Lorentzian path integral for quantum tunneling and WKB approximation for wave-function
PPrepared for submission
Lorentzian path integral for quantumtunneling and WKB approximationfor wave-function
Hiroki Matsui aa Department of Physics, Kyushu University, 744 Motooka, Nishi-Ku,Fukuoka 819-0395, Japan
E-mail: [email protected]
Abstract:
Recently, Lorentzian path integral formulation using Picard-Lefschetztheory has attracted much attention in quantum cosmology. In this work, we an-alyze tunneling transition probability in quantum mechanics using the Lorentzianpath integral and compare it with the WKB analysis of the Schrödinger equation.We find out that the saddle point action of the Picard-Lefschetz Lorentzian for-mulation are consistent with the exponent of the WKB wave function and thesemethods are surprisingly in perfect agreement in QM. Furthermore, we show thatthis Picard-Lefschetz Lorentzian method is equivalent to instanton method based onthe Euclidean path integral when the constraint equation (energy conservation of thesystem) is satisfied. These results suggest that the Picard-Lefschetz Lorentzian for-mulation is nothing more than the WKB approximation for wave-function. Finally,we propose and discuss a Lorentzian instanton formulation. a r X i v : . [ g r- q c ] F e b ontents Wave function distinguishes quantum and classical picture of physical systems, andcan be exactly calculated by solving Schrödinger equation for quantum mechanics(QM). In particular, quantum tunneling is one of the most important consequence ofthe wave function. The wave function inside potential barrier seeps out the barriereven if the kinetic energy is lower than the potential energy. As a result, the non-zeroprobability occurs outside the potential in the quantum system even if it is boundby the potential barrier in the classical system. Hence, the quantum tunneling is oneof the most important phenomena to describe a quantum nature of the system.The Feynman’s path integral [1] is a standard formulation for QM and quantumfield theory (QFT) which is equivalent with Schrödinger equation and defines quan-tum transition amplitude which is given by the integral over all paths weighted bythe factor e iS [ x ] / (cid:126) . In the path integral formulation, the transition amplitude from aninitial state x ( t i ) = x i to a final state x ( t f ) = x f is written by the functional integral, K ( x f ; x i ) = (cid:104) x f , t f | x i , t i (cid:105) = (cid:90) x ( t f )= x f x ( t i )= x i D x ( t ) exp (cid:18) iS [ x ] (cid:126) (cid:19) , (1.1)where we consider a unit mass particle whose the action S [ x ] is written by S [ x ] = (cid:90) d t (cid:18)
12 ˙ x − V ( x ) (cid:19) . (1.2)– 1 –n semi-classical regime the associated path integral can be given by the saddlepoint approximation which is dominated by the path δS [ x ] /δx ≈ . In particular,to describe the quantum tunneling in the Feynman’s formulation the Euclidean pathintegral method [2] is used. Performing the Wick rotation τ = it to Euclidean time,the dominant field configuration is given by solutions of the Euclidean equationsof motion imposed by a boundary condition. The instanton constructed by thehalf-bounce Euclidean solutions going from x i to x f [3] describes the tunneling eventacross a degenerate potential and the splitting energy. On the other hand, the bounceconstructed by the bouncing Euclidean solutions with x i = x f gives the vacuum decayratio from the local-minimum false vacuum [2]. The Euclidean instanton method isuseful tool for QFT [2–5] and even the gravity [6].However, the method is conceptually less straightforward. When a particle tun-nels through a potential barrier, we can accurately calculate the tunneling probabilityby using the Euclidean action S E [ x ] with the instanton solution, and in fact it workswell. However, when the particle is moving outside the potential, one needs to usethe Lorentzian/real-time path integral so that the instanton formulation lacks unityand it is unclear why this method works. In this perspective, the quantum tunnelingof the Lorentzian/real-time path integrals has been studied in recent years [7–19],where complex instanton solutions are considered. In addition, it has been arguedin Ref. [15] that the Euclidean-time instanton solutions for a rotated time t = τ e − iα close to Lorentzian-time describes something like a real time description of quantumtunneling. But, no satisfying approach has been found with so much work.Besides the extensions of the instanton method, a new tunneling approach hasbeen discussed by Ref. [20] for quantum cosmology where the Lorentzian path integralincludes a lapse integral and the saddle-point integration is performed by the Picard-Lefschetz theory. The gravitational amplitude using the Lorentzian path integral isgiven by G ( g f ; g i ) = (cid:90) D g µν exp (cid:18) iS [ g µν ] (cid:126) (cid:19) , (1.3)which derives solutions to the Wheeler-DeWitt equation under proper constraints [21]and S [ g µν ] is the Einstein-Hilbert action. For the mini-superspace model, the metricof the homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW)universe is written as d s = − N ( t ) q ( t ) d t + q ( t ) d Ω , (1.4)where dΩ is the unit -sphere metric and N ( t ) is the lapse function in the ADMformulation of general relativity. Following [22] the gravitational amplitude (1.3) canbe written by the path integral including the lapse integral, G ( q f ; q i ) = (cid:90) R d N (cid:90) D q ( t ) exp (cid:18) iS [ N, q ] (cid:126) (cid:19) . (1.5)– 2 –ince the action S [ N, q ] is quadratic, the functional integral (1.5) can be evaluatedby the semi-classical approximation, which leads to the following expression [20], G ( q f ; q i ) = (cid:114) πi (cid:126) (cid:90) R d NN / exp (cid:18) π iS [ N ] (cid:126) (cid:19) , (1.6)where S [ N ] is the semi-classical action, S = N Λ
