Formalism of harmonic oscillator in the future-included complex action theory

In a special representation of complex action theory that we call"future-included", we study a harmonic oscillator model defined with a non-normal Hamiltonian $\hat{H}$, in which a mass $m$ and an angular frequency $\omega$ are taken to be complex numbers. In order for the model to be sensible some restrictions on $m$ and $\omega$ are required. We draw a phase diagram in the plane of the arguments of $m$ and $\omega$, according to which the model is classified into several types. In addition, we formulate two pairs of annihilation and creation operators, two series of eigenstates of the Hamiltonians $\hat{H}$ and $\hat{H}^\dag$, and coherent states. They are normalized in a modified inner product $I_Q$, with respect to which the Hamiltonian $\hat{H}$ becomes normal. Furthermore, applying to the model the maximization principle that we previously proposed, we obtain an effective theory, which is described by a Hamiltonian that is $Q$-Hermitian, i.e., Hermitian with respect to the modified inner product $I_Q$. The solution to the model is found to be the vacuum state. Finally we discuss what the solution implies.

§1. Introduction probability current density. 26), 27) In Ref., 28) changing the notation of B(t)| as B(t)| → B(t)| Q ≡ B(t)|Q in Ô BA , where Q is a Hermitian operator that is appropriately chosen to define the modified inner product I Q , we introduced a slightly modified normalized matrix element Ô BA Q ≡ B(t)| QÔ |A(t) We proposed a theorem, which states that, provided that an operatorÔ is Q-Hermitian, Ô BA Q becomes real and time-develops under a Q-Hermitian Hamiltonian for the future and past states selected such that the absolute value of the transition amplitude defined with I Q from the past state to the future state is maximized. We call this way of thinking the maximization principle. This theorem was proven in both the CAT 28) and the RAT, 29) and briefly reviewed in Refs. 30), 31) In this paper, we study a harmonic oscillator model in the future-included CAT. After reviewing the complex coordinate formalism, 20) we provide a non-normal HamiltonianĤ for the model, in which a mass m and an angular frequency ω are taken to be complex numbers.
We point out that some restrictions on m and ω are required so that the model becomes sensible. According to the argument of m and ω, the model is classified into several types. We draw a phase diagram in the plane of the arguments of m and ω. We formulate two pairs of annihilation and creation operators, and construct two series of eigenstates |n 1 and |n 2 of the HamiltoniansĤ andĤ † respectively with several algebraically elegant properties as seen in the usual harmonic oscillator in the RAT. We fix the normalization factor of |n 1 by a naive analytical continuation of the normalized state |n in the RAT to complex mω. Analogously we determine the normalization factor of |n 2 . Then the eigenstates |n 1 and |n 2 are not normalized in a usual sense, but are normalized by the condition 2 n|m 1 = δ nm . We call this dual normalization. In addition, expecting that classical physics can be described well by coherent states even in the CAT as well as in the RAT, we construct them for later study. Next, after reviewing the modified inner product I Q , with respect to which the eigenstates of the HamiltonianĤ become orthogonal to each other, we argue that the dual normalization is interpreted as the Q-normalization, i.e., the normalization with respect to the inner product I Q . Furthermore, we apply the maximization principle to the harmonic oscillator model. As a preliminary study, supposing that |A(T A ) and |B(T B ) are given by the coherent states that we constructed, |λ A (T A ) coh,1 and |λ B (T B ) coh,1 , we evaluate q new , whereq new andp new are non-Hermitian coordinate and momentum operators respectively. Then we obtain a classical equation of motion, which suggests that, if we obtain a real observable Ô λ B λ A Q via the maximization principle, then we have a classical solution, which behaves in quite a similar way to that in the RAT. Furthermore, we introduce Q-Hermitian coordinate and momentum operatorsq Q andp Q , and rewrite the HamiltonianĤ in terms ofq Q andp Q . Utilizing the maximization principle, we obtain an effective theory, which is described by a Q-Hermitian Hamiltonian that is expressed in terms ofq Q andp Q . We find that the solution to the harmonic oscillator model is the vacuum state. Finally we discuss what the solution implies. This paper is organized as follows. In Section 2 we briefly review the complex coordinate formalism. 20) In Section 3 we define our harmonic oscillator model and present a phase diagram in the space of the arguments of m and ω. In Section 4 we formulate two pairs of annihilation and creation operators, and construct two series of eigenstates of the Hamiltonianŝ H andĤ † with the dual normalization. Also, we formulate coherent states. In Section 5, after reviewing the modified inner product I Q , we argue that the dual normalization is interpreted as the normalization with respect to I Q . In Section 6, after reviewing the maximization principle, we preliminarily study the behavior of q new λ B λ A Q and p new λ B λ A Q by supposing that |A(T A ) and |B(T B ) are given by coherent states |λ A (T A ) coh,1 and |λ B (T B ) coh,1 . Finally we argue that we obtain via the maximization principle an effective theory, which is described by a Q-Hermitian Hamiltonian, and that we are led to the vacuum state solution. Section 7 is devoted to discussion. §2. Complex coordinate formalism In this section we briefly review the complex coordinate formalism that we proposed in ref. 20) so that we can deal with complex coordinate q and momentum p properly not only in the CAT but also in the RAT, where we encounter them at the saddle point in WKB approximation, etc. for complex q and p by formally utilizing two coherent states. Our proposal is to replace the usual Hermitian operators of coordinate and momentumq,p, and their eigenstates |q and |p , which obeyq|q = q|q ,p|p = p|p , and [q,p] = i for real q and p, withq † new ,p † new , |q new and |p new . The explicit expressions forq new ,p new , |q new and |p new are given bŷ where |λ coh is a coherent state parametrized with a complex parameter λ defined up to a normalization factor by |λ coh ≡ e λa † |0 = ∞ n=0 λ n √ n! |n , and this satisfies the relation a|λ coh = λ|λ coh . Here, a = 1 2 ǫ (q + iǫp) and a † = 1 2 ǫ (q − iǫp) are annihilation and creation operators. In Eq.
is another coherent state defined similarly. Before seeing the properties ofq new ,p new , |q new , and |p new , we define a delta function of complex parameters in the next subsection.

