标度不变性

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The Wiener process is scale-invariant. 维纳过程具有标度不变性。

In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.

在物理学、数学和统计学中，Scale Invariance 标度不变性是物体或者物理定律的一种特征，如果长度、能量或者其他变量的标度与一个公因子相乘，而不发生改变，因此也就代表某种普遍性。

The technical term for this transformation is a dilatation (also known as dilation), and the dilatations can also form part of a larger conformal symmetry.

这种变换的专业名称是Dilatation 膨胀，膨胀也可以形成一个更大Conformal Symmetry 共形对称的一部分。

• In mathematics, scale invariance usually refers to an invariance of individual functions or curves. A closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the probability distributions of random processes to display this kind of scale invariance or self-similarity.
• In classical field theory, scale invariance most commonly applies to the invariance of a whole theory under dilatations. Such theories typically describe classical physical processes with no characteristic length scale.
• In quantum field theory, scale invariance has an interpretation in terms of particle physics. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.
• In statistical mechanics, scale invariance is a feature of phase transitions. The key observation is that near a phase transition or critical point, fluctuations occur at all length scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena. Such theories are scale-invariant statistical field theories, and are formally very similar to scale-invariant quantum field theories.
• Universality is the observation that widely different microscopic systems can display the same behaviour at a phase transition. Thus phase transitions in many different systems may be described by the same underlying scale-invariant theory.
• In general, dimensionless quantities are scale invariant. The analogous concept in statistics are standardized moments, which are scale invariant statistics of a variable, while the unstandardized moments are not.

• 在数学中，标度不变性通常指单个函数或曲线的不变性。与此密切相关的概念是Self-similarity 自相似性，其中函数或曲线在膨胀的离散子集下是不变的。随机过程的概率分布也可能表现出这种标度不变性或自相似性。
• 在Classical Field Theory 经典场论中，标度不变性最常用于整个理论在膨胀条件下的不变性。这些理论通常描述没有特征长度标度的经典物理过程。
• 在Quantum Field Theory 量子场论中，标度不变性可以用粒子物理学来解释。在标度不变的理论中，粒子相互作用的强度并不取决于所涉及粒子的能量。
• 在Statistical Mechanics 统计力学中，标度不变性是相变的一个特征。在相变或临界点附近，在所有长度标度上都出现了波动，因此，人们应该寻找一个明确的标度不变的理论来描述这一关键现象。这些理论是标度不变的统计场理论，在形式上与标度不变的量子场理论非常相似。
• Universality 普适性是指差异巨大的微观系统在相变时可以表现出相同的行为。因此，许多不同系统中的相变可以用相同的基本标度不变理论来描述。
• 一般来说，无量纲量是标度不变量。统计学中的类似概念是Standardized Moments 标准化矩，它是变量的标度不变统计量，而非标准化矩不是。

Scale-invariant curves and self-similarity 标度不变曲线与自相似性

In mathematics, one can consider the scaling properties of a function or curve f (x) under rescalings of the variable x. That is, one is interested in the shape of f (λx) for some scale factor λ, which can be taken to be a length or size rescaling. The requirement for f (x) to be invariant under all rescalings is usually taken to be

[math]\displaystyle{ f(\lambda x)=\lambda^{\Delta}f(x) }[/math]

for some choice of exponent Δ, and for all dilations λ. This is equivalent to f  being a homogeneous function of degree Δ.

在数学中，我们会考虑函数或曲线在变量x重新标度下的标度性质。也就是说，人们对某些标度因子λ 对应下f (λx)的形状感兴趣，这些标度因子可以被视为长度或大小的重新标度。对于某些选择的指数Δ和所有的膨胀λ，要求f (x) 在所有重新标度下保持不变需要满足：

[math]\displaystyle{ f(\lambda x)=\lambda^{\Delta}f(x) }[/math]

这等价于f 是一个次数为Δ的齐次函数。

Examples of scale-invariant functions are the monomials [math]\displaystyle{ f(x)=x^n }[/math], for which Δ = n, in that clearly

许多标变函数的实例是单项式：

[math]\displaystyle{ f(x)=x^n }[/math]，

其中 Δ = n，且有：

[math]\displaystyle{ f(\lambda x) = (\lambda x)^n = \lambda^n f(x)~. }[/math]

An example of a scale-invariant curve is the logarithmic spiral, a kind of curve that often appears in nature. In polar coordinates (r, θ), the spiral can be written as

一个标度不变曲线的例子是Logarithmic Spiral 对数螺线（等角螺线），这是一种在自然界中经常出现的曲线。在极坐标系中，螺旋线可以写成

[math]\displaystyle{ \theta = \frac{1}{b} \ln(r/a)~. }[/math]

Allowing for rotations of the curve, it is invariant under all rescalings λ; that is, θ(λr) is identical to a rotated version of θ(r).

在任意重新标度λ下，标度不变也允许曲线进行旋转；换句话说，θ(λr)与其旋转后的θ(r)一模一样。

Projective geometry 射影几何

The idea of scale invariance of a monomial generalizes in higher dimensions to the idea of a homogeneous polynomial, and more generally to a homogeneous function. Homogeneous functions are the natural denizens of projective space, and homogeneous polynomials are studied as projective varieties in projective geometry. Projective geometry is a particularly rich field of mathematics; in its most abstract forms, the geometry of schemes, it has connections to various topics in string theory.

