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第399行：
第399行： + 第417行：
第418行： + 第601行：
第603行： + 第606行：
第609行： − 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)。+ 第614行：
第617行： − 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。+ 第622行：
第625行： − 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。+ 第630行：
第633行： − 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。+ 第638行：
第641行： − 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。+ 第646行：
第649行： − 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。+ + 第656行：
第660行： − 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:+ 第664行：
第668行： − + 第672行：
第676行： − 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+ 第680行：
第684行： − + 第688行：
第692行： − 伯努利数和在哪里+ 第696行：
第700行： − + 第704行：
第708行： − 所以每个线性组合都可以被写成。+ 第712行：
第716行： − 或者简单地用 Abel-Plana 公式来计算每一个发散积分:+ 第720行：
第724行： − + 第728行：
第732行： − 这里 zeta 函数是赫尔维茨ζ函数，Beta 是正实数。+ 第736行：
第740行： − 这个“几何”类比是由，(如果我们使用矩形法)来计算积分的话:+ 第744行：
第748行： − + 第752行：
第756行： − 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)。+ 第768行：
第772行： − + 第776行：
第780行： − 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 > + − 第784行：
第787行： − 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+ 第792行：
第795行： − + 第800行：
第803行： − 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。+ 第920行：
第923行： + 第925行：
第929行： − 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。+ 第933行：
第937行： − 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。+ 第941行：
第945行： − 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。+ 第949行：
第953行： − 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。+ − + 第959行：
第963行： − 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:+ + + + + + + 第977行：
第987行： − 更深入的理解物理意义和一般化的 第983行：
第992行： − 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。+ + + − 这种方法涵盖了概念点，并给出了充分的计算实质+ + 第1,009行：
第1,021行： + 第1,027行：
第1,040行： + 第1,037行：
第1,051行： − + + − + + − + − 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 gauge theory#Quantum triviality|lattice Higgs theories]], but the nature of the quantum field theories associated with these remains an open question.<ref name="TrivPurs">{{cite journal| author=D. J. E. Callaway | year=1988
搬运“栗子CUGB”对第三、四、五、六小节的翻译校正
== Renormalized and bare quantities ==
== Renormalized and bare quantities ==
== 重整化的裸量 ==
The solution was to realize that the quantities initially appearing in the theory's formulae (such as the formula for the [[Lagrangian (field theory)|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 (field theory)|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.
=== Renormalization in QED ===
=== Renormalization in QED ===
=== 量子电动力学中的重整化 ===
[[Image:Counterterm.png|thumb|upright=1.1|Figure 3. The vertex corresponding to the {{math|''Z''<sub>1</sub>}} counterterm cancels the divergence in Figure 2.|链接=Special:FilePath/Counterterm.png]]
[[Image:Counterterm.png|thumb|upright=1.1|Figure 3. The vertex corresponding to the {{math|''Z''<sub>1</sub>}} counterterm cancels the divergence in Figure 2.|链接=Special:FilePath/Counterterm.png]]
== Regularization ==
== Regularization ==
== 正则化 ==
Since the quantity {{math|∞ − ∞}} is ill-defined, in order to make this notion of canceling divergences precise, the divergences first have to be tamed mathematically using the [[limit (mathematics)|theory of limits]], in a process known as [[regularization (physics)|regularization]] (Weinberg, 1995).
Since the quantity {{math|∞ − ∞}} is ill-defined, in order to make this notion of canceling divergences precise, the divergences first have to be tamed mathematically using the [[limit (mathematics)|theory of limits]], in a process known as [[regularization (physics)|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).
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{{math|∞ − ∞}}的定义是不明确的，为了使散度抵消的概念更加精确，散度首先必须使用极限理论在数学上被驯服，这一过程被称为正则化(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.
本质上任意修改圈被积函数，或调节器，可以使它们在高能量和动量下下降得更快，这样积分就会收敛。调节器有一个称为截止的特征能量标度；将这个截止值取为无穷大(或者，等价地，将相应的长度/时间标度取为零)，就可以恢复原来的积分。
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.
