arXiv:1305.5272v1 [quant-ph] 22 May 2013 otiuin fHiebr n Dirac. breaking and ground the Heisenberg in notable of most contributions is This it. cradle to sim- the ilarities structural unmistakable in bears was mechanics, it classical as of hand, born other the mechanics, dras- On quantum a modern concepts. is classical strangeness from quantum departure as tic regarded is and that entanglement the- much of quantum phenomenon them the of underlies set principle which that ory superposition differences the profound Thus the apart. as striking as are prdb ia’ eak nterl fteLagrangian the of role mechanics. the quantum in- on in was remarks Feynman’s which Dirac’s formulation is by integral this spired path of the of instance discovery approaches. compelling new especially seeking An in as well insight as under- invaluable mechanics our quantum providing of broadening structure in the of role standing important an played have aino nlg ihteHmlointer fclassi- of foun- theory a Hamiltonian mechanics.” on the words cal up with the built analogy In was of mechanics dation canonical regimes. “Quantum the dynamical Dirac, two in of the resemblance of remarkable structures a mechan- classical showed of ics formulation dynamics bracket quantum Poisson the of and relations fun- the commutation between damental correspondence the of discovery Dirac’s est arxi unu ehnc n h Liouville the and distribu- mechanics the to quantum system, in the respect of matrix with state density values the question characterising average in functions as tion quantities formulated the be may of gleaned dynamics quantities be both observable dy- All in can quantum observation. mechanics of following the classical analogs from in exist pictures there be- That namical case intermediate them. an picture tween Schr¨odinger Dirac and the Heisenberg with the pictures, mechanics, being quantum ones in principal known well the Dynamical course of regimes. struc- are two the pictures the in with pictures 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V ewl ics rcs formula- precise a discuss will we IV, itrsi Classical in Pictures l § I ewl s h interac- the use will we III § ewl rsn few a present will we V φ 4 2 defined on a Hilbert space of functions of the canonical Compared to Eq. (4), the representation in Eq. (8) coordinates (q, p). This wavefunction is postulated to has clearly shifted the origin of the time dependence of obey the evolution equation (t) to the dynamical variable while the phase space density is taken at the fixed initial time. This ∂φ(q, p,t) i = Lφˆ (q, p,t), (1) is of course the Heisenberg picture of the time evolu- ∂t tion of the system. The time-dependent form of the dy- where Lˆ = iL, with L is the standard Liouville operator namical variable Aˆ(t), on the other hand, is given by A[Q(t, Q0, P0), P(t, Q0, P0),t], exactly as expected. ∂ ∂ ∂ ∂ L = [ H(q, p) − H(q, p) ]. (2) We note that Eq. (8) can be more formally ob- i X ∂qi ∂pi ∂pi ∂qi tained from Eq. (5) by observing thatρ ˆ(t) = Here H(q, p) is the underlying classical Hamiltonian. exp[−iLtˆ ]ˆρ exp[iLtˆ ],ρ ˆ =ρ ˆ(0): The classical phase space density associated with φ is then given by ρ(q, p,t) = | φ |2, which equals the prob- (t) =tr[Aˆρˆ(t)] = tr{Aˆ exp[−iLtˆ ]ˆρ exp[iLtˆ ]} ability density that the system has coordinates and mo- =tr{exp[iLtˆ ]Aˆ exp[−iLtˆ ]ˆρ} = tr[Aˆ(t)ˆρ]. (9) menta (q, p). Using the postulated Eq. (1), we find that the resulting probability distribution obeys the standard Note that, by applying Eq. (6), we can verify that the Liouville equation: definition of Aˆ(t) resulting from this equation agrees with ∂ρ(q, p,t) what we deduced above from Eq. (8). = [L,ρ(q, p,t)]. (3) ∂t It should now be clear that the above transformations are formally analogous to their quantum counterparts as The average value of any dynamical quantity A(q, p) the two regimes rest on similar canonical structures. can now be expressed as At this juncture we can consider fluid flow as an exam- ple and notice that Eq. (8) corresponds to the Lagrangian (t)= dqdpA(q, p)ρ(q, p,t), (4) Z description whereby all dynamical quantities are averages over particle trajectories starting from an initial configu- or symbolically as ration. Equation (4), on the other hand, corresponds to (t) = tr[Aˆρˆ(t)]. (5) the Eulerian description whereby the phase space density is considered a function of time, from which dynamical It is clear from Eqs. (3) and (4) that the time depen- averages are obtained using