1,068
个编辑
更改
跳到导航
跳到搜索
第9行：
第9行： − 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. − − The technical term for this [[transformation (mathematics)|transformation]] is a '''dilatation''' (also known as '''dilation'''), and the dilatations can also form part of a larger [[conformal symmetry]]. − − *In mathematics, scale invariance usually refers to an invariance of individual [[function (mathematics)|functions]] or [[curve]]s. 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 distribution]]s of [[random process]]es 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 transition]]s. The key observation is that near a phase transition or [[critical point (thermodynamics)|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 theory|statistical field theories]], and are formally very similar to scale-invariant quantum field theories. − *[[universality (dynamical systems)|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 moment]]s, which are scale invariant statistics of a variable, while the unstandardized moments are not. 第31行：
第21行： − + 第62行：
第52行： − + − 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 [[scheme (mathematics)|schemes]], it has connections to various topics in [[string theory]]. − + − It is sometimes said that [[fractal]]s 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 {{mvar|λ}}, and even then a translation and rotation may have to be applied to match the fractal up to itself. 第78行：
第66行： − − Some fractals may have multiple scaling factors at play at once; such scaling is studied with [[multi-fractal analysis]]. − − Periodic [[External ray|external and internal rays]] are invariant curves . − + 第102行：
第86行： − 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]]. − Examples of scale-invariant distributions are the [[Pareto distribution]] and the [[Zipfian distribution]]. − + − '''[[Tweedie distributions]]''' are a special case of '''[[exponential dispersion model]]s''', a class of statistical models used to describe error distributions for the [[generalized linear model]] and characterized by [[Closure (mathematics)|closure]] under additive and reproductive convolution as well as under scale transformation.<ref name="Jørgensen1997">{{cite book |last=Jørgensen |first=B. |year=1997 |title=The Theory of Dispersion Models |publisher=Chapman & Hall |location=London |isbn=978-0412997112 }}</ref> 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 distribution]]s, and extreme stable distributions. − Consequent to their inherent scale invariance Tweedie [[random variable]]s ''Y'' demonstrate a [[variance]] var(''Y'') to [[mean]] E(''Y'') power law: − : <math>\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''',<ref name="Eisler2008">Eisler, Z.; Bartos, I.; Kertész, J. (2008). "Fluctuation scaling in complex systems: Taylor's law and beyond". ''Adv Phys''. '''57''' (1): 89–142. arXiv:0708.2053. Bibcode:2008AdPhy..57...89E. doi:10.1080/00018730801893043. S2CID 119608542.</ref> and in the ecology literature as [[Taylor's law]].<ref name="Kendal2011a">{{cite journal |last1=Kendal |first1=W. S. |last2=Jørgensen |first2=B. |year=2011 |title=Taylor's power law and fluctuation scaling explained by a central-limit-like convergence |journal=Phys. Rev. E |volume=83 |issue=6 |pages=066115 |doi=10.1103/PhysRevE.83.066115 |pmid=21797449 |bibcode = 2011PhRvE..83f6115K }}</ref> − + + − + − Random sequences, governed by the Tweedie distributions and evaluated by the [[Tweedie distributions|method of expanding bins]] exhibit a [[Logical biconditional|biconditional]] relationship between the variance to mean power law and power law [[autocorrelation]]s. The [[Wiener–Khinchin theorem]] further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest [[pink noise|''1/f'' noise]].<ref name="Kendal2011">{{cite journal |last1=Kendal |first1=W. S. |last2=Jørgensen |first2=B. |year=2011 |title=Tweedie convergence: A mathematical basis for Taylor's power law, 1/''f'' noise, and multifractality |journal=Phys. Rev. E |volume=84 |issue=6 |pages=066120 |doi=10.1103/PhysRevE.84.066120 |bibcode = 2011PhRvE..84f6120K |pmid=22304168|url=https://findresearcher.sdu.dk:8443/ws/files/55639035/e066120.pdf }}</ref> − + − The [[Tweedie distributions|'''Tweedie convergence theorem''']] provides a hypothetical explanation for the wide manifestation of fluctuation scaling and ''1/f'' noise.