Subsystems and Algebraic Factorization

In an unconstrained quantum field theory, a spatial cut often suggests $\mathcal H\simeq\mathcal H_A\otimes\mathcal H_{\bar A}$. Gauge constraints obstruct that canonical split because flux through the cut is measured from both sides. Gravity strengthens the obstruction: energy and other charges are encoded in boundary fields, while gauge-invariant excitations require relational or asymptotic dressing. Useful subsystems still exist, but only after declaring the boundary algebra, charge sector, reference frame, and approximation.

Required background. Variational and Symplectic Geometry supplies boundary symplectic structure and constraints. Equivalence Principles and Gravitational Measurement supplies operational region definitions.

Helpful background. Flat-Spacetime Quantum Field Theory supplies local algebras, Einstein Equations supplies gravitational Gauss laws, Quantum Information supplies entanglement and channels, and Quantum Gravity supplies the regime distinctions.

Why a tensor product fails

Gauss’s law in electrodynamics is $\nabla\cdot\mathbf E=\rho$. Integrating it over $A$ relates the charge inside to the normal electric flux through $\partial A$. The same flux is part of the exterior description with the opposite orientation. In a regulated theory it cannot be independently chosen on both factors, so the physical Hilbert space has the schematic form

$$\mathcal H_{\mathrm{phys}} \subseteq\bigoplus_q \mathcal H_A^{q}\otimes\mathcal H_{\bar A}^{-q},$$

where $q$ labels compatible boundary-flux sectors. This is a direct sum of matched sectors, not an unconstrained tensor product. Algebraically, boundary flux can lie in the center of the chosen regional observable algebra. Casini, Huerta, and Rosabal show how different center choices organize gauge-theory entanglement and why no unique field-coordinate factorization is forced Casini, Huerta, and Rosabal 2014, §§II–IV.

The regulator matters even without gauge symmetry: continuum local QFT algebras generally do not yield an ordinary density-matrix tensor product across a sharp cut. A lattice or a finite separating buffer makes the comparison precise. Gauge constraints add the shared-flux obstruction on top of that ultraviolet issue.

In gravity, the constraints relate bulk stress energy and momentum to geometric surface data. A regional specification may need an induced metric or area element, normal-frame data, surface charges, and the attachment of gravitational dressings. A transformation that vanishes at the cut can be gauge, while one that moves the cut or changes a surface charge can be a physical boundary symmetry. “The Hilbert space of the region” is therefore incomplete unless these choices are named.

Results and limitations as of 2026-08-20

The table compares controlled constructions rather than treating them as equivalent definitions of one exact object.

Formulation or result	Degrees of freedom and boundary data	Evidence or controlled result	Scope and limitation
Algebra with a center	Gauge-invariant bulk operators plus chosen boundary flux or charge center	Physical states decompose into superselection sectors; entropy separates sector weights and within-sector entropy	Center choice depends on the regional algebra; gravity may not admit a complete compactly supported algebra
Extended Hilbert space and edge modes	Independent boundary frames on each side, glued by a matching constraint	Restores a tensor-product kinematics before projection and accounts for boundary symplectic transformations Donnelly and Freidel 2016, §§2–4	Added edge variables are prescription-dependent until tied to operational boundary data; kinematical factorization is not full dynamics
Perturbative gravitational split	Matter states with a separating buffer and chosen weak metric dressings	Exterior metric matrix elements depend only on total Poincaré charges at first order Donnelly and Giddings 2018, §IV, PDF	Weak-field asymptotically flat construction; it does not control every nonlinear exterior observable or subsequent evolution
Relational finite region	Bulk fields located relative to material clocks, rods, or a finite boundary	Provides diffeomorphism-invariant regional questions when the reference map is invertible	Reference fluctuations and backreaction remain; multiple clock and boundary choices can yield inequivalent factorizations

This discussion reflects results available through 2026-08-20. The conservative position says there is no canonical exact spatial tensor product in gravity, but several boundary-conditioned algebras and approximate split structures are precise in controlled regimes. The permissive position treats edge modes or relational boundary frames as genuine subsystem degrees of freedom that may persist in a microscopic theory. These competing positions depend on whether the added boundary variables are regarded as physical observables, gauge-restoring bookkeeping, or an effective code subspace.

The unresolved issue is not whether constraints correlate the two sides—they do—but which construction survives a nonperturbative continuum limit and supports operationally complete observables. A parallel table row does not make a kinematical extension equal to a dynamical factorization theorem.

A finite Gauss-law model

Take two qubits, one assigned to each side of a cut, and impose the $\mathbb Z_2$ Gauss constraint

$$G=Z_AZ_B=+1.$$

The unconstrained tensor product has basis $\lvert00\rangle$, $\lvert01\rangle$, $\lvert10\rangle$, $\lvert11\rangle$. The physical subspace is

$$\mathcal H_{\mathrm{phys}}= \operatorname{span}\{\lvert00\rangle,\lvert11\rangle\},$$

which has dimension two, not four. The boundary flux measured by $Z_A$ equals that measured by $Z_B$ on physical states. Neither side can independently flip its qubit with $X_A$ or $X_B$, because either operation violates $G=+1$; only the bilocal product $X_AX_B$ preserves the constraint.

