Lieb-Robinson bounds for classical anharmonic lattice systems Hillel Raz Department of Mathematics University of California at Davis 9 0 Davis CA 95616, USA 0 Email: [email protected] 2 n and a J Robert Sims 0 3 Department of Mathematics University of Arizona ] h Tucson, AZ 85721, USA p Email: [email protected] - h t a Version: January 30, 2009 m [ 1 v 5 Abstract 2 0 We prove locality estimates, in the form of Lieb-Robinson bounds, for classical oscillator 0 systems defined on a lattice. Our results hold for the harmonic system and a variety of anhar- . 2 monic perturbations with long range interactions. The anharmonic estimates are applicable to 0 a special class of observables, the Weyl functions, and the bounds which follow are not only 9 independent of the volume but also the initial condition. 0 : v 1 Introduction i X r A notion of locality is crucial in rigorously analyzing most physical systems. Typically, sets of a local observables are associated with bounded regions of space, and one is interested in how these observables evolve dynamically with respect to the interactions governing the system. In relativistic theories, the evolution of a local observable remains local, i.e. the associated dynamics is restricted to a light cone. For non-relativistic models, such as those we will be considering in the present work, the dynamics does not preserve locality in the sense that, generically, an observable initially chosen localized to a particular site is immediately evolved into an observable dependent on all sites of the system. In 1972, Lieb and Robinson [8] explored a quasi-locality of the dynamics corresponding to non-relativistic quantum spin systems. Roughly speaking, a quantum spin system is described by a self-adjoint Hamiltonian, which describes the inteactions of the system, and its associated Heisenberg dynamics, see e.g. [2] for more details. The estimates they proved, which we will ∗Copyright (cid:13)c 2009 by theauthors. This papermay bereproduced, in its entirety,for non-commercial purposes. 1 refer to as Lieb-Robinson bounds, demonstrate that, up to exponentially small errors,the time evolution of a local observable remains essentially supported in a ball of radius proportional to v|t| for some v > 0. This quantity v defines a natural velocity of propagation, and it can be estimated in terms of the system’s free parameters,for example, the interactionstrength of the Hamiltonian. Themodelsanalyzedinthispaperwillcorrespondtoaclassicalsystemofoscillatorsevolving according to a Hamiltonian dynamics. Hamiltonians of this type have frequently appeared in the literature as their analysis provides an important means of studying the emergence of non- equilibrium phenomena in macroscopic systems. For example, rigorous results on the existence of the thermodynamic limit for these models date back to [9]. A notion of quasi-locality, sim- ilar to the Lieb-Robinson bounds mentioned above, for these oscillator systems was originally considered in 1978 by Marchioro et. al. in [15], and a recent generalization of these estimates appeared in [4]. Both of these results were obtained in the spirit of deriving an analogue of the Lieb-Robinson bounds found in [8]. Overthe pastfew yearsanumberofimportantimprovementsonthe originalLieb-Robinson bounds have appeared in the literature [5, 12, 6, 10, 11], see [14] for the most current review article. These new estimates have found a variety of intriguing applications [5, 3, 13, 7], but perhaps most interestingly for the present work, the results found in [11] establish bounds which are applicable beyond the context of quantum spin systems. In [11], the authors prove a version of the Lieb-Robinson bounds for quantum anharmonic lattice systems. Motivated by themethodologyintroducedin[11],wearereturningtotheclassicalsettingtore-derivedistinct estimates for anharmonic lattice systems. To express our locality results more precisely, we introduce the following notation. We will consider systems confined to a large but finite subset Λ ⊂ Zν; here ν ≥ 1 is an integer. With each site x∈Λ, we will associate an oscillator with coordinate q ∈R and momentum p ∈R. x x ThestateofthesysteminΛwillbedescribedbyasequencex={(q ,p )} ,andphasespace, x x x∈Λ i.e. the set of all such sequences, will be denoted by X . Λ A Hamiltonian, H, is a real-valued function on phase space. Typically the Hamiltonian of interest generates a flow, Φ , on phase space. Specifically, given H : X → R one defines, for t Λ any t∈R, a function Φ :X →X by setting Φ (x)={(q (t),p (t))} , the sequence whose t Λ Λ t x x x∈Λ components satisfy Hamilton’s equations: for any x∈Λ, ∂H q˙ (t) = (Φ (x)), x t ∂p x (1.1) ∂H p˙ (t) = − (Φ (x)), x t ∂q x with initial condition Φ (x)=x. 0 To measure the effects of this Hamiltonian dynamics on the system, one introduces observ- ables. An observable A is a complex-valued function of phase space. We will denote by A the Λ spaceofalllocalobservablesinΛ,i.e. thesetofallfunctionsA:X →C. AgivenHamiltonian, Λ H, generates a dynamics α on the space of local observables in the sense that, for any t ∈ R, t the dynamics α :A →A is defined by setting α (A)=A◦Φ . t Λ Λ t t Forthelocalityresultwewillpresent,thenotionofsupportofalocalobservableisimportant. Given A ∈ A , the support of A is defined to be the minimal set X ⊂ Λ for which A depends Λ only on those parameters q or p with x∈X. x x As in [15], see also [4], our locality result will be expressed in terms of the Poisson bracket between local observables. Here the Poisson bracket is the observable given by ∂A ∂B ∂A ∂B {A,B} = · − · , (1.2) ∂q ∂p ∂p ∂q x x x x x∈Λ X for sufficiently smooth observables A and B. 2 Observe that for disjoint subsets X,Y ⊂ Λ and observables A with support in X and B with support in Y, it is clear that {A,B} = 0. The quasi-locality question of interest in this context is: given a Hamiltonian H, its corresponding dynamics α , and a pair of observables t A and B with disjoint supports, is there a bound on the quantity {α (A),B} for small times t t? Physically, one expects that if the Hamiltonian is comprised of local interaction terms, then there should be a bound on the velocity of propagation through the system. Such intuition could be confirmed by establishing an estimate of the form |{α (A),B}(x)| ≤ Ce−µ(d(X,Y)−v|t|), (1.3) t where d(X,Y) denotes the distance between the supports of the local observables A and B. This bound demonstrates that for times t with |t| ≤ d(X,Y)/v, the Poisson bracket remains exponentiallysmall,andthenumberv >0appearingaboveisaboundonthesystem’svelocity. In proving estimates of the form (1.3), special attention must be given to the dependence of the constants C, µ, and v on the observables A and B, the initial condition x, and the free parameters in the Hamiltonian. Most crucially, these constants must be independent of the underlying volume Λ, so that they persist in the thermodynamic