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

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dc.creatorStaruszkiewicz, Andrzej
dc.date2004-06-30
dc.date.accessioned2026-07-07T04:17:07Z
dc.date.available2026-07-07T04:17:07Z
dc.descriptionLet 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 $(\hbar=1=c)(8πe^2)^{-1}\int dx g^{ik}\partial_i S\partial_k S$, 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 $S(u)=(1/4 π)\int dxδ(ux)S(x)$is smooth enough to be exponentiated. We prove that if $0<e^2<π$, 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 $C_1=-(1/2)M_{μν}M^{μν}$; $M_{μν}$ 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 $<u\mid \exp (-σC_1)\mid u>, σ> 0$, and show that the function $\exp (z)\cdot (1-z)\cdot \exp[-σz (2-z)], z= e^2/ π$, is an exact solution of this differential equation, recovering thus both the eigenvalue and the probability of occurrence of the bound state.
dc.description13 pages
dc.identifierhttps://arxiv.org/abs/hep-th/0406270
dc.identifierhttp://arxiv.org/abs/hep-th/0406270
dc.identifierActa Phys.Polon. B35 (2004) 2249-2260
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/52626
dc.subjectHigh Energy Physics - Theory
dc.titleA New Proof of Existence of a Bound State in the Quantum Coulomb Field
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