A New Proof of Existence of a Bound State in the Quantum Coulomb Field

Let S(x) be a massless scalar quantum field which lives on the three-dimensional hyperboloid $xx=(x^0{)}^2-(x^1{)}^2-(x^2{)}^2-(x^3{)}^2=-1.$ The classical action is assumed to be $(\hbar=1=c)(8\pi e^2)^{-1}\int dx g^{ik}\partial_i S\partial_k S$,where$e^2$isthecouplingconstant,$dx$ is the invariant measure on the de Sitter hyperboloid $xx=-1$ and $g_{ik}, i,k=1,2,3$,istheinternalmetriconthishyperboloid\mathrm{.}Let$u$ be a fixed four-velocity. The field $S(u)=(1\mathrm{/}4\pi )\int dx\delta (ux)S(x)$is smooth enough to be exponentiated. We prove that if $0<e^2<\pi$, then the state $\mathrm{\mid }u>=\exp (-iS(u))\mid 0>$, where $\mid 0>$ is the Lorentz invariant vacuum state, contains a normalizable eigenstate of the Casimir operator $C_1=-(1\mathrm{/}2)M_{\mu \nu }{M}^{\mu \nu }$; $M_{\mu \nu }$ are generators of the proper orthochronous Lorentz group. This theorem was first proven by the Author in 1992 in his contribution to the Czyz Festschrift, see Erratum {\it Acta Phys. Pol. B} {\bf 23}, 959 (1992). In this paper a completely different proof is given: we derive the partial, differential equation satisfied by the matrix element $<u\mid \exp (-\sigma C_1)\mid u>,\sigma >0$, and show that the function $\exp (z)\cdot (1-z)\cdot \exp [-\sigma z(2-z)],z=e^2\mathrm{/}\pi$, is an exact solution of this differential equation, recovering thus both the eigenvalue and the probability of occurrence of the bound state.