On the Quantum Field Theory of the Interaction between the Graviton and the Matter Field

Abstract

In order to quantize the gravitational field so as to be consistent with the general relativistic treatment, we mvestigate what physical etfects the quantizable and unquantizable parts of gravttat10nal field b ave, when the matter fields exist. To construct the Hamiltonian, we show that the mteraction Eamiltoman m the mteractton representation can be obtdined by the method extended from that of Yang-Feldman. The role of the non-linear part of gravitational field for such a problem IS also discussed. Only the potenttal, obtamed by elimmatmg the longitudinal and scalar parts of graVItational field with· the atd of the supplerr.entary condition, has an etfect upon the macroscopical phenomena and gives the equatton of two-body problem m general relativity found by Einstein, Infeld and Hotfmarm. On the other hand, while the pure transverse graviton does not contribute to the macro­ ncopical phenomena, 1t gives, m the region of high energy, the large damping etfect upon the mterac· tion between the elementary particles. § l. Introduction and summary Recendy, in connection with the difficulties encountered in the quantum field theory, the effect of the gravitational field upon the interaction between the elementaty patticles has been discussed by some authors. In order to discuss the gravitational field in the theory of quantized field, we must first consider what sort of the classical gravitational equation should be quantized, and secondly examine which patt of the gravitational field quantities can be quantized and what role another patt that cannot be quantized plays in the macroscopical phenomena. In spite of many attempts on the quantization of the gravitational field, there are neither necessaty reasons for the , quantization nor experimental ·evidences for the existence of the gravitational quanta (say, graviton). It will not be reasonable, therefore, to discuss only the quantum chatacter of gravitational field. In other words, the problem whether the field equations or field quantities that we quantize are appropriate, should be considered not merely by looking into the properties of graviton, but also by examining whether the procedure of quantization is compatible with the general relativistic treatment. For example, if we take Einstein's gravitational equation as the one that we quantize, the gravitational field must be quantized in such a way that the patt remaining unquantized can give the same physical effects as in the general relativity. In the linear approximation of Einstein's gravitational field equation, the metric