单项式标度不变性的概念在高维时推广到Homogeneous Polynomial 齐次多项式，更一般地推广到Homogeneous Function 齐次函数。齐次函数是射影空间的“土著”，齐次多项式在射影几何中作为Projective Varieties 射影簇进行研究。射影几何是数学中一个内容特别丰富的领域；在其最抽象的形式——Schemes 概型的几何学中，它与String Theory 弦理论中的各种主题都有联系。

Fractals 分形

A Koch curve is self-similar. 科赫雪花具有自相似性。

It is sometimes said that fractals are scale-invariant, although more precisely, one should say that they are self-similar. A fractal is equal to itself typically for only a discrete set of values λ, and even then a translation and rotation may have to be applied to match the fractal up to itself.

有时人们认为分形是标度不变的，尽管更准确地来说，应该说分形是自相似的。分形通常是在某个λ值的离散集合内等同于其本身，即使这样，有时也需要通过平移和旋转变换来实现。

Thus, for example, the Koch curve scales with ∆ = 1, but the scaling holds only for values of λ = 1/3ⁿ for integer n. In addition, the Koch curve scales not only at the origin, but, in a certain sense, "everywhere": miniature copies of itself can be found all along the curve.

因此，以∆ = 1的Koch Curve 科赫雪花缩放为例，但是该缩放只适用于λ = 1/3ⁿ，（n为整数）的值。此外，科赫雪花不仅在初始点，而且在某种意义上，在整条曲线上都可以找到其“缩影”。

Some fractals may have multiple scaling factors at play at once; such scaling is studied with multi-fractal analysis.

某些分形可能同时具有多个标度因子，可以应用Multi-Fractal Analysis 多重分形分析进行研究。

Periodic external and internal rays are invariant curves .

周期性外部和内部射线是不变的曲线。

Scale invariance in stochastic processes 随机过程中的标度不变性

If P(f ) is the average, expected power at frequency f , then noise scales as

[math]\displaystyle{ P(f) = \lambda^{-\Delta} P(\lambda f) }[/math]

with Δ = 0 for white noise, Δ = −1 for pink noise, and Δ = −2 for Brownian noise (and more generally, Brownian motion).

If is the average, expected power at frequency , then noise scales as

P(f) = \lambda^{-\Delta} P(\lambda f)

with = 0 for white noise, = −1 for pink noise, and = −2 for Brownian noise (and more generally, Brownian motion).

如果P(f )是频率f 处的平均期望幂，那么噪声依下式标度变化：

[math]\displaystyle{ P(f) = \lambda^{-\Delta} P(\lambda f) }[/math]

当Δ= 0时对应White noise 白噪声，Δ=-1时对应Pink noise 粉红噪声，Δ=-2时对应Brownian noise 布朗噪声（更一般的是Brownian motion 布朗运动）。

More precisely, scaling in stochastic systems concerns itself with the likelihood of choosing a particular configuration out of the set of all possible random configurations. More context is needed here. Probability and entropy are definitely related to choosing a particular configuration, but it is not obvious how scale invariance is connected to this. This likelihood is given by the probability distribution.

更准确地说，随机系统中的标度变化涉及从所有可能的随机排列中选择一个特定排列的可能性。这一可能性可由概率分布给出。此外还需要更多的背景内容。概率和熵必然与一个特定排列的选定有关，但标度不变性与它们之间的相互联系还不明显。

Examples of scale-invariant distributions are the Pareto distribution and the Zipfian distribution.

标度不变分布的例子还有Pareto distribution 帕累托分布和Zipfian distribution 齐夫分布。

Scale invariant Tweedie distributions 标度不变的Tweedie分布

Tweedie distributions are a special case of exponential dispersion models, a class of statistical models used to describe error distributions for the generalized linear model and characterized by closure under additive and reproductive convolution as well as under scale transformation. These include a number of common distributions: the normal distribution, Poisson distribution and gamma distribution, as well as more unusual distributions like the compound Poisson-gamma distribution, positive stable distributions, and extreme stable distributions. Consequent to their inherent scale invariance Tweedie random variables Y demonstrate a variance var(Y) to mean E(Y) power law:

[math]\displaystyle{ \text{var}\,(Y) = a[\text{E}\,(Y)]^p }[/math],

where a and p are positive constants. This variance to mean power law is known in the physics literature as fluctuation scaling, and in the ecology literature as Taylor's law.

Tweedie分布是Exponential Dispersion Models 指数弥散模型的一种特殊情况，是一类用于描述广义线性模型误差分布的统计模型，在可加卷积和再生卷积以及标度变换下具有闭包性。这包括一些常见的分布：正态分布、Poisson distribution 泊松分布和Gamma Distribution 伽玛分布，以及其他一些非同寻常的分布，如复合泊松-伽玛分布、正稳定分布和极端稳定分布。由于它们固有的标度不变性，Tweedie随机变量 y 显示方差var(Y)与均值E(Y)之间服从幂律关系：

[math]\displaystyle{ \text{var}\,(Y) = a[\text{E}\,(Y)]^p }[/math]，

其中a和p是正常数。这种方差-均值的幂律关系在物理学文献中称为Fluctuation Scaling 涨落标度，在生态学文献中称为Taylor's Law 泰勒定律。

Random sequences, governed by the Tweedie distributions and evaluated by the method of expanding bins exhibit a biconditional relationship between the variance to mean power law and power law autocorrelations. The Wiener–Khinchin theorem further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest 1/f noise.

