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What is the Mechanism giving mass to fundamental particles?

The discovery of the Higgs boson is an enormous clue about the mechanism for giving mass to fundamental particles, as conceived by Higgs, Brout, Englert, Guralnik, Hagen, and Kibble. What is this mechanism? It is a complicated mathematical theory, but its main idea can be understood with a simple analogy.

Imagine that a cocktail party full of physicists chattering quietly is like space filled with the Higgs field...