Lorentzian Path Integral for Quantum Tunneling and WKB Approximation for Wave-Function

Lorentzian path integral for quantum tunneling and WKB approximation for wave-function

Hiroki Matsuia,1

aCenter for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto, Japan

ABSTRACT

Recently, the Lorentzian path integral formulation using the Picard-Lefschetz theory has at- tracted much attention in quantum cosmology. In this paper, we analyze the tunneling amplitude in quantum mechanics by using the Lorentzian Picard-Lefschetz formulation and compare it with the WKB analysis of the conventional Schr¨odinger equation. We show that the Picard- Lefschetz Lorentzian formulation is consistent with the WKB approximation for wave-function and the Euclidean path integral formulation utilizing the solutions of the Euclidean constraint equation. We also consider some problems of this Lorentzian Picard-Lefschetz formulation and discuss a simpler semiclassical approximation of the Lorentzian path integral without integrating the lapse function. arXiv:2102.09767v3 [gr-qc] 23 Apr 2021

[email protected] 1 Intorduction

Wave function distinguishes quantum and classical picture of physical systems and can be exactly calculated by solving the Schrödingerequation for quantum mechanics (QM). In particular, quantum tunneling is one of the most important consequences of the wave function. The wave function inside the potential barrier seeps out the barrier even if the kinetic energy is lower than the potential energy. As a result, the non-zero probability occurs outside the potential in the quantum system even if it is bound by the potential barrier in the classical system. Hence, quantum tunneling is one of the most important phenomena to describe the quantum nature of the system. The Feynman’s path integral [1] is a standard formulation for QM and quantum field theory (QFT) which is equivalent with Schrödingerequation and defines quantum transition amplitude which is given by the integral over all paths weighted by the factor eiS[x]/~. In the path integral formulation, the transition amplitude from an initial state x(ti) = xi to a final state x(tf ) = xf is written by the functional integral,

Z x(tf )=xf iS[x] K(xf ; xi) = hxf , tf | xi, tii = Dx(t) exp , (1) x(ti)=xi ~ where we consider a unit mass particle whose the action S[x] is written by

Z 1 S[x] = dt x˙ 2 − V (x) . (2) 2

In a semi-classical regime, the associated path integral can be given by the saddle point approximation which is dominated by the path δS[x]/δx ≈ 0. In particular, to describe the quantum tunneling in Feynman’s formulation the Euclidean path integral method [2] is used. Performing the Wick rotation τ = it to Euclidean time, the dominant ﬁeld conﬁguration is given by solutions of the Euclidean equations of motion imposed by boundary conditions. The instanton constructed by the half-bounce Euclidean solutions going from xi to xf [3] describes the tunneling event across a degenerate potential and the splitting energy. On the other hand, the bounce constructed by the bouncing Euclidean solutions with xi = xf gives the vacuum decay ratio from the local-minimum false vacuum [2]. The Euclidean instanton method is useful tool for QFT [2–5] and even the gravity [6]. However, the method is conceptually less straightforward. When a particle tunnels through a potential barrier, we can accurately calculate the tunneling probability by using the Euclidean action SE[x] with the instanton solution, and in fact, it works well. However, when the particle is moving outside the potential, one needs to use the Lorentzian or real-time path integral so that the instanton formulation lacks unity and it is unclear why this method works. In

2 this perspective, the quantum tunneling of the Lorentzian or real-time path integrals has been studied in recent years [7–20], where complex instanton solutions are considered. It has been argued in Ref. [15, 20] that the Euclidean-time instanton solutions for a rotated time t = τe−iα close to Lorentzian-time describes something like a real-time description of quantum tunneling. Besides the extensions of the instanton method, a new tunneling approach has been proposed by Ref. [21] for quantum cosmology where the Lorentzian path integral includes a lapse integral and the saddle-point integration is performed by the Picard-Lefschetz theory (we call this method Lorentzian Picard-Lefschetz formulation). Feldbrugge et al. [21] showed that the Lorentzian path integral reduces to the Vilenkin’s tunneling wave function [22] by perfuming the integral over a contour. On other hand, Diaz Dorronsoro et al. reconsidered the Lorentzian path integral by integrating the lapse gauge over a different contour [23] and show that the Lorentzian path integral reduces to be the Hartle-Hawking’s no boundary wave function [24]. Both the tunneling or no-boundary wave functions can be derived as the Wentzel-Kramers-Brillouin (WKB) solutions of the Wheeler-DeWitt equation in mini-superspace model [25–27]. However, it is not fully understood why different wave functions can be obtained by different contours of the lapse integration in the Lorentzian path integral, and why the gravitational amplitude under the semi-classical approximation using the method of steepest descents or saddle-point corresponds to the WKB solution of the Wheeler-DeWitt equation [28–33] 1. In this paper, we consider the application of this Lorentzian path integral method to the tunneling amplitude of QM to discuss these conundrums without the complications associated with quantum gravity (QG). We will reconfirm that the conjecture of the tunneling or no- boundary wave functions based on the path integral of QG holds for QM as well, and show that the path integral (3) under the Lorentzian Picard-Lefschetz method corresponds to the WKB wave function of the Schrödingerequation. We will provide some examples in Section2 and Section3. Furthermore, we will discuss and confirm the relations between the Lorentzian Picard-Lefschetz formulation, and the WKB approximation for wave-function and the standard instanton method based on the Euclidean path integral. We also consider some problems of this Lorentzian Picard-Lefschetz formulation and discuss a simpler semiclassical approximation of the Lorentzian path integral. The present paper is organized as follows. In Section2 we introduce the Lorentzian path integral with the Picard-Lefschetz theory and apply this Lorentzian Picard-Lefschetz formulation to the linear, harmonic oscillator, and inverted harmonic oscillator models. In Section3 we review the Euclidean path integral, instanton and WKB approximation and consider their relations to the Lorentzian Picard-Lefschetz formulation. In Section4 we discuss a simpler semiclassical approximation method of the Lorentzian path integral without involving the lapse integral. Finally, in Section5 we conclude our work.