随机序列由Tweedie分布控制，并通过展开箱的方法进行评估，在方差-均值幂律和幂律自相关之间表现出双条件关系。Wiener–Khinchin Theorem 维纳-辛钦定理进一步表明，在这些条件下，对于任何具有方差-均值幂律的序列，也会出现1/f噪声。

The Tweedie convergence theorem provides a hypothetical explanation for the wide manifestation of fluctuation scaling and 1/f noise. It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a variance function that comes within the domain of attraction of a Tweedie model. Almost all distribution functions with finite cumulant generating functions qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form. Hence many probability distributions have variance functions that express this asymptotic behavior, and the Tweedie distributions become foci of convergence for a wide range of data types.

Tweedie Convergence Theorem Tweedie 收敛定理为涨落标度和1/f噪声的广泛出现提供了一个假设性解释。本质上，它要求任何一个可以渐近地显示方差-均值幂律的指数弥散模型，需要在Tweedie模型的吸引域内表达一个方差函数。几乎所有具有有限累积母函数的分布函数都符合指数弥散模型，而大多数指数弥散模型都表现出这种形式的方差函数。因此，许多概率分布都有表达这种渐近行为的方差函数，而Tweedie分布成为了不同数据类型收敛的焦点。

Much as the central limit theorem requires certain kinds of random variables to have as a focus of convergence the Gaussian distribution and express white noise, the Tweedie convergence theorem requires certain non-Gaussian random variables to express 1/f noise and fluctuation scaling.

正如中心极限定理要求某些类型的随机变量以高斯分布为收敛焦点并表示白噪声一样，Tweedie收敛定理要求某些非高斯随机变量来表达1/f噪声和涨落标度。

Cosmology 宇宙学

In physical cosmology, the power spectrum of the spatial distribution of the cosmic microwave background is near to being a scale-invariant function. Although in mathematics this means that the spectrum is a power-law, in cosmology the term "scale-invariant" indicates that the amplitude, P(k), of primordial fluctuations as a function of wave number, k, is approximately constant, i.e. a flat spectrum. This pattern is consistent with the proposal of cosmic inflation.

在Physical Cosmology 宇宙物理学，Cosmic Microwave Background 宇宙微波背景的空间分布功率频谱近似于标度不变函数。尽管在数学上这意味着该频谱服从幂律，但在宇宙学中“标度不变”一词表明，Primordial Fluctuations 原始涨落的振幅P(k)，作为波数k的函数，是近似常数，也就是一个平谱。这种模式与Cosmic Inflation 宇宙膨胀论的主张是一致的。

Scale invariance in classical field theory 经典场论中的标度不变性

Classical field theory is generically described by a field, or set of fields, φ, that depend on coordinates, x. Valid field configurations are then determined by solving differential equations for φ, and these equations are known as field equations.

经典场论一般用依赖于坐标x 的场或场集 φ 来描述。然后通过求解 φ 的微分方程来确定有效的场构型，这些方程被称为场方程。

For a theory to be scale-invariant, its field equations should be invariant under a rescaling of the coordinates, combined with some specified rescaling of the fields,

[math]\displaystyle{ x\rightarrow\lambda x~, }[/math]

[math]\displaystyle{ \varphi\rightarrow\lambda^{-\Delta}\varphi~. }[/math]

对于一个具有标度不变性的理论，它的场方程应该在坐标的缩放下保持不变，并结合特定的场的缩放，

[math]\displaystyle{ x\rightarrow\lambda x~, }[/math]，

[math]\displaystyle{ \varphi\rightarrow\lambda^{-\Delta}\varphi~. }[/math]。

The parameter Δ is known as the scaling dimension of the field, and its value depends on the theory under consideration. Scale invariance will typically hold provided that no fixed length scale appears in the theory. Conversely, the presence of a fixed length scale indicates that a theory is not scale-invariant.

参数 Δ 称为场的Scaling Dimension 标度维数，其大小取决于所考虑的理论。如果理论中没有固定长度的标度，标度不变性通常会成立。相反，如果存在固定的长度标度，则表明理论不具有标度不变性。

A consequence of scale invariance is that given a solution of a scale-invariant field equation, we can automatically find other solutions by rescaling both the coordinates and the fields appropriately. In technical terms, given a solution, φ(x), one always has other solutions of the form

标度不变性的一个结果是：给定一个标度不变性场方程的解，我们可以通过适当地缩放坐标和场自动地找到其他解。具体来说，给定一个解φ(x)，总有其他形式的解

[math]\displaystyle{ \lambda^{\Delta}\varphi(\lambda x) }[/math].

Scale invariance of field configurations 场结构中的标度不变性

For a particular field configuration, φ(x), to be scale-invariant, we require that

[math]\displaystyle{ \varphi(x)=\lambda^{-\Delta}\varphi(\lambda x) }[/math]

对于特定的场构型φ(x)，要具有标度不变性，我们就要满足：

[math]\displaystyle{ \varphi(x)=\lambda^{-\Delta}\varphi(\lambda x) }[/math]

where Δ is, again, the scaling dimension of the field.

其中 Δ 是场的标度维数。

We note that this condition is rather restrictive. In general, solutions even of scale-invariant field equations will not be scale-invariant, and in such cases the symmetry is said to be spontaneously broken.

我们注意到这个条件限制性很强。一般来说，即使是标度不变场方程的解也不是标度不变的，在这种情况下，对称性出现Spontaneously Broken 自发破缺。

Classical electromagnetism 经典电磁学

An example of a scale-invariant classical field theory is electromagnetism with no charges or currents. The fields are the electric and magnetic fields, E(x,t) and B(x,t), while their field equations are Maxwell's equations.

