论文标题
何时泊松和千而且托架相等？
When Poisson and Moyal Brackets are equal?
论文作者
论文摘要
In the phase space $\R^{2d}$, let us denote $\{A,B\}$ the Poisson bracket of two smooth classical observables and $\{A, B\}_\circledast $ their Moyal bracket, defined as the Weyl symbol of $i[ A, B]$, where $ \hat A$ is the Weyl quantization of $A$ and $[ \hat a，\ hat b] = \ hat a \ hat b- \ hat b \ hat a $（换向器）。 在本说明中，我们证明，如果相位空间上平稳的汉密尔顿$ h $ $ \ r^{2d} $，带有中等增长的衍生物，请满足 $ \ {a，h \} = \ {a，h \} _ \ circledast $对于任何平滑且有限的可观察到$ a $ then $ h $，$ h $最多必须是多项式的2。 这与多项式可观察物的量化有关的Groenewold-Van hove定理\ cite {gotay，groen，vhove}。
In the phase space $\R^{2d}$, let us denote $\{A,B\}$ the Poisson bracket of two smooth classical observables and $\{A, B\}_\circledast $ their Moyal bracket, defined as the Weyl symbol of $i[ A, B]$, where $ \hat A$ is the Weyl quantization of $A$ and $[ \hat A, \hat B]= \hat A \hat B- \hat B \hat A$ (commutator). In this note we prove that if a smooth Hamiltonian $H$ on the phase space $\R^{2d}$, with derivatives of moderate growth, satisfies $\{A,H\}= \{A, H\}_\circledast$ for any smooth and bounded observable $A$ then $H$ must be a polynomial of degree at most 2. This is related with the Groenewold-van Hove Theorem \cite{Gotay, Groen, vHove} concerning quantization of polynomial observables.
