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:<math>-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>-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>-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>
这个表达式中的各种{{math|''γ<sup>μ</sup>''}}因子是和狄拉克方程的协变公式一样的伽马矩阵; 它们与电子的自旋有关。{{mvar|e}}的因子为电耦合常数，{\displaystyle i\epsilon}提供了动量空间中绕极点积分轮廓的启发式定义。对于我们的目的来说，重要的部分是被积函数中三个主要因子对{{math|''q<sup>μ</sup>''}}的依赖，这三个因子来自圈中的两条电子线和光子线的传播子。
这是一个上面有两个{{math|''q<sup>μ</sup>''}}的幂的部分，在较大的{{math|''q<sup>μ</sup>''}}值时占优势(Pokorski 1987, p. 122)：
:<math>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 \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>
这个积分是发散且无限的，除非我们在能量和动量有限的时候截断它。
这个积分是发散且无限的，除非我们在能量和动量有限的时候截断它。
类似的环散度也出现在其他量子场论中。
== 重整化的裸量 ==
== 重整化的裸量 ==
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) ，代表着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所得的物理常数。如上所述，它们是裸量，并没有考虑虚粒子环效应对物理常数本身的影响。在其他情况中，这些影响还包括让经典电磁学理论家为难的电磁反作用量子对应物。一般来说，这些效应最初就会像考虑中的振幅一样发散; 所以有限的测量量通常意味着发散裸量。（OK）
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) ，代表着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所得的物理常数。如上所述，它们是裸量，并没有考虑虚粒子环效应对物理常数本身的影响。在其他情况中，这些影响还包括让经典电磁学理论家为难的电磁反作用量子对应物。一般来说，这些效应最初就会像考虑中的振幅一样发散; 所以有限的测量量通常意味着发散裸量。（OK）
因此，为了与现实接轨，这些公式必须以可测量的、重整化的量进行重写。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(这种定义下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)。剩下的涉及剩余裸量的拉格朗日部分，可以被重新解释为包含在发散图中，且正好抵消其他图发散现象的反项。（OK）
因此，为了与现实接轨，这些公式必须以可测量的、重整化的量进行重写。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(这种定义下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)。剩下的涉及剩余裸量的拉格朗日部分，可以被重新解释为包含在发散图中，且正好抵消其他图发散现象的反项。（OK）
=== 量子电动力学中的重整化 ===
=== 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|图3。对应于反项的顶点抵消了图2中的发散。|链接=Special:FilePath/Counterterm.png]]
例如，在量子电动力学的拉格朗日函数中
:<math>\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>\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>
磁场和耦合常数实际上是裸量（？），因此可见上面的下标如此{{mvar|B}}。通常，裸量相应的拉格朗日项是重整化项的倍数:
磁场和耦合常数实际上是裸量（？），因此可见上面的下标如此{{mvar|B}}。通常，裸量相应的拉格朗日项是重整化项的倍数:
:<math>\left(\bar\psi m \psi\right)_B = Z_0 \bar\psi m \psi</math>
:<math>\left(\bar\psi m \psi\right)_B = Z_0 \bar\psi m \psi</math>
:<math>\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>\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>\left(F_{\mu\nu}F^{\mu\nu}\right)_B = Z_3\, F_{\mu\nu}F^{\mu\nu}.</math>
:<math>\left(F_{\mu\nu}F^{\mu\nu}\right)_B = Z_3\, F_{\mu\nu}F^{\mu\nu}.</math>
通过 Ward-Takahashi 恒等式规范不变性，证明了我们可以重整共变导数的两个项在一起
通过 Ward-Takahashi 恒等式规范不变性，证明了我们可以重整共变导数的两个项在一起
:<math>\bar \psi (\partial + ieA) \psi</math>
:<math>\bar \psi (\partial + ieA) \psi</math>
(Pokorski 1987，第115页) ，这是实际上也是{{math|''Z''<sub>2</sub>}}所发生的; 与{{math|''Z''<sub>1</sub>}}相同。
(Pokorski 1987，第115页) ，这是实际上也是{{math|''Z''<sub>2</sub>}}所发生的; 与{{math|''Z''<sub>1</sub>}}相同。
拉格朗日函数中的一个项，例如图1所示的电子-光子相互作用，就可以被写出来
拉格朗日函数中的一个项，例如图1所示的电子-光子相互作用，就可以被写出来
:<math>\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 {{mvar|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 {{math|''Z''<sub>0</sub>}} and {{math|''Z''<sub>3</sub>}}).
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 ).
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 ).
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 ).
这个物理常数{{mvar|e}}，即电子的电荷，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征相等，第一个项就会给出我们在实验室中看到的相互作用(只要提供诸如磁矩的高阶修正，从环形图中就可以得到小的、有限的修正)。剩下的就是反项（？）了。如果理论是可重整化的(更多内容见下文) ，就像量子点动力学中一样，环路图的分叉部分都可以分解由成三个或更少分支（？）组成的部分，并且其拥有可以被第二项(或者类似的从{{math|''Z''<sub>0</sub>}} 和{{math|''Z''<sub>3</sub>}}得到的反项)抵消的代数形式。
as the regulator {{math|Λ → ∞}}. Based on this, he considered using the values of {{math|''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 [[Emilio Elizalde|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 [[Emilio Elizalde|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
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{{math|Λ → ∞}}函数正则化重整化之间的联系。在此基础上，他考虑利用{{math|''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
valid when {{math|''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.
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{{math|''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。
so the multi-loop integral will converge for big enough {{mvar|s}} using the Zeta regularization we can analytic continue the variable {{mvar|s}} to the physical limit where {{math|''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 {{math|''ζ''(−''m'')}}.
so the multi-loop integral will converge for big enough using the Zeta regularization we can analytic continue the variable {{mvar|s}} to the physical limit where {{math|''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 {{math|''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 .
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{{mvar|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'')}}。
== 态度以及解读 ==
== 态度以及解读 ==
量子电动力学和其他量子场论的早期规范者通常对这种（重整化的）处理方式不满意。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。
量子电动力学和其他量子场论的早期规范者通常对这种（重整化的）处理方式不满意。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。
=== 原理 ===
=== 原理 ===
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。