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 |$\frac{1}{\alpha(dt)}$|, 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, |$\alpha(dt)$| is a |$dt$|-dependent normalization factor, which is properly fixed later.” On the right-hand side of the first line of Eq. (39), |$\int_C' d\xi$| should be replaced with |$\frac{1}{\alpha(dt)} \int_{C'} d\xi$|. The factor |$\sqrt{\frac{2\pi i \hbar dt}{m}}$| in the second line of Eq. (39) should be deleted. The phrase “where |$\hat{H}$| is given by” below Eq. (39) should be replaced with the following phrase: “where we have taken |$\alpha(dt)= \sqrt{\frac{2\pi i \hbar dt}{m}}$| so that both sides of Eq. (39) correspond to each other in the vanishing limit of |$dt$|, and |$\hat{H}$| is given by”. The following sentence should be inserted below Eq. (40): “Then Eq. (39) is reduced to |$|\psi(t+dt) \rangle = e^{-\frac{i}{\hbar} \hat{H} dt}|\psi(t) \rangle$|.” 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, 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^†|.”
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.
I believe that the author of this note is confusing definite and indefinite integrals. Any indefinite integre 'I I can be reduced to the corresponding definite integral by the proper choice of the limits of integration and the constant of integration. The fact that [5] is a solution of [4] can be looked-up directly from any table of elementary integrals; see, for example, ref. 1. For finite values of z i t is easy to show that [3] and [4] are equivalent to the time-dependent Schrodinger's wave equation by taking the time derivative of both sides of [4]. Moreover, the limit z + 0 cannot be allowed, since from [3] this corresponds to an infinite energy E. The meaning of the unit of imaginary numbers in on the right-hand side of [4] is given immediately after this equation.
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.
1) Equation (2·8) should read к×vк=0. 2) The summand in the first term ori the right-hand side of Eq. (2·9) should be supplemented by a factor δ0,p+q. 3) Every summation symbol Σ in Eqs. (3·7) and (3·14) should be replaced by Σ′. 4) The inner product (p·s) in the expression for FII(3, 11) (p, q, r; s) in Eq. (3·25) should be followed by a factor √λqλrλs/λp in place of √λpλrλs/λp. 6) On the right-hand side of (4·22), the factor 2 should be dropped from the second term in the square bracket [⋯]. 7) The denominator (Zλ0)2 on the right-hand side of (4·23) should be replaced by (4λ0)2). 8) The first term in the square bracket [⋯] in Eq. (4·24) should read (1-λ02) in place of (1-λ0)2. 10) The last term on the right-hand side of Eq. (A·1) should be multiplied by 2.
We formulate a complex action theory which includes operators of coordinate and momentum q^ and p^ being replaced with non-hermitian operators q^new and p^new, and their eigenstates ${}_m