Relative entropy for $λφ^4$ in the Rindler wedge

Abstract: We consider the relative entropy between the vacuum and a coherent state in the Rindler wedge for an interacting $λφ<sup>4$ theory to first order in $λ$. We construct the perturbatively interacting Weyl algebra of the wedge, and employ Tomita--Takesaki modular theory and the Araki--Uhlmann formula to compute the relative entropy. We verify that the relative entropy reduces to the classical (interacting) boost Noether charge, analogously to the free theory, and that the Bekenstein bound holds.

• The paper computes relative entropy to first order in λ and shows that it equals 2π times the boost-weighted classical energy of a causal interacting solution in the Rindler wedge.
• It develops renormalized stress-tensor distributions and verifies that the resulting interacting Noether charges generate Poincaré transformations while preserving wedge-algebra locality.
• For excitations localized within a strip of width R, the positive energy representation directly yields the Bekenstein bound S_rel ≤ 2πRE, although all-orders validity and solution uniqueness remain open.

Motivation and setting

Relative entropy is one of the few entropic quantities in quantum field theory that is both operationally meaningful and UV-finite. For two faithful normal states $\omega$ and $\phi$ on a von Neumann algebra $\mathcal{A}$, the Araki–Uhlmann formula expresses it as $$S_\mathrm{rel}(\omega\Vert\phi) = -\omega(\ln \Delta_{\phi,\omega})$$, where $\Delta_{\phi,\omega}$ is the relative modular operator. When $\phi$ is a unitary excitation of a cyclic and separating vector state $\omega$, the relative modular operator satisfies $\Delta_{\phi,\omega} = U\Delta_\omega U^\dagger$, so the computation reduces to the expectation value of the modular Hamiltonian in the excited state. By the Bisognano–Wichmann theorem, for the Rindler wedge algebra and the Minkowski vacuum the modular Hamiltonian is $2\pi$ times the boost generator $\mathcal{M}^{01}$ — a result valid for any Wightman QFT, including interacting ones.

Despite this structural simplicity, explicit computations of relative entropy in perturbatively interacting, non-conformal theories have been scarce; prior results concerned CFTs perturbed by relevant operators, KMS states of free versus interacting scalars, or perturbative expansions around common states. The paper under review fills this gap by computing, to first order in the coupling $\phi$0, the relative entropy between the Minkowski vacuum $\phi$1 and an interacting coherent state $\phi$2 in the Rindler wedge $\phi$3 of massive $\phi$4 theory in $\phi$5 dimensions (2607.07810).

Classical theory and causal solutions

The authors first construct classical solutions of the nonlinear Klein–Gordon equation $\phi$6 with controlled support properties. While the free solution $\phi$7 has support contained in $\phi$8, naive perturbative corrections to the retarded, advanced, or time-symmetric solutions lose this property: already at first order their support can spread over all of spacetime. The key observation is that by suitably correcting the smearing function,

$\phi$9

one obtains a solution $\mathcal{A}$0 that solves the interacting equation to first order and retains the causal support property. Importantly, this solution is not unique: arbitrary multiples of $\mathcal{A}$1 and $\mathcal{A}$2 can be added without spoiling the equation of motion, the support property, or the scaling covariance $\mathcal{A}$3, $\mathcal{A}$4. This non-uniqueness is a genuine limitation that resurfaces in the interpretation of the final result. The section closes with the classical Noether charges (translations and boosts), which satisfy the Poincaré algebra, with positive energy density and $\mathcal{A}$5 for wedge-supported solutions.

Quantization, renormalization, and Noether charges

The interacting field operator is defined via the Bogoliubov map, $\mathcal{A}$6 with $\mathcal{A}$7, satisfying the normal-ordered interacting equation of motion. The stress tensor requires more care: the symmetrized product $\mathcal{A}$8 contains formally divergent terms such as $\mathcal{A}$9. The authors construct renormalized distributions $$S_\mathrm{rel}(\omega\Vert\phi) = -\omega(\ln \Delta_{\phi,\omega})$$0 and $$S_\mathrm{rel}(\omega\Vert\phi) = -\omega(\ln \Delta_{\phi,\omega})$$1 explicitly via dimensional regularization combined with Feynman parametrization, imposing the constraint that the resulting first-order stress tensor correction be conserved. The renormalized result involves nonlocal integral kernels $$S_\mathrm{rel}(\omega\Vert\phi) = -\omega(\ln \Delta_{\phi,\omega})$$2 and $$S_\mathrm{rel}(\omega\Vert\phi) = -\omega(\ln \Delta_{\phi,\omega})$$3 built from $$S_\mathrm{rel}(\omega\Vert\phi) = -\omega(\ln \Delta_{\phi,\omega})$$4 in momentum space, plus local counterterms fixed by mass and field-strength renormalization. Consistency is verified by showing that the interacting Noether charges $$S_\mathrm{rel}(\omega\Vert\phi) = -\omega(\ln \Delta_{\phi,\omega})$$5 generate the correct transformations, $$S_\mathrm{rel}(\omega\Vert\phi) = -\omega(\ln \Delta_{\phi,\omega})$$6, reproducing the Poincaré algebra in the adiabatic limit. A supporting argument for the cancellation of the $$S_\mathrm{rel}(\omega\Vert\phi) = -\omega(\ln \Delta_{\phi,\omega})$$7 contributions uses a decomposition $$S_\mathrm{rel}(\omega\Vert\phi) = -\omega(\ln \Delta_{\phi,\omega})$$8 with spacelike-separated supports; the generalization to curved spacetimes and higher orders is deferred to Hollands' general framework rather than proven here.

