，这些影响还包括电磁反作用的量子对应物，这让电磁学的经典理论家们非常恼火。一般来说，这些效应就像首先考虑的振幅一样发散; 所以有限的测量量，一般来说，意味着发散裸量。

To make contact with reality, then, the formulae would have to be rewritten in terms of measurable, *renormalized* quantities. The charge of the electron, say, would be defined in terms of a quantity measured at a specific kinematic *renormalization point* or *subtraction point* (which will generally have a characteristic energy, called the *renormalization scale* or simply the energy scale). The parts of the Lagrangian left over, involving the remaining portions of the bare quantities, could then be reinterpreted as counterterms, involved in divergent diagrams exactly *canceling out* the troublesome divergences for other diagrams.

To make contact with reality, then, the formulae would have to be rewritten in terms of measurable, renormalized quantities. The charge of the electron, say, would be defined in terms of a quantity measured at a specific kinematic renormalization point or subtraction point (which will generally have a characteristic energy, called the renormalization scale or simply the energy scale). The parts of the Lagrangian left over, involving the remaining portions of the bare quantities, could then be reinterpreted as counterterms, involved in divergent diagrams exactly canceling out the troublesome divergences for other diagrams.

因此，为了与现实接触，这些公式必须以可测量的、重整化的量重写。例如，电子的电荷可以用在特定运动学重整化点或减点测量的量来定义(通常具有一个特征能量，称为重整化标度或简称为能量标度)。剩下的部分，包括剩下的裸量的部分，可以被重新解释为反项，包含在发散图中，完全抵消了其他图的麻烦的分歧。

### Renormalization in QED

Figure 3. The vertex corresponding to the counterterm cancels the divergence in Figure 2.

图3。图2中对应于反项的顶点抵消了发散。

For example, in the Lagrangian of QED

For example, in the Lagrangian of QED

例如，在 QED 的拉格朗日函数中

- [math]\displaystyle{ \mathcal{L}=\bar\psi_B\left[i\gamma_\mu \left (\partial^\mu + ie_BA_B^\mu \right )-m_B\right]\psi_B -\frac{1}{4}F_{B\mu\nu}F_B^{\mu\nu} }[/math]

[math]\displaystyle{ \mathcal{L}=\bar\psi_B\left[i\gamma_\mu \left (\partial^\mu + ie_BA_B^\mu \right )-m_B\right]\psi_B -\frac{1}{4}F_{B\mu\nu}F_B^{\mu\nu} }[/math]

“数学”{ l } = bar psi _ b 左[ i gamma _ mu 左(部分 ^ mu + ie _ ba _ b _ mu 右)-m _ b 右] psi _ b-frac {1}{4} f { b mu } f ^ { mu } </math >

the fields and coupling constant are really *bare* quantities, hence the subscript B above. Conventionally the bare quantities are written so that the corresponding Lagrangian terms are multiples of the renormalized ones:

the fields and coupling constant are really bare quantities, hence the subscript above. Conventionally the bare quantities are written so that the corresponding Lagrangian terms are multiples of the renormalized ones:

磁场和耦合常数实际上是极小的数量，因此上面的下标是这样的。通常，赤裸的量被写成相应的拉格朗日项是重整化项的倍数:

- [math]\displaystyle{ \left(\bar\psi m \psi\right)_B = Z_0 \bar\psi m \psi }[/math]

[math]\displaystyle{ \left(\bar\psi m \psi\right)_B = Z_0 \bar\psi m \psi }[/math]

左边(bar psi，右边) _ b = z _ 0 bar psi，psi

- [math]\displaystyle{ \left(\bar\psi\left(\partial^\mu + ieA^\mu \right )\psi\right)_B = Z_1 \bar\psi \left (\partial^\mu + ieA^\mu \right)\psi }[/math]

[math]\displaystyle{ \left(\bar\psi\left(\partial^\mu + ieA^\mu \right )\psi\right)_B = Z_1 \bar\psi \left (\partial^\mu + ieA^\mu \right)\psi }[/math]

左(bar psi 左(partial ^ mu + ieA ^ mu right) psi 右) _ b = z _ 1 bar psi 左(partial ^ mu + ieA ^ mu right) psi

- [math]\displaystyle{ \left(F_{\mu\nu}F^{\mu\nu}\right)_B = Z_3\, F_{\mu\nu}F^{\mu\nu}. }[/math]

[math]\displaystyle{ \left(F_{\mu\nu}F^{\mu\nu}\right)_B = Z_3\, F_{\mu\nu}F^{\mu\nu}. }[/math]

[math]\displaystyle{ \left(F_{\mu\nu}F^{\mu\nu}\right)_B = Z_3\, F_{\mu\nu}F^{\mu\nu}. }[/math]

Gauge invariance, via a Ward–Takahashi identity, turns out to imply that we can renormalize the two terms of the covariant derivative piece

Gauge invariance, via a Ward–Takahashi identity, turns out to imply that we can renormalize the two terms of the covariant derivative piece

规范不变性，通过 Ward-Takahashi 恒等式，证明了我们可以重新整合共变导数的两个术语

- [math]\displaystyle{ \bar \psi (\partial + ieA) \psi }[/math]

[math]\displaystyle{ \bar \psi (\partial + ieA) \psi }[/math]

巴特普赛(部分 + 国际能源署)

together (Pokorski 1987, p. 115), which is what happened to *Z*_{2}; it is the same as *Z*_{1}.

together (Pokorski 1987, p. 115), which is what happened to ; it is the same as .

一起(Pokorski 1987，第115页) ，这是发生了什么; 它是相同的。

A term in this Lagrangian, for example, the electron-photon interaction pictured in Figure 1, can then be written

A term in this Lagrangian, for example, the electron-photon interaction pictured in Figure 1, can then be written

这个拉格朗日函数中的一个术语，例如，图1所示的电子-光子相互作用，就可以写出来

- [math]\displaystyle{ \mathcal{L}_I = -e \bar\psi \gamma_\mu A^\mu \psi - (Z_1 - 1) e \bar\psi \gamma_\mu A^\mu \psi }[/math]

[math]\displaystyle{ \mathcal{L}_I = -e \bar\psi \gamma_\mu A^\mu \psi - (Z_1 - 1) e \bar\psi \gamma_\mu A^\mu \psi }[/math]

< math > mathcal { l } _ i =-e bar psi gamma _ mu a ^ mu psi-(z _ 1-1) e bar psi gamma _ mu a ^ mu psi </math >

The physical constant e, the electron's charge, can then be defined in terms of some specific experiment: we set the renormalization scale equal to the energy characteristic of this experiment, and the first term gives the interaction we see in the laboratory (up to small, finite corrections from loop diagrams, providing such exotica as the high-order corrections to the magnetic moment). The rest is the counterterm. If the theory is *renormalizable* (see below for more on this), as it is in QED, the *divergent* parts of loop diagrams can all be decomposed into pieces with three or fewer legs, with an algebraic form that can be canceled out by the second term (or by the similar counterterms that come from *Z*_{0} and *Z*_{3}).

The physical constant , the electron's charge, can then be defined in terms of some specific experiment: we set the renormalization scale equal to the energy characteristic of this experiment, and the first term gives the interaction we see in the laboratory (up to small, finite corrections from loop diagrams, providing such exotica as the high-order corrections to the magnetic moment). The rest is the counterterm. If the theory is renormalizable (see below for more on this), as it is in QED, the divergent parts of loop diagrams can all be decomposed into pieces with three or fewer legs, with an algebraic form that can be canceled out by the second term (or by the similar counterterms that come from and ).

物理常数，也就是电子的电荷，可以用一些特定的实验来定义: 我们把重整化标度设置为与这个实验的能量特性相等，第一个项给出了我们在实验室中看到的相互作用(从环形图中可以得到小的、有限的修正，提供诸如磁矩的高阶修正)。剩下的就是反条件了。如果理论是可重整化的(更多内容见下文) ，就像 QED 中一样，环路图的分叉部分都可以分解成三个或更少的部分，并且可以用第二个项(或者来自和的类似的反项)抵消代数形式。

The diagram with the *Z*_{1} counterterm's interaction vertex placed as in Figure 3 cancels out the divergence from the loop in Figure 2.

