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
025019-2 COARSE GRAINED QUANTUM DYNAMICS PHYS. REV. D 98, 025019 (2018) physical time averaging, we will see a Born-Oppenheimer- defined on some lattice with spacing a. Now consider a type expansion emerge, in which non-Hamiltonian and measuring device which directly couples to ϕ, but has finite non-Markovian dynamics appear starting at second order in resolution in space and time. That is, if we write the field ϕ 1 Δ = EUV. We provide a simple expression for the time in the Schrödinger picture as evolution of R´enyi entropies of the IR density matrix. We will work two simple examples—a coupled spin system Xπ=a 1 and the Caldeira-Leggett model [51] at zero temperature— ϕðxÞ¼ pffiffiffiffiffiffiffiffiffiffiffiffi a eik·x þ H:c:; ð4Þ 2 πω k to see how this expansion plays out, and then describe the k¼−π=a ðkÞ master equation for scalar quantum field theories with † cubic interactions. Such master equations for low-spatial- then our measuring devices couple to ak, ak for Λ ≪ π momentum modes have been computed for four-dimen- jkj < a, and record the time of the measurement with sional theories via the influence functional approach [19], temporal accuracy δt. and our work further contributes a Born-Oppenheimer-like The Hilbert space can be broken up into framework and a time-averaging procedure for understand- ing the effects of finite time resolution. H H ⊗ H ¼ IR UV ð5Þ In the conclusions we will draw what lessons we can for gauge-gravity duality, relating the appearance of non- H † Λ H where IR is generated by ak for jkj < and UV is Hamiltonian and non-Markovian terms to nonlocal terms † generated by a for jkj > Λ. Note that we are not directly that emerge [36] in the Wilsonian approach of Refs. [34,35]. k breaking up the Hilbert space according to energy scale. We will argue that the natural framework for understanding First, for an interacting theory, spatial momentum and this work would be to use the results of Refs. [34,35] to energy will not be directly related. Second, we may be compute an influence functional [52,53] for the IR modes, interested in high-energy objects made up of many low- and note further that the logarithm of the influence functional energy quanta. After all, the physics of the sun is well is the natural extension to the Wilsonian effective action to described by the standard model cutoff at a TeV, even understanding finite-time processes. We will then provide 1054 3 some further speculations regarding the use of these results though its total mass is of order GeV. We imagine an experiment of the following form. Begin for understanding gauge-gravity duality. with the system in its exact ground state j0i, and act on it In the appendices we review some concepts which may O be unfamiliar for some of our audience (while being bread with some infrared operator IR. Let the resulting state and butter to others). First, we address a common confusion evolve in time, we encountered when discussing this work: textbook −i ψ ℏHtO 0 treatments of renormalization do not consider entanglement j ðtÞi ¼ e IRj i: ð6Þ between UV and IR modes. We review various approaches to renormalization of quantum field theories to explain Now compute the probability of measuring the IR d.o.f. in ∈ H how, in those cases, these modes are disentangled. Next, we some state jai IR. We are not making any measure- H discuss some issues with states with initial entanglement. ments in UV, so we should sum the probabilities over all H We then discuss an obstruction to computing the von possible final states in UV. The result is Neumann entropy for the IR density matrix, in perturbation X −i 2 ℏHtO 0 theory. Finally, we review the path-integral approach to Pða; tÞ¼ jhujhaje IRj ij computing the dynamics of density matrices, and in jui∈HUV particular the Feynman-Vernon influence functional. −i † i ¼ trP e ℏHtO j0ih0jO eℏHt Since the first version of this work appeared on the arXiv, a IR IR P ρ a number of interesting papers on related subjects have ¼ trHIR a IRðtÞð7Þ appeared, including Refs. [24–26,49,50,54–56]. Further, the results in this paper were further built on by two of us where Pa ¼jaihaj, and in Ref. [57]. −i i ρ ℏHt ψ 0 ψ 0 ℏHt IRðtÞ¼trHUV ½e j ð Þih ð Þje : ð8Þ II. DYNAMICS OF ρIR IN PERTURBATION THEORY More generally, the expectation value at time t of mea- H ρ surements of AIR acting on IR is hAi¼trA IRðtÞ. Based A. Motivation ρ on this we take IR to be the object of interest. Consider an interacting scalar field theory 3 1 1 1 λ One may, however, wish to restrict the Hilbert space to states 2 2 2 2 4 density ΛD D H ¼ πϕ þ ð∂ϕÞ þ m ϕ þ ϕ ð3Þ with low energy of order , where is the spacetime 2 2 2 4 dimension.
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Energy simplification. In general, we expect a local system to have a nested hierarchy of energy levels, as shown in Fig. 1, corresponding to different momentum modes of the fundamental fields. That being said, at this order of perturbation theory, we will find that we can also compute the master equation for cubic scalar quantum field theories. (3) Factorized initial states. Following much of the literature on open quantum systems, we will con-
ΔE sider initial states for which the UV and IR d.o.f. are UV ρ 0 not entangled, so that IRð Þ is a pure state. These have a master equation which is local in time. This is of course not the most general situation—small Δ excitations of the ground state such as Eq. (6) will E IR in general be highly entangled between the UV and FIG. 1. Cartoon of band structure: large jumps correspond to IR—but it can arise in interesting physical situations. UV quanta, small jumps to IR quanta. For example, if we prepare the state by measuring the IR with a nondegenerate Hermitian operator, the state will collapse to a product state. Another well- B. Setup studied situation is the “interaction quench,” in Wilson emphasized [40,42] that the energy spectrum for which λ is suddenly turned on at t ¼ 0. The quantum field theories has a hierarchical structure, as entanglement of spatial regions after a quench has illustrated in Fig. 1. In order to focus on the effects of been well studied, beginning with the pioneering this structure, we work with a simpler abstract model that work of Refs. [59–61]. captures it. That is, we consider a Hilbert space with product structure C. Perturbative calculation The calculation of the master equation for ρ ðtÞ can be H ¼ H ⊗ H ð9Þ IR IR UV done by, e.g., projection operator techniques (see Ref. [45]). To be self-contained, we will rederive results and a Hamiltonian of the form from the theory of open quantum systems in a manner ⊗ 1 1 ⊗ λ consistent with our approximations and perspective. H ¼ HIR þ HUV þ V ð10Þ Ψ 0 ψ We consider j ð Þi¼j IRijui,whereu labels eigen- states of H ; u is some particular state, possibly but H H UV where V acts on both IR and UV. not necessarily the ground state, while jψ i is taken to “ ” ρ IR Our goal is to compute the master equation for IRðtÞ, H be some arbitrary state in IR. In general, when the that is, the right-hand side of the expression initial state of a coupled system and environment is factorized between the two, the reduced density matrix ℏ∂ ρ L ρ i t IR ¼ ð Þm ð11Þ of the system satisfies a master equation which is localintime(seeRefs.[45,62,63] for discussion and L ρ ρ where, ð Þ represents a differential operator on IR, which references): depends on the full ρ through the initial conditions. ℏ∂ ρ ρ ρ γ ρ Recognizing that measurements in the IR theory will i t IRðtÞ¼½HeffðtÞ; IRðtÞ þ ifAðtÞ; IRðtÞg þ ½ IRðtÞ typically have limited time resolution, we will also deter- ≡ ρ Γ ρ mine the time evolution of a version of ρ that is coarse ½Heff; IRðtÞ þ ½ IRðtÞ : ð12Þ grained by time averaging. Γ labels the non-Hamiltonian part of the master equation We will assume the following: ρ (1) The interaction λV can be treated perturbatively. for IR, with Specifically we will work to order λ2. In perturbation X 1 † A t − h t L t L t ; theory starting from Fock space there is a tight ð Þ¼ 2 klð Þ l ð Þ kð Þ connection between momentum and energy. Note X k that perturbation theory for the density matrix itself γ ρ ρ † ½ IR ¼i hklðtÞLkðtÞ IRðtÞLl ðtÞ; ð13Þ can fail at long times, due to secular terms in the k perturbative expansion [58]. (2) The eigenvalue spacing for HUV, HIR can be where Lk are some set of operators that can depend on Δ ≫ Δ characterized by scales EUV EIR. This is a the initial state of the UV d.o.f. but act on the IR, and