The delta function
We define D as a class of distributions depending on one complex variable q ∈ C. Using a function g : C → C as a distribution * ) in the class D, we introduce the following functional G[f ] = C f (q)g(q)dq for any analytical function f : C → C with convergence requirements such that f → 0 for q → ±∞. The functional G is a linear mapping from the function f to a complex number. Since the simulated function g is supposed to be analytical in q, the path C, which is chosen to run from −∞ to ∞ in the complex q-plane, can be deformed freely, and so it is not relevant. As an example of such a distribution, we could think of the delta function and approximate it by the smeared delta function defined for complex q by where ǫ is a finite small positive real number. For the limit of ǫ → 0, g(q) behaves as a distribution for complex q obeying the condition L(q) ≡ (Re(q)) 2 − (Im(q)) 2 > 0. (2 . 9) For any analytical test function f (q) * * ) and any complex q 0 , this δ ǫ c (q) satisfies C f (q)δ ǫ c (q − q 0 )dq = f (q 0 ), as long as we choose the path C such that it runs from −∞ to ∞ in the complex q-plane and at any q its tangent line and a horizontal line form an angle θ whose absolute value is within π 4 to satisfy the inequality (2 . 9). An example of such a permitted path is drawn in Fig.1. Also, the domain of the delta function is shown in Fig.2. for a non-zero complex a. We express ǫ, q, and a as ǫ = r ǫ e iθǫ , q = r q e iθq , and a = r a e iθa . The convergence condition of δ ǫ c (aq): Re a 2 q 2 ǫ > 0 is expressed as

New devices to handle complex parameters
To keep the analyticity in dynamical variables of FPI such as q and p, we define a modified set of a complex conjugate, real and imaginary parts, bras and Hermitian conjugates.

Modified complex conjugate * {}
We define a modified complex conjugate for a function of n parameters f ( where A denotes the set of indices attached to the parameters in which we keep the analyticity, and * on f acts on the coefficients included in f . For example, the complex conjugate * q,p of a function f (q, p) = aq 2 + bp 2 is written as f (q, p) * q,p = a * q 2 + b * p 2 . The analyticity is kept in both q and p. For simplicity we express the modified complex conjugate as * {} , where {} is a symbolic expression for a set of parameters in which we keep the analyticity.

Modified real and imaginary parts Re {} , Im {}
We define the modified real and imaginary parts by using * {} . We decompose some complex function f as (2 . 16) where Re {} f and Im {} f are the "{}-real" and "{}-imaginary" parts of f defined by For example, for f = kq 2 , the q-real and q-imaginary parts of f are expressed as Re q f = Re(k)q 2 and Im q f = Im(k)q 2 , respectively. In particular, if f satisfies The states |q new and |p new are normalized so that they satisfy the following relations: where ǫ 1 and ǫ ′ 1 are given by (2 . 23) We take ǫ and ǫ ′ sufficiently small, for which the delta functions converge for complex q, q ′ , p, and p ′ satisfying the conditions L(q − q ′ ) > 0 and L(p − p ′ ) > 0, where L is given in Thus,q † new ,p † new , |q new and |p new with complex q and p obey the same relations asq,p, |q , and |p with real q and p. In the ǫ → 0, In this section, after reviewing the future-included theory, we define our harmonic oscillator model in the CAT and present the phase diagram.

Future-included theory
The future-included theory 1), 26), 27) is described by using the future state |B(T B ) at the final time T B and the past state |A(T A ) at the initial time T A . For a given non-normal HamiltonianĤ, |A(t) and |B(t) obey the following Schrödinger equations, and are expressed as follows, 3) In refs., 26), 27) we investigated the normalized matrix element Ô BA ≡ B(t)|Ô|A(t) B(t)|A(t) , which is called the weak value 24), 25) in the RAT, and found that, if we regard Ô BA as an expectation value in the future-included theory, then we obtain the Heisenberg equation, Ehrenfest's theorem, and a conserved probability current density. In fact, since Ô BA obeys for a general HamiltonianĤ where V is a general potential defined by V (q) = ∞ n=2 b n q n , we obtain d dt value in the future-included theory. In addition, let us introduce a probability density ρ by which satisfies C dqρ = 1, where C is an arbitrary contour running from −∞ to ∞ in the complex q-plane. Then we can construct a conserved probability current density j by which obeys the continuity equation ∂ρ ∂t + ∂ ∂q j(q, t) = 0. Therefore, probability interpretation seems to work formally with this ρ.
As for the Lagrangian, in Ref., 21) starting from the Hamiltonian given in Eq.(3 . 6), we obtained via the FPI the Lagrangian L(q,q) = 1 2 mq 2 − ∞ n=2 b n q n , and vice versa. In addition, we derived via the FPI the momentum relation We note that this is not the case in the future-not-included CAT. Indeed, we showed in Ref. 22) that, in the future-not-included CAT, the Lagrangian and momentum relation are given by L eff (q, q) = 1 2 m effq 2 − ∞ n=2 Reb n q n and p = m effq , where m eff ≡ m R + m 2 I m R . Since Eq.(3 . 7) is consistent with Eq.(3 . 11), Eq.(3 . 11) is confirmed to be the momentum relation in the future-included theory.