limit; once the existence of such a limit has been established. As we have mentioned before, bounds of the form (1.3) have appeared in the literature, see [15] and more recently [4], for a variety of different Hamiltonians. Both our approach and our estimates are distinct from those mentioned above. For example, we do not work with time invariant states, and our bounds are independent of the initial conditions. Our main goal is to provide a new method for establishing these estimates, and we are strongly motivated by the quantum techniques found in [11]. We begin by considering finite volume restrictions of the harmonic Hamiltonian, i.e. HΛ : h X →R is given by Λ ν HΛ(x) = p2 +ω2q2+ λ (q −q )2, (1.4) h x x j x x+ej x∈Λ j=1 X X wheree ,forj =1,...,ν, arethe cannonicalbasisvectorsinZν,andthe parametersω ≥0and j λ ≥0aretheon-siteandcouplingstrength,respectively. Asiswell-known,avarietyofexplicit j calculations may be performed for this harmonic Hamiltonian. Perhaps most importantly, for any integer L ≥ 1 and each subset Λ = (−L,L]ν ⊂ Zν, the flow Φh,L corresponding to HΛL L t h may be explicitly computed, see Section 2.1 for details. Once this is known, a locality estimate easily follows for a specific set of observables. We will equip the set of local observables A ΛL with the sup-norm, and we will say that A∈A is bounded if ΛL kAk = sup |A(x)| (1.5) ∞ x∈XΛL is finite. Furthermore, we will denote by A(1) the set of all A ∈ A for which: given any ΛL ΛL x∈Λ , ∂A ∈A , ∂A ∈A , and L ∂qx ΛL ∂px ΛL ∂A ∂A k∂Ak = sup max , < ∞. (1.6) ∞ ∂q ∂p x∈ΛL (cid:18)(cid:13) x(cid:13)∞ (cid:13) x(cid:13)∞(cid:19) (cid:13) (cid:13) (cid:13) (cid:13) We can now state our first result. (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) Theorem 1. Let X and Y be finite subsets of Zν and take L to be the smallest integer such 0 that X,Y ⊂ Λ . For any L ≥ L , denote by αh,L the dynamics corresponding to HΛL. For L0 0 t h any µ > 0 and any observables A,B ∈ A(1) with support of A in X and support of B in Y, ΛL0 there exist positive numbers C and v , both independent of L, such that the bound h αht,L(A),B ≤Ck∂Ak∞k∂Bk∞min(|X|,|Y|)e−µ(d(X,Y)−vh|t|) (1.7) ∞ (cid:13)n o(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 3 holds for all t∈R. Someadditionalcommentsareinorder. First,thequantityd(X,Y)appearingabovedenotes the distance between the sets X and Y, measured in the L1-sense, and for any Z ⊂ Λ , the L number |Z| is the cardinality of Z. Next, the fact that the bound (1.7) is true for any µ > 0 implies that the Poisson bracket above has arbitrarily fast exponential decay in space. To achievefaster decayin space,however,the numbers C andv increase. We describe anoptimal h harmonic velocity v (µ) in Section 2.2. h Onenoveltyofourapproachis thatthe bound in(1.7)is notonlyindependent ofthe length scale L, it is also independent of the initial condition x∈X . This fact remains true when we ΛL consider anharmonic perturbations, see Theorem 2 below, and thereby distinguishes our result from that of [15] and [4]. Ournextresult,Theorem2below,concernssinglesiteperturbationsoftheharmonicHamil- tonian. Tostatethisprecisely,fixafunctionV :R→R. Foranysitez ∈Zν defineV :X →R z ΛL by setting Vz(x)=V(qz). We considerfinite volume anharmonicHamiltoniansHΛL :XΛL →R of the form HΛL =HΛL + V . (1.8) h z zX∈ΛL InordertoproveTheorem2,weneedthefollowingassumptionsonV: V ∈C2(R), V′ ∈L1(R), V′′ ∈L∞(R), and κ = |r| V′(r) dr <∞. V Z (cid:12) (cid:12) Here V′ is the