The diagram with the counterterm's interaction vertex placed as in Figure 3 cancels out the divergence from the loop in Figure 2.

如图3所示，将反项的交互顶点放置在图3中的图抵消了图2中与循环的发散。

Historically, the splitting of the "bare terms" into the original terms and counterterms came before the renormalization group insight due to Kenneth Wilson.^{[20]} According to such renormalization group insights, detailed in the next section, this splitting is unnatural and actually unphysical, as all scales of the problem enter in continuous systematic ways.

Historically, the splitting of the "bare terms" into the original terms and counterterms came before the renormalization group insight due to Kenneth Wilson. According to such renormalization group insights, detailed in the next section, this splitting is unnatural and actually unphysical, as all scales of the problem enter in continuous systematic ways.

从历史上看，将“无条件术语”分解为原始术语和对位术语的做法，早于肯尼思 · 威尔逊对重整化群的洞察力。根据这些重整化群的见解，这种分裂是非自然的，实际上是非物理的，因为问题的所有尺度都是以连续的系统方式进入的。

### Running couplings

To minimize the contribution of loop diagrams to a given calculation (and therefore make it easier to extract results), one chooses a renormalization point close to the energies and momenta exchanged in the interaction. However, the renormalization point is not itself a physical quantity: the physical predictions of the theory, calculated to all orders, should in principle be *independent* of the choice of renormalization point, as long as it is within the domain of application of the theory. Changes in renormalization scale will simply affect how much of a result comes from Feynman diagrams without loops, and how much comes from the remaining finite parts of loop diagrams. One can exploit this fact to calculate the effective variation of physical constants with changes in scale. This variation is encoded by beta-functions, and the general theory of this kind of scale-dependence is known as the renormalization group.

To minimize the contribution of loop diagrams to a given calculation (and therefore make it easier to extract results), one chooses a renormalization point close to the energies and momenta exchanged in the interaction. However, the renormalization point is not itself a physical quantity: the physical predictions of the theory, calculated to all orders, should in principle be independent of the choice of renormalization point, as long as it is within the domain of application of the theory. Changes in renormalization scale will simply affect how much of a result comes from Feynman diagrams without loops, and how much comes from the remaining finite parts of loop diagrams. One can exploit this fact to calculate the effective variation of physical constants with changes in scale. This variation is encoded by beta-functions, and the general theory of this kind of scale-dependence is known as the renormalization group.

为了尽量减少环路图对给定计算的贡献(从而使得计算结果更容易提取) ，我们选择一个重整化点接近相互作用中交换的能量和动量。然而，重整化点本身并不是一个物理量: 理论的物理预测，计算到所有的阶，原则上应该独立于重整化点的选择，只要它在理论的应用范围内。重整化尺度的变化将简单地影响有多少结果来自没有循环的费曼图，有多少结果来自循环图剩余的有限部分。人们可以利用这一事实来计算物理常数随规模变化的有效变化。这种变化是由 β 函数编码的，这种尺度依赖的一般理论被称为重整化群。

Colloquially, particle physicists often speak of certain physical "constants" as varying with the energy of interaction, though in fact, it is the renormalization scale that is the independent quantity. This *running* does, however, provide a convenient means of describing changes in the behavior of a field theory under changes in the energies involved in an interaction. For example, since the coupling in quantum chromodynamics becomes small at large energy scales, the theory behaves more like a free theory as the energy exchanged in an interaction becomes large – a phenomenon known as asymptotic freedom. Choosing an increasing energy scale and using the renormalization group makes this clear from simple Feynman diagrams; were this not done, the prediction would be the same, but would arise from complicated high-order cancellations.

Colloquially, particle physicists often speak of certain physical "constants" as varying with the energy of interaction, though in fact, it is the renormalization scale that is the independent quantity. This running does, however, provide a convenient means of describing changes in the behavior of a field theory under changes in the energies involved in an interaction. For example, since the coupling in quantum chromodynamics becomes small at large energy scales, the theory behaves more like a free theory as the energy exchanged in an interaction becomes large – a phenomenon known as asymptotic freedom. Choosing an increasing energy scale and using the renormalization group makes this clear from simple Feynman diagrams; were this not done, the prediction would be the same, but would arise from complicated high-order cancellations.

通俗地说，粒子物理学家经常说某些物理“常数”随着相互作用的能量而变化，尽管事实上，重整化标度才是独立量。然而，这种运行确实提供了一种方便的手段来描述场理论在相互作用所涉及的能量变化下的行为变化。例如，由于量子色动力学中的耦合在大能量尺度下变小，该理论表现得更像一个自由理论，因为在相互作用中交换的能量变大了---- 这种现象被称为渐近自由。选择一个递增的能量尺度并使用重整化群，可以从简单的费曼图中清楚地看出这一点; 如果不这样做，预测结果将是一样的，但是会出现复杂的高阶取消。

For example,

For example,

比如说,

- [math]\displaystyle{ I=\int_0^a \frac{1}{z}\,dz-\int_0^b \frac{1}{z}\,dz=\ln a-\ln b-\ln 0 +\ln 0 }[/math]

[math]\displaystyle{ I=\int_0^a \frac{1}{z}\,dz-\int_0^b \frac{1}{z}\,dz=\ln a-\ln b-\ln 0 +\ln 0 }[/math]

< math > i = int _ 0 ^ a frac {1}{ z } ，dz-int _ 0 ^ b frac {1}{ z } ，dz = ln a-ln b-ln 0 + ln 0 </math >

is ill-defined.

is ill-defined.

是不明确的。

To eliminate the divergence, simply change lower limit of integral into ε_{a} and ε_{b}:

To eliminate the divergence, simply change lower limit of integral into and :

为了消除散度，只需将积分的下限改为和:

- [math]\displaystyle{ I=\ln a-\ln b-\ln{\varepsilon_a}+\ln{\varepsilon_b} = \ln \tfrac{a}{b} - \ln \tfrac{\varepsilon_a}{\varepsilon_b} }[/math]

[math]\displaystyle{ I=\ln a-\ln b-\ln{\varepsilon_a}+\ln{\varepsilon_b} = \ln \tfrac{a}{b} - \ln \tfrac{\varepsilon_a}{\varepsilon_b} }[/math]

“ i = ln a-ln b-ln { varepsilon _ a } + ln { varepsilon _ b } = ln tfrac { a }{ b }-ln varepsilon _ a }{ varepsilon _ b } </math >

Making sure 模板:Sfrac → 1, then *I* = ln 模板:Sfrac.

Making sure → 1}}, then ln .}}

确保→1} ，然后 ln. }}

## Regularization

Since the quantity ∞ − ∞ is ill-defined, in order to make this notion of canceling divergences precise, the divergences first have to be tamed mathematically using the theory of limits, in a process known as regularization (Weinberg, 1995).

Since the quantity is ill-defined, in order to make this notion of canceling divergences precise, the divergences first have to be tamed mathematically using the theory of limits, in a process known as regularization (Weinberg, 1995).

由于这个量是不明确的，为了使取消发散的概念更加精确，首先必须用极限理论在数学上驯服这些发散，这个过程被称为正则化(Weinberg，1995)。

An essentially arbitrary modification to the loop integrands, or *regulator*, can make them drop off faster at high energies and momenta, in such a manner that the integrals converge. A regulator has a characteristic energy scale known as the cutoff; taking this cutoff to infinity (or, equivalently, the corresponding length/time scale to zero) recovers the original integrals.

An essentially arbitrary modification to the loop integrands, or regulator, can make them drop off faster at high energies and momenta, in such a manner that the integrals converge. A regulator has a characteristic energy scale known as the cutoff; taking this cutoff to infinity (or, equivalently, the corresponding length/time scale to zero) recovers the original integrals.