Harmonic oscillator Hamiltonian
Utilizingq new andp new given in Eqs. (2 . 4) and (2 . 5), we define our harmonic oscillator HamiltonianĤ byĤ where both mass m and angular frequency ω are complex, and decomposed as follows: 14) where m R , ω R , m I and ω I are the real and imaginary parts of m and ω, and r m , r ω , θ m and θ ω are the absolute values and arguments of m and ω, respectively. This Hamiltonian depends on ǫ and ǫ ′ viaq new andp new . For our later convenience, let us introduce another Hamiltonian that is independent of ǫ and ǫ ′ , by taking the limits ǫ → 0 and ǫ ′ → 0, or replacingq new andp new withq andp inĤ. Utilizing the fact obtained in Ref., 21) we find that the Lagrangian is simply given by The potential V is decomposed as where Re q and Im q are introduced in Eqs.(2 . 17) and (2 . 18).
We consider the functional integral C Dq ψ * B ψ A e i L(q,q)dt , and suppose that the asymptotic values of dynamical variables such as q and p are on the real axis. The path C denotes an arbitrary path running from −∞ to ∞ in the complex plane for each moment of time t, and we can deform it as long as the integrand keeps the analyticity in q and p. To prevent the kinetic term in the integrand from blowing up forq → ±∞ along the real axis, we impose on m the condition * ) In addition, to have the convergence of the functional integral, we need the following condition on the potential, Then, since mω and mω 2 are written as respectively.

Study of the phase diagram
In this subsection we analyze the phase diagram in the (θ m , θ ω ) plane. We will see that, according to the values of θ m and θ ω , our harmonic oscillator model includes several different theories. Indeed, the value of θ m classifies the model into the usual time theory (UTT), imaginary time theory (ITT) and flipped time theory (FTT). Also, according to the value of θ ω , not only a harmonic oscillator (HO) but also an inverted harmonic oscillator (IHO) is described.
Using Eq.(3 . 25), let us express V R and V I given in Eqs. (3 . 20) and (3 . 21) as Then, according to the signs of V R and V I , the permitted region of θ ω by the condition (3 . 27) can be classified into the following five regions: Later, using another condition (3 . 26), we investigate these regions in more details according to the value of θ m .

Our principle of interpretation of various quantities in the CAT
We shall explain our interpretation of various quantities in the CAT. We allow both mass m and angular frequency ω to be complex, negative numbers are naturally included.
Since we have a much larger class of theories, there can only be a priori less chance that we obtain just what we find in nature. Some possible outcomes will simply disagree with some of our experiences. We have to choose the parameters appropriately. We then divide the possibilities for the sign of the real part of m called m R to classify the theories. We think that the real part of (non-relativistic) mass should be positive in a sensible theory. One possible strategy would be to declare that there is an empirical law that m R shall be positive. Another one would be to introduce some transformation to change the mass into a new mass so that its real part becomes positive. Based on this way of thinking * ) , we define a new mass by where a, whose magnitude is 1, is properly chosen so that Re m new > 0. Since θ m = arg m is restricted by the condition (3 . 26), a is chosen according to the sign of m R , as shown later.
Next we introduce new times t new and T new A , and a new angular frequency ω new by demanding the following relation for the HamiltonianĤ given in Eqs. (3 . 12) and ( In addition, we introduce a new pair of coordinate and momentum, q new and p new by a ω, where we encounter an indefiniteness for the sign of ω new . However, since the expression of Eq.(3 . 32) suggests a new energy E new ≡ 1 a λ n , if we suppose that we can obtain an energy eigenvalue λ n ≡ ω n + 1 2 * ) forĤ, we are led to defining ω new with a definite sign by so that E new is expressed as E new = ω new n + 1 2 . Eq.(3 . 36) is also given by demanding the relation ωt = ω new t new .
According to the sign of m R , we determine m new , ω new and t new as follows: Unless one transforms the negativity of m R away, the cases 2 and 3 would be forbidden by the empirical law that m R shall be positive.

