标度不变性

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The Wiener process is scale-invariant.

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.

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.

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.

• 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.

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

f(\lambda x)=\lambda^{\Delta}f(x)

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

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

Examples of scale-invariant functions are the monomials f(x)=x^n, for which , 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

An example of a scale-invariant curve is the logarithmic spiral, a kind of curve that often appears in nature. In polar coordinates , 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).

Allowing for rotations of the curve, it is invariant under all rescalings ; that is, is identical to a rotated version of .

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.

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齐次多项式，更一般地推广到齐次函数。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——概型的几何学中，它与弦理论中的各种主题都有联系。

Fractals 分形学

A Koch curve is self-similar.

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.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 , and even then a translation and rotation may have to be applied to match the fractal up to itself.

有时人们说分形是尺度不变的，尽管更准确地说，我们应该说它们是自相似的。一个分形通常只对一个离散的值集等于它自己，即使在这种情况下，平移和旋转也可能被用来匹配这个分形本身。

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.

Thus, for example, the Koch curve scales with , but the scaling holds only for values of for integer . 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，举例来说，Koch 曲线的尺度为，但是该尺度只适用于整数的值。此外，科氏曲线不仅在起源处，而且在某种意义上说，“到处”都是科氏曲线的缩影: 曲线上到处都可以找到科氏曲线本身的缩影。

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

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，这种标度是用多重分形分析研究的。

Periodic external and internal rays are invariant curves .

Periodic external and internal rays are invariant curves .

周期性的外部和内部射线是不变的曲线。

Scale invariance in stochastic processes

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).

如果是平均值，期望功率的频率，那么噪声标度为: p (f) = lambda ^ {-Delta } p (lambda f) ，其中白噪声 = 0，粉红噪声 =-1，布朗噪声 =-2(更一般的是布朗运动)。

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.

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，随机系统的标度关系到从所有可能的随机配置集中选择一个特定配置的可能性。这里需要更多的背景知识。概率和熵肯定与选择一个特定的配置有关，但是尺度不变性和这个联系并不明显。这种可能性是由美国概率分布研究所给出的。

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

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

尺度不变分布的例子有帕累托分布分布和 Zipfian 分布。

Scale invariant Tweedie distributions

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 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:

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

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 分布是指数离散模型的一个特例，它是一类统计模型，用于描述在加性和再生卷积以及尺度变换下广义线性模式和拥有属性闭包的误差分布。这些分布包括一些常见的分布: 正态分布，泊松分佈和伽玛分布，以及更不寻常的分布，如复合泊松-伽马分布，正稳定分布和极端稳定分布。由于它们固有的尺度不变性，Tweedie 随机变量 y 证明了一个方差 var (y)到平均 e (y)幂定律: : text { var } ，(y) = a [ text { e } ，(y)] ^ p，其中 a 和 p 是正常数。这种均值幂律的变化在物理学文献中称为涨落标度，在生态学文献中称为泰勒定律。

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.

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，并用扩箱法进行评价，表现出方差到均方幂律和幂律自相关的双重关系。维纳-钦钦定理进一步暗示，对于任何在这些条件下表现出方差到平均幂律的序列，也会表现出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.

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 收敛定理为涨落标度和1/f 噪声的广泛表现提供了一个假设性的解释。本质上，它要求任何渐近显示出均值幂律方差的指数离散模型都必须表示出一个方差函数，这个方差函数来自于 Tweedie 模型的吸引域。几乎所有具有有限累积量母函数的分布函数都可以称为指数离散模型，大多数指数离散模型都具有这种形式的方差函数。因此，许多概率分布具有表示这种渐近性的方差函数，而 Tweedie 分布成为各种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.

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.

正如中心极限定理需要某些类型的随机变量作为聚焦点来收敛正态分布和表示白噪声一样，Tweedie 收敛定理需要某些非高斯随机变量来表示1/f 噪声和波动尺度。

Cosmology

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.

