A Remark on the Takahashi-Umezawa-Katayama Theory of Interaction Representation : Theory of Quantization of Field Part I

The general theory of interaction representation proposed by Takahashi and Umezawa is re-examined by our improved method for a simple case of interaction Lagrange function with the fourth derivative. This example shows that the expansion with respect to the interaction parameter g is impossible for some cases. In the case of quantization by one harmonic oscillator (Heisenberg-Pauli's quantization of field), an interaction Hamiltonian H' can be determined, but is proved to be valid only up to its first order of the interaction parameter g, while in the case of quantization of field by two harmonic oscillators (Pais-Uhlenbeck's quantization of field), the Hamiltonian is valid up to any order of g. Thus the general proof of the existence of the G-function given by Takahashi and Umezawa becomes doubtful. To quantize the field, our method used in this paper will be more general than that of Takahashi and Umezawa