36 + N (cid:18) − Λ( q i + q f )2 + 3 K (cid:19) + 1 N (cid:18) −
34 ( q f − q i ) (cid:19) . (1.7)By imposing the initial condition q i = 0 [23–25], the gravitational amplitude (1.6)corresponds to the cosmological wave function from nothing [26] although the lapseintegral does not converge [22, 27]. However, one can overcome this mathematicalproblem by using Picard-Lefschetz theory which provides a unique way to find a com-plex integration contour based on the steepest descent or ascent flows, and proceedwith such oscillatory integral as, (cid:90) R d x exp (cid:18) iS [ x ] (cid:126) (cid:19) = (cid:88) σ n σ (cid:90) C d x exp (cid:18) iS [ x σ ] (cid:126) (cid:19) . (1.8)Based on the Lorentzian path integral using the Picard-Lefschetz theory, Feldbruggeet al. [20] showed that the gravitational amplitude (1.6) reduces to the Vilenkin’stunneling wave function [28] by perfuming the integral over a contour R = (0 , ∞ ) .On other hand, Diaz Dorronsoro et al. reconsidered the gravitational amplitude (1.3)by integrating the lapse over a different contour R = ( −∞ , ∞ ) [29] and show thatthe gravitational amplitude (1.6) reduces to be the Hartle-Hawking’s no boundarywave function [24]. Both the tunneling or no-boundary wave functions can be derivedas some solutions of the Wheeler-DeWitt equation in mini-superspace model [30–32].Therefore, it is strange why the different integration paths of the lapse N yielddifferent wave functions and why the gravitational amplitude (1.3) using the Picard-Lefshetz theory to the lapse integral is exactly consistent with some solutions of theWheeler-DeWitt equation. These issues were discussed in Ref. [33] where the authorconsidered the tunneling/no boundary wave function in terms of the Wheeler-DeWittequation under semiclassical approximation and also perturbation problems, andsuggested that there is no advantage to using the functional integral with the Picard-Lefshetz method because the lapse dependence for the path integral drops out of thesemi-classical action when one uses the constraint equation [33]. Therefore, it is stillunclear why it is necessary to perform the path integral involving the lapse function.To clarify these issues we consider the Lorentzian path integral for QM includingthe lapse N integral K ( x f ; x i ) = (cid:90) C d N (cid:90) x ( t f )= x f x ( t i )= x i D x exp (cid:18) iS [ N, x ] (cid:126) (cid:19) , (1.9) The field perturbation issues for the tunneling or no-boundary wave functions in the Lorentzianpath integral have been recently discussed in Refs [34–45]. – 3 –here we extend the lapse integration range from real R to complex C , and discussquantum tunneling process in this formulation. We will show that applying thePicard-Lefshetz theory to the path integral (1.9) for the quantum tunneling thetransition amplitude corresponds to the wave function of the Schrödinger equationin Wentzel-Kramers-Brillouin (WKB) approximation and provide some examples inSection 2 and Section 3. The exact reasons why the two approaches correspond willbe discussed later, but in brief they are as follows: Applying the Picard-Lefshetzmethod to the integration of the lapse N corresponds to taking a semi-classical pathsuch that the energy conservation of the system is satisfied. Therefore, applying thePicard-Lefshetz theory to the path integral (1.9) necessarily corresponds to the WKBapproximation to the wave function. Furthermore, the extrema of N correspondingto the saddle points of the Lorentzian path integral (1.9) with the Picard-Lefshetztheory deviate from N = − i and do not always reproduce the amplitude based on theEuclidean path integral. We discuss the relations between the Lorentzian formulationand the Euclidean path integral with instanton and WKB approximation. We alsoconsider some problems of the Lorentzian formulation and finally discuss a newLorentzian approach without the lapse integral.The present paper is organized as follows. In Section 2 we introduce the Lorentzianpath integral with the Picard-Lefshetz theory and apply this path integral formula-tion to the linear, harmonic oscillator and inverted harmonic oscillator models. InSection 3 we review the Euclidean path integral, instanton and WKB approximationand consider their relations to the Lorentzian path integral formulation. In Section 4we discuss a new Lorentzian instanton formulation which can describe the real-timedynamics and does not involve the lapse