关于量子Poincaré-Cartan积分不变量

On Quantal Poincaré-Cartan Integral Invariant

基于相空间Ｇｒｅｅｎ函数的生成泛函，导出了普遍情况下正规Ｌａｇｒａｎｇｅ量系统和奇异 Ｌａｇｒａｎｇｅ量系统的量子Ｐｏｉｎｃａｒé－Ｃａｒｔａｎ（ＰＣ）积分不变量．证明了该不变量与量子正则方程等价．当变换的Ｊａｃｏｂｉ行列式不为１时，仍可导出量子ＰＣ积分不变量；这与量子Ｎｏｅｔｈｅｒ定理不同．并将量子ＰＣ积分不变量与经典情况作了对比．结果表明：经典和量子ＰＣ积分不变量成立的条件和表达式均不同．

Based on the phase-space generating funtional of green function, the quantal Poincere- Cartan integrel invariant (QPCII) for a system with a regular and a singular Lagrangian are deduced respectively, the equivalence between the QPCII and quantum canonical equation is verified. For the case in which the Jacobian of the transformation does not equal 1, QPCII still can be derived, which is different from Noether theorem at the quantum level. The comparison of QPCII with the classical results is discussed. The result shows that their requirements and expressions are different.