Time-Ordered Evolutions on the Feynman-Dyson Hilbert Space

Abstract

In this work, we consider hyperbolic and parabolic evolution problems on the Feynman-Dyson Hilbert space, FD2 circle times. We use the possible opportunities given in FD2 circle times to find solutions for both homogeneous and non-homogeneous cases. Therefore, we first focus the structure of the Feynman-Dyson Hilbert space from a mathematical perspective in terms of the construction of this space and the lifting of operator theory to this time-ordered setting. We then observe that FD2 circle times allows operators acting at different times to commute, while maintain their relative position on paper. We also deal with a time-ordered version of the Hille-Yosida theorem for semigroups of operators. This approach has the added advantage of requiring the weakest known domain and continuity conditions. We show these advantages for the generic classes of time-dependent homogeneous hyperbolic and parabolic problems. We also see that the theory has advantages for operators with no time dependence.