Semi-Classical Asymptotic Evaluation of Feynman Path Integrals.

Complex (semi-)classical paths, or instantons, form an integral part of our understanding of quantum physics. Whereas real classical paths describe classically allowed transitions in the real-time Feynman path integral, classically forbidden evolution is captured by complex semi-classical paths or instantons. In this paper, we uncover the rich, intricate nature of complex semi-classical paths and interference in the Feynman propagator of a non-relativistic quantum particle in both a smooth Woods-Saxon and a discontinuous Heaviside step potential. We demonstrate that the complex semi-classical paths are connected to caustics and may cease to exist as naive solutions to the boundary value problem when the semi-classical path encounters singularities of the potential. We generalise complex semi-classical paths to equivalence classes. Using this generalisation, we track the contribution of complex semi-classical paths beyond these singularity crossings and identify the instanton responsible for the quantum reflection. Whereas most complex contributions to the path integral are small, we demonstrate that in some cases the contribution of the complex semi-classical path is unsuppressed and persists into the semi-classical limit. Conversely, the Heaviside step potential includes a real classical path that does not contribute in the semi-classical limit. Finally, we develop methods to detect the presence of complex semi-classical paths from propagation amplitudes. The structure of complex semi-classical paths and methods developed here generalises to a large set of problems in real-time quantum physics.Complex (semi-)classical paths, or instantons, form an integral part of our understanding of quantum physics. Whereas real classical paths describe classically allowed transitions in the real-time Feynman path integral, classically forbidden evolution is captured by complex semi-classical paths or instantons. In this paper, we uncover the rich, intricate nature of complex semi-classical paths and interference in the Feynman propagator of a non-relativistic quantum particle in both a smooth Woods-Saxon and a discontinuous Heaviside step potential. We demonstrate that the complex semi-classical paths are connected to caustics and may cease to exist as naive solutions to the boundary value problem when the semi-classical path encounters singularities of the potential. We generalise complex semi-classical paths to equivalence classes. Using this generalisation, we track the contribution of complex semi-classical paths beyond these singularity crossings and identify the instanton responsible for the quantum reflection. Whereas most complex contributions to the path integral are small, we demonstrate that in some cases the contribution of the complex semi-classical path is unsuppressed and persists into the semi-classical limit. Conversely, the Heaviside step potential includes a real classical path that does not contribute in the semi-classical limit. Finally, we develop methods to detect the presence of complex semi-classical paths from propagation amplitudes. The structure of complex semi-classical paths and methods developed here generalises to a large set of problems in real-time quantum physics.

Annals of PhysicsApr 20, 20261 citation

Path integral predictions for pre-asymptotic false vacuum decay

Joshua Lin, Bruno Scheihing-Hitschfeld, Thomas Steingasser

When tunneling occurs out of generic initial states, a significant fraction of probability is lost at early times during which the dynamics is governed by excited resonance states. However, first-principles analyses based on path integrals have only captured the leading asymptotic behavior during which the tunneling rate is dominated by the false vacuum contribution. In this work, we discuss the behavior in the pre-asymptotic regime from a first-principles path integral perspective. We demonstrate how the relevant expressions can be evaluated systematically through semi-classical methods in the recently developed steadyon picture. This approach allows one to trace the role of the relevant physical scales, making transparent the underlying assumptions and approximations and offering a clear path to establishing a systematically improvable framework to evaluate tunneling rates non-perturbatively.When tunneling occurs out of generic initial states, a significant fraction of probability is lost at early times during which the dynamics is governed by excited resonance states. However, first-principles analyses based on path integrals have only captured the leading asymptotic behavior during which the tunneling rate is dominated by the false vacuum contribution. In this work, we discuss the behavior in the pre-asymptotic regime from a first-principles path integral perspective. We demonstrate how the relevant expressions can be evaluated systematically through semi-classical methods in the recently developed steadyon picture. This approach allows one to trace the role of the relevant physical scales, making transparent the underlying assumptions and approximations and offering a clear path to establishing a systematically improvable framework to evaluate tunneling rates non-perturbatively.

