We can consider a dynamical theory to be a "transport theory" if it can be described entirely by a series of continuity equations of the form:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot \left({\rho \otimes \vec{v}}\right) = \sigma$$

where $\rho$ is a tensor of arbitrary order, $\vec{v}$ represents a generalized velocity, and $\sigma$ represents a source term.

The Groenewold theorem appears to directly imply that there is no valid quantum transport theory for a probability density over phase space; in vastly simplified terms, the canonical commutation relations prevent the validity of any probability that a particle is in position $q$ while possessing momentum $p$. Consequently, deformation quantization must be employed, the probability density must be changed into a Wigner function, and most "classical transport" aspects of the theory are lost–unless, perhaps, if one considers the existence of the Wigner function to be "real" rather than a mathematical tool.

Alternatively, the Madelung equations describe the flow of a probability density function over position space that should be equivalent to the evolution of a wavefunction in position space in the Schrödinger picture. These do appear to be formulated in "classical transport" form, albeit with a highly complicated source term called the quantum potential.

Since both formulations should be theoretically equivalent, it appears as if the issues caused by the Groenewold theorem disappear once one "integrates" over the momentum coordinates of the Wigner function. That being said, the effects of the uncertainty principle don't seem to pop up explicitly in the Madelung formulation. In short,

1. How do the effects of the uncertainty principle appear in the Madelung formulation of quantum mechanics?

2. Is there a "classical transport" version of the Groenewold-Moyal picture in which the quantity $\rho$ being advected is the Wigner function $\psi$?

I apologize if this question is too long-winded/broad; I'd be happy to delete/edit as needed.

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    $\begingroup$ Related; covers compressible phase space quantum flows. $\endgroup$ Jun 25, 2018 at 19:38
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    $\begingroup$ The above link takes you to a well-known advection for the Wigner function, so "almost yes" to your 2: Euler compressible flow in phase-space, therefore no trajectories, unlike Bohemian trajectories in Madelung. But I have no clue what/why you are asking about the UP, as time dependence does not directly impact it... are you asking how S enters it differently than ρ ? $\endgroup$ Jun 26, 2018 at 0:08
  • $\begingroup$ NB I certainly wouldn't wish to leave you with the mis-impression that Madelung flow is compressible, however. It is not! $\nabla\cdot {\mathbf u}\neq 0$. $\endgroup$ Jun 26, 2018 at 15:57
  • $\begingroup$ Thank you for the helpful comments @CosmasZachos! Your linked answer certainly covers question 2. Regarding 1, the intent was to figure out whether or not the uncertainty principle was encoded directly into the Madelung equations via the quantum potential or if it entered in as a separate auxiliary condition. $\endgroup$ Jun 26, 2018 at 16:01
  • $\begingroup$ Madelung is all about flow.... The basic Richardson UP argument inequality works identically for the radius-and-phase change of variables of the wave function, here, so nothing new.... $\endgroup$ Jun 26, 2018 at 16:11

1 Answer 1


Here is an oversimplified but perhaps helpful way to think about it:

  1. On one hand, one may consider the Heisenberg equations of motion $$\frac{df}{dt}~=~ \frac{1}{i\hbar} [f\stackrel{\star}{,}H]+\frac{\partial f}{\partial t}\tag{1}$$ for an observable $f$ in the Heisenberg picture. In particular, one has Liouville eq.
    $$0~=~\frac{d\rho }{dt}~=~ \frac{1}{i\hbar} [\rho \stackrel{\star}{,}H]+\frac{\partial \rho }{\partial t}\tag{2}$$ for the density operator (since states don't evolve in the Heisenberg picture). The Groenewold-Moyal star product $\star$ helps in both case to systematically organize the non-commutativity and HUP of observables. See also this & this related Phys.SE posts.

  2. On the other hand, the Madelung equations descend from the TDSE for the wavefunction $\psi$ in the Schrödinger picture. Concerning non-commutativity of operators and HUP, the Madelung variables $\rho$ and ${\bf u}$ offer no systematic independent framework/platform beyond the underlying wavefunction $\psi$, TDSE & Schrödinger picture.


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