Oliver Heaviside (1850 - 1925) was a self-taught physicist and mathematician who introduced complex numbers into circuit theory, invented the Laplace-transform approach to solving differential equations, first wrote Maxwell's equations in the form we now recognize, pioneered vector calculus... and incidentally discovered gauge invariance! [It took a long time to emerge in its modern form. For a thorough history, check here.]



To explore broken symmetries, we will eventually use the good old Klein-Gordon equation which has the advantage of being both relativistic and simple.




Yang-Mills Fields: In 1954, Yang and Mills showed how to construct a gauge-invariant theory of massless field bosons with an internal quantum number comparable to charge. Imagine photons existed as three types, γ+, γ0 and γ- for example! Yang-Mills fields have been the basis of the Standard Model, needed to describe both the strong and weak interaction fields, which are “non-Abelian.”





Aharonov-Bohm Effect


Yakir Aharonov (1932 - )

A topological (Berry) phase.