Dimensional regularization
Encyclopedia
In theoretical physics
, dimensional regularization is a method introduced by Giambiagi and Bollini for 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.
Dimensional regularization writes a Feynman integral as an integral depending on the spacetime dimension d and the squared distances (xi−xj)2 of the spacetime points xi, ... 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.
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.
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.
For example, the volume
of a unit (d − 1)-sphere is
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 volume 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 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.
If one wishes to evaluate a loop integral which is logarithmically divergent in four dimensions, like
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.
This gives
Emilio Elizalde has shown that both 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.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, dimensional regularization is a method introduced by Giambiagi and Bollini for regularizing
Regularization (physics)
-Introduction:In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator...
integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s in the evaluation of Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s; in other words, assigning values to them that are meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
s of an auxiliary complex parameter d, called (somewhat confusingly) the dimension.
Dimensional regularization writes a Feynman integral as an integral depending on the spacetime dimension d and the squared distances (xi−xj)2 of the spacetime points xi, ... appearing in it. In Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, 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
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
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
Bernstein–Sato polynomial
In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by and , . It is also known as the b-function, the b-polynomial, and the Bernstein polynomial, though it is not related to the Bernstein polynomials used in approximation...
to carry out the analytic continuation.
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.
For example, the volume
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
of a unit (d − 1)-sphere is
where Γ is the gamma functionGamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
when d is a positive integer, so in dimensional regularization it is common to say that this is the volume 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 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.
If one wishes to evaluate a loop integral which is logarithmically divergent in four dimensions, like
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.
This gives
Emilio Elizalde has shown that both Zeta regularization
Zeta function regularization
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators...
and dimensional regularization are equivalent since they use the same principle of using analytic continuation in order for a series or integral to converge.