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Local approximations of global Hamiltonian from inclusion of algebras

Yidong Chen, Nima Lashkari, Kwing Lam Leung

TL;DR

The authors address reconstructing or approximating the global Minkowski Hamiltonian of QFTs from strictly local, algebraic data by leveraging L^2-nuclearity and the characteristic functions of algebra inclusions. They introduce a regulator framework that uses inclusions of ball-shaped regions and coarse-graining maps to define local generators $\tilde{H}_R(z)$, connecting local modular data to the global dynamics. In conformal field theory on the Lorentzian cylinder, they rigorously derive ball-based reconstructions of the global Hamiltonian, showing that $2\pi H = \cosh(2\pi u)\tilde{H}_u - \sinh(2\pi u)K_u$ and that a local expression $2\pi H_z = \frac{1}{R}(\mathcal{E}_z(R\partial_R K_{B(R)}) + K_B(R))$ reproduces the global generator in appropriate limits. The paper then generalizes to QFT, proposing a family of local approximations $H_z$ and $H_p$ that can be tuned to capture infrared data or conformal perturbation theory corrections, with broad implications for quantum chaos diagnostics, entanglement bootstrap, and holographic setups. Overall, it provides a principled, local-data–driven route to extending local observations to global dynamics in both flat and curved spacetimes.

Abstract

We write down the global Hamiltonian of conformal field theory (CFT) in finite volume in terms of the modular Hamiltonian of the vacuum reduced to a local ball-shaped region, and use it to propose local approximations to the global Minkowski Hamiltonian in quantum field theory (QFT). The proposed Hamiltonians are motivated by the operator-algebraic property of nuclearity. They are constructed from the characteristic functions of inclusion of algebras and can be viewed as regulators of the modular Hamiltonian of local algebras of QFT

Local approximations of global Hamiltonian from inclusion of algebras

TL;DR

The authors address reconstructing or approximating the global Minkowski Hamiltonian of QFTs from strictly local, algebraic data by leveraging L^2-nuclearity and the characteristic functions of algebra inclusions. They introduce a regulator framework that uses inclusions of ball-shaped regions and coarse-graining maps to define local generators , connecting local modular data to the global dynamics. In conformal field theory on the Lorentzian cylinder, they rigorously derive ball-based reconstructions of the global Hamiltonian, showing that and that a local expression reproduces the global generator in appropriate limits. The paper then generalizes to QFT, proposing a family of local approximations and that can be tuned to capture infrared data or conformal perturbation theory corrections, with broad implications for quantum chaos diagnostics, entanglement bootstrap, and holographic setups. Overall, it provides a principled, local-data–driven route to extending local observations to global dynamics in both flat and curved spacetimes.

Abstract

We write down the global Hamiltonian of conformal field theory (CFT) in finite volume in terms of the modular Hamiltonian of the vacuum reduced to a local ball-shaped region, and use it to propose local approximations to the global Minkowski Hamiltonian in quantum field theory (QFT). The proposed Hamiltonians are motivated by the operator-algebraic property of nuclearity. They are constructed from the characteristic functions of inclusion of algebras and can be viewed as regulators of the modular Hamiltonian of local algebras of QFT
Paper Structure (20 sections, 20 theorems, 150 equations, 2 figures)

This paper contains 20 sections, 20 theorems, 150 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Regulating Modular Hamiltonian
  3. QFT Hamiltonian from Half-space
  4. CFT Hamiltonian from a ball
  5. Local approximation to global QFT Hamiltonian
  6. Discussion
  7. Operator Algebraic Structures of $L^2$-Nuclearity
  8. Common cyclic and separating vectors
  9. Characteristic functions of von Neumann algebra inclusions
  10. Relative characteristic functions of von Neumann algebra Inclusions
  11. $L^2$-Nuclearity Condition
  12. $L^2$-Nuclearity and discrete spectrum in CFT
  13. Nuclear norm
  14. Conformal field theory calculation
  15. Conformal symmetry on Lorentzian cylinder
  16. ...and 5 more sections

Key Result

Theorem 1

For the inclusion of wedges $W(a)\subset W(0)$, for all complex $z$ with $\Im(z)\in (-1/2,0)$ we have

Figures (2)

  • Figure 1: (a) Euclidean modular flow $\Delta_W^\alpha$ is a rotation by angle $\alpha$ on the Euclidean plane around $x=0$. (b) $\Delta_{W(a)}^{-\alpha}$ rotates back by the same angle, but around $x=a$. (c) The new result is a Euclidean strip of width $a\sin(2\pi\alpha)$. (c) The end point of $W(0)$ (red-dot) has moved in the $x^1$ direction by $a\sin(2\pi\alpha)$ in the Euclidean time direction combined with the translation by $a(1-\cos(2\pi \alpha))$.
  • Figure 2: (a) Extension of CFT from Minkowski space to Lorentzian cylinder. The blue and red lines are surfaces of constant $x^0$ and $r$ in Minkowski coordinates, and the range of $\theta\in(0,\pi)$. (b) The inclusion of concentric balls $B(e^{-2\pi (s+ u)})\subset B(e^{-2\pi u})$ with $s>0$ on Lorentzian cylinder. (c) The inclusion of concentric balls in Minkowski space.

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 20 more