most recent 30 from http://mathoverflow.net 2013-06-19T07:44:32Z http://mathoverflow.net/feeds/user/8560 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35054/characterizing-the-harmonic-oscillator-creation-and-annihilation-operators-in-a-r/35908#35908 Zhang Jiang 2010-08-17T22:48:56Z 2010-08-17T22:48:56Z

<p>As a physicist let me try to give my opinion on this question. First, the polynomial representation you mentioned is called the Fock-Bargmann representation (in physics literiture) which can be found in Bargmann's paper in 60's (Bargmann was Einstein's assistant). This representation is useful for its convinience in dealing with Gaussian states. Second, while the statement that $H$ is invariant under ratation is probabaly ture, $V_{n+2m}$ does change under such rotation. What happens here is that $H$ has some degeneracy (the energy quantum for each mode $k$ is the same), rotations in the degenerate subspace don't change $H$. Third, may be the most convinient way to study the changing of states under rotation is to use the so called coherent state (eigenstates of the annihilation operators) instead of number states (eigenstates of $H$). Such coherent states which form an over complete non-orthogonal basis can be parametrized by a vector $\vec{\alpha}$, and such vectors rotate just like $C$. </p>