Fixing the boundary sector $Z_A=Z_B=+1$ leaves the one-dimensional state $\lvert00\rangle$, which trivially factors. Allowing both sectors admits superpositions such as

$$\lvert\psi\rangle=\alpha\lvert00\rangle+\beta\lvert11\rangle.$$

The sector label is shared data. An extended Hilbert-space construction can duplicate it into two edge variables and impose equality only when gluing; an algebraic construction can retain it as a center. These prescriptions reproduce the same matching constraint but assign entropy differently unless the center and state preparation are fixed.

Now change the boundary condition by permitting a charged endpoint at the cut. Then a single-side flip can be accompanied by a boundary-charge flip, and the regional algebra expands. The obstruction is therefore robust, while the minimal boundary data depends on which charged boundary excitations are admissible.

Approximate gravitational splitting

In a weak-field asymptotically flat region, surround the region of interest by a finite buffer and choose a common exterior dressing. At first order, interior information can then vary while exterior metric matrix elements depend only on the total Poincaré generators. The statement concerns charge matrix elements within the chosen state space; equality of expectation values alone would be too weak to ensure indistinguishability of superpositions. It defines a perturbative gravitational split, analogous to a quantum code subspace, rather than an exact factorization of the full Hilbert space.

The leading construction uses dressing freedom to remove dependence on the source’s higher multipoles from the specified exterior matrix elements. Arbitrary dressings need not have this property. Nonlinear corrections, a larger exterior algebra, and evolution that lets radiation escape require further analysis; first-order existence alone supplies no all-order error bound. At finite boundaries, corner symmetries and normal-frame choices add further data.

A useful subsystem claim should therefore specify

$$(\mathcal A_A,\ \mathcal A_{\bar A},\ \mathcal Z_{\partial A},\ \mathcal H_{\mathrm{code}},\ \varepsilon),$$

where the algebras, shared center, restricted state space, and accuracy $\varepsilon$ are explicit. Without this tuple, “factorization” can mean incompatible things.

Interpretation and adverse checks

Changing the cut, center, dressing, or boundary condition is an adverse check. Quantities advertised as universal should be invariant under these changes or have a transformation law that explains the difference. Entanglement entropy itself can contain prescription-dependent edge contributions, while relative entropy or mutual information between well-separated gauge-invariant algebras may be more robust.

No factorization obstruction by itself proves that information is delocalized in an operationally accessible way. Shared charges can correlate descriptions without allowing an observer to reconstruct the interior state. Conversely, an approximate split does not erase the exact constraints; it limits the observables and states for which their boundary imprint is indistinguishable.

Exercises

Exercise. In the two-qubit model, show that a general physical state has the form $\alpha\lvert00\rangle+\beta\lvert11\rangle$. Compute the entropy obtained by tracing over qubit $B$ in the extended tensor product, and compare it with the entropy after fixing the $Z=+1$ boundary sector.

Solution

The condition $Z_AZ_B=+1$ removes $\lvert01\rangle$ and $\lvert10\rangle$, leaving the stated superposition with $\lvert\alpha\rvert^2+\lvert\beta\rvert^2=1$. Tracing over $B$ gives

$$\rho_A=\lvert\alpha\rvert^2\lvert0\rangle\!\langle0\rvert +\lvert\beta\rvert^2\lvert1\rangle\!\langle1\rvert,$$

so

$$S_A=-\lvert\alpha\rvert^2\log\lvert\alpha\rvert^2 -\lvert\beta\rvert^2\log\lvert\beta\rvert^2.$$

If the $Z=+1$ sector is fixed, only $\lvert00\rangle$ remains and the within-sector entropy is zero. The difference is the classical uncertainty over the shared boundary sector. It shows why an entropy statement must say whether sectors are summed, fixed, or represented by edge variables.

What could change these conclusions

The conclusions could change after a nonperturbative construction proves or refutes a gravitational split, a continuum calculation stabilizes an edge algebra under regulator changes, an operational protocol selects among center choices, or evidence after 2026-08-20 changes the relative standing of the formulations.

References

• Casini, Horacio, Marina Huerta, and José Alejandro Rosabal. “Remarks on Entanglement Entropy for Gauge Fields.” Physical Review D 89 (2014): 085012. DOI.
• Donnelly, William, and Laurent Freidel. “Local Subsystems in Gauge Theory and Gravity.” Journal of High Energy Physics 2016, no. 9 (2016): 102. DOI. Open PDF.
• Donnelly, William, and Steven B. Giddings. “Gravitational Splitting at First Order: Quantum Information Localization in Gravity.” Physical Review D 98 (2018): 086006. DOI. Open PDF.