  The quantum system of a massless charged scalar field with a self-interaction
is investigated. By introducing a spacial cut-off function, the Hamiltonian of
the system is realized as a linear operator on a boson Fock space. It is proven
that the Hamiltonian strongly commutes with the total charge operator. This
fact implies that the state space of the charged scalar field is decomposed
into the infinite direct sum of fixed total charge spaces. Moreover, under
certain conditions, the Hamiltonian is bounded below, self-adjoint and has a
ground ground state for an arbitrarily coupling constant. A relation between
the total charge of the ground state and a number operator bound is also
revealed.