1The ﬁeld perturbation issues for the tunneling or no-boundary wave functions in the Lorentzian path integral have been discussed in Refs [32–44].

3 2 Lorentzian path integral with Picard-Lefschetz theory

We introduce the Lorentzian path integral for QM and apply the method of steepest descents or saddle-point utilizing the Picard-Lefschetz theory to the Lorentzian path integral. The Lorentzian path integral for QM is given by,

Z Z x(t1)=x1 iS[N, x] K(xf ; xi) = DN(t) Dx(t) exp , (3) x(t0)=x0 ~ where N(t) is the lapse function. From here we ﬁx the gauge: N(t) = N = const.. Extending the lapse function N from real R to complex C enable to consider classically prohibited evolution of the particles where N = 1 corresponds to moving along the real-time whereas N = −i corresponds to the Euclidean time. The action S[N, x] is written as

Z tf x˙ 2 S[N, x] = dtN(t) 2 − V (x) + E , (4) ti 2N(t) where V (x) is the potential and E is the energy of the system. We will discuss the linear, harmonic oscillator, inverted harmonic oscillator and double well models for QM. Fig.1 shows the potential V (x) for these models. From (4) we derive the following constraint equation and equations of motion [33],

x˙ 2 δS[x, N]/δN = 0 =⇒ + N 2V (x) = N 2E, (5) 2 δS[x, N]/δx = 0 =⇒ x¨ = −N 2V 0(x), (6)

It should be emphasized that the definition of the Lorentzian path integral (3) is not necessarily the same as the original path integral (1). However, there is a correspondence between these formulations from the definition of path integral, Z −iH(tf −ti) iS[x]/~ hxf | e | xii = Dx(t) e , (7) as the Euclidean path integral formulation which is given by the standard Wick rotation t → −iτ. The Lorentzian path integral (3) is given by the transformation of t → Nt and can be regarded as a complex-time formulation of the path integral by assuming N to be complex. Note that there is no obvious choice of integration contours in the complex-time path integral. However, as will be shown later, we can solve this problem by using the method of steepest descents or saddle-point method [28–31]. Moreover, the Picard-Lefschetz theory allows us to develop this argument more mathematically and rigorously. This Picard-Lefschetz theory provides a unique way to find a complex integration contour based on the steepest descent path

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-�� -� -� � � � -�� -�� -�� Figure 1: These ﬁgures show the potential V (x) for the linear, harmonic oscillator, inverted harmonic oscillator, and double well models. In this paper, we consider the Lorentzian path integral for these potentials.

(Lefschetz thimbles Jσ) and proceed with such oscillatory integral as [45], Z Z iS[x] X iS[xσ] dx exp = nσ dx exp . (8) R ~ σ Jσ ~

where nσ is the intersection number of the Lefschetz thimbles Jσ and steepest ascent path Kσ. In this section, we consider the application of this Lorentzian Picard-Lefschetz method to the tunneling transition of QM. 2

2.1 Quantum tunneling with Lorentzian path integral

For simplicity, let us consider the linear potential V = V0 − Λx with Λ > 0 which corresponds to the no-boundary proposal with the Lorentzian path integral for QG [21]. Therefore, we have

2The Lorentzian Picard-Lefschetz method assumes the semi-classical approximation, and other methods utilizing the Picard-Lefschetz theory might provide a complete analysis of the path integral for QM and QFT (see e.g., [46–48]).

5 the following action, Z 1 x˙ 2 S[x, N] = dtN 2 − V0 + Λx + E , (9) 0 2N whose classical solution is given as

Λ 1 x = N 2t2 + − N 2Λ + x − x t + x . (10) s 2 2 1 0 0

Following [21, 28], we can evaluate the Lorentzian path integral (3) under the semi-classical approximation. We assume the full solution x(t) = xs(t) + Q(t) where Q(t) is the Gaussian ﬂuctuation around the semi-classical solution (10). By substituting it for the action (9) and integrating the path integral over Q(t), 3 we have the following oscillatory integral,

r Z 1 dN iS0[N] K(x1; x0) = 1/2 exp , (11) 2πi~ C N ~ where

Z 1 2 x˙ s S0[N] = dtN 2 − V0 + Λxs + E 0 2N Λ2N 3 Λ (x − x )2 = − − N − (x + x ) − E + V + 1 0 . (12) 24 2 1 0 0 2N

Thus, we can calculate the transition amplitude by only performing the integration of the lapse function. Although it is generally diﬃcult to handle such oscillatory integrals, the Picard- Lefschetz theory deals with such integrals. The Picard-Lefschetz theory complexiﬁes the vari- ables and selects a complex path such that the original integral does not change formally via an extension of Cauchy’s integral theorem, and especially pass the saddle points known as the

Lefschetz thimbles Jσ.