<ref name="Jørgensen1994">Jørgensen, B.; Martinez, J. R.; Tsao, M. (1994). "Asymptotic behaviour of the variance function". ''Scand J Statist''. '''21''' (3): 223–243. JSTOR 4616314.</ref> It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a [[natural exponential family|variance function]] that comes within the [[Attractor|domain of attraction]] of a Tweedie model. Almost all distribution functions with finite [[cumulant|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 expansion|asymptotic behavior]], and the Tweedie distributions become foci of convergence for a wide range of data types.<ref name="Kendal2011" /> − '''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<ref name="Jørgensen1994" />。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<ref name="Kendal2011" />。 − Much as the [[central limit theorem]] requires certain kinds of random variables to have as a focus of convergence the [[normal distribution|Gaussian distribution]] and express [[white noise]], the Tweedie convergence theorem requires certain non-Gaussian random variables to express ''1/f'' noise and fluctuation scaling.<ref name="Kendal2011" />+ + − + − 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, {{math|''P''(''k'')}}, of [[primordial fluctuations]] as a function of [[wave number]], {{mvar|k}}, is approximately constant, i.e. a flat spectrum. This pattern is consistent with the proposal of [[cosmic inflation]]. − + − [[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 equation]]s. 第153行：
第128行： − − 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. − 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 第164行：
第136行： − + 第176行：
第148行： − 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]]. − + − An example of a scale-invariant classical field theory is [[electromagnetic field|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. 第215行：
第183行： − + − Another example of a scale-invariant classical field theory is the massless [[scalar field theory|scalar field]] (note that the name [[scalar (physics)|scalar]] is unrelated to scale invariance). The scalar field, {{math|''φ''('''''x''''', ''t'')}} is a function of a set of spatial variables, '''''x''''', and a time variable, {{mvar|t}}. 第246行：
第213行： − + 第289行：
第256行： − For a QFT to be scale-invariant, its coupling parameters must be independent of the energy-scale, and this is indicated by the vanishing of the beta-functions of the theory. Such theories are also known as [[Renormalization group|fixed points]] of the corresponding renormalization group flow.<ref name=":0">J. Zinn-Justin (2010) Scholarpedia article "Critical Phenomena: field theoretical approach".</ref>+ − − 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<ref name=":0" />。 − + − A simple example of a scale-invariant QFT is the quantized electromagnetic field without charged particles. This theory actually has no coupling parameters (since [[photon]]s are massless and non-interacting) and is therefore scale-invariant, much like the classical theory. − However, in nature the electromagnetic field is coupled to charged particles, such as [[electron]]s. The QFT describing the interactions of photons and charged particles is [[quantum electrodynamics]] (QED), and this theory is not scale-invariant. We can see this from the [[beta-function#Quantum electrodynamics|QED beta-function]]. This tells us that the [[electric charge]] (which is the coupling parameter in the theory) increases with increasing energy. Therefore, while the quantized electromagnetic field without charged particles '''is''' scale-invariant, QED is '''not''' scale-invariant. − + − Free, massless [[scalar field (quantum field theory)|quantized scalar field theory]] has no coupling parameters. Therefore, like the classical version, it is scale-invariant. In the language of the renormalization group, this theory is known as the [[Gaussian fixed point]]. − − However, even though the classical massless ''φ''<sup>4</sup> 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''. − − Even though the quantized massless ''φ''<sup>4</sup> 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. − + − + − + − + − + − 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 {{mvar|D}} spatial dimensions, the corresponding statistical field theory is formally