Theory Including Future Not Excluded -- Formulation of Complex Action Theory II --

Abstract

We study a complex action theory (CAT) whose path runs over not only past but also future. We show that if we regard a matrix element defined in terms of the future state at time $T_B$ and the past state at time $T_A$ as an expectation value in the CAT, then we are allowed to have the Heisenberg equation, the Ehrenfest's theorem and the conserved probability current density. In addition we show that the expectation value at the present time $t$ of a future-included theory for large $T_B-t$ and large $t- T_A$ corresponds to that of a future-not-included theory with a proper inner product for large $t- T_A$. Hence the CAT with future explicitly present in the formalism and influencing in principle the past is not excluded phenomenologically, because the effects are argued to be very small in the present era. Furthermore we explicitly derive the Schr\"{o}dinger equation and Hamiltonian for the future state via path integral, and confirm that the Hamiltonian is given by the Hermitian conjugate of the Hamiltonian for the past state.