Now let us integrate the lapse N integral along the Lefschetz thimbles Jσ, and we obtain

r Z X 1 dN iS0[N] K(x ; x ) = n exp . (13) 1 0 σ 2πi N 1/2 σ ~ Jσ ~

Since the lapse integral (13) can be approximately estimated based on the saddle points Ns

3We used the following path integral formulation,

Z X[1]=0 i Z 1 1 r m DX(t) exp dt mx˙ 2 = . X[0]=0 ~ 0 2 2πi~

6 and solving ∂S0[N]/∂N = 0, the saddle-points of the action S0[N] are given by, √ 2 N = a (Λx + E − V )1/2 + a (Λx + E − V )1/2 , (14) s 1 Λ 0 0 2 1 0 where a1, a2 ∈ {−1, 1}. The four saddle points (14) correspond to the intersection of the steepest descent path Jσ (Lefschetz thimbles) and steepest ascent path Kσ where Re [iS0 (N)] decreases and increases monotonically on Jσ and Kσ. The saddle-point action S0[Ns] evaluated at Ns is given by √ 2 2 h i S [N ] = a (Λx + E − V )3/2 + a (Λx + E − V )3/2 . (15) 0 s 1 3Λ 0 0 2 1 0 Thus, using the saddle-point approximation we can get the following result,

r Z 2 X iθσ 1 exp (iS0[Ns]/~) 1 ∂ S0[Ns] 2 K(x1; x0) ≈ nσe dR exp − R 2πi 1/2 2 ∂N 2 σ ~ Ns Jσ ~ v u X iθσ 1 iS0[Ns] ≈ nσe u exp , (16) t 2 iN ∂ S0[Ns] ~ σ s ∂N 2

4 where we expand S0[N] around a saddle point Ns as follows,

iS [N] iS [N] 1 ∂2S [N ] i ∂3S [N] 0 = 0 − 0 s R2 + 0 (N − N )3 + ... (17) 2 3 s ~ ~ N=Ns 2~ ∂N 6~ ∂N N=Ns

However, the original lapse N integral range is not specified and this fact leads to indefinite- ness in the results of the Lorentzian Picard-Lefschetz formulation (13). For the no-boundary proposal [24] using the Lorentzian path integral of QG, Feldbrugge et al. [21] considered a complex contour in R = (0, ∞) and showed that the gravitational amplitude reduces to the Vilenkin’s tunneling wave function [49]. On other hand, Diaz Dorronsoro et al. [23] reconsidered a different contour in R = (−∞, ∞) and showed that the gravitational amplitude reduces to be the Hartle-Hawking’s no boundary wave function [24]. From here we will show that the Lorentzian Picard-Lefschetz formulation (13) for QM reproduces the same result.

Let us consider a simple case with x0 = 0, E = 0 and x1 > V0/Λ. Only one Lefschetz thimble can be chosen in the integration domain R = (0, ∞). Fig.2 discribes Re [ iS0 (N)] in the complex plane where we set V = 3, Λ = 3 and x = 3, where the upper right ﬁgure suggests that a 0 1 √ q 2 Lefschetz thimble thorough N4 = − 3 i − 2 can be only chosen in R = (0, ∞). Thus, if ∂2S [N] 4 iθσ 0 i(2θσ +α) We introduced N − Ns ≡ Re and Arg ∂N 2 = α. Thus, we get e = i and θσ = N=Ns π/4 − α/2.

7 3

1 N2 N3 ▲ ▲

0 s

-1 ▲▲ N1 N4

-2

-3 -3 -2 -1 0 1 2 3 3 3

2 2

1 N2 1 N2 N3 ▲ ▲N3 ▲ ▲

0 0

N1 N4 -1 ▲▲ -1 ▲▲ N1 N4

-2 -2

-3 -3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

Figure 2: These four figures show Re [iS0 (N)] in the complex plane where we set V0 = 3, Λ = 3 and x1 = 3. The x-axis in these figures corresponds to the real axis of the complex lapse N and the y-axis to its imaginary axis. The blue dashed line shows the corresponding the Lefschetz √ √ √ q 2 q 2 q 2 thimbles with the saddle-points; N1 = − 3 i + 2 , N2 = 3 i − 2 , N3 = 3 i + 2 , √ q 2 N4 = − 3 i − 2 . The upper right figure consider N = (0, ∞) and a Lefschetz thimble with N4 can be only chosen. The lower figures take N = (−∞, ∞) and we can chose two different contours where one pass all saddle points N1,2,3,4 and another pass lower two saddle-points N1,4. We note that N2,4 corresponds to the exponent of the WKB wave function and this point will be discussed in Section3.

8 we consider the positive lapse N = (0, ∞), we obtain the following result, √ i π ! e 4 2 2i h 3/2 3/2i K(x1) ≈ exp − (−V0) − (Λx1 − V0) . (18) 1/2 1/4 1/4 3Λ 2 V0 (Λx1 − V0) ~

2 For the purposes of the later discussion in Section3 we consider the case with V0 = Λ /2 and x1 = V0/Λ = Λ/2 and the transition amplitude is written as

Λ2 K(x1) ≈ exp − . (19) 3~ which corresponds to the Euclidean path integral in Section3. On the other hand, by integrating the complex lapse integral along R = (−∞, ∞) and through the four saddle points, we obtain