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, , of primordial fluctuations as a function of wave number, , is approximately constant, i.e. a flat spectrum. This pattern is consistent with the proposal of cosmic inflation.

在21物理宇宙学，宇宙微波背景辐射空间分布的功率谱接近于尺度不变函数。尽管在数学上这意味着光谱是一个幂定律，但在宇宙学中“尺度不变量”这一术语表明，原始涨落的振幅，作为波数的函数，是近似常数，即。一个平坦的光谱。这种模式与宇宙膨胀的提议是一致的。

Scale invariance in classical field theory

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.

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]

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,

x\rightarrow\lambda x~,

\varphi\rightarrow\lambda^{-\Delta}\varphi~.

对于一个理论是标度不变的，它的场方程应该在坐标的重新标度下保持不变，并结合一些特定的重新标度，: x right tarrow lambda x ~ ，: varphi right tarrow lambda ^ {-Delta } varphi ~ 。

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.

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尺，尺度不变性通常会成立。相反，一个固定长度尺度的存在表明一个理论不是尺度不变的。

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

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

尺度不变性的一个结果是，给定一个尺度不变场方程的解，我们可以通过合适地重新标度坐标和场来自动地找到其他解。用专业术语来说，给定一个解 φ (x) ，总会有其它形式的解

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

\lambda^{\Delta}\varphi(\lambda x).

lambda ^ { Delta } varphi (lambda x).

Scale invariance of field configurations

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

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

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

\varphi(x)=\lambda^{-\Delta}\varphi(\lambda x)

对于特定的字段配置 φ (x) ，要使其具有尺度不变性，我们需要: varphi (x) = lambda ^ {-Delta } varphi (lambda x)

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

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.

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相当有限制性。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，对称性被称为自发破缺。

= 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.

比例不变的经典场论的一个例子是没有电荷和电流的电磁学。电场是电场和磁场，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.

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

\nabla^2 \mathbf{E} = \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2}

\nabla^2\mathbf{B} = \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2}

where c is the speed of light.

没有电荷和电流，这些场方程采用波动方程的形式: nabla ^ 2 mathbf { e } = frac {1}{ c ^ 2} frac { partial ^ 2 mathbf { e }{ partial t ^ 2} : nabla ^ 2 mathbf { b } = frac {1}{ c ^ 2} frac { b }{ partial t ^ 2}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]

These field equations are invariant under the transformation

x\rightarrow\lambda x,

t\rightarrow\lambda t.

这些字段方程在变换下是不变的: x right tarrow lambda x，: t right tarrow lambda。

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.

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.

给出了麦克斯韦方程组 e (x，t)和 b (x，t)的解，证明了 e (λx，λt)和 b (λx，λt)也是解。

Massless scalar field theory

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.

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, is a function of a set of spatial variables, x, and a time variable, .

标度不变的经典场论的另一个例子是无质量标量场(注意标量的名称与尺度不变性无关)。标量场，是一组空间变量 x 和一个时间变量 x 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]

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

\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi = 0,

and is invariant under the transformation

x\rightarrow\lambda x,

t\rightarrow\lambda t.

首先考虑线性理论。像上面的电磁场方程一样，这个理论的运动方程也是一个波动方程: frac {1}{ c ^ 2} frac { partial ^ 2 varphi }{ partial t ^ 2}-nabla ^ 2 varphi = 0，并且在变换下是不变的: x tarrow lambda x，: t right tarrow da t。

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.

The name massless refers to the absence of a term \propto m^2\varphi 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, is physically equivalent to a fixed length scale through

L=\frac{\hbar}{mc},

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

名称 massless 是指在字段方程中没有一个 propto m ^ 2 varphi 项。这样的术语通常被称为“质量”术语，它打破了上述变换下的不变性。在相对论场理论中，质量尺度在物理上等同于一个固定长度的尺度: l = frac { hbar }{ mc } ，因此大质量纯量场理论不具有尺度不变性也就不足为奇了。

φ⁴ theory

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.

