Modular Flow Reconstruction and Bulk Modular Evolution

Boundary modular flow can reconstruct an operator in an entanglement wedge by resolving it into modular frequencies. The construction is relative to a boundary region, reference state, operator algebra, and semiclassical code sector. It does not make a generic modular Hamiltonian local or produce a state-independent bulk operator outside those domains.

Required background. Tomita–Takesaki Modular Operators and Flow supplies the algebraic flow. Boundary Relative Entropy and Bulk Modular Data supplies the holographic modular relation.

Helpful background. Nonlocal Modular Hamiltonians and Limits of Explicit Control states the generic obstruction. Causal Wedges and Subregion Reconstruction fixes the bulk region. Entanglement Spectra and Modular Spectral Data supplies spectral diagnostics, and Modular Analyticity and Chaos Bounds fixes the analytic strip.

First application. Reconstruct a wedge operator using modular evolution for a region with a known boundary modular generator.

Modular evolution and frequency

For an algebra $\mathcal A_A$ and cyclic separating reference state $\sigma$, modular flow is

$$O_A(s)=e^{isK_A^\sigma}O_Ae^{-isK_A^\sigma}.$$

The modular-frequency component is

$$O_A(\omega)=\int_{-\infty}^{\infty}ds\,e^{-i\omega s}O_A(s), \qquad [K_A^\sigma,O_A(\omega)]=\omega O_A(\omega),$$

in a domain where the transform is defined distributionally. Except for special regions and states—such as a vacuum half-space or ball—the generator is nonlocal and usually unknown in closed form.

The JLMS operator relation implies, within the code subspace, that the boundary and bulk modular actions agree on wedge operators up to the central geometric term and controlled corrections. This motivates

$$e^{isK_A^{\mathrm{CFT}}}O_a e^{-isK_A^{\mathrm{CFT}}} \simeq e^{isK_a^{\mathrm{bulk}}}O_a e^{-isK_a^{\mathrm{bulk}}}.$$

The equality is about action on code states, not an unrestricted equality of operators on the full CFT Hilbert space.

Modular-frequency reconstruction

For a bulk point $X$ in the entanglement wedge, a perturbative scalar can be represented schematically as

$$\phi(X)=\int_A d^dx\int_{-\infty}^{\infty}ds\, f_A(X\mid x,s)\,O_A(x,s)+\text{corrections}.$$

The kernel is built from bulk–boundary modular-frequency correlators and their inverse on the chosen operator subspace. This extends ordinary causal smearing when $X$ lies beyond the causal wedge but inside the entanglement wedge Faulkner and Lewkowycz 2017, §§2–4.

For a ball in the vacuum, modular flow is geometric: it is generated by the conformal Killing field preserving the causal diamond. The kernel can then be checked against AdS–Rindler mode reconstruction. For a generic excited state, modular evolution is nonlocal; the formal representation remains useful only if the correlator kernel, inverse, and operator domain are controlled.

Region and state dependence

The same logical bulk operator can have different boundary representatives for overlapping regions. Their agreement is required only on the code subspace. State dependence enters through the reference modular operator and through which QES defines the wedge. A large deformation that changes the wedge cannot be treated by holding the original modular kernel fixed.

Modular zero modes also require care. Components with $\omega=0$ can lie in a center or commute with the modular generator, making the inverse kernel nonunique. A gauge condition or choice of zero-mode frame is part of the reconstruction.

Adversarial controls

Assume locality of $K_A$. Insert the vacuum-ball kernel into a generic excited state. Failure of the modular correlators to satisfy the geometric flow exposes the unsupported step.

Invert outside the sampled subspace. A correlator matrix with null directions cannot reconstruct those operator components. Regularizing the inverse must come with an error norm and cutoff dependence.

Leave the code sector. If backreaction changes the extremal surface or brings in new light degrees of freedom, the original wedge algebra and kernel no longer license the representation.

Evidence ceiling

Modular-frequency methods give a perturbative representation of wedge operators when the reference modular data, code subspace, correlator inverse, and surface branch are controlled. They do not establish a universal local modular Hamiltonian, exact finite-N bulk locality, or a single boundary representative valid for every state and region.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Faulkner, Thomas, and Aitor Lewkowycz. “Bulk Locality from Modular Flow.” Journal of High Energy Physics 2017, 151 (2017). DOI; arXiv:1704.05464.
• Jafferis, Daniel L., Aitor Lewkowycz, Juan Maldacena, and S. Josephine Suh. “Relative Entropy Equals Bulk Relative Entropy.” Journal of High Energy Physics 2016, 004 (2016). DOI; arXiv:1512.06431.