√ √ 2 2i 3/2 3/2 2 2i 3/2 3/2 − 3Λ [(−V0) +(Λx1−V0) ] 3Λ [(−V0) −(Λx1−V0) ] K(x1) ≈ C1 e ~ + C2 e ~ √ √ (20) 2 2i 3/2 3/2 2 2i 3/2 3/2 3Λ [(−V0) +(Λx1−V0) ] − 3Λ [(−V0) −(Λx1−V0) ] + C3 e ~ + C4 e ~ , where C is the prefactor at these saddle points, and this Lorentzian amplitude corresponds to the result of Diaz Dorronsoro et al. [23]. Strangely, therefore, all the saddle points contribute to the Lorentzian transition amplitude. Fortheremore, as pointed out in Ref [35] choosing a diﬀerent contour in R = (−∞, ∞) leads to the diﬀerent transition amplitude,

√ √ 2 2i 3/2 3/2 2 2i 3/2 3/2 − 3Λ [(−V0) +(Λx1−V0) ] − 3Λ [(−V0) −(Λx1−V0) ] K(x1) ≈ C1 e ~ + C4 e ~ . (21)

In Fig.2 the lower figures consider N = (−∞, ∞) and we can chose two different contours where one pass all saddle points N1,2,3,4 and another pass lower two saddle-points N1,4. The ambiguity of the lapse integral (11) is still obscure and it seems that the lapse function N is a gauge degree of freedom of time. Also, depending on the semiclassical approximation of the action, it can be shown that the gauge degrees of freedom are over-integrated. Let us consider the Lorentzian path integral (3) and take a different semi-classical approximation to the action (4). In the previous discussion, the action was semi-classically approximated by the solution of the equation of motion (6), but now let us consider the semi-classical approximation of the action (4) by the constraint equation (5). Thus, by solving the constraint equation (5) forx ˙ and substituting in the action (4), we can get the following semi-classical action,

Z 1 Z x1 dx Z x1 p S0 = dt[2N(E − V )] = 2N(E − V ) = ± dx 2(E − V ), (22) 0 x0 x˙ x0

9 where it is important to note that this semi-classical action is diﬀerent from S0[N](12), cancels and does not have the contribution of N [33]. 5

Furthermore, strangely, the saddle-point of the semi-classical action S0[N](12) is consistent with S0 (22). In fact, in the linear potential, the sem-classical action is given by

Z x1 p Z x1 p S0 = ± dx 2(E − V ) = ± dx 2(E − V0 + Λx) √x0 x0 2 2 h i = ± (Λx + E − V )3/2 − (Λx + E − V )3/2 , (24) 3Λ 0 0 1 0 which corresponds to the saddle-point action S0[Ns](15). In Section3 we will discuss these coincidences in detail. We note that the two saddle points (14) corresponds to the WKB approximation, but other two saddle-points do not since the semi-classical action S0[Ns] is non-zero in the limit x1 → x0.

2.2 Harmonic oscillator and inverted harmonic oscillator models

In the previous subsection, we applied the Lorentzian path integral formulation to the linear potential and discuss some problems with the ambiguity of the lapse function. Let us put these issues aside and consider this Lorentzian formulation to the harmonic oscillator and inverted harmonic oscillator models, which are well known in QM. 1 2 2 First, let us consider the harmonic oscillator model with V = V0 + 2 Ω x . For simplicity, we consider the zero-energy system with E = 0 and the solution of the equations of motion is given by

xs = x0 cos (ΩNt) − x0 cot (ΩN) sin (ΩNt) + x1 csc (ΩN) sin (ΩNt) , (25) where we set x(0) = x0 and x(1) = x1. By applying the semi-classical approximation to the Lorentzian path integral (3) as well as the linear potential case, we obtain the following integral,

r Z 1 dN iS0[N] K(x1; x0) = 1/2 exp , (26) 2πi~ C N ~ 5By using the Lorentzian path integral (3) and integrating the lapse N, the semi-classical path integral diverges,

√ ∞ Z R x Z iS0 ±i 1 dx 2(E−V )/~ K(xf ; xi) ≈ dN exp ≈ e x0 dN → ∞ . (23) C ~ 0

10 saddle Figure 3: These ﬁgures show 3D-plot of Re iS0 (N) where we take V0 = 1 and x1 = 1 for harmonic and inverted harmonic√ oscillator models. The triangles describe the corresponding√ −1 −1 saddle-points Ns = ±i sinh 1/ 2 for the harmonic oscillator model and Ns = ±i sin 1/ 2 for the inverted harmonic oscillator model. The Lefschetz thimbles Jσ on the lapse integral is taken along the imaginary y-axis in the harmonic oscillator whereas it is taken along the real x-axis for the inverse harmonic oscillator. where

Z 1 2 x˙ s 1 2 2 S0[N] = dtN 2 − V0 − Ω xs 0 2N 2 (27) 1 = − NV + x2 + x2 Ω cot (ΩN) − x x Ω csc (ΩN) . 0 2 0 1 0 1

When we take x0 = 0 and Ω = 1, the semi-classical action S0[N] reads,

1 S [N] = −NV + x2 cot (N) . (28) 0 0 2 1

By solving ∂S0[N]/∂N = 0, the saddle-points are given as,

2 2 x1 sin (Ns) = − ⇐⇒ 2V0 s s (29) 2 2 −1 x1 −1 x1 Ns = ±i sinh + 2πc1, π ± i sinh + 2πc1 , 2V0 2V0

saddle where c1 ∈ Z. In Fig.3 we show the contour plot of Re iS0 (N) in the complex plane where we set V0 = 1 and x1 = 1 for the harmonic and inverted harmonic oscillator models. Although there are many saddle points, we will simply consider the contour corresponding to c1 = 0 in the integration domain R = (0, ∞).