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4 ρ σ 0 σ 0 † 0 hkl is a Hermitian matrix. Here k, l are arbitrary indices ðtÞ¼TrUV ðtÞ¼TrUVUðt; Þ ð ÞU ðt; Þ L X that index the k; they do not have to have any 0 † 0 particular relation to the UV or IR Hilbert spaces. ¼ hujUðt; ÞjuijIRihIRjhujU ðt; Þjui This is almost the Kossakowski-Lindblad equation for Xu † Markovian dynamics [64,65].However,Markovian ¼ hujUðt; 0Þjuiρð0ÞhujU ðt; 0Þjuið17Þ dynamics requires that the eigenvalues of h be positive, u and this condition is well known to fail in general.5 where u and IR are the UVand IR parts of the factorized Thus Eq. (12) can be non-Markovian even if it looks j i j i initial state, u is a basis for the UV Hilbert space with one local in time, because there can be history dependence j i basis element being u , and ρ 0 IR IR is the initial hidden in the operators A and γ diagnosed by the j i ð Þ¼j ih j reduced density matrix for the IR modes. So the Kraus breakdown of positive definiteness of h [45,62,63]. ij operators can be taken to be We will construct Eq. (12) to second order in perturba- tion theory, using the fact that the finite-time evolution of 0 ρ ρ Ku ¼hujUðt; Þjuið18Þ IR, simply denoted as hereafter, has a Kraus representa- tion (see for example Refs. [62,63]). Thus with u indexing UV d.o.f., and jui being the UV part of the X initial state. Thus the Kraus operators are treated as ρ ≡ ρ ρ 0 † IRðtÞ ðtÞ¼ KαðtÞ ð ÞKαðtÞð14Þ functions of u that are initial-state dependent, but we have α suppressed the state dependence in our notation. We can rewrite the master equation (12) and (13) in terms in terms of certain operators Kα that can be derived from the of the Kraus operators. First, we can perturbatively expand time evolution.6 In our example, the density matrix σðtÞ ≡ Ψ Ψ H H ρ ρð0Þ λρð1Þ λ2ρð2Þ j ðtÞih ðtÞj in the full Hilbert space IR × UV satisfies ðtÞ¼ ðtÞþ ðtÞþ ðtÞþ ; unitary evolution: λ ð1Þ λ2 ð2Þ Heff ¼ HIR þ Heff þ Heff þ ; λ ð1Þ λ2 ð2Þ σðtÞ¼UðtÞσð0ÞU†ðtÞ; ð15Þ A ¼ A þ A þ ; γ ¼ λγð1Þ þ λ2γð2Þ þ ð19Þ where Here ρð0Þ is the density matrix of the initial IR state evolved R t in time by the IR Hamiltonian HIR. The master equation for − −iλ dt0V ðt0Þ UðtÞ¼e iðHIRþHUVÞtTe 0 I ; ð16Þ time evolution can be determined by representing ρðtÞ in the Kraus representation and expanding Ku in the pertur- bation. We find to Oðλ2Þ VI is the perturbation in the interaction picture, and T is the time-ordering operator. The time evolution of ρðtÞ in our K ¼ exp f−iðH þ iAÞtg; case is u X eff † γ ¼ i∂t KuðtÞρð0ÞKuðtÞ; ð20Þ 4Such state dependence is not usually discussed in Wilsonian u≠u renormalization: in particle physics examples, one usually as- sumes that the UV theory is in the ground state. More generally, where Ku is a partial matrix element for transitions between the Born-Oppenheimer discussion in Appendix A shows that an initial UV state u to itself; γ controls transitions out of u even the effective Hamiltonian (A6) depends on the state of the UV modes. into other UV states, while A describes the associated loss 5 ⊗ H In the Kossakowski-Lindblad equation, following from the of unitarity in the subspace jui IR. The sum in the assumption that the time evolution of ρ is described by a expression for γ runs over the part of the UV Hilbert space completely positive dynamical semigroup, h, L are time that is orthogonal to the initial state jui. independent. However, a more general definition of Marko- Since K ≠ is nonvanishing only at OðλÞ and higher, vian behavior includes divisible dynamical maps [66],in u u γ λ2γð2Þ which hk, Lk can be time dependent, but the eigenvalues of h ¼ þ The form of the time-local master equation remain positive. (and direct computation) also shows that A ¼ λ2Að2Þ þ 6 ρ 0 → The Kraus representation guarantees that the map ð Þ Thus, to order OðλÞ we simply get a correction to the ρðtÞ is “completely positive.” This representation is possible when the initial state is disentangled between the IR and the UV. effective Hamiltonian in the master equation: For intermediate times t0, the state will be entangled, and the map 1 from ρðt0Þ → ρðtÞ will not be completely positive. This may Hð Þ ¼hujVjui: ð21Þ include t0 arbitrarily close to t, as diagnosed by the nonpositivity eff of the eigenvalues of h; this nonpositivity for infinitesimal time 2 ð2Þ ð2Þ evolution means that the evolution will not be Markovian, as At order Oðλ Þ, A ; γ can be written in the form (13) by entanglement has been generated. choosing the indices k, l to each run over the composite
025019-5 AGON, BALASUBRAMANIAN, KASKO, and LAWRENCE PHYS. REV. D 98, 025019 (2018) index k; l ¼ um with the index u ≠ u running over a basis Expression in terms of UV correlation functions: There for the part of the UV Hilbert space which is orthogonal to is an elegant expression for the various terms in the master the initial state, and m ∈ f1; 2g. With this notation, we can equation in terms of correlation functions of UV operators. write the operators on the right-hand side of Eq. (13) that Let us write γ define A and as X λV ¼ ΦaOa; ð27Þ Lu1 ¼hujVjui; ð22Þ Z a t 0 0 − Lu2 ¼ dt hujVIðt tÞjui; ð23Þ O Φ 0 where a, a are sets of UV and IR operators respectively. Using the path-integral formalism in Appendix D, and where V and VI are the interactions in the Schrodinger and defining the connected two-point function interaction pictures respectively. The effective Hamiltonian and other operators that govern the master equation at this W 0 00 OI 0 OI 00 − OI 0 OI 00 Gabðt ;t Þ¼h aðt Þ bðt Þi h aðt Þih bðt Þi; ð28Þ order can be written in terms of Lum: X ð2Þ i † † OI τ OI τ ρ 0 ρ 0 7 H − h 1 2 L L 2 − L L 1 ; where h ð Þi ¼ TrUV ð Þ UVð Þ with UVð Þ¼juihuj, eff ¼ 2 u ;u ð u1 u u2 u Þ ð24Þ u≠u we find that Eq. (13) can be written in terms of the following set of operators8: 1 X ð2Þ † † A − 1 2 2 1 ¼ 2 hu ;u ðLu1Lu þ Lu2Lu Þ; ð25Þ ≠ 2 0 u u h ; 0 λ δ m − m ; X am bm ¼ abj j γð2Þ ρð0Þ † ρð0Þ † Φ 0 ¼ i hu1;u2ðLu1 Lu2 þ Lu2 sLu1Þ; ð26Þ La1 ¼ að Þ; ≠ Z u u t τ W τ Φ τ − La2 ¼ d Gabðt; Þ bð tÞ; ð29Þ 0 0 ð0Þ 0 where hum;u0m0 ¼ δðu; u Þjm − m j and ρ ðtÞ is the density matrix of the initial IR state evolved in time by the IR Hamiltonian HIR. and Z X t ð2Þ τ W τ ΦI τ − ΦI 0 − W τ ΦI 0 ΦI τ − H ¼ i d ðGabð ;tÞ að tÞ bð Þ Gabðt; Þ að Þ bð tÞÞ; 0 ab Z X t ð2Þ τ W τ ΦI τ − ΦI 0 W τ ΦI 0 ΦI τ − A ¼ d ðGabð ;tÞ að tÞ bð ÞþGabðt; Þ að Þ bð tÞÞ; 0 ab Z X t γð2Þ τ W τ ΦI 0 ρð0Þ ΦI τ − W τ ΦI τ − ρð0Þ ΦI 0 ¼ i d ðGabð ;tÞ bð Þ ðtÞ að tÞþGabðt; Þ bð tÞ ðtÞ að ÞÞ: ð30Þ 0 ab Non-Markovianity: A natural question is whether the evolution, packaged in this form, is Markovian. It is well known to those who study coarse-grained quantum systems that it is not, in general. Assume the particularly simple case that V 0 Γ commutes with HIR but not with HUV (when ½V;HUV ¼ , vanishes). A short calculation of the non-Hamiltonian part of the master equation gives X † 1 † h 0 L ρ t L − L L 0 ; ρ um;um um ð Þ um0 2 f um um g u≠u;m;m0 X 2 sinðE tÞ 1 ¼ uu hujVjuiρðtÞhujV†jui − fhujV†juihujVjui; ρðtÞg ð31Þ E 2 u≠u uu
− ≥ Δ ≠ where Euu ¼ Eu Eu, and Eu is the energy of jui with respect to HUV. In general, if Euu EUV for u u, the sine term Δ will lead to oscillations at the scale EUV. Since the matrix elements in the sum are therefore not positive definite, we know on general grounds [62,63] that the evolution is not Markovian.
7The super index I refers to the fact that those operators are described in the interaction picture of quantum mechanics. 8This framework generalizes easily to include an arbitrary initial density matrix for the UV d.o.f., again assuming the density matrix for the full system is factorized between UV and IR at t ¼ 0.