对环路被积函数或调节器进行任意的修改，可以使它们在高能量和动量下下降得更快，从而使被积函数收敛。调节器具有一个称为截止值的特征能量刻度; 将这个截止值取至无穷大(或者，等效地，将相应的长度/时间刻度取为零)恢复原始积分。

With the regulator in place, and a finite value for the cutoff, divergent terms in the integrals then turn into finite but cutoff-dependent terms. After canceling out these terms with the contributions from cutoff-dependent counterterms, the cutoff is taken to infinity and finite physical results recovered. If physics on scales we can measure is independent of what happens at the very shortest distance and time scales, then it should be possible to get cutoff-independent results for calculations.

With the regulator in place, and a finite value for the cutoff, divergent terms in the integrals then turn into finite but cutoff-dependent terms. After canceling out these terms with the contributions from cutoff-dependent counterterms, the cutoff is taken to infinity and finite physical results recovered. If physics on scales we can measure is independent of what happens at the very shortest distance and time scales, then it should be possible to get cutoff-independent results for calculations.

当调节器到位，并且截止值有限时，积分中的发散项就会变成有限但是依赖于截止项的项。在用截止相关反项的贡献抵消这些项之后，截止项被取到无穷远处，并得到有限的物理结果。如果我们可以测量的尺度上的物理是独立于在最短的距离和时间尺度上发生的事情，那么就有可能得到截止独立的计算结果。

Many different types of regulator are used in quantum field theory calculations, each with its advantages and disadvantages. One of the most popular in modern use is *dimensional regularization*, invented by Gerardus 't Hooft and Martinus J. G. Veltman,^{[21]} which tames the integrals by carrying them into a space with a fictitious fractional number of dimensions. Another is *Pauli–Villars regularization*, which adds fictitious particles to the theory with very large masses, such that loop integrands involving the massive particles cancel out the existing loops at large momenta.

Many different types of regulator are used in quantum field theory calculations, each with its advantages and disadvantages. One of the most popular in modern use is dimensional regularization, invented by Gerardus 't Hooft and Martinus J. G. Veltman, which tames the integrals by carrying them into a space with a fictitious fractional number of dimensions. Another is Pauli–Villars regularization, which adds fictitious particles to the theory with very large masses, such that loop integrands involving the massive particles cancel out the existing loops at large momenta.

量子场论计算中使用了许多不同类型的调节器，各有优缺点。在现代应用中最流行的是维度正则化，由 Gerardus’ t Hooft 和马丁纽斯·韦尔特曼发明，它通过将积分带入一个虚拟的分数维空间来驯服积分。另一个是泡利-维拉正则化，它在理论中加入了具有很大质量的虚构粒子，使得包含大质量粒子的环积分在很大动量时抵消了现有的环积分。

Yet another regularization scheme is the *lattice regularization*, introduced by Kenneth Wilson, which pretends that hyper-cubical lattice constructs our space-time with fixed grid size. This size is a natural cutoff for the maximal momentum that a particle could possess when propagating on the lattice. And after doing a calculation on several lattices with different grid size, the physical result is extrapolated to grid size 0, or our natural universe. This presupposes the existence of a scaling limit.

Yet another regularization scheme is the lattice regularization, introduced by Kenneth Wilson, which pretends that hyper-cubical lattice constructs our space-time with fixed grid size. This size is a natural cutoff for the maximal momentum that a particle could possess when propagating on the lattice. And after doing a calculation on several lattices with different grid size, the physical result is extrapolated to grid size 0, or our natural universe. This presupposes the existence of a scaling limit.

另一个正则化方案是由 Kenneth Wilson 提出的格子正则化方案，它假装超立方格子以固定的网格大小构造我们的时空。这个尺寸是粒子在晶格上传播时所能拥有的最大动量的自然截止值。在不同网格大小的格子上进行计算之后，物理结果被外推到网格大小0，或者我们的自然宇宙。这预先假定存在一个标度极限。

A rigorous mathematical approach to renormalization theory is the so-called causal perturbation theory, where ultraviolet divergences are avoided from the start in calculations by performing well-defined mathematical operations only within the framework of distribution theory. In this approach, divergences are replaced by ambiguity: corresponding to a divergent diagram is a term which now has a finite, but undetermined, coefficient. Other principles, such as gauge symmetry, must then be used to reduce or eliminate the ambiguity.

A rigorous mathematical approach to renormalization theory is the so-called causal perturbation theory, where ultraviolet divergences are avoided from the start in calculations by performing well-defined mathematical operations only within the framework of distribution theory. In this approach, divergences are replaced by ambiguity: corresponding to a divergent diagram is a term which now has a finite, but undetermined, coefficient. Other principles, such as gauge symmetry, must then be used to reduce or eliminate the ambiguity.

一个严格的重整化理论的数学方法是所谓的因果摄动理论，其中紫外线差异是避免从一开始的计算，通过执行良好定义的数学运算只在分布理论的框架内。在这种方法中，发散被模糊性所代替: 对应于发散图的项现在有一个有限的，但是未确定的系数。其他原则，例如规范对称，必须用来减少或消除模糊性。

### Zeta function regularization

Julian Schwinger discovered a relationship^{[citation needed]} between zeta function regularization and renormalization, using the asymptotic relation:

Julian Schwinger discovered a relationship between zeta function regularization and renormalization, using the asymptotic relation:

朱利安·施温格利用渐近关系发现了 Ζ函数正规化和重整化之间的关系:

- [math]\displaystyle{ I(n, \Lambda )= \int_0^{\Lambda }dp\,p^n \sim 1+2^n+3^n+\cdots+ \Lambda^n \to \zeta(-n) }[/math]

[math]\displaystyle{ I(n, \Lambda )= \int_0^{\Lambda }dp\,p^n \sim 1+2^n+3^n+\cdots+ \Lambda^n \to \zeta(-n) }[/math]

I (n，Lambda) = int _ 0 ^ { Lambda } dp，p ^ n sim 1 + 2 ^ n + 3 ^ n + cdots + Lambda ^ n to zeta (- n) </math >

as the regulator Λ → ∞. Based on this, he considered using the values of *ζ*(−*n*) to get finite results. Although he reached inconsistent results, an improved formula studied by Hartle, J. Garcia, and based on the works by E. Elizalde includes the technique of the zeta regularization algorithm

as the regulator . Based on this, he considered using the values of to get finite results. Although he reached inconsistent results, an improved formula studied by Hartle, J. Garcia, and based on the works by E. Elizalde includes the technique of the zeta regularization algorithm

作为调节器。在此基础上，他考虑使用值得到有限的结果。虽然他得到了不一致的结果，一个改进的公式研究哈特尔，j. 加西亚，并基于 e. Elizalde 的工作，包括技术 zeta 正则化算法

- [math]\displaystyle{ I(n, \Lambda) = \frac{n}{2}I(n-1, \Lambda) + \zeta(-n) - \sum_{r=1}^{\infty}\frac{B_{2r}}{(2r)!} a_{n,r}(n-2r+1) I(n-2r, \Lambda), }[/math]

[math]\displaystyle{ I(n, \Lambda) = \frac{n}{2}I(n-1, \Lambda) + \zeta(-n) - \sum_{r=1}^{\infty}\frac{B_{2r}}{(2r)!} a_{n,r}(n-2r+1) I(n-2r, \Lambda), }[/math]

< math > i (n，Lambda) = frac { n }{2} i (n-1，Lambda) + zeta (- n)-sum { r = 1} ^ { infty } frac { b _ {2r }{(2r) ! }A _ { n，r }(n-2r + 1) i (n-2r，Lambda) ，</math >

where the *B'*s are the Bernoulli numbers and

where the Bs are the Bernoulli numbers and

伯努利数和在哪里

- [math]\displaystyle{ a_{n,r}= \frac{\Gamma(n+1)}{\Gamma(n-2r+2)}. }[/math]

[math]\displaystyle{ a_{n,r}= \frac{\Gamma(n+1)}{\Gamma(n-2r+2)}. }[/math]

< math > a _ { n，r } = frac { Gamma (n + 1)}{ Gamma (n-2r + 2)}。 </math >

So every *I*(*m*, Λ) can be written as a linear combination of *ζ*(−1), *ζ*(−3), *ζ*(−5), ..., *ζ*(−*m*).

