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Second Quantization for the Kepler Problem

John C. Baez

TL;DR

The paper develops a geometric and algebraic framework linking the Kepler problem to second quantization by exploiting the hidden $\mathrm{SU}(2)\times\mathrm{SU}(2)$ symmetry of bound states, recasting hydrogen as solutions of the left-handed Weyl equation on $\mathbb{R}\times S^3$, and then formulating a fermionic Fock space that corresponds to a massless spin-$\tfrac{1}{2}$ quantum field on the Einstein universe. It establishes a unitary equivalence between the single-particle hydrogen problem and the Weyl field, and lifts this to the many-particle level, enabling a field-theoretic description of multi-electron atoms. A Madelung-type Hamiltonian is constructed from Duflo-corrected angular momenta to reproduce the qualitative subshell filling rules of the periodic table. This work connects atomic structure to conformal geometry and quantum field theory on curved spacetime, offering a novel perspective on the periodicity of elements via second quantization on $\mathbb{R}\times S^3$.

Abstract

The Kepler problem concerns a point particle in an attractive inverse square force. After a brief review of the classical and quantum versions of this problem, focused on their hidden $\text{SU}(2) \times \text{SU}(2)$ symmetry, we discuss the quantum Kepler problem for a spin-$\frac{1}{2}$ particle. We show that the Hilbert space $\mathcal{H}$ of bound states for this problem is unitarily equivalent, as a representation of $\text{SU}(2) \times \text{SU}(2)$, to the Hilbert space of solutions of the Weyl equation on the spacetime $\mathbb{R} \times S^3$. This equation describes a massless left-handed spin-$\frac{1}{2}$ particle. We then form the fermionic Fock space on $\mathcal{H}$ and show this is unitarily equivalent to the Hilbert space of a massless left-handed spin-$\frac{1}{2}$ free quantum field on $\mathbb{R} \times S^3$, again as representations of $\text{SU}(2) \times \text{SU}(2)$. By modifying the Hamiltonian of this free field theory, we obtain the well-known "Madelung rules". These give a reasonable approximation to the observed filling of subshells as we consider elements with more and more electrons, and match the rough overall structure of the periodic table.

Second Quantization for the Kepler Problem

TL;DR

The paper develops a geometric and algebraic framework linking the Kepler problem to second quantization by exploiting the hidden symmetry of bound states, recasting hydrogen as solutions of the left-handed Weyl equation on , and then formulating a fermionic Fock space that corresponds to a massless spin- quantum field on the Einstein universe. It establishes a unitary equivalence between the single-particle hydrogen problem and the Weyl field, and lifts this to the many-particle level, enabling a field-theoretic description of multi-electron atoms. A Madelung-type Hamiltonian is constructed from Duflo-corrected angular momenta to reproduce the qualitative subshell filling rules of the periodic table. This work connects atomic structure to conformal geometry and quantum field theory on curved spacetime, offering a novel perspective on the periodicity of elements via second quantization on .

Abstract

The Kepler problem concerns a point particle in an attractive inverse square force. After a brief review of the classical and quantum versions of this problem, focused on their hidden symmetry, we discuss the quantum Kepler problem for a spin- particle. We show that the Hilbert space of bound states for this problem is unitarily equivalent, as a representation of , to the Hilbert space of solutions of the Weyl equation on the spacetime . This equation describes a massless left-handed spin- particle. We then form the fermionic Fock space on and show this is unitarily equivalent to the Hilbert space of a massless left-handed spin- free quantum field on , again as representations of . By modifying the Hamiltonian of this free field theory, we obtain the well-known "Madelung rules". These give a reasonable approximation to the observed filling of subshells as we consider elements with more and more electrons, and match the rough overall structure of the periodic table.

Paper Structure

This paper contains 13 sections, 2 theorems, 159 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Note on units
  3. Acknowledgements
  4. The hydrogen atom --- ignoring spin
  5. Digression on the Duflo isomorphism
  6. The hydrogen atom --- with spin
  7. Spinors on the 3-sphere
  8. Digression on quaternions
  9. The Weyl equation on the Einstein universe
  10. Digression on the conformal group
  11. Second quantization
  12. Multi-electron atoms --- the Madelung rules
  13. Proofs

Key Result

Theorem 1

The operator $F \colon L^2(S^3) \otimes \mathbb{C}^2 \to \mathsf{H}$ is a unitary equivalence between the representation $\rho$ of $\operatorname{SU}(2) \times \operatorname{SU}(2)$ on $L^2(S^3) \otimes \mathbb{C}^2$ and the representation $\rho_\mathsf{H}$ of this group on $\mathsf{H}$, and $F \sla

Figures (3)

  • Figure 1: The Madelung rules, from ChemistryGod.
  • Figure 2: Periodic table illustrating the Madelung rules.
  • Figure 3: The Madelung rules for elements 1 to 120, and the energies $E_{n,\ell}$.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • proof
  • proof