Mathematics – Differential Geometry
Scientific paper
Feb 2006
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006cqgra..23..737p&link_type=abstract
Classical and Quantum Gravity, Volume 23, Issue 3, pp. 737-751 (2006).
Mathematics
Differential Geometry
1
Scientific paper
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.
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