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.
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.
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 the complex action theory whose path runs over not only past but also future, we study a normalized matrix element of an operator |O^| defined in terms of the future state at the latest time |TB| and the past state at the earliest time |TA| 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^| is |Q|-Hermitian, i.e., Hermitian with regard to the proper inner product, the normalized matrix element becomes real and time-develops under a |Q|-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 |Q|-Hermitian Hamiltonian in terms of |Q|-Hermitian coordinate and momentum operators, and construct a conserved probability current density.
We formulate a complex action theory which includes operators of coordinate and momentum q and p being replaced with non-hermitian operators qnew and pnew, and their eigenstates m new q| and m new p| with complex eigenvalues q and p. Introducing a philosophy of keeping the analyticity in path integration variables, we define a modified set of complex conjugate, real and imaginary parts, hermitian conjugates and bras, and explicitly construct qnew, pnew, m new q| and m new p| by formally squeezing coherent states.We also pose a theorem on the relation between functions on the phase space and the corresponding operators.Only in our formalism can we describe a complex action theory or a real action theory with complex saddle points in the tunneling effect etc. in terms of bras and kets in the functional integral.Furthermore, in a system with a non-hermitian diagonalizable bounded Hamiltonian, we show that the mechanism to obtain a hermitian Hamiltonian after a long time development proposed in our letter 11) works also in the complex coordinate formalism.If the hermitian Hamiltonian is given in a local form, a conserved probability current density can be constructed with two kinds of wave functions.
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^, which is defined in terms of the future and past states with a proper inner product IQ that makes a given Hamiltonian normal. The theorem states that, provided that the operator O^ is Q-Hermitian, i.e. Hermitian with regard to the proper inner product IQ, the normalized matrix element becomes real and time-develops under a Q-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^| defined in terms of the future state at the final time |TB| and the fixed past state at the initial time |TA| 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 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^†|.”