An example of a scale-invariant classical field theory is electromagnetism with no charges or currents. The fields are the electric and magnetic fields, E(x,t) and B(x,t), while their field equations are Maxwell's equations.

标度不变的经典场论的一个实例是没有电荷和电流的电磁学。场是电场和磁场，E(x,t) 和 B(x,t)，而它们的场方程是麦克斯韦方程组。

With no charges or currents, these field equations take the form of wave equations

[math]\displaystyle{ \nabla^2 \mathbf{E} = \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} }[/math]

[math]\displaystyle{ \nabla^2\mathbf{B} = \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2} }[/math]

where c is the speed of light.

在没有电荷或电流的情况下，这些场方程采用波动方程的形式：

[math]\displaystyle{ \nabla^2 \mathbf{E} = \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} }[/math]

[math]\displaystyle{ \nabla^2\mathbf{B} = \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2} }[/math]

其中 c 是光速。

These field equations are invariant under the transformation

[math]\displaystyle{ x\rightarrow\lambda x, }[/math]

[math]\displaystyle{ t\rightarrow\lambda t. }[/math]

这些场方程在进行如下变换下是不变的：

[math]\displaystyle{ x\rightarrow\lambda x, }[/math]，

[math]\displaystyle{ t\rightarrow\lambda t. }[/math]。

Moreover, given solutions of Maxwell's equations, E(x, t) and B(x, t), it holds that E(λx, λt) and B(λx, λt) are also solutions.

此外，已知E(x, t)和B(x, t)是麦克斯韦方程组的解，则可以认为E(λx, λt)和B(λx, λt)也是解。

Massless scalar field theory 无质量标量场理论

Another example of a scale-invariant classical field theory is the massless scalar field (note that the name scalar is unrelated to scale invariance). The scalar field, φ(x, t) is a function of a set of spatial variables, x, and a time variable, t.

标度不变经典场论的另一个例子是无质量标量场(注意名称“标量”与标度不变性无关)。标量场φ(x, t)是一组空间变量 x 和一个时间变量 t 的函数。

Consider first the linear theory. Like the electromagnetic field equations above, the equation of motion for this theory is also a wave equation,

[math]\displaystyle{ \frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi = 0, }[/math]

and is invariant under the transformation

[math]\displaystyle{ x\rightarrow\lambda x, }[/math]

[math]\displaystyle{ t\rightarrow\lambda t. }[/math]

首先考虑线性理论。像上述的电磁场方程一样，这个理论的运动方程也是一个波动方程：

[math]\displaystyle{ \frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi = 0, }[/math]，

并且在进行如下变换时是不变的：

[math]\displaystyle{ x\rightarrow\lambda x, }[/math]

[math]\displaystyle{ t\rightarrow\lambda t. }[/math]。

The name massless refers to the absence of a term [math]\displaystyle{ \propto m^2\varphi }[/math] in the field equation. Such a term is often referred to as a `mass' term, and would break the invariance under the above transformation. In relativistic field theories, a mass-scale, m is physically equivalent to a fixed length scale through

[math]\displaystyle{ L=\frac{\hbar}{mc}, }[/math]

and so it should not be surprising that massive scalar field theory is not scale-invariant.

无质量是指在场方程中没有[math]\displaystyle{ \propto m^2\varphi }[/math]项。这一项通常称为“质量”项，它会破坏上述变换下的不变性。在Relativistic Field Theories 相对论场理论中，质量标度m在物理上等同于一个固定的长度标度：

[math]\displaystyle{ L=\frac{\hbar}{mc}, }[/math]，

因此质量标量场理论不具有标度不变性也就不足为奇了。

φ⁴ theory φ⁴ 理论

The field equations in the examples above are all linear in the fields, which has meant that the scaling dimension, Δ, has not been so important. However, one usually requires that the scalar field action is dimensionless, and this fixes the scaling dimension of φ. In particular,

[math]\displaystyle{ \Delta=\frac{D-2}{2}, }[/math]

where D is the combined number of spatial and time dimensions.

上面例子中的场方程在场中都是线性的，这意味着标度维数Δ并不是那么重要。然而，通常要求标量场的作用是无量纲的，这就固定了φ的标度维数。特别是：

[math]\displaystyle{ \Delta=\frac{D-2}{2}, }[/math]

其中D是空间维数和时间维数的总和。

Given this scaling dimension for φ, there are certain nonlinear modifications of massless scalar field theory which are also scale-invariant. One example is massless φ⁴ theory for D=4. The field equation is

[math]\displaystyle{ \frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi+g\varphi^3=0. }[/math]。

已知φ的标度维数，则无质量标量场理论的某些非线性修正也是标度不变的。例如，D=4的无质量φ⁴theory φ⁴理论。场方程是：

[math]\displaystyle{ \frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi+g\varphi^3=0. }[/math]。

(Note that the name φ⁴ derives from the form of the Lagrangian, which contains the fourth power of φ.