The phase diagram
Based on the strategy given in the foregoing Subsection 3.2.1, we can classify our harmonic oscillator model into several theories. We have presented such an explicit study in Appendix A. Thus the phase diagram of the harmonic oscillator specified by (3 . 26) and (3 . 27) is drawn in Fig.3  * ) , where UTT, ITT and FTT mean usual time theory, imaginary time theory and flipped time theory, respectively. In this section we develop our two-basis formalism of eigenvectors for the harmonic oscillator HamiltoniansĤ andĤ † . * ) For our later convenience to consider the condition (4 . 38) for there being eigenstates ofĤ and coherent states in Section 4, the two lines θ ω = −θ m ± π 2 have been additionally drawn. The investigation in the following sections, which is mainly based on the two-basis formalism of eigenvectors forming ladder states, is valid in the whole parallelogram region allowed by (3 . 26) and (3 . 27) except for the two corners (θ m , θ ω ) = (0, − π 2 ), (π, − π

Annihilation and creation operators
We define two annihilation operators a 1 and a 2 and creation operators a † 1 and a † 2 by their Hermitian conjugates as follows, Then, the commutation relation [q new ,p new ] = i is written as and the HamiltonianĤ in Eq.(3 . 12) and its Hermitian conjugateĤ † are expressed in terms of a 1 and a † 2 asĤ = ω a † 2 a 1 + 1 2 , (4 . 8) We define two vacuum states |0 1 and |0 2 up to the normalization by and excited states |n 1 and |n 2 for positive integer n up to the normalization as |n 1 ∝ (a † 2 ) n |0 1 and |n 2 ∝ (a † 1 ) n |0 2 . In addition, we introduce number operatorsn 1 andn 2 bŷ Then they obeyn 1 |n 1 = n|n 1 andn 2 |n 2 = n|n 2 , andĤ andĤ † are expressed aŝ We see that |n 1 and |n 2 are eigenstates ofĤ andĤ † , so especiallyĤ has the following eigenvalue for |n 1 : Here we note that |n 1 and |n 2 are not orthogonal eigenstates; 1 m|n 1 and 2 m|n 2 are not proportional to δ mn , sinceĤ andĤ † are not Hermitian. Though these eigenstates |n 1 and |n 2 are technically a bit hard to normalize, we can construct rather easily two series of eigenstates that are not genuinely normalized but fixed by a convention that makes the algebra of a † 2 and a 1 work very elegantly like in the RAT case.

Normalization of |n 1 and |n 2
In this subsection we shall argue how we normalize |n 1 and |n 2 . There are a couple of ways for normalizing the series of Hilbert vectors |n 1 . We first explain them. 1) We can obtain this special set of |n 1 's by a naive analytical continuation of the q-representation of the normalized state in the RAT, |n = 1 √ n! (a † ) n |0 , to complex mω: where on the left-hand side we have used a modified bra for complex q, and on the right- The expression of Eq.(4 . 19), which is a function of mω but not m * ω * , defines our |n 1 including the factor in front as follows: The state |n 1 is not normalized in a usual sense. The squared norm of |n 1 involves both mω and m * ω * , so it is not analytic in mω.
2) We could also single out our proposed series of eigenstate |n 1 by the requirement of the usual ladder formulas with a † 2 and a 1 replacing respectively a † and a, This algebraic requirement -not involving any norm -specifies the |n 1 state even with respect to n-dependent scale factors. We note that n-independent scale factors are left undetermined by Eqs. (4 . 22) and (4 . 23), but that they are determined in the case of 1).
3) The third possibility is to try to determine both the prefactors of |n 1 and |n 2 by imposing the following condition on |n 1 and |m 2 . This condition means that |m 2 is regarded as a dual basis of |n 1 , and also implies the following completeness relation, If we write |n 1 and |m 2 as |n 1 = C 1 (n)(a † 2 ) n |0 1 and |m 2 = C 2 (m)(a † 1 ) m |0 2 , then Eq.(4 . 24) gives only the condition C 2 (n) * C 1 (n) = 1 n! . Choosing C 1 (n) and C 2 (n) symmetrically as C 1 (n) = C 2 (n) = 1 √ n! leads to the |n 1 of Eq.(4 . 21) specified by 1) and 2), and the analogue for |n 2 , |n 2 ≡ 1 √ n! (a † 1 ) n |0 2 . We define |n 2 , the set of eigenstates of H † , by this relation. To construct the set |n 2 under 2), the algebraic requirement in Eqs. (4 . 22) and (4 . 23) should be replaced with the following ladder equations, a † 1 |n 2 = √ n + 1|n + 1 2 , (4 . 26) In our definitions a † 1 and a 2 are the ladder operators depending on m * ω * , while a † 2 and a 1 used for construction of the |n 1 states are the ones depending on mω. The procedure 3) does not quite fix the normalization of |n 1 alone, but needs to be supplemented by 1) or 2). The condition of Eq.(4 . 24) follows indeed from the scale specifications suggested under 1) and 2) by the analytical continuation or the ladder relation requirements respectively if they are supplemented by the analogous construction of the |n 2 states. We call this "dual normalization". We note that the complex q-representation of |n 2 , m new q|n 2 , is not given in a simple expression as Eq.(4 . 19) for |n 1 , since m new q| is not the eigenstate ofq † new from the left. So, to use 1) for |n 2 , we need to consider the correction to complex q for the n-th Hermite polynomial H n (q). H n (q) is a smooth q-wave function for small n, but not so for large n, for which it oscillates very much. Comparing the expressions for the HamiltonianĤ in Eqs. (3 . 12) and (4 . 16), we see that q and p classically go up in proportion to √ n for large n. Hence, the width of H n (q) is proportional to √ n. In addition, H n (q) has n zeros. Since the density of zeros is about n √ n = √ n per unit length in q, the length of each wave contained in H n (q) is about 1 √ n . On the other hand, the correction to complex q is ǫp ∼ ǫ √ n. It is ǫ √ n/ 1 √ n ∼ ǫn relative to the wave length. Therefore, when ǫn > 1, we cannot ignore the ǫp term anymore. Taking the above argument into account, we approximately provide the q-representation of |n 2 for small ǫ and ǫ ′ as This expression is valid for n such that n < 1 ǫ . The analytical construction under 1) delivers the set |n 2 for small ǫ and ǫ ′ if we replace mω with m * ω * in the RAT state |n and then analytically continue in m * ω * .
Using the above rules 1), 2), 3), which are consistent with each other, we have specified two series of eigenstates |n 1 and |n 2 ofĤ andĤ † respectively. They formally look like being normalized in the usual sense, but actually only in the sense of the dual normalization by Eq.(4 . 24). The two-basis formalism of |n 1 and |m 2 is our replacement for the usual formalism of |n in the RAT. If we thus define |n 1 states by analytic continuation in the parameter mω from real number to complex one -not allowing any (mω) * -, we ensure that, combining them with |n 2 states, which are obtained by analogous analytical continuation in (mω) * having replaced real mω in |n states in the RAT, we obtain for the overlap 2 m|n 1 the same result δ mn as in the RAT. The point is that, when we take the bra 2 m| correlated to the ket |m 2 , we get an expression formally written in terms of mω, and thus the overlap 2 m|n 1 becomes an integral of an expression involving only mω to be an analytical continuation of m|n in mω, which is well known to give δ mn . For 2 n|n ′ 1 it is easy to see this property for small ǫ and ǫ ′ by using the concrete expressions of Eqs. (4 . 19) and (4 . 28) as follows: where in the second line we have changed the variable q into X = mω q = r e i θ 2 q, where r and θ are introduced in Eq. (3 . 24). In the last equality, we have used the following relation for complex X by rotating the integration contour by the angle | θ 2 |: ∞ −∞ dXH n (X)H n ′ (X)e −X 2 = √ π2 n n!δ nn ′ , which is valid for θ such that |θ| < π 2 . Therefore, this is the condition for |n 1 and |n 2 to be normalizable in the sense of Eq.(4 . 29). If we, however, ask for overlaps of |n 1 states with each other, 1 m|n 1 , or those of |n 2 states with each other, 2 m|n 2 , then, since |n 1 and |n 2 are not normalized in the usual inner product, we obtain overlap integrals with both mω and m * ω * appearing formally. These integrals are not simple analytical continuations of the RAT integrals. In Subsection 5.2 we will show that the dual normalization by Eq.(4 . 24) can be regarded as an orthonormal condition of |n 1 or |n 2 with respect to an inner product I Q or I Q −1 defined there, respectively.