11 q 2 −1 x1 Thus, we take one saddle-point Ns = −i sinh and the saddle-point action is evalu- 2V0 ated as 1 S [N ] = −N V + x2 cot (N ) 0 s s 0 2 1 s s √ s 2 2 −1 x1 ix1 V0 x1 = iV0 sinh + 2 + , (30) 2V0 2 V0 and we obtain the following transition amplitude,

  s √ s  2 2 −1 −1 x1 x1 V0 x1 K(x1) ≈ exp  V0 sinh + 2 +  . (31) ~ 2V0 2 V0

For the purposes of the later discussion in Section3 let us consider the speciﬁc case which satisfy √ 2 (e − 1) V0 x1 = √ , (32) 2e and the transition amplitude reads

2 4 (1 − 4e − e ) V0 K(x1) ≈ exp . (33) 4~ e2

1 2 2 Next, let us consider the inverted harmonic oscillator model with V = V0 − 2 Ω x . For simplicity, we consider the zero-energy system with E = 0 and the solution of the system is given by e−ΩNt −x e2ΩNt + x e2ΩNt+ΩN + x e2ΩN − x eΩN x = 0 1 0 1 . (34) s e2ΩN − 1 By taking semi-classical approximation to the Lorentzian path integral (3) we get the following semi-classical action,

Z 1 2 x˙ s 1 2 2 S0[N] = dtN 2 − V0 − Ω xs 0 2N 2 (35) 1 = − NV + Ω x2 + x2 coth(NΩ) − x x Ω csch(NΩ) . 0 2 0 1 0 1

We set x0 = 0 and Ω = 1, and S0[N] reads,

1 S [N] = −NV + x2 coth (N) . (36) 0 0 2 1

12 By solving ∂S0[N]/∂N = 0, the corresponding saddle-points are given by,

2 2 x1 sinh (Ns) = − ⇐⇒ 2V0 s s (37) 2 2 −1 x1 −1 x1 Ns = ±i sin + 2iπc2, iπ ± i sin + 2iπc2 , 2V0 2V0 where c2 ∈ Z. As before, we consider the contour corresponding to c2 = 0 in R = (0, ∞). q 2 −1 x1 Hence, we take Ns = −i sin and the saddle-point action S0[Ns] is evaluated as 2V0

1 S [N ] = −N V + x2 coth (N ) 0 s s 0 2 1 s s √ s 2 2 −1 x1 ix1 V0 x1 = iV0 sin + 2 − . (38) 2V0 2 V0

For the later discussion in Section3, let us consider the following case,

p x1 = 2V0 sin(1), (39) and the transition amplitude reads V0 + V0 sin(1) cos(1) K(x1) ≈ exp − . (40) ~

3 Euclidean path integral, instanton and WKB approximation

In this section, we review the Euclidean path integral, instanton and WKB approximation for the quantum tunneling and discuss their relations to the Lorentzian Picard-Lefschetz formulation (13).

3.1 Euclidean path integral and instanton

The evolution of the wave function in the classical regime can be approximated by saddle points δS[x]/δx ≈ 0 which satisfy the classical equation of motion. On the other hand, for quantum tunneling path which is the classically forbidden region the transition amplitude is approximately given by the saddle points of the Euclidean path integral which is derived by the solution of the Euclidean equations of motion. We can easily show that setting N = −i the Lorentzian path integral (3) corresponds to the Euclidean amplitude. In fact, the reparameterized action S[x, N] is given by t → Nt in S[x]

13 and the Euclidean action SE[x] is given by t → −iτ. Therefore, the action S[x, −i] represents the Euclidean action SE[x],

Z tf x˙ 2 iS[x, −i] = − dt + V (x) (41) ti 2 2 ! Z τf 1 dx ⇐⇒ iS[x] ≡ −SE[x] = − dτ + V (x) , (42) τi 2 dτ where the potential changes sign V (x) → −V (x) in SE[x]. From (42) we derive the following Euclidean equations of motion,

d2x = V 0(x). (43) dτ 2 However, the extrema of N corresponding to the saddle points of the Lorentzian path integral (3) using the Picard-Lefschetz theory deviate from N = −i and do not reproduce the transition amplitude based on the Euclidean path integral. From here we show some examples to clarify this fact. Let us consider the linear potential V = V0 − Λx with Λ > 0 for simplicity. Note that solving the Euclidean equations of motion and substituting the solutions for the Euclidean action SE[x] leads to the instanton amplitude. Therefore, the oscillatory integral (11) is consistent with the Euclidean amplitude except for the lapse integral. Thus, by taking N = −i we have the Euclidean transition amplitude for the linear potential,

r −1 r −1 iS0[−i]/~ −SE /~ K(x1; x0) = e = e , (44) 2π~ 2π~ where SE is

( 2 ) Z τf =1 1 dx S = dτ s + V − Λx E 2 dτ 0 s τi=0 (45) Λ2 Λ (x − x )2 = − − (x + x ) + V + 1 0 . 24 2 1 0 0 2

For simplicity, we consider x0 = 0 and the transition amplitude is given by

2 2 1 Λ Λx1 x1 K(x1) ≈ exp − − − + V0 + , (46) ~ 24 2 2 which is not consistent with the result (18) of the Lorentizan path integral. As we will see later, this result is not compatible with the WKB approximation either. The reason is that the semi-classical solution (10) with N = −i does not satisfy the constraint equation (5), which is the law of conservation of energy. Thus, when the constraint