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− Evolution of UV-IR entanglement: One of the motiva- 1 − e iðEuuþEijÞt tions for this work is to understand the structure of hijLu;2jji¼ hijhujVjuijjið37Þ iðEuu þ EijÞ entanglement between the UV and IR. It is straightforward to see that entanglement evolves precisely because of the where Eij ¼ Ei − Ej. The first term will, in general, survive non-Hamiltonian part of the evolution. For reasons we time averaging. discuss in Appendix C (see also Ref. [7]), the von Neumann We find by construction that to second order in pertur- entropies are difficult to compute in perturbation theory, but bation theory in V, the time-averaged evolution equation9 the R´enyi entropies for ρ takes the form ln TrρnðtÞ S ðtÞ¼− ð32Þ i∂ ρðtÞ¼½H ; ρðtÞ þ ifA; ρðtÞg þ γðtÞ: ð39Þ n n − 1 t eff are easily seen to satisfy the equation The time-averaged operators are most easily written in the basis of eigenstates of HIR, and are ρn−1 Γ dSnðtÞ in Tr½ ðtÞ ðtÞ X X ¼ : ð33Þ 1 hu; ijVju; ji dt n − 1 Trρn t ð2Þ − ð Þ H ¼ 2 hujVjuijiihjj ≠ ðEuu þ EijÞ u u ij hu; ijVju; ji D. Time averaging þ jiihjjhujVjui ; ð40Þ ðEuu − EijÞ Realistic apparatuses have limited accuracy in specifying X X the time that a given measurement takes place. To find the 1 hu; ijVju; ji Að2Þ ¼ − hujVjuijiihjj probability of a given outcome, one should average Pða; tÞ, 2 ≠ ij ðEuu þ EijÞ the probability of outcome a at a time t, over a time interval u u τ hu; ijVju; ji determined by an appropriate window function fδtð ;tÞ − jiihjjhujVjui ; ð41Þ where t is the peak of the window function and δt is the ðEuu − EijÞ width. A typical example is a Gaussian X X 2 hu; ijVju; ji 0 1 γð Þ ¼ jiihjjρð ÞhujVjui −ðτ−tÞ2=δt2 f δ ðτ − tÞ¼pffiffiffi e : ð34Þ ≠ ij ðEuu þ EijÞ g; t πδt u u hu; ijVju; ji − hujVjuiρð0Þjiihjj : ð42Þ With this normalization, the sum of Eq. (7) over all possible ðEuu − EijÞ orthogonal outcomes (a) is equal to 1. Given a time- dependent function FðtÞ, we denote the time average as Effective theories via Born-Oppenheimer expansion: Z These operators can be written in the form (13) and (26). Let us choose k; l ¼ um with ðu ≠ u, m ∈ f1; 2gÞ, F t dτfδ τ;t F τ : ð Þ¼ tð Þ ð Þ ð35Þ 0 0 and hum;u0m0 ¼ δðu; u Þjm − m j. Then define In the case of the Gaussian window function, this expres- 9Up to second order in perturbation theory we find that sion can be written in Fourier space as Eq. (39) is valid, namely, the time average of the operator Z products appearing in the master equation equals the product of dω 2 2 FðtÞ¼ pffiffiffiffiffiffi e−ω δt =4eiωtF˜ ðωÞ: ð36Þ their time averages. Of course, this is not generally the case. For 2π example, the time average of a product of functions has the closed expression As expected, there is a sharp exponential cutoff X∞ 2 ¯ ¯ −1 ¯ δt n dnFðtÞ dnGðtÞ for ω > δt . FðtÞGðtÞ¼F¯ ðtÞG¯ ðtÞþ ; ð38Þ 2nn! dtn dtn In applying this averaging to Eq. (12), we will consider n¼1 Δ ≪ δ −1 ≡ ≪ Δ EIR t Ec EUV. Thus we will throw away Δ when a Gaussian window function is considered. If the time terms in Eq. (12) which have frequencies of Oð EUVÞ as variation of F, G is slow compared to δt, with characteristic these will be exponentially suppressed after time averaging. frequency Ω, then the average of the product is the product of the O Δ k O Ωδ 2 We will, however, keep terms of order ½ð EIR=EcÞ . averages up to corrections of order ðð tÞ Þ. However, when We now wish to compute a master equation for the time- the functions F, G both have fast oscillatory behaviors charac- ω averaged density matrix ρðtÞ The nontrivial time depend- terized by a frequency those corrections can add up to an exponentially large factor and then the leading term on the right- ence of the terms in Eq. (12) arises from Lu;2 in Eq. (23).If hand side of Eq. (38) will not be a good approximation to its left- iωt we study a matrix element of L2 in the basis jii of IR hand side. Consider for example the case F ¼ e F0 and G ¼ −iωt eigenstates with energies Ei, we find that e G0 where F0 and G0 are constants.
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Lu;1 ¼hujVjui; E. Examples X We will work through two simple quantum-mechanical − jiihijhujVjuijjihjj Lu;2 ¼ i : ð43Þ examples capturing our hierarchy of energy levels, in order Eu;u þ Eij ij to build up our intuition for the different possible dynamics of ρðtÞ. In the first example of coupled spins, the non- It is then easy to show that Eqs. (12) and (13) reduce to Hamiltonian contributions will vanish upon time averaging. Eq. (39) with the operators defined as in Eqs. (40)–(42).We The second example is the well-studied case of coupled can write Lu;2 in a more basis-independent form by linear oscillators [51]; we will work with a different expanding the denominator in a power series in spectrum and quantum state for the “bath,” highlighting ðEij=EuuÞ and noting that EijjiiOijhjj¼½HIR; jiiOijhjj : the differences between our results and those in Ref. [51]. Finally, we will give the time-averaged master equation for zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{k times X∞ … … a scalar QFT with cubic self-couplings, which to second hujVjui ½ ½Vu;HIR ; ;HIR L 2 ¼ −i − i ð44Þ order in perturbation theory can be computed with the u; E kþ1 uu k¼1 Euu formulas given. This problem makes contact with the H holographic setting of Ref. [36]. where Vu ¼hujVjui is an operator acting on IR. The expansion in IR operators of increasingly high dimension weighted by inverse powers of EUV is what we would 1. Coupled spins expect from a good effective field theory, and is a central First consider an IR spin coupled to k ¼ 1 M UV consequence of the hierarchical nature of the spectrum of spins, all in the 2j þ 1-dimensional irreducible representa- 2 H the full (unreduced) theory. tion of SUð Þ with spin j. Thus the Hilbert space is IR ¼ Leading-order approximation: The master equation to H , H ¼⊕ H . We take the Hamiltonian to be Δ Δ jIR UV k jUV;k the leading order in EIR= EUV is z † † † H ¼ −μ BS ; X V V 1 X V V ;H − V ;H V IR IR IR ð2Þ − u u − u½ u IR ½ u IR u H ¼ 2 XM E 2 E z u≠u uu u≠u ð uuÞ H − μ BS ; UV ¼ UV;k UV;k 1 k¼1 O þ 3 ; XM Euu λ λ⃗ ⃗ V ¼ SIR · SUV;k; ð46Þ X † 1 1 ½VuV ;H 1 k¼ Að2Þ ¼ − u IR þ O ; 2 E2 E3 u≠u uu uu where S⃗are the usual spin operators, satisfying S ;S ½ i j ¼ X 0 † 0 † ℏϵ V ;H ρð ÞV V ρð Þ V ;H i ijkSk and B is a fixed constant (a magnetic field). We γð2Þ ½ u IR u þ u ½ u IR ¼ 2 μ ≪ μ take IR UV, so that this system has the hierarchical ≠ Euu u u structure of energy levels we discussed above. 1 We can rewrite the interaction term as þ O : ð45Þ E3 uu X λ X X λV ¼ λ Sz Sz þ S− Sþ þ Sþ S− To leading order in 1=E , the evolution of ρ is completely IR UV;k 2 IR UV;k IR UV;k uu k k k Hamiltonian. This is consistent with our discussion of the Born-Oppenheimer approximation in Appendix A3, and ð47Þ with the results of Refs. [46,47]. Our results display the where S ¼ S iS are the raising and lowering oper- kind of decoupling that occurs in Wilsonian renormaliza- x y ators in the basis of S eigenstates. We write states in the tion: the effects of transitions to excited states of the UV Q z basis jj ;m i jj m i where j is the total angular d.o.f. are suppressed by powers of 1=E . IR IR k UV;k UV;k UV m S One may ask whether the time averaging we have momentum and is the eigenvalue of z. It is straightfor- H H H λV implemented leads to a Markovian master equation. ward to see that the ground state of ¼ IR þ UV þ is independent of λ to all orders in perturbation theory: Once again, this will not happen beyond the leading order Y in EIR=EUV for which the evolution is Hamiltonian. If we j0i¼jj ;m¼ j i jj ;m¼ j i: ð48Þ ∈ R IR IR UV UV consider the restricted case ½HIR;Vu ¼auVu, au ,we k 2 find Að Þ ¼ γð2Þ ¼ 0, so that the evolution is not only Thus, it is natural to consider an initial state of the form Y Markovian but Hamiltonian. Outside of this approximation, − ψ 0 jIR m 0 O 1 2 j ð Þi ¼ CmðS−;IRÞ j i¼jjIR;mi jjUV;m¼ jUVi Lu2 at order ð =EuuÞ is not proportional to Lu1, so that the k negative eigenvalue of hui;uj will contribute to Eq. (13).To go further we must examine more specific cases. ð49Þ
025019-8 COARSE GRAINED QUANTUM DYNAMICS PHYS. REV. D 98, 025019 (2018) which results from perturbing the ground state by an action 2. Linear oscillators − − of the operator ðS Þj m. H IR Following Refs. [44,51], we consider IR as the Hilbert The terms in the non-time-averaged equation of motion H space of a simple harmonic oscillator, and UV as a bath of (12) to second order are harmonic oscillators, with the Hamiltonian comprising a 2 2 2 linear coupling between them: X X 2λ ℏ g j H ¼ H − λℏ g j Sz − k k eff IR k k IR ðμ − μ ÞB P2 1 k k UV;k IR Ω2 2 HIR ¼ þ M X ; 1 − μ − μ − þ 2M 2 × ð cos ½ð UV;k IRÞBt ÞSIR;SIR; X p2 1 X λ2ℏ2g2j H ¼ k þ m ω2x2 ; Að2Þ − k k sin μ − μ Bt S− Sþ ; UV 2m 2 k k k ¼ 2 μ − μ ½ð UV;k IRÞ IR IR k k k ð UV;k IRÞB X λV ¼ C x X; ð54Þ X 2iλ2ℏ2g2j k k γð2Þ k k μ − μ k ¼ μ − μ sin ½ð UV;k IRÞBt k ð UV;k IRÞB where we take Cl ∼ OðλÞ. Of course, this can be solved þ ρð0Þ − × SIR ðtÞSIR; ð50Þ exactly by a change of variables. However, in the spirit of this paper we are interested in the dynamics of the “bare” variable ρð0Þ where is the density matrix of the initial IR state X, to which we imagine our measuring devices couple. evolved in time by the IR Hamiltonian HIR. In order to match what we expect from a quantum field γ A, can be written in the form (13) if the indices are theory calculation such as that outlined in the Introduction, expanded to ðkmÞ where k labels the UV oscillators and we will take the frequencies ω ≫ Ω. Furthermore, we will m ∈ f1; 2g. With this notation, k assume that the UV oscillators are in an eigenstate of HUV 0 (such as the ground state) at leading order in perturbation h 0 ¼ δ jm − m j; km;lm klrffiffiffiffi theory. The resulting system then differs from those studied jk þ in Refs. [44,51]. Those works considered the oscillators xk L 1 ¼ ℏ S ; k 2 IR to be some “environment,” with a spectrum designed n o h i iðμUV;k−μIRÞBt ðμUV;k−μIRÞBt phenomenologically to model quantum Brownian motion pffiffiffiffiffiffiffi exp − 2 sin 2 þ or dissipation. The environment contains oscillators with L 2 ℏ 2j S : k ¼ k μ − μ IR ð UV;k IRÞB arbitrarily low frequency, to model dissipation of energy and phase coherence into an environment, over time scales ð51Þ long compared to a given experiment. Furthermore, we are In this example, the non-Hamiltonian terms Að2Þ; γð2Þ are most interested in the UV oscillators that are initially in rapidly oscillating, and vanish after time averaging. For their ground state; thus, our treatment is closest to the zero- completeness we compute the effective Hamiltonian for the temperature limit of Refs. [44,51]. In this case, some time-averaged equation at Oðλ2Þ: approximations made in those works fail. Finally, we implement time averaging differently, by directly averaging X the density matrix over a coarse-graining kernel. The net H H − λℏ g j Sz eff ¼ IR k k IR result is a qualitatively different master equation for the k 2 X 2 density matrix. 