similar to a {{mvar|D}}-dimensional CFT. The scaling dimensions in such problems are usually referred to as [[critical exponent]]s, 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. − + − An example that links together many of the ideas in this article is the phase transition of the [[Ising model]], a simple model of [[ferromagnet]]ic 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 {{mvar|D}}-dimensional periodic lattice. Associated with each lattice site is a [[magnetic moment]], or [[spin (physics)|spin]], and this spin can take either the value +1 or −1. (These states are also called up and down, respectively.) − − 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, {{math|''T<sub>c</sub>''}} , [[spontaneous magnetization]] is said to occur. This means that below {{math|''T<sub>c</sub>''}} the spin-spin interaction will begin to dominate, and there is some net alignment of spins in one of the two directions. − 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 {{mvar|r}}. This has the generic behaviour: − :<math>G(r)\propto\frac{1}{r^{D-2+\eta}},</math> − for some particular value of <math>\eta</math>, which is an example of a critical exponent. 第351行：
第300行： − + − The fluctuations at temperature {{math|''T<sub>c</sub>''}} 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 (quantum field theory)|scalar field theory]]. − In this context, {{math|''G''(''r'')}} is understood as a [[correlation function]] of scalar fields, − :<math>\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. 第366行：
第311行： − From the above, one sees that the critical exponent, {{mvar|η}}, for this phase transition, is also an '''anomalous dimension'''. This is because the classical dimension of the scalar field, − :<math>\Delta=\frac{D-2}{2}</math> − is modified to become − :<math>\Delta=\frac{D-2+\eta}{2},</math> − where {{mvar|D}} is the number of dimensions of the Ising model lattice. 第382行：
第322行： − 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. − Note that for dimension {{math|''D'' ≡ 4−''ε''}}, {{mvar|η}} can be calculated approximately, using the '''epsilon expansion''', and one finds that − :<math>\eta=\frac{\epsilon^2}{54}+O(\epsilon^3)</math>. 第395行：
第330行： − In the physically interesting case of three spatial dimensions, we have {{mvar|ε}}=1, and so this expansion is not strictly reliable. However, a semi-quantitative prediction is that {{mvar|η}} is numerically small in three dimensions. − − 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 model (physics)|minimal models]], a family of well-understood CFTs, and it is possible to compute {{mvar|η}} (and the other critical exponents) exactly, − :<math>\eta_{_{D=2}}=\frac{1}{4}</math>. 第406行：
第337行： − + − The anomalous dimensions in certain two-dimensional CFTs can be related to the typical [[fractal dimension]]s 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 2''d'' critical Ising model and the more general 2''d'' critical [[Potts model]]. Relating other 2''d'' CFTs to SLE is an active area of research. − + − A phenomenon known as [[universality (dynamical systems)|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. 第420行：
第347行： − 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 (thermodynamics)|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. − 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: − * [[Avalanche]]s 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 outage]]s 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.{{Citation needed|date=February 2017}} − * 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 [[dielectric]]s, 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 [[molecule]]s in [[Solution (chemistry)|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). 第444行：
第361行： − 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. − + − + − Under certain circumstances, [[fluid mechanics]] is a scale-invariant classical field theory. The fields are the velocity of the fluid flow, <math>\mathbf{u}(\mathbf{x},t)</math>, the fluid density, <math>\rho(\mathbf{x},t)</math>, and the fluid pressure, <math>P(\mathbf{x},t)</math>. These fields must satisfy both the [[Navier–Stokes equation]] and the [[continuity equation#Fluid dynamics|continuity equation]]. For a [[Newtonian fluid]] these take the respective forms − :<math>\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>\frac{\partial \rho}{\partial t}+\nabla\cdot \left(\rho\mathbf{u}\right)=0</math> − where <math>\mu</math> is the [[dynamic viscosity#Viscosity .28dynamic viscosity.29: .CE.BC|dynamic viscosity]]. 第462行：