integral. Finally, in Section 5 we draw theconclusion of our work. In this section we introduce the Lorentzian path integral for QM and apply this pathintegral formulation to the Picard-Lefshetz theory. The action (1.2) for QM can bewritten as S [ x ( t ) , N ( t )] = (cid:90) t f t i dtN ( t ) (cid:18) ˙ x N ( t ) − V ( x ) + E (cid:19) , (2.1)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 classi-cally prohibited evolution of the particles where N = 1 corresponds to moving alongthe real time whereas N = − i corresponds to the imaginary time and the action S [ x, N ] represents the Euclidean action S E [ x ] . We will discuss the linear, harmonicoscillator, inverted harmonic oscillator and double well models for QM. Fig. 1 showsthe potential V ( x ) for these models. – 4 – ��� - ��� - ��� ��� ��� ��� ��� - ������ � � ( � ) - � - � � � ���������������� � � ( � ) - � - � � � � - ��� - ������������ � � ( � ) - ��� - ��� - ��� ��� ��� ��� ������������������ � � ( � ) Figure 1 . The above figures show the potential V ( x ) for the linear, harmonic oscillator, in-verted harmonic oscillator and double well models. In this paper we consider the Lorentzianand Euclidean path integral for these potential. From (2.1) we derive the following constraint equation and equations of motion, δS [ x, N ] /δN = 0 = ⇒ ˙ x N V ( x ) = N E, (2.2) δS [ x, N ] /δx = 0 = ⇒ ¨ x = − N V (cid:48) ( x ) , (2.3)where E is the energy of the system. For simplicity, let us consider the linear potential V = V − Λ x with Λ > which cor-responds to the no-boundary proposal with the Lorentzian path integral for quantumgravity [20]. Thus, we have the following action, S [ x, N ] = (cid:90) dtN (cid:18) ˙ x N − V + Λ x + E (cid:19) , (2.4)where we take t i,f = 0 , and x i,f = x , and the classical solution for the action (2.4)is given by x s = Λ2 N t + (cid:18) − N Λ + x − x (cid:19) t + x . (2.5)– 5 –ollowing [20, 22], we can evaluate the Lorentzian path integral (1.9) under thesemi-classical approximation. We assume the full solution x ( t ) = x s ( t ) + Q ( t ) where Q ( t ) is the Gaussian fluctuation around the semi-classical solution. By substitutingit for the action (2.4) and integrating the path integral over Q ( t ) , we have thefollowing oscillatory integral, K ( x ; x ) = (cid:114) πi (cid:126) (cid:90) C d NN / exp (cid:18) iS [ N ] (cid:126) (cid:19) , (2.6)where S [ N ] = (cid:90) dtN (cid:18) ˙ x s N − V + Λ x s + E (cid:19) = − Λ N − N (cid:18) − Λ2 ( x + x ) − E + V (cid:19) + ( x − x ) N . (2.7)Therefore, we can calculate the transition amplitude by performing the integrationof the lapse function. Although it is generally difficult to handle such oscillatoryintegrals, the Picard-Lefschetz theory deals with such integrals. The Picard-Lefschetztheory complexifies the variables and selects a complex path such that the originalintegral does not change formally via an extension of Cauchy’s integral theorem, andespecially pass the saddle points known as the Lefschetz thimbles J σ . Now let usintegrate the lapse N integral along the Lefschetz thimbles J σ , and we get K ( x ; x ) = (cid:88) σ n σ (cid:114) πi (cid:126) (cid:90) J σ d NN / exp (cid:18) iS [ N ] (cid:126) (cid:19) . (2.8)The lapse integral (2.8) can be approximately estimated based on the saddle points N s and solving ∂S [ N ] /∂N = 0 , the saddle-points of the action S [ N ] are given by, N s = a √ (cid:2) (Λ x + E − V ) / + a (Λ x + E − V ) / (cid:3) , (2.9)where a , a ∈ {− , } . The four saddle points (2.9) correspond to the intersectionof the steepest descent path J σ (Lefschetz thimbles) and steepest ascent path K σ where Re [ iS ( N )] decreases and increases monotonically on J σ and K σ , and n σ isthe intersection number. The saddle-point action S [ N s ] evaluated at N s is given by S [ N s ] = a √ (cid:104) (Λ x + E − V ) / + a (Λ x + E − V ) / (cid:105) . (2.10) We used the following path integral formulation, (cid:90) X [1]=0 X [0]=0 D X ( t ) exp (cid:18) i (cid:126) (cid:90) d t m ˙ x (cid:19) = (cid:114) m πi (cid:126) . – 6 –hus, using the saddle-point approximation we can get the following result, K ( x ; x ) ≈ (cid:88) σ n σ e iθ σ (cid:114) πi (cid:126) exp ( iS [ N s ] / (cid:126) ) N / s (cid:90) J σ d R exp (cid:18) − (cid:126) (cid:12)(cid:12)(cid:12)(cid:12) ∂ S [ N s ] ∂N (cid:12)(cid:12)(cid:12)(cid:12) R (cid:19) ≈ (cid:88) σ n σ e iθ σ (cid:118)(cid:117)(cid:117)(cid:116) iN s (cid:12)(cid:12)(cid:12) ∂ S [ N s ] ∂N (cid:12)(cid:12)(cid:12) exp (cid:18) iS [ N s ] (cid:126) (cid:19) , (2.11)where we expand S [ N ] around a saddle point N s as follows, iS [ N ] (cid:126) = iS [ N ] (cid:126) (cid:12)(cid:12)(cid:12) N = N s − (cid:126) (cid:12)(cid:12)(cid:12)(cid:12) ∂ S [ N s ] ∂N (cid:12)(cid:12)(cid:12)(cid:12) R + i (cid:126) ∂ S [ N ] ∂N (cid:12)(cid:12)(cid:12) N = N s ( N − N s ) + . . . (2.12)The second derivative is given by ∂ S [ N ] ∂N (cid:12)(cid:12)(cid:12) N = N s = 2 a N s ( E − V ) / (Λ x + E − V ) / . (2.13)However, the original lapse N integral range is not specified and this fact leadsto indefiniteness in the results of the Lorentzian formulation. For the no-boundaryproposal [24] using