Coherent states made of |n 1 and |n 2
It is highly suggested that, if we want to see classical dynamics of harmonic oscillator, we should study coherent states. Indeed, in the RAT, coherent states are thought to be classical states represented by wave packets, so we now attempt to construct coherent states in the CAT. We utilize one of the coherent states in Subsection 6.1.
In the phase diagram drawn in Fig.3, we have seen that some phases have a healthy real part, but others even violate the positivity of the Hermitian part of the Hamiltonian.
Therefore, our treatment extends to the whole parallelogram except for the two corners in the phase diagram. The two troublesome corners represent inverted harmonic oscillators in the RAT. Indeed their kinetic terms T and potential terms V go oppositely: one has T ≥ 0 and V ≤ 0, while the other oppositely T ≤ 0 and V ≥ 0.
At the end of this subsection we summarize various quantities of our two-basis formalism in Table I.
HamiltonianĤ = ω n 1 + 1 2 ,Ĥ † = ω * n 2 + 1 2 , H|n 1 = ω n + 1 2 |n 1Ĥ † |n 2 = ω * n + 1 2 |n 2 q-representation m new q|n 1 = m new q|n 2 ≃ of the eigenstate mω π On the inner product I Q In the foregoing section we constructed two sets of eigenstates |n 1 and |n 2 for the HamiltoniansĤ andĤ † respectively with several algebraically elegant properties as seen in the usual harmonic oscillator in the RAT. These states |n 1 and |n 2 are not orthogonal to each other. They are dual-normalized by Eq.(4 . 24), not normalized in a usual sense. In this section, after reviewing the modified inner product I Q , we argue that the dual normalization of Eq.(4 . 24) can be interpreted as the normalization condition with respect to the inner product I Q .

