dimensional regularization
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__NOTOC__ In
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, dimensional regularization is a method introduced by Giambiagi and Bollini as well as – independently and more comprehensively – by 't Hooft and Veltman for regularizing
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
s in the evaluation of
Feynman diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s; in other words, assigning values to them that are
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
s of a complex parameter ''d'', the analytic continuation of the number of spacetime dimensions. Dimensional regularization writes a
Feynman integral The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional i ...
as an integral depending on the spacetime dimension ''d'' and the squared distances (''x''''i''−''x''''j'')2 of the spacetime points ''x''''i'', ... appearing in it. In
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, the integral often converges for −Re(''d'') sufficiently large, and can be
analytically continued In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ne ...
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 Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
to obtain physical quantities. 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. Although the method is most well understood when poles are subtracted and ''d'' is once again replaced by 4, it has also led to some successes when ''d'' is taken to approach another integer value where the theory appears to be strongly coupled as in the case of the Wilson–Fisher fixed point. A further leap is to take the interpolation through fractional dimensions seriously. This has led some authors to suggest that dimensional regularization can be used to study the physics of crystals that macroscopically appear to be
fractals In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
. It has been argued that
Zeta regularization Zeta (, ; uppercase Ζ, lowercase ζ; grc, ζῆτα, el, ζήτα, label=Demotic Greek, classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived fr ...
and dimensional regularization are equivalent since they use the same principle of using analytic continuation in order for a series or integral to converge.


Example

Suppose one wishes to dimensionally regularize a loop integral which is logarithmically divergent in four dimensions, like :I = \int\frac\frac. First, write the integral in a general non-integer number of dimensions d = 4 - \varepsilon, where \varepsilon will later be taken to be small,I = \int\frac\frac.If the integrand only depends on p^2, we can apply the formula\int d^dp \, f(p^2) = \frac \int_0^\infty dp \, p^ f(p^2).For integer dimensions like d = 3, this formula reduces to familiar integrals over thin shells like \int_0^\infty dp \, 4 \pi p^2 f(p^2). For non-integer dimensions, we ''define'' the value of the integral in this way by analytic continuation. This givesI = \int_0^\infty \frac \frac \frac = \fracm^ = \frac-\frac\left(\ln \frac+\gamma\right)+ \mathcal(\varepsilon).Note that the integral again diverges as \varepsilon \rightarrow 0, but is finite for arbitrary small values \varepsilon \neq 0.


Notes


References

* * * {{DEFAULTSORT:Dimensional Regularization Quantum field theory Summability methods