有了调节器，并且截止值是有限的，积分中的发散项就变成了有限的，且与截止相关的项。在用依赖截止的反项抵消这些项后，截止到无穷大，并恢复有限的物理结果。如果我们可以测量的标度上的物理现象与在最短距离和时间尺度上发生的事情无关，那么就有可能得到与截止无关的计算结果。
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, 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，它们各有优缺点。在现代应用中最流行的是由Gerardus 't Hooft和Martinus J. G. Veltman发明的量纲正则化[21]，它通过将积分带入一个虚构的分数维数的空间来驯服积分。另一种是保利-维拉斯正则化，它以非常大的质量将虚构的粒子添加到理论中，这样涉及大质量粒子的圈积分在大动量中抵消了现有的圈。
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.
另一种正则化方案是肯尼斯·威尔逊(Kenneth Wilson)提出的晶格正则化，它假设超立方晶格以固定的网格大小构建我们的时空。这一网格大小是粒子在晶格上传播时所能拥有的最大动量的自然截止。在对几个网格大小不同的网格进行计算后，物理结果外推到网格大小为0的情况，或是我们的自然宇宙。这以标度极限的存在为先决条件。
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.
重正化理论的一个严格的数学方法是因果摄动理论，其中紫外散度从计算的开始就可以避免，只需要在分布理论的框架内进行定义良好的数学运算。在这种方法中，散度可以由模糊度代替：这个对应于散度图的术语是一个有限的，但未确定的系数。之后其他原理，如规范对称，必须用来减少或消除模糊度。
=== Zeta function regularization ===
=== Zeta function regularization ===
=== Zeta函数正则化 ===
[[Julian Schwinger]] discovered a relationship{{citation needed|date=June 2012}} between [[zeta function regularization]] and renormalization, using the asymptotic relation:
[[Julian Schwinger]] discovered a relationship{{citation needed|date=June 2012}} 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:
Julian Schwinger discovered a relationship between zeta function regularization and renormalization, using the asymptotic relation:
朱利安·施温格使用渐近关系作为调节器（其中Λ → ∞）：
<math> I(n, \Lambda )= \int_0^{\Lambda }dp\,p^n \sim 1+2^n+3^n+\cdots+ \Lambda^n \to \zeta(-n)</math>
<math> 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 >
<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 . 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
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
发现了Zeta函数正则化重整化之间的联系。在此基础上，他考虑利用ζ(−n)的值来得到有限的结果。尽管他得出的结果不一致，但是由Hartle, J. Garcia研究的改进公式，并基于E. Elizalde的工作，依然囊括了zeta正则化算法的技术
<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>
<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>
< 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 >
<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 Bs are the Bernoulli numbers and
where the Bs are the Bernoulli numbers and
其中B代表伯努利数，并且
<math>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>
< math > 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 can be written as a linear combination of .
So every can be written as a linear combination of .
所以每个{{math|''I''(''m'', Λ)}}都可以写成{{math|''ζ''(−1), ''ζ''(−3), ''ζ''(−5), ..., ''ζ''(−''m'')}}。
Or simply using Abel–Plana formula we have for every divergent integral:
Or simply using Abel–Plana formula we have for every divergent integral:
或者简单地对每一个发散积分使用阿贝尔-普拉纳公式：
<math> \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> \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 >
<math> \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>
valid when , 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.
当m>0时成立，这里的Zeta函数是赫尔维茨函数，其中β是一个正实数。
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:
“几何”的类比由下式给出，(如果我们使用矩形法)来计算积分：
<math> \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>
< math > 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).
使用赫尔维茨Zeta正则化加上步骤h的矩形法(此处h不要与普朗克常数混淆)。
<math> \sum_{n=0}^{\infty} \frac{1}{n+a}= - \psi (a)+\log (a) </math>
<math> \sum_{n=0}^{\infty} \frac{1}{n+a}= - \psi (a)+\log (a) </math>
1}{ n + a } =-psi (a) + log (a) </math >
<math> \sum_{n=0}^{\infty} \frac{1}{n+a}= - \psi (a)+\log (a) </math>
since for the Harmonic series <math> \sum_{n=0}^{\infty} \frac{1}{an+1} </math> in the limit <math> a \to 0 </math> we must recover the series <math> \sum_{n=0}^{\infty}1 =1/2 </math>
since for the Harmonic series <math> \sum_{n=0}^{\infty} \frac{1}{an+1} </math> in the limit <math> a \to 0 </math> we must recover the series <math> \sum_{n=0}^{\infty}1 =1/2 </math>
因为对于调和级数<math> \sum_{n=0}^{\infty} \frac{1}{an+1} </math>在a趋近于零处，我们必须恢复级数<math> \sum_{n=0}^{\infty}1 =1/2 </math>
For multi-loop integrals that will depend on several variables <math>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>\int d \Omega</math> by a sum so we have only a divergent integral, that will depend on the modulus <math>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>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
For multi-loop integrals that will depend on several variables <math>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>\int d \Omega</math> by a sum so we have only a divergent integral, that will depend on the modulus <math>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>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<math>k_1, \cdots, k_n</math>的多圈积分，我们可以将变量转换为极坐标，然后用一个和替换角度上的积分<math>\int d \Omega</math>，因此我们只有一个发散积分，它取决于模<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，多圈积分的主要思想是将因子<math>F(q_1,\cdots,q_n)</math>替换为超球坐标{{math|''F''(''r'', Ω)}}，使紫外重叠散度编码在变量{{mvar|r}}中。为了正则化这些积分，需要一个调节器，对于多圈积分的情况，这些调节器可以被视为：
<math> \left (1+ \sqrt{q}_{i}q^{i} \right )^{-s} </math>
<math> \left (1+ \sqrt{q}_{i}q^{i} \right )^{-s} </math>
< math > 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 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 .
