Momentum-space entanglement and renormalization in quantum field theory

TL;DR

The paper develops a momentum-space perspective on quantum-field-theoretic entanglement by defining a low-energy density matrix ρ(μ) for modes with |p|<μ and relating it to the Wilsonian action S_W(μ). It introduces and computes momentum-space entanglement observables, notably the entanglement entropy S(μ) and mutual information I between momentum subsets, using perturbation theory in scalar theories φ^n to reveal how entanglement decays as momentum scales separate and how decoupling emerges in this framework. The authors derive explicit leading-order expressions for S(μ) and I in φ^3 and φ^4 theories across dimensions, analyze massless limits, large-μ behavior, and potential divergences, and discuss the range of entanglement in momentum space via single-mode and two-mode information, with implications for holography and tensor-network approaches. The work provides a quantitative bridge between renormalization and quantum information in QFT, offering a framework to quantify how ultraviolet degrees of freedom influence infrared physics and suggesting directions for further links to AdS/CFT and coarse-grained representations like MERA.

Abstract

The degrees of freedom of any interacting quantum field theory are entangled in momentum space. Thus, in the vacuum state, the infrared degrees of freedom are described by a density matrix with an entanglement entropy. We derive a relation between this density matrix and the conventional Wilsonian effective action. We argue that the entanglement entropy of and mutual information between subsets of field theoretic degrees of freedom at different momentum scales are natural observables in quantum field theory and demonstrate how to compute these in perturbation theory. The results may be understood heuristically based on the scale-dependence of the coupling strength and number of degrees of freedom. We measure the rate at which entanglement between degrees of freedom declines as their scales separate and suggest that this decay is related to the property of decoupling in quantum field theory.

Figures (4)

• Figure 1: Leading contributions to $S(\mu)$ for $\phi^3$ theory in 1+1 dimensions. Full result for $S(\mu)$ is proportional to $\lambda^2 ( \log ( 1/ \lambda^2) + 1)$ times bottom function plus $\lambda^2$ times top function.
• Figure 2: ( A) Integration regions for $\phi^3$ theory in 2+1 dimensions. ( B) The function $F(x)$ appearing in the entanglement entropy for $\phi^4$ theory in $1+1$ dimensions.
• Figure 3: Ratio of first and second terms in (\ref{['der']}) vs $\mu = (\mu_2 - \mu_1)/m$ for ( A) $\mu_1 = 1$ and ( B) $\mu_1 = 4$. This is a measure of the range of entanglement in $\phi^4$ theory in $1+1$ dimensions. We have taken the mass to be $m=1$.
• Figure 4: Single-mode entanglement entropy vs magnitude of mode momentum for $\phi^3$ field theory in 1+1 (bottom), 2+1 (middle), and 3+1 (top) dimensions. The entropies are normalized by their values at $p=0$.