14 3 3

2 2

1 1 ▲ ▲ ▲

0 0

▲▲ -1 ★★ -1 ★★▲▲▲

-2 -2

-3 -3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

saddle Figure 4: The left figure shows Re iS0 (N) in complex plane where we set V0 = 3, Λ = 3 and x1 = 3 and the star expresses N = −i. On the other hand, in the right figure we set V0 = 9/2, Λ = 3 and x1 = 3/2 and this figure shows that the saddle points of the action (15) is Ns also coincides with the Euclidean saddle point N = −i. equation (5) is actually satisfied the result of the Euclidean path integral coincides with the 2 Lorentzian Picard-Lefschetz formulation (18). For instance, when we take V0 = Λ /2 and x1 = V0/Λ = Λ/2 and the transition amplitude is given by

Λ2 K(x1) ≈ exp − , (47) 3~ which is consistent with the result (19) in the Lorentzian Picard-Lefschetz formulation. In this case the saddle points of the action (15), which is Ns = ±i also coincides with the Euclidean saddle point N = −i. In Fig.4 we show this correspondence. Next, let us discuss the well-known double well potential,

λ V (x) = x2 − a22 , (48) 4 and consider the quantum tunneling from x0 = −a to x1 = a. The usual method for ﬁnding the Euclidean (instanton) solution in this case, is to obtain it directly from the Euclidean energy conservation rather than solving the Euclidean equations of motion. For simplicity, we set

15 E = 0 and the Euclidean energy conservation gives,

2 r 1 dx dx λ − V (x) = 0 =⇒ = ± x2 − a2 . (49) 2 dτ dτ 2

Thus, we can get the following instanton solution interpolating between −a and a, ω x(τ) = ± a tanh (τ − τ ) , (50) 2 0 √ 3 where ω = 2λa and τ0 is an integration constant. The plus and minus classical solutions are the instanton and anti-instanton. The instanton corresponds to the particle initially sitting on the maximum of −V (x) at x = −a, passing x = 0 for a very short time and ending up at the other maximum of −V (x) at x = a. From the instanton the Euclidean action is given by √ 2 2λ a3 S = , (51) E 3 whose the transition amplitude K(a) is consistent with the WKB approximation of the wave function. For instance, if we extend this instanton as follows, ω x(t) = a tanh (iNt − τ ) , (52) 2 0 and apply it to the Lorentzian Picard-Lefschetz formulation, it returns the same result. As already discussed, a saddle-point approximation of the action by a solution of the Euclidean equations of motion does not give the correct transition amplitude. The correct result is obtained when the solution of the Euclidean equation of motion satisﬁes the energy conservation. Since the instanton solution is derived from the energy conservation in Euclidean form, it necessarily corresponds to the saddle point of the Lorentzian path integral. It is important to note that the solutions of the Euclidean equation of motion do not necessarily correspond to the correct semiclassical saddle point solutions. It is physically meaningful only if the solutions satisfy the constraint equation (5). We can see the same relations with the harmonic oscillator and inverted harmonic oscillator models. By taking N = −i we have the Euclidean transition amplitude for the harmonic and inverted harmonic oscillator, H 1 1 2 2 K (x1; x0) ≈ exp − V0 + x0 + x1 Ω coth Ω − x0x1Ω csch Ω , (53) ~ 2 I 1 1 2 2 K (x1; x0) ≈ exp − V0 + x0 + x1 Ω cot Ω − x0x1Ω csc Ω , (54) ~ 2

H I where we denote that K (x1; x0), K (x1; x0) are the transition amplitudes for the harmonic

16 and inverted harmonic oscillator are not consistent with the Lorentizan formulations (31). As discussed previously, when we take x0 = 0 and Ω = 1 and choose x1 which satisﬁes the Euclidean energy conservation, √ 2 H (e − 1) V0 I p x = √ , x = 2V0 sin(1), (55) 1 2e 1 the Euclidean transition amplitudes are consistent with the results (33) and (40) of the Lorentzian formulation,

2 4 H (1 − 4e − e ) V0 K (x1) ≈ exp , (56) 4~e2 I V0 + V0 sin(1) cos(1) K (x1) ≈ exp − . (57) ~

3.2 WKB approximation of Schr¨odingerequation

Finally, we will discuss WKB approximation of the Schrödingerequation and discuss the correspondence to the Lorentzian and Euclidean path integral formulation. The WKB approximation (or WKB method) is one of the semi-classical approximation methods for the Schrödinger equation. For the Schrödingerequation, which is the fundamental equation of QM and reads,

2 d2 Hˆ Ψ(x) = −~ + V (x) Ψ(x) = E Ψ(x), (58) 2 dx2 where Hˆ is the Hamiltonian, we assume that the solution is in the form of exp ( i S[x]) and ~ i (S0[x]+~S1[x]+··· ) expanded as a perturbation series of ~. By substituting Ψ(x) ≈ e ~ in the Schr¨odingerequation (58) we obtain the following equations,

1 dS 2 dS i d dS 0 + V − E = 0, 1 = ln 0 ..., (59) 2 dx dx 2 dx dx where S0 is the dominant contribution of the WKB wave function and can also be obtained by the constraint equation (5) as already discussed in Section2. We note that the Schr¨odinger equation (58) and wave function do not have the contribution of N even if the semi-classical action includes the lapse function N. In the leading order of the WKB approximation the wave function is given by

x c i Z p Ψ(x) ≈ 1/4 exp ± dx 2(E − V (x)) . (60) |2(E − V (x))| ~ x0

17 For the linear potential the wave function in the WKB approximation is given by

√ 2 2i 3/2 3/2 c1 + [(Λx0+E−V0) −(Λx1+E−V0) ] Ψ(x) ≈ e 3Λ~ |2(E − V (x))|1/4 √ (61) 2 2i 3/2 3/2 c2 − [(Λx0+E−V0) −(Λx1+E−V0) ] + e 3Λ~ , |2(E − V (x))|1/4 where the exponent of the above WKB wave function agrees with the two saddle-point actions

