By complex phase space I mean $z=q + i\,p \in \mathbb{C}^{n}$ and $z = q -i \, p \in \mathbb{C}^{n}$ and the Hamiltonian $H(z,\bar{z})$ is a function of $z, \bar{z} \in \mathbb{C}^{n}$.

  • $\begingroup$ I found this "An almost complex structure integrates to a complex structure if the half-dimensional subspace is closed under the Lie bracket." on wikipedia. $\endgroup$ – v217 Oct 14 '16 at 12:40
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    $\begingroup$ Complex and holomorphic models penetrate all areas of physics. $\endgroup$ – Qmechanic Oct 14 '16 at 16:43
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    $\begingroup$ So $q$ and $p$ would be the canonical variables? $\endgroup$ – valerio Oct 15 '16 at 17:41
  • $\begingroup$ @valerio92 yes. $\endgroup$ – v217 Oct 16 '16 at 11:03
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    $\begingroup$ There are many physical problems where the hamiltonian can be reduced to a function of the variables $\alpha q \pm i \beta p$. I don't know if you could be interested in this. $\endgroup$ – valerio Oct 16 '16 at 21:00

What follows doesn't completely answer your question but hopefully it can be useful.

A classical example of an hamiltonian $H(p,q)$ which can be reduced to the form $H(z_+,z_-)$ with $z_{\pm}=\alpha q \pm i \beta p$ is the harmonic oscillator. The hamiltonian of an harmonic oscillator is

$$H(p,q)=\frac 1 {2m} [p^2 + m \omega q^2]$$

Using the identity

$$p^2+m \omega q^2 = (p+i q \sqrt{m \omega})(p-i q \sqrt{m \omega})$$

we see that we can write the hamiltonian as

$$H(p,q) = \frac 1 {2m} (p+i q \sqrt{m \omega})(p-i q \sqrt{m \omega})\\= \left(\frac p {\sqrt{2m}} + i q \sqrt{\frac \omega {2}}\right) \left(\frac p {\sqrt{2m}} - i q \sqrt{\frac \omega {2}}\right)$$

and defining

$$z_{\pm} = \left(\frac p {\sqrt{2m}} \pm i q \sqrt{\frac \omega {2}}\right)$$

we obtain

$$H(z_+,z_-) = z_+ z_-$$

The same trick can be used whenever $p,q$ appear as couples of quadratic terms in the hamiltonian. In quantum mechanics, this trick leads to the definition of the creation and annihilation operators.


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