So every can be written as a linear combination of .

所以每个线性组合都可以被写成。

Or simply using Abel–Plana formula we have for every divergent integral:

Or simply using Abel–Plana formula we have for every divergent integral:

或者简单地用 Abel-Plana 公式来计算每一个发散积分:

- [math]\displaystyle{ \zeta(-m, \beta )-\frac{\beta ^{m}}{2}-i\int_ 0 ^{\infty}dt \frac{ (it+\beta)^{m}-(-it+\beta)^{m}}{e^{2 \pi t}-1}=\int_0^\infty dp \, (p+\beta)^m }[/math]

[math]\displaystyle{ \zeta(-m, \beta )-\frac{\beta ^{m}}{2}-i\int_ 0 ^{\infty}dt \frac{ (it+\beta)^{m}-(-it+\beta)^{m}}{e^{2 \pi t}-1}=\int_0^\infty dp \, (p+\beta)^m }[/math]

{2}-i int _ 0 ^ { infty } dt frac {(it + beta) ^ { m }-(- it + beta) ^ { m }{ m }{ e ^ {2 pi }-1} = int _ 0 ^ infty dp，(p + beta) ^ m </math >

valid when *m* > 0, Here the zeta function is Hurwitz zeta function and Beta is a positive real number.

valid when , Here the zeta function is Hurwitz zeta function and Beta is a positive real number.

这里 zeta 函数是赫尔维茨ζ函数，Beta 是正实数。

The "geometric" analogy is given by, (if we use rectangle method) to evaluate the integral so:

The "geometric" analogy is given by, (if we use rectangle method) to evaluate the integral so:

这个“几何”类比是由，(如果我们使用矩形法)来计算积分的话:

- [math]\displaystyle{ \int_0^\infty dx \, (\beta +x)^m \approx \sum_{n=0}^\infty h^{m+1} \zeta \left( \beta h^{-1} , -m \right) }[/math]

[math]\displaystyle{ \int_0^\infty dx \, (\beta +x)^m \approx \sum_{n=0}^\infty h^{m+1} \zeta \left( \beta h^{-1} , -m \right) }[/math]

< math > int _ 0 ^ infty dx，(beta + x) ^ m approx sum _ { n = 0} ^ infty h ^ { m + 1} zeta left (beta h ^ {-1} ,-m right) </math >

Using Hurwitz zeta regularization plus the rectangle method with step h (not to be confused with Planck's constant).

Using Hurwitz zeta regularization plus the rectangle method with step h (not to be confused with Planck's constant).

使用 Hurwitz zeta 正则化加上步骤 h 的矩形法方程(不要与 Planck 常数混淆)。

The logarithmic divergent integral has the regularization

The logarithmic divergent integral has the regularization

对数发散积分具有正则化

- [math]\displaystyle{ \sum_{n=0}^{\infty} \frac{1}{n+a}= - \psi (a)+\log (a) }[/math]

[math]\displaystyle{ \sum_{n=0}^{\infty} \frac{1}{n+a}= - \psi (a)+\log (a) }[/math]

1}{ n + a } =-psi (a) + log (a) </math >

since for the Harmonic series [math]\displaystyle{ \sum_{n=0}^{\infty} \frac{1}{an+1} }[/math] in the limit [math]\displaystyle{ a \to 0 }[/math] we must recover the series [math]\displaystyle{ \sum_{n=0}^{\infty}1 =1/2 }[/math]

since for the Harmonic series [math]\displaystyle{ \sum_{n=0}^{\infty} \frac{1}{an+1} }[/math] in the limit [math]\displaystyle{ a \to 0 }[/math] we must recover the series [math]\displaystyle{ \sum_{n=0}^{\infty}1 =1/2 }[/math]

因为对于调和数列 < math > sum { n = 0} ^ { infty } frac {1}{ an + 1} </math > 在极限 < math > a 到0 </math > 我们必须恢复数列 < math > sum { n = 0} ^ { infty > 1 = 1/2 </math >

For multi-loop integrals that will depend on several variables [math]\displaystyle{ k_1, \cdots, k_n }[/math] we can make a change of variables to polar coordinates and then replace the integral over the angles [math]\displaystyle{ \int d \Omega }[/math] by a sum so we have only a divergent integral, that will depend on the modulus [math]\displaystyle{ r^2 = k_1^2 +\cdots+k_n^2 }[/math] and then we can apply the zeta regularization algorithm, the main idea for multi-loop integrals is to replace the factor [math]\displaystyle{ F(q_1,\cdots,q_n) }[/math] after a change to hyperspherical coordinates *F*(*r*, Ω) so the UV overlapping divergences are encoded in variable r. In order to regularize these integrals one needs a regulator, for the case of multi-loop integrals, these regulator can be taken as

For multi-loop integrals that will depend on several variables [math]\displaystyle{ k_1, \cdots, k_n }[/math] we can make a change of variables to polar coordinates and then replace the integral over the angles [math]\displaystyle{ \int d \Omega }[/math] by a sum so we have only a divergent integral, that will depend on the modulus [math]\displaystyle{ r^2 = k_1^2 +\cdots+k_n^2 }[/math] and then we can apply the zeta regularization algorithm, the main idea for multi-loop integrals is to replace the factor [math]\displaystyle{ F(q_1,\cdots,q_n) }[/math] after a change to hyperspherical coordinates so the UV overlapping divergences are encoded in variable . In order to regularize these integrals one needs a regulator, for the case of multi-loop integrals, these regulator can be taken as

对于依赖于多个变量的多循环积分，我们可以将变量改为极坐标，然后用一个和来代替角度上的积分，这样我们只有发散积分,这取决于模数 < math > r ^ 2 = k _ 1 ^ 2 + cdots + k _ n ^ 2 </math > ，然后我们可以应用 zeta 正则化算法，多循环积分的主要思想是将因子 < math > f (q _ 1，cdots，q _ n) </math > 变换为超球坐标后，将 UV 重叠发散编码为可变的。为了使这些积分正则化，需要一个调节器，对于多环积分，这些调节器可以看作是

- [math]\displaystyle{ \left (1+ \sqrt{q}_{i}q^{i} \right )^{-s} }[/math]

[math]\displaystyle{ \left (1+ \sqrt{q}_{i}q^{i} \right )^{-s} }[/math]

< math > left (1 + sqrt { q } _ { i } q ^ { i } right) ^ {-s } </math >

so the multi-loop integral will converge for big enough s using the Zeta regularization we can analytic continue the variable s to the physical limit where *s* = 0 and then regularize any UV integral, by replacing a divergent integral by a linear combination of divergent series, which can be regularized in terms of the negative values of the Riemann zeta function *ζ*(−*m*).

so the multi-loop integral will converge for big enough using the Zeta regularization we can analytic continue the variable to the physical limit where 0}} and then regularize any UV integral, by replacing a divergent integral by a linear combination of divergent series, which can be regularized in terms of the negative values of the Riemann zeta function .

因此，多回路积分将收敛到足够大的使用 Zeta 正则化，我们可以解析变量继续到物理极限0}的地方，然后正则化任何紫外积分，通过替换发散积分的一个线性组合的发散级数，这可以正则化的黎曼ζ函数的负值。

## Attitudes and interpretation

The early formulators of QED and other quantum field theories were, as a rule, dissatisfied with this state of affairs. It seemed illegitimate to do something tantamount to subtracting infinities from infinities to get finite answers.

The early formulators of QED and other quantum field theories were, as a rule, dissatisfied with this state of affairs. It seemed illegitimate to do something tantamount to subtracting infinities from infinities to get finite answers.

量子电动力学和其他量子场论的早期公式制定者通常对这种状态不满意。为了得到有限的答案而从无限中减去无限，这似乎是不合理的。

Freeman Dyson argued that these infinities are of a basic nature and cannot be eliminated by any formal mathematical procedures, such as the renormalization method.^{[22]}^{[23]}

Freeman Dyson argued that these infinities are of a basic nature and cannot be eliminated by any formal mathematical procedures, such as the renormalization method.