time independent (equiva- dence of (t) originates inρ ˆ(t). Thus the descrip- lently, initial-time) expressions of those variables. tion given in Eqs. (3-5) is in the Schr¨odinger picture since The foregoing discussion has established the two prin- the time dependence of average values originates in the cipal dynamical pictures in classical mechanics. Evi- quantity that represents the state of the system. dently other equivalent pictures can similarly be defined. To construct the corresponding Heisenberg de- We will consider Dirac’s interaction picture in the follow- scription, we first consider the particle trajectories ing section. [Q(t, Q0, P0), P(t, Q0, P0)] obeying Hamilton’s equa- tions of motion d(Q, P) III. INTERACTION PICTURE IN CLASSICAL i = −Lˆ(Q, P), (6) dt MECHANICS and subject to the initial conditions (Q0, P0) at time t = 0. We can then relate the phase space density at The interaction picture in classical mechanics is con- time t to its initial distribution using these trajectories: structed in analogy with quantum mechanics. Consider a splitting of Lˆ into two pieces, Lˆ = Lˆ0 +LˆI , corresponding ρ[Q, P,t]= ρ[Q0(t, Q, P), P0(t, Q, P), 0], (7) to splitting the Hamiltonian into “free” and “interaction” parts, Hˆ = Hˆ + Hˆ . Then, defining the interaction pic- where we have inverted the trajectory equations in writ- 0 I ture quantities asρ ˆI (t) = exp[iLˆ0t]ˆρ(t) exp[−iLˆ0t] and ing the initial coordinates (Q0, P0) in terms of their cor- ˆ ˆ ˆ ˆ responding values at time t. Equation (7) is a conse- LI (t) = exp[iL0t]LI exp[−iL0t], we find quence of Liouville’s theorem expressing the constancy ∂ρˆI (t) of the phase space density along a given trajectory. i = LˆI (t)ˆρI (t), (10) At this point we can substitute the right-hand side ∂t of Eq. (7) in Eq. (4), change variables of integration to while the expectation value of A is given by (Q0, P0) (remembering that the Jacobian equals unity due to Liouville’s theorem), and arrive at (t) = tr[AˆI ρˆI (t)], (11)
To complete the analogy, we can rewrite Eq. (10) by IV. SENSITIVITY TO INITIAL CONDITIONS IN QUANTUM DYNAMICS defining an evolution operator Uˆ(t) such thatρ ˆI (t) = Uˆ(t)ˆρI (0). Then Classical chaos is a long term dynamical instability ∂Uˆ(t) which causes initially close trajectories to diverge ex- i = Lˆ (t)Uˆ(t), (12) ∂t I ponentially and eventually lose memory of their initial conditions.8 It is ubiquitous among nonlinear systems of subject to Uˆ(0) = 1. In strict analogy with quantum more than one degree of freedom. Chaotic behavior is mechanics, we can construct a perturbative solution to measured in terms of the rate of trajectory divergence Eq. (12) using Dyson’s formula:6 mentioned above, and when present is characterized by an effectively random behavior for sufficiently long times. t ˆ ˆ Because the quantitative measure of chaos is based on U(t)= T exp[−i dτLI (τ)], (13) classical trajectories, it cannot be directly applied to Z0 quantum dynamics. As will be seen below, it turns out where T is the time ordering operator. that the above-mentioned instability is absent in quan- To see an illustration of the foregoing formulation, tum dynamics.9 On the other hand, there are a number consider the case of motion in one dimension with of characteristic behaviors exhibited by quantum systems 2 H(q,p) = p /2m + V (q). Then Lˆ0 = −i(p/m)∂/∂q whose classical versions are chaotic. These are a subject ′ 10,11 and LˆI = iV (q)∂/∂p. We also have LˆI (t) = of active study often referred to as “quantum chaos.” ′ 7 exp[iLˆ0t][iV (q)∂/∂p] exp[−iLˆ0t], which leads to The aim of this section is to demonstrate that the cor- respondence detailed in the previous sections allows a ′ precise measure of chaos in quantum dynamics by anal- LˆI (t)= iV (q + pt/m)[∂/∂p − (t/m)∂/∂q]. (14) ogy to classical dynamic. As a simple application, let us consider an initial distribu- tion of particles of density f(q), all moving with momen- ′ tum p0, and subject to a constant force F = −V (q) start- A. Chaos in Classical Dynamics ing at t = 0. Then the initial density is given by f(q)δ(p− p0), so thatρ ˆI (t) = exp[iLˆ0t] exp[−iLtˆ ]f(q)δ(p − p0). We start by outlining the quantitative definition of Next, we use the Baker-Hausdorff identity exp(A + chaos in classical mechanics.8,12,13 Consider a classical B) = exp(A) exp(B) exp(−[A, B]/2), where exp([A, B]) system of N degrees of freedom described by 2N canon- commutes wit both A and B, to find that ical variables {qi,pi}, i = 1,...,N