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

\Delta=\frac{D-2}{2},

where 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，，并没有那么重要。但是，通常需要标量场操作是无量纲的，这修正了。特别是: Delta = frac { D-2}{2} ，其中是空间维度和时间维度的组合数。

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]

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

\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi+g\varphi^3=0.

给定这个尺度维数，无质量纯量场理论的非线性修正也是尺度不变的。一个例子是 = 4的无质量 φ < sup > 4 理论。场方程是: frac {1}{ c ^ 2} frac { partial ^ 2 varphi }{ partial t ^ 2}-nabla ^ 2 varphi + g varphi ^ 3 = 0。

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

(Note that the name 4 derives from the form of the Lagrangian, which contains the fourth power of .)

(注意，名称4来源于拉格朗日函数的形式，它包含. 的四次方。)

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]

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

x\rightarrow\lambda x,

t\rightarrow\lambda t,

\varphi (x)\rightarrow\lambda^{-1}\varphi(x).

当 = 4时。3空间维和1时间维) ，标量场标度维为 = 1。字段方程在变换下是不变的: x right tarrow lambda x，: t right tarrow lambda t，: varphi (x) right tarrow lambda ^ {-1} varphi (x)。

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.

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

关键在于参数必须是无量纲的，否则就会在理论中引入一个固定长度的标度: 对于4理论，这只是 = 4的情况。注意，在这些转换下，函数的参数是不变的。

Scale invariance in quantum field theory

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.

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.

量子场论的尺度依赖性表现为它的耦合参数依赖于给定物理过程的能量尺度。这种对能量的依赖性由重整化群描述，并被编码在理论的 beta 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.

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.J. Zinn-Justin (2010) Scholarpedia article "Critical Phenomena: field theoretical approach".

对于具有尺度不变性的 QFT，其耦合参数必须与能量尺度无关，这可以用理论中 β 函数的消失来表示。这种理论也被称为相应重整化群流的不动点。Zinn-Justin (2010) Scholarpedia 文章“批判现象: 场域理论方法”。

Quantum electrodynamics

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.

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 QFT 的一个简单例子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) ，因此是尺度不变的，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.

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力学理论是量子电动力学的，而且这个理论不是标度不变的。我们可以从 QED beta 函数中看到这一点。这告诉我们电荷(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尺度不变的，QED 却不是尺度不变的。

Massless scalar field theory

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.

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、无质量的量子化纯量场理论没有耦合参数。因此，像经典的版本一样，它是尺度不变的。在重整化群的语言中，这个理论被称为高斯不动点。

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.

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，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 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.

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.

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

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.

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，尺度不变量子图几乎都是不变的，对此类量子图的研究是共形场论的。CFT 中的算子有一个定义良好的尺度维度，类似于上面讨论的经典场的尺度维度 something。然而，CFT 中算子的尺度维数与经典理论中场的尺度维数不同。在 CFT 中出现的附加贡献被称为异常尺度维数。

Scale and conformal anomalies

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.

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，即使相应的经典场论是尺度不变的(或共形不变的) ，量子场论的耦合参数也可以是尺度相关的。如果是这种情况，经典尺度(或共形)不变性被称为反常的。一个经典的尺度不变场理论，其中尺度不变性被量子效应打破，提供了早期宇宙近乎指数膨胀的解释，称为宇宙暴涨，只要该理论可以通过摄动理论研究。

Phase transitions

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21统计力学，当一个系统经历相变时，它的波动用一个尺度不变的统计场理论来描述。对于一个处于平衡状态的系统(即。时间无关)的空间维度，相应的统计场理论形式上类似于一维 CFT。这类问题中的标度维数通常被称为临界指数，原则上可以在适当的 CFT 中计算这些指数。

The Ising model

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.)

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 -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例子是伊辛模型的相变，这是一个铁磁物质的简单模型。这是一个统计力学的模型，也有一个描述的共形场论。该系统由一系列晶格位点组成，这些位点构成了一个维的周期晶格。与每个晶格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上升和下降。)

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.

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, , spontaneous magnetization is said to occur. This means that below the spin-spin interaction will begin to dominate, and there is some net alignment of spins in one of the two directions.

关键点在于伊辛模型具有自旋-自旋相互作用，使其能量上有利于两个相邻的自旋对齐。另一方面，热涨落通常在自旋排列中引入一种随机性。在某些临界温度下，据说会发生自发磁化。这意味着在自旋-自旋相互作用下将开始占主导地位，并且在两个方向之一存在一些自旋的网络排列。

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.

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 . This has the generic behaviour:

G(r)\propto\frac{1}{r^{D-2+\eta}},

for some particular value of \eta, 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的一个例子就是距离分开的自旋之间的相互关系。这有一个通用的行为: : g (r) propto frac {1}{ r ^ { D-2 + eta }} ，对于某个特定的 eta 值，这是一个临界指数的例子。

CFT description

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.

The fluctuations at temperature 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.

温度的波动是尺度不变的，因此相变时的伊辛模型可以用尺度不变的统计场理论来描述。事实上，这个理论就是 Wilson-Fisher 的定点，一个特殊的尺度不变纯量场理论。

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.

In this context, is understood as a correlation function of scalar fields,

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

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

在这个上下文中，被理解为标量场的相关函数(量子场论) ，: langle phi (0) phi (r) rangle to frac {1}{ r ^ { D-2 + eta }。现在我们可以把已经看到的一些想法组合在一起。

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.

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

\Delta=\frac{D-2}{2}

is modified to become

\Delta=\frac{D-2+\eta}{2},

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

从上面，我们可以看到，对于这种相变，临界指数也是一个反常的维度。这是因为标量场的经典维数: Delta = frac { D-2}{2}被修改为: Delta = frac { D-2 + eta }{2} ，其中是 Ising 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，共形场论中的这个反常维度与伊辛模型相变的一个特定临界指数维度是相同的。

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].

Note that for dimension , can be calculated approximately, using the epsilon expansion, and one finds that

\eta=\frac{\epsilon^2}{54}+O(\epsilon^3).

注意，对于维度，可以使用 epsilon 展开式近似地计算，并且可以发现: eta = frac { epsilon ^ 2}{54} + o (epsilon ^ 3)。

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.

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

在物理上有趣的三维空间情况下，我们有 = 1，因此这种扩展严格来说是不可靠的。然而，半定量预测在三维空间中数值较小。

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].

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 (and the other critical exponents) exactly,

\eta_{_{D=2}}=\frac{1}{4}.

另一方面，在二维情况下，伊辛模型是完全可解的。特别地，它等价于一个极小模型，一个很好理解的 CFTs 族，并且可以精确地计算(和其他临界指数) : eta _ { _ { d = 2} = frac {1}{4}。

Schramm–Loewner evolution

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.

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 CFTs 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-洛沃纳演化(SLE)定义的。正如我们上面所看到的，CFTs 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。例子包括二维临界 Ising 模型和更一般的二维临界 Potts 模型。关联其他2 d cft 与系统性红斑狼疮是一个积极的研究领域。

Universality

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.

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.

在各种各样的物理系统中都可以看到一种被称为普遍性的现象。它表达了不同的微观物理学可以在相变过程中产生相同的标度行为的观点。普适性的一个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.

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学完全不同，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理论计算这些指数。关键的观察结果是，在相变或临界点，在所有长度尺度上都会出现波动，因此我们应该寻找一个尺度不变的统计场理论来描述这种现象。从某种意义上说，普适性就是观察到这样的尺度不变理论相对较少。

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).

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).

由同一尺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的例子有:

• 沙堆中的雪崩。发生雪崩的可能性与雪崩的规模成幂次比例，而且可以看到雪崩发生在所有规模的范围内。
• 互联网网络中断的频率，与规模和持续时间有关。
• 引用期刊论文的频率，在所有论文的引用网络中考虑，作为一篇论文引用次数的函数。
• 从钢铁、岩石到纸张等材料的裂纹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，与尺寸比例成幂律关系。
• 电介质的电击穿，类似裂缝和眼泪。
• 流体穿过无序介质(如石油穿过裂缝性岩层)或水穿过滤纸(如在色谱法中)的过程。幂律标度将流速与裂缝分布联系起来。
• 分子在溶液中的扩散和扩散限制聚集现象。
• 在受重力作用而摇动的骨料混合物中，不同大小的岩石的分布。

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.

