Theory including future not excluded: Formulation of complex action theory II

Abstract

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 |$\hat{H}_B$| has been derived for the future state |$| B(t) \rangle$| via the Feynman path integral in the complex action theory. In the fifth line below Eq. (5.8), where |$f(D D^\dagger)^{-1}$| should have been replaced with |$( f(D) f(D)^\dagger )^{-1}$|⁠, we have chosen |$f(D)$| such that |$(P^\dagger)^{-1} ( f(D) f(D)^\dagger )^{-1} P^\dagger = F(\hat{H}^\dagger)$|⁠, which is rewritten as |$( f(D) f(D)^\dagger )^{-1} = F(D^\dagger)$|⁠. However, this relation does not stand, because the left-hand side is Hermitian, while the right-hand side is not Hermitian. Accordingly, the expression |$Q'=F(\hat{H}^\dagger) Q$| below Eq. (5.8), which was introduced based on the above relation, has to be corrected. In addition, the next statement, “|$F(\hat{H}^\dagger) Q \simeq F(\hat{H}_\text{eff}^\dagger) Q$| for the restricted subspace,” is not right. This is because, for any reasonable function |$h$| and any state |$|A(t) \rangle = \sum_i a_i(t) | \lambda_i \rangle$| that obeys the Schrödinger equation |$i\hbar \frac{d}{dt} | A (t) \rangle = \hat{H} | A (t) \rangle$|⁠, the following relation holds for large |$t-T_A$|⁠: |$h(\hat{H}) | A (t) \rangle\simeq h(\hat{H}_\text{eff} + i B \Lambda_A ) | \tilde{A} (t) \rangle\equiv \tilde{h}(\hat{H}_\text{eff} ) | \tilde{A} (t) \rangle$|⁠, where we have used the automatic Hermiticity mechanism and introduced |$| \tilde A(t) \rangle \equiv \sum_{i \in A} a_i(t) | \lambda_i \rangle$|⁠, |$\Lambda_A\equiv\sum_{i \in A} | \lambda_i \rangle \langle \lambda_i |_Q$|⁠, and another function |$\tilde{h}$| such that |$\tilde{h}(\text{Re}\,\lambda_i ) = h(\text{Re}\,\lambda_i + iB )$|⁠. Similarly, the statement “|$Q_2 = F(\hat{H}_{\text{eff}}^\dagger) Q$| for the restricted subspace” given in Eq. (5.6) has to be corrected. A |$dt$|-dependent normalization factor, say |$\frac{1}{\alpha(dt)}$|⁠, 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 |$\ldots$| complex plane” below Eq. (A.2): “In addition, |$\alpha(dt)$| is a |$dt$|-dependent normalization factor, which is properly fixed later.” The factor |$\sqrt{\frac{2\pi i \hbar dt}{m}}$| in the second line of Eq. (A.4) should be deleted. The following sentences should be inserted after the phrase “where |$\ldots$| Eq. (3.7)” below Eq. (A.4): “Here we have taken |$\alpha(dt)= \sqrt{\frac{2\pi i \hbar dt}{m}}$| 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 |$|\psi(t+dt) \rangle = e^{-\frac{i}{\hbar} \hat{H} dt}|\psi(t) \rangle$|⁠.” The next sentence, “Thus we have found that |$\ldots$| Eq. (A.2),” below Eq. (A.4) should be replaced with “Thus we have derived the Schrödinger equation and found that |$\ldots$| 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 |$\frac{1}{\alpha(dt)^*}$|⁠, |$\frac{1}{\alpha(-dt')^*}$|⁠, and |$\frac{1}{\alpha(-dt)}$| 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, |$\hat{H}_B$| is given |$\ldots$||$\hat{H}^\dagger$|⁠,” should be replaced with “Indeed, we obtain the Schrödinger equation |$|B (t-dt) \rangle = e^{\frac{i}{\hbar} \hat{H}_B dt}|B (t) \rangle$|⁠, where |$\hat{H}_B$| is given |$\ldots$||$\hat{H}^\dagger$|⁠.”