Narrow Results

The historical and geometrical origin of Gauge Transformation and Yang's phase loop of gauge theory are discussed. In the present talk, we present the following points: 1. Parallel transport of a vector; 2. Weyl 1918 gauge transformation 3. Concept of non-integrable phase factor; 4. Berry's quantum geometrical phase; 5. Parallel transport of quantum state vector produces the phase physics.

The relationship between local Weyl scaling invariant models and local dilatation invariant actions is critically scrutinized. While actions invariant under local Weyl scalings can be constructed in a straightforward manner, actions invariant under local dilatation transformations can only be achieved in a very restrictive case. The invariant couplings of matter fields to an Abelian vector field carrying a nontrivial scaling weight can be easily built, but an invariant Abelian vector kinetic term can only be realized when the local scale symmetry is spontaneously broken.