維度正規化

在量子場論中，維度正規化是一種正規化辦法。Giambiagi、Bollini、 傑拉德·特·胡夫特和馬丁紐斯·韋爾特曼都提出了這個辦法。物理學家使用維度正規化來計算費曼圖的積分。積分的值是d的亞純函數；d是時空的維度。

其他應用

Wilson–Fisher固定點、分形維數

舉例

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

若d = 4 − ε，上文的積分是

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

有人認為Ζ函數正規化和維度正規化是等效的等同因為解析延拓。

閱讀

• 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 (不活躍 2020-01-22), （原始內容存檔於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, （原始內容存檔於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