# coherent states phase-space topology

Quantum mechanics can be formulated in various different ways. One of these is the so called phase space formulation, where we use quasi-probability distribution functions. The most recognized is the Wigner function, but one may also encounter Husimi $Q$ function or Glauber-Sudarshan $P$ function, where we make use of overcompletness property of coherent states.

Phase space of coherent states associated with Heisenberg-Weyl Lie group is isomorphic to complex plane. However, different systems may posses symmetries of arbitrary Lie group (especially $\mathfrak{su}$) and phase space topology differing from the complex plane Trimborn 1, Trimborn 2.

My question is specific for cold atom gases: If You make mean-field approximation of the Hamiltonian (quantum) with dynamical group $\mathfrak{su}(n)$ is the "classical" phase space flat (like for complex plane) or curved? Is it possible to have chaos in two-dimensional systems with non-flat geometry unlike Wikpedia?

• What is your notion of flatness for the phase space? – ACuriousMind Jan 1 '15 at 23:35
• non-compact topology of complex plane – WoofDoggy Jan 2 '15 at 13:45
• Soo...you call any phase space that is not the complex plane with the usual topology non-flat? – ACuriousMind Jan 2 '15 at 14:00