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=∂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.
In the complex action theory (CAT) we explicitly examine how the momentum and Hamiltonian are defined from the Feynman path integral (FPI) point of view based on the complex coordinate formalism of our foregoing paper. After reviewing the formalism briefly, we describe in FPI with a Lagrangian the time development of a ξ-parametrized wave function, which is a solution to an eigenvalue problem of a momentum operator. Solving this eigenvalue problem, we derive the momentum, Hamiltonian, and Schrodinger equation. Oppositely, starting from the Hamiltonian we derive the Lagrangian in FPI, and we are led to the momentum relation again via the saddle point for p. This study confirms that the momentum and Hamiltonian in the CAT have the same forms as those in the real action theory. We also show the third derivation of the momentum relation via the saddle point for q.
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 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 |TA| and some specific time, say, the present time |t|. Besides such a future-not-included theory, we can consider the future-included theory, in which not only the past state |∣A(TA)⟩| at the initial time |TA| but also the future state |∣B(TB)⟩| at the final time |TB| 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.
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 TB and the past state at time TA 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 TB−t and large t−TA corresponds to that of a future-not-included theory with a proper inner product for large t−TA. This correspondence principle suggests that the future-included theory is not excluded phenomenologically.
In Ref. [1] we have found errata, which are mainly a result of the corrections to Ref. [2]. They do not influence the result of the manuscript. A |dt|-dependent normalization factor, say |α(dt)1|, should be inserted on the right-hand sides of Eq. (34) and of the first line of Eq. (37). The following sentence should be inserted after the sentence “|C| is an arbitrary ... complex plane” below Eq. (34): “In addition, |α(dt)| is a |dt|-dependent normalization factor, which is properly fixed later.” On the right-hand side of the first line of Eq. (39), |∫C′dξ| should be replaced with |α(dt)1∫C′dξ|. The factor |m2πiℏdt| in the second line of Eq. (39) should be deleted. The phrase “where |H^| is given by” below Eq. (39) should be replaced with the following phrase: “where we have taken |α(dt)=m2πiℏdt| so that both sides of Eq. (39) correspond to each other in the vanishing limit of |dt|, and |H^| is given by”. The following sentence should be inserted below Eq. (40): “Then Eq. (39) is reduced to |∣ψ(t+dt)⟩=e−ℏiH^dt∣ψ(t)⟩|.” The following sentence should be added after the last sentence of Sect. 3.1: “In addition, we have derived the Schrödinger equation. Such a derivation of the Schrödinger equation is well known in the RAT [3].”
In Ref. [1] we have found errata. They are composed of two parts: one part is for the body, which is also explained in our recently published book [2], while the other part is for the appendix, which is mainly a result of the corrections to Ref. [3]. They do not influence the result of the manuscript. Rather, the latter part provides us a new additional result: the Schrödinger equation described with the Hamiltonian |H^B| has been derived for the future state |∣B(t)⟩| via the Feynman path integral in the complex action theory. In the fifth line below Eq. (5.8), where |f(DD†)−1| should have been replaced with |(f(D)f(D)†)−1|, we have chosen |f(D)| such that |(P†)−1(f(D)f(D)†)−1P†=F(H^†)|, which is rewritten as |(f(D)f(D)†)−1=F(D†)|. However, this relation does not stand, because the left-hand side is Hermitian, while the right-hand side is not Hermitian. Accordingly, the expression |Q′=F(H^†)Q| below Eq. (5.8), which was introduced based on the above relation, has to be corrected. In addition, the next statement, “|F(H^†)Q≃F(H^eff†)Q| for the restricted subspace,” is not right. This is because, for any reasonable function |h| and any state |∣A(t)⟩=∑iai(t)∣λi⟩| that obeys the Schrödinger equation |iℏdtd∣A(t)⟩=H^∣A(t)⟩|, the following relation holds for large |t−TA|: |h(H^)∣A(t)⟩≃h(H^eff+iBΛA)∣A~(t)⟩≡h~(H^eff)∣A~(t)⟩|, where we have used the automatic Hermiticity mechanism and introduced |∣A~(t)⟩≡∑i∈Aai(t)∣λi⟩|, |ΛA≡∑i∈A∣λi⟩⟨λi∣Q|, and another function |h~| such that |h~(Reλi)=h(Reλi+iB)|. Similarly, the statement “|Q2=F(H^eff†)Q| for the restricted subspace” given in Eq. (5.6) has to be corrected. A |dt|-dependent normalization factor, say |α(dt)1|, should be inserted on the right-hand sides of Eq. (A.2) and of the first line of Eq. (A.4). The following sentence should be inserted after the sentence “|C| is an arbitrary |…| complex plane” below Eq. (A.2): “In addition, |α(dt)| is a |dt|-dependent normalization factor, which is properly fixed later.” The factor |m2πiℏdt| in the second line of Eq. (A.4) should be deleted. The following sentences should be inserted after the phrase “where |…| Eq. (3.7)” below Eq. (A.4): “Here we have taken |α(dt)=m2πiℏdt| so that both sides of Eq. (A.4) correspond to each other in the vanishing limit of |dt|. Then Eq. (A.4) is reduced to |∣ψ(t+dt)⟩=e−ℏiH^dt∣ψ(t)⟩|.” The next sentence, “Thus we have found that |…| Eq. (A.2),” below Eq. (A.4) should be replaced with “Thus we have derived the Schrödinger equation and found that |…| Eq.(A.2).” The following sentence should be added after the above replaced sentence: “Such a derivation of the Schrödinger equation is well known in the real action theory [4].” Factors |α(dt)∗1|, |α(−dt′)∗1|, and |α(−dt)1| should be inserted on the right-hand side of the equation in the second sentence of the last paragraph of the appendix, on the right-hand sides of Eqs (A.5) and (A.6), respectively. The second sentence below Eq. (A.6), “Indeed, |H^B| is given |…||H^†|,” should be replaced with “Indeed, we obtain the Schrödinger equation |∣B(t−dt)⟩=eℏiH^Bdt∣B(t)⟩|, where |H^B| is given |…||H^†|.”