弗里曼戴森认为，这些无穷大是一个基本的性质，不能消除任何正式的数学程序，如重整化方法。

Dirac's criticism was the most persistent.^{[24]} As late as 1975, he was saying:^{[25]}

Dirac's criticism was the most persistent. As late as 1975, he was saying:

狄拉克的批评是最持久的。直到1975年，他还在说:

- Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics is a good theory and we do not have to worry about it any more.' I must say that I am very dissatisfied with the situation because this so-called 'good theory' does involve neglecting infinities which appear in its equations, ignoring them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves disregarding a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!

Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics is a good theory and we do not have to worry about it any more.' I must say that I am very dissatisfied with the situation because this so-called 'good theory' does involve neglecting infinities which appear in its equations, ignoring them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves disregarding a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!

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”确实涉及到忽视方程式中出现的无穷大，以一种武断的方式忽视它们。这根本不是合理的数学。明智的数学涉及到忽略一个小量——不要仅仅因为它无限大而你不想要它就忽略它！

Another important critic was Feynman. Despite his crucial role in the development of quantum electrodynamics, he wrote the following in 1985:^{[26]}

Another important critic was Feynman. Despite his crucial role in the development of quantum electrodynamics, he wrote the following in 1985:

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:

- The shell game that we play is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.

The shell game that we play is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.

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迷糊的过程！不得不求助于这样的骗术使我们无法证明量子电动力学理论在数学上是自洽的。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合理的。

Feynman was concerned that all field theories known in the 1960s had the property that the interactions become infinitely strong at short enough distance scales. This property called a Landau pole, made it plausible that quantum field theories were all inconsistent. In 1974, Gross, Politzer and Wilczek showed that another quantum field theory, quantum chromodynamics, does not have a Landau pole. Feynman, along with most others, accepted that QCD was a fully consistent theory.^{[citation needed]}

Feynman was concerned that all field theories known in the 1960s had the property that the interactions become infinitely strong at short enough distance scales. This property called a Landau pole, made it plausible that quantum field theories were all inconsistent. In 1974, Gross, Politzer and Wilczek showed that another quantum field theory, quantum chromodynamics, does not have a Landau pole. Feynman, along with most others, accepted that QCD was a fully consistent theory.

费曼担心，在20世纪60年代所有已知的场理论都有这样的特性: 在足够短的距离尺度上，相互作用会变得无限强烈。这个性质被称为兰道极，使得量子场理论全部不一致的说法似乎是可信的。1974年，Gross，Politzer 和 Wilczek 证明了另一个量子场论，量子色动力学，并没有兰道极点。费曼和其他大多数人一样，承认 QCD 是一个完全一致的理论。

The general unease was almost universal in texts up to the 1970s and 1980s. Beginning in the 1970s, however, inspired by work on the renormalization group and effective field theory, and despite the fact that Dirac and various others—all of whom belonged to the older generation—never withdrew their criticisms, attitudes began to change, especially among younger theorists. Kenneth G. Wilson and others demonstrated that the renormalization group is useful in statistical field theory applied to condensed matter physics, where it provides important insights into the behavior of phase transitions. In condensed matter physics, a *physical* short-distance regulator exists: matter ceases to be continuous on the scale of atoms. Short-distance divergences in condensed matter physics do not present a philosophical problem since the field theory is only an effective, smoothed-out representation of the behavior of matter anyway; there are no infinities since the cutoff is always finite, and it makes perfect sense that the bare quantities are cutoff-dependent.

The general unease was almost universal in texts up to the 1970s and 1980s. Beginning in the 1970s, however, inspired by work on the renormalization group and effective field theory, and despite the fact that Dirac and various others—all of whom belonged to the older generation—never withdrew their criticisms, attitudes began to change, especially among younger theorists. Kenneth G. Wilson and others demonstrated that the renormalization group is useful in statistical field theory applied to condensed matter physics, where it provides important insights into the behavior of phase transitions. In condensed matter physics, a physical short-distance regulator exists: matter ceases to be continuous on the scale of atoms. Short-distance divergences in condensed matter physics do not present a philosophical problem since the field theory is only an effective, smoothed-out representation of the behavior of matter anyway; there are no infinities since the cutoff is always finite, and it makes perfect sense that the bare quantities are cutoff-dependent.

直到20世纪70年代和80年代，这种普遍的不安在文本中几乎是普遍存在的。然而，从20世纪70年代开始，受到重整化群理论和有效场理论工作的启发，尽管狄拉克和其他各种人---- 他们都属于老一辈---- 从未收回他们的批评，但态度开始改变，尤其是在年轻的理论家中。肯尼斯·威尔森和其他人证明了重整化群在统计领域理论应用于凝聚态物理学中是有用的，它提供了对相变行为的重要见解。在21凝聚态物理学，存在一个物理的短距离调节器: 物质不再是连续的原子规模。凝聚态物理学中的短距离分歧并不构成哲学问题，因为场论只是对物质行为的一种有效的、平滑的表示; 因为截止值总是有限的，所以没有无限性，而且光裸量是截止值依赖的，这是完全合理的。

If QFT holds all the way down past the Planck length (where it might yield to string theory, causal set theory or something different), then there may be no real problem with short-distance divergences in particle physics either; *all* field theories could simply be effective field theories. In a sense, this approach echoes the older attitude that the divergences in QFT speak of human ignorance about the workings of nature, but also acknowledges that this ignorance can be quantified and that the resulting effective theories remain useful.

If QFT holds all the way down past the Planck length (where it might yield to string theory, causal set theory or something different), then there may be no real problem with short-distance divergences in particle physics either; all field theories could simply be effective field theories. In a sense, this approach echoes the older attitude that the divergences in QFT speak of human ignorance about the workings of nature, but also acknowledges that this ignorance can be quantified and that the resulting effective theories remain useful.

如果 QFT 能一直保持到普朗克长度以下(在那里它可能会产生弦论、因果集合论或其他不同的理论) ，那么粒子物理学中的短距离分歧可能也不存在真正的问题; 所有场论都可能只是有效场论。在某种意义上，这种方法呼应了以前的观点，即量子力学中的分歧说明了人类对自然规律的无知，但也承认这种无知是可以量化的，由此产生的有效理论仍然是有用的。

Be that as it may, Salam's remark^{[27]} in 1972 seems still relevant

Be that as it may, Salam's remark in 1972 seems still relevant

尽管如此，萨拉姆1972年的言论似乎仍然有意义

- Field-theoretic infinities — first encountered in Lorentz's computation of electron self-mass — have persisted in classical electrodynamics for seventy and in quantum electrodynamics for some thirty-five years. These long years of frustration have left in the subject a curious affection for the infinities and a passionate belief that they are an inevitable part of nature; so much so that even the suggestion of a hope that they may, after all, be circumvented — and finite values for the renormalization constants computed — is considered irrational. Compare Russell's postscript to the third volume of his autobiography
*The Final Years, 1944–1969*(George Allen and Unwin, Ltd., London 1969),^{[28]}p. 221:

Field-theoretic infinities — first encountered in Lorentz's computation of electron self-mass — have persisted in classical electrodynamics for seventy and in quantum electrodynamics for some thirty-five years. These long years of frustration have left in the subject a curious affection for the infinities and a passionate belief that they are an inevitable part of nature; so much so that even the suggestion of a hope that they may, after all, be circumvented — and finite values for the renormalization constants computed — is considered irrational. Compare Russell's postscript to the third volume of his autobiography The Final Years, 1944–1969 (George Allen and Unwin, Ltd., London 1969), p. 221:

场论的无穷大首次出现在洛伦兹计算电子自质量的过程中，它在经典电动力学中已经存在了七十年，在量子电动力学中已经存在了35年。这么多年的挫折使得这个课题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; 以至于即使有希望它们毕竟可以被绕过ーー计算出的重整化常数的有限值ーー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，第221页:

- In the modern world, if communities are unhappy, it is often because they have ignorances, habits, beliefs, and passions, which are dearer to them than happiness or even life. I find many men in our dangerous age who seem to be in love with misery and death, and who grow angry when hopes are suggested to them. They think hope is irrational and that, in sitting down to lazy despair, they are merely facing facts.