2λ ℏj g 0 − k S− Sþ O λ3 As stated, we assume thatQ at t ¼ , the UVoscillators are in μ − μ IR IR þ ð Þ B k k IR an energyeigenstatejui¼ ljnli.Inthiscase,thefirst-order z z 2 shift of the Hamiltonian vanishes, because the expectation ¼ −μ˜BS þ βðS Þ − Eg ð52Þ IR IR value of xl vanishes in energy eigenstates of the harmonic – where oscillator. Using Eqs. (40) (42) we find that the operators in the second-order time-averaged equation (39) are ℏλ X ℏ2λ2 X 2 μ˜ μ − − gkjk ¼ IR gkjk ; 1 X 2 1 2 ℏ B B μ − μ 2 ð nl þ ÞCl 2 k k k IR Hð Þ − X − ; ¼ 2 ω2 − Ω2 2 ω ℏλ2 X 2 l mlð l Þ M l gkjk β ¼ ; 1 X 2 1 2 B μ − μ 2 ð nl þ ÞCl k k IR Að Þ − X; P ; ¼ 2 2 ω ω2 − Ω2 f g ℏ3 1 X 2 l Mml lð l Þ − jðj þ Þ gkjk Eg ¼ : ð53Þ X 2 1 2 B μ − μ 2 ð nl þ ÞCl 0 0 k k IR γð Þ i Pρð Þ t X Xρð Þ t P : ¼ 2 ω ω2 − Ω2 ð ð Þ þ ð Þ Þ l Mml lð l Þ Heff is related to HIR by renormalization of the magnetic moment, coupling β, and vacuum energy. ð55Þ
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At this order, the Hamiltonian is changed by a shift in the proportional to X in L2. In general, their master equation is oscillator frequency and the ground-state energy. We can also not Markovian, unless we were to take the limit rewrite A, γ in the form (13) if we let m runfrom1to2and kT → ∞, ηkT finite (cf. Ref. [45]). define As discussed in Ref. [57], this model captures some essential features of local quantum field theories, if we X 2 ð2nl þ 1ÞCl choose CðωÞ appropriately. In particular, there can be h12 h21 ; ¼ ¼ 2 ω2 − Ω2 divergences when the number of states grows sufficiently l mlð l Þ rapidly with energy. h11 ¼ h22 ¼ 0; ð56Þ 3. Scalar QFT with cubic self-coupling L1 ¼ X; At second order in perturbation theory, it is straightfor- P − ward to apply our formulas to scalar quantum field theories. L2 ¼ i X þ i ω : ð57Þ M l We give a brief description here of the cubic theory in d spatial dimensions. A fuller account of our computation, Thus the eigenvalues of the h matrix are h12.Aswe ij and an interpretation of the resulting divergences, can be explained before, the lack of positive definiteness implies that found in Ref. [57]. Here our goal is to demonstrate features the evolution of ρ is not Markovian beyond the leading order of the master equation also found in a holographic context in Ω=ωl. in Ref. [36]. Before continuing, it is worth comparing the form of our Consider the Lagrangian master equation to that of Caldeira and Leggett [51]. This arises when 1 1 L ∂ϕ 2 − 2ϕ2 − g ϕ3 (1) xl describes a continuous spectrum of oscillators, ¼ 2 ð Þ 2 m 3! : ð62Þ for which X Z The master equation for the quartic theory in four dimen- 2 2 ClfðωlÞ → dωρðωÞCðωÞ fðωÞ; ð58Þ sions was computed in Ref. [19], via the Feynman-Vernon l influence functional. We wish to consider Eq. (62) in light of our more abstract conceptualization; in addition, with and ml ¼ m, with our Hamiltonian regulator, we will find some dimension- dependent issues that would be absent in four dimensions. 2mηω2 ρC2ðωÞ¼ θðΛ − ωÞð59Þ The scalar field in the interaction picture can be π decomposed as follows: Λ η Z where is some UV cutoff, and is a phenom- d d k ⃗ ⃗ ϕ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ik·x⃗ † −ik·x⃗ enologically determined coefficient. ðx; tÞ¼ faIRðkÞe þ aIRðkÞe g ⃗ Λ 2π d2ω (2) Furthermore, the oscillators xl are placed at finite jkj< ð Þ ðkÞ Z temperature T ≫ Λ. ddk In this case they derived a master equation [Eq. (5.12) in þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ⃗ Λ 2π d2ω Ref. [51]] which can be rewritten in the form (12)–(13) with M>jkj> ð Þ ðkÞ the indices i; j ∈ f1; 2g and10 ik⃗·x⃗ † −ik⃗·x⃗ × faUVðkÞe þ aUVðkÞe g ϕ ϕ ηΛ ¼ IRðx; tÞþ UVðx; tÞð63Þ h12 ¼ h21 ¼ ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ℏ 2 where ωðkÞ¼ k⃗ þ m2, Λ is the spatial coarse-graining L1 ¼ X; ð60Þ scale, and M is the cutoff. The dual conjugate momentum is ℏ 2 Z kT d L2 ¼ −i X − P þ X; ð61Þ d kð−iωðkÞÞ ⃗ † ⃗ 2MΛ Λ πðx; tÞ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fa ðkÞeik·x⃗− a ðkÞe−ik·x⃗g d IR IR jk⃗j<Λ ð2πÞ 2ωðkÞ η ω Λ Z where is a function of Cð Þ and the UV cutoff, and is a ddkð−iωðkÞÞ UV energy scale that accounts for the frequency renorm- þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ⃗ Λ 2π d2ω alization. Note the relative factor of −i in the coefficient of M>jkj> ð Þ ðkÞ P, as well as the additional temperature-dependent term ik⃗·x⃗− † −ik⃗·x⃗ × faUVðkÞe aUVðkÞe g π π ¼ IRðx; tÞþ UVðx; tÞ: ð64Þ 10In fact the master equation in Ref. [51], which yields the operators L we report, is missing a term of order kT=Λ; this term i The normalized single-particle momentum eigenstates are is argued to be small even in the kT ≫ Λ limit. We discussed this pffiffiffiffiffiffiffiffi 2ω † 0 † δd − 0 further in Ref. [57]. jki¼ kakj i, where ½ak;ak0 ¼ ðk k Þ.
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H There is thus a decomposition of the Hilbert space (1) Creation of a single particle in UV: This means that ϕ2 the relevant components of IR will be two nearly H H ⊗ H H ¼ IR UV: ð65Þ collinear particles in IR with total momentum ⃗ ⃗ ⃗ k1;IR þ k2;IR ¼ kUV. Both IR and UV momenta must We can split the Hamiltonian accordingly into H ¼ HIR þ have magnitudes close to the scale Λ that splits IR HUV þ gV where from UV. (2) Creation of two particles in H : The matrix 1 1 g UV H ∂ϕ 2 − m2ϕ2 − ϕ3 ; elements that contribute will have two excitations IR;UV ¼ 2 ð IR;UVÞ 2 IR;UV 3! IR;UV in H with nearly back-to-back momenta that sum g g UV ϕ ϕ2 ϕ2 ϕ to a momentum with magnitude below Λ. There is a gV ¼ 2 IR UV þ 2 IR UV: ð66Þ much larger set of possibilities: almost any magni- In essence, each oscillator with UV momentum acts as a tude of UV momentum will be allowed, and for each separate harmonic oscillator. value k there will be a sphere in phase space of ∼ d−1 We will take the initial state to be of the form volume k of possible UV momenta. Ψ ψ 0 0 The importance of each type of term depends on the IR j i¼j iIRj iUV, where j iUV is the vacuum with respect momenta, and on the number of dimensions. For low to HIR. With gV defined as above there are two classes of matrix elements that contribute to Eqs. (30) and (40)–(42), enough IR momenta, only the second type of term can γ contribute. For simplicity, we will focus on this possibility. i.e., to Heff, A and that control the time evolution of the IR density matrix. A straightforward application of our formalism yields
Z Z t d d 0 ð2Þ i d k d k 0 0 H ðtÞ¼− dτ ½h0jVjkk ihkk jVIð−τÞj0i − H:c: ; 4 0 2ω 2ω 0 Z Zuv k k 1 t d d 0 ð2Þ d k d k 0 0 A ¼ − dτ ½h0jVjkk ihkk jVIð−τÞj0iþH:c: ; 4 0 2ω 2ω 0 Z Z uv k k t d d 0 ð2Þ i d k d k 0 ð0Þ 0 γ ¼ dτ ½h0jVjkk iρ hkk jVIð−τÞj0iþH:c: : ð67Þ 2 0 uv 2ωk 2ωk0 The integral of the time-dependent matrix element is Z Z Z t d t d x − 0 − ω ω τ τ 0 −τ 0 λ iðkþk Þ·x τϕ −τ ið kþ k0 Þ d uvhkk jVð Þj iuv ¼ e d irð Þe 0 ð2πÞd 0 Z Z d d iðω −ω −ω 0 Þt † −i ω ω ω t d x d p a ð1 − e p k k Þ a− 1 − e ð pþ kþ k0 Þ − λ −iðkþk0Þ·x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p pð Þ ipx ¼ i d e þ e : ð2πÞ ir d ωk þ ωk0 − ωp ωk þ ωk0 þ ωp ð2πÞ 2ωp ð68Þ
It is clear that H, A, γ will have time dependence on scales of order the UV momenta, which range from Λ to M. Next, let us consider the time-averaged quantities Hð2Þ and Að2Þ. This amounts to dropping the rapidly oscillating exponential terms in Eq. (68) to find Z Z Z Z t d d 0 d d d k d k 2 d x d p † τ 0 0 0 −τ 0 → − λ ϕ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − ipx d h jVjkk ihkk jVð Þj i i d irðxÞ ðap þ a pÞe 0 uv 2ωk 2ωk0 ð2πÞ ir d ð2πÞ 2ωp Z d d k ωk þ ωp−k × 2ω 2 2 uv k 2ωp−k½ðωk þ ωp−kÞ − ωp Z Z d d ω 2 d x d pi p † − λ ϕ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − − ipx d irðxÞ ða p apÞe ð2πÞ ir d ð2πÞ 2ωp Z ddk 1 × 2ω 2 2 : ð69Þ uv k 2ωp−k½ðωk þ ωp−kÞ − ωp
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Note that the second and third lines, proportional to −iλ2, Again, it is clear that the additional terms (higher-order in ⃗2 2 ⃗2 2 will contribute to H, while the fourth and fifth lines, ∇ =M , ∇ =Λ ) lead to spatial nonlocality on scales of 2 proportional to −λ , will contribute to A, γ. order Λ, M. The terms of the form fπ; ϕg are clearly Let us first examine the correction to the Hamiltonian. If analogous to the fX; Pg terms which appear in Að2Þ for the ⃗ ω ≪ Λ ≪ jpj; p M, we can expand the integral linearly coupled oscillator. Z We can see from Eqs. (71) and (74) that we also have d ω ω d k k þ p−k new divergences in high enough dimension. In prior studies 2ω 2 2 uv k 2ωp−k½ðωk þ ωp−kÞ − ωp of quartic scalar field theory [19], the non-Hamiltonian terms in the master equation (derived from the influence p⃗2 Λ 1 d functional) had no divergences. There are two differences ¼ FðM; Þ þ h1 Λ2 þ ð70Þ here. We look at more general spacetime dimensions, and we adopt a spatial regulator appropriate to our Hamiltonian where treatment, after the fashion of Ref. [67]. This was discussed 8 in more depth in Ref. [57]. > d−3 <> Λ d<3; The upshot is that we have non-Markovian behavior, M indicating the development of entanglement between the F ¼ F0 lnðΛÞ d ¼ 3; ð71Þ > UV and IR, and nonlocality on the order of the cutoff Λ. : d−3 3 M d> ; This is precisely the structure hinted at in Ref. [36].