the Lorentzian path integral, Feldbrugge et al. [20] considered acomplex contour in R = (0 , ∞ ) and showed that the gravitational amplitude (1.6)using the Picard-Lefschetz theory reduces to the Vilenkin’s tunneling wave func-tion [26]. On other hand, Diaz Dorronsoro et al. [29] reconsidered a different contourin R = ( −∞ , ∞ ) and showed that the gravitational amplitude (1.6) reduces to bethe Hartle-Hawking’s no boundary wave function [24]. From here we will show thatthe lapse integral (2.6) for QM with the Picard-Lefschetz theory reproduces the sameresult.Let us consider a simple case with x = 0 , E = 0 and x > V / Λ . We will showthat only one Lefschetz thimble can be chosen in the integration domain R = (0 , ∞ ) .Fig. 2 discribes Re [ iS ( N )] in the complex plane where we set V = 3 , Λ = 3 and x = 3 . In Fig. 2 the upper right figure suggests that a Lefschetz thimble thorough N = − (cid:113) (cid:0) i − √ (cid:1) can be only chosen in R = (0 , ∞ ) . Thus, if we consider thepositive lapse N = (0 , ∞ ) , we obtain the following result, K ( x ) ≈ e i π / V / (Λ x − V ) / exp (cid:32) − √ i (cid:126) (cid:104) ( − V ) / − (Λ x − V ) / (cid:105)(cid:33) . (2.14)For the purposes of the later discussion in Section 3 we consider the case with V =Λ / and x = V / Λ = Λ / and the transition amplitude is written as K ( x ) ≈ exp (cid:18) − Λ (cid:126) (cid:19) . (2.15) We introduced N − N s ≡ Re iθ σ and Arg (cid:18) ∂ S [ N ] ∂N (cid:12)(cid:12)(cid:12) N = N s (cid:19) = α . Thus, we get e i (2 θ σ + α ) = i and θ σ = π/ − α/ . – 7 – ▲▲ ▲▲▲ N s N N N - - - - - - ▲ ▲▲ ▲ N N N N - - - - - - ▲ ▲▲ ▲▲▲ N N N N - - - - - - Figure 2 . These four figures show Re [ iS ( N )] in the complex plane where we set V = 3 , Λ = 3 and x = 3 . The x -axis in these figures corresponds to the real axis of the complexlapse N and the y -axis to its imaginary axis. The blue dashed line shows the correspondingthe Lefschetz thimbles with the saddle-points; N = − (cid:113) (cid:0) i + √ (cid:1) , N = (cid:113) (cid:0) i − √ (cid:1) , N = (cid:113) (cid:0) i + √ (cid:1) , N = − (cid:113) (cid:0) i − √ (cid:1) . The upper right figure consider N = (0 , ∞ ) anda Lefschetz thimble with N can be only chosen. The lower figures take N = ( −∞ , ∞ ) andwe can chose two different contours where one pass all saddle points N , , , and anotherpass lower two saddle-points N , . We note that N , corresponds to the exponent of theWKB wave function and this point will be discussed in Section 3. We will see later that this result corresponds to the Euclidean path integral in Sec-tion 3.On the other hand, by integrating the complex lapse integral along R = ( −∞ , ∞ ) – 8 –nd through the four saddle points, we can obtain K ( x ) ≈ C exp (cid:32) − √ i (cid:126) (cid:104) ( − V ) / + (Λ x − V ) / (cid:105)(cid:33) + C exp (cid:32) √ i (cid:126) (cid:104) ( − V ) / − (Λ x − V ) / (cid:105)(cid:33) + C exp (cid:32) √ i (cid:126) (cid:104) ( − V ) / + (Λ x − V ) / (cid:105)(cid:33) + C exp (cid:32) − √ i (cid:126) (cid:104) ( − V ) / − (Λ x − V ) / (cid:105)(cid:33) , (2.16)where C is the prefactor at these saddle points, and this Lorentzian amplitude cor-responds to the result of Diaz Dorronsoro et al. [29]. Strangely, therefore, all thesaddle points contribute to the Lorentzian transition amplitude. Fortheremore, aspointed out in Ref [35] choosing a different contour in R = ( −∞ , ∞ ) leads to thedifferent transition amplitude, K ( x ) ≈ C exp (cid:32) − √ i (cid:126) (cid:104) ( − V ) / + (Λ x − V ) / (cid:105)(cid:33) + C exp (cid:32) − √ i (cid:126) (cid:104) ( − V ) / − (Λ x − V ) / (cid:105)(cid:33) . (2.17)In Fig. 2 the lower figures consider N = ( −∞ , ∞ ) and we can chose two differentcontours where one pass all saddle points N , , , and another pass lower two saddle-points N , .The ambiguity of the lapse integral (2.6) is still obscure. In fact, it seems thatthe lapse function N is a gauge degree of freedom of time, and it seems to be unneces-sary to integrate if the gauge is perfectly fixed. Also, depending on the semiclassicalapproximation of the action, it can be shown that the gauge degrees of freedom areover-integrated. Let us consider the Lorentzian path integral (1.9) and take the semi-classical approximation to the action (2.1). In the previous discussion, the actionwas semi-classically approximated by the solution of the equation of motion (2.3),but now let us consider semi-classical approximation of the action (2.1) by the con-straint equation (2.2). Therefore, by solving the constraint equation (2.2) for ˙ x andsubstituting in the action (2.1), we can get the following semi-classical action, S = (cid:90) dt [2 N ( E − V )] = (cid:90) x x dx ˙ x N ( E − V ) = ± (cid:90) x x dx (cid:112) E − V ) , (2.18)where it is important to note that this semi-classical action is different from S [ N ] (2.7),cancels and does not have the contribution of N [33]. By using the Lorentzian pathintegral (1.9) and integrating the lapse function N , the path integral diverges K ( x f ; x i ) ≈ (cid:90) C d N exp (cid:18) iS (cid:126) (cid:19) ≈ e ± i (cid:82) x x dx √ E − V ) / (cid:126) (cid:90) ∞ d N → ∞ , (2.19)– 9 –hich suggests