and Hamiltonian H(q, p). The trajectory divergence behavior of a sys- ˆ ˆ exp[iL0t] exp[−iLt] = exp(−Ft∂/∂p) tem is best measured by following the evolution of an × exp[(Ft2/2m)∂/∂q], (15) infinitesimal 2N-dimensional sphere in phase space ini- tially centered at point (q0, p0). Such a sphere will be which we apply to the evaluation of ρI (q,p,t): distorted in the course of time, stretching in some di- rections and contracting in others, while maintaining a 2 ρI (q,p,t) = exp(−Ft∂/∂p) exp[(Ft /2m)∂/∂q]f(q) fixed 2N-dimensional. The details of these deformations Ft2 are fully conveyed by the Jacobian of the flow, namely × δ(p − p0)= f(q + )δ[p − (p0 + Ft)]. (16) the 2N × 2N, nonsingular block matrix 2m ∂q(t, q0, p0)/∂q ∂q(t, q0, p0)/∂p Finally, we can find ρ(q,p,t) from Eq. (16) usingρ ˆ(t) = T (t)= 0 0 , (18) ∂p(t, q0, p0)/∂q0 ∂p(t, q0, p0)/∂p0 exp(−Lˆ0t)ˆρI (t):
pt Ft2 where [q(t, q0, p0), q(t, q0, p0)] is the phase-space tra- ρ(q,p,t)= f(q + + )δ[p − (p0 + Ft)], (17) jectory that starts from (q , p ). We shall refer to this m 2m 0 0 Jacobian as the sensitivity matrix. It has a unit determi- which corresponds to the phase space density displaced nant by Liouville’s theorem. from position q to q + pt/m + Ft2/2m with momentum A long-time exponential growth in any of the singular p0+Ft, exactly as expected under the action of a constant values of this sensitivity matrix signals chaotic behav- force. ior. Thus we consider the Hermitian matrix ln(T †T /2t), The above calculations serve to illustrate the formal whose eigenvalues as t → ∞ constitute the Lyapunov structure of the interaction picture in classical mechan- spectrum, or indices, for the system; the dagger here ics. As an application of the classical-quantum dynami- represents the transpose of a matrix. These indices rep- cal picture analogy that shows its utility in dealing with resent exponential growth rates in certain phase-space difficult questions, we will discuss the precise meaning directions, and they occur in pairs of equal magnitude of sensitivity to initial conditions and chaos in quantum and opposite signs. For a non-chaotic system, all indices dynamics in the following section. vanish, while for a chaotic system, a number of indices 4
(≤ N − 1) can be positive. The sum of such positive B. Sensitivity Operator in Quantum Mechanics indices, h, is known as the metric or KS entropy, after Kolmogorov and Sinai,14 and is an important information theoretical invariant quantity associated with a dynami- Using the quantum-classical analogies described above, cal system. Thus a system is chaotic if, and only if, h> 0. we will next develop a quantum counterpart of the sensi- As mentioned above, the long-term behavior of a chaotic tivity matrix for quantum dynamics. First we note that system is effectively random even though the system is the definition of the latter in Eq. (18) is in the Heisen- governed by fully deterministic equations of motion. How berg representation of classical mechanics inasmuch as such an outcome is possible is best understood in terms the time dependence is carried by the dynamical vari- of the concept of algorithmic information content, which able. Thus we construct the analogous object in quan- we discuss next. tum mechanics by applying Dirac’s correspondence rules to Eq. (18):
In order to characterize the information content of a −i [qˆ(t), pˆ ] −[qˆ(t), qˆ ] system, Solomonoff, Kolmogorov, and Chaitin, indepen- Tˆ(t)= 0 0 . (19) dently and nearly simultaneously, introduced the idea of ~ [pˆ(t), pˆ0] −[pˆ(t), qˆ0] algorithmic information content, defined as the length of the most economical description of the system.15 We shall refer to this matrix of operators as the sensi- This seemingly ambiguous characterization can be made tivity operator. The elements of this object constitute mathematically precise, and as such provides a rigorous a set of 4N 2 Hermitian operators, so that its expecta- definition of a random sequence as one which does not tion value for a quantum stateρ ˆ, i.e., tr[Tˆ(t)ˆρ], results allow a significantly shorter description. Equivalently, a in a real sensitivity matrix T (t) which is the counter- random sequence is one that cannot be algorithmically part of the classical object. The Lyapunov indices can compressed. As an example, we note that the seemingly then be calculated as in the classical case. In particular, random digits in the decimal (or binary) representation any exponential growth in time occurring in