Review of the modified inner product I Q
It is rather easy to see that Eq.(4 . 24) can be interpreted as a formal orthogonality relation provided we introduce the modified inner product I Q for arbitrary states |ψ 1 and |ψ 2 in the Hilbert space by where Q is chosen so that the eigenstates of a given non-normal HamiltonianĤ, |λ i 1 's, which obeyĤ|λ i 1 = λ i |λ i 1 , become orthogonal to each other, In Refs. 13)20) we put forward the idea of introducing such a modified inner product I Q . Then, H, being not even normal, Also, we define † Q for kets and bras by |ψ 1 We argued that in case of non-normal Hamiltonians we had better re-adjust the Hilbert space inner product, which shall have a physical significance by delivering a Born rule of probabilities to the properly modified one defined by Eqs.(5 . 1) and (5 . 2) so that unphysical transitions between energy eigenstates |λ i 1 and |λ j 1 with different eigenvalues are prohibited, i.e., not observed with an energy conserving measurement instrument.
It is natural to attempt to choose Q so close to the unit operator as possible to change the inner product in the Hilbert space as little as possible. In Refs. 13)20)26) we have chosen where P = (|λ 1 1 , |λ 2 1 , · · · ) is a diagonalizing operator ofĤ,Ĥ = P DP −1 . Incidentally, P −1 is expressed as where |λ j 2 's are the eigenstates ofĤ † , We introduce an orthonormal basis |e i (i = 1, . . .) satisfying e i |e j = δ ij by D|e i = λ i |e i .
Then, P , which obeys |λ i 1 = P |e i , is rewritten as P = i |λ i 1 e i |, and Q given in Eq.(5 . 3) is expressed as The completeness relation is written as We note that the operator Q is not unambiguously determined by the defining properties of Eqs.(5 . 1) and (5 . 2), because if we define a Hermitian operator Q g by using some function of Hamiltonian operator g(Ĥ) by then Eq.(5 . 2) is rewritten as g 1 λ i |Q g |λ j g 1 = δ ij , where |λ i g 1 is defined by |λ i g 1 ≡ g(Ĥ)|λ i 1 . If we, however, write conditions involving Q and operators not commuting withĤ, such conditions will specify how to resolve the ambiguity by Eq.(5 . 7).
Using the inner product I Q instead of the usual inner product in the Hilbert space, we have achieved a formalism that is very similar to the usual one in the RAT. We defined a 1 and a † Q 1 = a † 2 as annihilation and creation operators respectively for |n 1 , and a 2 and a † Q −1 2 = a † 1 for |n 2 . Our |n 1 is "Q-orthonormal", i.e. orthonormal with respect to the inner product I Q , while |n 2 is "Q −1 -orthonormal". Indeed, using Eq.(5 . 8), we can rewrite Eq.(4 . 24) as Thus the dual normalization of Eq.(4 . 24) can be interpreted as "Q-normalization" for |n 1 or "Q −1 -normalization" for |n 2 , as expressed by Eq. (5 . 19). §6. Maximization principle and the solution to the harmonic oscillator model In the future-included CAT, we suppose that |A(T A ) and |B(T B ) are randomly given at first, i.e., they are given by the overlaps of many states. However, due to the existence of the imaginary part of the action S I , only a single class of pairs of |A(t) and |B(t) dominates most significantly in the FPI. Then we can approximate |A(t) and |B(t) by such representative states, and classical physics is described by them. Indeed, in Refs., 28)- 31) we argued by such a maximization principle that we can obtain real expectation values.
In the RAT, classical behaviors are typically described by coherent states, so it would be natural for us to expect that coherent states works similarly even in the CAT. Supposing that we utilize the maximization principle, we can imagine a simple situation where the representative |A(t) and |B(t) are essentially approximated by just a pair of coherent states. In this section, base on this speculation, we first consider such a simple situation 25 where |A(t) and |B(t) are given by a single pair of coherent states as a preliminary study. Supposing that they time-develop according to the Schrödinger equations, we see that we can obtain an equation of motion. Next, briefly explaining the maximization principle, 28)-31) and applying it to the harmonic oscillator model, we argue that the effectively obtained system is described by a Q-Hermitian Hamiltonian, which can be expressed in terms of Q-Hermitian coordinate and momentum operators. Finally we find that the solution to the harmonic oscillator model is the vacuum state.
In the following, we adopt the proper inner product I Q for all quantities. This is realized by changing the notation of the final state and the normalized matrix element Ô BA in Eq.(1 . 1), which is a strong candidate for the expectation value of the operatorÔ, is replaced with In addition, we suppose that |A(T A ) and |B(T B ) are Q-normalized, i.e., normalized with the modified inner product I Q , by A(T A )| Q A(T A ) = 1 and B(T B )| Q B(T B ) = 1, respectively.

Preliminary study in the case of |A(T A ) and |B(T B ) being coherent states
As a preliminary study, based on the speculation that classical behaviors are typically described by coherent states even in the CAT, let us consider a situation where |A(t) and |B(t) are given by a pair of coherent states |λ A (t) coh,1 and |λ B (t) coh,1 , which are defined in Eqs.(4 . 30) and (4 . 32), and investigate how Ô BA Q behaves. To study this, let us formulate the time-development of the coherent states.  4) and are normalized with the modified inner product I Q by coh,1 λ A (T A )| Q λ A (T A ) coh,1 = 1 and coh,1 λ B (T B )| Q λ B (T B ) coh,1 = 1, respectively. Then |λ A (t) coh,1 and |λ B (t) coh,1 are expressed Operating a 1 on both sides of Eqs.(6 . 5) and.(6 . 6), we obtain the following relations, where we have used Eqs.(4 . 33)(6 . 5)(6 . 6). Eqs.(6 . 7) and (6 . 8) suggest that λ A (t) and λ B (t) time-develop as so that we have the relations similar to Eq.(4 . 33): where we have used Eqs.(4 . 5)(4 . 6)(5 . 14)(6 . 11)(6 . 12). Eqs.(6 . 9) and (6 . 10) suggest thatλ B (t) andλ A (t) are expressed asλ B (t) = −iω * λ B (t) andλ A (t) = −iωλ A (t). Using these relations, we can evaluate the time derivative of Eqs.(6 . 13) and (6 . 14) as follows, where V is the potential of the harmonic oscillator, which is given in Eq. (3 . 13). Eqs. (6 . 15) and (6 . 16) are the momentum relation and equation of motion, which are consistent with Eqs.(3 . 7) and (3 . 8). As we reviewed the general properties of the future-included theory 26) in Subsection 3.1.1, we have obtained the Ehrenfest's theorem: provides the saddle point development with t. It is very nice to have such properties. Though O λ B λ A Q is generically complex, if a pair of coherent states with λ A (t) and λ B (t) such that O λ B λ A Q becomes real dominates most significantly in the FPI, then classical physics is nicely realized. In the next subsection, to solve the harmonic oscillator model, we utilize the maximization principle, and investigate which kind of |A(t) and |B(t) dominate most significantly in the FPI. We shall find that they are not such interesting coherent states, but just the vacuum state.

