Constraining Quantum Fields using Modular Theory

TL;DR

This work develops a modular-theory–based framework to constrain quantum fields by exploiting the monotonicity of information-theoretic measures built from the relative modular operator $\Delta^A_{\Omega|\Phi}$. It defines Petz divergences and a Sandwiched Rényi divergence in quantum field theory, proves their monotonicity under region enlargement, and links them to Euclidean correlation functions, enabling new, ultraviolet-finite inequalities for $2n$-point functions. A central conjecture is that the second null derivative of the sandwiched Rényi divergence is non-negative, generalizing the quantum null energy condition to the Rényi family; evidence is provided in free and conformal theories. The framework also suggests potential holographic applications, where monotonicity-based constraints could bound bulk effective field theories and illuminate generalized second laws in out-of-equilibrium settings.

Abstract

Tomita-Takesaki modular theory provides a set of algebraic tools in quantum field theory that is suitable for the study of the information-theoretic properties of states. For every open set in spacetime and choice of two states, the modular theory defines a positive operator known as the relative modular operator that decreases monotonically under restriction to subregions. We study the consequences of this operator monotonicity inequality for correlation functions in quantum field theory. We do so by constructing a one-parameter Renyi family of information-theoretic measures from the relative modular operator that inherit monotonicity by construction and reduce to correlation functions in special cases. In the case of finite quantum systems, this Renyi family is the sandwiched Renyi divergence and we obtain a new simple proof of its monotonicity. Its monotonicity implies a class of constraints on correlation functions in quantum field theory, only a small set of which were known to us. We explore these inequalities for free fields and conformal field theory. We conjecture that the second null derivative of Renyi divergence is non-negative which is a generalization of the quantum null energy condition to the Renyi family.

Figures (3)

• Figure 1: The Euclidean path integrals that prepare (a) excited state $|\Phi\rangle$, (b) vacuum density matrix on region $A$, (c) the density matrix of $A$ in the excited state. (d) The operator $\omega^{\frac{1-n}{2n}}\phi\omega^{\frac{1-n}{2n}}$ corresponds to a path-integral on a wedge of angular size $2\pi/n$. (e) The correlator that appears in the definition of the sandwiched Rényi divergence.
• Figure 2: (a) The first derivative in shape deformation of the sandwiched Rényi divergence for $n=4$. The commutator is represented with the charge written as an integral on the codimension one surface encircling the pair $\Phi_0^\dagger\Phi_0$. (b) Same commutator with the charges written on a different surface, (c) A term that appears at the second order in deformation which is reflection symmetric around $\theta=\pi/2$. Note that $P^{(1)}$ is the same as $P$ but rotated by $\pi/2$.
• Figure 3: (top) The first and the second spatial derivatives of the sandwiched Rényi divergence for $n=2$ to $n=26$ are non-negative and consistent with monotonicity and our proof. (bottom) The third and the fourth spatial derivatives are plotted for $n=2$ to $n=16$. For large enough $n$, the fourth derivative becomes negative.