Lesson plan
Dimensional regularization
__NOTOC__ In theoretical physics, dimensional regularization is a method introduced by Juan José Giambiagi and Carlos Guido Bollini as well as – independently and more comprehensively – by Gerard 't Hooft and Martinus J. G. Veltman for regularizing integrals in the evaluation of Feynman diagrams; in other words, assigning values to them that are meromorphic functions of a complex parameter d, the analytic continuation of the number of spacetime dimensions. Dimensional regularization writes a Feynman integral 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, 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. Pavel Etingof 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.…
Article summary
__NOTOC__ In theoretical physics, dimensional regularization is a method introduced by Juan José Giambiagi and Carlos Guido Bollini as well as – independently and more comprehensively – by Gerard 't Hooft and Martinus J. G. Veltman for regularizing integrals in the evaluation of Feynman diagrams; in other words, assigning values to them that are meromorphic functions of a complex parameter d, the analytic continuation of the number of spacetime dimensions. Dimensional regularization writes a Feynman integral 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, 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. Pavel Etingof 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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