✎ Phase Space Quantum Mechanics [The Physics Travel Guide]

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====== Phase Space Quantum Mechanics ======

<tabbox Intuitive> 
<blockquote>Ever since Werner Heisenberg’s 1927 paper on uncertainty, there has
been considerable hesitancy in simultaneously considering positions and
momenta in quantum contexts, since these are incompatible observables. [...] However, they too are wrong. Quantum mechanics (QM) can be consistently
and autonomously formulated in phase space, with c-number position
and momentum variables simultaneously placed on an equal footing, in a way
that fully respects Heisenberg’s principle. [...] The net result is that quantum mechanics works smoothly and consistently
in phase space, where position coordinates and momenta blend together closely
and symmetrically. Thus, sharing a common arena and language with classical
mechanics [14], QMPS connects to its classical limit more naturally and intuitively
than in the other two familiar alternate pictures, namely, the standard
formulation through operators in Hilbert space, or the path integral formulation. [...]

Still, as every physics undergraduate learns early on, classical phase space is
built out of “c-number” position coordinates and momenta, x and p, ordinary
commuting variables characterizing physical particles; whereas such observables
are usually represented in quantum theory by operators that do not commute.
How then can the two be reconciled? The ingenious technical solution to this
problem was provided by Groenewold in 1946, and consists of a special binary
operation, the ⋆-product (see Star Product), which enables x and p to maintain
their conventional classical interpretation, but which also permits x and p to
combine more subtly than conventional classical variables; in fact to combine
in a way that is equivalent to the familiar operator algebra of Hilbert space
quantum theory. Nonetheless, expectation values of quantities measured in the lab (observables)
are computed in this picture of quantum mechanics by simply taking
integrals of conventional functions of x and p with a quasi-probability density
in phase space, the Wigner function — essentially the density matrix in this
picture. But, unlike a Liouville probability density of classical statistical mechanics,
this density can take provocative negative values and, indeed, these can
be reconstructed from lab measurements [11].
<cite>[[https://arxiv.org/abs/1104.5269|Quantum Mechanics in Phase space]] by Curtright and Zachos</cite>

</blockquote>

<blockquote>Classical statistical mechanics is a ' crypto-deterministic'
theory, where each element of the probability distribution of the dynamical
variables specifying a given system evolves with time according to deterministic
laws of motion; the whole uncertainty is contained in the form of the initial distributions.
A theory based on such concepts could not give a satisfactory account of such
non-deterministic effects as radioactive decay or spontaneous emission (cf. Whittaker
(2)). Classical statistical mechanics is, however, only a special case in the general
theory of dynamical statistical (stochastic) processes. In the general case, there is
the possibility of 'diffusion' of the probability 'fluid', so that the transformation with
time of the probability distribution need not be deterministic in the classical sense.
In this paper, we shall attempt to interpret quantum mechanics as a form of such
a general statistical dynamics.

<cite>[[https://www.cambridge.org/core/services/aop-cambridge-core/content/view/9D0DC7453AD14DB641CF8D477B3C72A2/S0305004100000487a.pdf/quantum_mechanics_as_a_statistical_theory.pdf|Quantum Mechanics as a statistical theory]] by J. E. Moyal</cite>
</blockquote>

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**Reading Recommendations**

* [[https://arxiv.org/abs/1104.5269|Quantum Mechanics in Phase space]] by Curtright and Zachos
  
<tabbox Concrete> 
**Time Evolution**

The time evolution in the phase space formulation of quantum mechanics is described by the vonNeumann equation

$$ \frac{\partial f}{\partial t} = -\frac{1}{i \hbar} (f \star H - H \star f ). $$

For the Wigner function, this equation reads

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

where $\{\{ ,\}\}$ denotes the Moyal bracket and $\{ ,\}$ the [[advanced_notions:poisson_bracket|Poisson bracke]]t.

In the limit, $\hbar \to 0$ the von Neumann equation reduces to the [[theorems:liouvilles_theorem|Liouville equation]].

The difference between the von Neumann equation and the Liouville equation is that in the former the density of points in phase space is not conserved. Formulated differently, the probability fluid is diffusive and compressible.

----

**Reading Recommendations**

* See also the corresponding chapter in Ballentine's Quantum Mechanics book, and also 
  * [[http://aapt.scitation.org/doi/pdf/10.1119/1.16475|Canonical transformation in quantum mechanics]] Y. S. Kim and E. Wigner
  * [[https://arxiv.org/abs/1104.5269|A Concise Treatise on Quantum Mechanics in Phase Space]] by Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos,  World Scientific, 2014. 
  * [[https://en.wikipedia.org/wiki/Phase_space_formulation]]

----

**Important Papers**

* [[https://www.cambridge.org/core/services/aop-cambridge-core/content/view/9D0DC7453AD14DB641CF8D477B3C72A2/S0305004100000487a.pdf/quantum_mechanics_as_a_statistical_theory.pdf|Quantum Mechanics as a statistical theory]] by J. E. Moyal
<tabbox Abstract> 
In the phase space formulation, the transition from classical mechanics to quantum mechanics is known as [[advanced_tools:quantization|deformation quantization.]]

The map between function in the phase space formulation of quantum mechanics and the operators in the corresponding Hilbert space is known as Wigner–Weyl transform.

<tabbox Why is it interesting?> 
<blockquote>The chief advantage of the phase space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space".[5] This formulation is statistical in nature and offers logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two (cf. classical limit). <cite>https://en.wikipedia.org/wiki/Phase_space_formulation</cite></blockquote>

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