Reality from maximizing overlap in the future-included real action theory

Keiichi Nagao,Holger Bech Nielsen

doi:10.1093/ptep/ptx102

Abstract

In the future-included real action theory whose path runs over not only past but also future, we demonstrate a theorem, which states that the normalized matrix element of a Hermitian operator $\hat{\cal O}$ defined in terms of the future state at the final time $T_B$ and the fixed past state at the initial time $T_A$ becomes real for the future state selected such that the absolute value of the transition amplitude from the past state to the future state is maximized. This is a special version of our previously proposed theorem for the future-included complex action theory. We find that though the maximization principle leads to the reality of the normalized matrix element in the future-included real action theory, it does not specify the future and past states so much as in the case of the future-included complex action theory. In addition, we argue that the normalized matrix element seems to be more natural than the usual expectation value. Thus we speculate that the functional integral formalism of quantum theory could be most elegant in the future-included complex action theory.

Highlights

Quantum theory is usually formulated so that in the path integral the time integration is performed over the period between the initial time TA and some specific time, say, the present time t
In Ref.21) we proposed a theorem that states that, provided that an operator Ois Q-Hermitian, i.e., Hermitian with regard to the proper inner product IQ which makes the given Hamiltonian normal by using an appropriately chosen Hermitian operator Q, the normalized matrix element defined with IQ becomes real and time-develops under a Q-Hermitian Hamiltonian for the past and future states selected such that the absolute value of the transition amplitude defined with IQ from the past state to the future state is maximized
We showed that, provided that |A(t) and |B(t) time-develop according to the Schrodinger equation with a given Hermitian diagonalizable Hamiltonian Hand are normalized at the initial time TA and at the final time TB respectively, O BA becomes real for the given state |A(t) and the chosen state |B(t) max such that the absolute value of the transition amplitude

Summary

Introduction

Quantum theory is usually formulated so that in the path integral the time integration is performed over the period between the initial time TA and some specific time, say, the present time t. We find that though the maximization principle leads to the reality of the normalized matrix element in the future-included RAT, it does not specify the future and past states so much as in the case of the future-included CAT. Arguing that the normalized matrix element O BA seems to be more natural than the usual expectation value O AA, we speculate that the functional integral formalism of quantum theory could be most elegant in the future-included CAT.

Results

Conclusion