  How to include spacetime translations in fibre bundle gauge theories has been
a subject of controversy, because spacetime symmetries are not internal
symmetries of the bundle structure group. The standard method for including
affine symmetry in differential geometry is to define a Cartan connection on an
affine bundle over spacetime. This is equivalent to (1) defining an affine
connection on the affine bundle, (2) defining a zero section on the associated
affine vector bundle, and (3) using the affine connection and the zero section
to define an "associated solder form," whose lift to a tensorial form on the
frame bundle becomes the solder form. The zero section reduces the affine
bundle to a linear bundle and splits the affine connection into translational
and homogeneous parts; however it violates translational equivariance / gauge
symmetry. This is the natural geometric framework for Einstein-Cartan theory as
an affine theory of gravitation. The last section discusses some alternative
approaches that claim to preserve translational gauge symmetry.
