Studying the time development of the expectation value in the future-not-included complex action theory, we point out that the momentum relation (the relation analogous to $p=\frac{\partial L}{\partial \dot{q}}$ ), which was derived via the Feynman path integral and was shown to be correct in the future-included theory in our previous papers, is not valid in the future-not-included theory. We provide the correct momentum relation in the future-not-included theory, and argue that the future-not-included classical theory is described by a certain real action. In addition, we provide another way to understand the time development of the future-not-included theory by utilizing the future-included theory. Furthermore, properly applying the method used in our previous paper to the future-not-included theory by introducing a formal Lagrangian, we derive the correct momentum relation in the future-not-included theory.Studying the time development of the expectation value in the future-not-included complex action theory, we point out that the momentum relation (the relation analogous to $p=\frac{\partial L}{\partial \dot{q}}$), which was derived via the Feynman path integral and was shown to be correct in the future-included theory in our previous papers, is not valid in the future-not-included theory. We provide the correct momentum relation in the future-not-included theory, and argue that the future-not-included classical theory is described by a certain real action. In addition, we provide another way to understand the time development of the future-not-included theory by utilizing the future-included theory. Furthermore, properly applying the method used in our previous paper to the future-not-included theory by introducing a formal Lagrangian, we derive the correct momentum relation in the future-not-included theory.

arXiv:1304.4017Progress of Theoretical and Experimental PhysicsJul 1, 20131 citation

Future Work

Massimo Melucci

The Kluwer international series on information retrievalJan 1, 20150 citations

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

Keiichi Nagao, Holger Bech Nielsen

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 | $\hat{\mathcal O}$ | 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.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 |$\hat{\mathcal O}$| 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.

arXiv:1705.01585Progress of Theoretical and Experimental PhysicsJun 30, 20171 citation

Comment

V. Hizhnyakov

Springer eBooksJan 1, 19760 citations

Comment

Y. R. Shen

Springer eBooksJan 1, 19760 citations

Correspondence between future-included and future-not-included theories

Keiichi Nagao, HOLGER BECH NIELSEN

We briefly review the correspondence principle proposed in our previous paper, which claims 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 complex action theory whose path runs over not only past but also future, 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 $t-T_A$ . This correspondence principle suggests that the future-included theory is not excluded phenomenologically.We briefly review the correspondence principle proposed in our previous paper, which claims 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 complex action theory whose path runs over not only past but also future, 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 $t-T_A$. This correspondence principle suggests that the future-included theory is not excluded phenomenologically.

arXiv:1211.7269Nov 30, 20124 citations

Studying the time development of the expectation value in the future-not-included complex action theory, we point out that the momentum relation (the relation analogous to

p=\frac{\partial L}{\partial \dot{q}}

), which was derived via the Feynman path integral and was shown to be correct in the future-included theory in our previous papers, is not valid in the future-not-included theory. We provide the correct momentum relation in the future-not-included theory, and argue that the future-not-included classical theory is described by a certain real action. In addition, we provide another way to understand the time development of the future-not-included theory by utilizing the future-included theory. Furthermore, properly applying the method used in our previous paper to the future-not-included theory by introducing a formal Lagrangian, we derive the correct momentum relation in the future-not-included theory.

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{\mathcal 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.

We briefly review the correspondence principle proposed in our previous paper, which claims 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 complex action theory whose path runs over not only past but also future, 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

. This correspondence principle suggests that the future-included theory is not excluded phenomenologically.