Momentum Space Entanglement from the Wilsonian Effective Action

The entanglement between momentum modes of a quantum field theory at different scales is not as well studied as its counterpart in real space, despite the natural connection with the Wilsonian idea of integrating out the high-momentum degrees of freedom. We push such a connection further by developing a novel method to calculate the Rényi and entanglement entropies between slow and fast modes, which is based on the Wilsonian effective action at a given scale. This procedure is applied to the perturbative regime of some scalar theories, comparing the lowest-order results with those from the literature and interpreting them in terms of Feynman diagrams. Our method is also easily generalized to higher-order or nonperturbative calculationsa and it has the advantage of avoiding the matrix diagonalizations of other techniques. Finally, we also use a quantum information point of view of the renormalization group to derive a remarkable property of theories at a RG fixed point: they have no entanglement between momentum scales. Our results pave the way for further exploring the relation between renormalization and entanglement, including the role played by the latter in determining the phase structure of the underlying theories.