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.

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，有效场论是一种近似，或者说是一种有效的理论，用于基础的物理理论，比如量子场论或者统计力学模型。一个有效的场论包括适当的自由度来描述在选定的长度尺度或能量尺度下发生的物理现象，而忽略在较短距离上的子结构和自由度(或者等效地，在较高的能量上)。直观上，一个人可以用较短的长度尺度对潜在理论的行为进行平均，从而得出一个希望成为较长长度尺度下的简化模型。有效的领域理论通常最好的时候有一个大分离的长度尺度的兴趣和长度尺度的基本动态。有效的场理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，并允许处理耗散和辐射效应。

The renormalization group

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，有效场理论的讨论背景下的重整化群(RG) ，其中积分出短距离自由度的过程是系统化的。虽然这种方法不够具体，不足以实际构建有效的领域理论，但通过 RG 分析，对其有用性的粗略理解变得清晰起来。通过对对称性的分析，这种方法也印证了构造有效场理论的主要技术。如果在微观理论中有一个单一的质量尺度 m，那么有效场理论可以看作是在1/M 中的展开。建立一个精确到1/M 次方的有效场理论需要在1/M 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 something 1的过程是有用的。由于有效场理论在小尺度上是不可重整的，所以它们不一定是可重整的。事实上，一个有效场论所需要的每一个1/M 次序中不断扩大的参数数量意味着它们通常不具有与量子电动力学相同的可重整化性，后者只需要重整化两个参数。

Examples of effective field theories

Fermi theory of beta decay

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]

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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.

这个理论假定了参与这些反应的四个费米子之间的点状相互作用。这个理论在现象学上取得了巨大的成功，并最终被理解为产生于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道的规范玻色子 w < sup > ± 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。这样的分离，超过3个数量级，还没有在任何其他情况下达到。

BCS theory of 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.

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称为声子的晶格振动相互作用。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。这些对的长度尺度远远大于声子的波长，因此可以忽略声子的动力学，建立一个理论，即两个电子在某一点有效地相互作用。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

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. 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.

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.

有效的场理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

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.

• In particle physics the effective field theory of QCD called chiral perturbation theory has had better success. 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.

• 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.

• 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.

• 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).

• Much of condensed matter physics consists of writing effective field theories for the particular property of matter being studied.

• Hydrodynamics can also be treated using Effective Field Theories

See also

• Form factor (quantum field theory)

• Renormalization group

• Quantum field theory

• Quantum triviality

• Ginzburg–Landau theory

External links

• 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. S2CID 13843227.

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

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