Eigenvalues and eigenvectors of the interaction term in local quantum field theory

A basic feature of nonrelativistic quantum mechanics is the existence of one representation (the momentum representation) which diagonalizes the free part of the Hamiltonian, a second representation (the position representation) which diagonalizes the interaction part of the Hamiltonian, and a unitary transformation (Fourier transform) which connects these two representations. In local Lagrangian field theory the free-particle representation which diagonalizes the free part of the Hamiltonian is well known, but the representation which diagonalizes the interaction part of the Hamiltonian has not been systematically studied. In what follows, this representation is explicitly constructed and it is shown that there is no unitary transformation connecting it with the free-particle representation. In fact, this representation space is not even a Hilbert space, in the sense that to define a meaningful norm seems impossible.