Local approximations of global Hamiltonian from inclusion of algebras

1. Problem Statement

In Quantum Field Theory (QFT), local algebras of observables associated with causally complete regions (such as balls or wedges) are typically Type III$_1$ von Neumann algebras. A defining feature of these algebras is that their modular Hamiltonians ($K = -\log \rho$) have a continuous spectrum covering the entire real line and lack normalizable eigenvectors. Consequently, standard information-theoretic quantities, such as the modular spectral form factor ($\text{tr}(\Delta^{iT})$), diverge.

Furthermore, in Algebraic Quantum Field Theory (AQFT), the global Hamiltonian (which generates time translations for the entire spacetime) is not directly accessible to a local observer restricted to a finite region. While the Bisognano-Wichmann theorem relates the modular Hamiltonian of a Rindler wedge to the boost generator, reconstructing the global Minkowski or Cylindrical Hamiltonian from purely local data in a general QFT (especially away from the UV fixed point) remains a challenge.

The authors aim to:

1. Construct a local approximation of the global Hamiltonian using only the data of the inclusion of local algebras.
2. Provide a regulator for the modular Hamiltonian and the spectral form factor that relies on local data rather than global UV cutoffs.
3. Connect these constructions to $L^2$-nuclearity, an operator-algebraic condition characterizing the density of states.

2. Methodology

The paper employs tools from Operator Algebraic QFT, specifically the theory of modular inclusions and characteristic functions.

• Inclusion of Algebras: The authors consider a nested sequence of ball-shaped regions $B(e^{-2\pi s}R) \subset B(R)$ (or wedges in Minkowski space). Let $B_s \subset B_0$ denote this inclusion.
• Characteristic Functions: They define the characteristic function of the inclusion as:
  $$T_R(z) = \Delta_{B(R)}^{iz} \Delta_{B(e^{-2\pi s}R)}^{-iz}$$
  where $\Delta$ is the modular operator and $z$ is a complex parameter. This function encodes the relative modular structure between the two regions.
• Derivation of the Effective Hamiltonian: By differentiating the characteristic function with respect to the scaling parameter $s$ at $s=0$, they define a self-adjoint operator $\tilde{H}_R(z)$:
  $$\tilde{H}_R(z) = \frac{-1}{2\sin(2\pi i z)} \partial_s \log \left( T_R(z) T_R(-z)^{-1} \right) \Big|_{s=0}$$
• Coarse-Graining Map: The authors identify that $\tilde{H}_R(z)$ acts as a low-pass filter (or coarse-graining map) on the operator $R \partial_R K_B(R)$. This map, denoted $E_z$, averages the operator over modular time:
  $$E_z(X) = \frac{1}{2\sinh(2\pi z)} \int_{-z}^{z} dt \, e^{itK} X e^{-itK}$$
• Perturbation Theory: They apply conformal perturbation theory to study how these approximations behave when the theory is deformed away from the Conformal Field Theory (CFT) fixed point by a relevant operator.

3. Key Contributions and Results

A. Reconstruction of the Global Hamiltonian in CFT

In a vacuum CFT on a Lorentzian cylinder, the authors prove that the local data of the inclusion of balls allows for the exact reconstruction of the global Hamiltonian $H$.

• They show that $\tilde{H}_u$ (where $u$ parametrizes the radius) is independent of $z$ and relates to the global Hamiltonian via:
  $$2\pi H = \cosh(2\pi u) \tilde{H}_u - \sinh(2\pi u) K_u$$
  where $K_u$ is the modular Hamiltonian of the ball of radius $R=e^{-2\pi u}$.
• This result demonstrates that the global Hamiltonian is encoded in the local modular data of the inclusion, provided one accounts for the dilatation (scaling) of the region.

B. Local Approximations for General QFT

For general QFTs (not necessarily conformal), the authors propose a local approximation for the global Minkowski Hamiltonian:
$$2\pi H_z = \frac{1}{R} \left( \tilde{H}_R(z) + K_{B(R)} \right)$$

• Positivity and Regulation: While $\tilde{H}_R(z)$ is positive, the addition of $K_{B(R)}$ does not guarantee positivity. However, the operator serves as a regulator for the modular Hamiltonian.
• Spectral Form Factor: The operator $\tilde{H}_R(z)$ can be used to regulate the divergent modular spectral form factor. The trace $\text{tr}(e^{-\beta \tilde{H}_R(z)})$ is finite under the assumption of $L^2$-nuclearity.
• Optimization: The authors suggest that different choices of the parameter $z$ (or a probability distribution $p(z)$ over $z$) correspond to different "filters" of the infrared physics. One can optimize $p(z)$ to match specific physical constraints, such as the first-order correction in conformal perturbation theory.

C. Connection to Nuclearity and Chaos

• $L^2$-Nuclearity: The finiteness of the trace $\text{tr}(e^{-\beta \tilde{H}_R(-i/4)})$ is equivalent to the $L^2$-nuclearity condition. This condition ensures that the growth of the density of states is bounded, linking local algebraic inclusions to the global thermodynamic properties of the theory.
• Quantum Chaos: The regulated spectral form factor derived from these operators is expected to exhibit the universal "ramp and plateau" behavior characteristic of chaotic systems. This provides a local definition of quantum chaos that does not rely on inaccessible degrees of freedom far from the observer.
• Holography: The work connects to the "double-cone" regulator proposed by Saad, Shenker, and Stanford for eternal black holes. The authors' operator $\tilde{H}_R(z)$ effectively implements a similar coarse-graining of the modular Hamiltonian but derived purely from local algebraic inclusions.

D. Perturbative Analysis

The authors analyze the behavior of their proposed Hamiltonian under conformal perturbations ($H = H_0 + \lambda V$).

• They derive the first-order correction to the local approximation.
• They show that while the approximation matches the UV fixed point (CFT) Hamiltonian, the infrared corrections depend on the choice of the filter $z$.
• They argue that for chaotic CFTs, the eigenstate thermalization hypothesis (ETH) implies that the local approximation's spectrum remains stable and does not behave erratically across microstates.

4. Significance and Implications

1. Local-to-Global Reconstruction: The paper provides a rigorous operator-algebraic mechanism to reconstruct global time evolution (the Hamiltonian) from purely local data (modular Hamiltonians of nested regions). This challenges the notion that global symmetries are fundamentally inaccessible to local observers.
2. Regulation of Divergences: It offers a new, physically motivated regulator for Type III$_1$ algebras that does not require an ad-hoc UV cutoff but instead uses the geometric scaling of the region (inclusion of algebras).
3. Entanglement Bootstrap: The work generalizes the "entanglement bootstrap" philosophy (classifying phases via reduced states) to gapless systems. It suggests that the global Hamiltonian of a theory can be viewed as a "fictitious" Hamiltonian in a purified state constructed from local data, maximizing specific entropic measures.
4. Cosmological and Black Hole Applications: The framework is applicable to spacetimes without global Killing vectors (e.g., de Sitter space or black hole interiors). It provides a canonical way to define a global Hamiltonian for an observer restricted to a causal patch, potentially resolving ambiguities in defining vacuum states in curved spacetime.
5. Quantum Chaos: It offers a pathway to define and measure quantum chaos (via spectral form factors) using only local observables, which is crucial for understanding the emergence of spacetime and gravity in holographic contexts.

In summary, the paper establishes a deep link between the inclusion structure of local algebras and the global dynamics of QFT, providing a powerful tool for regulating divergences and reconstructing global physics from local quantum information.