S0[Ns](15) of the Lorentzian path integral. In the Lorentzian Picard-Lefschetz formulation (13), four-saddle points dominate the path integral, and only one saddle point contributes when the lapse integral is defined as positive. On the other hand, in the WKB analysis, there is no uncertainty of the lapse function, and the exponents of the wave function can be either positive or negative, depending on the initial conditions. Thus, the positive and negative lapse function in the Lorentzian Picard-Lefschetz formulation (13) could be generally considered. On the other hand, for the harmonic and inverted harmonic oscillator potentials where we take x0 = 0 and E = 0, the exponents of the WKB wave function are given by s √ s 2 2 −1 x1 ix1 V0 x1 S0 = ±iV0 sinh ± 2 + , (62) 2V0 2 V0 s √ s 2 2 −1 x1 ix1 V0 x1 S0 = ±iV0 sin ± 2 − , (63) 2V0 2 V0 which are exactly consistent with the saddle points of the Lorentzian formulation (30) and (38). Finally, we comment the double well potential (4) with zero-energy system E = 0 and the corresponding semi-classical action is given by √ Z x1=a r 3 λ 2 2 2 2 2λ a S0 = ± dx − (x − a ) = ±i , (64) x0=−a 2 3 which is consistent with the Euclidean action SE utilizing the instanton (51). In summary the Lorentzian Picard-Lefschetz formulation (13) including the lapse N integral and using the Picard-Lefschetz theory, and the Euclidean formulation utilizing instanton are nothing more than the WKB analysis of the Schrödingerequation. The reason why these approaches correspond is as follows: Applying the method of steepest descents or saddle-point using the Picard-Lefschetz theory to the integration of the lapse N corresponds to taking semi-classical contours such that the constraint equation (5) is satisfied. Therefore, the Lorentzian Picard-Lefschetz formulation (13) corresponds to the WKB approximation to the wave function whose S0 is given by the constraint equation (5). Conversely, in order for the Euclidean path integral SE to be the correct semiclassical approximation, the solutions of the Euclidean equation of motion must satisfy the constraint equation (5).

18 4 Quantum tunneling with Lorentzian instanton

In this section, we will introduce a new instanton method based on the previous discussions. As discussed in Section3, the Euclidean saddle-point action corresponding to the exponent of the WKB wave function does not consist only of the simple solutions of the equation of motion. The solutions of the Euclidean equation of motion must satisfy the constraint equation (5) in the Euclidean form. The Lorentzian Picard-Lefschetz formulation [21] constructs the semi- classical transition amplitude by finding saddle points on the lapse integration which implies δS[x, N]/δN = 0. Since δS[x, N]/δN = 0 corresponds to the constraint equation (5), the transition amplitude becomes consistent with the WKB approximation for wave-function. Hence, we can expect to obtain the saddle-point action corresponding to the WKB approximation by finding the Lorentzian solution from the constraint equation (5) and substituting it into the Lorentzian action. We will now briefly introduce the method and call it Lorentzian instanton formulation.

Let us write the Lorentzian path integral including lapse function NL again,

Z xf iS[x] Z tf x˙ 2 K(xf ; xi) = Dx exp ,S[x] = dtNL 2 − V (x) + E , (65) xi ~ ti 2NL which ﬁxes the lapse NL and does not integrate. From (65) we derive the constraint equation and the equations of motion,

x˙ 2 δS[x]/δN = 0 =⇒ + N 2V (x) = N 2E, (66) L 2 L L 2 0 δS[x]/δx = 0 =⇒ x¨ = −NLV (x). (67)

As is well known in analytical mechanics, the equation of motion (66) is obtained by diﬀer- entiating the constraint equation (67). Therefore, only the constraint equation is considered.

Solving the constraint equation (67) for x with the initial condition x(ti) = x0, we get one semi-classical solution. Then, we impose the ﬁnal condition x(tf ) = x1 on the solution and determine the lapse function NL on the complex path. Thus, we can get the Lorentzian real-time solution even for quantum tunneling and construct the path integral (65) from the Lorentzian classical solution as well as the instanton method based on the Euclidean path integral. From here, we will show that this formulation is consistent with Lorentzian Picard-Lefschetz formulation (13) and WKB approximation for the wave function. For simplicity, let us consider the linear potential V = V0 − Λx and assume ti = 0 and tf = 1. The constraint equation (67) with the initial condition x(ti = 0) = x0 gives the following classical solution,