that the gauge volume is not properly treated. Furthermore, strangely,the saddle-point of the semi-classical action S [ N ] (2.7) is consistent with S (2.18).In fact, in the linear potential, the sem-classical action is given by S = ± (cid:90) x x dx (cid:112) E − V ) = ± (cid:90) x x dx (cid:112) E − V + Λ x )= ± √ (cid:104) (Λ x + E − V ) / − (Λ x + E − V ) / (cid:105) , (2.20)which corresponds to the saddle-point action S [ N s ] (2.10). In Section 3 we willdiscuss these coincidences in detail. We note that the two saddle points (2.9) cor-responds to the WKB, but other two saddle-points do not since the semi-classicalaction S [ N s ] is non-zero in the limit x → x . In the previous subsection, we applied the Lorentzian path integral formulation to thelinear 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 harmonicoscillator and inverted harmonic oscillator models, which are well known in QM.First, let us consider the harmonic oscillator model with V = V + Ω x . Forsimplicity, we consider the zero-energy system with E = 0 and the solution of theequations of motion is given by x s = x cos (Ω N t ) − x cot (Ω N ) sin (Ω N t ) + x csc (Ω N ) sin (Ω N t ) , (2.21)where we set x (0) = x and x (1) = x . By applying the semi-classical approximationto the Lorentzian path integral (1.9) as well as the linear potential case, we get thefollowing integral, K ( x ; x ) = (cid:114) πi (cid:126) (cid:90) C d NN / exp (cid:18) iS [ N ] (cid:126) (cid:19) , (2.22)where S [ N ] = (cid:90) dtN (cid:18) ˙ x s N − V −
12 Ω x s (cid:19) = − N V + 12 (cid:0) x + x (cid:1) Ω cot (Ω N ) − x x Ω csc (Ω N ) . (2.23)For simplicity, we take x = 0 and Ω = 1 , and the semi-classical action S [ N ] reads, S [ N ] = − N V + 12 x cot ( N ) . (2.24)– 10 – ▲ ▲▲ ▲▲ - - - - - - ▲▲▲▲▲▲▲▲ - - - - - - Figure 3 . These left and right figures show Re (cid:2) iS saddle0 ( N ) (cid:3) in the complex plane where weset V = 1 and x = 1 for harmonic and inverted harmonic oscillator models. The trianglesdescribe the corresponding saddle-points N s = ± i sinh − (cid:113) , ± π ± i sinh − (cid:113) for theharmonic oscillator and N s = ± i sin − (cid:113) , ± iπ ± i sin − (cid:113) , iπ − i sin − (cid:113) , − iπ + i sin − (cid:113) for the inverted harmonic oscillator. The Lefschetz thimbles J on the lapseintegral is taken along the imaginary y -axis in the harmonic oscillator model. On the otherhand, it is taken along the real x -axis for the inverse harmonic oscillator model. By solving ∂S [ N ] /∂N = 0 , the saddle-points are given as, sin ( N s ) = − x V ⇐⇒ N s = ± i sinh − (cid:115) x V + 2 πc , π ± i sinh − (cid:115) x V + 2 πc , (2.25)where c ∈ Z . In Fig. 3 we show the contour plot of Re (cid:2) iS saddle0 ( N ) (cid:3) in the complexplane where we set V = 1 and x = 1 for the harmonic and inverted harmonicoscillator models. It is clearly suggested that there are many saddle points, but forsimplicity, we will consider the contour corresponding to c = 0 in the integrationdomain R = (0 , ∞ ) . Thus, we take one saddle-point N s = − i sinh − (cid:113) x V and thesaddle-point action is evaluated as S [ N s ] = − N s V + 12 x cot ( N s )= iV sinh − (cid:115) x V + ix √ V (cid:115) x V . (2.26)– 11 –hus, we obtain the following transition amplitude, K ( x ) ≈ exp − (cid:126) V sinh − (cid:115) x V + x √ V (cid:115) x V . (2.27)For the purposes of the later discussion in Section 3 let us consider the specific casewhich satisfy x = ( e − √ V √ e , (2.28)and the transition amplitude reads K ( x ) ≈ exp (cid:20) (1 − e − e ) V (cid:126) e (cid:21) . (2.29)Next, let us consider the inverted harmonic oscillator model with V = V − Ω x .For simplicity, we consider the zero-energy system with E = 0 and the action (2.4)gives the equations of motion whose the solution of x is given by x s = e − Ω Nt (cid:0) − x e Nt + x e Nt +Ω N + x e N − x e Ω N (cid:1) e N − . (2.30)By taking semi-classical approximation to the Lorentzian path integral (1.9) we getthe following semi-classical action, S [ N ] = (cid:90) dtN (cid:18) ˙ x s N − V −
12 Ω x s (cid:19) = − N V + 12 Ω (cid:0) x + x (cid:1) coth( N Ω) − x x Ω csch ( N Ω) . (2.31)For simplicity, we set x = 0 and Ω = 1 , and S [ N ] reads, S [ N ] = − N V + 12 x coth ( N ) . (2.32)By solving ∂S [ N ] /∂N = 0 , the corresponding saddle-points are given by, sinh ( N s ) = − x V ⇐⇒ N s = ± i sin − (cid:115) x V + 2 iπc , iπ ± i sin − (cid:115) x V + 2 iπc , (2.33)where c ∈ Z . As before, we will consider the contour corresponding to c = 0 inthe integration domain R = (0 , ∞ ) . Hence, we take N s = − i sin − (cid:113) x V and thesaddle-point action S [ N s ] is evaluated as S [ N s ] = − N s V + 12 x coth ( N s )= iV sin − (cid:115) x V + ix √ V (cid:115) − x V . (2.34)– 12 –s before and for the later discussion in Section 3, let us consider the following case, x = (cid:112) V sin(1) , (2.35)and the transition amplitude reads K ( x ) ≈ exp (cid:18) − V + V sin(1) cos(1) (cid:126) (cid:19) . (2.36) In this section we review the Euclidean path integral, instanton and WKB approxi-mation for the quantum tunneling and discuss their relations to the Lorentzian pathintegral formulation (1.9).
