维度正规化

在量子场论中，维度正规化是一种正规化办法。Giambiagi、Bollini、^[1][2] 杰拉德·特·胡夫特和马丁纽斯·韦尔特曼^[3]都提出了这个办法。物理学家使用维度正规化来计算费曼图的积分。积分的值是d的亚纯函数；d是时空的维度。

其他应用

Wilson–Fisher固定点（英语：Wilson–Fisher fixed point）、分形维数^[4]

举例

${\displaystyle \int {\frac {d^{d}p}{(2\pi )^{d}}}{\frac {1}{\left(p^{2}+m^{2}\right)^{2}}},}$

若d = 4 − ε，上文的积分是

${\displaystyle \int _{0}^{\infty }{\frac {dp}{(2\pi )^{4-\varepsilon }}}{\frac {2\pi ^{(4-\varepsilon )/2}}{\Gamma \left({\frac {4-\varepsilon }{2}}\right)}}{\frac {p^{3-\varepsilon }}{\left(p^{2}+m^{2}\right)^{2}}}={\frac {2^{\varepsilon -4}\pi ^{{\frac {\varepsilon }{2}}-1}}{\sin \left({\frac {\pi \varepsilon }{2}}\right)\Gamma \left(1-{\frac {\varepsilon }{2}}\right)}}m^{-\varepsilon }={\frac {1}{8\pi ^{2}\varepsilon }}-{\frac {1}{16\pi ^{2}}}\left(\ln {\frac {m^{2}}{4\pi }}+\gamma \right)+{\mathcal {O}}(\varepsilon ).}$

有人认为Ζ函数正规化和维度正规化是等效的等同因为解析延拓。^[5]

参考文献

1. Bollini 1972, p. 20.
2. Bietenholz, Wolfgang; Prado, Lilian. Revolutionary physics in reactionary Argentina. Physics Today. 2014-02-01, 67 (2): 38–43. Bibcode:2014PhT....67b..38B. ISSN 0031-9228. doi:10.1063/PT.3.2277.
3. Hooft, G. 't; Veltman, M., Regularization and renormalization of gauge fields, Nuclear Physics B, 1972, 44 (1): 189–213, Bibcode:1972NuPhB..44..189T, ISSN 0550-3213, doi:10.1016/0550-3213(72)90279-9
4. Le Guillo, J.C.; Zinn-Justin, J. Accurate critical exponents for Ising-like systems in non-integer dimensions. Journal de Physique. 1987, 48 [2020-03-07]. （原始内容存档于2020-07-10）.
5. A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti and S. Zerbini, Analytic Aspects of Quantum Field, World Scientific Publishing, 2003, ISBN 981-238-364-6

阅读

• Bollini, Carlos; Giambiagi, Juan Jose, Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter., Il Nuovo Cimento B (1971-1996) (Il Nuovo Cimento B), 1972, 12 (1): 20–26 [2020-03-07], doi:10.1007/BF02895558 (不活跃 2020-01-22), （原始内容存档于2017-10-16）
• Etingof, Pavel, Note on dimensional regularization, Quantum fields and strings: a course for mathematicians, Vol. 1,(Princeton, NJ, 1996/1997), Providence, R.I.: Amer. Math. Soc.: 597–607, 1999 [2020-03-07], ISBN 978-0-8218-2012-4, MR 1701608, （原始内容存档于2019-05-29）
• Hooft, G. 't; Veltman, M., Regularization and renormalization of gauge fields, Nuclear Physics B, 1972, 44 (1): 189–213, Bibcode:1972NuPhB..44..189T, ISSN 0550-3213, doi:10.1016/0550-3213(72)90279-9