In the modern world, if communities are unhappy, it is often because they have ignorances, habits, beliefs, and passions, which are dearer to them than happiness or even life. I find many men in our dangerous age who seem to be in love with misery and death, and who grow angry when hopes are suggested to them. They think hope is irrational and that, in sitting down to lazy despair, they are merely facing facts.

在现代社会，如果社区不幸福，那往往是因为他们有无知、习惯、信仰和激情，这些东西对他们来说比幸福甚至生命更重要。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，当有人向他们提出希望时，他们就会生气。他们认为希望是非理性的，坐下来懒洋洋地绝望，他们只是面对事实。

In QFT, the value of a physical constant, in general, depends on the scale that one chooses as the renormalization point, and it becomes very interesting to examine the renormalization group running of physical constants under changes in the energy scale. The coupling constants in the Standard Model of particle physics vary in different ways with increasing energy scale: the coupling of quantum chromodynamics and the weak isospin coupling of the electroweak force tend to decrease, and the weak hypercharge coupling of the electroweak force tends to increase. At the colossal energy scale of 10^{15} GeV (far beyond the reach of our current particle accelerators), they all become approximately the same size (Grotz and Klapdor 1990, p. 254), a major motivation for speculations about grand unified theory. Instead of being only a worrisome problem, renormalization has become an important theoretical tool for studying the behavior of field theories in different regimes.

In QFT, the value of a physical constant, in general, depends on the scale that one chooses as the renormalization point, and it becomes very interesting to examine the renormalization group running of physical constants under changes in the energy scale. The coupling constants in the Standard Model of particle physics vary in different ways with increasing energy scale: the coupling of quantum chromodynamics and the weak isospin coupling of the electroweak force tend to decrease, and the weak hypercharge coupling of the electroweak force tends to increase. At the colossal energy scale of 10^{15} GeV (far beyond the reach of our current particle accelerators), they all become approximately the same size (Grotz and Klapdor 1990, p. 254), a major motivation for speculations about grand unified theory. Instead of being only a worrisome problem, renormalization has become an important theoretical tool for studying the behavior of field theories in different regimes.

在 QFT 中，一个物理常数的值，一般来说，取决于我们选择的重整化点的尺度，在能量尺度变化的情况下，研究重整化群物理常数的运行变得非常有趣。粒子物理标准模型中的耦合常数随着能量的增加而以不同的方式变化: 量子色动力学的耦合和电弱力的弱同位旋耦合趋于减小，电弱力的弱超荷耦合趋于增加。在10 < sup > 15 GeV 的巨大能量范围内(远远超出我们现有的粒子加速器的能量范围) ，它们都变得大致相同(Grotz 和 Klapdor 1990，p. 254) ，这是推测大统一理论的主要动机。重整化已经不再是一个令人担忧的问题，而是成为研究不同区域中场理论行为的一个重要理论工具。

If a theory featuring renormalization (e.g. QED) can only be sensibly interpreted as an effective field theory, i.e. as an approximation reflecting human ignorance about the workings of nature, then the problem remains of discovering a more accurate theory that does not have these renormalization problems. As Lewis Ryder has put it, "In the Quantum Theory, these [classical] divergences do not disappear; on the contrary, they appear to get worse. And despite the comparative success of renormalisation theory, the feeling remains that there ought to be a more satisfactory way of doing things."^{[29]}

If a theory featuring renormalization (e.g. QED) can only be sensibly interpreted as an effective field theory, i.e. as an approximation reflecting human ignorance about the workings of nature, then the problem remains of discovering a more accurate theory that does not have these renormalization problems. As Lewis Ryder has put it, "In the Quantum Theory, these [classical] divergences do not disappear; on the contrary, they appear to get worse. And despite the comparative success of renormalisation theory, the feeling remains that there ought to be a more satisfactory way of doing things."

如果一个具有重整化特征的理论(例如:。量子电动力学(QED)只能理性地解释为一种有效场理论，即。作为反映人类对自然规律的无知的近似值，那么发现一个更精确的理论，而不存在这些重整化问题的问题仍然存在。正如路易斯 · 莱德所说，“在量子理论中，这些[经典]分歧并没有消失; 相反，它们似乎变得更糟。尽管重整化理论相对成功，但人们仍然认为，应该有一种更令人满意的做事方式。”

## Renormalizability

From this philosophical reassessment, a new concept follows naturally: the notion of renormalizability. Not all theories lend themselves to renormalization in the manner described above, with a finite supply of counterterms and all quantities becoming cutoff-independent at the end of the calculation. If the Lagrangian contains combinations of field operators of high enough dimension in energy units, the counterterms required to cancel all divergences proliferate to infinite number, and, at first glance, the theory would seem to gain an infinite number of free parameters and therefore lose all predictive power, becoming scientifically worthless. Such theories are called *nonrenormalizable*.

From this philosophical reassessment, a new concept follows naturally: the notion of renormalizability. Not all theories lend themselves to renormalization in the manner described above, with a finite supply of counterterms and all quantities becoming cutoff-independent at the end of the calculation. If the Lagrangian contains combinations of field operators of high enough dimension in energy units, the counterterms required to cancel all divergences proliferate to infinite number, and, at first glance, the theory would seem to gain an infinite number of free parameters and therefore lose all predictive power, becoming scientifically worthless. Such theories are called nonrenormalizable.

从这种哲学上的重新评估，一个新的概念自然而然地随之而来: 可重整性的概念。并非所有的理论都适合上述方式的重整化，反项的有限供应和所有数量在计算结束时成为截止独立的。如果拉格朗日函数包含能量单位中足够高维的场算子的组合，那么消除所有分歧所需的对位项就会扩散成无限数，而且，乍一看，这个理论似乎获得了无限多的自由参数，因此失去了所有的预测能力，在科学上变得毫无价值。这样的理论被称为不可重整性理论。

The Standard Model of particle physics contains only renormalizable operators, but the interactions of general relativity become nonrenormalizable operators if one attempts to construct a field theory of quantum gravity in the most straightforward manner (treating the metric in the Einstein–Hilbert Lagrangian as a perturbation about the Minkowski metric), suggesting that perturbation theory is useless in application to quantum gravity.

The Standard Model of particle physics contains only renormalizable operators, but the interactions of general relativity become nonrenormalizable operators if one attempts to construct a field theory of quantum gravity in the most straightforward manner (treating the metric in the Einstein–Hilbert Lagrangian as a perturbation about the Minkowski metric), suggesting that perturbation theory is useless in application to quantum gravity.

粒子物理学的标准模型只包含可重整化的算符，但是如果一个人试图以最直接的方式构建量子引力场理论(把 Einstein-Hilbert Lagrangian 中的度量当作闵可夫斯基度量的扰动) ，那么广义相对论的相互作用就成为不可重整化的算符，这表明摄动理论在量子引力的应用中是无用的。

However, in an effective field theory, "renormalizability" is, strictly speaking, a misnomer. In nonrenormalizable effective field theory, terms in the Lagrangian do multiply to infinity, but have coefficients suppressed by ever-more-extreme inverse powers of the energy cutoff. If the cutoff is a real, physical quantity—that is, if the theory is only an effective description of physics up to some maximum energy or minimum distance scale—then these additional terms could represent real physical interactions. Assuming that the dimensionless constants in the theory do not get too large, one can group calculations by inverse powers of the cutoff, and extract approximate predictions to finite order in the cutoff that still have a finite number of free parameters. It can even be useful to renormalize these "nonrenormalizable" interactions.

