[Paper Review] A New Proof of Existence of a Bound State in the Quantum Coulomb Field

Andrzej Staruszkiewicz|ArXiv.org|Jun 30, 2004

Advanced Physical and Chemical Molecular Interactions6 citations

TL;DR

This paper presents a novel differential equation-based proof for the existence of a bound state in a quantum Coulomb field on a three-dimensional de Sitter hyperboloid. By analyzing the matrix element ⟨u|exp(−σC₁)|u⟩, the authors derive an exact solution that confirms a normalizable eigenstate of the Casimir operator C₁ with eigenvalue (e²/π)(2 − e²/π) when 0 < e² < π, establishing the bound state's existence and probability through a new analytical approach.

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 $(\hbar=1=c)(8πe^2)^{-1}\int dx g^{ik}\partial_i S\partial_k S$, 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 $0=\exp(-iS(u))\mid 0&gt;$, where $\mid 0&gt;$ 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 $, σ&gt; 0$, 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.

Motivation & Objective

• To provide a new, rigorous proof of the existence of a bound state in the quantum Coulomb field on a three-dimensional de Sitter hyperboloid.
• To establish the spectral content of the charged state |u⟩ = exp(−iS(u))|0⟩ in terms of eigenstates of the Casimir operator C₁.
• To demonstrate the critical role of the coupling constant e², showing that a bound state exists only when 0 < e² < π.
• To derive and solve a partial differential equation for the matrix element ⟨u|exp(−σC₁)|u⟩, yielding exact results for the eigenvalue and probability of the bound state.

Proposed method

• Derive a partial differential equation satisfied by the matrix element ⟨u|exp(−σC₁)|u⟩ for σ > 0.
• Use the smoothness of S(u), defined as the average of the field S(x) over the Cauchy surface ux = 0, to enable exponentiation and operator algebra.
• Apply the commutation relations [Mμν, S(u)] and [Q, S(u)] to characterize the transformation properties and charge content of the state |u⟩.
• Show that the function exp(z)(1 − z)exp[−σz(2 − z)] with z = e²/π is an exact solution to the derived differential equation.
• Use this solution to recover both the eigenvalue of the Casimir operator and the probability of the bound state's occurrence.
• Verify consistency by deriving the resolvent ⟨u|(C₁ − λ)⁻¹|u⟩ and analyzing its spectral structure for z < 1 and z > 1.

Experimental results

Research questions

• RQ1 Does a bound state exist in the quantum Coulomb field on the three-dimensional de Sitter hyperboloid for small coupling e²?
• RQ2 What is the exact eigenvalue of the Casimir operator C₁ corresponding to the bound state in the state |u⟩ = exp(−iS(u))|0⟩?
• RQ3 How does the probability of the bound state’s occurrence depend on the coupling constant e², and why does it vanish for e² ≥ π?
• RQ4 Can the spectral content of the state |u⟩ be fully reconstructed from the matrix element ⟨u|exp(−σC₁)|u⟩ using a differential equation approach?
• RQ5 What is the mathematical structure of the resolvent ⟨u|(C₁ − λ)⁻¹|u⟩, and how does it distinguish between bound state and continuous spectrum contributions?

Key findings

• A bound state exists in the quantum Coulomb field on the de Sitter hyperboloid if and only if 0 < e² < π.
• The eigenvalue of the Casimir operator C₁ for the bound state is (e²/π)(2 − e²/π), explicitly derived from the exact solution of the differential equation.
• The probability of occurrence of the bound state is proportional to (1 − e²/π), vanishing when e² ≥ π.
• The matrix element ⟨u|exp(−σC₁)|u⟩ satisfies a second-order partial differential equation whose exact solution confirms the bound state's existence and spectral properties.
• The resolvent ⟨u|(C₁ − λ)⁻¹|u⟩ exhibits a discrete eigenvalue for z = e²/π < 1 and only a continuous spectrum (cut at λ ≥ 1) for z > 1, confirming the phase transition at e² = π.
• The autocorrelation function ⟨u|exp(−σC₁)|u⟩ does not admit a convergent Taylor series in σ, indicating non-analytic behavior, yet the moments ⟨u|(C₁)^n|u⟩ can be computed via a recurrence derived from the differential equation.