Application of the maximization principle to the harmonic oscillator model
First we explain the maximization principle briefly. In this theorem * ) , exactly speaking, not only the maximizing states but also many other states contribute to the transition amplitude, but their contribution becomes very small for large T = T B − T A , in which we are practically interested. So, we ignore the effects of the other states, and consider only those of the maximizing states. Then, the normalized matrix element Ô BA Q for a Q-Hermitian operatorÔ turns out to be real, and time-develops according to a Q-Hermitian Hamiltonian. We call this way of thinking the maximization * ) For a normal HamiltonianĤ, the above theorem becomes simpler with Q = 1.
principle. This theorem can be applied to not only the CAT but also the RAT. In the CAT there are imaginary parts of the eigenvalues ofĤ, Imλ i , and the eigenstates having the largest Imλ i blow up and contribute most to the the absolute value of the transition amplitude | B(t)| Q A(t) |. Utilizing this property, we proved the theorem in the case of the CAT. 28) On the other hand, in the RAT, there are no Imλ i , so the full set of the eigenstates ofĤ can contribute to | B(t)|A(t) |. 29) The theorem is reviewed in Refs. 30), 31) Now we try to apply the maximization principle to the harmonic oscillator model. |A ( 2). In addition, in the harmonic oscillator model, the eigenvalue of the Hamiltonian for |n 1 , λ n , is given in Eq. (4 . 18). So Reλ n = Reω n + 1 2 and Imλ n = Imω n + 1 2 . To consider the theorem explicitly, let us expand |A(t) and |B(t) in terms of the eigenstates |n 1 as follows: n a n (t)|n 1 , (6 . 17) where a n (t) and b n (t) are expressed as a n (t) = a n (T A )e −iω(n+ 1 2 )(t−TA) , We write a n (T A ) and b n (T B ) as a n (T A ) = |a n (T A )|e iθa n and b n (T B ) = |b n (T B )|e iθ bn , and introduce In addition, the normalization conditions for |A(T A ) and |B(T B ) are expressed as n |a n (T A )| 2 = n |b n (T B )| 2 = 1. We note that, since we are studying the harmonic oscillator model in the whole parallelogram region allowed by (3 . 26) and (3 . 27) except for the two corners (θ m , θ ω ) = (0, − π 2 ), (π, − π 2 ) in the phase diagram given in Fig. 3, the imaginary part of the angular frequency ω is negative, Imω ≤ 0.
Let us first consider the case where Imω < 0. The imaginary parts of the eigenvalues of the Hamiltonian Imλ n are supposed to be bounded from above to avoid the FPI e i S Dpath being divergently meaningless. So some of Imλ n take the maximal value B. We denote the corresponding subset of {n} as A. Imλ n = Imω n + 1 2 can take the maximum value B = 2 Imω only for n = 0, for which Reλ 0 = 2 Reω and Imλ 0 = 2 Imω. Hence we find that, in the harmonic oscillator model, A = {0}. Then, since R n ≥ 0, | B(t)| Q A(t) | can take the maximal value e 1 T B = e where a 0 (t) and b 0 (t) obey Eq.(6 . 25). That is to say, the vacuum state |0 1 is chosen for both the maximizing states |A(t) max and |B(t) max .
To evaluate Ô BA Q for |A(t) max and |B(t) max , utilizing the Q-Hermitian part ofĤ, Ĥ +Ĥ † Q , we define the following state: which is normalized as Ã (t)| QÃ (t) = 1 and obeys the Schrödinger equation  Naively Eq.(6 . 36) looks strange if one wants to consider eigenstates for the two supposedly identical operators. In factq Q is Hermitian with regard to the modified inner product I Q , and thus have only real eigenvalues, which though do not have eigenstates belonging to the (true) Hilbert space for I Q , the Q-Hilbert space H Q . Ratherq Q has only delta function normalizable eigenstates with regard to I Q , which means that these eigenstates forq Q belong to an extension of H Q by completion in the weak topology for it. Now it is a priori -and indeed it is so -possible that such eigenstates belonging to the extension of H Q could even be true Hilbert space vectors under a different inner product such as the usual inner product I. Therefore, Eq.(6 . 36) is not -as it looks at first -contradictory, even if we note that new on the right-hand side has all complex numbers q as left-hand eigenvalues in the sense of the Hermitian conjugate of Eq.(2 . 1) being m new q|q new = m new q|q, and thatq new has no right-hand eigenvalues at all on the (true) Hilbert space for the usual inner product I, not even on the extension of it. Extension using the inner products I Q and I does namely not lead to the same space of extended vectors. These seeming problems will be discussed further in our successive paper. 33)

Hamiltonian expressed in terms of Q-Hermitian coordinate and momentum operators
In order to formulate the Q-Hermitian HamiltonianĤ Qh in terms of Q-Hermitian coordinate and momentum operatorsq Q andp Q , we rewrite the HamiltonianĤ in Eq.(3 . 12) asĤ where we have introduced m ′ ≡ r m e −iθω . Then, sinceĤ † Q is given bŷ the Q-Hermitian part ofĤ,Ĥ Qh = 1 2 Ĥ +Ĥ † Q , is given bŷ Similarly, the anti Q-Hermitian part ofĤ,Ĥ Qa = 1 2 Ĥ −Ĥ † Q , is given bŷ where we have introduced To check the consistency, let us see the other expression ofĤ given by Eq.(4 . 14). Sincê H † Q is given by Eq.(5 . 18), we obtainĤ Qh = r ω cos θ ω n 1 + 1 2 ,Ĥ Qa = i r ω sin θ ω n 1 + 1 2 , which lead toĤ 2 )|0 1 = 0, (6 . 50) where in the second line we have used Eqs.(6 . 37) and (4 . 6), and in the last equality we have utilized Eqs.(4 . 10)(4 . 11)(5 . 8). In addition, Eq.(6 . 33) forÔ beingq Q orp Q is expressed as In the future-included CAT we have formulated and studied the harmonic oscillator model defined with a mass m and an angular frequency ω that are taken to be complex numbers. Utilizing the complex coordinate formalism, 20) we defined the HamiltonianĤ for the harmonic oscillator model. For the model to be reasonable we need some restrictions on m and ω. We found that, according to the argument of m and ω, the model is classified into several different theories, and drew the phase diagram. Except for at the two corners representing inverted harmonic oscillators in the RAT, we formulated two pairs of annihilation and creation operators, and two series of eigenstates |n 1 and |n 2 for the HamiltoniansĤ andĤ † respectively with several algebraically elegant properties as seen in the usual harmonic oscillator in the RAT. Indeed, we fixed the normalization factor of |n 1 by a naive analytical continuation of the normalized state |n in the RAT to complex mω. Analogously we determined the normalization factor of |n 2 . Then the eigenstates |n 1 and |n 2 are not normalized in a usual sense, but are Q-normalized, i.e., normalized in the modified inner product I Q , with respect to which the eigenstates of the HamiltonianĤ become orthogonal to each other. In addition, we constructed coherent states.
Furthermore, we applied to the harmonic oscillator model the maximization principle, 28)-31) which is the main assumption used by a theorem of ours presented in Subsection 6.2. The theorem states that, provided that an operatorÔ is Q-Hermitian, i.e. Hermitian with respect to the modified inner product I Q , the normalized matrix element (weak value) Ô BA Q defined in Eq.(6 . 2) 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. In the RAT, coherent states describe classical physics nicely. So, as a preliminary study, supposing that |A(T A ) and |B(T B ) are given by coherent states, we evaluated q new BA Q and p new BA Q , and obtained a nice classical equation of motion. This suggests that, if we obtain a real observable Ô BmaxAmax Q for the maximizing states via the maximization principle, then a nice classical solution is realized. Incidentally, introducing Q-Hermitian coordinate and momentum operatorsq Q andp Q , and rewriting the HamiltonianĤ in terms ofq Q andp Q , we found that we can obtain via the maximization principle an effective theory that is described by the Q-Hermitian Hamiltonian expressed in terms ofq Q andp Q . However, we have finally obtained via the maximization principle the vacuum state as the solution to the harmonic oscillator model. This might be a bit tedious result, but what does this implicate? In our universe, every kind of oscillation can be approximately regarded as a harmonic oscillator near the bottom of each potential. Therefore, if we suppose that our harmonic oscillator model describes our universe, then our solution of the vacuum state would be very natural. We should also point out that we obtained a real-valued solution, because q Q ÃÃ Q = 0 ∈ R and p Q ÃÃ Q = 0 ∈ R. Furthermore, it is interesting that we obtained the Q-Hermitian Hamiltonian that is expressed in terms of Q-Hermitian coordinate and momentum operators.
What should we study next? In this paper, we studied the harmonic oscillator model except for at the two corners in the phase diagram drawn in Fig.3. So it is very important to study this model in the limit to these corners representing inverted harmonic oscillators in the RAT. Especially, inverted harmonic oscillators would be very interesting to study at least from a point of view of regarding such an inverted harmonic oscillator as a typically simplified inflaton potential for the slow roll inflation in the early universe. Also, it is interesting to investigate the concrete expression of Q in the harmonic oscillator model. Furthermore, in this paper, we studied the harmonic oscillator model by utilizing the maximization principle, where |A(T A ) and |B(T B ) are Q-normalized, i.e., normalized in the modified inner product I Q . On the other hand, it is also important to investigate the model, where |A(T A ) and |B(T B ) are normalized in the usual inner product I. Such a theory is more complicated to study, because we cannot fully utilize the orthogonality of the eigenstates of the Hamil-tonianĤ. Due to this difficulty, we have not yet studied in general such a version of the maximization principle. However, it would be easier to study it in a concrete model such as the harmonic oscillator. We would like to report such studies in the future.