# 有效场论

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In physics, an **effective field theory** is a type of approximation, or effective theory, for an underlying physical theory, such as a quantum field theory or a statistical mechanics model. An effective field theory includes the appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale or energy scale, while ignoring substructure and degrees of freedom at shorter distances (or, equivalently, at higher energies). Intuitively, one averages over the behavior of the underlying theory at shorter length scales to derive what is hoped to be a simplified model at longer length scales. Effective field theories typically work best when there is a large separation between length scale of interest and the length scale of the underlying dynamics. Effective field theories have found use in particle physics, statistical mechanics, condensed matter physics, general relativity, and hydrodynamics. They simplify calculations, and allow treatment of dissipation and radiation effects.^{[1]}^{[2]}

In physics, an effective field theory is a type of approximation, or effective theory, for an underlying physical theory, such as a quantum field theory or a statistical mechanics model. An effective field theory includes the appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale or energy scale, while ignoring substructure and degrees of freedom at shorter distances (or, equivalently, at higher energies). Intuitively, one averages over the behavior of the underlying theory at shorter length scales to derive what is hoped to be a simplified model at longer length scales. Effective field theories typically work best when there is a large separation between length scale of interest and the length scale of the underlying dynamics. Effective field theories have found use in particle physics, statistical mechanics, condensed matter physics, general relativity, and hydrodynamics. They simplify calculations, and allow treatment of dissipation and radiation effects.

在物理学中， 有效场论Effective field theory是一种有效的近似理论，用于基础的物理理论，比如量子场论或者统计力学模型理论。有效场论用适当的自由度来描述特定距离尺度或能量尺度下发生的物理现象，而忽略在较小尺度上的子结构和自由度(或者相仿地，在较高的能量上)。直观地说，一个人可以用较短的长度尺度对潜在理论的结果取平均，从而得出一个在较长长度尺度下的简化模型。当研究者感兴趣的尺度与相互作用的基本尺度存在较大差异时，有效场论是最实用的。有效场论已经在粒子物理学、统计力学、凝聚态物理学、广义相对论和流体力学中得到了应用。它们简化了计算，并可以处理耗散和辐射效应。

## The renormalization group重整化群

Presently, effective field theories are discussed in the context of the renormalization group (RG) where the process of *integrating out* short distance degrees of freedom is made systematic. Although this method is not sufficiently concrete to allow the actual construction of effective field theories, the gross understanding of their usefulness becomes clear through an RG analysis. This method also lends credence to the main technique of constructing effective field theories, through the analysis of symmetries. If there is a single mass scale **M** in the *microscopic* theory, then the effective field theory can be seen as an expansion in **1/M**. The construction of an effective field theory accurate to some power of **1/M** requires a new set of free parameters at each order of the expansion in **1/M**. This technique is useful for scattering or other processes where the maximum momentum scale **k** satisfies the condition **k/M≪1**. Since effective field theories are not valid at small length scales, they need not be renormalizable. Indeed, the ever expanding number of parameters at each order in **1/M** required for an effective field theory means that they are generally not renormalizable in the same sense as quantum electrodynamics which requires only the renormalization of two parameters.

Presently, effective field theories are discussed in the context of the renormalization group (RG) where the process of integrating out short distance degrees of freedom is made systematic. Although this method is not sufficiently concrete to allow the actual construction of effective field theories, the gross understanding of their usefulness becomes clear through an RG analysis. This method also lends credence to the main technique of constructing effective field theories, through the analysis of symmetries. If there is a single mass scale M in the microscopic theory, then the effective field theory can be seen as an expansion in 1/M. The construction of an effective field theory accurate to some power of 1/M requires a new set of free parameters at each order of the expansion in 1/M. This technique is useful for scattering or other processes where the maximum momentum scale k satisfies the condition k/M≪1. Since effective field theories are not valid at small length scales, they need not be renormalizable. Indeed, the ever expanding number of parameters at each order in 1/M required for an effective field theory means that they are generally not renormalizable in the same sense as quantum electrodynamics which requires only the renormalization of two parameters.

目前，有效场论是在 重整化群Renormalization group（RG）的背景下讨论的，重整化群使短距离自由度的积分过程变得系统化。尽管这种方法不够具体，无法实际构建有效场论，但通过RG分析，对其有用性的总体理解变得清晰。通过对对称性的分析，该方法也为构造有效场论的主要技术提供了依据。如果微观理论中只有一个质量尺度M，因此，有效场论可以看作是1/M的展开式。建立精确到1/M幂次的有效场理论需要在1/M阶展开的每一阶上都有一组新的自由参数。这种方法对于散射或其他最大动量标度k满足条件k/M≪1的过程是有用的。由于有效场论在小尺度下是无效的，所以它们不必是可重整化的。事实上，随着阶次升高，有效场论要求的参数数目不断增加，这意味着它们通常不像只需要两个参数即可重整化的量子电动力学那样可重整化。

## Examples of effective field theories有效场理论实例

### Fermi theory of beta decay贝塔衰变的费米理论

The best-known example of an effective field theory is the Fermi theory of beta decay. This theory was developed during the early study of weak decays of nuclei when only the hadrons and leptons undergoing weak decay were known. The typical reactions studied were:

The best-known example of an effective field theory is the Fermi theory of beta decay. This theory was developed during the early study of weak decays of nuclei when only the hadrons and leptons undergoing weak decay were known. The typical reactions studied were:

有效场理论最著名的例子是贝塔衰变费米理论。这个理论是在早期研究弱衰变核时发展起来的，当时物理学家只知道经历弱衰变的强子和轻子。研究的典型反应有:

- [math]\displaystyle{ \lt math\gt 《数学》 \begin{align} \begin{align} 开始{ align } n & \to p+e^-+\overline\nu_e \\ n & \to p+e^-+\overline\nu_e \\ N & to p + e ^-+ overline nu _ e \mu^- & \to e^-+\overline\nu_e+\nu_\mu. \mu^- & \to e^-+\overline\nu_e+\nu_\mu. Mu ^-& to e ^-+ overline nu _ e + nu _ mu. \end{align} \end{align} 结束{ align } }[/math]

</math>

数学

This theory posited a pointlike interaction between the four fermions involved in these reactions. The theory had great phenomenological success and was eventually understood to arise from the gauge theory of electroweak interactions, which forms a part of the standard model of particle physics. In this more fundamental theory, the interactions are mediated by a flavour-changing gauge boson, the W^{±}. The immense success of the Fermi theory was because the W particle has mass of about 80 GeV, whereas the early experiments were all done at an energy scale of less than 10 MeV. Such a separation of scales, by over 3 orders of magnitude, has not been met in any other situation as yet.

This theory posited a pointlike interaction between the four fermions involved in these reactions. The theory had great phenomenological success and was eventually understood to arise from the gauge theory of electroweak interactions, which forms a part of the standard model of particle physics. In this more fundamental theory, the interactions are mediated by a flavour-changing gauge boson, the W^{±}. The immense success of the Fermi theory was because the W particle has mass of about 80 GeV, whereas the early experiments were all done at an energy scale of less than 10 MeV. Such a separation of scales, by over 3 orders of magnitude, has not been met in any other situation as yet.

这一理论假定参与这些反应的四个费米子之间的点状相互作用，在现象学上取得了巨大的成功，成为描述弱电相互作用的规范理论，它构成了粒子物理学标准模型的一部分。在这个更基本的理论中，相互作用是由一个可以改变味的规范玻色子w±介导的。费米理论的巨大成功是因为 w 粒子的质量约为80gev，而早期的实验都是在能量小于10mev 的情况下进行的。这种差距超过了3个数量级，其他任何实验都难以达到。。

### BCS theory of superconductivityBCS超导理论

Another famous example is the BCS theory of superconductivity. Here the underlying theory is of electrons in a metal interacting with lattice vibrations called phonons. The phonons cause attractive interactions between some electrons, causing them to form Cooper pairs. The length scale of these pairs is much larger than the wavelength of phonons, making it possible to neglect the dynamics of phonons and construct a theory in which two electrons effectively interact at a point. This theory has had remarkable success in describing and predicting the results of experiments on superconductivity.

Another famous example is the BCS theory of superconductivity. Here the underlying theory is of electrons in a metal interacting with lattice vibrations called phonons. The phonons cause attractive interactions between some electrons, causing them to form Cooper pairs. The length scale of these pairs is much larger than the wavelength of phonons, making it possible to neglect the dynamics of phonons and construct a theory in which two electrons effectively interact at a point. This theory has had remarkable success in describing and predicting the results of experiments on superconductivity.

