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Non-Linear Operator-valued Elliptic Flows with Application to Quantum Field Theory

Jean-Bernard Bru, Nathan Metraud

TL;DR

The paper addresses nonlinear differential equations on operator spaces by introducing a novel elliptic operator-valued flow with equations $\partial_t \Upsilon_t = 16 D_t D_t^{\ast}$ and $\partial_t D_t = -2( \Upsilon_t D_t + D_t \Upsilon_t^{\top})$, under a lower bound $\Upsilon_0 \ge -\mu \mathbf{1}$. It proves well posedness in strong and Schatten topologies, establishes ellipticity via constant of motion invariants and analyzes infinite-time asymptotics, including exponential convergence when $D_0$ is Hilbert-Schmidt. A primary application is to diagonalize quadratic fermionic Hamiltonians, yielding an $\mathrm{N}$-diagonal limit Hamiltonian and sharpening previous results on fermionic quadratic models. The work thus provides a rigorous mathematical framework for operator-valued flows with potential implications for quantum field theory and quantum statistical mechanics.

Abstract

Differential equations on spaces of operators are very little developed in Mathematics, being in general very challenging. Here, we study a novel system of such (non-linear) differential equations. We show it has a unique solution for all times, for instance in the operator or Hilbert-Schmidt norm topologies. This system presents remarkable ellipticity properties that turn out to be crucial for the study of the infinite-time limit of its solution, which is proven under relatively weak, albeit probably not necessary, hypotheses on the initial data. This system of differential equations is the elliptic counterpart of an hyperbolic flow applied to quantum field theory to diagonalize Hamiltonians that are quadratic in the bosonic field. In a similar way, this elliptic flow, in particular its asymptotics, has application in quantum field theory: it can be used to diagonalize Hamiltonians that are quadratic in the fermionic field while giving new explicit expressions and properties of these pivotal Hamiltonians of quantum field theory and quantum statistical mechanics.

Non-Linear Operator-valued Elliptic Flows with Application to Quantum Field Theory

TL;DR

The paper addresses nonlinear differential equations on operator spaces by introducing a novel elliptic operator-valued flow with equations and , under a lower bound . It proves well posedness in strong and Schatten topologies, establishes ellipticity via constant of motion invariants and analyzes infinite-time asymptotics, including exponential convergence when is Hilbert-Schmidt. A primary application is to diagonalize quadratic fermionic Hamiltonians, yielding an -diagonal limit Hamiltonian and sharpening previous results on fermionic quadratic models. The work thus provides a rigorous mathematical framework for operator-valued flows with potential implications for quantum field theory and quantum statistical mechanics.

Abstract

Differential equations on spaces of operators are very little developed in Mathematics, being in general very challenging. Here, we study a novel system of such (non-linear) differential equations. We show it has a unique solution for all times, for instance in the operator or Hilbert-Schmidt norm topologies. This system presents remarkable ellipticity properties that turn out to be crucial for the study of the infinite-time limit of its solution, which is proven under relatively weak, albeit probably not necessary, hypotheses on the initial data. This system of differential equations is the elliptic counterpart of an hyperbolic flow applied to quantum field theory to diagonalize Hamiltonians that are quadratic in the bosonic field. In a similar way, this elliptic flow, in particular its asymptotics, has application in quantum field theory: it can be used to diagonalize Hamiltonians that are quadratic in the fermionic field while giving new explicit expressions and properties of these pivotal Hamiltonians of quantum field theory and quantum statistical mechanics.
Paper Structure (14 sections, 31 theorems, 251 equations, 1 figure)

This paper contains 14 sections, 31 theorems, 251 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Elliptic Operator-Valued Flow
  3. Notation and Preliminary Definitions
  4. Well-Posedness of the Operator-Valued Flow
  5. Ellipticity and Asymptotics of the Flow
  6. Application of the Flow in Quantum Field Theory
  7. Technical Results on the Flow
  8. Local Solution to the Flow -- Strong Operator Topology
  9. Global Solution to the Flow -- Strong Operator Topology
  10. Extension to the Schatten Topology
  11. Constants of Motion
  12. Elliptic Properties of the Flow
  13. Asymptotics of the Elliptic Flow
  14. Illustration of the Flow on Scalar Fields

Key Result

Theorem 2.2

Assume $\Upsilon _{0}=\Upsilon _{0}^{\ast }\geq -\mu \mathbf{1}$ with $\mu \in \mathbb{R}$ and $D_{0}\in \mathcal{B}\left( \mathfrak{h}\right)$ ($D_{0}\neq 0$). Then, there exists a unique solution $\Delta \equiv (\Delta _{t})_{t\geq 0},D\equiv (D_{t})_{t\geq 0}$ of strongly continuous mappings on $

Figures (1)

  • Figure 1: Time evolution for $f$ and $g$, with parameters $\alpha=0.99$ and $\beta = 0.07$.

Theorems & Definitions (38)

  • Remark 2.1
  • Theorem 2.2: Well-posedness of the flow -- Strong topology
  • Theorem 2.3: Well-posedness of the flow -- Hilbert-Schmidt topology
  • Theorem 2.4: Elliptic operator-valued flow
  • Theorem 2.5: Asymptotics of the operator-valued flow
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8: Diagonalization of quadratic Hamiltonians -- QuadraticFermionic
  • Lemma 3.1: Local solution to (\ref{['initial value problemlocal']})
  • Lemma 3.2: Existence of $D$ for small times
  • ...and 28 more