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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。

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）

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。

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）

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

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

For example, in the Lagrangian of QED

For example, in the Lagrangian of QED

<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>

“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 >  

<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>

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:

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:

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>

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

<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(\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>

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

<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>

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

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>

巴特普赛(部分 + 国际能源署)

<math>\bar \psi (\partial + ieA) \psi</math>

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

together (Pokorski 1987, p.&nbsp;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。

(Pokorski 1987，第115页) ，这是实际上也是{{math|''Z''₂}}所发生的; 与{{math|''Z''₁}}相同。

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

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>

< 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 , 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 ).

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。

这个物理常数，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''₀}}得到的反项)抵消的代数形式。（OK）