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arXiv:1211.7269

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

Keiichi Nagao, HOLGER BECH NIELSEN

Published Nov 30, 2012• 4 citations
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Abstract

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.

DOI
10.48550/arxiv.1211.7269

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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 TBT_BTB​ and the past state at time TAT_ATA​ 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 ttt of a future-included theory for large TB−tT_B-tTB​−t and large t−TAt-T_At−TA​ corresponds to that of a future-not-included theory with a proper inner product for large t−TAt-T_At−TA​. This correspondence principle suggests that the future-included theory is not excluded phenomenologically.
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 TB and the past state at time TA as an expectation value in the CAT, then we are allowed to have the Heisenberg equation, 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 TB − t and large t − TA corresponds to that of a future-not-included theory with a proper inner product for large t − TA. 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 Hamiltonian for the future state via a path integral, and confirm that it is given by the Hermitian conjugate of the Hamiltonian for the past state.
In the future-included complex and real action theories whose paths run over not only the past but also the future, we briefly review the theorem on the normalized matrix element of an operator O^\hat{\cal O}O^, which is defined in terms of the future and past states with a proper inner product IQI_QIQ​ that makes a given Hamiltonian normal. The theorem states that, provided that the operator O^\hat{\cal O}O^ is QQQ-Hermitian, i.e. Hermitian with regard to the proper inner product IQI_QIQ​, the normalized matrix element becomes real and time-develops under a QQQ-Hermitian Hamiltonian for the past and future states selected such that the absolute value of the transition amplitude from the past state to the future state is maximized. Discussing what the theorem implicates, we speculate that the future-included complex action theory would be the most elegant quantum 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 |O^\hat{\mathcal O}O^| defined in terms of the future state at the final time |TBT_BTB​| and the fixed past state at the initial time |TAT_ATA​| 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 quantum theory its action is usually taken to be real, but we can consider another theory whose action is complex. In addition, in the Feynman path integral, the time integration is usually performed over the period between the initial time |TAT_ATA​| and some specific time, say, the present time |ttt|⁠. Besides such a future-not-included theory, we can consider the future-included theory, in which not only the past state |∣A(TA)⟩| A(T_A) \rangle∣A(TA​)⟩| at the initial time |TAT_ATA​| but also the future state |∣B(TB)⟩| B(T_B) \rangle∣B(TB​)⟩| at the final time |TBT_BTB​| is given at first, and the time integration is performed over the whole period from the past to the future. Thus quantum theory can be classified into four types, according to whether its action is real or not, and whether the future is included or not. We argue that, if a theory is described with a complex action, then such a theory is suggested to be the future-included theory, rather than the future-not-included theory. Otherwise persons living at different times would see different histories of the universe.
In the complex action theory whose path runs over not only past but also future, we study a normalized matrix element of an operator |O^\hat {\mathcal O}O^| defined in terms of the future state at the latest time |TBT_BTB​| and the past state at the earliest time |TAT_ATA​| with a proper inner product that makes normal a given Hamiltonian that is non-normal at first. We present a theorem that states that, provided that the operator |O^\hat {\mathcal O}O^| is |QQQ|-Hermitian, i.e., Hermitian with regard to the proper inner product, the normalized matrix element becomes real and time-develops under a |QQQ|-Hermitian Hamiltonian for the past and future states selected such that the absolute value of the transition amplitude from the past state to the future state is maximized. Furthermore, we give a possible procedure to formulate the |QQQ|-Hermitian Hamiltonian in terms of |QQQ|-Hermitian coordinate and momentum operators, and construct a conserved probability current density.
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=∂L∂q˙p=\frac{\partial L}{\partial \dot{q}}p=∂q˙​∂L​), 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.