| Popis: |
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. |
