Chapter 18 Conformal Invariance

At the beginning of the semester we motivated our investigation of symmetries by illustrating that, given differential equations which were symmetric, the solutions had to transform into each other under the symmetries as a representation of the symmetry. The first illustrations considered Schrödinger equations with symmetric potentials, such as the electrons in the spherically symmetric potential of an atom, having wavefunctions transforming under the rotation group. The Laplacian is invariant under both rotations and translations, but the source of the potential may be taken as invariant rotationally but not translationally. Under translations, we might make a connection of the wave function at ~x for an atom with a nucleus at ~y with the wave function at ~x+~a for an atom with a nucleus at ~y+~a, as the physics is translationally invariant if we translate both the point of evaluation and the boundary conditions, or sources. Thus the way symmetries act on solutions depend on both the differential operator having the symmetry and the boundary conditions or sources having the symmetry. The electric field of a point charge at the center of a conducting cube would not have a full rotational symmetry, but would behave as a representation of the symmetry group of the cube. It is obvious