Diffeomorphism Group Representations in Nonrelativistic and Relativistic Quantum Theory: Some Future Directions

Diffeomorphism groups and their unitary representations unify the description of a wide variety of diverse nonrelativistic (Galilean) quantum systems. In recent work, we have suggested an approach to relativistic quantum field theory that begins with hierarchies of diffeomorphism group representations and the corresponding current densities. We introduce (noncovariant) creation and annihilation fields that intertwine the representations, and restore the (relativistic) spacetime symmetry by redefining local quantum fields and the Hamiltonian operator in terms of the intertwining fields and current densities. Here we outline the main ideas, and point to next steps and possible future directions of this line of inquiry.