Algebraic Factorization of Radiation and Gravity

Gauge and gravitational constraints generally prevent the physical Hilbert space from factorizing into independent spatial tensor factors. Radiation entropy is nevertheless well defined when the bath is nongravitating and regulated, or when one specifies a von Neumann algebra, its center, edge extension, and gravitational dressing. Changing that choice changes the entropy being computed.

Required background. Why Continuum QFT Does Not Factorize Naively supplies the local-algebra problem. The Information Problem: Assumptions and Observables supplies the black-hole deployment.

Helpful background. Gauge Constraints, Centers, and Edge Data gives the finite-cut model. Crossed-Product Gravitational Algebras and Generalized-Entropy Terms gives a gravitational construction. Fine-Grained, Coarse-Grained, and Algebraic Entropy fixes terminology.

Constraint-induced direct sums

In a lattice gauge theory cut into $R$ and $\bar R$, Gauss’s law correlates the electric flux through the boundary. The physical Hilbert space has the schematic decomposition

$$\mathcal H_{\mathrm{phys}} =\bigoplus_q \mathcal H_R^{\,q}\otimes \mathcal H_{\bar R}^{\,-q},$$

not a single tensor product. The boundary flux operator is central in the gauge-invariant regional algebra. A state decomposes with probabilities $p_q$, and an electric-center entropy takes the form

$$S_R =-\sum_q p_q\log p_q +\sum_q p_q\,S(\rho_R^q) +S_{\mathrm{edge}},$$

where the edge term depends on the chosen extension and gauge group. Different center choices are different observables, not computational gauges Casini, Huerta, and Rosabal 2014.

Gravity sharpens this issue: the Hamiltonian is a boundary charge on constraints, and a bulk excitation must be gravitationally dressed to an anchor. A dressed “inside” operator can change asymptotic metric data, so an exactly commuting inside–outside pair is not automatic.

Application: cutoff factor versus physical algebra

Consider a nongravitating lattice bath joined to a gravitating region. At the regulated level,

$$\mathcal H_{\mathrm{kin}} =\mathcal H_{\mathrm{bath}}\otimes\mathcal H_{\mathrm{grav,kin}},$$

and a bath interval $R$ has an ordinary reduced density matrix. After imposing a shared gauge constraint at the interface, however,

$$\mathcal H_{\mathrm{phys}} =\bigoplus_E \mathcal H_R^{E}\otimes\mathcal H_{\mathrm{rest}}^{-E}.$$

If the bath truly carries no dynamical gravity and the interface charge is treated as external data, the tensor-factor entropy is a controlled model observable. If the bath gravitates, integrating over the interface charge adds a center and changes the entropy. The same formal symbol $S(R)$ cannot be transported between these cases without stating the algebra.

Algebraic entropy can instead be defined from the state restricted to a von Neumann algebra $\mathcal A_R$. In type-III continuum QFT the von Neumann entropy itself diverges, while relative entropy is often finite. Crossed-product constructions can convert certain gravitational algebras to semifinite ones and relate their entropy to generalized-entropy terms, but this requires a specified modular flow and state.

Anchor and Gauss-law stress test

Move the dressing anchor from the bath boundary to the black-hole boundary while keeping the nominal bulk excitation fixed. Compute its commutator with asymptotic charges. If the regional algebra changes, so does the purported factorization. Next retain the Gauss constraint rather than working only in the kinematic Hilbert space; any independent tensor factors that disappear were regulator extensions.

This does not invalidate every bath entropy. It identifies the hypotheses under which it is physical: a nongravitating bath, fixed interface data, or a declared algebra-with-center prescription.

Scope and handoff

Algebraic nonfactorization does not by itself solve the information problem; the correlations and dynamics still must be computed. It does invalidate arguments that assume independent interior, black-hole, and radiation factors globally without a dressing or center prescription. Code-subspace representatives are developed next on Interior Reconstruction, State Dependence, Recovery, and Scrambling.

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

• Casini, H., M. Huerta, and J. A. Rosabal. “Remarks on Entanglement Entropy for Gauge Fields.” Physical Review D 89 (2014): 085012. DOI.
• Donnelly, W., and A. C. Wall. “Entanglement Entropy of Electromagnetic Edge Modes.” Physical Review Letters 114 (2015): 111603. DOI.
• Giddings, S. B., and D. Marolf. “A Global Picture of Quantum de Sitter Space.” Physical Review D 76 (2007): 064023. DOI.