The evolution of the wave function in the classical regime can be approximated bysaddle points δS [ x ] /δx ≈ which satisfy classical equation of motion. On the otherhand, through quantum tunneling path which is the classically forbidden region thetransition amplitude is approximately given by the saddle points of the Euclideanpath 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 (1.9)corresponds to the Euclidean amplitude. In fact, the reparameterized action S [ x, N ] is given by t → N t in S [ x ] and the Euclidean action S E [ x ] is given by t → − iτ .Thus, the action S [ x, − i ] represents the Euclidean action S E [ x ] , iS [ x, − i ] = − (cid:90) t f t i dt (cid:18) ˙ x V ( x ) (cid:19) (3.1) ⇐⇒ iS [ x ] ≡ − S E [ x ] = − (cid:90) τ f τ i d τ (cid:32) (cid:18) dxdτ (cid:19) + V ( x ) (cid:33) , (3.2)where the potential energy changes sign V ( x ) → − V ( x ) in S E [ x ] . From (3.2) wederive the following Euclidean equations of motion, d xdτ = V (cid:48) ( x ) . (3.3)However, the extrema of N corresponding to the saddle points of the Lorentzianpath integral (1.9) using the Picard-Lefshetz theory deviate from N = − i and donot reproduce the transition amplitude based on the Euclidean path integral. Fromhere we show some example to clarify this fact. Let us consider the linear potential V = V − Λ x with Λ > for simplicity. Note that solving the Euclidean equations– 13 – ▲▲ ▲▲▲ ★★ - - - - - - ★★ ▲▲▲▲ - - - - - - Figure 4 . The left figure shows Re (cid:2) iS saddle0 ( N ) (cid:3) in complex plane where we set V = 3 , Λ = 3 and x = 3 and the star expresses N = − i . On the other hand, in the right figurewe set V = 9 / , Λ = 3 and x = 3 / and this figure shows that the saddle points of theaction (2.10) is N s also coincides with the Euclidean saddle point N = − i . of motion and substituting the solutions for the Euclidean action S E [ x ] leads to theinstanton amplitude. Therefore, the oscillatory integral (2.6) is consistent with theEuclidean amplitude except for the lapse integral. Thus, by taking N = − i we havethe Euclidean transition amplitude for the linear potential, K ( x ; x ) = (cid:114) − π (cid:126) e iS [ − i ] / (cid:126) = (cid:114) − π (cid:126) e − S E / (cid:126) , (3.4)where S E is S E = (cid:90) τ f =1 τ i =0 d τ (cid:40) (cid:18) dx s dτ (cid:19) + V − Λ x s (cid:41) = − Λ − Λ2 ( x + x ) + V + ( x − x ) . (3.5)For simplicity, we consider x = 0 and the transition amplitude is given by K ( x ) ≈ exp (cid:20) − (cid:126) (cid:18) − Λ − Λ x V + x (cid:19)(cid:21) , (3.6)which is not consistent with the result (2.14) of the Lorentizan path integral. As wewill see later, this result is not compatible with the WKB approximation either.The reason is that the semi-classical solution (2.5) with N = − i does not satisfythe constraint equation (2.2), which is the law of conservation of energy. Thus, when– 14 –he constraint equation (2.2) is actually satisfied the result of the Euclidean pathintegral coincides with the Lorentzian formulation (2.14). For instance, if we take V = Λ / and x = V / Λ = Λ / and the transition amplitude is given by K ( x ) ≈ exp (cid:18) − Λ (cid:126) (cid:19) , (3.7)which is consistent with the result (2.15) in the Lorentzian formulation. In this casethe saddle points of the action (2.10), which is N s = ± i also coincides with theEuclidean saddle point N = − i . In Fig. 4 we show this correspondence.Next, let us discuss the well-known double well potential, V ( x ) = λ (cid:0) x − a (cid:1) , (3.8)and consider the quantum tunneling from x = − a to x = a . The ususal method forfinding the Euclidean (instanton) solution in this case is to obtain it directly fromthe Euclidean energy conservation rather than solving the Euclidean equations ofmotion. For simplicity, we set E = 0 and the Euclidean energy conservation gives, (cid:18) dxdτ (cid:19) − V ( x ) = 0 = ⇒ dxdτ = ± (cid:114) λ (cid:0) x − a (cid:1) . (3.9)Thus, we can get the following instanton solution interpolating between − a and a , x ( τ ) = ± a tanh ω τ − τ ) , (3.10)where ω = √ λa and τ is an integration constant. The plus and minus classicalsolutions are the instanton and anti-instanton. The instanton corresponds to theparticle initially sitting on the maximum of − V ( x ) at x = − a , passing x = 0 for avery short time and ending up at the other maximum of − V ( x ) at x = a . From theinstanton the Euclidean action is given by S E = 2 √ λ a , (3.11)whose the transition amplitude K ( a ) is consistent with the WKB approximation ofthe wave function. For instance, if we extend this instanton as follows, x ( t ) = a tanh ω iN t − τ ) , (3.12)and apply it to the Lorentzian formulation, it returns the same result. As alreadydiscussed, a saddle-point approximation of the action by a solution of the Euclideanequations of motion does not give the correct transition amplitude. The correct resultis obtained when the solution of the Euclidean equation of motion satisfies the energy– 15 –onservation. Since the instanton solution is derived from the energy conservation inEuclidean form, it necessarily corresponds to the saddle point of the path integral.We can see the same relations with the harmonic oscillator and inverted harmonicoscillator models. By taking N = − i we have the Euclidean transition amplitude forthe harmonic and inverted harmonic oscillator, K H ( x ; x ) ≈ exp (cid:20) − (cid:126) (cid:18) V + 12 (cid:0) x + x (cid:1) Ω coth Ω − x x Ω csch Ω (cid:19)(cid:21) , (3.13) K I ( x ; x ) ≈ exp (cid:20) − (cid:126) (cid:18) V + 12 (cid:0) x + x (cid:1) Ω cot Ω − x x Ω csc Ω (cid:19)(cid:21) , (3.14)where we denote that K H ( x ; x ) , K I ( x ; x ) are the transition amplitudes for theharmonic and inverted harmonic oscillator are not consistent with the Lorentizanformulations (2.27). As discussed previously, when we take x = 0 and Ω = 1 andchoose x which satisfies the Euclidean energy conservation, x H = ( e − √ V √ e , x I = (cid:112) V sin(1) , (3.15)the Euclidean transition amplitudes are consistent