# 有效场论

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.

## 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.

## 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:

\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 } }

[/itex]

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.

### 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.

### 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.

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.

### Other examples其他例子

Presently, effective field theories are written for many situations.

Presently, effective field theories are written for many situations.

• 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

## References参考

1. 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)
2. 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)
3. 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)
4. [1]
5. 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)
6. 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)
7. 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)
8. 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)
9. 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)

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

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

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