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Existence, degeneracy and stability of ground states by logarithmic Sobolev inequalities on Clifford algebras

Fabio E. G. Cipriani

TL;DR

The paper develops a noncommutative, Clifford-algebra framework to study ground states via logarithmic Sobolev inequalities. By embedding energy forms into Dirichlet-form theory on a standard Clifford representation and leveraging noncommutative entropy tools, it proves existence and finite degeneracy of ground states and establishes stability under certain unbounded perturbations. It then connects these abstract results to physical Hamiltonians for spin-1/2 Dirac particles, providing an infinitesimal, LSI-based route to existence and uniqueness of the ground state for Gross-type models. The approach yields quantitative degeneracy bounds, stability estimates, and a rigorous link between entropy, energy, and ground-state structure in a noncommutative quantum field setting.

Abstract

We prove existence and finite degeneracy of ground states of energy forms satisfying logarithmic Sobolev inequalities with respect to non vacuum states of Clifford algebras. We then derive the stability of the ground state with respect to certain unbounded perturbations of the energy form. Finally, we show how this provides an infinitesimal approach to existence and uniqueness of the ground state of Hamiltonians considered by L. Gross in QFT, describing spin $1/2$ Dirac particles subject to interactions with an external scalar field.

Existence, degeneracy and stability of ground states by logarithmic Sobolev inequalities on Clifford algebras

TL;DR

The paper develops a noncommutative, Clifford-algebra framework to study ground states via logarithmic Sobolev inequalities. By embedding energy forms into Dirichlet-form theory on a standard Clifford representation and leveraging noncommutative entropy tools, it proves existence and finite degeneracy of ground states and establishes stability under certain unbounded perturbations. It then connects these abstract results to physical Hamiltonians for spin-1/2 Dirac particles, providing an infinitesimal, LSI-based route to existence and uniqueness of the ground state for Gross-type models. The approach yields quantitative degeneracy bounds, stability estimates, and a rigorous link between entropy, energy, and ground-state structure in a noncommutative quantum field setting.

Abstract

We prove existence and finite degeneracy of ground states of energy forms satisfying logarithmic Sobolev inequalities with respect to non vacuum states of Clifford algebras. We then derive the stability of the ground state with respect to certain unbounded perturbations of the energy form. Finally, we show how this provides an infinitesimal approach to existence and uniqueness of the ground state of Hamiltonians considered by L. Gross in QFT, describing spin Dirac particles subject to interactions with an external scalar field.

Paper Structure

This paper contains 13 sections, 13 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Energy and Dirichlet forms on Clifford algebras
  3. Standard form of Clifford algebras
  4. Energy and Dirichlet forms
  5. Tools of noncommutative integration
  6. Tools of noncommutative integration
  7. Normal functionals vanishing in probability
  8. Relative uniform integrability of states
  9. Norm convergence to zero, relative vanishing and uniform integrability
  10. Uniform integrability of sublevel sets of relative entropy
  11. Ground states by logarithmic Sobolev inequalities
  12. Perturbations and stability of ground states
  13. On existence and uniqueness of the ground state of physical Hamiltonians

Key Result

Lemma 3.1

Let $\varphi,\psi\in M_{*+}$ be positive functionals and assume $\psi$ to be faithful. Then a linear, densely defined, positive, symmetric, closable operator $(R_\varphi,D(R_\varphi))$ is given by

Theorems & Definitions (35)

  • Remark 2.1
  • Example 2.2
  • Lemma 3.1
  • proof
  • Example 3.2
  • Definition 3.3: Relative vanishing for sequence of functionals
  • Lemma 3.4
  • proof
  • Definition 3.5: Relative uniform integrability of normal functionals
  • Theorem 3.6
  • ...and 25 more