第374行： − − 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>P=c_s^2\rho,</math> − where <math>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>x\rightarrow\lambda x,</math> − :<math>t\rightarrow\lambda^2 t,</math> − :<math>\rho\rightarrow\lambda^{-1} \rho,</math> − :<math>\mathbf{u}\rightarrow\mathbf{u}.</math> − Given the solutions <math>\mathbf{u}(\mathbf{x},t)</math> and <math>\rho(\mathbf{x},t)</math>, we automatically have that − <math>\lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2 t)</math> and <math>\lambda\rho(\lambda\mathbf{x},\lambda^2 t)</math> are also solutions. 第485行：
第387行： − + − {{Main article|Scale space}} − 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.<ref name=Lin13PONE>[https://dx.doi.org/10.1371/journal.pone.0066990 Lindeberg, T. (2013) Invariance of visual operations at the level of receptive fields, PLoS ONE 8(7):e66990.]</ref> − Detecting local maxima over scales of normalized derivative responses provides a general framework for obtaining scale invariance from image data.<ref name="Lindeberg1998">Lindeberg, Tony (1998). "Feature detection with automatic scale selection". ''International Journal of Computer Vision''. '''30''' (2): 79–116. doi:10.1023/A:1008045108935. S2CID 723210.</ref><ref name=Lin14CompVis>T. Lindeberg (2014) [http://www.csc.kth.se/~tony/abstracts/Lin14-ScSel-CompVisRefGuide.html "Scale selection", Computer Vision: A Reference Guide, (K. Ikeuchi, Editor), Springer, pages 701-713.]</ref> − Examples of applications include [[blob detection]], [[corner detection]], [[ridge detection]], and object recognition via the [[scale-invariant feature transform]]. − + − + − *[[Inverse square potential]] − *[[Power law]] − *[[Scale-free network]] − * − + + − + + − * 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]] − [[Category:Scale-invariant systems| ]] − − Category:Symmetry − Category:Scaling symmetries − Category:Conformal field theory − Category:Critical phenomena − − − 范畴: 对称范畴: 标度对称范畴: 共形场论范畴: 临界现象 − − <noinclude> − − <small>This page was moved from [[wikipedia:en:Scale invariance]]. Its edit history can be viewed at [[标度对称性/edithistory]]</small></noinclude>
无编辑摘要
维纳过程具有标度不变性。|链接=Special:FilePath/Wiener_process_animated.gif]]
维纳过程具有标度不变性。|链接=Special:FilePath/Wiener_process_animated.gif]]
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。
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。
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。
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。
* 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。
* 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。
* 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==
==标度不变曲线与自相似性==
In mathematics, one can consider the scaling properties of a [[function (mathematics)|function]] or [[curve]] {{math|''f'' (''x'')}} under rescalings of the variable {{mvar|x}}. That is, one is interested in the shape of {{math|''f'' (''λx'')}} for some scale factor {{mvar|λ}}, which can be taken to be a length or size rescaling. The requirement for {{math|''f'' (''x'')}} to be invariant under all rescalings is usually taken to be
In mathematics, one can consider the scaling properties of a [[function (mathematics)|function]] or [[curve]] {{math|''f'' (''x'')}} under rescalings of the variable {{mvar|x}}. That is, one is interested in the shape of {{math|''f'' (''λx'')}} for some scale factor {{mvar|λ}}, which can be taken to be a length or size rescaling. The requirement for {{math|''f'' (''x'')}} to be invariant under all rescalings is usually taken to be
:<math>f(\lambda x)=\lambda^{\Delta}f(x)</math>
:<math>f(\lambda x)=\lambda^{\Delta}f(x)</math>
在任意重新标度{{mvar|λ}}下，标度不变也允许曲线进行旋转；换句话说，{{math|''θ''(''λr'')}}与其旋转后的{{math|''θ''(''r'')}}一模一样。
在任意重新标度{{mvar|λ}}下，标度不变也允许曲线进行旋转；换句话说，{{math|''θ''(''λr'')}}与其旋转后的{{math|''θ''(''r'')}}一模一样。
===Projective geometry 射影几何===
=== 射影几何===
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。
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===
===分形===
[[File:Kochsim.gif|thumb|right|250px|A [[Koch curve]] is [[self-similar]].
