在量子場論中,維度正規化是一種正規化辦法。Giambiagi、Bollini、[1][2] 傑拉德·特·胡夫特和馬丁紐斯·韋爾特曼[3]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。
![{\displaystyle \int {\frac {d^{d}p}{(2\pi )^{d}}}{\frac {1}{\left(p^{2}+m^{2}\right)^{2}}},}](//wikimedia.org/api/rest_v1/media/math/render/svg/86d6194ac167c5a41ee513f9e49b3741128d0976)
若d = 4 − ε,上文的積分是
![{\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 ).}](//wikimedia.org/api/rest_v1/media/math/render/svg/6d8da5285068b47a6db4f9e001444333d127548e)
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。[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
- 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