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 .
所以多圈积分在足够大的s时收敛，使用正则化我们可以继续分析变量{{mvar|s}}直到{{math|''s'' {{=}} 0}}的物理极限，然后正则化任何紫外积分，通过用发散级数的线性组合替换发散积分，它可以正则化为黎曼ζ函数的负值{{math|''ζ''(−''m'')}}。
== Renormalizability ==
== Renormalizability ==
== 可重整性 ==
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 [[dimensional analysis|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 [[dimensional analysis|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.
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.
从这一哲学的重新评价中，一个新的概念自然地产生了：即可重整性。不是所有的理论都能以上述的方式重整化，且在计算结束时，有限的反项和所有的量变得截止无关。如果拉格朗日算子包含能量单位足够高维的场算符组合，抵消所有散度所需要的反项激增到无穷多个。乍一看这个理论似乎获得了无数的自由参数，然而却因此失去了所有的预测能力，也就在科学上变得毫无价值。这样的理论被称为不可重整的理论。
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.
粒子物理的标准模型只包含可重整算子，但如果有人试图以最直接的方式构建量子引力场理论(将爱因斯坦-希尔伯特拉格朗日公式中的度规视为对闵可夫斯基度规的扰动)，广义相对论的相互作用就会成为不可重整化的算子，这表明微扰理论在量子引力中的应用并不令人满意。
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，严格来说，“重整化性”是一个误称。在非重整有效场理论中，拉格朗日算子的各项确实可以增加到无穷，但系数会被越来越极端的能量截止逆幂所抑制。如果截止是一个真实的物理量，也就是说，如果这个理论仅仅是对某些最大能量或最小距离尺度下的物理的有效描述，那么这些额外的项就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，我们可以通过截止的逆幂来分组计算，在包含有限数量自由参数的截止中提取有限阶的近似预测。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.
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.
在有效场论中，当能量尺度比截止小得多时，非重整相互作用迅速变弱。经典例子是弱核力的费米理论，这是一种非重整有效理论，其截止可与W粒子的质量相当。这一事实也提供了一种可能的解释—为什么我们看到几乎所有粒子相互作用都可以用重整化理论来描述。可能存在于统一场论或普朗克尺度上的任何其他物质在我们能观测到的领域中都变得太弱了。只有一个例外：引力，它极其微弱的相互作用被巨大质量的恒星和行星的存在放大了。
== Renormalization schemes ==
== Renormalization schemes ==
== 重整化方案 ==
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:
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，为了抵消费曼图计算中超出树图的散度而引入的反项必须使用一组重整化条件来解决。常用的重整化方案包括：
* [[Minimal subtraction scheme|Minimal subtraction (MS) scheme]] and the related modified minimal subtraction (MS-bar) scheme
* [[Minimal subtraction scheme|Minimal subtraction (MS) scheme]] and the related modified minimal subtraction (MS-bar) scheme
最小减法(MS)方案和相关的改进最小减法(MS-bar)方案；
* [[On shell renormalization scheme|On-shell scheme]]
* [[On shell renormalization scheme|On-shell scheme]]
在壳方案
== Renormalization in statistical physics ==
== Renormalization in statistical physics ==
== 重整化在统计物理中的应用 ==
===History===
===History===
=== 历史 ===
A deeper understanding of the physical meaning and generalization of the
A deeper understanding of the physical meaning and generalization of the
A deeper understanding of the physical meaning and generalization of the
A deeper understanding of the physical meaning and generalization of the
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.<ref>[[Leo Kadanoff|L.P. Kadanoff]] (1966): "Scaling laws for Ising models near <math>T_c</math>", ''Physics (Long Island City, N.Y.)'' '''2''', 263.</ref> 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.<ref>[[Leo Kadanoff|L.P. Kadanoff]] (1966): "Scaling laws for Ising models near <math>T_c</math>", ''Physics (Long Island City, N.Y.)'' '''2''', 263.</ref> 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.
