On the Analytic Behaviour of Dyson Transformation Function

Abstract

The detailed properties of the Dyson transformation function, which represents the time development of an initial free state at infinite past due to an interaction, adiabatically switched in and off according to a factor exp( -alII/if), (a>O), were examined in special connection with bound states. The tansforma­ tion function is unitary and the power expansion in the interaction Hamiltonian is finite and regular in the limit a->O, except those matrix elements with respect to the lowest energy free state of the system in question. An initial free state except the lowest energy one is transformed at a finite time t into the corresponding unbound eigenstate of the total Hamiltonian with the same energy. Transitions to bound states are due to those initial states whose energy falls in the interval of the order e from that of the lowest energy state. § 1. Introduction and summary By the elaborate work of Dyson, based upon the covariant fonnalism of the field theory due to Tomonaga, Schwinger and Feynmann, the mathematical method of dealing with non-stationary problems perturbation theoretically seems to have been completely established. On the other hand, there remains unsolved the problem to establish a for­ mulation of the method of treating bound stationary states, which cannot be treated with perturbation theory, upon the correct field theoretical grounds_ In this connection, it may be of considerable value to examine the properties of the transformation function, which played an important role in Dyson formalism and yet has been treated rather superficially so far, in connection with bound states. Some worksl ) have been made on this line. However, definite conclusions seem to have not yet been drawn. In this paper we shall be chiefly concerned, with the fonnal mathematical character of the transformation function. We begin with the derivation of the equations for unbound and bound stationary state wave functions without use of perturbation theory and define from them the so called Heisenberg-M~ller's S matrix,2) say SH' representing the asymptotic behaviour of the unbound stationary states. The non-stationary solution of the system starting from a given initial free state at a time t = to can be expressed by means of the usual unitary transformation function U (t, to)' given in power series in the interaction Hamiltonian. In the limit to--oo, the time integrations appearing in U(t, to)~a can be perfonned formally unless ~ a is the lowest energy free state of the system in question, * and lim U(t, to) ~a coincides with the corresponding unbound stationary state 1F a with the 10+-00