any of the of the number π are very highly compressible and far components of this matrix signals chaotic behavior. As from random since this irrational number (hence also its seen in the following, such behavior does not occur in various representations) can be defined by a short sen- quantum mechanics. tence. In the case of an infinite sequence, its algorithmic ˆ −i complexity k is defined to be the ratio of the shortest Consider a typical element of T (t), say ~ [ˆqi(t), pˆj (0)], description to length of its subsequences in the limit of and a quantum stateρ ˆ. Letq ¯i(t) = tr[ˆqi(t)]ˆρ]andp ¯j(0) = longer and longer subsequences. If k> 0, the sequence is tr[ˆpj(0)ˆρ] be the corresponding mean values of the canon- said to be complex and is considered random. In partic- ical coordinates. We also recall the (generalized) Heisen- 2 ular, such a sequence cannot be defined by a finite one. berg inequality |tr[A,ˆ Bˆ]ˆρ| ≤ 4tr[Aˆ2ρˆ]tr[Bˆ2ρˆ], which we apply with Aˆ =q ˆi(t) − q¯i(t) and Bˆ =p ˆj (0) − p¯j (0). The The relevance of the above ideas arises from a profound result is theorem that shows that the output of a dynamical sys- tem is complex if, and only if, its metric entropy h is ˆ 2 positive, in which case its algorithmic complexity and |Tij (t)| = |tr[ˆqi(t)−q¯i(t), pˆj (0)−p¯j(0)]ˆρ|≤ ~δqˆi(t)δpˆj (0), metric entropy are equal, i.e., h = k.16 In other words, (20) the long-time output of a chaotic system (with k> 0) is where δqˆi(t) stands for the square root of the variance of indistinguishable from random. In practical terms, this the observableq ˆi(t). outcome can be understood by considering the fact that Inequality (20) establishes the lack of unbounded any initial conditions can only be specified with finite growth, exponential or otherwise, in the sensitivity op- accuracy, say to 100 decimal places, which will progres- erator for bounded quantum states. It is worth pointing sively be eroded in the course of time (due to the sen- out here that sensitivity to initial conditions and chaos sitivity condition) and totally lost in finite time (due to are concepts that are appropriate to bounded systems. the exponential character of the sensitivity to initial con- Since the right-hand side of (20) is related to variances ditions), resulting in output which is unrelated to the of bounded canonical coordinates, we conclude that the initial conditions. sensitivity operator in quantum mechanics is bounded and cannot exhibit sensitivity to initial conditions. It is worth emphasizing that the foregoing character- While there are other indications such as the discrete ization of a chaotic system is tied to its long-time be- nature of the spectra of bounded quantum systems sug- havior and is best understood in information theoretical gesting the same conclusion as above, we note the great terms. Note also that the definition of chaos in terms generality of the foregoing reasoning and its close con- of the complexity of its dynamical output is general and nection with the original definition of chaos in classical directly applicable to quantum systems. mechanics. 5
V. CONCLUDING REMARKS emergence as ~ → 0 leads to its chaotic classical counter- part? The answer lies in the fact that chaos and sensi- tivity to initial conditions are long-time properties, and require the t → ∞ limit which may not commute with Dynamical picture similarities are part of the land- the ~ → 0 limit. Indeed the so-called quantum kicked scape of quantum-classical analogies and play an im- rotor,11 whose classical version exhibits chaos, does ap- portant practical and conceptual role in extending our pear to be sensitive to initial conditions for a finite period understanding of quantum mechanics. These analogies of time. Its long-time behavior, however, is perfectly reg- notwithstanding, there are important structural differ- ular. This example underscores the key role of the t → ∞ ences between quantum and classical dynamics. One limit in characterizing chaotic behavior. More generally, such difference is evident from our discussion of sensi- it serves as a warning that, despite strong similarities tivity to initial conditions. Considering the quantum- between quantum and classical dynamics, the transition classical transition, how do we reconcile the lack of such across their boundary is anything but smooth and must sensitivity for a quantum system, where ~ 6= 0, with its be handled carefully.
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