One of the great crimes against humanity occurs each year in introductory quantum mechanics courses when students are introduced to an limit, sometimes decorated with words involving “the correspondence principle”. The problem isn’t with the content per se, but with the suggestion that this somehow gives a satisfying answer to why quantum mechanics looks like classical mechanics on large scales.
Sometimes this limit takes the form of a path integral, where the transition probability for a particle to move from position to in a time is
where is the integral over all paths from to , and is the action for that path ( being the Lagrangian corresponding to the Hamiltonian ). As , the exponent containing the action spins wildly and averages to zero for all paths not in the immediate vicinity of the classical path that make the action stationary.
Other times this takes the form of Ehrenfest’s theorem, which shows that the expectation values of functions of position and momentum follow the classical equations of motion. For any operator without explicit time dependence, we take the expectation value of the Heisenberg equation of motion to get . So we can derivea the Hamiltonian equations of motion for the expectation values:
This is often augmented with some hand-wavy discussion of sharply peaked wavepackets and higher derivatives of potentials, by which one argues that the wavefunctions of systems that are large compared to should have simultaneously well-defined values for and . (Or rather that the -sized error from the uncertainty principle is small compared to the system.)
This sort of reasoning is an atrocity committed against the mind, not because it’s not useful, but because people are often given the impression that this is a reasonably complete explanation for how classical mechanics arises out of quantum mechanics. Inevitably, the careful student experiences a healthy sense of confusion and dissatisfaction when learning this material, but they are encouraged to suppress these intuitions. Hey, von Neumann made these kind of arguments and he’s a hell of a lot smarter than you and me, right? Once they get in a habit of suppressing their BS-meter, things go bad quickly.
Unfortunately, even von Neumann is fallible. The proper way to teach the above material is to learn it for what it is: It clearly points toward an important insight, but ultimately cannot explain the quantum-classical transition on its own.
Now, I can’t give you a complete understanding of the the quantum-classical transition. If I could, I’d just write that down, post it to the arXiv, and go get a well-paying job in finance or something. But what I can do is point out what the heck is wrong with the above. And we know there has to be something wrong with the above; it’s just a simple limit, and yet all the crazy confusion with measurements and probabilities and Wigner’s friend is suddenly supposed to evaporate?
The most glaring problem is that the state spaces of classical and quantum mechanics are completely different, so you can’t have a simple limiting procedure unless you describe how you’re going to map one onto the other. Let’s compare with a much nicer case like the limit in which special relativity reduces to Galilean kinematics. In this case we need to take a configuration space (points in ), a set of trajectories (time-like worldlines in ), and a set of group transformation (the Lorentz transformation), and map it to their limit as . This yields points in , time-monotonic worldlines in , and the Gallilean transformation, respectively. The mapping is intuitive and unambiguous. If we want, we can add dynamics and get the limit of relativistic mechanics to Newtonian mechanics.
However, the quantum-to-classical case is vastly more complicated because the quantum state space (Hilbert space) is way bigger than classical configuration space (or phase space). Exponentially so! It might be hoped that the accessible state space of quantum mechanics becomes the same size as classical mechanics in the limit. Shouldn’t sufficiently narrow wavepackets stay narrow?
Nope, that’s not good enough; see Figure. Wavepackets get grossly distorted — producing macroscopic coherence — on a time scale given by the Lyapunov exponent of chaotic systems. (This applies to almost everything besides a harmonic oscillator.) Even an extremely macroscopic variable like the orientation of Hyperion, a moon of Saturn, fails to maintain a narrow wavepacket on human timescales . Of course we only see Hyperion oriented in one, unpredictable way, so the quantum stochasticityb has not been beaten back by taking . Any explanation of the quantum-classical limit worth its salt is going to need to handle this one-to-many nature of quantum time evolution.
We might not be able to fully understand the quantum-classical limit in this post, but we can at least say a few things about what sorts of state spaces we ought to be considering. The exponential size of the quantum state space (Hilbert space), and the fact that it’s not much smaller than the probabilistic extension to the space of density matrices, is a clue that maybe we shouldn’t be looking to limit toward classical configuration space or phase space. Rather, let’s look at classical probability distributions over phase space. This is consistent with the above observation that quantum mechanics remains stochastic even for vanishing , since we expect our classical probability distribution to gain entropy as time goes on (which couldn’t be captured by a limit involving just classical points in phase space).
In other words, the simple derivation of the limit given in introductory quantum classes break down because of what is essentially an anomalyc . The classical dynamics have a “symmetry” — determinism — that is not enjoyed by the quantum dynamics even in the limit.
