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

TL;DR

This paper provides a new proof demonstrating the existence of a bound state in the quantum Coulomb field, specifically for a massless scalar field on a hyperboloid, using differential equations and exact solutions.

Contribution

The paper introduces a completely different proof for the existence of a bound state in the quantum Coulomb field, previously proven in 1992, by deriving and solving a differential equation for matrix elements.

Findings

Proves the existence of a bound state for e^2 < π.

Derives a differential equation for matrix elements involving the Casimir operator.

Identifies an exact solution for the differential equation, confirming the bound state and its probability.

Abstract

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 $e^2$ is the coupling constant, $dx$ is the invariant measure on the de Sitter hyperboloid $xx=-1$ and $g_{ik}, i,k=1,2,3$, is the internal metric on this hyperboloid. Let $u$ be a fixed four-velocity. The field $S(u)=(1/4 \pi)\int dx\delta(ux)S(x)$is smooth enough to be exponentiated. We prove that if $0<e^2<\pi$, then the state $|u>=\exp(-iS(u))\mid 0>$, where $\mid 0>$ is the Lorentz invariant vacuum state, contains a normalizable eigenstate of the Casimir operator $C_1=-(1/2)M_{\mu\nu}M^{\mu\nu}$; $M_{\mu\nu}$ 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 $\exp (z)\cdot (1-z)\cdot \exp[-\sigma z (2-z)], z= e^2/ \pi$, is an exact solution of this differential equation, recovering thus both the eigenvalue and the probability of occurrence of the bound state.