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Phenomena was released theatrically in Italy on 31 January 1985 with a 116-minute running time. [24] This version of Phenomena is often referred to as the "integral cut". [24] A shorter version of the film was prepared for international release that had a 110-minute running time. [24]
One-loop diagrams are usually computed as the integral over one independent momentum that can "run in the cycle". The Casimir effect , Hawking radiation and Lamb shift are examples of phenomena whose existence can be implied using one-loop Feynman diagrams, especially the well-known "triangle diagram":
QED was designed to be a popular science book, written in a witty style, and containing just enough quantum-mechanical mathematics to allow the solving of very basic problems in quantum electrodynamics by an educated lay audience. It is unusual for a popular science book in the level of mathematical detail it goes into, actually allowing the ...
The remaining two half-lines in the two X s can be linked to each other in two ways, so that the total number of ways to form the diagram is 4 × 3 × 4 × 3 × 2 × 2, while the denominator is 4! × 4! × 2!. The total symmetry factor is 2, and the contribution of this diagram is divided by 2.
If A(p) and B(p) are linear functions of p, then the last integral can be evaluated using substitution. More generally, using the Dirac delta function δ {\displaystyle \delta } : [ 2 ]
For example in QED, the tensor indices of the integral may be contracted with Gamma matrices, and identities involving these are needed to evaluate the integral. In QCD, there may be additional Lie algebra factors, such as the quadratic Casimir of the adjoint representation as well as of any representations that matter (scalar or spinor fields ...
In theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini [1] as well as – independently and more comprehensively [2] – by 't Hooft and Veltman [3] 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 ...
Fikhtengol'ts's books on analysis are widely used in Middle and Eastern European, as well as Chinese universities, due to their exceptionally detailed and well-organized presentation of material on mathematical analysis. For unknown reasons, these books have not gained the same level of fame in universities in other parts of the world.
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