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The scale-dependence of a [[quantum field theory]] (QFT) is characterised by the way its [[coupling constant|coupling parameters]] depend on the energy-scale of a given physical process. This energy dependence is described by the [[renormalization group]], and is encoded in the [[beta-function]]s of the theory.

The scale-dependence of a [[quantum field theory]] (QFT) is characterised by the way its [[coupling constant|coupling parameters]] depend on the energy-scale of a given physical process. This energy dependence is described by the [[renormalization group]], and is encoded in the [[beta-function]]s of the theory.

量子场论(QFT)的标度依赖性的特征是其耦合参数依赖于给定物理过程的能量标度。这种能量依赖由重正化群描述，并编码在理论的β函数中。

量子场论(QFT)的标度依赖性的特征是其耦合参数依赖于给定物理过程的能量标度。这种能量依赖由重正化群描述，并编码在理论的'''Beta-function β函数'''中。

For a QFT to be scale-invariant, its coupling parameters must be independent of the energy-scale, and this is indicated by the vanishing of the beta-functions of the theory. Such theories are also known as [[Renormalization group|fixed points]] of the corresponding renormalization group flow.<ref>[[Jean Zinn-Justin|J. Zinn-Justin]] (2010)  Scholarpedia article [http://www.scholarpedia.org/article/Critical_Phenomena:_field_theoretical_approach "Critical Phenomena: field theoretical approach"].</ref>

For a QFT to be scale-invariant, its coupling parameters must be independent of the energy-scale, and this is indicated by the vanishing of the beta-functions of the theory. Such theories are also known as [[Renormalization group|fixed points]] of the corresponding renormalization group flow.<ref>[[Jean Zinn-Justin|J. Zinn-Justin]] (2010)  Scholarpedia article [http://www.scholarpedia.org/article/Critical_Phenomena:_field_theoretical_approach "Critical Phenomena: field theoretical approach"].</ref>

However, in nature the electromagnetic field is coupled to charged particles, such as [[electron]]s. The QFT describing the interactions of photons and charged particles is [[quantum electrodynamics]] (QED), and this theory is not scale-invariant. We can see this from the [[beta-function#Quantum electrodynamics|QED beta-function]]. This tells us that the [[electric charge]] (which is the coupling parameter in the theory) increases with increasing energy. Therefore, while the quantized electromagnetic field without charged particles '''is''' scale-invariant, QED is '''not''' scale-invariant.

However, in nature the electromagnetic field is coupled to charged particles, such as [[electron]]s. The QFT describing the interactions of photons and charged particles is [[quantum electrodynamics]] (QED), and this theory is not scale-invariant. We can see this from the [[beta-function#Quantum electrodynamics|QED beta-function]]. This tells us that the [[electric charge]] (which is the coupling parameter in the theory) increases with increasing energy. Therefore, while the quantized electromagnetic field without charged particles '''is''' scale-invariant, QED is '''not''' scale-invariant.

然而在自然界中，电磁场是与带电粒子耦合的，比如电子。描述光子和带电粒子相互作用的量子场论是量子电动力学，而这个理论并不是标度不变的。我们可以从量子电动力学的β函数中得到这一认识。这就告诉我们电荷(在理论上是耦合参数)随着能量的增加而增加。因此，尽管没有带电粒子的量子化电磁场是标度不变的，量子电动力学却不是标度不变的。

然而在自然界中，电磁场是与带电粒子耦合的，比如电子。描述光子和带电粒子相互作用的量子场论是量子电动力学（QED），而这个理论并不是标度不变的。我们可以从量子电动力学的β函数中得到这一认识。这就告诉我们电荷(在理论上是耦合参数)随着能量的增加而增加。因此，尽管没有带电粒子的量子化电磁场是标度不变的，量子电动力学却不是标度不变的。

===Massless scalar field theory 无质量标量场理论===

===Massless scalar field theory 无质量标量场理论===

Free, massless [[scalar field (quantum field theory)|quantized scalar field theory]] has no coupling parameters. Therefore, like the classical version, it is scale-invariant. In the language of the renormalization group, this theory is known as the [[Gaussian fixed point]].

Free, massless [[scalar field (quantum field theory)|quantized scalar field theory]] has no coupling parameters. Therefore, like the classical version, it is scale-invariant. In the language of the renormalization group, this theory is known as the [[Gaussian fixed point]].

Free, massless quantized scalar field theory has no coupling parameters. Therefore, like the classical version, it is scale-invariant. In the language of the renormalization group, this theory is known as the Gaussian fixed point.

自由的、无质量的'''Quantized Scalar Field Theory 量子化标量场理论'''没有耦合参数。因此，像经典的版本一样，它是标度不变的。在重整化群的范畴中，这个理论称做'''Gaussian Fixed Point 高斯定点'''。

 

However, even though the classical massless ''φ''⁴ theory is scale-invariant in ''D''=4, the quantized version is '''not''' scale-invariant. We can see this from the [[beta-function]] for the coupling parameter, ''g''.

However, even though the classical massless ''φ''⁴ theory is scale-invariant in ''D''=4, the quantized version is '''not''' scale-invariant. We can see this from the [[beta-function]] for the coupling parameter, ''g''.

However, even though the classical massless φ4 theory is scale-invariant in D=4, the quantized version is not scale-invariant. We can see this from the beta-function for the coupling parameter, g.

然而，尽管经典的无质量''φ''⁴理论在''D''=4时是标度不变的，但量子化的版本却不是如此。我们可以从耦合参数''g''的β函数中看出这一点。

 

然而，即使经典的无质量 φ4理论在 d = 4中是标度不变的，量化后的理论也不是标度不变的。我们可以从耦合参数 g 的 beta 函数中看到这一点。

Even though the quantized massless ''φ''⁴ is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the Gaussian fixed point. One example is the '''Wilson-Fisher fixed point''', below.

Even though the quantized massless ''φ''⁴ is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the Gaussian fixed point. One example is the '''Wilson-Fisher fixed point''', below.

Even though the quantized massless φ4 is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the Gaussian fixed point. One example is the Wilson-Fisher fixed point, below.

虽然量子化无质量''φ''⁴不是标度不变的，但除了高斯定点外，确实存在标度不变的量子化标量场理论。例如：'''Wilson-Fisher Fixed Point 威尔逊-费雪定点'''

 

尽管量子化的无质量 φ4不具有尺度不变性，但除了高斯不动点理论外，还存在尺度不变量子化标量场理论。一个例子是下面的 Wilson-Fisher 定点。

===Conformal field theory 共形场论===

===Conformal field theory 共形场论===