含有高次微商的量子理论

这篇短文比较了两种含有高次微商的量子理论。一个是在某些情形下可用的,它将变数q表为许多适合二阶方程的Q的线性组合,而在量子化时,各个Q分别地被量子化。另一个是先将q的运动方程正则化,再引入量子条件。我们证明了两个理论,无论就各种量的对易关系而言,或就总哈密顿而言,是等效的。

HIGH ORDER EQUATIONS IN QUANTUM THEORY

The purpose of this short paper is to compare the two existing theories of quantization of equations of motion containing high derivatives. As well known, when the order of the derivatives of field quantities q are finite, it is possible in certain cases to express q as a linear combination of quantities Q, each of which satisfies an equation of the second order. Quantization proceeds as if the various Q are independent. On the other hand, one may, following Ostro-gradski, put the equations of motion for the variables q in canonical form and then perform a subsequent quantization. (Such a theory was also discussed by the author in an earlier paper.) It is obviously worthwhile to see if the two theories are identical.In the following it is proved that the above two theories are in fact identical, both with respect to the commutation rules between q, Q and their derivatives and with respect to the total Hamiltonian.The importance of the above result lies in:(i) just as in the first theory, the Hamiltonian in the second theory is also not positive definite. In other words, possibilities of obtaining in the second theory positive definite Hamiltonians for field equatiens of certain types are found not to exist.(ii) just as in the second theory, q and its first few derivatives commute. This implies that in extending the first theory formally to cases containing derivatives of infinite order, q and all its derivatives of finite order commute. Since this is an undesirable feature, formal extension of the two theories to cases containing derivatives of infinite order seems difficult.