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.]

Write Maxwell's equations in terms of A and the scalar potential φ. Choose the Lorentz gauge, and the result is two covariant wave equations for the two potentials. Adding a mass term to these equations destroys gauge invariance.

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.”

Yakir Aharonov (1932 - )

A topological (Berry) phase.

David Bohm (1917 - 1992)

Symmetry breaking by perturbation.  To every symmetry corresponds a degeneracy. For example, for a spherically symmetric physical system, there is no dependence on quantum number m_ℓ... there is a (2ℓ + 1) degeneracy in the energy levels observed. The classic example is Zeeman splitting. Putting the system in a magnetic field introduces a term into the Hamiltonian that defines a specific axis in space, and the spherical symmetry is broken, resulting in the breaking of the degeneracy.

Spontaneous symmetry breaking. The system has a hidden rotational symmetry about the vertical axis, but when the particle winds up in its ground state of gravitational potential energy, and we have only this one system to study, we would not be aware of the underlying symmetry because it is automatically broken when the particle minimizes its potential energy. Perturbation of the particle would only result in oscillations about its equilibrium position and would not show any hint of the underlying symmetry.

The basic idea of the Higgs mechanism is that massless particles can couple to the Higgs field, which through spontaneous symmetry breaking has a non-zero vacuum expectation value. And this coupling is formally gauge invariant! Thus, the coupling term appears in the dynamical equations in the same spot a mass term would occupy, but unlike the mass term, which breaks gauge invariance, the new term is fully gauge invariant and the theory remains renormalizable!

Peter Higgs (1929 - )