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| {{Renormalization and regularization}}
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| In [[theoretical physics]], '''dimensional regularization''' is a method introduced by [[Juan José Giambiagi|Giambiagi]] and Bollini <ref>Bollini 1972, p. 20.</ref> for [[regularization (physics)|regularizing]] [[integral]]s in the evaluation of [[Feynman diagram]]s; in other words, assigning values to them that are [[meromorphic function]]s of an auxiliary complex parameter ''d'', called (somewhat confusingly) the dimension.
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| Dimensional regularization writes a [[Feynman integral]] as an integral depending on the spacetime dimension ''d'' and the squared distances (''x''<sub>''i''</sub>−''x''<sub>''j''</sub>)<sup>2</sup> of the spacetime points ''x''<sub>''i''</sub>, ... appearing in it. In [[Euclidean space]], the integral often converges for −Re(''d'') sufficiently large, and can be [[analytically continued]] from this region to a meromorphic function defined for all complex ''d''. In general, there will be a pole at the physical value (usually 4) of ''d'', which needs to be canceled by [[renormalization]] to obtain physical quantities.
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| {{harvtxt|Etingof|1999}} showed that dimensional regularization is mathematically well defined, at least in the case of massive Euclidean fields, by using the [[Bernstein–Sato polynomial]] to carry out the analytic continuation.
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| There is a tradition of confusing the parameter ''d'' appearing in dimensional regularization, which is a complex number, with the dimension of spacetime, which is a fixed positive integer (such as 4). The reason is that if ''d'' happens to be a positive integer, then the formula for the dimensionally regularized integral happens to be correct for spacetime of dimension ''d''.
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| For example, the [[Sphere#Generalization_to_other_dimensions|surface area]] of a unit (''d'' − 1)-sphere is <math>\frac{2\pi^{d/2}}{\Gamma\left(\frac{d}{2}\right)}</math> where Γ is the [[gamma function]] when ''d'' is a positive integer, so in dimensional regularization it is common to say that this is the surface area of a sphere in ''d'' dimensions even when ''d'' is not an integer. This is a useful mnemonic for remembering the formulas in dimensional regularization, but is otherwise meaningless{{Citation needed|date=November 2012}} as there is no such thing as a sphere in non-integral dimensions. This failure to distinguish between the dimension of spacetime and the formal parameter ''d'' has led to a lot of meaningless speculation about (non-existent) spacetimes of non-integral dimension.{{Citation needed|date=November 2012}}
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| If one wishes to evaluate a loop integral which is logarithmically divergent in four dimensions, like
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| :<math>\int\frac{d^dp}{(2\pi)^d}\frac{1}{\left(p^2+m^2\right)^2},</math>
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| one first rewrites the integral in some way so that the number of variables integrated over does not depend on ''d'', and then we formally vary the parameter ''d'', to include non-integral values like ''d'' = 4 − ''ε''.
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| This gives
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| :<math>\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(\frac{\pi\varepsilon}{2}) \Gamma(1-\frac{\varepsilon}{2})}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).</math>
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| [[Emilio Elizalde]] has shown that [[Zeta function regularization|Zeta regularization]] and dimensional regularization are equivalent since they use the same principle of using analytic continuation in order for a series or integral to converge.
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| ==See also==
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| * [[Regularization (physics)]]
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| * [[Pauli–Villars regularization]]
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| ==References==
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| {{Reflist}}
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| *{{Citation |last1=Bollini | first1=Carlos | last2=Giambiagi | first2=Juan Jose | title=Dimensional Renormalization: The Number of Dimensions
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| as a Regularizing Parameter. | publisher= Il Nuovo Cimento B | volume=12| pages=20-26 | year=1972}}
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| *{{Citation | last1=Etingof | first1=Pavel | title=Quantum fields and strings: a course for mathematicians, Vol. 1,(Princeton, NJ, 1996/1997) | url=http://www.math.ias.edu/QFT/fall/ | publisher=Amer. Math. Soc. | location=Providence, R.I. | isbn=978-0-8218-2012-4 | mr=1701608 | year=1999 | chapter=Note on dimensional regularization | pages=597–607}}
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| *{{Citation | last1=Hooft | first1=G. 't | last2=Veltman | first2=M. | title=Regularization and renormalization of gauge fields | url=http://www.sciencedirect.com/science/article/B6TVC-4719KYC-173/2/d6e222ccf600624017080c27f2774d62 | doi= 10.1016/0550-3213(72)90279-9 | year=1972 | journal=Nuclear Physics B | issn=0550-3213 | volume=44 | issue=1 | pages=189–213 |bibcode = 1972NuPhB..44..189T }}
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| *{{Citation | last1=[[Hagen Kleinert|Kleinert]] | first1=H. | last2=Schulte-Frohlinde | first2=V. | title=Critical Properties of φ<sup>4</sup>-Theories | url=http://www.worldscibooks.com/physics/4733.html |ISBN= 978-981-02-4659-4 | year=2001 | pages=1–474}}, Paperpack ISBN 978-981-02-4659-4 (also available [http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist&details=6 online]). Read Chapter 8 for Dimensional Regularization.
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| {{DEFAULTSORT:Dimensional Regularization}}
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| [[Category:Quantum field theory]]
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| [[Category:Summability methods]]
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