Phase spaces which are not cotangent bundles can be realized in mechanical systems with phase space constraints . The phase space given by Arnold: the two sphere $S^2$ can be mechanically realized as the reduced dynamics of an energy hypersurface of a two dimensional isotropic harmonic oscillator:

$ p_1^2+p_2^2+q_1^2+q_2^2= E$

We observe that the Hamiltonian generates a constant rotation rate in the $(p,q)$ planes, namely:

$ (p_i(t)+iq_i(t)) = exp(-iE_it) (p_i(0)+iq_i(0)) $

Thus we may choose to look at the system from the point of view of a "rotating system in phase space" in which the vector in the $(p_1, q_1)$ plane is always in the direction of $q_1$. Of course, we cannot do that on both planes because we have only one degree of freedom. Thus we are left with:

$p_2^2+q_1^2+q_2^2 = E$,

which is just the equation of a two-sphere. Thus the reduced dynamics of a constant energy hypersurface is on a two-sphere.

The symplectic form has to be proportional the area of the sphere, because it is the volume form of the sphere and a two sphere has only one volume form.

This approach gives us a very big bonus upon quantization. It is well known that from the quantization of a sphere we get spin quantization. From the point of view of the isotropic oscillator for $E = 2j \hbar$, ($j$ is half integral), this quantization corresponds to the following energies of the individual oscillators: $(2j, 0), (2j-1, 1), .,.,., (0,2j)$. As can be seen there are exactly (2j+1) states as in the spin system.

The full theory of quantizations allows to write the corresponding wave functions also in the coordinates of the two sphere. Thus, we actually quantized the isotropic oscillator using spin quantization.

The equivalence of this procedure to the standard quantization of the isotropic harmonic oscillator is a very celebrated theorem by Guillemin and Sternberg called "Quantization commutes with reduction". Actually, this is the principle we apply when we quantize gauge theories (although there is no formal proof for the infinite dimensional case). You can find on the net numerous works on this subject.

This post imported from StackExchange Physics at 2015-03-30 13:53 (UTC), posted by SE-user David Bar Moshe