[[File:Kochsim.gif|thumb|right|250px|A [[Koch curve]] is [[self-similar]].
科赫雪花具有自相似性。|链接=Special:FilePath/Kochsim.gif]]
科赫雪花具有自相似性。|链接=Special:FilePath/Kochsim.gif]]
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{{mvar|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。
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{{mvar|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。
因此，以{{math|∆ {{=}} 1}}的'''Koch Curve 科赫雪花'''缩放为例，但是该缩放只适用于{{math|''λ'' {{=}} 1/3<sup>''n''</sup>}}，（{{mvar|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”。
因此，以{{math|∆ {{=}} 1}}的'''Koch Curve 科赫雪花'''缩放为例，但是该缩放只适用于{{math|''λ'' {{=}} 1/3<sup>''n''</sup>}}，（{{mvar|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”。
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。
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。
周期性外部和内部射线是不变的曲线。
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==
==随机过程中的标度不变性==
If {{math|''P''(''f'' )}} is the [[expectation value|average, expected]] power at frequency {{mvar|f }}, then noise scales as
If {{math|''P''(''f'' )}} is the [[expectation value|average, expected]] power at frequency {{mvar|f }}, then noise scales as
:<math>P(f) = \lambda^{-\Delta} P(\lambda f)</math>
:<math>P(f) = \lambda^{-\Delta} P(\lambda f)</math>
当{{mvar|Δ}}= 0时对应'''White noise 白噪声'''，{{mvar|Δ}}=-1时对应'''Pink noise 粉红噪声'''，{{mvar|Δ}}=-2时对应'''Brownian noise 布朗噪声'''（更一般的是'''Brownian motion 布朗运动'''）。
当{{mvar|Δ}}= 0时对应'''White noise 白噪声'''，{{mvar|Δ}}=-1时对应'''Pink noise 粉红噪声'''，{{mvar|Δ}}=-2时对应'''Brownian noise 布朗噪声'''（更一般的是'''Brownian motion 布朗运动'''）。
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。
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。
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'''。
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===
===标度不变的Tweedie分布===
'''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<ref name="Jørgensen1997" />。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：
'''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<ref name="Jørgensen1997">{{cite book |last=Jørgensen |first=B. |year=1997 |title=The Theory of Dispersion Models |publisher=Chapman & Hall |location=London |isbn=978-0412997112 }}</ref>。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>\text{var}\,(Y) = a[\text{E}\,(Y)]^p</math>，
<math>\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'''<ref name="Eisler2008" />，在生态学文献中称为'''Taylor's Law 泰勒定律<ref name="Kendal2011a" />'''。
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'''<ref name="Eisler2008">Eisler, Z.; Bartos, I.; Kertész, J. (2008). "Fluctuation scaling in complex systems: Taylor's law and beyond". ''Adv Phys''. '''57''' (1): 89–142. arXiv:0708.2053. Bibcode:2008AdPhy..57...89E. doi:10.1080/00018730801893043. S2CID 119608542.</ref>，在生态学文献中称为'''Taylor's Law 泰勒定律'''<ref name="Kendal2011a">{{cite journal |last1=Kendal |first1=W. S. |last2=Jørgensen |first2=B. |year=2011 |title=Taylor's power law and fluctuation scaling explained by a central-limit-like convergence |journal=Phys. Rev. E |volume=83 |issue=6 |pages=066115 |doi=10.1103/PhysRevE.83.066115 |pmid=21797449 |bibcode = 2011PhRvE..83f6115K }}</ref>。
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<ref name="Kendal2011" />。
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<ref name="Kendal2011">{{cite journal |last1=Kendal |first1=W. S. |last2=Jørgensen |first2=B. |year=2011 |title=Tweedie convergence: A mathematical basis for Taylor's power law, 1/''f'' noise, and multifractality |journal=Phys. Rev. E |volume=84 |issue=6 |pages=066120 |doi=10.1103/PhysRevE.84.066120 |bibcode = 2011PhRvE..84f6120K |pmid=22304168|url=https://findresearcher.sdu.dk:8443/ws/files/55639035/e066120.pdf }}</ref>。
'''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<ref name="Jørgensen1994">Jørgensen, B.; Martinez, J. R.; Tsao, M. (1994). "Asymptotic behaviour of the variance function". ''Scand J Statist''. '''21''' (3): 223–243. JSTOR 4616314.</ref> 。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<ref name="Kendal2011" />。
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<ref name="Kendal2011" />。
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<ref name="Kendal2011" />。