Fourier transform of V′. Und(cid:12)(cid:12)cer th(cid:12)(cid:12)ese assumptions, we prove a locality result analogoustoTheorem1. AsisdiscussedinSection2.3,seealsotheproofinSection3,aspecific class ocf observables, the Weyl functions, are particularlywell-suited for our considerations,and they are defined as follows. For any function f : Λ → C, the Weyl function generated by f, L denoted by W(f), is the observable W(f):X →C given by ΛL [W(f)](x) = exp i Re[f(x)]q +Im[f(x)]p . (1.9) x x " # xX∈ΛL Clearly,if f is supportedin X ⊂Λ , then W(f) is supportedin X as well. Moreover,it is easy L to see that for any function f :Λ →C, kW(f)k =1. Our next result is L ∞ Theorem 2. Let V :R→R satisfy V ∈C2(R), V′ ∈L1(R), V′′ ∈L∞(R), and κ , as defined V above, is finite. Take X and Y to be finite subsets of Zν and let L to be the smallest integer 0 such that X,Y ⊂ΛL0. For any L≥L0, denote by αLt the dynamics corresponding to HΛL. For any µ>0 and any functions f,g :Λ →C with support of f in X and support of g in Y, there L0 exist positive numbers C and v , both independent of L, such that the bound ah αLt (W(f)),W(g) ∞ ≤Ckfk∞kgk∞min(|X|,|Y|)e−µ(d(X,Y)−vah|t|) (1.10) holds for all(cid:13)(cid:13)t(cid:8)∈R. (cid:9)(cid:13)(cid:13) The assumptions on V aboveare sufficient to imply that V is bounded. For this reason,our resultsdonotapplytomoresubstantialperturbations,e.g.,thoseoftheformV (x)=q4. In[4], z z theauthorsdoconsider,forexample,quarticperturbations. Theyprovethat,forreasonabletime invariant states and a set of initial conditions of full measure, after a time t local perturbations ofthermalequilibriumareexponentiallysmallinlog2(t)atadistancelargerthattlogα(t). This is insufficient to conclude the existence of a finite velocity v >0 as we have discussed above. It is an interesting question to determine whether or not this genuinely describes the behavior of such systems. We will not answer this question in the present work. 4 The paper is organized as follows. In Section 2, we discuss our results concerning the Har- monicHamiltonianandproveTheorem1. Usinganinterpolationargument,weproveTheorem2 inSection3. ThisresultdemonstratesthatourlocalityboundsforWeylfunctionsarepreserved under certain single-site anharmonic perturbations. In Section 4, we generalize Theorem 2 to cover a wide class of multi-site perturbations. Finally, Section 5 contains a variety of useful solution estimates used throughout the paper. 2 The Harmonic Hamiltonian ThemaingoalofthissectionistoproveTheorem1. Fortheconvenienceofthereader,webegin with a subsection describing some basic features of the harmonic Hamiltonian. In particular, we reintroduce the Hamiltonian and find an explicit expression for the corresponding flow. In the subsections which follow, we prove two locality estimates. The first is valid for a general class of smooth and bounded observables. The next holds for a special class of observables,the Weyl functions. This latter result will be particularly useful in subsequent sections. 