另一个著名的例子是超导现象的 BCS 理论。这里的基本理论是金属中的电子与声子相互作用。声子在一些电子之间引起吸引力的相互作用，导致它们形成库珀对。库珀对的长度比声子的波长大得多，因此可以忽略声子的动力学，建立两个电子在同一点上有效相互作用的理论。这个理论在描述和预测超导现象的实验结果方面取得了显著的成功。

### Effective Field Theories in Gravity重力中的有效场理论

General relativity itself is expected to be the low energy effective field theory of a full theory of quantum gravity, such as string theory or Loop Quantum Gravity. The expansion scale is the Planck mass.

General relativity itself is expected to be the low energy effective field theory of a full theory of quantum gravity, such as string theory or Loop Quantum Gravity. The expansion scale is the Planck mass.

广义相对论General relativity本身有望成为完整的量子引力理论的低能有效场论，如弦论或回圈量子重力理论。膨胀尺度是普朗克质量。

Effective field theories have also been used to simplify problems in General Relativity, in particular in calculating the gravitational wave signature of inspiralling finite-sized objects.^{[3]} The most common EFT in GR is "Non-Relativistic General Relativity" (NRGR),^{[4]}^{[5]}^{[6]} which is similar to the post-Newtonian expansion.^{[7]} Another common GR EFT is the Extreme Mass Ratio (EMR), which in the context of the inspiralling problem is called EMRI.

Effective field theories have also been used to simplify problems in General Relativity, in particular in calculating the gravitational wave signature of inspiralling finite-sized objects. The most common EFT in GR is "Non-Relativistic General Relativity" (NRGR), which is similar to the post-Newtonian expansion. Another common GR EFT is the Extreme Mass Ratio (EMR), which in the context of the inspiralling problem is called EMRI.

有效场论也被用来简化广义相对论中的问题，特别是在计算有限大小的物体的引力波特征时。GR 中最常见的 EFT 是“非相对论广义相对论”(NRGR) ，它类似于后牛顿力学近似方法。另一个常见的 GR EFT 是极端质量比(EMR) ，在激励问题的背景下被称为 EMRI。

### Other examples其他例子

Presently, effective field theories are written for many situations.

Presently, effective field theories are written for many situations.

目前，有效场理论是针对多种情况而编写的。

- One major branch of nuclear physics is quantum hadrodynamics, where the interactions of hadrons are treated as a field theory, which should be derivable from the underlying theory of quantum chromodynamics. Quantum hadrodynamics is the theory of the nuclear force, similarly to quantum chromodynamics being the theory of the strong interaction and quantum electrodynamics being the theory of the electromagnetic force. Due to the smaller separation of length scales here, this effective theory has some classificatory power, but not the spectacular success of the Fermi theory.
- 量子物理的一个主要分支是量子强子动力学，其中强子的相互作用被视为场理论，它应该从量子色动力学的基础理论中衍生出来。量子强子动力学是核力的理论，类似于量子色动力学是[[强相互作用]的理论，量子电动力学是[[电磁力]的理论。由于长度尺度的分离较小，这一有效理论具有一定的分类能力，但没有费米理论的惊人成功。
- In particle physics the effective field theory of QCD called chiral perturbation theory has had better success.
^{[8]}This theory deals with the interactions of hadrons with pions or kaons, which are the Goldstone bosons of spontaneous chiral symmetry breaking. The expansion parameter is the pion energy/momentum.

在粒子物理中， QCD中称为[[手征微扰理论]的有效场论有更好的表现成功。这一理论研究强子s与πs或kaons的相互作用，它们是自发手征对称性破坏的金石玻色子s。膨胀参数是pion能量/动量。

- For hadrons containing one heavy quark (such as the bottom or charm), an effective field theory which expands in powers of the quark mass, called the heavy quark effective theory (HQET), has been found useful.

对于含有一个重的夸克的强子s（例如底或粲），一种以夸克质量为幂展开的有效场论，称为[[重夸克有效理论]（HQET）。

- For hadrons containing two heavy quarks, an effective field theory which expands in powers of the relative velocity of the heavy quarks, called non-relativistic QCD (NRQCD), has been found useful, especially when used in conjunctions with lattice QCD.
- 对于含有两个重夸克的强子，以重夸克的相对速度为幂展开的有效场论很实用，称为非相对论性QCD（NRQCD），特别是在与晶格QCD结合时。
- For hadron reactions with light energetic (collinear) particles, the interactions with low-energetic (soft) degrees of freedom are described by the soft-collinear effective theory (SCET).

