Physical interpretation of differences between classical and quantum ensemble dynamics

The Groenewold-Moyal (phase space) picture of quantum mechanics describes the evolution of a probability density corresponding to a wavefunction that evolves as described by Schrödinger's equation. The central equation for Groenewold-Moyal quantum dynamics is the Moyal equation;

$$\frac{\partial \rho}{\partial t} = -\{\{ \rho,H\}\} = \frac{2}{\hbar}\rho\sin\left(\frac{\hbar}{2}\left(\overleftarrow{\partial_x}\overrightarrow{\partial_p}- \overleftarrow{\partial_p}\overrightarrow{\partial_x}\right)\right)H = -\left[\rho,H\right] + O(\hbar^2)$$

where $\{\{.,.\}\}$ is the Moyal bracket, $\left[.,.\right]$ is the Poisson bracket, $H$ is the Hamiltonian and $\rho$ is the phase space probability density.

The classical analogue, Liouville's equation, is simply $$\frac{\partial \rho}{\partial t} = -\left[\rho,H\right].$$ This paints a very clear mathematical picture of the correspondence between classical and quantum dynamics, namely that for $\hbar\rightarrow0$ we recover classical dynamics from the quantum picture.

This, however, opens up a few questions about the physical interpretation of what quantum mechanics "is" relative to classical mechanics in this picture. For example,

1. 
   

   Is there any physical reason why the "deformation" of the Poisson bracket into the Moyal bracket is specifically sinusoidal? Does it directly come from some fundamental assumption in the derivation of the Moyal equation?

   
   


2. 
   

   What is the physical role of $\hbar$ in this formulation of quantum mechanics? How would changing $\hbar$ change the evolution of the phase space probability density in a semi-intuitive sense?

   
   
   

Answer

Frankly, I am confused as to why you suspect you have to revisit QM interpretations in this picture; but your two circumscribed questions are answerable.

1. No physical reason--mere mathematical convenience. See Refs 1 & 2. The sine comes from the fact the relevant Groenewold star product is the exponential of the PB, a matter of coincidence, built into the Wigner map taking you from Hilbert space operators to phase-space c-number variables. Under this map, the quantum commutator antisymmetrizes these exponentials into a sine function of PBs. But other strictly mathematically equivalent star products, such as the one in Husimi's picture (cf. eqn. (124) of Ref. 1) will correspond to very different, in general messier, brackets. They all expand past the PB to $O(\hbar^2)$ terms including the strictly quantum deformation features, which are only neat powers of the sine expansion for the Groenewold-Moyal product... and awful messes in (most) other of the half-dozen pictures I know. It is these terms that specify the dynamics' departure from classical flows, as you may notice on the entertaining movies of Ref. 2, and give QM its distinctive diffusive and compressible flavor.


2. $~\hbar$ is a dimensionfull parameter, and, as such bimodal. Qualitatively, there are two cases: zero (classical), and nonzero (QM). Typically, it normalizes the phase space cell area, or the PB, or... into dimensionless entities. That is, you observe its (QM) effects for variables $xp/\hbar$ which are not enormous-ginormous... microscopic effects where $\hbar \partial_x \partial_p$ are not infinitesimal; when such action variables are enormous, as in a speeding locomotive or a mosquito, the QM deformation correction effects are normally invisible. The actual size of $\hbar$ only matters in the scale of objects falling into this microscopic quantum ambit. You might be invited to visualize a world of larger $\hbar$, so larger Bohr radius, so larger atoms, etc... The WP movies would look pretty much the same. Liouville's theorem is violated.

References:

1. Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014. The PDF file is available here.


2. WP article.