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.

关键的观察结果是，对于所有这些不同的系统来说，它们的行为类似于相变，并且可以用统计力学的语言和尺度不变的统计场理论来描述它们。

= = 牛顿流体力学没有外力 =

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.

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

\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)

\frac{\partial \rho}{\partial t}+\nabla\cdot \left(\rho\mathbf{u}\right)=0

where \mu is the dynamic viscosity.

在一定条件下，流体力学是一种尺度不变的经典场论。流场包括流体流动速度 mathbf { u }(mathbf { x } ，t)、流体密度 rho (mathbf { x } ，t)和流体压力 p (mathbf { x } ，t)。这些场必须同时满足 Navier-Stokes 方程和连续性方程。对于牛顿流体，它们有各自的形式: rho frac { partial mathbf { u }{ partial t } + rho mathabla nabla mathbf { u } =-nabla p + mu left (nabla ^ 2 mathbf { u } + frac {1} nabla left (nabla cbf { u }{ u }右)) : frac { partial rho }{ partial t } + nabla cabla cleft (rho mathbf { u }右) = 0其中 mu 是动力粘度。

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.

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

P=c_s^2\rho,

where c_s is the speed of sound in the fluid. Given this equation of state, Navier–Stokes and the continuity equation are invariant under the transformations

x\rightarrow\lambda x,

t\rightarrow\lambda^2 t,

\rho\rightarrow\lambda^{-1} \rho,

\mathbf{u}\rightarrow\mathbf{u}.

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

为了推导这些方程的尺度不变性，我们指定了一个状态方程，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和它所处的条件。例如，我们考虑满足以下条件的等温理想气体: p = c _ s ^ 2 rho，其中 c _ s 是流体中声速。给定这个状态方程，Navier-Stokes 和连续性方程在变换下是不变的: x right tarrow lambda x，: t right tarrow lambda ^ 2 t，: rho right tarrow lambda ^ {-1} rho，: mathbf { u } right tarrow mathbf { u }。给定解 mathbf { u }(mathbf { x } ，t)和 rho (mathbf { x } ，t) ，我们自动有 lambda mathbf { u }(lambda mathbf { x } ，lambda ^ 2 t)和 lambda rho (lambda mathbf { x } ，lambda ^ 2 t)也是解。

Computer vision

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.

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.Lindeberg, T. (2013) Invariance of visual operations at the level of receptive fields, PLoS ONE 8(7):e66990. Detecting local maxima over scales of normalized derivative responses provides a general framework for obtaining scale invariance from image data.T. Lindeberg (2014) "Scale selection", Computer Vision: A Reference Guide, (K. Ikeuchi, Editor), Springer, pages 701-713. 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，由于透视图像映射和物体在世界上有不同的物理大小，缩放变换产生。在这些领域中，图像尺度不变性是指当图像域的局部尺度改变时保持不变的图像数据的局部图像描述符或视觉表示。视觉操作在感受野水平的不变性，PLoS ONE 8(7) : e66990。检测标准化导数响应的局部极大值为从图像数据中获取尺度不变性提供了一个通用框架。林德伯格(2014)“比例选择”，《计算机视觉: 参考指南》 ，(k. Ikeuchi，编辑) ，Springer，701-713页。应用的例子包括斑点检测、角检测、脊线检测和通过尺度不变特征转换识别系统进行的目标识别。

See also

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

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

• 逆平方势
• 无尺度网络定律

Further reading

• 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.

• 广泛讨论量子和统计领域理论中的尺度不变性，临界现象和 epsilon 展开及相关主题的应用。

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

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