We develop some formalism which is very general Feynman path integral in the case of the action which is allowed to be complex. The major point is that the effect of the imaginary part of the action (mainly) is to determine which solution to the equations of motion gets realized. We can therefore look at the model as a unification of the equations of motion and the ``initial conditions". We have already earlier argued for some features matching well with cosmology coming out of the model. A Hamiltonian formalism is put forward, but it still has to have an extra factor in the probability of a certain measurement result involving the time after the measurement time. A special effect to be discussed is a broadening of the width of the Higgs particle. We reach crudely a changed Breit-Wigner formula that is a normalized square root of the originally expected one
We develop some formalism which is very general Feynman path integral in the case of the action which is allowed to be complex. The major point is that the effect of the imaginary part of the action (mainly) is to determine which solution to the equations of motion gets realized. We can therefore look at the model as a unification of the equations of motion and the ``initial conditions". We have already earlier argued for some features matching well with cosmology coming out of the model. A Hamiltonian formalism is put forward, but it still has to have an extra factor in the probability of a certain measurement result involving the time after the measurement time. A special effect to be discussed is a broadening of the width of the Higgs particle. We reach crudely a changed Breit-Wigner formula that is a normalized square root of the originally expected one
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.
Abstract We study the periodic complex action theory (CAT) by imposing a periodic condition in the future-included CAT where the time integration is performed from the past to the future, and extend a normalized matrix element of an operator O^, which is called the weak value in the real action theory, to another expression ⟨O^⟩periodictime. We present two theorems stating that ⟨O^⟩periodictime becomes real for O^ being Hermitian with regard to a modified inner product that makes a given non-normal Hamiltonian H^ normal. The first theorem holds for a given period tp in a case where the number of eigenstates having the maximal imaginary part B of the eigenvalues of H^ is just one, while the second one stands for tp selected such that the absolute value of the transition amplitude is maximized in a case where B ≤ 0 and |B| is much smaller than the distances between any two real parts of the eigenvalues of H^. The latter proven via a number-theoretical argument suggests that, if our universe is periodic, then even the period could be an adjustment parameter to be determined in the Feynman path integral. This is a variant type of the maximization principle that we previously proposed.
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.
It is shown that the transactional interpretation of quantum mechanics being referred back to Feynman-Wheeler's time reversal symmetric radiation theory has reminiscences to our complex action model. In this complex action model the initial conditions are in principle even calculable. Thus it philosophically points towards superdeterminism, but really the Bell theorem problem is solved in our model of complex action by removing the significance of signals running slower than by light velocity. Our model as earlier published predicts that LHC should have some failure before reaching to have produced as many Higgs-particles as would have been produced the SSC accelerator. In the present article, we point out that a cardgame involving whether to restrict LHC-running as we have proposed to test our model will under all circumstances be a success.
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 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^†|.”