d and F0, h1 are dimensionless constants. The neglected III. DISCUSSION AND CONCLUSIONS terms include both higher orders in p⃗2 as well as terms suppressed by powers of ðΛ=MÞ2. These terms give A. Relation to the holographic renormalization group corrections to the Hamiltonian of the form The results in Ref. [36] indicate that the “IR region” of an Z AdS geometry, i.e., the interior region far from the hd 2 spacetime boundary, functions as an open quantum system. ð2Þ ∝ λ2 d Λ ϕ2 1 π∇⃗ ϕ H d xFðM; Þ 0 þ Λ2 þ ð72Þ However, following Refs. [34,35], this work computed the Feynman path integral assuming vacuum boundary con- ⃗2 2 ditions in the far past and far future, integrating out d.o.f. where the dots indicate terms of higher order in ∇ =M , beyond some radial position. As we note in Appendix A, ⃗2 2 ∇ =Λ . It is thus clear that we will get terms with spatial this only makes sense if we have foreknowledge of the nonlocalities on scales Λ, M. Note that for d ≥ 3, we find d.o.f. we are integrating out. Such foreknowledge for divergent contributions to the mass, consistent with stan- interacting systems makes sense if either dard treatments of scalar QFT. 2 (1) we are working with renormalized variables in Next consider corrections to Að Þ. In this case, which the Hamiltonian is block diagonal between Z low and high energies, and wish to only measure ddk 1 these redefined variables in the low-energy Hilbert 2ω 2 2 uv k 2ωp−k½ðωk þ ωp−kÞ − ωp space, or (2) we are computing scattering amplitudes for asymp- p⃗2 G M; Λ 1 h˜ d totic states of well-separated particles. ¼ ð Þ þ 1 Λ2 þ ð73Þ The supergravity modes inside a radial cutoff in AdS=CFT clearly do not correspond to the renormalized variables in where point 1. As for point 2, in global AdS coordinates there are 8 no good well-separated asymptotic states in AdS, as all > Λd−4 4 <> d< ; excitations oscillate on time scales of order RAdS. More M generally, the analog of S-matrix elements, for which an G ¼ G0 lnðΛÞ d ¼ 4; ð74Þ > Lehmann-Symanzik-Zimmermann (LSZ)-type reduction : d−3 4 M d> ; applies, are correlators of local CFT operators, dual to non-normalizable modes supported near the AdS boundary ˜ d and G0, h1 are dimensionless coefficients. These will lead [68–70]. to corrections of the form Nonetheless, we can already learn something from Z Ref. [36]. The first point is that the holographic ˜ d 2 2 h1 ⃗2 Wilsonian action is nonlocal in time. Thus, this action Að Þ ∝ λ ddxG M; Λ ϕπ ϕ∇ π : : : ð Þ þ Λ2 þ H c þ cannot describe purely Hamiltonian dynamics, exactly as expected for an open quantum system. The time scale ð75Þ which describes mixing between the IR and the UV is of
025019-12 COARSE GRAINED QUANTUM DYNAMICS PHYS. REV. D 98, 025019 (2018) order the cutoff. This is exactly what we found for the the IR, the correct object to coarse grain is not a transition density matrix dynamics in Sec. II. The non-time-averaged amplitude but the density matrix for the system. The master equation has oscillations at time scales of order computation of the density matrix in path-integral language Δ EUV, which become time independent upon time averag- goes back to Feynman and Vernon [52]. Let us state the ing. Similarly, UV-IR entanglement evolves on time scales essential formulas here; the derivation, and a perturbation Δ of order EUV. theory calculation relevant for Sec. II C, can be found in As we will discuss below, for open quantum systems the Appendix D. correct analog of the Wilsonian action is the logarithm of We assume following Ref. [52] that the density matrix at the Feynman-Vernon influence functional, computed via time t ¼ 0 is factorized between the IR and the UV: that is, path-integral techniques as we describe in Appendix B. there is no initial entanglement. Let capital letters X, Y Applying this approach to the AdS=CFT correspondence denote quantum-mechanical variables describing the IR, with a bulk radial cutoff faces the challenge because the and lowercase letters x, y denote quantum variables gauge theory interpretation of such a cutoff remains unclear describing the UV. We assume the action can be written as [36–38]. However, in the limit that we can study small quantum fluctuations in anti–de Sitter space, it would be of _ _ δ great interest to compute the dynamics of a density matrix S½X; x ¼SIR½X; X þSUV½x; x þ S½x; X : ð76Þ for scalar fields supported in the region r Z ρ 0 0 ; 0 0 0 ρ 0 0 0 ðX; Y; tÞ¼ dX dY JðX; Y; t X Y ; Þ initðX ;Y ; Þ; Z YðtÞ¼Y;XðtÞ¼X i −i JðX; Y; t; X0Y0; 0Þ¼ DXDYeℏSIR½X ℏSIR½Y F½XðtÞ;YðtÞ ; ð77Þ Yð0Þ¼Y0;Xð0Þ¼X0 ρ where init is the initial density matrix for the IR d.o.f., and the influence functional Z 0 0 0 0 F½XðtÞ;YðtÞ ¼ dx dy dxρUðx ;y; 0Þ Z yðtÞ¼x;xðtÞ¼x i −i i δ −iδ × DxDyeℏSU½x ℏSU½y þℏ S½x;X ℏ S½y;Y ð78Þ yð0Þ¼y0;xð0Þ¼x0 contains the dependence on the initial state of the UV d.o.f., this cutoff makes less sense, and indeed the authors of as well as the interactions between the UV and IR d.o.f. In Refs. [16,19,20,48] coarse grained with respect to spatial general this cannot be written in the form F½X G½Y . The momenta. However, in most physical processes we will influence functional encodes the same data as the terms also have finite accuracy in determining the times at which γ Heff;A; in the master equation (see Appendix D). we prepare and measure the system, and temporal coarse The point of stating these well-known results is to graining is required. It would be of great interest to find a emphasize that the correct analog of the Wilsonian effective simple path-integral implementation of the time averaging action, in the case that the IR and UVare entangled and one that we discussed in this paper. is asking questions about finite-time processes, is the influence functional. This point has been made eloquently C. Additional questions in a number of papers, including Refs. [19,20,48]. Strongly coupled systems: In using the phrase “coarse However, this approach has not been applied to holographic graining” in our perturbative treatment, we implied that we Wilsonian renormalization. It should be. could assign an energy scale to the d.o.f. we were tracing The Wilsonian approach to renormalizing the path out, corresponding to a short distance scale. This makes integral is based on the Euclidean path integral, and the sense in weakly coupled quantum field theories, in which coarse graining is over Euclidean space (that is, the cutoff the energy and momenta of single quanta are tied together is placed on Euclidean momenta). In a real-time context and there is some meaning to these single quanta. In 025019-13 AGON, BALASUBRAMANIAN, KASKO, and LAWRENCE PHYS. REV. D 98, 025019 (2018) strongly coupled systems, especially those without quasi- In holographic theories, black holes are dual to high- particle excitations, this relation breaks down. It would be energy states with thermal behavior. In closed quantum interesting to study our coarse graining in such examples, systems, the eigenstate thermalization hypothesis [74,75] either analytically or numerically. states that a class of quantum operators will have expect- Cosmological perturbations: Primordial non- ation values and correlation functions which appear to be Gaussianities in cosmic microwave background fluctua- thermal. In many examples, these are local operators tions and large-scale structure measure correlations supported in a spatial subregion of the system, and the between quantum fluctuations at different scales, induced excited quantum state is strongly entangled between the by interactions in the inflaton sector. We expect the initial subregion and its complement so that the reduced density state to have a degree of quantum entanglement between matrix looks approximately thermal (see Ref. [76] for a scales, following the discussions in this paper, which recent discussion, and further references.) Of course, this is should help seed the classical correlations one actually not the only way to decompose the Hilbert space such that observes. the state is entangled between the components. The Some discussion of entanglement between scales during stretched horizon appears at some radius in the AdS–black inflation appears in Ref. [23]. In this work, the entangle- hole geometry, whose value should be dual to some scale in ment between short- and long-wavelength modes is used to the field theory dynamics. Studying entanglement between justify a Lindblad equation describing Markovian evolution d.o.f. at different scales could shed light on this system.11 for the long-wavelength modes, based on an argument that the Hubble scale sets a natural time scale for the decay of ACKNOWLEDGMENTS correlations of short-wavelength modes. It would be interesting to perform a more quantitative, first-principles