2.1 Some basics For anyintegerL≥1,we considersubsetsΛ =(−L,L]ν ⊂Zν andthe finite volume harmonic L Hamiltonian HΛL :X →R given by h ΛL ν HΛL(x)= p2 +ω2q2+ λ (q −q )2. (2.1) h x x j x x+ej xX∈ΛL Xj=1 Here,for eachj =1,...,ν, the e arethe canonicalbasisvectorsinZν, ω ≥0,andλ ≥0. The j j modelin(2.1)isdefinedwithperiodicboundaryconditions,inthesensethatq =q x+ej x−(2L−1)ej if x∈Λ but x+e ∈/ Λ . L j L Our first task is to provide an explicit expression for the flow corresponding to (2.1). In doing so,wewill fix anintegervalue ofL≥1 anddropits dependence ina varietyofquantities toeasenotation. Givenanyx∈X andt∈R,thecomponentsofΦh(x)={(q (t),p (t))} ΛL t x x x∈ΛL satisfy the following coupled system of differential equations: for each x∈Λ and t∈R, L q˙ (t) = 2p (t) x x ν p˙ (t) = −2ω2q (t)−2 λ 2q (t)−q (t)−q (t) (2.2) x x j x x+ej x−ej j=1 X (cid:0) (cid:1) with initial condition {(q (0),p (0))} = x. Introducing Fourier variables, the system de- x x x∈ΛL finedby(2.2)decoupleswhichleadsto anexactsolution. This isthe contentofLemma3found below. Before stating Lemma 3, it is useful to introduce some additional notation. Fourier sums will be defined via the set Λ∗ given by L xπ Λ∗ = : x∈Λ . L L L n o Note that Λ∗ ⊂ (−π,π]ν and |Λ∗| =|Λ |= (2L)ν. The following functions play an important L L L role in our calculations. Suppose ω >0 and take γ :Λ∗ →R to be given by L ν γ(k)= ω2+4 λ sin2(k /2), (2.3) j j v u j=1 u X t 5 and for each m∈{−1,0,1} and any t∈R, set h(m) :Λ →R to be t L 1 ei(k·x−2γ(k)t) h(−1)(x)=Im , t |Λ | γ(k)  L k∈Λ∗ XL   1 h(0)(x)=Re ei(k·x−2γ(k)t) , (2.4) t |Λ |  L k∈Λ∗ XL   1 h(1)(x)=Im γ(k)ei(k·x−2γ(k)t) . t |Λ |  L k∈Λ∗ XL   Each of these functions depend on the length scale L, however, we are suppressing that depen- dence. Lemma 3. Suppose ω > 0. For any x ∈ X and t ∈ R, the mapping Φh : X → X ΛL t ΛL ΛL is well-defined. In particular, for each x ∈ Λ and t ∈ R, the components of Φh(x) = L t {(q (t),p (t))} are given by x x x∈ΛL q (t)= q (0)h(0)(x−y)−p (0)h(−1)(x−y) (2.5) x y t y t yX∈ΛL and p (t)= q (0)h(1)(x−y)+p (0)h(0)(x−y). (2.6) x y t y t yX∈ΛL Here, if necessary, the function values h(m)(x−y) are defined by periodic extension, and we t regard x={(q (0),p (0))} . x x x∈ΛL Proof. Taking a second derivative of (2.2), we find that for each x∈Λ and any t∈R, L ν q¨ (t)=−4ω2q (t)−4 λ 2q (t)−q (t)−q (t) x x j x x+ej x−ej j=1 X (cid:0) (cid:1) (2.7) ν p¨ (t)=−4ω2p (t)−4 λ 2p (t)−p (t)−p (t) . x x j x x+ej x−ej j=1 X (cid:0) (cid:1) For any k ∈Λ∗ and t∈R, set L 1 1 Q (t) = e−ik·xq (t) and P (t) = e−ik·xp (t). (2.8) k x k x |Λ | |Λ | L xX∈ΛL L xX∈ΛL p p Inserting (2.7) into the second derivative of (2.8), we find an equivalent system of uncoupled differential equations. In fact, for each k ∈Λ∗ and any t∈R, L ν Q¨ (t)=−4ω2Q (t)−4 λ 2−eikj −e−ikj Q (t)=−4γ(k)2Q (t) k k j k k j=1 X (cid:0) (cid:1) (2.9) ν P¨ (t)=−4ω2P (t)−4 λ 2−eikj −e−ikj P (t)=−4γ(k)2P (t), k k j k k j=1 X (cid:0) (cid:1) where γ is as in (2.3). The solution of (2.9) is given by Q (t)=C e−2iγ(k)t+C e2iγ(k)t k k −k (2.10) P (t)=D e−2iγ(k)t+D e2iγ(k)t, k k −k 6 where −k is defined to be the element of Λ∗ whose components are given by L −k , if |k |<π, (−k) = j j j π, otherwise. (cid:26) Therelationshipbetweenthecoefficientsin(2.10)aboveisderivedusingthefactthattheinitial condition is real-valued, e.g., Q (0)=Q (0) and Q˙ (0)=Q˙ (0). k −k k −k Using Fourier inversion, we recover the components of the flow from (2.10). In fact, 1 q (t) = eik·xQ (t) x k |Λ | L k∈Λ∗ XL p1 = C ei(k·x−2γ(k)t)+C e−i(k·x−2γ(k)t), (2.11) k k |Λ | L k∈Λ∗ XL p and similarly, we find that 1 p (t) = D ei(k·x−2γ(k)t)+D e−i(k·x−2γ(k)t). (2.12) x k k |Λ | L k∈Λ∗ XL p To express these solutions explicitly in terms of the initial condition, we observe that Q (0)=C +C and P (0)=D +D , (2.13) k k −k k k −k and introduce 1 γ(k) 1 γ(k) B = P (0)−i Q (0) with B = P (0)+i Q (0). (2.14) k k k k −k −k 2γ(k) r 2 2γ(k) r 2 It is easypto see