A complexified path integral for a system of harmonic oscillators Original Research Article

A functional is a function from the space of functions to a number field. We constructed an integral theory for a functional, using double extensions of real, complex number field in nonstandard arguments. In this paper, we consider a good system of infinite harmonic oscillators. Our functional integral implies that the propagator of the system is represented as the Riemann zeta function. The result is a direct generalization of our previous work [T. Nitta, Complexification of the propagator for the harmonic oscillator, in: Topics in Contemporary Differential Geometry, Complex Analysis and Mathematical Physics, World Scientific, 2006, pp. 261–268]. We remark that the variables are not only real numbers but also complex numbers. We will assume that the reader is familiar with Nonstandard Analysis.