We thank Alberto Guijosa, Matthew Headrick, Robert analysis of entanglement between scales in some specific Konick, Sergei Khlebnikov, Sung-Sik Lee, Hong Liu, Emil model, following the discussion here. For example, as we Martinec, Greg Moore, Vadim Oganesyan, Anatoli have noted, even when correlation functions are essentially Polkovnikov, and Stephen Shenker for useful questions, local in time, the dynamics of long-wavelength modes can conversations, and (in some cases) encouragement. We still fail to be Markovian. thank Bei-Lok Hu for pointing us to Ref. [58], and Holographic renormalization: In our setup, the evolution Vladimir Rosenhaus for pointing out some mistakes in ρ Δ equation for is local on scales larger than EUV. This is in the first draft of this work. This project arose from accord with the discussion of holographic gauge theories in discussions during the KITP workshop on “Bits, Branes, Refs. [34–36,38], in which the Wilsonian effective action of and Black Holes.” During this time it was therefore a strongly coupled field theory was nonlocal on the scale of supported in part by the National Science Foundation the cutoff, reflecting the propagation of excitations into and under Grant No. NSF PHY11-25915. Much of the work back out of the UV region. However, string theory suggests on this paper was done at the 2014 Aspen Center for that there are other nongravitational theories in which the Physics (ACP) workshops “New Perspectives in time scale over which excitations are supported in the UV Thermalization” and “Emergent Spacetime in String becomes arbitrarily large. One example is little string theory: Theory”; the A. C. P. is supported by National Science in the holographic dual, massless excitations propagating Foundation Grant No. PHY-1066293. A. L. and C. A. are into the UV region take an infinite time to reach the supported in part by DOE Grant No. DE-SC0009987; C. A. “boundary.” This is tied to the exponential (Hagedorn) was also supported in part by the National Science growth of states at high energies in this theory. It would Foundation via CAREER Grant No. PHY10-53842 ρ be interesting to explore the dynamics of IR in this setting. awarded to M. Headrick. V. B. was supported by DOE More generally we would like a more precise under- Grant No. DE-FG02-05ER-41367. standing of the relationship between the framework in this paper and that of Wilsonian renormalization in holographic gauge theories, in which one “integrates out” a section of the APPENDIX A: RELATION TO AND geometry [30,34–36]. For example, this could cast an DIFFERENCES FROM WILSONIAN interesting light on black hole entropy. There is evidence RENORMALIZATION that bulk quantum corrections to the entanglement entropy of Textbook treatments of Wilsonian renormalization quantum fields between the interior and exterior of a explicitly disentangle IR and UV d.o.f. via a change of “stretched horizon” outside the black hole are mapped to variables. This point of view is important for computing the the Wald entropy of the black hole, using the renormalized low-energy spectrum and the S matrix of asymptotic states gravitational action (see Refs. [71,72] and references therein); and there are conjectures that the full Bekenstein- 11On a related note, Ref. [71] studied the progressive con- Hawking/Waldentropyoftheblackholecanbeconsideredas tribution of longer and longer wavelengths of bulk fields to the an entanglement entropy (see for example Ref. [73]). black hole entanglement entropy. 025019-14 COARSE GRAINED QUANTUM DYNAMICS PHYS. REV. D 98, 025019 (2018) with low energies and long wavelengths. Below we contrast include terms describable as a renormalized Hamiltonian, this approach with the work in this paper. together with a nontrivial “influence functional” which encodes the time development of entanglement between the 1. Path-integral approach IR and UV d.o.f. We discuss this in Appendix D. A further issue arises from the fact that higher-derivative The standard discussion of Wilsonian renormalization interactions are generically induced. These will include (cf. Refs. [41,77]) begins with a Euclidean path integral ̈ terms that are functions of ϕs and higher derivatives still, Z arising from nonlocalities on the scale of the running −SΛ ½ϕ 12 Z ¼ DϕðxÞe 0 ðA1Þ cutoff. In general, the Wilsonian action SΛ will not have an interpretation as an action that can be derived from the Legendre transform of a Hamiltonian, unless one adds for a field theory with UV cutoff Λ0. One breaks up ϕðxÞ into ϕ ¼ ϕ þ ϕ , where the “slow” fields ϕ are supported Stückelberg fields. Such a procedure amounts to adding the s f s short-distance d.o.f. back in. The interpretation of these on Euclidean momenta jkj < Λ and the “fast” fields ϕ are f higher-derivative terms is clear in the holographic picture of supported on Euclidean momenta Λ < k < Λ0. We coarse j j Wilsonian renormalization [36]: they reflect the fact that the grain the theory by integrating over ϕ to find f IR d.o.f. comprise an open quantum system, and that there Z are memory effects on the time scale of the UV dynamics. −SΛ½ϕs Z ¼ Dϕse : ðA2Þ 2. Hamiltonian approach For long-wavelength questions, we can work with this There is an alternative literature on Hamiltonian latter presentation. approaches to renormalization, pioneered originally by When Z represents the partition function in a classical Wilson [39,40], and applied first to a model of pion- equilibrium statistical physics problem, the interpretation is nucleon scattering and later to the Kondo problem, imple- clear: SΛ will represent the spatially coarse-grained mented by a successive diagonalization of d.o.f. with a classical Hamiltonian of the system. If we want to compute hierarchy of energy scales. In these models the d.o.f. of equal-time correlators in the analytical continuation to real some quantum field are coupled through a localized defect. time, we need to impose periodic boundary conditions in At each step one diagonalizes the Hamiltonian of the high- the integral over the high frequencies, while fixing the IR energy d.o.f. coupled to the defect, and works in the ground modes on two sides of a cut in time [7]. This yields an states of these d.o.f. This diagonalization mixes the (iso) effective action and an associated density matrix that can be spin states of the defect with excitations of the high-energy used to compute equal time correlation functions. But when modes of the quantum field: at each step, the low-energy Z represents the quantum-mechanical vacuum-vacuum spin d.o.f. becomes more delocalized. transition amplitude computed via Euclidean continuation, Variants for interacting quantum fields (without a defect) the decimation procedure above fixes both the initial and can be found in, for example, Refs. [42,78]. The authors of final state of the short-wavelength d.o.f. For the inclusive Ref. [42] removed divergences by making a transformation finite-time probabilities discussed in the Introduction, to “band-diagonal” form in which the Hamiltonian has no this procedure is not appropriate, beyond the leading order matrix elements between states with an energy difference in a Born-Oppenheimer approximation. Note that when larger than some value. The authors of Ref. [78] imple- describing scattering of initially well-separated particle mented partial diagonalization of the Hamiltonian by states into final states of the same form, the interactions removing only the matrix elements between the IR band are effectively inoperative at early and late times, and the and the high-energy d.o.f. assumption that the short-distance modes are in their As in Wilson’s work, the goal of the Hamiltonian ground state is essentially correct. This standard treatment approach to renormalization is to extract the spectrum. is designed to produce transition amplitudes between initial To do so, we reorganize the theory in terms of effective low- and final asymptotic states where only the low-energy energy d.o.f. where the original low-frequency components modes are excited. By contrast, we are interested in finite- of the Hilbert space are appropriately dressed by the high- time questions for states that have UV-IR entanglement, frequency components so as to partially diagonalize the including the natural ground states of interacting theories, Hamiltonian between the UV and the IR. If one is studying and states produced by the action of coarse-grained thermodynamics at low temperatures, or the dynamics operators. of quasiparticles built from the renormalized variables, For the inclusive finite-time questions we are discussing the low-energy Hilbert space can be treated as a closed in this paper, the decimation procedure is best applied to quantum system. Similarly, such an approach is also spatial momenta, in the real-time path integral developed by Feynman and Vernon [52] for density matrices. In this 12A related discussion, which partially inspired this paper, can case, the analog of the Wilsonian effective action will be found in Ref. [7]. 