(注意，φ4的名称来自拉格朗日量的形式，它包含φ的四次幂)

When D=4 (e.g. three spatial dimensions and one time dimension), the scalar field scaling dimension is Δ=1. The field equation is then invariant under the transformation

[math]\displaystyle{ x\rightarrow\lambda x, }[/math]

[math]\displaystyle{ t\rightarrow\lambda t, }[/math]

[math]\displaystyle{ \varphi (x)\rightarrow\lambda^{-1}\varphi(x). }[/math]

当D=4(如三维空间维数和一维时间维数)时，标量场标度维数为Δ=1。场方程在进行如下变换下是不变的：

[math]\displaystyle{ x\rightarrow\lambda x, }[/math]

[math]\displaystyle{ t\rightarrow\lambda t, }[/math]

[math]\displaystyle{ \varphi (x)\rightarrow\lambda^{-1}\varphi(x). }[/math]

The key point is that the parameter g must be dimensionless, otherwise one introduces a fixed length scale into the theory: For φ⁴ theory, this is only the case in D=4. Note that under these transformations the argument of the function φ is unchanged.

关键是参数g必须是无量纲的，否则就会引入一个固定的长度标度到理论中：对于φ⁴理论，只有在D=4时才会出现这种情况。注意，在这些变换下，函数φ的参数是不变的。

Scale invariance in quantum field theory 量子场论中的标度不变性

The scale-dependence of a quantum field theory (QFT) is characterised by the way its 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-functions 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 fixed points of the corresponding renormalization group flow.

对于具有标度不变性的量子场论（QFT），其耦合参数必须与能量标度无关，这由理论中β函数的消失来表示。这类理论也被称为相应重整化群流的固定点。

Quantum electrodynamics 量子电动力学

A simple example of a scale-invariant QFT is the quantized electromagnetic field without charged particles. This theory actually has no coupling parameters (since photons are massless and non-interacting) and is therefore scale-invariant, much like the classical theory.

标度不变量子场论的一个简单实例是没有带电粒子的量子化电磁场。这个理论实际上没有耦合参数(因为光子是无质量和非相互作用的) ，因此是标度不变的，这很像经典理论。

However, in nature the electromagnetic field is coupled to charged particles, such as electrons. 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 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（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

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 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φ⁴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。

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φ⁴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。

Conformal field theory 共形场论

Scale-invariant QFTs are almost always invariant under the full conformal symmetry, and the study of such QFTs is conformal field theory (CFT). Operators in a CFT have a well-defined scaling dimension, analogous to the scaling dimension, ∆, of a classical field discussed above. However, the scaling dimensions of operators in a CFT typically differ from those of the fields in the corresponding classical theory. The additional contributions appearing in the CFT are known as anomalous scaling dimensions.

在完全共形对称条件下，标度不变的量子场论几乎总是不变的，对此类量子场论的研究就是共形场论（CFT）。共形场论中的算子具有定义明确的标度维数，类似于前面所讨论的经典场标度维数 ∆。然而，共形场论中算子的标度维数与经典理论中场的标度维数不同。在共形场论中出现的附加贡献称做Anomalous Scaling Dimensions 异常标度维数。

Scale and conformal anomalies 标度与共形异常

The φ⁴ theory example above demonstrates that the coupling parameters of a quantum field theory can be scale-dependent even if the corresponding classical field theory is scale-invariant (or conformally invariant). If this is the case, the classical scale (or conformal) invariance is said to be anomalous. A classically scale invariant field theory, where scale invariance is broken by quantum effects, provides an explication of the nearly exponential expansion of the early universe called cosmic inflation, as long as the theory can be studied through perturbation theory.

上面的φ⁴理论例子表明，量子场论的耦合参数可以是标度依赖的，即使相应的经典场论是标度不变(或共形不变)。如果是这种情况，则称经典标度(或共形)不变性为异常。经典的标度不变场论，当量子效应打破其中的标度不变性，可以为接近指数级膨胀的早期宇宙提供了一种解释，即为Cosmic Inflation 宇宙膨胀，只要该理论可以通过Perturbation Theory 微扰理论研究。

Phase transitions 相变

In statistical mechanics, as a system undergoes a phase transition, its fluctuations are described by a scale-invariant statistical field theory. For a system in equilibrium (i.e. time-independent) in D spatial dimensions, the corresponding statistical field theory is formally similar to a D-dimensional CFT. The scaling dimensions in such problems are usually referred to as critical exponents, and one can in principle compute these exponents in the appropriate CFT.

In statistical mechanics, as a system undergoes a phase transition, its fluctuations are described by a scale-invariant statistical field theory. For a system in equilibrium (i.e. time-independent) in spatial dimensions, the corresponding statistical field theory is formally similar to a -dimensional CFT. The scaling dimensions in such problems are usually referred to as critical exponents, and one can in principle compute these exponents in the appropriate CFT.

在统计力学中，当某个系统经历相变时，其波动可以用标度不变的统计场论来描述。对于在D空间维度中处于平衡状态(即时间无关)的系统，相应的统计场论形式上类似于D维共形场论。这类问题中的标度维数通常称为Critical Exponents 临界指数，原则上可以在适当的共形场论中计算这些指数。

The Ising model 伊辛模型

An example that links together many of the ideas in this article is the phase transition of the Ising model, a simple model of ferromagnetic substances. This is a statistical mechanics model, which also has a description in terms of conformal field theory. The system consists of an array of lattice sites, which form a D-dimensional periodic lattice. Associated with each lattice site is a magnetic moment, or spin, and this spin can take either the value +1 or −1. (These states are also called up and down, respectively.)