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.
凝聚态物理学对重整化过程的物理意义和推广提供了更深入的理解，它超越了传统重整化理论的膨胀群。Leo P. Kadanoff在1966年的论文中提出了“块区自旋”重整群。分块思想是一种将理论中远距离的分量定义为较短距离分量的集合的方法。
This approach covered the conceptual point and was given full computational substance<ref name=Wilson1975 /> in the extensive important contributions of [[Kenneth G. Wilson|Kenneth Wilson]]. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the [[Kondo effect|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<ref name=Wilson1975 /> in the extensive important contributions of [[Kenneth G. Wilson|Kenneth Wilson]]. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the [[Kondo effect|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.
这种方法涵盖了概念，另外Kenneth Wilson在他的大量杰出工作中给出了完整的计算内容[20]。威尔逊思想的力量在1974年通过对一个长期存在的问题——近藤问题（或称康多问题），的建设性迭代重整化解决方案得到了证明，在此之前，他的新方法在1971年的二阶相变理论和临界现象的开创性发展也得到了证明。1982年，鉴于威尔逊杰出的贡献，他被授予诺贝尔奖。
This approach covered the conceptual point and was given full computational substance
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===
===Principles===
=== 原理 ===
In more technical terms, let us assume that we have a theory described
In more technical terms, let us assume that we have a theory described
whole description of the physics of the system.
whole description of the physics of the system.
在更专业地来说，让我们假设我们有一个由状态变量<math>\{s_i\}</math>和耦合常数<math>\{J_k\}</math>的某个函数<math>Z</math>描述的理论。这个函数可以是配分函数、作用函数、哈密顿函数等等。它必须包含整个系统的物理描述。
'''renormalizable'''.
'''renormalizable'''.
现在我们考虑状态变量<math>\{s_i\}\到\{\tilde s_i\}</math>到\{{tilde s_i\}的某种分块变换，<math>\tilde s_i</math>的数目必须小于<math>s_i</math>的数目。现在让我们尝试仅根据{\displaystyle {\tilde <math>s_i</math>来重写<math>Z</math>函数。如果这可以通过参数的某种变化实现，则{\displaystyle<math>\{J_k\}\改为\{\tilde J_k\}</math>，则该理论是可重整化的。
set of fixed points.
set of fixed points.
系统在大尺度上可能的宏观状态是由这组固定点给出的。
===Renormalization group fixed points===
===Renormalization group fixed points===
=== 重整化群的固定点 ===
The most important information in the RG flow is its '''fixed points'''. A fixed point is defined by the vanishing of the [[beta function (physics)|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 most important information in the RG flow is its '''fixed points'''. A fixed point is defined by the vanishing of the [[beta function (physics)|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.
重整化群流中最重要的内容是它的固定点。固定点是由与流相关的β函数的消失来定义的。然后根据定义，重整化群的固定点是标度不变的。在许多物理领域内，标度不变性扩大为正形不变性。然后在固定点处符合共形场论。
The ability of several theories to flow to the same fixed point leads to [[Universality (dynamical systems)|universality]].
The ability of several theories to flow to the same fixed point leads to [[Universality (dynamical systems)|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 gauge theory#Quantum triviality|lattice Higgs theories]], but the nature of the quantum field theories associated with these remains an open question.<ref name="TrivPurs">{{cite journal| author=D. J. E. Callaway | year=1988
几种理论都可以流动到同一固定点的性质产生了普遍性。如果这些固定点与自由场论相对应，那么这个理论就表现出了量子的平凡性。在格子希格斯理论的研究中出现了许多固定点，但与之相关的量子场论的本质仍然是一个悬而未决的问题。
| title=Triviality Pursuit: Can Elementary Scalar Particles Exist?| journal=[[Physics Reports]]
| title=Triviality Pursuit: Can Elementary Scalar Particles Exist?| journal=[[Physics Reports]]