PHYSICAL REVIEW D 98, 025019 (2018) Coarse grained quantum dynamics Cesar Agon,1,2 Vijay Balasubramanian,3,4,5 Skyler Kasko,6 and Albion Lawrence2 1C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York 11794, USA 2Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02454, USA 3David Rittenhouse Laboratory, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 4CUNY Graduate Center, Initiative for the Theoretical Sciences, New York, New York 10016, USA 5Theoretische Natuurkunde, Vrije Universiteit Brussel, and International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium 6Deptartment of Physics, University of California, Santa Barbara, California 93106, USA (Received 8 June 2018; published 26 July 2018) Inspired by holographic Wilsonian renormalization, we consider coarse graining a quantum system divided between short-distance and long-distance degrees of freedom (d.o.f.), coupled via the Hamiltonian. Observations using purely long-distance observables are described by the reduced density matrix that arises from tracing out the short-distance d.o.f. The dynamics of this density matrix is non-Hamiltonian and nonlocal in time, on the order of some short time scale. We describe this dynamics in a model system with a simple hierarchy of energy gaps ΔEUV > ΔEIR, in which the coupling between high- and low-energy d.o.f. is treated to second order in perturbation theory. We then describe the equations of motion under suitable time averaging, reflecting the limited time resolution of actual experiments, and find an expansion of the master equation in powers of ΔEIR=ΔEUV, after the fashion of effective field theory. The failure of the system to be Hamiltonian or even Markovian appears at higher orders in this ratio. We compute the evolution of the density matrix in three specific examples: coupled spins, linearly coupled simple harmonic oscillators, and an interacting scalar quantum field theory. Finally, we argue that the logarithm of the Feynman-Vernon influence functional is the correct analog of the Wilsonian effective action for this problem. DOI: 10.1103/PhysRevD.98.025019 I. INTRODUCTION However, there are many contexts in which one does not do this, or even wish to: Quantum entanglement has emerged as a central concept (1) As argued in Ref. [7], integrating out large Euclidean in the study of the underpinnings of gauge-gravity duality. momenta in a path integral leads to a reduced density The prescription of Ryu and Takayanagi [1,2], and its time- matrix for the IR modes, and the higher-derivative dependent generalization [3], encodes the entanglement terms are precisely the sign of entanglement between entropy between spatial regions in the field theory in the the UV and IR. area of minimal or extremal surfaces in the dual spacetime. (2) The entanglement spectrum of a reduced density Through this, there are good arguments that spatial con- matrix for low-momentum modes can be a useful nectedness in the bulk encodes quantum entanglement of way to characterize the long-wavelength behavior of – disjoint regions on the boundary [4 6]. a lattice theory [8]. On the other hand, the partitioning of a quantum field (3) There is a venerable history of treating “slow” theory according to spatial or spacetime scales is funda- variables (defined in various ways) as an open mental to our physical understanding of quantum field quantum system interacting with “fast modes” to theory, via the renormalization group. In textbook treat- provide a microscopic underpinning of stochastic ments of renormalization one chooses variables so as to and hydrodynamic equations. For classic work see disentangle the “UV” and “IR” degrees of freedom (d.o.f.). Refs. [9–13]. Some recent work (hardly an exhaus- tive list!) includes Refs. [14–18]. (4) Fluctuations of the cosmic microwave background radiation are analyzed by momentum scale, and Published by the American Physical Society under the terms of different momentum modes are entangled [19–25]. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to (5) It is useful to treat the IR region of jets in high- the author(s) and the published article’s title, journal citation, energy particle collisions as an open quantum and DOI. Funded by SCOAP3. system, cf. Ref. [26] and the references therein. 