However, in an effective field theory, "renormalizability" is, strictly speaking, a misnomer. In nonrenormalizable effective field theory, terms in the Lagrangian do multiply to infinity, but have coefficients suppressed by ever-more-extreme inverse powers of the energy cutoff. If the cutoff is a real, physical quantity—that is, if the theory is only an effective description of physics up to some maximum energy or minimum distance scale—then these additional terms could represent real physical interactions. Assuming that the dimensionless constants in the theory do not get too large, one can group calculations by inverse powers of the cutoff, and extract approximate predictions to finite order in the cutoff that still have a finite number of free parameters. It can even be useful to renormalize these "nonrenormalizable" interactions.

然而，在有效场理论中，严格地说，“可重整性”是一个用词不当。在不可重整化的有效场理论中，拉格朗日项确实可以乘以无穷大，但系数却被截断能量的更极端的反幂所抑制。如果截止值是一个真实的物理量，也就是说，如果这个理论只是一个有效的物理描述，能量达到一定的最大值或最小距离尺度，那么这些附加项可以代表真实的物理相互作用。假设理论中的无量纲常数不会变得太大，人们可以通过截止函数的反幂对计算进行分组，并提取截止函数中仍然有有限个自由参数的有限阶的近似预测。甚至可以对这些“不可重整化”的交互进行重整化。

Nonrenormalizable interactions in effective field theories rapidly become weaker as the energy scale becomes much smaller than the cutoff. The classic example is the Fermi theory of the weak nuclear force, a nonrenormalizable effective theory whose cutoff is comparable to the mass of the W particle. This fact may also provide a possible explanation for *why* almost all of the particle interactions we see are describable by renormalizable theories. It may be that any others that may exist at the GUT or Planck scale simply become too weak to detect in the realm we can observe, with one exception: gravity, whose exceedingly weak interaction is magnified by the presence of the enormous masses of stars and planets.^{[citation needed]}

Nonrenormalizable interactions in effective field theories rapidly become weaker as the energy scale becomes much smaller than the cutoff. The classic example is the Fermi theory of the weak nuclear force, a nonrenormalizable effective theory whose cutoff is comparable to the mass of the W particle. This fact may also provide a possible explanation for why almost all of the particle interactions we see are describable by renormalizable theories. It may be that any others that may exist at the GUT or Planck scale simply become too weak to detect in the realm we can observe, with one exception: gravity, whose exceedingly weak interaction is magnified by the presence of the enormous masses of stars and planets.

有效场理论中不可重整化的相互作用随着能量尺度的变小而迅速减弱。经典的例子是弱核力的费米理论，这是一个不可重整化的有效理论，其截止点相当于 w 粒子的质量。这一事实也可能提供一个可能的解释，为什么我们看到的几乎所有粒子相互作用都可以用可重整化理论来描述。也许存在于内脏尺度或者普朗克尺度的其他粒子变得太弱，以至于我们无法在我们可以观察到的范围内探测到，只有一个例外: 引力，它极其微弱的相互作用被巨大的恒星和行星的存在放大了。

## Renormalization schemes

In actual calculations, the counterterms introduced to cancel the divergences in Feynman diagram calculations beyond tree level must be *fixed* using a set of * renormalisation conditions*. The common renormalization schemes in use include:

In actual calculations, the counterterms introduced to cancel the divergences in Feynman diagram calculations beyond tree level must be fixed using a set of renormalisation conditions. The common renormalization schemes in use include:

在实际计算中，为了消除费曼图计算中超出树层的偏差而引入的反项必须使用一组重整化条件来确定。目前常用的重整化方案包括:

- Minimal subtraction (MS) scheme and the related modified minimal subtraction (MS-bar) scheme

## Renormalization in statistical physics

### History

A deeper understanding of the physical meaning and generalization of the

A deeper understanding of the physical meaning and generalization of the

更深入的理解物理意义和一般化的

renormalization process, which goes beyond the dilatation group of conventional *renormalizable* theories, came from condensed matter physics. Leo P. Kadanoff's paper in 1966 proposed the "block-spin" renormalization group.^{[30]} The *blocking idea* is a way to define the components of the theory at large distances as aggregates of components at shorter distances.

renormalization process, which goes beyond the dilatation group of conventional renormalizable theories, came from condensed matter physics. Leo P. Kadanoff's paper in 1966 proposed the "block-spin" renormalization group. The blocking idea is a way to define the components of the theory at large distances as aggregates of components at shorter distances.

重整化过程超越了传统的可重整化理论的扩张集团，它来自于凝聚态物理学。1966年，Leo p. Kadanoff 的论文提出了“块旋转”重整化群。模块化思想是将理论中大距离的组件定义为短距离的组件聚集体的一种方法。

This approach covered the conceptual point and was given full computational substance^{[20]} in the extensive important contributions of Kenneth Wilson. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the Kondo problem, in 1974, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and critical phenomena in 1971. He was awarded the Nobel prize for these decisive contributions in 1982.

This approach covered the conceptual point and was given full computational substance

这种方法涵盖了概念点，并给出了充分的计算实质

### Principles

In more technical terms, let us assume that we have a theory described

by a certain function [math]\displaystyle{ Z }[/math] of the state variables

[math]\displaystyle{ \{s_i\} }[/math] and a certain set of coupling constants

[math]\displaystyle{ \{J_k\} }[/math]. This function may be a partition function,

an action, a Hamiltonian, etc. It must contain the

whole description of the physics of the system.

Now we consider a certain blocking transformation of the state

variables [math]\displaystyle{ \{s_i\}\to \{\tilde s_i\} }[/math],

the number of [math]\displaystyle{ \tilde s_i }[/math] must be lower than the number of

[math]\displaystyle{ s_i }[/math]. Now let us try to rewrite the [math]\displaystyle{ Z }[/math]

function *only* in terms of the [math]\displaystyle{ \tilde s_i }[/math]. If this is achievable by a

certain change in the parameters, [math]\displaystyle{ \{J_k\}\to \{\tilde J_k\} }[/math], then the theory is said to be

**renormalizable**.

The possible

macroscopic states of the system, at a large scale, are given by this

set of fixed points.

### Renormalization group fixed points

The most important information in the RG flow is its **fixed points**. A fixed point is defined by the vanishing of the beta function associated to the flow. Then, fixed points of the renormalization group are by definition scale invariant. In many cases of physical interest scale invariance enlarges to conformal invariance. One then has a conformal field theory at the fixed point.

The ability of several theories to flow to the same fixed point leads to universality.

If these fixed points correspond to free field theory, the theory is said to exhibit quantum triviality. Numerous fixed points appear in the study of lattice Higgs theories, but the nature of the quantum field theories associated with these remains an open question.^{[31]}

