Field operators and their spectral properties in finite-dimensional quantum field theory

Abstract

In Ref. 1 we have considered the finite-dimensional quantum mechanics. There the quantum mechanical space of states wasV=C r. It is known that the second quantization of this space is the space of square-summable functions of finite number of variables(L 2(Rr,dx)) (Segal isomorphism). Creation and annihilation operators were introduced in Ref. 1, and the former coincided with the usual position and momentum operators in the conventional quantum mechanics. In this paper we shall investigate the spectral properties of field operators. We shall show that the isomorphism between the exponential ofV andL 2(Rr,dx) can be understood as the decomposition by generalized eigenvectors of field operators (“Fourier transform”).