将本文中的许多观点联系在一起的一个实例是伊辛模型的相变，这是一个关于铁磁物质的简单模型。还是一个具有共形场论描述的统计力学模型。该系统由一系列格子点位组成，这些点位构成了一个D维的周期格子。与每个格子位置相关联的是磁矩或自旋，这个自旋可以取 +1或-1。(这些状态也分别称为向上和向下。)

The key point is that the Ising model has a spin-spin interaction, making it energetically favourable for two adjacent spins to be aligned. On the other hand, thermal fluctuations typically introduce a randomness into the alignment of spins. At some critical temperature, T_c , spontaneous magnetization is said to occur. This means that below T_c the spin-spin interaction will begin to dominate, and there is some net alignment of spins in one of the two directions.

关键地是，伊辛模型具有自旋-自旋相互作用，这使得两个相邻的自旋在能量上更有利于排列。另一方面，热波动通常会给自旋的排列带来随机性。在某些临界温度(Tc)下，就会发生Spontaneous Magnetization 自发磁化。这意味着在临界温度以下，自旋-自旋相互作用将开始占据主导地位，并且在两个方向中的任一方向上存在部分自旋的净排列。

An example of the kind of physical quantities one would like to calculate at this critical temperature is the correlation between spins separated by a distance r. This has the generic behaviour:

[math]\displaystyle{ G(r)\propto\frac{1}{r^{D-2+\eta}}, }[/math]

for some particular value of [math]\displaystyle{ \eta }[/math], which is an example of a critical exponent.

在这个临界温度下，人们想要计算的物理量之一是存在距离的自旋之间的相互关系。此处通式为：

[math]\displaystyle{ G(r)\propto\frac{1}{r^{D-2+\eta}}, }[/math]，

对于某个特定的[math]\displaystyle{ \eta }[/math]值，这是一个临界指数的例子。

CFT description 共形场论描述

The fluctuations at temperature T_c are scale-invariant, and so the Ising model at this phase transition is expected to be described by a scale-invariant statistical field theory. In fact, this theory is the Wilson-Fisher fixed point, a particular scale-invariant scalar field theory.

在临界温度处的波动是标度不变的，因此相变时的伊辛模型可以用标度不变的统计场论来描述。事实上，这个理论就是威尔逊-费雪定点，一个特殊的标度不变标量场理论。

In this context, G(r) is understood as a correlation function of scalar fields,

[math]\displaystyle{ \langle\phi(0)\phi(r)\rangle\propto\frac{1}{r^{D-2+\eta}}. }[/math]

Now we can fit together a number of the ideas seen already.

此处，G(r)理解为标量场的相关函数，

[math]\displaystyle{ \langle\phi(0)\phi(r)\rangle\propto\frac{1}{r^{D-2+\eta}}. }[/math]。

现在我们可以把已经看到的一些想法联系起来。

From the above, one sees that the critical exponent, η, for this phase transition, is also an anomalous dimension. This is because the classical dimension of the scalar field,

[math]\displaystyle{ \Delta=\frac{D-2}{2} }[/math]

is modified to become

[math]\displaystyle{ \Delta=\frac{D-2+\eta}{2}, }[/math]

where D is the number of dimensions of the Ising model lattice.

由上可知，这种相变的临界指数也是异常维数。这是因为标量场的经典维数:

[math]\displaystyle{ \Delta=\frac{D-2}{2} }[/math]

修正为：

[math]\displaystyle{ \Delta=\frac{D-2+\eta}{2}, }[/math]

其中D 是伊辛模型格子的维数。

So this anomalous dimension in the conformal field theory is the same as a particular critical exponent of the Ising model phase transition.

So this anomalous dimension in the conformal field theory is the same as a particular critical exponent of the Ising model phase transition.

因此，共形场论中的这个异常维数与伊辛模型相变的特定临界指数是相同的。

Note that for dimension D ≡ 4−ε, η can be calculated approximately, using the epsilon expansion, and one finds that

[math]\displaystyle{ \eta=\frac{\epsilon^2}{54}+O(\epsilon^3) }[/math].

对于维度D ≡ 4−ε，可以使用epsilon展开式近似地计算η，并且可以发现：

[math]\displaystyle{ \eta=\frac{\epsilon^2}{54}+O(\epsilon^3) }[/math]。

In the physically interesting case of three spatial dimensions, we have ε=1, and so this expansion is not strictly reliable. However, a semi-quantitative prediction is that η is numerically small in three dimensions.

在物理上很有趣的三维空间情况下，我们有ε=1，因此这种膨胀并不严格可靠。然而，半定量的预测是η在三维上的数值很小。

On the other hand, in the two-dimensional case the Ising model is exactly soluble. In particular, it is equivalent to one of the minimal models, a family of well-understood CFTs, and it is possible to compute η (and the other critical exponents) exactly,

[math]\displaystyle{ \eta_{_{D=2}}=\frac{1}{4} }[/math].

另一方面，在二维情况下，伊辛模型是完全可解的。特别地，它等价于Minimal Model 最小模型之一，即一组很好理解的共形场论，并且可以精确地计算η(和其他临界指数)，

[math]\displaystyle{ \eta_{_{D=2}}=\frac{1}{4} }[/math]

Schramm–Loewner evolution 施拉姆—洛纳演化

The anomalous dimensions in certain two-dimensional CFTs can be related to the typical fractal dimensions of random walks, where the random walks are defined via Schramm–Loewner evolution (SLE). As we have seen above, CFTs describe the physics of phase transitions, and so one can relate the critical exponents of certain phase transitions to these fractal dimensions. Examples include the 2d critical Ising model and the more general 2d critical Potts model. Relating other 2d CFTs to SLE is an active area of research.

