The vector potential allows one to define a new kind of derivative, such
that the derivative of the fieldψhas the same homogeneous transformation
properties underGasψitself:
Definition(Covariant derivative). Given a connectionA, the associated co-
variant derivative in theμdirection is the operator
DAμ=∂
∂xμ−ieAμ(x)With this definition, the effect of a gauge transformation isDAμψ→(
DAμ−ie∂φ
∂xμ)
eieφ(x)ψ=eieφ(x)(
DAμ+ie∂
∂xμ−ie∂
∂xμ)
ψ=eieφ(x)DAμψIf one replaces derivatives by covariant derivatives, terms in a Hamiltonian such
as
|∇ψ|^2 =∑^3
j=1∂ψ
∂xj∂ψ
∂xjwill become
∑^3
j=1(DjAψ)(DAjψ)which will be invariant under the infinite dimensional groupG. The procedure
of starting with a theory of complex fields, then introducing a connection while
changing derivatives to covariant derivatives in the equations of motion is called
the “minimal coupling prescription.” It determines how a theory of complex free
fields describing charged particles can be turned into a theory of fields coupled
to a background electromagnetic field, in the simplest or “minimal” way.
45.2 Curvature, electric and magnetic fields
While the connection or vector potentialAis the fundamental geometrical quan-
tity needed to construct theories with gauge symmetry, one often wants to work
instead with certain quantities derived fromAthat are invariant under gauge
transformations. To a mathematician this is the curvature of a connection, to
a physicist these are the electric and magnetic field strengths derived from a
vector potential. The definition is:
Definition(Curvature of a connection, electromagnetic field strengths). The
curvature of a connectionAμis given by
Fμν=i
e[DAμ,DνA]which can more explicitly be written
Fμν=∂Aν
∂xμ−
∂Aμ
∂xν