I would like to analyse the semiclassical and nonrelativistic limit of the Klein-Gordon equation, \begin{equation} \frac{1}{c^2} \partial_t^2 \phi - \Delta \phi + \frac{M^2 c^2}{\hbar^2} \phi =0. \label{KGEscalar} \end{equation}

Therefore I used a Wigner transformation and passed to the limit $\epsilon = \frac{1}{c^2}=\hbar \to 0$. The evolution equation for the Wigner density $\omega^{\epsilon}$ in the limit which I obtained in both cases is \begin{equation} \frac{\partial}{\partial t} \omega^{0} =0. \end{equation} This means that the "quasi probability" to detect a particle at a specific location in the phase space is constant, i.e. it is zero when we demand any normalization conditions.

I mean this is not surprisingly when we consider the Klein-Gordon equation above, if we let $\epsilon \to 0$ this of course would lead to problems. But this seems really strange to me from a physical point of view. In the nonrelativistic limit we have that the rest energy become unbounded and therefore can cause problems. But what happens in the semiclassical limit, i.e. $\hbar \to 0$? At the moment I'm not sure how to intepret this result.

To be clear, I defined the Wigner transformation for $L^2$ functions $f, g$ as

\begin{equation} \omega^{\epsilon} (f, g) (x, \xi) := \int _{\mathbb{R}} f(x-\frac{v} {2} \epsilon) g(x-\frac{v} {2} \epsilon) e^{i v \xi } d v. \end{equation}

Then choose $f=g= \phi$. The evolution equation for $\omega^{\epsilon}_q (\phi, \phi) $ can easily be obtained by a Fourier transformation $\mathscr{F} (x \mapsto \zeta) $and reads \begin{equation} \partial_t \omega^{\epsilon}_q - i \zeta \int _{-\frac{1}{2} }^{\frac{1}{2} } \nabla \lambda_q (\xi - s \epsilon \zeta) ds=0. \end{equation} Where $\lambda_q $ are the eigenvalues of the associated symbol of the differential operator of the Klein Gordon equation. The constants $c$ and $\hbar$ enter in the symbol.

  • $\begingroup$ Frankly, I have no idea where the last equation emerged from... the single time derivative? the absence of a star product? As I indicate in my answer, it is so much easier to look at all the plane-wave solutions of the K-G equations than pretend to solve it from first principles in phase space... $\endgroup$ – Cosmas Zachos Nov 19 '18 at 23:22
  • $\begingroup$ The single time derivative simply comes from expressing the KGE as a first order system of equations. The last equation come from manipulating the expressions defined above and computing the eigenvalues of the associated symbol of the differential operator. I would like to understand the limits of the KGE in the framework of pseudo differential operators, therefore I chose the approach above $\endgroup$ – Hamilcar Nov 19 '18 at 23:31
  • $\begingroup$ Should be linear in $\omega$... actually the star convolution should vanish before any limit.... $\endgroup$ – Cosmas Zachos Nov 20 '18 at 4:01
  • $\begingroup$ I'm not sure which convolution operation you mean, but when one is passing to the limit $\epsilon \to 0$, the evolution equation is in general non linear, e.g. when one is considering the Schrödinger equation with a harmonic oszillator potential instead $\endgroup$ – Hamilcar Nov 20 '18 at 8:30
  • $\begingroup$ Moyal’s equation is linear. Be more explicit in your ill defined evolution equation. $\endgroup$ – Cosmas Zachos Nov 20 '18 at 12:13

I would go to the nonrelativistic limit in a different way. Substitute $\Psi = e^{-mc^2/\hbar} \phi$ and neglect the second time derivative of $\phi$. The result is the Schrödinger equation, so it is sufficient to study the Schrödinger equation in the semiclassical limit.


Assuming you are considering the plane-wave solution of the K-G equation, take it to be in 1+1 for computational simplicity, $$ \phi \propto \exp \left (-it\sqrt{k^2 + M^2c^2/\hbar^2} +ikx\right ), $$ the (real) Wigner transform of its density matrix would be proportional to the unavoidable $$ \int dy e^{-iyp} e^{-ik(x-y\hbar/2)+ik(x+y\hbar/2 ) } \propto \delta (p-\hbar k).$$

The $k\mapsto -k$ case is, equivalently, also a solution. c does not enter because the time dependence washes out (energy-momentum conservation), while $\hbar$ may be absorbed into the units of the initial (conserved) wavenumber k. As un-normalizable plane waves, issues of normalization are moot. Perhaps you wish to provide more clearly and explicitly your normalization considerations.

In consequence, as you might expect for a free wave, the Wigner function is nothing but conservation of momentum, in any limit. You never had to take such.

It is independent of time, of course, and a plain flat line parallel to the x-axis, as in the non-relativistic case.


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