某些二维共形场论的异常维数可能与随机游动的典型分形维数有关，其中随机游动是通过施拉姆-洛纳演化(SLE)定义的。正如我们上面所看到的，共形场论描述了相变的物理过程，因此我们可以把某些相变的临界指数与这些分形维数联系起来。例如，二维临界伊辛模型和更一般的二维临界波茨模型。将其他二维共形场论与施拉姆-洛纳演化联系起来是一个活跃的研究领域。

Universality 普适性

A phenomenon known as universality is seen in a large variety of physical systems. It expresses the idea that different microscopic physics can give rise to the same scaling behaviour at a phase transition. A canonical example of universality involves the following two systems:

• The Ising model phase transition, described above.
• The liquid-vapour transition in classical fluids.

普适性的现象存在于许多不同的物理系统中。它表达了不同的微观物理过程可以在相变中产生相同的标度行为的观点。普适性的典型例子涉及以下两个系统:

• 伊辛模型相变，如上所述。
• 经典流体中的液-气转变。

Even though the microscopic physics of these two systems is completely different, their critical exponents turn out to be the same. Moreover, one can calculate these exponents using the same statistical field theory. The key observation is that at a phase transition or critical point, fluctuations occur at all length scales, and thus one should look for a scale-invariant statistical field theory to describe the phenomena. In a sense, universality is the observation that there are relatively few such scale-invariant theories.

尽管这两个系统的微观物理过程完全不同，但它们的临界指数却是相同的。此外，我们可以使用相同的统计场论计算这些指数。在相变或临界点，值得我们注意的是，在所有长度标度上都会出现波动，因此我们应该寻找一个标度不变的统计场论来描述这种现象。从某种意义上说，普适性就是指观察到这样的标度不变理论相对稀少。

The set of different microscopic theories described by the same scale-invariant theory is known as a universality class. Other examples of systems which belong to a universality class are:

• Avalanches in piles of sand. The likelihood of an avalanche is in power-law proportion to the size of the avalanche, and avalanches are seen to occur at all size scales.
• The frequency of network outages on the Internet, as a function of size and duration.
• The frequency of citations of journal articles, considered in the network of all citations amongst all papers, as a function of the number of citations in a given paper.
• The formation and propagation of cracks and tears in materials ranging from steel to rock to paper. The variations of the direction of the tear, or the roughness of a fractured surface, are in power-law proportion to the size scale.
• The electrical breakdown of dielectrics, which resemble cracks and tears.
• The percolation of fluids through disordered media, such as petroleum through fractured rock beds, or water through filter paper, such as in chromatography. Power-law scaling connects the rate of flow to the distribution of fractures.
• The diffusion of molecules in solution, and the phenomenon of diffusion-limited aggregation.
• The distribution of rocks of different sizes in an aggregate mixture that is being shaken (with gravity acting on the rocks).

由同一标度不变理论描述的不同微观理论的集合被称为普适性类。属于普适性类别的其他系统有:

• 沙堆中的塌落现象。发生塌落的可能性与塌落的规模服从幂律，而且可以看到塌落发生在所有不同的尺度上。
• 互联网网络中断的频率，是其规模和持续时间的函数。
• 期刊论文引用的频率（在所有论文的所有引用网络中考虑），是任一篇给定论文引用次数的函数。
• 从钢铁、岩石再到纸张等材料的裂缝和撕裂的形成和扩展。撕裂方向的变化，或破裂表面的粗糙度，与尺度成幂律关系。
• 电介质的电击穿现象，类似于裂缝和撕裂。
• 流体通过无序介质的渗透，如石油通过破碎的岩层，或水通过滤纸，如色谱法。幂律标度变化将流速与裂缝的分布联系起来。
• 分子在溶液中的扩散和Diffusion-limited Aggregation 扩散限制聚集现象。
• 在受重力作用而震动混杂的混合物中，不同大小的岩石碎块的分布。

The key observation is that, for all of these different systems, the behaviour resembles a phase transition, and that the language of statistical mechanics and scale-invariant statistical field theory may be applied to describe them.

最关键的是，对于所有这些不同的系统来说，它们的行为都类似于相变，并且可以用统计力学的方式和标度不变的统计场论来描述。

Other examples of scale invariance 标度不变性的其他实例

Newtonian fluid mechanics with no applied forces 无应力牛顿流体力学

Under certain circumstances, fluid mechanics is a scale-invariant classical field theory. The fields are the velocity of the fluid flow, [math]\displaystyle{ \mathbf{u}(\mathbf{x},t) }[/math], the fluid density, [math]\displaystyle{ \rho(\mathbf{x},t) }[/math], and the fluid pressure, [math]\displaystyle{ P(\mathbf{x},t) }[/math]. These fields must satisfy both the Navier–Stokes equation and the continuity equation. For a Newtonian fluid these take the respective forms

[math]\displaystyle{ \rho\frac{\partial \mathbf{u}}{\partial t}+\rho\mathbf{u}\cdot\nabla \mathbf{u} = -\nabla P+\mu \left(\nabla^2 \mathbf{u}+\frac{1}{3}\nabla\left(\nabla\cdot\mathbf{u}\right)\right) }[/math]

[math]\displaystyle{ \frac{\partial \rho}{\partial t}+\nabla\cdot \left(\rho\mathbf{u}\right)=0 }[/math]

where [math]\displaystyle{ \mu }[/math] is the dynamic viscosity.

