Gauge symmetry respectively gauge invariance plays a crucial role in
modern quantum field theory. Historically, the first locally gauge-invariant equations
of motion are Maxwell’s original equations for the electrical charges and currents and
the electromagnetic fields created by them. In order to keep the relation to classical
mechanics as close as possible, this chapter starts with a critical examination of global
gauge invariance, basing on Helmholtz’s interpretation of the potential energy as
“disposable work storage”. Then, Helmholtz’s explorations of the relationships
between forces and energies are sketched and extended. Their generalization to
quantum mechanics leads to the gauge invariance of the stationary and timedependent
Schrödinger equations. An important application is the Ehrenberg-Siday-
Aharonov-Bohm effect and its gravito-electromagnetic analog. For the sake of the
unity of physics, it is shown, how the (gravito-)electromagnetic equations can be
deduced. Here, Newton’s imaginations about the field of gravity are also exploited.
Keywords: Ehrenberg-Siday-Aharonov-Bohm effect, Energy, Field, Force, Gauge
invariance, Gauge symmetry, Gravito-electromagnetism, Helmholtz, Maxwell’s
equations, Newton, Potential Energy, Schrödinger equation, Unity of physics,
Work storage.