[Other parts in this series: .] 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 (6)   where the Moyal bracket is a different binary operator on the space of functions over phase space. I’ll just quote the definition up from the Wikipedia pageThe notation here is , with Sine (or any other analytic function) of the partial derivative operator being understood in terms of a power series.d  , (7)   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 (essentiallyThere 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.e  ) 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). Footnotes (↵ returns to text) 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… Continue reading

Suggestions for my new format for estimating square roots? Still want it to be visual. #esumating #squareroots #visualmath #maths #teach180 #secondarymath #mathchat