非对易平面上自由薛定谔方程的对称性
Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane
摘要：
研究了非对易平面上自由薛定谔方程的所有对称性。这些对称变换形成了一个无限维的外尔代数，它自然地出现在一个由伽利略助推和动量生成的二维海森堡代数中。这些无限高的对称性对于构造非相对论相互作用的高自旋理论是有用的。一个有限维的子代数是由薛定谔代数，其中除了伽利略生成元，还包含伸缩和扩展。我们考虑了在约化和扩展相空间中对称生成元的量子化，并讨论了这两种方法之间的关系。
英文摘要：
We study all the symmetries of the free Schrödinger equation in the non-commutative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schrödinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.
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