025019-15 AGON, BALASUBRAMANIAN, KASKO, and LAWRENCE PHYS. REV. D 98, 025019 (2018) 0 appropriate for S-matrix elements of well-separated par- variable through the dependence in HYðx Þ. Meanwhile 0 “ ” ticles. In this case, the initial state lies in the low-energy jx ix indicates a state in the slow Hilbert space indexed by Hilbert space, and the final state will as well. In this way, if the slowly changing value x0. one studies the scattering into final states of well-separated To lowest order in the Born-Oppenheimer approxi- particles (or low-energy bound states), one can work mation ΔEx=ΔEY ≪ 1, the leading n ¼ 0 term satisfies entirely within the closed quantum system of the low- the time-dependent Schrödinger equation with effective energy Hilbert space. Hamiltonian The calculations we describe in this paper, however, assume that measuring devices couple to the bare variables Heff ¼ Hx þ HBerry þ E0ðxÞðA6Þ with some finite spatial resolution, and that measurements ’ are made at finite time. In this setting, low- and high- where HBerry are the additional terms induced by Berry s momentum modes cannot be easily separated, and thus the phase [43,80,81].13 In this approximation, the reduced measurable d.o.f. form an open quantum system. density matrix for the slow d.o.f. can be written as Z ρ 0 00ψ 0 ψ 00 0 0; 0 0 00 00 3. Born-Oppenheimer approximation IR ¼ dx dx 0ðx Þ 0ðx ÞtrY½jx ixj x iYYh ;x jxhx j : Our treatment of long-wavelength modes is closest to the A7 Born-Oppenheimer approximation. In textbook form ð Þ [43,79], one separates the quantum-mechanical d.o.f. into This describes a mixed state if Fðx0;x00Þ¼ “fast variables” Y and “slow variables” x. To implement the tr ½jx0; 0i hx00; 0j is not factorizable in x0 and x00. approximation, one considers the case that eigenstates of x Y YY Nonetheless, its evolution is unitary, with Hamiltonian form a (possibly overcomplete) basis of the Hilbert space H , in this approximation. The failure of unitarity—that described by the slow d.o.f., and considers Hamiltonians of eff is, the status of the IR d.o.f. as an open quantum system— the form will appear at higher orders in the Born-Oppenheimer approximation, for finite-time processes. This includes H ¼ H þ H ðxÞ: ðA3Þ x Y processes like recoil of the heavy d.o.f. This has been dis- cussed in the classical limit of the IR d.o.f. in Refs. [46,47]; For example, x could be the positions of heavy nuclei, and corrections to the leading adiabatic limit lead to friction and Y the positions of electrons moving in the backgrounds of dissipation. these nuclei. This framework is essentially what we desire. However, Consider x to take some frozen value and treat it as a we are interested in the more general case of systems for background field. Then the Hilbert space of the “fast” d.o.f. which the coupling between IR and UV d.o.f. cannot be can be written in eigenstates n; x j i simply expressed in terms of an IR operator which can be diagonalized. An example of this is the Hamiltonian for two H x n; x E x n; x : A4 Yð Þj i¼ nð Þj i ð Þ coupled spins, Δ Δ Let Ex, EYðxÞ be the gap between eigenvalues of Hx, −μ z − μ z λ⃗ ⃗ H ¼ LBSL HBSH þ SL · SH ðA8Þ HYðxÞ. The simplest version of the Born-Oppenheimer Δ ∼ − ≫ Δ approximation works when EY EnðxÞ EmðxÞ Ex, ⃗2 2 where μL ≪ μH, and the total spin SL=ℏ ¼ jLðjL þ 1Þ is for all x where the wave function of the slow d.o.f. has ⃗ appreciable support. Let E0ðxÞ be the instantaneous ground not too large. There is no basis which diagonalizes SL.On ≫ 1 state. One can write the general wave function as the other hand, for jL , or for long-wavelength modes Z in a spin chain, there is a semiclassical limit in which the spin can be treated as a semiclassical variable. Ψ 0ψ 0 0 0; 0 j ðtÞi ¼ dx ðx ;tÞjx ixj x iY Z X APPENDIX B: ENTANGLED INITIAL STATES 0δψ 0 ; 0 þ dx nðx; tÞjx ixjn x iY; ðA5Þ n>0 When the initial state is entangled between the UV and IR, the evolution of the IR density matrix is harder to where the subscripts x and Y on the kets indicate states in characterize. (See Sec. 4 of Ref. [63] for a preliminary the“slow” and “fast” Hilbert spaces labeled by the indicated discussion of this case.) Let us consider the specific initial quantum numbers, while ψ and δψ n are the weights of the state linear combination defining the full state. A state like 0 “ ” jn; x iY indicates that the fast modes Y are in an energy 13In the case that there are N near-degenerate eigenstates with 0 “ ” eigenstate of HYðx Þ with quantum number n. This fast energies close to E0, ψ 0 is replaced by an N-component wave eigenstate depends on the“frozen” value x0 of the slow function, with a non-Abelian UðNÞ Berry phase [82]. 025019-16 COARSE GRAINED QUANTUM DYNAMICS PHYS. REV. D 98, 025019 (2018) 1 This can be difficult to compute in practice. A simpler set of jΨð0Þi ¼ pffiffiffi ðjχiju1iþjζiju2iÞ ðB1Þ 2 quantities to calculate are the R´enyi entropies for ρðtÞ: n where u1 2 are eigenstates of H with eigenvalues E1 2, ρ j ; i UV ; − ln Tr ðtÞ χ ζ H SnðtÞ¼ : ðC2Þ and j i; j i are states in IR which we will take to be n − 1 linearly independent. The initial density matrix is If the resulting expression yields a smooth n → 1 limit, one 1 may use these to compute the von Neumann entropy. ρð0Þ¼ ðjχihχjþjζihζjÞ: ðB2Þ 2 We must take some care when computing S in pertur- bation theory, due to the logarithm. If the unperturbed As we will see, the complication will arise because each density matrix has zero eigenvalues and the perturbation is term will evolve differently, in a fashion dependent on the sufficiently generic, we expect the full density matrix to λ UVeigenstates they are coupled to. At zeroth order in , the have eigenvalues that scale as λp. Thus, there will be terms density matrix is simply that scale as λp ln λ in the von Neumann entropy, and perturbation theory will break down: this fact was dis- 0 1 − ρð Þ iHIRt χ χ ζ ζ iHIRt cussed in Ref. [7]. While the R´enyi entropies for fixed ðtÞ¼2 e ðj ih jþj ih jÞe integer n>1 can have good analytic expansions in λ,itis 1 χ χ ζ ζ straightforward to see that the λ → 0, n → 1 limits will not ¼ ðj ðtÞIih ðtÞIjþj ðtÞIih ðtÞIjÞ ðB3Þ 2 commute. For a simple example, consider the density − 0 matrix ψ iHIRt ψ 0 ρð Þ where j ðtÞIi¼e j ð Þi. ðtÞ evolves by ℏ∂ ρð0Þ ρð0Þ Hamiltonian evolution, i t ðtÞ¼½HIR; ðtÞ . 1 − aλ 0 At first order in λ, a calculation identical to those of ρ ¼ ; ðC3Þ 0 λ Sec. II yields a ℏ∂ ρð1Þ ρð1Þ for which i t ðtÞ¼½HIR; ðtÞ V11; χ t χ t V22; ζ t ζ t 1 þ½ j ð ÞIih ð ÞIjþ½ j ð ÞIih ð ÞIj n n Sn ¼ − ln½ð1 − aλÞ þðaλÞ Þ −iE21t − 1 þ½V12e ; jζðtÞ ihχðtÞ j n I I 1 iE21t ðn−1Þ lnð1−aλÞ ðn−1Þ lnðaλÞ þ½V21e ; jχðtÞ ihζðtÞ j ðB4Þ ¼ ln½ð1 − aλÞe þðaλÞe Þ : I I n − 1 − ∼ ðC4Þ where Vij ¼huijVjuji, and E21 ¼ E2 E1 EUV. There is no obvious sense in which the evolution is Markovian. The time averaging of the first-order evolution Note that in the cases we are studying, these entropies equation (B4) is straightforward: we simply drop the final capture both the degree to which the initial IR density 1 matrix is in a mixed state, as well as any entanglement that line, which oscillates rapidly at a time scale of order =EUV. The resulting equation is arises from time evolution of the coupled system. Therefore, the most interesting question for us is the 1 1 evolution of these quantities with time. Focusing on the ℏ∂ ρ χ χ ζ ζ i t ðtÞ¼½Heff;1; 2 j ðtÞih ðtÞj þ ½Heff;2; 2 j ðtÞih ðtÞj R´enyi entropies with integer n (so that we are sure to work with well-defined quantities), we find ðB5Þ dS ðtÞ in Tr½ρn−1ðtÞi∂ ρðtÞ where n ¼ t : ðC5Þ dt n − 1 TrρnðtÞ Heff;i ¼ HIR þhuijVjuii: ðB6Þ If we insert Eq. (12), the contributions from Heff will This is not a Hamiltonian evolution. vanish, due to the cyclicity of the trace, so that dS ðtÞ in Tr½ρn−1ðtÞΓðtÞ APPENDIX C: UV-IR ENTANGLEMENT n ¼ : ðC6Þ dt n − 1 TrρnðtÞ Having computed the density matrix, we can ask how entangled the systems become with time. The most robust Thus we see that the non-Hamiltonian components of the quantity to compute is the von Neumann entropy time evolution specified by Γ in Eq. (12) are precisely responsible for producing UV-IR entanglement as time − ρ ρ SðtÞ¼ Tr ðtÞ ln ðtÞ: ðC1Þ passes. 