hep-thCross-listedarXiv:2511.08669Nov 13, 2025

An integration by parts formula for Feynman path integrals

Daisuke Fujiwara

We are concerned with rigorously defined, by time slicing approximation method, Feynman path integral $\int_{\Omega_{x,y}} F(\gamma) e^{i\nu S(\gamma)} {\cal D}(\gamma)$ of a functional $F(\gamma)$ , cf. [13]. Here $\Omega_{x,y}$ is the set of paths $\gamma(t)$ in R $^d$ starting from a point $y \in$ R $^d$ at time $0$ and arriving at $x\in$ R $^d$ at time $T$ , $S(\gamma)$ is the action of $\gamma$ and $\nu=2\pi h^{-1}$ , with Planck's constant $h$ . Assuming that $p(\gamma)$ is a vector field on the path space with suitable property, we prove the following integration by parts formula for Feynman path integrals: $ \int_{\Omega_{x,y}}DF(\gamma)[p(\gamma)]e^{i\nu S(\gamma)} {\cal D}(\gamma) $ $ = -\int_{\Omega_{x,y}} F(\gamma) {\rm Div}\, p(\gamma) e^{i\nu S(\gamma)} {\cal D}(\gamma) -i\nu \int_{\Omega_{x,y}} F(\gamma)DS(\gamma)[p(\gamma)]e^{i\nu S(\gamma)}{\cal D}(\gamma). $ (1) Here $DF(\gamma)[p(\gamma)]$ and $DS(\gamma)[p(\gamma)]$ are differentials of $F(\gamma)$ and $S(\gamma)$ evaluated in the direction of $p(\gamma)$ , respectively, and ${\rm Div}\, p(\gamma)$ is divergence of vector field $p(\gamma)$ . This formula is an analogy to Elworthy's integration by parts formula for Wiener integrals, cf. [1]. As an application, we prove a semiclassical asymptotic formula of the Feynman path integrals which gives us a sharp information in the case $F(\gamma^*)=0$ . Here $\gamma^*$ is the stationary point of the phase $S(\gamma)$ .We are concerned with rigorously defined, by time slicing approximation method, Feynman path integral $\int_{\Omega_{x,y}} F(\gamma) e^{i\nu S(\gamma)} {\cal D}(\gamma)$ of a functional $F(\gamma)$, cf. [13]. Here $\Omega_{x,y}$ is the set of paths $\gamma(t)$ in R$^d$ starting from a point $y \in$ R$^d$ at time $0$ and arriving at $x\in$ R$^d$ at time $T$, $S(\gamma)$ is the action of $\gamma$ and $\nu=2\pi h^{-1}$, with Planck's constant $h$. Assuming that $p(\gamma)$ is a vector field on the path space with suitable property, we prove the following integration by parts formula for Feynman path integrals: $ \int_{\Omega_{x,y}}DF(\gamma)[p(\gamma)]e^{i\nu S(\gamma)} {\cal D}(\gamma) $ $ = -\int_{\Omega_{x,y}} F(\gamma) {\rm Div}\, p(\gamma) e^{i\nu S(\gamma)} {\cal D}(\gamma) -i\nu \int_{\Omega_{x,y}} F(\gamma)DS(\gamma)[p(\gamma)]e^{i\nu S(\gamma)}{\cal D}(\gamma). $ (1) Here $DF(\gamma)[p(\gamma)]$ and $DS(\gamma)[p(\gamma)]$ are differentials of $F(\gamma)$ and $S(\gamma)$ evaluated in the direction of $p(\gamma)$, respectively, and ${\rm Div}\, p(\gamma)$ is divergence of vector field $p(\gamma)$. This formula is an analogy to Elworthy's integration by parts formula for Wiener integrals, cf. [1]. As an application, we prove a semiclassical asymptotic formula of the Feynman path integrals which gives us a sharp information in the case $F(\gamma^*)=0$. Here $\gamma^*$ is the stationary point of the phase $S(\gamma)$.

Journal of the Mathematical Society of JapanOct 1, 20139 citations

We are concerned with rigorously defined, by time slicing approximation method, Feynman path integral

\int_{\Omega_{x,y}} F(\gamma) e^{i\nu S(\gamma)} {\cal D}(\gamma)

of a functional

F(\gamma)

, cf. [13]. Here

\Omega_{x,y}

is the set of paths

\gamma(t)

in R

^d

starting from a point

y \in

^d

at time

0

and arriving at

x\in

^d

at time

T

S(\gamma)

is the action of

\gamma

and

\nu=2\pi h^{-1}

, with Planck's constant

h

. Assuming that

p(\gamma)

is a vector field on the path space with suitable property, we prove the following integration by parts formula for Feynman path integrals: $ \int_{\Omega_{x,y}}DF(\gamma)[p(\gamma)]e^{i\nu S(\gamma)} {\cal D}(\gamma) $ $ = -\int_{\Omega_{x,y}} F(\gamma) {\rm Div}\, p(\gamma) e^{i\nu S(\gamma)} {\cal D}(\gamma) -i\nu \int_{\Omega_{x,y}} F(\gamma)DS(\gamma)[p(\gamma)]e^{i\nu S(\gamma)}{\cal D}(\gamma). $ (1) Here

DF(\gamma)[p(\gamma)]

and

DS(\gamma)[p(\gamma)]

are differentials of

F(\gamma)

and

S(\gamma)

evaluated in the direction of

p(\gamma)

, respectively, and

{\rm Div}\, p(\gamma)

is divergence of vector field

p(\gamma)

. This formula is an analogy to Elworthy's integration by parts formula for Wiener integrals, cf. [1]. As an application, we prove a semiclassical asymptotic formula of the Feynman path integrals which gives us a sharp information in the case

F(\gamma^*)=0

. Here

\gamma^*

is the stationary point of the phase

S(\gamma)