A Diagonalization Method of a Hamiltonian in Second-Quantization. II

Abstract

A method to obtain all the operators diagonalizing a given Hamiltonian in second-quan-tization theory is presented. The importance not only of the dynamical commutation relations but also of the kinematical and mutual commutation relations is pointed out. Different classes of Hamiltonians (with examples), programs, degree of a program, order of calculation are introduced. Finally, the practical utilization of the method is given. § 1. Introduction Let us consider the following problem: H is a given Hamiltonian expressed In terms of normal products of annihilation and creation operators in second-quantization and we propose to investigate all the operators 0 (k) satisfying [0 (k), HJ-= (j)kO (k), (1·1) where [, J-means the commutator, (j)k is a c-number, k is a parameter which stands for momentum, spin, isospin, mass, etc. We call 0 (k) an "eigen-operator" (physical operators)**) of Hand (j)k the corresponding eigen value of H. In what follows, (+) means the hermitic conjugate (or the complex conjugate for. a c-number). Let us also call kinematical commutation relations the commutation relations between the same kind of operators 0 (k) and its hermitic conjugate 0+ (k), we have