Λ p x (t) = x + N 2t2 ± N t 2E − 2V + 2Λx . (68) L 0 2 L L 0 0

19 By imposing the ﬁnal condition x(tf = 1) = x1 on the solution we can determine the lapse function NL and obtain, √ 2 N = ∓ (Λx + E − V )1/2 ± (Λx + E − V )1/2 , (69) L Λ 0 0 1 0 1/2 1/22 2 xL(t) = x0 + (Λx0 + E − V0) ± (Λx1 + E − V0) t √ 2 t p − (Λx + E − V )1/2 ± (Λx + E − V )1/2 2E − 2V + 2Λx . (70) Λ 0 0 1 0 0 0 where NL corresponds to the four saddle points (14) of the Lorentzian Picard-Lefschetz formulation and xL(t) is given by imposing NL to the Lorentzian solution (68). Note that plus sign solution in Eq. (70) is non-trivial since it is non-zero in the limit x1 → x0. Here, the degree of freedom of lapse is uniquely determined. Thus, the semi-classical action S[xL] is given by √ 2 2 h i S[x ] = ± (Λx + E − V )3/2 ± (Λx + E − V )3/2 . (71) L 3Λ 0 0 1 0 which is consistent with the semi-classical action for the WKB wave function and saddle- point action (15) of the Lorentzian Picard-Lefschetz formulation. By developing the saddle point method where x = xL + δx is decomposed as the Lorentzian classical solutions and the ﬂuctuation around them, the Lorentzian transition amplitude can be approximately given by 6

iS[x ] Z δx(1)=0 i δ2S[x] K(x ; x ) ' exp L Dδx exp δx2 1 0 2 ~ δx(0)=0 2~ δx x=xL Z δx(1)=0 Z 1 iS[xL] i 2 00 2 ' exp Dδx exp dt δx˙ − V (xL)δx (72) ~ δx(0)=0 2~NL 0 r 1 iS[x ] ' exp L , 2πi~NL ~

6 00 When V (xL) is non-zero, as the usual instanton in QFT [5], we can expand the ﬂuctuation and get the following expression,

Z δx(1)=0 Z 1 2 iS[xL] i d 00 2 K(xf ; xi) ' exp Dδx exp dtNL 2 2 − V (xL) δx ~ δx(0)=0 2~ 0 NLdt Z ∞ ∞ ! iS[xL] −1 X ' exp dc exp c2 λ n 2 iN n n ~ −∞ ~ L n=0 r iS[xL] Y 2π iNL ' exp ~ , λ ~ n n where λn are the eigenvalues and we omitted the normalization. For the Euclidean path integral where NL = −i, when all eigenvalues in a saddle point solution are positive, the ﬂuctuation around the saddle point increases the action and the saddle point approximation is correct. On the other hand, the negative eigenvalues reduce the action, and such a solution would be discarded [50]. A similar argument can be applied here.

20 00 where V (xL) is zero for the linear potential. Note that this method still has a problem how to select the correct Lorentzian solutions. Let us compare this result with the real-time path integral for the linear potential. Following Feynman’s famous textbook [51], the real-time transition amplitude is approximately given by

r 2 2 1 i (x1 − x0) Λ(x1 + x0) Λ K(x1; x0) ' exp + − , (73) 2πi~ ~ 2 2 24 which does not agree with the Lorentzian formulation (72). However, in this saddle point approximation the classical path solution does not respect the constraint equation (6) and if we explicitly write the energy of the system and usually think about that, x1,0 is restricted by E as classical mechanics. By imposing the constraint (6) on the real-time classical solution and setting x0 = 0 and V0 = E = 0 for simplicity, the real-time transition amplitude (73) agrees with the Lorentzian amplitude (72),

r 2 1 4ix1 K(x1; 0) ' exp . (74) 2πi~ 3~

As discussed in Section3, the Lorentzian transition amplitudes, which approximate the saddle point from the constraint equation (6) in correspondence with the WKB analysis, would be more accurate than the real-time amplitude (73) if the energy of the system is ﬁxed.

5 Conclusion

In this paper, we have analyzed the tunneling transition amplitude for QM using the Lorentzian Picard-Lefschetz formulation (13) for QM and compare it with the WKB analysis of the conventional Schrödingerequation. In the literature [21, 23, 28–35] the gravitational transition amplitude using the method of steepest descents and the Picard-Lefschetz theory corresponds to the WKB solution of the Wheeler-DeWitt equation. We have shown that they are agreement for the linear, harmonic oscillator, inverted harmonic oscillator, and double well models in QM. This correspondence has been summarized by Eq. (15) and Eq. (24) for the linear potential. For the harmonic oscillator or inverted harmonic oscillator models Eq. (30) and Eq. (62) or Eq. (38) and Eq. (63) have shown this correspondence. The two saddle points of the Lorentzian Picard- Lefschetz formulation (13) corresponds to the exponents of the WKB wave function whereas the others are conjugate. These results suggest that the Lorentzian Picard-Lefschetz formulation is consistent with the WKB analysis of the Schrödingerequation for QM. In Section3 and Section4 we have argued why the Lorentzian Picard-Lefschetz formulation is consistent with the WKB analysis of the conventional Schrödinger equation. After all, applying the saddle- point method of action to satisfy the constraint equation (6) leads to the correct semiclassical

21 approximation of the path integral. From this perspective, in Section4 we have provided a simpler semi-classical approximation way of the Lorentzian path integral without integrating the lapse function. On the other hand, in the Lorentzian Picard-Lefschetz formulation (13), the range of the lapse integral is controversial in Refs [21, 23, 34, 35], but we can consider negative range of the lapse from the correspondence of the WKB approximation. However, extending the integration range to the negative domain provides the ambiguity of the paths of steepest descent. Furthermore, depending on the semiclassical approximation of the action, the lapse degrees of freedom are over-integrated. In addition to these theoretical shortcomings, the deﬁnition of the Lorentzian path integral (3) is not necessarily the same as the original path integral (1). There is still no complete discussion of these identities and correspondences. In this paper, we will not consider further, but this approach still needs to be discussed.

Acknowledgment

We thank K. Yamamoto for stimulating discussions and valuable comments. We also thank S. Kanno, A. Matsumura, and H. Suzuki for helpful discussions.

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