===Cosmology 宇宙学===<!-- This section is linked from [[Cosmic inflation]] -->
=== 宇宙学===
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{{math|''P''(''k'')}}，作为波数{{mvar|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。
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{{math|''P''(''k'')}}，作为波数{{mvar|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==
==经典场论中的标度不变性==
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。
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。
<math>\varphi\rightarrow\lambda^{-\Delta}\varphi~.</math>。
<math>\varphi\rightarrow\lambda^{-\Delta}\varphi~.</math>。
参数 {{mvar|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。
参数 {{mvar|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。
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
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>\lambda^{\Delta}\varphi(\lambda x)</math>.
:<math>\lambda^{\Delta}\varphi(\lambda x)</math>.
===Scale invariance of field configurations 场结构中的标度不变性===
===场结构中的标度不变性===
For a particular field configuration, ''φ''(''x''), to be scale-invariant, we require that
For a particular field configuration, ''φ''(''x''), to be scale-invariant, we require that
:<math>\varphi(x)=\lambda^{-\Delta}\varphi(\lambda x)</math>
:<math>\varphi(x)=\lambda^{-\Delta}\varphi(\lambda x)</math>
其中 {{mvar|Δ}} 是场的标度维数。
其中 {{mvar|Δ}} 是场的标度维数。
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'''。
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 ===
===经典电磁学 ===
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。
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。
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'''(λ'''x''', λ''t'')和'''B'''(λ'''x''', λ''t'')也是解。
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'''(λ'''x''', λ''t'')和'''B'''(λ'''x''', λ''t'')也是解。
===Massless scalar field theory 无质量标量场理论===
===无质量标量场理论===
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{{math|''φ''('''''x''''', ''t'')}}是一组空间变量 '''''x''''' 和一个时间变量 {{mvar|t}} 的函数。
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{{math|''φ''('''''x''''', ''t'')}}是一组空间变量 '''''x''''' 和一个时间变量 {{mvar|t}} 的函数。
因此质量标量场理论不具有标度不变性也就不足为奇了。
因此质量标量场理论不具有标度不变性也就不足为奇了。
====φ<sup>4</sup> theory φ<sup>4</sup> 理论====
====φ<sup>4</sup> 理论====
The field equations in the examples above are all [[linear]] in the fields, which has meant that the [[scaling dimension]], {{mvar|Δ}}, has not been so important. However, one usually requires that the scalar field [[action (physics)|action]] is dimensionless, and this fixes the [[scaling dimension]] of {{mvar|φ}}. In particular,
The field equations in the examples above are all [[linear]] in the fields, which has meant that the [[scaling dimension]], {{mvar|Δ}}, has not been so important. However, one usually requires that the scalar field [[action (physics)|action]] is dimensionless, and this fixes the [[scaling dimension]] of {{mvar|φ}}. In particular,
:<math>\Delta=\frac{D-2}{2},</math>
:<math>\Delta=\frac{D-2}{2},</math>
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。
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。
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<ref name=":0">J. Zinn-Justin (2010) Scholarpedia article "Critical Phenomena: field theoretical approach".</ref>。
===Quantum electrodynamics 量子电动力学===
===量子电动力学===
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。
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。
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。
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===
===无质量标量场理论===
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'''。
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'''。
然而，尽管经典的无质量''φ''<sup>4</sup>理论在''D''=4时是标度不变的，但量子化的版本却不是如此。我们可以从耦合参数''g''的β函数中看出这一点。
然而，尽管经典的无质量''φ''<sup>4</sup>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。
虽然量子化无质量''φ''<sup>4</sup>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'''。
虽然量子化无质量''φ''<sup>4</sup>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===
===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). [[operator (physics)|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 dimension]]s.