在一定条件下，流体力学是一种标度不变的经典场论。流场包括流体流动速度[math]\displaystyle{ \mathbf{u}(\mathbf{x},t) }[/math]、流体密度[math]\displaystyle{ \rho(\mathbf{x},t) }[/math]和流体压力[math]\displaystyle{ P(\mathbf{x},t) }[/math]。这些场必须同时满足Navier–Stokes equation 纳维-斯托克斯方程和Continuity Equation 连续性方程。对于Newtonian Fluid 牛顿流体，它们有各自的形式：

[math]\displaystyle{ \rho\frac{\partial \mathbf{u}}{\partial t}+\rho\mathbf{u}\cdot\nabla \mathbf{u} = -\nabla P+\mu \left(\nabla^2 \mathbf{u}+\frac{1}{3}\nabla\left(\nabla\cdot\mathbf{u}\right)\right) }[/math]

[math]\displaystyle{ \frac{\partial \rho}{\partial t}+\nabla\cdot \left(\rho\mathbf{u}\right)=0 }[/math]

其中[math]\displaystyle{ \mu }[/math]是Dynamic Viscosity 动态黏度。

In order to deduce the scale invariance of these equations we specify an equation of state, relating the fluid pressure to the fluid density. The equation of state depends on the type of fluid and the conditions to which it is subjected. For example, we consider the isothermal ideal gas, which satisfies

[math]\displaystyle{ P=c_s^2\rho, }[/math]

where [math]\displaystyle{ c_s }[/math] is the speed of sound in the fluid. Given this equation of state, Navier–Stokes and the continuity equation are invariant under the transformations

[math]\displaystyle{ x\rightarrow\lambda x, }[/math]

[math]\displaystyle{ t\rightarrow\lambda^2 t, }[/math]

[math]\displaystyle{ \rho\rightarrow\lambda^{-1} \rho, }[/math]

[math]\displaystyle{ \mathbf{u}\rightarrow\mathbf{u}. }[/math]

Given the solutions [math]\displaystyle{ \mathbf{u}(\mathbf{x},t) }[/math] and [math]\displaystyle{ \rho(\mathbf{x},t) }[/math], we automatically have that [math]\displaystyle{ \lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2 t) }[/math] and [math]\displaystyle{ \lambda\rho(\lambda\mathbf{x},\lambda^2 t) }[/math] are also solutions.

为了推导这些方程的尺度不变性，我们指定一个状态方程，将流体压力与流体密度联系起来。状态方程取决于流体的类型及其所处的条件。例如，我们考虑等温理想气体，它满足：

[math]\displaystyle{ P=c_s^2\rho, }[/math]，

其中[math]\displaystyle{ c_s }[/math]是流体中声速。给定这个状态方程，纳维-斯托克斯方程和连续性方程在进行如下变换时是不变的：

[math]\displaystyle{ x\rightarrow\lambda x, }[/math]

[math]\displaystyle{ t\rightarrow\lambda^2 t, }[/math]

[math]\displaystyle{ \rho\rightarrow\lambda^{-1} \rho, }[/math]

[math]\displaystyle{ \mathbf{u}\rightarrow\mathbf{u}. }[/math].

已知解[math]\displaystyle{ \mathbf{u}(\mathbf{x},t) }[/math]和[math]\displaystyle{ \rho(\mathbf{x},t) }[/math]，我们自然可以得到[math]\displaystyle{ \lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2 t) }[/math]和[math]\displaystyle{ \lambda\rho(\lambda\mathbf{x},\lambda^2 t) }[/math]也是解。

Computer vision 计算机视觉

模板:Main article In computer vision and biological vision, scaling transformations arise because of the perspective image mapping and because of objects having different physical size in the world. In these areas, scale invariance refers to local image descriptors or visual representations of the image data that remain invariant when the local scale in the image domain is changed. Detecting local maxima over scales of normalized derivative responses provides a general framework for obtaining scale invariance from image data. Examples of applications include blob detection, corner detection, ridge detection, and object recognition via the scale-invariant feature transform.

在计算机视觉和生物视觉中，由于图像的透视映射和世界上物体的物理尺寸不同而产生了标度变换。在这些领域中，标度不变性是指当图像域的局部尺度发生变化时，图像数据的图像描述或视觉表达效果保持不变。在归一化导数响应的尺度上检测局部极大值为从图像数据中获取标度不变性提供了一个通用框架。应用的例子包括Blob Detection 斑点检测、Corner Detection 角点检测、Ridge Detection 脊线检测和通过Scale-Invariant Feature Transform 标度不变特征变换进行的目标识别。

See also 另见

• Inverse square potential
• Power law
• Scale-free network

• 逆平方势
• 幂律
• 无尺度网络定律

Further reading 拓展阅读

• Zinn-Justin, Jean (2002). Quantum Field Theory and Critical Phenomena. Oxford University Press.  Extensive discussion of scale invariance in quantum and statistical field theories, applications to critical phenomena and the epsilon expansion and related topics.
• DiFrancesco, P.; Mathieu, P.; Senechal, D. (1997). Conformal Field Theory. Springer-Verlag. 
• Mussardo, G. (2010). Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics. Oxford University Press. 

• Extensive discussion of scale invariance in quantum and statistical field theories, applications to critical phenomena and the epsilon expansion and related topics.

Category:Symmetry Category:Scaling symmetries Category:Conformal field theory Category:Critical phenomena

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