So instead of trying to derive Hamilton’s equations,
let’s look to Louiville’s equation,
where the Poisson brackets define a binary operator on the space of functions over phase space:
(Louiville’s equation of course just reduces to Hamilton’s equations when the position and momentum of the particle are known with certainty, .) Luckily, quantum mechanics comes equipped with a phase space representation, where the density matrix is replaced by a Wigner function (previous posts: 1, 2) and the von Neumann equation is replaced with the Moyal equation
since I actually have embarrassingly little intuition for the Moyal bracket other than the fact that it clearly reduces to the Poisson bracket as . It’s the Moyal bracket (not the Poisson bracket as erroneously envisioned by Dirac) that takes the role of the quantum commutator within the phase-space formulation; the Poisson bracket is only recovered after .
OK, now we are actually getting somewhere. Rather than have two theories that look completely foreign to each other — wavefunctions versus points, operators versus functions — quantum and classical mechanics are starting to look pretty similar, at least in terms of the basic mathematical objects. (Interpretation, measurement, and all that crud still must be dealt with, but we will not do so today.) The state space of both consists of functions over phase space, the dynamics of both are expressed in terms of a binary bracket operator with a Hamiltonian, and the Hamiltonian are now (essentiallye ) the same in each theory!
But we aren’t done achieving even our modest goals, and the reason is this: although the state space of quantum and classical mechanics are both normalized functions over phase space, these state spaces are not exactly the same. In the classical case, is always positive but in the quantum case the Wigner function is generally not. Furthermore, this is not a restriction that goes away in the limit. In this limit, the Wigner function for a coherent state (i.e., Gaussian wavepacket) approaches a delta function (which can be interpreted as an distribution as we expect), but for a superposition of two states does not approach the sum of two delta function. Rather, there are fine oscillations — sub- structure — that become more extreme as approaches zero (see Figure). Some quantum states have sensible classical analogs, but some, like this grossly non-classical superposition, do not.
However, this is exactly what we should expect. We already know that it should be possible, at least in principle, to create macroscopically quantum states, and we expect any derivation of a classical limit to break down here. So what is the missing ingredient that usually leads quantum mechanics to look classical for systems small compared to , but that is not a strict mathematical inevitability? Decoherence. Indeed, isolated systems will generally produce grossly non-classical states, as evidenced by negativity in their Wiger function, even when they are initialized with nice coherent wavepacket states. But this is almost always destroyed by including even miniscule interactions with an environment . In fact, for the simplest cases of ideal quantum Brownian motion one can show that all initial states become exactly positive in finite time !
This suggests that we were right to dispense with the restriction to pure states when seeking a quantum-classical limit (motivated above by the observation that quantum mechanics is “intrinsically” stochastic, whatever that means). The decoherence that looks necessary to recover classical mechanics in the limit of quantum mechanics can only be obtained if we allow for open systems, which necessarily means dealing with density matrices rather than just pure states.
OK, but isn’t it disappointing that understanding this limit, as opposed to the limit of special relativity, appears to require a messy discussion of decoherence in particular systems with particular interactions? And we’re still confused about all that business with measurement and probabilities that mysteriously disappear in classical mechanics, right? And couldn’t we keep considering larger and larger systems until we got to the whole universe, prohibiting any reliance on open system dynamics? Yes, yes, and most definitely yes.
Edit 2016-2-7: Philosopher of science Joshua Rosaler has a recent article making similar arguments in Topoi: ‘Formal’ Versus ‘Empirical’ Approaches to Quantum–Classical Reduction (PDF.) A video of his talk on this at PI available here. See also his reference to the work of Batterman, with more technical discussion of the oscillatory singular nature of the limit.
Edited 2016-12-12: (added Berry quote).
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- Here, assume for simplicity that we have a Hamiltonian operator that can be expressed as a power series in the operators and , i.e., . (This is possible for any Hamiltonian that corresponds to a classical analytic Hamiltonian function on phase space , although there may be many quantum Hamiltonian operators for each classical Hamiltonian function on account of operator ordering.) We can then calculate that , using . Likewise, . In particular, keep in mind that is just some power series in the operators and .↵
- There is obviously a bunch of subtlety bundled up in distinction between stochastic and deterministic for quantum mechanics. We’re just operating intuitively here and won’t solve these issue.↵
- Wikipedia has a neat, strictly classical example: “Perhaps the first known anomaly was the dissipative anomaly in turbulence: time-reversibility remains broken (and energy dissipation rate finite) at the limit of vanishing viscosity.”↵
- The notation here is , with Sine (or any other analytic function) of the partial derivative operator being understood in terms of a power series.↵
- There are some deep issues here with operator ordering and the many-to-one nature of the quantum dynamics to classical dynamics. For quadratic Hamiltonians, the quantum and classical theories differ only by a constant offset, but for higher order Hamiltonians things can be more complicated. We are putting these aside and just trying to make quantum states one-to-one with classical states.↵