025019-17 AGON, BALASUBRAMANIAN, KASKO, and LAWRENCE PHYS. REV. D 98, 025019 (2018) Let us focus on the particular case that the initial IR state forward in time. The density matrix should express the is an energy eigenstate jii of the IR Hamiltonian, and work probability that the final IR state is jψi. This can be to Oðλ2Þ. Since Γ is nonvanishing only at Oðλ2Þ, we can written as ρ evolve ðtÞ with HIR alone, and it will remain pure. X −1 † Therefore we can replace ρn → ρ with n>1. Then, Pψ ¼ hujhψjUðt; 0Þj0ih0jUðt; 0Þ jψijui: ðD1Þ using our known expressions for Γ, we find u X ω ψ 0 0 dS ðtÞ 2n sin uu;i jt The amplitude hujh jUðt; Þj i can be expressed as a path n ¼ jhu; ijVju; jij2; 0 dt n − 1 ω integral with the boundary conditions at times ;t inte- u≠u;j≠i uu;i j grated against the wave function for j0i; hujhψj. Similarly, Z † t dS ðt0Þ the amplitude h0jUðt; 0Þ jψijui would be represented as SðtÞ¼ dt0 n 0 dt the complex conjugate of this path integral: when the system enjoys time reversal invariance, this can be 2n X 1 − cos ω t ¼ uu;i j jhu; ijVju; jij2; described in terms of paths propagating backwards in time. n − 1 ω2 u≠u;j≠i uu;i j The result is a path integral over paths moving forward then 2n X 1 backwards in time, with the UV d.o.f. at t set equal and SðtÞ¼ jhu; ijVju; jij2: ðC7Þ summed over. This is reminiscent of the Schwinger- n − 1 ω2 u≠u;j≠i uu;i j Keldysh formalism for “in-in” expectation values [85–88]; indeed, if one was to sum the above expression The R´enyi entropies thus vary on the time scale of the UV over all final states, one would arrive at the path-integral ≥ 0 d.o.f. Note that SnðtÞ always; thus the time average is expression for such expectation values. nonvanishing and also time independent (because the To make this discussion more explicit, consider a system oscillations are at the UV time scale and the IR state is H for which IR describes the states of a particle with position an eigenstate of the unperturbed Hamiltonian). H X, and UV the states of a particle with position x.We consider an action of the form APPENDIX D: PATH-INTEGRAL FORMALISM _ The time evolution of the reduced density matrix has a S½x; X ¼SU½x;_ x þSI½X; X þδS½x; X ðD2Þ path-integral formalism going back to Feynman and Vernon [52,53], which points to an avenue for a systematic where we write the interaction δS in the form computation of higher-order corrections. Related results Z X t appear in the literature (see Ref. [45] for a discussion and 0 UV IR δS½x; X ¼−λ dt Oa Φa references). 0 Many readers may be familiar with this formalism in the a Z X t context of quantum Brownian motion [51] and quantum 0 ¼ −λ dt Oa½x Φa½X : ðD3Þ dissipation [44,83,84]; the coupled oscillator model (54) is a a 0 classic example to which this formalism has been applied. As we discussed in Sec. III D 2,the“bath” of oscillators xl in We are assuming the interaction term depends on the these models contains a continuum of oscillators down to low coordinates only and not on the velocities; thus the frequencies, with a spectrum designed so that energy is correction to the action is minus the correction to the dissipated into the bath without returning to the observed Hamiltonian. We have factored out a small dimensionless system over the lifetime of the experiment. Furthermore the parameter λ ≪ 1 to better organize a perturbative treatment bath is typically taken to be at finite temperature. In our of the system. As in the previous section, we also assume discussion, the “bath” consists of d.o.f. with high frequencies, that the initial state of the system can be described by a which are generally in the ground state at leading order in λ. factorized density matrix 1. Review of the influence functional σ ; ; 0 ρ ;0 ρ ;0 ðx; X y; Y t ¼ Þ¼ IRðX; Y Þ UVðx; y Þ: ðD4Þ We wish to compute the density matrix starting with some known state j0i of the full system that evolves The density matrix for the IR d.o.f. X is Z ρ 0 0 ; 0 0 0 ρ 0 0 0 ðX; Y; tÞ¼ dX dY JðX; Y; t X Y ; Þ IRðX ;Y ; Þ; Z YðtÞ¼Y;XðtÞ¼X i −i JðX; Y; t; X0Y0; 0Þ¼ DXDYeℏSI ½X ℏSI ½Y F½XðtÞ;YðtÞ ; ðD5Þ Yð0Þ¼Y0;Xð0Þ¼X0 025019-18 COARSE GRAINED QUANTUM DYNAMICS PHYS. REV. D 98, 025019 (2018) ρ where IR is the initial density matrix for the IR d.o.f., and the influence functional Z F 0 0 ρ 0 0 0 ½XðtÞ;YðtÞ ¼ dx dy dx UVðx ;y; Þ Z yðtÞ¼x;xðtÞ¼x i −i i δ −iδ × DxDyeℏSU½x ℏSU½y þℏ S½x;X ℏ S½y;Y ðD6Þ yð0Þ¼y0;xð0Þ¼x0 contains the dependence on the initial state of the UV d.o.f., PW ðt0;t00Þ¼Tr OI ðt0ÞOI ðt00Þρ ð0Þ; ðD13Þ as well as the interactions between the UVand IR d.o.f. We a;b UV a b UV F O λ2 will now pass to computing to order ð Þ. and T˜ denotes time antiordering. Note that the time-ordered two-point function, the antitime-ordered two-point func- 2. Perturbation theory for the influence functional tion, and the Wightman function appear for essentially the same reason that they do in the Schwinger-Keldysh We expand Eq. (D6) to second order in δS½x; X − formalism for in-in expectation values. Finally, note that δS½y; Y to find all of the operators should be understood as being in the ð0Þ ð1Þ 2 ð2Þ interaction picture. F½X; Y ¼F þ λF ½X; Y þλ F ½X; Y ðD7Þ 2 At Oðλ Þ, we can exponentiate the OðλÞ term, to arrive at using the representation (D3). Up to first order, we find ð1Þ F ¼ eλF ð1 þ λ2F˜ ð2ÞÞþOðλ3Þ: ðD14Þ F ð0Þ ¼ 1; Z The effect is to shift PF, P˜ F, and PW to GF, G˜ F, GW, where −i X t F ð1Þ ¼ dt0hOI ðt0ÞiðΦ ½X − Φ ½Y Þ; ℏ a a a GF;W t; t0 PF;W t; t0 − O t O t0 D15 a 0 a;b ð Þ¼ a;b ð Þ h að Þih bð Þi ð Þ ˜ F where and G ¼ðGFÞ as before. OI 0 ≡ OI 0 ρ 0 h aðt Þi TrUVð aðt Þ UVð ÞÞ 3. Relating the influence functional OI 0 − ρ ≡ OI 0 − to the master equation ¼ TrUVð aðt tÞ UVðtÞÞ h aðt tÞit: ðD8Þ If we insert Eq. (D14) into Eq. (D5), F ð1Þ can clearly be The superscript I denotes the interaction picture, and ρUðtÞ absorbed into a shift in the action of the form evolves as Z t δSð1Þ X − dt0 λ O t0 Φ X t0 : D16 ∂ ρ ρ ½ ¼ 0 h að Þi a½ ð Þ ð Þ i t UVðtÞ¼½HUV; UVðtÞ ðD9Þ 0 in this expression. As a shift in the Hamiltonian, this is identical to the result The second-order correction is (21) derived via operator methods, if the initial state of the UV d.o.f. is the pure state u . Z j i λ2 X t Next, by taking the time derivative of Eq. (D5) using F ð2Þ − 0 00 F 0 00 ΦI 0 ΦI 00 ¼ 2 dt dt Pa;bðt ;t Þ a½Xðt Þ b½Xðt Þ Eq. (D14), we can write the master equation to this order in 2ℏ 0 a;b Z terms of UV Green functions: λ2 X t − 0 00 ˜ F 0 00 ΦI 0 ΦI 00 2 2 1 1 2 dt dt Pa;bðt ;t Þ a½Yðt Þ b½Yðt Þ ∂ ρð Þ ρð Þ ð Þ ρð Þ 2ℏ 0 i t ðtÞ¼½HIR; þ½H ; a;b Z Z t 2 X 0 λ t iλ dτGW τ;t Φ 0 ρð Þ t Φ τ − t 0 00 W 0 00 ΦI 0 ΦI 00 þ abð Þ bð Þ ð Þ að Þ þ 2 dt dt Pa;bðt ;t Þ a½Xðt Þ b½Yðt Þ : 0 ℏ 0 Z a;b t þ iλ dτGW ðt; τÞΦ ðτ − tÞρð0ÞðtÞΦ ð0Þ ðD10Þ ab b a Z0 t − λ τ W τ Φ 0 Φ τ − ρð0Þ where i d Gabðt; Þ að Þ bð tÞ ðtÞ Z0 t PF ðt0;t00Þ¼Tr TðOI ðt0ÞOI ðt00ÞÞρ ð0Þ; ðD11Þ − λ τ W τ ρð0Þ Φ τ − Φ 0 a;b UV a b UV i d Gabð ;tÞ ðtÞ að tÞ bð Þ: 0 ˜ F 0 00 ˜ OI 0 OI 00 ρ 0 Pa;bðt ;t Þ¼TrUVTð aðt Þ bðt ÞÞ UVð Þ; ðD12Þ ðD17Þ 025019-19 AGON, BALASUBRAMANIAN, KASKO, and LAWRENCE PHYS. REV. D 98, 025019 (2018) This can be written in the form (12)–(13) if we write 4. Criteria for Markovian behavior ∈ 1 2 Lk ¼ Lam, m f ; g, and set The expressions for Hð2Þ, Að2Þ, and Γð2Þ above indicate a necessary condition for time-local, Markovian evolution, λ2δ − 0 W ham;bm0 ¼ abjm m j; namely that Gab (the unordered Wightman function) falls off rapidly for jτ − tj ≥ δt. In typical quantum systems, this L 1 ¼ Φ ð0Þ; a Z a requires the following (see for example Refs. [89,90]): t τ W τ Φ τ − (1) The operators O should have matrix elements La2 ¼ d Gabðt; Þ bð tÞðD18Þ 0 between the initial UV state jui and a set of UV energy levels that are finely spaced by the inverse of (where again τ is a complex parameter which factors out of a time scale tP much larger than the scale of the t the master equation, but is included so that Lai all have the experiment; at scales of order P one expects same dimension).14 Finally, the operators in Eq. (12) then quasiperiodic behavior characteristic of Poincar´e become recurrences. (2) The matrix elements of O contributing to the Z Γ t correlation function should have a finite width ð2Þ τ W τ Φ τ − Φ 0 H ¼ i d ðGabð ;tÞ að tÞ bð Þ in energy, leading to exponential falloff at a time 0 scale Γ−1. For finite-temperature correlators, where − GW ðt; τÞΦ ð0ÞΦ ðτ − tÞÞ; the Boltzmann factor cuts off large-energy states, Z ab a b t this may be of order the inverse temperature. ð2Þ τ W τ Φ τ − Φ 0 A ¼ d ðGabð ;tÞ að tÞ bð Þ Local correlators alone are not sufficient to guarantee 0 behavior that is Markovian in the strict sense of the þ GW ðt; τÞΦ ð0ÞΦ ðτ − tÞÞ; Z ab a b dynamical map being divisible. As an example, for a t Brownian particle coupled to a spectrum of harmonic γð2Þ τ W τ Φ 0 ρð0Þ Φ τ − ¼ i d ðGabð ;tÞ bð Þ ðtÞ að tÞ 0 oscillators at finite temperature TB [51], correlators fall off on a time scale of order T−1. 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