## See also

## References

- ↑ See e.g., Weinberg vol I, chapter 10.
- ↑ 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). - ↑ H. Bethe (1947). "The Electromagnetic Shift of Energy Levels".
*Physical Review*.**72**(4): 339–341. Bibcode:1947PhRv...72..339B. doi:10.1103/PhysRev.72.339. - ↑ Schwinger, J. (1948). "On quantum-electrodynamics and the magnetic moment of the electron".
*Physical Review*.**73**(4): 416–417. Bibcode:1948PhRv...73..416S. doi:10.1103/PhysRev.73.416. - ↑ Schwinger, J. (1948). "I. A covariant formulation".
*Physical Review*. Quantum Electrodynamics.**74**(10): 1439–1461. Bibcode:1948PhRv...74.1439S. doi:10.1103/PhysRev.74.1439. - ↑ Schwinger, J. (1949). "II. Vacuum polarization and self-energy".
*Physical Review*. Quantum Electrodynamics.**75**(4): 651–679. Bibcode:1949PhRv...75..651S. doi:10.1103/PhysRev.75.651. - ↑ Schwinger, J. (1949). "III. The electromagnetic properties of the electron radiative corrections to scattering".
*Physical Review*. Quantum Electrodynamics.**76**(6): 790–817. Bibcode:1949PhRv...76..790S. doi:10.1103/PhysRev.76.790. - ↑ Feynman, Richard P. (1948). "Space-time approach to non-relativistic quantum mechanics" (PDF).
*Reviews of Modern Physics*.**20**(2): 367–387. Bibcode:1948RvMP...20..367F. doi:10.1103/RevModPhys.20.367. - ↑ Feynman, Richard P. (1948). "A relativistic cut-off for classical electrodynamics" (PDF).
*Physical Review*.**74**(8): 939–946. Bibcode:1948PhRv...74..939F. doi:10.1103/PhysRev.74.939. - ↑ Feynman, Richard P. (1948). "A relativistic cut-off for quantum electrodynamics" (PDF).
*Physical Review*.**74**(10): 1430–1438. Bibcode:1948PhRv...74.1430F. doi:10.1103/PhysRev.74.1430. - ↑ Tomonaga, S. (August 1, 1946). "On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields".
*Progress of Theoretical Physics*. Oxford University Press (OUP).**1**(2): 27–42. Bibcode:1946PThPh...1...27T. doi:10.1143/ptp.1.27. ISSN 1347-4081. - ↑ Koba, Z.; Tati, T.; Tomonaga, S.-i. (October 1, 1947). "On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. II: Case of Interacting Electromagnetic and Electron Fields".
*Progress of Theoretical Physics*. Oxford University Press (OUP).**2**(3): 101–116. Bibcode:1947PThPh...2..101K. doi:10.1143/ptp/2.3.101. ISSN 0033-068X. - ↑ Koba, Z.; Tati, T.; Tomonaga, S.-i. (December 1, 1947). "On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. III: Case of Interacting Electromagnetic and Electron Fields".
*Progress of Theoretical Physics*. Oxford University Press (OUP).**2**(4): 198–208. Bibcode:1947PThPh...2..198K. doi:10.1143/ptp/2.4.198. ISSN 0033-068X. - ↑ Kanesawa, S.; Tomonaga, S.-i. (March 1, 1948). "On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. [IV]: Case of Interacting Electromagnetic and Meson Fields".
*Progress of Theoretical Physics*. Oxford University Press (OUP).**3**(1): 1–13. doi:10.1143/ptp/3.1.1. ISSN 0033-068X. - ↑ Kanesawa, S.; Tomonaga, S.-i. (June 1, 1948). "On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields V: Case of Interacting Electromagnetic and Meson Fields".
*Progress of Theoretical Physics*. Oxford University Press (OUP).**3**(2): 101–113. Bibcode:1948PThPh...3..101K. doi:10.1143/ptp/3.2.101. ISSN 0033-068X. - ↑ Koba, Z.; Tomonaga, S.-i. (September 1, 1948). "On Radiation Reactions in Collision Processes. I: Application of the "Self-Consistent" Subtraction Method to the Elastic Scattering of an Electron".
*Progress of Theoretical Physics*. Oxford University Press (OUP).**3**(3): 290–303. Bibcode:1948PThPh...3..290K. doi:10.1143/ptp/3.3.290. ISSN 0033-068X. - ↑ Tomonaga, Sin-Itiro; Oppenheimer, J. R. (July 15, 1948). "On Infinite Field Reactions in Quantum Field Theory".
*Physical Review*. American Physical Society (APS).**74**(2): 224–225. Bibcode:1948PhRv...74..224T. doi:10.1103/physrev.74.224. ISSN 0031-899X. - ↑ Dyson, F. J. (1949). "The radiation theories of Tomonaga, Schwinger, and Feynman".
*Phys. Rev*.**75**(3): 486–502. Bibcode:1949PhRv...75..486D. doi:10.1103/PhysRev.75.486. - ↑ Peskin, Michael E.; Schroeder, Daniel V. (1995).
*An Introduction to Quantum Field Theory*. Reading: Addison-Wesley. Chapter 10. https://archive.org/details/introductiontoqu0000pesk. - ↑
^{20.0}^{20.1}Wilson, Kenneth G. (October 1, 1975). "The renormalization group: Critical phenomena and the Kondo problem".*Reviews of Modern Physics*. American Physical Society (APS).**47**(4): 773–840. Bibcode:1975RvMP...47..773W. doi:10.1103/revmodphys.47.773. ISSN 0034-6861. - ↑ 't Hooft, G.; Veltman, M. (1972). "Regularization and renormalization of gauge fields".
*Nuclear Physics B*.**44**(1): 189–213. Bibcode:1972NuPhB..44..189T. doi:10.1016/0550-3213(72)90279-9. hdl:1874/4845. - ↑ Dyson, F. J. (February 15, 1952). "Divergence of Perturbation Theory in Quantum Electrodynamics".
*Physical Review*. American Physical Society (APS).**85**(4): 631–632. Bibcode:1952PhRv...85..631D. doi:10.1103/physrev.85.631. ISSN 0031-899X. - ↑ Stern, A. W. (November 7, 1952). "Space, Field, and Ether in Contemporary Physics".
*Science*. American Association for the Advancement of Science (AAAS).**116**(3019): 493–496. Bibcode:1952Sci...116..493S. doi:10.1126/science.116.3019.493. ISSN 0036-8075. PMID 17801299. - ↑ P.A.M. Dirac, "The Evolution of the Physicist's Picture of Nature," in Scientific American, May 1963, p. 53.
- ↑ Kragh, Helge;
*Dirac: A scientific biography*, CUP 1990, p. 184 - ↑ Feynman, Richard P.;
*QED: The Strange Theory of Light and Matter*, Penguin 1990, p. 128 - ↑ Isham, C. J.; Salam, Abdus; Strathdee, J. (May 15, 1972). "Infinity Suppression in Gravity-Modified Electrodynamics. II".
*Physical Review D*. American Physical Society (APS).**5**(10): 2548–2565. Bibcode:1972PhRvD...5.2548I. doi:10.1103/physrevd.5.2548. ISSN 0556-2821. - ↑ Russell, Bertrand.
*The Autobiography of Bertrand Russell: The Final Years, 1944-1969*(Bantam Books, 1970) - ↑ Ryder, Lewis.
*Quantum Field Theory*, page 390 (Cambridge University Press 1996). - ↑ L.P. Kadanoff (1966): "Scaling laws for Ising models near [math]\displaystyle{ T_c }[/math]",
*Physics (Long Island City, N.Y.)***2**, 263. - ↑ D. J. E. Callaway (1988). "Triviality Pursuit: Can Elementary Scalar Particles Exist?".
*Physics Reports*.**167**(5): 241–320. Bibcode:1988PhR...167..241C. doi:10.1016/0370-1573(88)90008-7.

## Further reading

### General introduction

- DeDeo, Simon;
*Introduction to Renormalization*(2017). Santa Fe Institute Complexity Explorer MOOC. Renormalization from a complex systems point of view, including Markov Chains, Cellular Automata, the real space Ising model, the Krohn-Rhodes Theorem, QED, and rate distortion theory.

- Delamotte, Bertrand (2004). "A hint of renormalization".
*American Journal of Physics*.**72**(2): 170–184. arXiv:hep-th/0212049. Bibcode:2004AmJPh..72..170D. doi:10.1119/1.1624112. Unknown parameter`|s2cid=`

ignored (help)

- Baez, John;
*Renormalization Made Easy*, (2005). A qualitative introduction to the subject.

Category:Concepts in physics

分类: 物理概念

- Blechman, Andrew E.;
*Renormalization: Our Greatly Misunderstood Friend*, (2002). Summary of a lecture; has more information about specific regularization and divergence-subtraction schemes.

Category:Particle physics

类别: 粒子物理学

- Cao, Tian Yu; Schweber, Silvan S. (1993). "The conceptual foundations and the philosophical aspects of renormalization theory".
*Synthese*.**97**: 33–108. doi:10.1007/BF01255832. Unknown parameter`|s2cid=`

ignored (help)

Category:Quantum field theory

范畴: 量子场论

- Shirkov, Dmitry;
*Fifty Years of the Renormalization Group*, C.E.R.N. Courrier 41(7) (2001). Full text available at :*I.O.P Magazines*.

Category:Renormalization group

类别: 重整化群

- E. Elizalde;
*Zeta regularization techniques with Applications*.

Category:Mathematical physics

类别: 数学物理

This page was moved from wikipedia:en:Renormalization. Its edit history can be viewed at 重整化/edithistory