Interacting Weyl algebra

The wedge algebra is generated by interacting Weyl operators $$S_\mathrm{rel}(\omega\Vert\phi) = -\omega(\ln \Delta_{\phi,\omega})$$9 with $\Delta_{\phi,\omega}$0. Although $\Delta_{\phi,\omega}$1 involves integrations over the entire causal past of $\Delta_{\phi,\omega}$2 through $\Delta_{\phi,\omega}$3, the authors verify microcausality directly: $\Delta_{\phi,\omega}$4 and $\Delta_{\phi,\omega}$5 commute when $\Delta_{\phi,\omega}$6 and $\Delta_{\phi,\omega}$7 are spacelike separated, because the relevant distributional combination vanishes whenever the future cone of one support does not meet the past cone of the other. This confirms, in a concrete perturbative model, the general construction of interacting algebras due to Dutsch and Fredenhagen. The commutant is correspondingly generated by Weyl operators supported in the left wedge.

Relative entropy

With the Bisognano–Wichmann identification $\Delta_{\phi,\omega}$8, the relative entropy reduces to $\Delta_{\phi,\omega}$9. Evaluating this requires the vacuum expectation value of products of interacting Weyl operators, computed using Wick's theorem and normal-ordering combinatorics, together with the boost action on test functions generated by $\phi$0. After substantial manipulations — including an extended Green's identity (derived in an appendix) that handles functions which are only spatially compact — the result takes the manifestly positive form

$\phi$1

where $\phi$2 is the classical energy density of the interacting theory and $\phi$3 is precisely the causal interacting solution constructed earlier. Since $\phi$4 implies $\phi$5, positivity follows immediately. The main structural conclusion is that the free-field pattern — relative entropy equals the classical boost Noether charge evaluated on the classical solution corresponding to the coherent excitation — survives interactions at first order, even though identifying the correct classical solution required the full computation, since causality alone does not select it uniquely.

Bekenstein bound

Because the same computation with $\phi$6 replaced by $\phi$7 yields the energy expectation value $\phi$8, the Bekenstein bound follows trivially: if $\phi$9, then

$\omega$0

This is considerably simpler than the general proofs based on estimates for relative modular Hamiltonians (Longo's theorem, or the Hollands–Longo bounds for approximately localized excitations), and it holds here because the excitation is exactly localized and its entropy is given by a positive local energy density weighted by $\omega$1. Note that this is distinct from Casini's formulation of the Bekenstein bound, which reduces to positivity of relative entropy and therefore holds unconditionally.

Limitations and open questions

Several caveats qualify these results. First, everything is computed only to first order in $\omega$2; the all-orders formula $\omega$3 is stated as a conjecture, and since the causal interacting solution is not unique beyond leading order, the conjecture cannot even be made precise without further work determining $\omega$4 exactly. Second, the renormalized stress tensor contains arbitrary constants ($\omega$5, $\omega$6) ultimately fixed by mass and wavefunction renormalization, so the numerical content of the entropy depends on this renormalization scheme. Third, the derivation assumes exact compact support of the excitation within the wedge (or a strip of width $\omega$7); extensions to soft modes or asymptotic states would require relaxing this assumption. Finally, the Poincaré algebra of the charges holds only in the adiabatic limit, whose existence is guaranteed for commutators but not necessarily for the charges themselves.

Conclusion

This paper provides the first explicit computation of relative entropy between the vacuum and a coherent state in a perturbatively interacting, non-conformal QFT. The central result — that the relative entropy equals $\omega$8 times the classical boost charge evaluated on the causal interacting classical solution, implying the Bekenstein bound — extends the known free-field structure to $\omega$9 theory at order $\Delta_{\phi,\omega} = U\Delta_\omega U^\dagger$0, at the cost of a technically involved renormalization of composite operators in the stress tensor. The framework opens concrete follow-up computations: Petz–Rényi entropies and $\Delta_{\phi,\omega} = U\Delta_\omega U^\dagger$1-divergences, capacity of entanglement and higher modular Hamiltonian moments, Yukawa-type theories with fermions, and coherent excitations of states with known modular Hamiltonians on bifurcate Killing horizons such as the Hartle–Hawking and generalized Unruh states, where analogous entropy–area relations could be established.