with the results (2.29) and (2.36)of the Lorentzian formulation, K H ( x ) ≈ exp (cid:18) (1 − e − e ) V (cid:126) e (cid:19) , (3.16) K I ( x ) ≈ exp (cid:18) − V + V sin(1) cos(1) (cid:126) (cid:19) . (3.17) Finally, in this subsection we will discuss WKB approximation of the Schrödingerequation and discuss the correspondence to the Lorentzian and Euclidean path in-tegral 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, ˆ H Ψ( x ) = (cid:18) − (cid:126) d dx + V ( x ) (cid:19) Ψ( x ) = E Ψ( x ) , (3.18)where ˆ H is the Hamiltonian, we assume that the solution is in the form of exp ( i (cid:126) S [ x ]) and expanded as a perturbation series of (cid:126) . By substituting Ψ( x ) ≈ e i (cid:126) ( S [ x ]+ (cid:126) S [ x ]+ ··· ) in the Schrödinger equation we obtain the following equations, (cid:18) dS dx (cid:19) + V − E = 0 , dS dx = i ddx (cid:18) ln dS dx (cid:19) . . . , (3.19)– 16 –here S is the dominant contribution of the WKB wave function and can also beobtained by the constraint equation (2.2) as already discussed in Section 2. We notethat the Schrödinger equation 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 Ψ( x ) ≈ c | E − V ( x )) | / exp (cid:20) ± i (cid:126) (cid:90) xx dx (cid:112) E − V ( x )) (cid:21) . (3.20)For the linear potential the wave function in the WKB approximation is given by Ψ( x ) ≈ c | E − V ( x )) | / exp (cid:32) + 2 √ i (cid:126) (cid:104) (Λ x + E − V ) / − (Λ x + E − V ) / (cid:105)(cid:33) + c | E − V ( x )) | / exp (cid:32) − √ i (cid:126) (cid:104) (Λ x + E − V ) / − (Λ x + E − V ) / (cid:105)(cid:33) , (3.21)where the exponent of the above WKB wave function agrees with the two saddle-pointactions S [ N s ] (2.10) of the Lorentzian path integral. In the Lorentzian path integralmethod, four saddle points dominates the path integral, and only one saddle pointcontributes when the lapse integral is defined as positive. On the other hand, in theWKB analysis, there is no uncertainty of the lapse function, and the exponents of thewave function can be either positive or negative, depending on the initial conditions.Therefore, in the Lorentzian path integral method, the positive and negative lapsefunction would be generally considered.On the other hand, for the harmonic and inverted harmonic oscillator potentialswhere we take x = 0 and E = 0 , the exponents of the WKB wave function are givenby S = ± iV sinh − (cid:115) x V ± ix √ V (cid:115) x V , (3.22) S = ± iV sin − (cid:115) x V ± ix √ V (cid:115) − x V , (3.23)which are exactly consistent with the saddle points of the Lorentzian formulation (2.26)and (2.34). Finally, we comment the double well potential (2.1) with zero-energy sys-tem E = 0 and the corresponding semi-classical action is given by S = ± (cid:90) x = ax = − a dx (cid:114) − λ x − a ) = ± i √ λ a , (3.24)which is consistent with the Euclidean action S E with the instanton (3.11). Insummary the Lorentzian formulation (2.8) including the lapse N integral and usingthe Picard-Lefschetz theory, and the Euclidean formulation with the instanton arenothing more than the WKB analysis of the Schrödinger equation.– 17 – Quantum tunneling with Lorentzian instanton
In this section, we will introduce a new instanton method based on the previous dis-cussions. As discussed in Section 3, the Euclidean saddle-point action correspondingto the exponent of the WKB wave function does not consist only of the simple solu-tions of the equation of motion. The solutions of the Euclidean equation of motionmust satisfy the constraint equation (2.2) in the Euclidean form. The Lorentzianpath integral method using the Picard-Lefschetz theory [20] constructs the semi-classical transition amplitude by finding saddle points on the lapse integration whichimplies δS [ x, N ] /δN = 0 . Since δS [ x, N ] /δN = 0 corresponds to the constraintequation (2.2), the transition amplitude becomes consistent with the WKB approx-imation. Therefore, we can expect to obtain the saddle point action correspondingto the WKB approximation by finding the Lorentzian solution from the constraintequation (2.2) and substituting it into the action. We will now briefly introduce themethod and call it Lorentzian instanton formulation .Let us write the Lorentzian path integral including lapse function N L again, K ( x f ; x i ) = (cid:90) x f x i D x exp (cid:18) iS [ x ] (cid:126) (cid:19) , S [ x ] = (cid:90) t f t i dtN L (cid:18) ˙ x N L − V ( x ) + E (cid:19) , (4.1)which fixes the lapse N L and does not integrate. From (4.1) we derive the constraintequation and the equations of motion in a way of the analytical mechanics, δS [ x ] /δN L = 0 = ⇒ ˙ x N L V ( x ) = N L E, (4.2) δS [ x ] /δx = 0 = ⇒ ¨ x = − N L V (cid:48) ( x ) . (4.3)As is well known in analytical mechanics, the equation of motion (4.2) is ob-tained by differentiating the constraint equation (4.3). Therefore, only the constraintequation is considered. Solving the constraint equation (4.3) for x with the initialcondition x ( t i ) = x , we get a semi-classical solution. Then, we impose the finalcondition x ( t f ) = x on the solution and determine the lapse function N L on thecomplex path. Thus, we can get the Lorentzian real-time solution even for quantumtunneling and construct the path integral (4.1) from the Lorentzian classical solutionas well as the instanton method based on the Euclidean path integral.From here we will show that this formulation is consistent with the Lorentzianpath integral using Picard-Lefschetz theory and WKB approximation. For simplicity,let us consider the linear potential V = V − Λ x and asuume t i = 0 and t f = 1 . Theconstraint equation (4.3) with the initial condition x ( t i = 0) = x gives the followingclassical solution, x L ( t ) = x + Λ2 N L t ± N L t (cid:112) E − V + 2Λ x . (4.4)– 18 –y imposing the final condition x ( t f = 1) = x on the solution we can determine thelapse function N L and obtain, N L = ∓ √ (cid:2) (Λ x + E − V ) / ± (Λ x + E − V ) / (cid:3) , (4.5) x L ( t ) = x + (cid:2) (Λ x + E − V ) / ± (Λ x + E − V ) / (cid:3) t − √ t Λ (cid:2) (Λ x + E − V ) / ± (Λ x + E − V ) / (cid:3) (cid:112) E − V + 2Λ x . (4.6)where N L corresponds to the four saddle points (2.9) of the Lorentzian formulationand x L ( t ) is given by imposing N L to the Lorentzian solution (4.4). We note thatplus sign solution in Eq. (4.6) is non-trivial since it is non-zero in the limit x → x .Here, the degree of freedom of lapse is uniquely determined. Thus, the semi-classicalaction S [ x L ] is given by S [ x L ] = ± √ (cid:104) (Λ x + E − V ) / ± (Λ x + E − V ) / (cid:105) . (4.7)which is consistent with the exponent for the WKB wave function and semi-classicalsaddle-point action (2.10) of the Lorentzian path integral. By developing the saddlepoint method where x = x L + δx is decomposed as the Lorentzian classical solu-tions and the fluctuation around them, the Lorentzian transition amplitude can beapproximately given by K ( x ; x ) (cid:39) exp (cid:18) iS [ x L ] (cid:126) (cid:19) (cid:90) δx (1)=0 δx (0)=0 D δx exp (cid:18) i (cid:126) δ S [ x ] δx (cid:12)(cid:12)(cid:12) x = x L δx (cid:19) (cid:39) exp (cid:18) iS [ x L ] (cid:126) (cid:19) (cid:90) δx (1)=0 δx (0)=0 D δx exp (cid:18) i (cid:126) N L (cid:90) dt (cid:0) δ ˙ x − V (cid:48)(cid:48) ( x L ) δx (cid:1)(cid:19) (cid:39) (cid:114) πi (cid:126) N L exp (cid:18) iS [ x L ] (cid:126) (cid:19) , (4.8) When V (cid:48)(cid:48) ( x L ) is non-zero, as the usual instanton in QFT [5], we can expand the fluctuationand get the following expression, K ( x f ; x i ) (cid:39) exp (cid:18) iS [ x L ] (cid:126) (cid:19) (cid:90) δx (1)=0 δx (0)=0 D δx exp (cid:18) i (cid:126) (cid:90) dtN L (cid:18) d N L dt − V (cid:48)(cid:48) ( x L ) (cid:19) δx (cid:19) (cid:39) exp (cid:18) iS [ x L ] (cid:126) (cid:19) (cid:90) ∞−∞ dc n exp (cid:32) − (cid:126) iN L ∞ (cid:88) n =0 c n λ n (cid:33) (cid:39) exp (cid:18) iS [ x L ] (cid:126) (cid:19) (cid:89) n (cid:114) π (cid:126) iN L λ n , where λ n are the eigenvalues and we omitted the normalization. For the Euclidean path integralwhere N L = − i , when all eigenvalues in a saddle point solution are positive, the fluctuation aroundthe saddle point increases the action and the saddle point approximation is correct. On the otherhand, the negative eigenvalues reduce the action, and such a solution would be discarded [46]. Asimilar argument can be applied here. – 19 –here V (cid:48)(cid:48) ( x L ) is zero for the linear potential. Note that this method still has aproblem how to select the correct Lorentzian solutions. Let us compare this resultwith the real-time path integral for the linear potential. Following Feynman’s famoustextbook [47] on quantum mechanics and path integrals, the real-time transitionamplitude is approximately given by K ( x ; x ) (cid:39) (cid:114) πi (cid:126) exp (cid:20) i (cid:126) (cid:26) ( x − x ) x + x )2 − Λ (cid:27)(cid:21) , (4.9)which does not agree with the Lorentzian formulation (4.8). However, in this saddlepoint approximation the classical path solution does not respect the constraint equa-tion (2.3) and if we explicitly write the energy of the system and usually think aboutthat, x , is restricted by E as classical mechanics. By imposing the constraint (2.3)on the real-time classical solution and setting x = 0 and V = E = 0 for simplicity,the real-time transition amplitude (4.9) agrees with the Lorentzian amplitude (4.8), K ( x ; 0) (cid:39) (cid:114) πi (cid:126) exp (cid:18) ix (cid:126) (cid:19) . (4.10)As discussed in Section 3, the Lorentzian transition amplitudes, which approximatethe saddle point from the constraint equation (2.3) in correspondence with the WKBanalysis, may be more accurate than the real-time amplitude (4.9). In this paper,we will not discuss further, but this technique is still well worth considering. In this paper, we have analyzed the tunneling transition probability in QM usingthe Lorentzian path integral with the Picard-Lefschetz theory and compare it withthe WKB analysis of the conventional Schrödinger equation. As a result, we haveshown that they are in perfect agreement for the linear, harmonic oscillator, invertedharmonic oscillator and double well models. This correspondence has been givenby Eq. (2.10) and Eq. (2.20) for the linear potential. For the harmonic oscillatoror inverted harmonic oscillator models Eq. (2.26) and Eq. (3.22) or Eq. (2.34) andEq. (3.23) have shown this correspondence. The two saddle points in the Lorentzianpath integral (2.6) corresponds to the exponents of the WKB wave function whereasthe others do not. These results suggest that the Picard-Lefschetz Lorentzian for-mulation is consistent with the WKB analysis of the Schrödinger equation.On the other hand, in the Picard-Lefschetz Lorentzian formulation, the rangeof the lapse integral is controversial in Refs [20, 29, 34, 35], but it seems to be pos-sible to consider negative range of the lapse from the correspondence of the WKBapproximation. However, if the lapse integration is performed over positive and neg-ative domain, there exit different integration paths in the Picard-Lefschetz method.– 20 –urthermore, we have shown that depending on the semiclassical approximation ofthe action, the lapse gauge degrees of freedom can be over-integrated. Due to thesetheoretical shortcomings, it may not be necessary to perform the lapse integrationin the Lorentzian path integral. We have shown that this method is equivalent tothe Euclidean path integral with instanton when the constraint equation is satisfied,and finally proposed a Lorentzian instanton formulation which is an extension of theEuclidean instanton formulation to Lorentzian path integral and remove the ambi-guity of the integral. However, this method still has some problems such as how toselect correct Lorentzian solutions.
Acknowledgments
We thank K. Yamamoto for stimulating discussions and valuable comments. We alsothanks S. Kanno, A. Matsumura and H. Suzuki for helpful discussions.
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