Symmetries of Field Configurations and No-Go Theorems

We recap the theory of diffeomorphisms and covariances acting on a space and their induced action on fields, and how this leads to their decomposition into Fourier modes and harmonics. We show that a careful consideration of this theory reveals some unexpected results. For a start, harmonics are distinguished by their ‘quantum numbers’, but this analysis is valid for classical fields, so that classical fields can be written as a linear sum of modes with ‘quantum numbers’. Furthermore, certain models of compactification, as described in previous works by the author, are found to exploit a loophole in a key ‘no-go’ theorem, allowing the Poincaré group and an internal O(N ) symmetry group to be embedded in a larger symmetry group. This embedding is achieved without the need for the fermionic generators of supersymmetry and it is manifest in the ‘decompactification limit’. Away from this limit – for a realistic universe – the homogeneous part of this larger symmetry group is non-linearly realised. The translation part of the original symmetry group is realised as additional rotational transformations on the compact factor space.