对于与光能（共线）粒子的强子反应，用软共线有效理论（SCET）描述了与低能（软）自由度的相互作用。

- Much of condensed matter physics consists of writing effective field theories for the particular property of matter being studied.
- 许多凝聚态物理都是为所研究的物质的特殊性质建立有效理论。
- Hydrodynamics can also be treated using Effective Field Theories
^{[9]}

[流体力学]也可以使用有效场论进行处理

## See also参见

形状因子（量子场论）

重整化群

量子场论

量子平凡性

金茨堡-兰道理论

## References参考

- ↑ Galley, Chad R. (2013). "Classical Mechanics of Nonconservative Systems" (PDF).
*Physical Review Letters*.**110**(17): 174301. doi:10.1103/PhysRevLett.110.174301. PMID 23679733. Archived from the original (PDF) on 2014-03-03. Retrieved 2014-03-03. Unknown parameter`|s2cid=`

ignored (help) - ↑ Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2014). "Radiation reaction at the level of the action".
*International Journal of Modern Physics A*.**29**(24): 1450132. arXiv:1402.2610. doi:10.1142/S0217751X14501322. Unknown parameter`|s2cid=`

ignored (help) - ↑ Goldberger, Walter; Rothstein, Ira (2004). "An Effective Field Theory of Gravity for Extended Objects".
*Physical Review D*.**73**(10). arXiv:hep-th/0409156. doi:10.1103/PhysRevD.73.104029. Unknown parameter`|s2cid=`

ignored (help) - ↑ [1]
- ↑ Kol, Barak; Smolkin, Lee (2008). "Non-Relativistic Gravitation: From Newton to Einstein and Back".
*Classical and Quantum Gravity*.**25**(14): 145011. arXiv:0712.4116. doi:10.1088/0264-9381/25/14/145011. Unknown parameter`|s2cid=`

ignored (help) - ↑ Porto, Rafael A (2006). "Post-Newtonian corrections to the motion of spinning bodies in NRGR".
*Physical Review D*.**73**(104031): 104031. arXiv:gr-qc/0511061. doi:10.1103/PhysRevD.73.104031. Unknown parameter`|s2cid=`

ignored (help) - ↑ Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2013). "Theory of post-Newtonian radiation and reaction".
*Physical Review D*.**88**(10): 104037. arXiv:1305.6930. doi:10.1103/PhysRevD.88.104037. Unknown parameter`|s2cid=`

ignored (help) - ↑ Leutwyler, H (1994). "On the Foundations of Chiral Perturbation Theory".
*Annals of Physics*.**235**: 165–203. arXiv:hep-ph/9311274. doi:10.1006/aphy.1994.1094. Unknown parameter`|s2cid=`

ignored (help) - ↑ Endlich, Solomon; Nicolis, Alberto; Porto, Rafael; Wang, Junpu (2013). "Dissipation in the effective field theory for hydrodynamics: First order effects".
*Physical Review D*.**88**(10): 105001. arXiv:1211.6461. doi:10.1103/PhysRevD.88.105001. Unknown parameter`|s2cid=`

ignored (help)

## External links外部链接

- Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (1998). "Effective Field Theory". arXiv:hep-ph/9806303.

- Hartmann, Stephan (2001). "Effective Field Theories, Reductionism and Scientific Explanation" (PDF).
*Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics*.**32**(2): 267–304. doi:10.1016/S1355-2198(01)00005-3.

- Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (1997). "Aspects of Heavy Quark Theory".
*Annual Review of Nuclear and Particle Science*.**47**: 591–661. arXiv:hep-ph/9703290. doi:10.1146/annurev.nucl.47.1.591. Unknown parameter`|s2cid=`

ignored (help)

- Effective field theory (Interactions, Symmetry Breaking and Effective Fields - from Quarks to Nuclei. an Internet Lecture by Jacek Dobaczewski)

模板:Industrial and applied mathematics

Category:Quantum field theory

范畴: 量子场论

Category:Statistical mechanics

类别: 统计力学

Category:Renormalization group

类别: 重整化群

Category:Chemical physics

类别: 化学物理

Category:Nuclear physics

类别: 核物理学

Category:Condensed matter physics

类别: 凝聚态物理学

This page was moved from wikipedia:en:Effective field theory. Its edit history can be viewed at 有效场论/edithistory