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

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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 ampli- tude 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 integrat- ing 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 Schr¨odingerequation for quantum mechanics (QM). In par- ticular, 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 Schr¨odingerequation 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 j 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 Schr¨odingerequation. 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 formula- tion 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 t 2 Z f x_ 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 nec- essarily 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 j e j 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 4 � ��� � ��� � ) ) � � ( ( ��� � � � ��� � ��� -� -��� -��� -��� ��� ��� ��� ��� -� -� � � � � � ��� ��� ��� ��� ) ) � � ( ��� ( ��� � � -��� ��� -��� ��� -� -� � � � -��� -��� -��� ��� ��� ��� ��� � � 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.
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