Scale-invariant QFTs are almost always invariant under the full [[conformal symmetry]], and the study of such QFTs is [[conformal field theory]] (CFT). [[operator (physics)|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 dimension]]s.
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'''。
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===
=== 标度与共形异常===
The φ<sup>4</sup> 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 [[conformal anomaly|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 [[Inflation (cosmology)|cosmic inflation]], as long as the theory can be studied through [[perturbation theory]].<ref name=":1">{{cite journal|last=Salvio, Strumia|title=Agravity|journal=JHEP |volume=2014 |issue=6|pages=080|date=2014-03-17|url=http://inspirehep.net/record/1286134|arxiv = 1403.4226|bibcode = 2014JHEP...06..080S|doi=10.1007/JHEP06(2014)080}}</ref>
The φ<sup>4</sup> 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 [[conformal anomaly|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 [[Inflation (cosmology)|cosmic inflation]], as long as the theory can be studied through [[perturbation theory]].
上面的φ<sup>4</sup>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<ref name=":1" />。
上面的φ<sup>4</sup>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<ref name=":1">{{cite journal|last=Salvio, Strumia|title=Agravity|journal=JHEP |volume=2014 |issue=6|pages=080|date=2014-03-17|url=http://inspirehep.net/record/1286134|arxiv = 1403.4226|bibcode = 2014JHEP...06..080S|doi=10.1007/JHEP06(2014)080}}</ref>。
==Phase transitions 相变==
==相变==
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{{mvar|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{{mvar|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。
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{{mvar|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{{mvar|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===
===伊辛模型===
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{{mvar|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。)
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{{mvar|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。)
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。
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。
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：
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：
对于某个特定的<math>\eta</math>值，这是一个临界指数的例子。
对于某个特定的<math>\eta</math>值，这是一个临界指数的例子。
====CFT description 共形场论描述====
====共形场论描述====
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。
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。
此处，{{math|''G''(''r'')}}理解为标量场的相关函数，
此处，{{math|''G''(''r'')}}理解为标量场的相关函数，
现在我们可以把已经看到的一些想法联系起来。
现在我们可以把已经看到的一些想法联系起来。
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:
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:
其中{{mvar|D}} 是伊辛模型格子的维数。
其中{{mvar|D}} 是伊辛模型格子的维数。
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。
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。
对于维度{{math|''D'' ≡ 4−''ε''}}，可以使用epsilon展开式近似地计算{{mvar|η}}，并且可以发现：
对于维度{{math|''D'' ≡ 4−''ε''}}，可以使用epsilon展开式近似地计算{{mvar|η}}，并且可以发现：
<math>\eta=\frac{\epsilon^2}{54}+O(\epsilon^3)</math>。
<math>\eta=\frac{\epsilon^2}{54}+O(\epsilon^3)</math>。
在物理上很有趣的三维空间情况下，我们有{{mvar|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。
在物理上很有趣的三维空间情况下，我们有{{mvar|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。
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)，
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>\eta_{_{D=2}}=\frac{1}{4}</math>
<math>\eta_{_{D=2}}=\frac{1}{4}</math>
===Schramm–Loewner evolution 施拉姆—洛纳演化===
===施拉姆—洛纳演化===
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。
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==
==普适性==
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:
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:
* 经典流体中的液-气转变。
* 经典流体中的液-气转变。
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。