2470-0010=2018=98(2)=025019(22) 025019-1 Published by the American Physical Society AGON, BALASUBRAMANIAN, KASKO, and LAWRENCE PHYS. REV. D 98, 025019 (2018) One important setting where choosing variables to “dis- acts precisely to describe the transfer of these modes entangle” the UVand IR d.o.f. can obscure the physics is in the between the IR and UV regions on time scales of order −1 AdS=CFT correspondence. In this case, there is ample Λ . In other words, they take care of the fact that the IR 1 evidence that spacetime scale in a quantum field theory region comprises an open quantum system. (QFT) is related via gauge-gravity duality to the radial In this work we will study simple quantum systems direction in the dual asymptotically anti–de Sitter (AdS) which capture the spirit of the split between infrared and space [27,28], with the region of anti–de Sitter space close ultraviolet modes seen in quantum field theories. We will to the boundary dual to the UV region of the quantum field focus on theories with a hierarchical structure of energy Δ ≪ Δ theory (i.e., “scale-radius duality”). Various prescriptions levels governed by level splittings EIR EUV. This have emerged for relating the radial evolution of bulk fields structure, the underpinning of the Born-Oppenheimer to the renormalization group flow of the dual field theory approximation, is the basis of Wilson’s pioneering work [29–35]. Of these, the Wilsonian prescription of Refs. [34,35] [39–42], and provides a conceptual underpinning for lends itself most readily to a finite-N generalization [36].In effective field theory [43]. this scheme, the IR and UV regions are clearly entangled [36]. As we will argue, experiments with limited spatial In this paper we will explore such open quantum systems resolution are described by an “IR density matrix,” a from the quantum-mechanical/quantum-field-theoretic reduced density matrix which arises from tracing out point of view, with the eventual aim of shedding light short-distance modes. However, realistic experiments also on scale-radius duality. Before doing this, let us recall the have limited resolution in time, so we will implement a discussion in Ref. [36]. straightforward, physical time-averaging procedure to The AdS=CFT correspondence states that the dual of a d- describe them. We will compute the master equation dimensional large-N conformal field theory [N could be the describing the time evolution of the time-averaged IR rank of a gauge group, or the central charge of a two- density matrix. We will find that the master equation can Δ Δ dimensional conformal field theory (CFT)] is string- or M- be organized in a power series in ð EIR= EUVÞ after the fashion of effective field theory, for which we can begin to theory in AdSdþ1 × X, where X is some space with constant positive curvature. For CFTs on R1;d−1, one considers a identify parallels with the discussion in Ref. [36]. Poincar´e patch of anti–de Sitter space, with coordinates The study of open quantum systems is a well-developed subject (see the reviews [20,44,45], e.g., the treatment of dr2 r2 fast or ultraviolet modes as an environment for slow, ds2 ¼ R2 þ dx2: ð1Þ r2 R2 d infrared modes. Our work contributes an abstract treatment 2 that leads to an effective-field-theory-like expansion of the Here dxd is the flat metric on d-dimensional Minkowski master equation for reduced density matrices of subsys- space, and R is the radius of curvature of AdSd.To tems.2 We focus on this abstract language for two reasons. implement a renormalization group flow after the fashion First, it highlights essential physics—the presence of a of Wilson, the authors of Refs. [34,35] proposed the hierarchy of energy scales. Second, an abstract approach is following. The cutoff Λ is associated with a definite radial 2 best suited to our goal of understanding gauge-gravity coordinate, rΛ ¼ R Λ. One breaks up the path integral over duality, for which the variables that appear in a path- fields propagating on AdSd into modes with r>rΛ and integral approach to the gauge theory have by themselves r<rΛ, interprets the path integral for r<rΛ with fixed no clear dual (to begin with they are not even gauge fields at r ¼ rΛ as the generating functions of correlators in invariant). the cutoff theory, and integrates this over the field values at In the following section, we will embark on our r ¼ rΛ weighted by the path integral over the fields computation of the master equation in perturbation theory for r>rΛ. for a simple quantum system motivated by the essential In this procedure, nontrivial operators are induced at the structure of quantum field theories. After implementing a cutoff even when the theory is an unperturbed conformal field theory [36]. In particular, at a given cutoff Λ, one 1Note that the relationship between this holographic cutoff and induces terms in the Wilsonian action of the form any factorization of the Hilbert space is an open question (see for Z example Ref. [37] for a discussion). In large-N vector models d d dual to higher-spin theories in anti–de Sitter space, the associated ΔSΛ ¼ d xd yγabðx − y; ΛÞOaðxÞObðyÞð2Þ cutoff appears to be a point-splitting cutoff on gauge-invariant bilocal operators [38].