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Spectral analysis of a massless charged scalar field with cutoffs. (English) Zbl 1487.47041

Summary: A quantum system of a massless charged scalar field with a self-interaction is investigated. By introducing a spacial cut-off function, a Hamiltonian of the quantum system is realized as a linear operator on a boson Fock space. Under certain conditions, it is proven that the Hamiltonian is bounded below, self-adjoint and has a ground state for an arbitrary coupling constant. Moreover 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. A total charge of an eigenstate is discussed.

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis