Momentum relation and classical limit in the future-not-included complex action theory

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 $p=\frac{\partial L}{\partial \dot{q}}$), 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.


§1. Introduction
Complex action theory (CAT) is one of the attempts to extend quantum theories by allowing their action to be complex. CAT has recently been studied with the expectation that the imaginary part of the action would give some falsifiable predictions. 1), 2), 3), 4) So far, various interesting suggestions have been made for Higgs mass, 5) quantum mechanical philosophy, 6), 7), 8) some fine-tuning problems, 9), 10) black holes, 11) de Broglie-Bohm particles and a cut-off in loop diagrams. 12) Related to CAT, integration contours in the complex plane 13) , 14) complex Langevin equations 15) and complexified solution sets 16) 17) have also been studied.
In ref., 18) in a system with a non-Hermitian diagonalizable bounded HamiltonianĤ, introducing a proper inner product * ) and considering the long time development of some states, we effectively obtained a Hermitian Hamiltonian. We note thatĤ is generically non-Hermitian, so it does not belong to the class of PT-symmetric non-Hermitian Hamiltonians which has been intensively studied recently. 22), 23), 19), 20), 21) For details of PT-symmetric non-Hermitian Hamiltonians, see the reviews 24), 25), 26), 27) and the references therein. In addition, non-Hermitian time-dependent Hamiltonians are studied in ref. 28) In ref., 29) introducing various mathematical tools such as a modified set of complex conjugate, real and imaginary parts, Hermitian conjugates and bras, complex delta function etc., we explicitly constructed non-Hermitian operators of coordinate and momentum,q new andp new , and the eigenstates of their Hermitian conjugates |q new and |p new for complex q and p by utilizing coherent states of harmonic oscillators. Indeed, |q , which obeysq|q = q|q , is defined only for real q, i.e. the eigenvalue of the Hermitianq, so q is not allowed to be complex unlessq is extended to a non-Hermitian operator. Only in our complex coordinate formalism can we deal with complex q and p. This formalism would be a part of proof of consistency in using complex q and p in contours of integration for WKB (Wentzel-Kramers-Brillouin) approximation, etc.
in the usual real action theory (RAT). Using this formalism in ref., 30) we explicitly examined the momentum and Hamiltonian in the CAT via the Feynman path integral (FPI). We studied the time development of some ξ-parametrized state, which is a solution to a kind of eigenvalue problem for a momentum operator. Finding the value of ξ that gives the largest contribution in FPI, we derived the momentum relation p = mq and Hamiltonian.
The future-included theory, i.e. the theory including not only a past time but also a future time as an integration interval of time, was studied in ref., 1) whose authors introduced the future state |B(T B ) at the final time T B = ∞ in addition to the past state |A(T A ) at the initial time T A = −∞. The states |A(T A ) and |B(T B ) time-develop according * ) Similar inner products were also studied in refs. 19), 20), 21) 2 to the non-Hermitian HamiltoniansĤ andĤ B =Ĥ † , respectively. The authors of ref. 1) speculated a correspondence of the future-included theory to the future-not-included one, i.e. Ô BA ≃ Ô AA , where Ô BA ≡ B(t)|Ô|A(t) B(t)|A(t) , Ô AA ≡ A(t)|Ô|A(t) A(t)|A(t) , and t is the present time. In the RAT the matrix element Ô BA is called the weak value, 31) and has been intensively studied. For details of the weak value, see the reviews 32), 33), 34), 35) and the references therein. In refs. 36), 37) we investigated Ô BA carefully, and found that if we regard it as an expectation value, then we obtain the Heisenberg equation, Ehrenfest's theorem, and a conserved probability current density. This result strongly suggests that we can regard Ô BA as the expectation value in the future-included theory. Furthermore, using both the complex coordinate formalism 29) and the automatic hermiticity mechanism, 18), 29) i.e., a mechanism to obtain a Hermitian Hamiltonian after a long time development, we obtained a correspondence principle that Ô BA for large T B − t and large t − T A is almost equivalent to Ô AA Q ′ for large t − T A , where Q ′ is a Hermitian operator which is used to define a proper inner product. Thus the future-included theory is not excluded, although it looks exotic.
As for the momentum relation, in ref. 36 Thus we are motivated to examine the momentum relation in the future-not-included theory. In this paper, studying the time development of Ô AA , we argue that the momentum relation in the future-not-included theory is not given by p = mq but by another expression p = m effq , where m eff is a certain real mass. Moreover, since the effect of the anti-Hermitian part of the Hamiltonian is suppressed in the classical limit, we claim that classical theory in the future-not-included theory is described by the real part of the non-Hermitian Hamiltonian, or a certain real action S eff . In addition, we present another way to understand the time development of the future-not-included theory by utilizing the future-included theory.
Furthermore, we discuss how we can utilize the method studied in ref. 30) to obtain the correct momentum relation in the future-not-included theory. In the method, we analyze the time development of ξ-parametrized state in a transition amplitude from initial time to final time, where the present time t is supposed to be between the initial and final times.
This is the case for the future-included theory, but not for the future-not-included theory.
Therefore, to properly apply the method to the future-not-included theory, we introduce a formal Lagrangian by rewriting the transition amplitude in the future-not-included theory, , into an expression such as B(t)|A(t) , which is the transition amplitude in the future-included theory. We argue that using this formal Lagrangian in the method we obtain p = m effq , the correct momentum relation in the future-not-included theory.
This paper is organized as follows. In section 2 we review the complex coordinate formalism proposed in ref. 29) In section 3, following ref., 30) we explain the method used to derive the momentum relation p = mq via the Feynman path integral. In section 4, based on ref., 36) we show that Ô BA behaves as if it were the expectation value of some operatorÔ in the future-included theory. Also, we obtain the relation p new BA = m d dt q new BA , which is consistent with the momentum relation obtained in ref. 30) In section 5, studying O AA , we obtain the momentum relation in the future-not-included theory, p = m effq . Moreover, we argue that the classical theory is described by a certain real action S eff . Furthermore, we provide another way to understand the time development of the future-not-included theory by making use of the future-included theory. In section 6 we apply the method of ref. 30) to the future-not-included theory properly by introducing the formal Lagrangian, and derive the momentum relation in the future-not-included theory, which is consistent with that derived in section 5. 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. 29) so that we can deal with complex coordinate q and momentum p properly not only in the CAT but also in a real action theory (RAT), where we encounter them at the saddle point in WKB approximation, etc.

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 define the following functional 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 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 9) * ) For simplicity we have replaced the parameters mω and m ′ ω ′ used in ref. 29) with 1 ǫ and ǫ ′ . * * ) We recently noticed that another complex distribution was introduced in ref. 38) It is different from ours in the following points: the complex distribution in ref., 38) where g(q) is supposed to have poles, is not well defined by g(q) alone, but needs the indication of which side of the poles the path C passes through. On the other hand, in our complex distribution we assume not the presence of poles of g(q) but f not being a bounded entire function.
where ǫ is a finite small positive real number. For the limit of ǫ → 0, g(q) converges in the distribution sense for complex q obeying the condition L(q) ≡ (Re(q)) 2 − (Im(q)) 2 > 0.
(2 . 10) For any analytical test function f (q) * ) and any complex q 0 , this δ ǫ c (q) satisfies as long as we choose the path C such that it runs from −∞ to ∞ in the complex 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 . 10). An example permitted path is shown in Fig.1, and the domain of the delta function is shown in Fig.2.  for non-zero complex a. We express ǫ, q, and a as ǫ = r ǫ e iθǫ , q = re iθ , and a = r a e iθa . The convergence condition of δ ǫ c (aq): Re a 2 q 2 ǫ > 0 is expressed as (2 . 14) For q, ǫ, and a such that eqs.(2 . 13)(2 . 14) are satisfied, δ ǫ c (aq) behaves well as a delta function of aq, and we obtain the relation where we have introduced an expression

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 ({a i } i=1,...,n ) by 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 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

Modified bras m | and {} |, and modified Hermitian conjugate † {}
For some state |λ with some complex parameter λ, we define a modified bra m λ| by so that it preserves the analyticity in λ. In the special case of λ being real it becomes a normal bra. In addition we define a slightly generalized modified bra {} | and a modified Hermitian The states |q new and |p new are normalized so that they satisfy the following relations: 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 eq.(2 . 10). These conditions are satisfied only when q and q ′ or p and p ′ are on the same paths respectively. Eqs.(2 . 22)(2 . 23) represent the orthogonality relations for |q new and |p new , and we have the following relations for complex q and p: 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 limits of ǫ → 0 and This theorem is understood by noticing that such a matrix element can be expressed as the summation of the products of factors made of q ′ , p ′ , q ′′ , p ′′ or their differential operators and distributions. Thus, we do not have to worry about the anti-Hermitian terms inq new ,q † new , p new andp † new , provided that we are satisfied with the result in the distribution sense. §3. Deriving the momentum relation via Feynman path integral We briefly explain how we derived the momentum relation in ref. The FPI in the CAT is described with the following Lagrangian -a typical example for a system with a single degree of freedom -: where V (q) = ∞ n=2 b n q n is a potential term. For our later convenience we decompose V and L as V = V R + iV I and L = L R + iL I , where V R , V I , L R and L I are given by Here, Re q and Im q are as introduced in eqs.(2 . 19)(2 . 20), and we have decomposed m into its real and imaginary parts as m = m R + im I . We consider the functional integral C e i L(q,q)dt Dq by discretizing the time direction , where dt is assumed to be a small quantity. Since we use the Schrödinger representation for wave functions, to avoid the confusion with the Heisenberg representation we introduce the notations q t ≡ q(t) and q t+dt ≡ q(t + dt), which we regard as independent variables. We suppose that the asymptotic values of dynamical variables such as q and p are on the real axis, while parameters such as m and b n are complex in general.
The path C denotes an arbitrary path running from −∞ to ∞ in the complex plane, 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 the condition m I ≥ 0 on m. In FPI the time development of some wave function m new q t |ψ(t) at some time t to t + dt is described by where L(q,q) is given by eq.(3 . 1), and C is an arbitrary path running from −∞ to ∞ in the complex plane. In ref., 30) to derive the momentum relation p = ∂L ∂q , we considered some wave function m new q t |ξ that obeys where ξ is any number. Since the set {|ξ } is an approximately reasonable basis which has roughly completeness 1 ≃ C dξ|ξ m anti ξ| and orthogonality m anti where m anti ξ| is a dual basis of |ξ , we can expand the wave function m new q t |ψ(t) into a linear combination of m new q t |ξ as Then, solving eq.(3 . 7), we obtain Since m new q t+dt |ψ(t + dt) | ξ is equal to the linear combination of δ c (q t+dt − ξ) and its derivative, only the component with ξ = q t+dt contributes to m new q t+dt |ψ(t + dt) . Thus, we have obtained the momentum relation in the sense of eq.(3 . 7): Furthermore, we can estimate the right-hand side of eq.(3 . 6) explicitly as follows: whereĤ is given byĤ Thus, starting from eq.(3 . 6), we have found that the HamiltonianĤ has the same form as that in the RAT.  30) we derive the Lagrangian and momentum relation. We analyze the transition amplitude from an initial state |i at time t i to a final state |f at time t f , which is written as where L(p j , q j ,q j ) is given by where in the second equality we have introduced q i = q 1 and q f = q N . We perform the following Gaussian integral around the saddle point p j = mq j , where L(q j , q j ) = 1 2 mq j 2 − V (q j ). Thus, we have obtained the momentum relation (3 . 10) and the Lagrangian (3 . 1). §4. Properties of the future-included theory

Future-included theory
Improving the definition given in ref., 1) based on the complex coordinate formalism, 29) in ref. 36) we introduced |A(t) and |B(t) by where path(t) = q means the boundary condition at the present time t, and T A and T B are taken as −∞ and ∞ respectively. |A(t) and |B(t) are supposed to time-develop according to whereĤ B =Ĥ † . The authors of ref. 1) speculated that the following matrix element * ) of some operatorÔ corresponds to the expectation value in the future-not-included theory, i.e. Ô BA ≃ Ô AA . In refs. 36), 37) we investigated Ô BA carefully. Using both the complex coordinate formalism 29) and the automatic hermiticity mechanism, 18), 29) i.e., a mechanism to obtain the Hermitian Hamiltonian after a long time development, we obtained a correspondence principle that Ô BA for large T B − t and large t − T A is almost equivalent to Ô AA Q ′ for large t − T A , where Q ′ is a Hermitian operator which is used to define a proper inner product. * * ) We note that Ô BA is not an expectation value but a matrix element in the usual sense.
But in ref. 36) we found that if we regard it as an expectation value in the future-included theory, then we obtain the Heisenberg equation, Ehrenfest's theorem and a conserved probability * ) In the RAT the matrix element Ô BA is called the weak value 31) and has been intensively studied. For details of the weak value, see the reviews 32), 33), 34), 35) and the references therein. * * ) For simplicity, in this paper we are not concerned with the proper inner product, which is defined by making the Hamiltonian normal, since it does not have an essential role in this study. current density. This result strongly suggests that we can regard Ô BA as an expectation value in the future-included theory.

The Heisenberg equation and Ehrenfest's theorem
In ref. 36) we defined the Heisenberg operator, whereĤ is given in eq.(3 . 12) and t ref is some reference time chosen arbitrarily such that This Heisenberg operator, which appears in the numerator of Ô BA as In addition, since Ô BA obeys we obtain we express Ô AA as where we have introduced the Heisenberg operatorÔ f ni This operatorÔ f ni H (t, t 0 ) obeys the slightly modified Heisenberg equation, whereĤ h andĤ a are the Hermitian and anti-Hermitian parts ofĤ respectively. We note that eq.(5 . 4) is more complicated than the Heisenberg equation in the future-included theory, eq.(4 . 8). In addition, |A(t) N obeys the slightly modified Schrödinger equation,

Classical limit of the future-not-included theory
As we pointed out in refs., 18), 29) eqs.(5 . 4)(5 . 5) suggest that the effect of the anti-Hermitian part of the non-Hermitian HamiltonianĤ disappears in the classical limit, though the theory is defined withĤ at the quantum level. To see this in terms of the expectation value Ô AA , utilizing eq.(5 . 5) we give the following expression, where F (Ô,Ĥ a )(t), a quantum fluctuation term given by disappears in the classical limit, so we have used the relation F (Ô,Ĥ a ) AA ≃ 0. Substitutingq new andp new forÔ in eq.(5 . 6), we obtain On the other hand, we do not encounter such a contradiction for eq.(5 . 11) in the future-not-included theory, because m eff is real. Therefore, we conclude that the momentum relations in the future-included and future-not-included theories are given by eqs.(3 . 10)(5 . 11), respectively. Then, one may question why the method of ref. 30) for deriving eq.(3 . 10), which was explained in section 3, does not work in the future-not-included theory. Later, in section 6, we will come back to this point and explain that the method works even in the future-not-included theory, and provides eq.(5 . 11), if it is properly applied to the future-not-included theory. Combining eq.(5 . 8) with eq.(5 . 9), we obtain Ehrenfest's theorem, 12) which suggests that the classical theory of the future-not-included theory is described not by a full action S, but S eff defined by Here we note that L eff is different from L R given in eq.(3 . 4). Thus, we claim that the classical theory of the future-not-included theory is described by δS eff = 0. Then the momentum relation given in eq.(5 . 11) is rewritten as p = ∂L eff ∂q . This is quite in contrast to the classical theory of the future-included theory, which would be described by δS = 0, where S = T B T A dtL, and the momentum relation given by eq.(3 . 10). In addition, the classical Hamiltonian in the future-not-included theory is given by where H R is the q-real part of the classical Hamiltonian H ≡ 1 2m p 2 + V (q), which is given by replacingq new andp new with q and p respectively inĤ. In refs. 18), 29) introducing a proper inner product so that the eigenstates ofĤ are orthogonal to each other and considering a long time development, we obtained a Hermitian Hamiltonian. But now without using the automatic hermiticity mechanism we have obtained a real Hamiltonian in the classical limit. This is an intriguing property of the future-not-included theory, though restricted to the classical limit. We make a comparison between the future-included and future-not-included theories in Table I. Table I. Comparison between the future-included and future-not-included theories future-included theory future-not-included theory Another method for seeing the time development of Ô AA by re-choosing the B state The quantity Ô BA in the future-included theory behaves as an expectation value, despite looking like a matrix element, and it time-develops according to the very simple expression of eq.(4 . 9). On the other hand, the expectation value Ô AA in the future-not-included theory time-develops in a more complicated way at the quantum level with the additional term Ô ,Ĥ a − Ĥ a AA , as seen in eq.(5 . 6). Hence, we are motivated to study whether we can simplify the description of the time development of Ô AA by rewriting it formally in the expression of the future-included theory and utilizing the simple time development of the future-included theory. Even if we cannot make it simpler, it would be interesting to reproduce and understand the time development of the future-not-included theory from a different point of view via the future-included theory. At the least, this would become a consistency check of the theory, and we could claim that the future-included theory can be used as a mathematical tool to compute the time development of Ô AA . Therefore, in this subsection, we try to describe the time development of the expectation value of the future-not-included theory Ô AA by making use of the future-included theory.
We begin by putting the condition on the B state at some time t. * ) We call this "re-choosing" the B state. Expressing the B state re-chosen at t as |B t (t ′ ) , where t ′ is a formal time to allow the time-development as a B state, we have the following relation for the time t: Then eq.(4 . 6) is rewritten as for each t. In a realistic future-included theory it would be a very strange accident to have the relation of eq.(5 . 16) even at one time. Hence, the re-choosing cannot be taken seriously.
We just look for some formal rule to use the future-included theory as long as possible but to obtain the future-not-included theory as our result.
The re-chosen B state |B t (t ′ ) obeys Since eq.(5 . 20) provides the expression we obtain For t ′ = t this is expressed as where the left-hand side is rewritten as Next, we calculate the time derivative of Ô AA , is formally a good classical solution in the future-included theory for each t ′ as long as the equation of motion is considered. Indeed, the second term of eq.(5 . 28) is expressed as On the other hand, the first term of eq.(5 . 28) does not become a simple expression. We can rewrite this by utilizing eq.(5 . 26) as follows: (5 . 30) Substituting eqs.(5 . 29)(5 . 30) for eq.(5 . 28), we obtain eq.(5 . 6). Thus, we have shown that we can derive the time development of Ô AA , the expectation value in the future-not-included theory, by making use of the future-included theory. In particular, we have explicitly seen that it is the first term of eq.(5 . 28) that provides the anti-commutator term, which disappears in the classical limit, besides the commutator [Ô, H a ] A(t)A(t) . As a result, this method is not so simple, but it is interesting in the sense that this provides another way to understand the time development of the future-not-included theory. Indeed, we have seen that the time development of Ô AA is expressed as the simple time development of Ô BA and a slightly complicated correction due to the formal re-choosing of the B state. §6. Reconsideration of the method for deriving the momentum relation via the Feynman path integral in the future-not-included theory In the foregoing sections we have seen that the momentum relation of eq.(3 . 10) derived via FPI in ref. 30) is valid in the future-included theory, because it is consistent with eq.(4 . 10), which was derived by looking at the time development of q new BA in the future-included theory. In eq.(5 . 11) we obtained another momentum relation in the future-not-included theory by analyzing the time development of q new AA . Now, one might question why the method of ref. 30) for deriving the momentum relation via FPI, which was reviewed in section 3, is not valid in the future-not-included theory. The reason is as follows: In the method of ref., 30) we analyzed the time development of a ξ-parametrized state in a transition amplitude from the initial time t i to the final time t f , where the present time t is supposed to be between t i and t f . Such a transition amplitude is similar to that in the future-included theory, which is written as where the present time t is between T A and T B . On the other hand, in the future-not-included theory the transition amplitude is given by so we have to consider a path starting from the initial time T A to the present time t, and also that going backward from t to T A . In this section we discuss how to apply the method of ref. 30) for deriving the momentum relation via the Feynman path integral, which was reviewed in section 3, to the future-not-included theory.

Formal Lagrangian in the future-not-included theory
To apply the method of ref. 30) to the future-not-included theory, we formally rewrite the transition amplitude A(t)|A(t) into another expression similar to B(t)|A(t) , and introduce a formal Lagrangian L formal . We argue that using this formal Lagrangian L formal in place of the original Lagrangian L in the method of ref. 30) we obtain the momentum relation in the future-not-included theory, eq.(5 . 11).
In the future-not-included theory, we can rewrite eq.(6 . 2) as the following path integral At an intermediate time t ′ such that T A < t ′ < t, we would be allowed to use a kind of futureincluded formulation, because it looks as if there is a future for t ′ . But for the present time t there is no future but only the past, so we have to be careful about quantities at the time t, especiallyq, etc. Therefore, we transform I into an expression like a transition amplitude from the time T A to T B by inverting the time direction of the transition amplitude from T A to t so that t becomes an intermediate time.
For this purpose we express S T A to t (q) * q as where in the second equality we have changed the variable by t ′′ = −t ′ + 2t, (6 . 5) and introduced the formal coordinate q formal by q formal (t ′′ , t) ≡ q(−t ′′ + 2t) = q(t ′ ), (6 . 6) which has the time dependence of not only t but also t ′′ and suggests Then I is written as where C ′′ is a contour of q formal (t ′′ , t), which is obtained by a reflection of C at t in the time direction, and J is given by Using the relation we obtain a simple expression for J, We note that the time 2t − T B is not so far from T A because we suppose T B ≃ −T A ≃ ∞.
Expressing q ′ (t ′ ) for T A ≤ t ′ ≤ t as q formal (t ′ , t) formally, we can rewrite the integral I as ×ψ A (q formal (T B , t), 2t − T B ) * q formal ψ A (q formal (T A , t), T A ), (6 . 12) where ǫ(t) is a step function defined as 1 for t > 0 and −1 for t < 0, and we have introduced the formal Lagrangian L formal by Here, m formal (t ′ − t) and V formal (q formal (t ′ , t), t ′ − t) are the formal mass and potential given by m formal (t ′ − t) ≡ m R − iǫ(t ′ − t)m I , (6 . 14) V formal (q formal (t ′ , t), t ′ − t) ≡ V R (q formal (t ′ , t)) − iǫ(t ′ − t)V I (q formal (t ′ , t)). (6 . 15) In eq.(6 . 12) we have defined L formal by extracting the factor −ǫ(t ′ − t), which is caused by the time reflection of eq.(6 . 5). L formal looks like a non-translational invariant Lagrangian depending on t ′ , and t is just a selected point in time. Therefore, we normally have to think of t ′ as the time when using L formal .
One may think that the transition amplitude of eq.(6 . 2) can be expressed as

22
where θ(t) = 1 2 (ǫ(t) + 1) is a step function defined as 1 for t > 0 and 0 for t < 0, and L formal, 2 is given by L formal, 2 (q(t ′ ),q(t ′ ), t ′ − t) ≡ Re q L(q(t ′ ),q(t ′ )) − iǫ(t ′ − t)Im q L(q(t ′ ),q(t ′ )). (6 . 17) We might think that this rewriting is also good for our purpose, but this is not the case, since in eq.(6 . 16) only the half of the original path, i.e. the path going from T A to t, is mapped onto the time interval [T A , T B ] over which L formal, 2 is time-integrated.

Momentum relation in the future-not-included theory
Since we have found the formal Lagrangian L formal , we try to obtain the momentum relation in the future-not-included theory by replacing L with L formal in the method of ref. 30) Then we obtain the formal momentum p formal (t ′ , t): p formal (t ′ , t) = ∂L formal (q formal (t ′ , t), ∂ t ′ q formal (t ′ , t), t ′ − t) ∂(∂ t ′ q formal (t ′ , t)) = m formal (t ′ − t)∂ t ′ q formal (t ′ , t).
(6 . 18) Since ∂ t ′ q formal could jump up around t ′ = t, we take the time average of this around t ′ = t, expecting a finite observation time. Thus, the time derivative of q(t), which is given in eq.(6 . 7), is evaluated as where in the first equality we have used the relation ∂ ∂t q formal (t ′ , t) | t ′ =t = 0, (6 . 20) which holds because q formal (t ′ , t) is independent of t for t ′ < t and is supposed to be smooth.
In the second equality we have changed the expression into the time average of ∂ t ′ q formal around t ′ = t. In the third and fourth equalities we have used eq.(6 . 18), and supposed that p formal changes very little near t ′ = t, and m eff and p(t) are given by eq.(5 . 10) and p(t) ≡ p formal (t, t). (6 . 21)