高斯自由场

在量子场论中，高斯自由场（Gaussian Free Field）是最简单的场论之一。这个也称为无质量玻色子场论。

介绍

高斯自由场的泛函积分是

${\displaystyle Z=\int D\phi \exp(-\int _{R^{d}}d^{d}x\ {\frac {1}{2}}\phi ^{2})}$ 

${\displaystyle \phi }$ 是高斯自由场。${\displaystyle \phi (x)=f(x)}$ 的概率是

${\displaystyle P(\phi =f)={\frac {1}{Z}}\exp(-\int {\frac {1}{2}}f^{2})}$ 

属性

• 高斯场论是一个共形场论
• 高斯场论描述一个无质量的玻色子
• ${\displaystyle m=0}$ 的Klein-Gordon场论
• ${\displaystyle d=1}$ 的高斯场论描述维纳过程

应用

• 随机矩阵
• 随机漫步（格子）
• 双体模型
• Schramm–Loewner演进

参考文献

• Ding, J.; Lee, J. R.; Peres, Y. (2012), "Cover times, blanket times, and majorizing measures", Annals of Mathematics, 175: 1409–1471, arXiv:1004.4371, doi:10.4007/annals.2012.175.3.8
• Dubédat, J. (2009), "SLE and the free field: Partition functions and couplings", J. Amer. Math. Soc., 22: 995–1054, arXiv:0712.3018, Bibcode:2009JAMS...22..995D, doi:10.1090/s0894-0347-09-00636-5
• Kenyon, R. (2001), "Dominos and the Gaussian free field", Annals of Probability, 29 (3): 1128–1137, arXiv:math-ph/0002027, doi:10.1214/aop/1015345599, MR 1872739
• Peres, Y. (2001), "An Invitation to Sample Paths of Brownian Motion" (PDF), Lecture notes at UC Berkeley
• Rider, B.; Virág, B. (2007), "The noise in the Circular Law and the Gaussian Free Field", International Mathematics Research Notices: article ID rnm006, 32 pages, MR 2361453
• Sheffield, S. (2005), "Local sets of the Gaussian Free Field", Talks at the Fields Institute, Toronto, on September 22–24, 2005, as part of the "Percolation, SLE, and related topics" Workshop.
• Sheffield, S. (2007), "Gaussian free fields for mathematicians", Probability Theory and Related Fields, 139: 521–541, arXiv:math.PR/0312099, doi:10.1007/s00440-006-0050-1, MR 2322706
• Friedli, S.; Velenik, Y. (2017). Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction. Cambridge: Cambridge University Press. ISBN 9781107184824.