Infinite Multiplets and the Hydrogen Atom

Section I of this report deals with attempts to construct infinite-component wave equations with mass spectra that are free of unphysical features. An example is given of a second-order relativistic wave equation whose mass spectrum, both the discrete and the continuous part, is precisely the same as that of the nonrelativistic hydrogen atom. Section II deals with the nonrelativistic hydrogen atom, without any approximations. It is shown that the Schrödinger equation is equivalent to a nonrelativistic analog of Majorana-Nambu wave equations. The Hilbert space of all the states, including the continuum, is profitably and economically used as a representation space for a unitary, irreducible representation of the group $\mathrm{SO}(4,2)\mathrm{}$. The dipole operator is an $\mathrm{SO}(4,2)\mathrm{}$ generator. The theory is generalized to arbitrary frames of reference by application of Galilei transformations. The generators of Galilei transformations also belong to the $\mathrm{SO}(4,2)\mathrm{}$ algebra. What results is a field theory of the hydrogen atom. The exact electromagnetic interaction, including all multipoles, takes the form of a local interaction between the electromagnetic field and the infinite-component hydrogen field. In Sec. III the significance of all this for hadron physics is discussed.