重整化群

方程

g(μ) = G−1( (μ/M)d G(g(M)) ) ,

g(κ) = G−1( (κ/μ)d G(g(μ)) ) = G−1( (κ/M)d G(g(M)) )

块自旋

1. ${\displaystyle T=0}$ ${\displaystyle J\to \infty }$ 。从宏观上来看，温度对系统的影响变得可以忽略不计。这时系统处于铁磁相。
2. ${\displaystyle T\to \infty }$ ${\displaystyle J\to 0}$ 。与第1种情形正好相反，温度对系统的影响占据了主导，系统在宏观上变得无序。
3. ${\displaystyle T=T_{c}}$ ${\displaystyle J=J_{c}}$ 。在这一特定的状态上，改变系统的标度不改变系统的物理性质，因为系统处于分形态上。这对应居里相变，这个点称为临界点

举例计算

${\displaystyle {\mathcal {L}}(\phi )={m^{2} \over 2}\phi ^{2}+{1 \over 2}(\partial _{\mu }\phi )^{2}+{\lambda \over 4!}\phi ^{4}.}$

${\displaystyle Z=\int {\mathcal {D}}\phi \exp \left[-\int d^{(d)}x\left({m^{2} \over 2}\phi ^{2}+{1 \over 2}(\partial _{\mu }\phi )^{2}+{\lambda \over 4!}\phi ^{4}\right)\right].}$

${\displaystyle Z=\int \left[{\mathcal {D}}\phi \right]_{\Lambda }\exp \left[-\int d^{(d)}x\left({m^{2} \over 2}\phi ^{2}+{1 \over 2}(\partial _{\mu }\phi )^{2}+{\lambda \over 4!}\phi ^{4}\right)\right].}$

${\displaystyle 0

${\displaystyle {\hat {\phi }}(p)={\begin{cases}\phi (p),&{\mbox{if }}b\Lambda \leqslant |p|<\Lambda \\0,&{\mbox{if }}|p|
${\displaystyle \phi (p)={\begin{cases}0,&{\mbox{if }}b\Lambda \leqslant |p|<\Lambda \\\phi (p),&{\mbox{if }}|p|

${\displaystyle Z=\int \left[{\mathcal {D}}\phi \right]_{b\Lambda }\int {\mathcal {D}}{\hat {\phi }}\exp \left[-\int d^{(d)}x\left({m^{2} \over 2}(\phi +{\hat {\phi }})^{2}+{1 \over 2}(\partial _{\mu }\phi +\partial _{\mu }{\hat {\phi }})^{2}+{\lambda \over 4!}(\phi +{\hat {\phi }})^{4}\right)\right].}$

${\displaystyle Z=\int \left[{\mathcal {D}}\phi \right]_{b\Lambda }e^{-\int d^{(d)}x{\mathcal {L}}(\phi )}\int {\mathcal {D}}{\hat {\phi }}\exp \left[-\int d^{(d)}x\left({m^{2} \over 2}{\hat {\phi }}^{2}+{1 \over 2}(\partial _{\mu }{\hat {\phi }})^{2}+\lambda ({1 \over 6}\phi ^{3}{\hat {\phi }}+{1 \over 4}\phi ^{2}{\hat {\phi }}^{2}+{1 \over 6}\phi {\hat {\phi }}^{3}+{1 \over 4!}{\hat {\phi }}^{4})\right)\right].}$

${\displaystyle Z=\int \left[{\mathcal {D}}\phi \right]_{b\Lambda }\exp {\left[-\int d^{(d)}x{\mathcal {L}}_{\textrm {eff}}(\phi )\right]},}$

${\displaystyle {\mathcal {L}}_{\textrm {eff}}(\phi )}$  不同于${\displaystyle {\mathcal {L}}(\phi )}$ ，因为${\displaystyle \lambda ,\phi }$  改变了。 上面的 Z 陈述一个。若 ${\displaystyle x'=bx,p'={p \over b},|p|<\Lambda }$ .

${\displaystyle \int d^{(d)}x{\mathcal {L}}_{\textrm {eff}}(\phi )=\int d^{(d)}x'b^{-d}\left[{1 \over 2}(1+\Delta Z)b^{2}(\partial '_{\mu }\phi )^{2}+{1 \over 2}(m^{2}+\Delta m^{2})\phi ^{2}+{1 \over 4!}(\lambda +\Delta \lambda )\phi ^{4}+\Delta Bb^{4}(\partial '_{\mu }\phi )^{4}+\Delta C\phi ^{6}+...\right]}$

${\displaystyle \phi '=[b^{(2-d)}(1+\Delta Z)]^{1/2}\cdot \phi }$
${\displaystyle m'^{2}=(1+\Delta Z)^{-1}(m^{2}+\Delta m^{2})b^{-2}}$
${\displaystyle \lambda '=(1+\Delta Z)^{-2}(\lambda +\Delta \lambda )b^{d-4}}$
${\displaystyle B'=(1+\Delta Z)^{-2}(B+\Delta B)b^{d}}$
${\displaystyle C'=(1+\Delta Z)^{-3}(C+\Delta C)b^{2d-6}}$

${\displaystyle \int d^{(d)}x{\mathcal {L}}_{\textrm {eff}}(\phi )=\int d^{(d)}x'\left[{1 \over 2}(\partial '_{\mu }\phi ')^{2}+{1 \over 2}m'^{2}\phi '^{2}+{1 \over 4!}\lambda '\phi ^{4}+\Delta B(\partial '_{\mu }\phi ')^{4}+\Delta C'\phi '^{6}+...\right]}$

${\displaystyle m'^{2}=m^{2}b^{-2},\lambda '=\lambda b^{d-4},B'=Bb^{d},C'=Cb^{2d-6}}$

三种耦合

• 无关耦合（irrelevant）：耦合减少了
• 相关耦合（relevant）：耦合增加了
• 边缘耦合（marginal）：耦合不变

${\displaystyle d=4}$ ，因为${\displaystyle b<1}$ 所以B和C是无关的，m是相关的，并且${\displaystyle \lambda }$ 是边缘的。

扩展阅读

相关著作

• T. D. Lee 李政道; Particle physics and introduction to field theory, Harwood academic publishers, 1981, [ISBN 3-7186-0033-1]. 是总结
• L. Ts. Adzhemyan, N.V.Antonov and A. N. Vasiliev; The Field Theoretic Renormalization Group in Fully Developed Turbulence; Gordon and Breach, 1999. [ISBN 90-5699-145-0].
• Vasil'ev, A. N.; The field theoretic renormalization group in critical behavior theory and stochastic dynamics; Chapman & Hall/CRC, 2004. [ISBN 9780415310024] (Self-contained treatment of renormalization group applications with complete computations);
• Zinn-Justin, Jean; Quantum field theory and critical phenomena, Oxford, Clarendon Press (2002), ISBN 0-19-850923-5 (a very thorough presentation of both topics);
• The same author: Renormalization and renormalization group: From the discovery of UV divergences to the concept of effective field theories, in: de Witt-Morette C., Zuber J.-B. (eds), Proceedings of the NATO ASI on Quantum Field Theory: Perspective and Prospective, June 15–26, 1998, Les Houches, France, Kluwer Academic Publishers, NATO ASI Series C 530, 375-388 (1999) [ISBN ]. Full text available in PostScript页面存档备份，存于互联网档案馆）.
• Kleinert, H. and Schulte Frohlinde, V; Critical Properties of φ4-Theories, World Scientific (Singapore, 2001); Paperback ISBN 981-02-4658-7. Full text available in PDF页面存档备份，存于互联网档案馆）.

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