Momentum-space entanglement and renormalization in quantum field theory

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 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.

Left-right entanglement entropy of boundary states

“…Our approach generalized the intuition developed in quantum mechanics of tracing over a set of harmonic oscillators for a system of many coupled oscillators. This approach is, perhaps, more akin to recent investigation of momentum entanglement [30][31][32]. However, it is quite natural, in the context of 2d CFT to trace over left-or right-moving modes.…”

Holographic geometry of cMERA for quantum quenches and finite temperature

“…See e.g. [18][19][20][21][22][23][24][25][26] for other interesting developments in this topic. Refer also [27] for another interesting connection between the holographic emergent metric and the renormalization group flow.…”

Holographic mutual information for singular surfaces

“…In order to gain some information about the field content of the corresponding CFT, one needs to probe the theory by means of more powerful entanglement measures. In particular mutual information is such a measure which is defined for two disjoint entangling regions A 1 and A 2 as follows 5) where S A 1 ∪A 2 is the entanglement entropy for the union of two entangling regions.…”

“…We will study crease entangling regions in section 4. 5 In order to clarify the definition of these surfaces consider a d dimensional flat space time, i.e., R 1,d−1 , as follows…”

“…In the context of quantum field theories (QFTs), the notion of entanglement can reflect various features of the theory, depending on how the Hilbert space of the theory is decomposed. This leads to different types of entanglement in QFTs including those which are denoted in the literature by spatial (or geometric) entanglement [1][2][3][4], momentum space entanglement [5], and field space entanglement [6][7][8]. Here we focus on geometric entanglement, which has been widely studied during recent years, in order to extend some recent developments in the context of corner contributions to different entanglement measures.…”