Momentum-space entanglement and renormalization in quantum field theory

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 Wilsonian effective action obtained by integrating out degrees of freedom with spatial momentum above some scale. 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.

Figure 1

Leading contributions to $S(\mu )$ for ${\varphi }^3$ 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 ${\varphi }^3$ theory in $2+1$ dimensions. (B) The function $F(x)$ appearing in the entanglement entropy for ${\varphi }^4$ theory in $1+1$ dimensions.

Figure 3

Ratio of first and second terms in Eq. (53) 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 ${\varphi }^4$ 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 ${\varphi }^3$ field theory in $1+1$ (bottom), $2+1$ (middle) and $3+1$ (top) dimensions. The entropies are normalized by their values at $p=0$.