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。
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:
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:
* 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。
* 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。
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。
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==
== 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===
===无应力牛顿流体力学===
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>\mathbf{u}(\mathbf{x},t)</math>、流体密度<math>\rho(\mathbf{x},t)</math>和流体压力<math>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：
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>\mathbf{u}(\mathbf{x},t)</math>、流体密度<math>\rho(\mathbf{x},t)</math>和流体压力<math>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：
其中<math>\mu</math>是'''Dynamic Viscosity 动态黏度'''。
其中<math>\mu</math>是'''Dynamic Viscosity 动态黏度'''。
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：
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：
:已知解<math>\mathbf{u}(\mathbf{x},t)</math>和<math>\rho(\mathbf{x},t)</math>，我们自然可以得到<math>\lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2 t)</math>和<math>\lambda\rho(\lambda\mathbf{x},\lambda^2 t)</math>也是解。
:已知解<math>\mathbf{u}(\mathbf{x},t)</math>和<math>\rho(\mathbf{x},t)</math>，我们自然可以得到<math>\lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2 t)</math>和<math>\lambda\rho(\lambda\mathbf{x},\lambda^2 t)</math>也是解。
===Computer vision 计算机视觉===
=== 计算机视觉===
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<ref name="Lin13PONE" />。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<ref name="Lindeberg1998" /><ref name="Lin14CompVis" />。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。
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<ref name=Lin13PONE>[https://dx.doi.org/10.1371/journal.pone.0066990 Lindeberg, T. (2013) Invariance of visual operations at the level of receptive fields, PLoS ONE 8(7):e66990.]</ref> 。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<ref name="Lindeberg1998">Lindeberg, Tony (1998). "Feature detection with automatic scale selection". ''International Journal of Computer Vision''. '''30''' (2): 79–116. doi:10.1023/A:1008045108935. S2CID 723210.</ref><ref name=Lin14CompVis>T. Lindeberg (2014) [http://www.csc.kth.se/~tony/abstracts/Lin14-ScSel-CompVisRefGuide.html "Scale selection", Computer Vision: A Reference Guide, (K. Ikeuchi, Editor), Springer, pages 701-713.]</ref>。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==
==参见==
* 逆平方势
* 逆平方势
* 幂律
* 幂律
* 无尺度网络定律
* 无尺度网络定律
==References 参考文献==
==参考文献==
{{Reflist}}
{{Reflist}}
==Further reading 拓展阅读==
==拓展阅读==
*{{cite book |last=Zinn-Justin |first=Jean |title=Quantum Field Theory and Critical Phenomena |publisher=Oxford University Press |year=2002 }} Extensive discussion of scale invariance in quantum and statistical field theories, applications to critical phenomena and the epsilon expansion and related topics.
*{{cite book |last=Zinn-Justin |first=Jean |title=Quantum Field Theory and Critical Phenomena |publisher=Oxford University Press |year=2002 }} Extensive discussion of scale invariance in quantum and statistical field theories, applications to critical phenomena and the epsilon expansion and related topics.
*{{cite book |first1=P. |last1=DiFrancesco |first2=P. |last2=Mathieu |first3=D. |last3=Senechal |title=Conformal Field Theory |publisher=Springer-Verlag |year=1997 }}
*{{cite book |first1=P. |last1=DiFrancesco |first2=P. |last2=Mathieu |first3=D. |last3=Senechal |title=Conformal Field Theory |publisher=Springer-Verlag |year=1997 }}
*{{cite book |first=G. |last=Mussardo |title=Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics |publisher=Oxford University Press |year=2010 }}
*{{cite book |first=G. |last=Mussardo |title=Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics |publisher=Oxford University Press |year=2010 }}
== 编者推荐 ==
== 编者推荐 ==