that p i γ(k) Q (0) = B − B and P (0) = B + B , (2.15) k k −k k k −k 2γ(k) r 2 (cid:0) (cid:1) (cid:0) (cid:1) and therefore, p iB γ(k) k C = and D = B . (2.16) k k k 2γ(k) r 2 Plugging this into (2.11), we findpthat 1 iB iB q (t) = k ei(k·x−2γ(k)t)− k e−i(k·x−2γ(k)t) (2.17) x |Λ | 2γ(k) 2γ(k) L k∈Λ∗ XL p 1 p p = Q (0)ei(k·x−2γ(k)t)+Q (0)e−i(k·x−2γ(k)t) k k 2 |Λ | L k∈Λ∗ XL p i P (0) P (0) + k ei(k·x−2γ(k)t)− k e−i(k·x−2γ(k)t) 2 |Λ | γ(k) γ(k) L k∈Λ∗ XL 1 p 1 P (0) = Re Q (0)ei(k·x−2γ(k)t) − Im k ei(k·x−2γ(k)t) . k |Λ | |Λ | γ(k) L kX∈Λ∗L h i L kX∈Λ∗L (cid:20) (cid:21) p p 7 Moreover,one finds that 1 Re Q (0)ei(k·x−2γ(k)t) = q (0)Re ei(k·(x−y)−2γ(k)t) (2.18) k y |Λ | h i L yX∈ΛL h i while p P (0) 1 1 Im k ei(k·x−2γ(k)t) = p (0)Im ei(k·(x−y)−2γ(k)t) . (2.19) y γ(k) |Λ | γ(k) (cid:20) (cid:21) L yX∈ΛL (cid:20) (cid:21) With the functions h(m), as definedpin (2.4), we conclude that t q (t)= q (0)h(0)(x−y)−p (0)h(−1)(x−y), (2.20) x y t y t yX∈ΛL as claimed in (2.5). A similar calculation yields (2.6). Since the functions h(m) are real valued, t so too are the solutions q (t) and p (t). This proves Lemma 3. x x Remark 4. An analogue of (2.5) and (2.6) holds in the event that ω = 0. This is seen by proceeding as in the proof of Lemma 3 and observing that now γ(0)=0, but γ(k)6=0 for k 6=0. For k 6=0, the formulas above are correct, and a simple calculation shows that, in this case, Q (t)=Q (0)+2P (0)t 0 0 0 (2.21) P (t)=P (0), 0 0 similarto(2.10). Oneeasilyseesthattheequations(2.5)and(2.6)stillholdwiththeconvention that 2t 1 ei(k·x−2γ(k)t) h(−1)(x)=− +Im . (2.22) t |Λ | |Λ | γ(k)  L L k∈ΛX∗L\{0}   We end this subsection with the following crucial estimate which was proven in [11]. Lemma 5. Fix L ≥ 1 and consider the functions h(m) as defined in (2.4) for m ∈ {−1,0,1}. t For any µ>0, the bounds h(t0)(x) ≤e−µ(|x|−cω,λmax(µ2,e(µ/2)+1)|t|) h(cid:12)(cid:12)(−1)(x)(cid:12)(cid:12)≤c−1e−µ(|x|−cω,λmax(µ2,e(µ/2)+1)|t|) (2.23) (cid:12)t (cid:12) ω,λ (cid:12)(cid:12)(cid:12) h(t1)(x)(cid:12)(cid:12)(cid:12)≤cω,λeµ/2e−µ(|x|−cω,λmax(µ2,e(µ/2)+1)|t|) (cid:12) (cid:12) hold for all t ∈ R and(cid:12) x ∈ Λ(cid:12) . Here |x| = ν |x | and one may take c = (ω2 + (cid:12) (cid:12)L j=1 i ω,λ 4 ν λ )1/2. j=1 j P PWe refer the interested reader to Lemma 3.7 of [11] for the proof. Moreover, we stress that Lemma 5 is valid for all ω ≥0. 2.2 A general locality estimate Our first locality bound for the harmonic Hamiltonian follows directly from Lemma 3 and Lemma 5. We state this as Theorem 6 below. As we will see, Theorem 1 is an immediate (1) consequence of Theorem 6. Recall that we have defined A to be the set of observables ΛL A∈A for which: given any x∈Λ , ∂A ∈A , ∂A ∈A , and ΛL L ∂qx ΛL ∂px ΛL ∂A ∂A k∂Ak = sup max , < ∞. (2.24) ∞ ∂q ∂p x∈ΛL (cid:18)(cid:13) x(cid:13)∞ (cid:13) x(cid:13)∞(cid:19) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 8 Theorem 6. Let X and Y be finite subsets of Zν and take L to be the smallest integer such 0 that X,Y ⊂ Λ . For any L ≥ L , let αh,L denote the dynamics corresponding to HΛL. For L0 0 t h any µ>0 and any observables A,B ∈A(1) with support of A in X and support of B in Y, the ΛL0 bound αht,L(A),B ≤Ck∂Ak∞k∂Bk∞ e−µ(d(x,y)−cω,λmax(µ2,e(µ/2)+1)|t|) (2.25) ∞ (cid:13)n o(cid:13) x∈XX,y∈Y (cid:13) (cid:13) holds f(cid:13)or all t∈R. H(cid:13)ere ν d(x,y)= min|x −y +2Lη | (2.26) j j j j=1ηj∈Z X is the distance on the torus and the constants may be taken as C = (2+c eµ/2+c−1) with ω,λ ω,λ c =(ω2+4 ν λ )1/2. ω,λ j=1 j Proof. The PoiPsson bracket is easy to calculate. In fact, for any x∈X , ΛL ∂ ∂B ∂ ∂B αh,L(A),B (x)= A Φh,L(x) · (x) − A Φh,L(x) · (x). (2.27) t ∂q t ∂p ∂p t ∂q y y y y hn oi yX∈Y (cid:16) (cid:17) (cid:16) (cid:17) By the chain rule, ∂ ∂A ∂q ∂A ∂p A Φh,L(x) = Φh,L(x) · x(t) + Φh,L(x) · x(t) (2.28) ∂q t ∂q t ∂q ∂p t ∂q y x y x y (cid:16) (cid:17) xX∈X (cid:16) (cid:17) (cid:16) (cid:17) and a similar formula holds for ∂ A Φh,L(x) . Now estimating (2.27), we find that ∂py t (cid:16) (cid:17) αh,L(A),B ≤ k∂Ak k∂Bk h(−1)(x−y) + 2 h(0)(x−y) + h(1)(x−y) , t ∞ ∞ t t t ∞ (cid:13)n o(cid:13) x∈XX,y∈Y (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:13) (cid:13) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (2.29(cid:12)) (cid:13) (cid:13) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) using Lemma 3. The bound in (2.25) now follows from Lemma 5. FromTheorem6,andspecificallythe bound(2.25),wesee thatforanyµ>0,the harmonic velocity v is essentially described by h 2 v (µ)=c max ,e(µ/2)+1 . (2.30) h ω,λ µ (cid:18) (cid:19) In fact, given (2.25) for some µ>0, it is easy to see that for any 0<ǫ<1, e−µd(x,y) ≤e−ǫµd(X,Y) min(|X|,|Y|) e−µ(1−ǫ)d(0,z), (2.31) x∈XX,y∈Y zX∈ΛL where we have set d(X,Y) = min d(x,y). Thus, Theorem 1 is a simple consequence of x∈X,y∈Y Theorem 6. It is interesting to note that for any L the quantity e−µ(1−ǫ)d(0,z) ≤ e−µ(1−ǫ)|z|, (2.32) zX∈ΛL zX∈Zν where |z| denotes the L1-metric on Zν. Given this and the fact that, for sufficiently large L, the distanced(X,Y)agreeswiththeL1-distancebetweenX andY,itisclearthatthe estimate proven in Theorem 1 is genuinely independent of the length scale L. 9 Sincetheboundsarevalidforanyµ>0,Theorem6demonstratesarbitrarilyfastexponential decay in space with a velocity that depends on µ. Typically, however, one is interested in the bestpossibleestimatesonv givensomedecayrate. Inthissense,theoptimalharmonicvelocity, h as described by (2.30), occurs when the equation µ =e(µ/2)+1 (2.33) 2 holds. It is easy to see that the solution to (2.33), denoted by µ , satisfies 1/2 < µ < 1, and 0 0 therefore the corresponding velocity v (µ )≤4c . h 0 ω,λ 2.3 The harmonic evolution of Weyl functions InpreparationforourargumentsinSections3and4,wewillnowpresentadifferentproofofour locality result, analogous to Theorem 6, valid for Weyl functions. Recall that a Weyl function is an observable, generated by a function f :Λ →C, with the form L [W(f)](x) = exp i Re[f(x)]q +Im[f(x)]p . (2.34) x x " # xX∈ΛL One important property of the Weyl functions is typically referred to as the Weyl relation. We state this as Proposition 7. Proposition 7 (Weyl Relation). Let f,g :Λ →C. We have that L {W(f),W(g)}=−Im[hf,gi]W(f)W(g). (2.35) where the inner product is taken in ℓ2(Λ ). L Proof. A direct calculation yields ∂ ∂ ∂ ∂ {W(f),W(g)} = W(f) W(g)− W(f) W(g) ∂q ∂p ∂p ∂q x x x x xX∈ΛL = (−Re[f(x)]Im[g(x)]+Im[f(x)]Re[g(x)])W(f)W(g). xX∈ΛL Noting that Im[hf,gi] = Im f(x)g(x) (cid:2)xX∈ΛL (cid:3) = (−Im[f(x)]Re[g(x)]+Re[f(x)]Im[g(x)]) xX∈ΛL proves the proposition. Another useful property of the Weyl functions is that the harmonic dynamics leaves this class of observables invariant. This important fact, which follows immediately from Lemma 3, is the contentofthe nextproposition. Beforestatingthis, itisconvenienttointroducenotation for the convolution of two functions f,g :Λ →C, L (f ∗g)(x)= f(y)g(x−y), (2.36) yX∈ΛL where, if necessary,g(x−y) is calculated by periodic extension. 10