# Nonlocal scalar field in an external potential: WKB approximation

Valeri P. Frolov<EMAIL_ADDRESS>Theoretical Physics Institute,
University of Alberta, Edmonton, Alberta, Canada T6G 2E1

###### Abstract

We consider a nonlocal theory of a scalar massive field in a flat spacetime
background in the presence of an external potential and construct WKB
solutions for this theory. We use a model in which the kinetic part of the
scalar field action is modified by changing $\Box$ to $\Box f(\Box)$ operator.
We discuss conditions when the corresponding form factor $f$ is chosen so that
the theory does not contain new unphysical degrees of freedom. We applied the
obtained WKB solutions for study energy levels of the field trapped by a one-
dimensional potential and the probability of the barrier penetration. This
allows us to illustrate how the effects of the nonlocality change the known
results obtained for the local field theory.

##  I Introduction

The idea of nonlocality is quite old in the theoretical physics. Nonlocal
modifications of the field theory were discussed already in the publications
Wataghin (1934); Yukawa (1950a, b). Even if one works with a local quantum
field theory, already at the one loop level its effective action is nonlocal.
This nonlocality is connected with the vacuum polarization and particle
creation effects and it reflects a fact that a vacuum is a physical medium and
in this sense it behaves as a condensed matter with very special properties. A
nonlocality that we consider in this paper is of a different origin. We assume
that a field theory is described by a nonlocal action already at the tree
level. In particular this means that its interaction with external sources as
well as a self-interaction is nonlocal.

Usually such a nonlocal modification contains a scale parameter which
determines either the energy at which the effects of the nonlocality become
important or the corresponding spacetime length when it happens. Nonlocal
theories were discussed in the beginning of seventies of the past century by
Efimov Efimov (1968, 1972, 1974); Alebastrov and Efimov (1974). The
nonlocality is often introduced to modify high energy properties of the theory
and to improve its ultraviolet behavior. More recently the interest to
nonlocal theories increased. This was mainly stimulated by the development of
the string theory. Nonlocal fields naturally arise in the string theory and in
the theories with a noncommutive geometry (see e.g. Witten (1986); Frampton
and Okada (1988); Eliezer and Woodard (1989); Kostelecky and Samuel (1990);
Harms and Leblanc (1995); Tseytlin (1995); Calcagni and Modesto (2014, 2015)
and references therein). Nonlocal effects in quantum gravity were widely
discussed recently. A comprehensive review of modifications of gravity
involving a minimal length scale and related references can be found in
Hossenfelder (2013).

There is a subclass of the non-local field theories sometimes called ghost-
free theories. An effective action in these models contains an infinite number
of derivatives so that the corresponding field equations are effectively
nonlocal. These models have been widely discussed recently and they have
rather ”nice properties”. In these theories the nonlocality is introduced in
such a way that it preserves local Lorentz invariance and it does not
introduce new unphysical degrees of freedom (ghosts). In these models the
ultraviolet (UV) behaviour of the theory at short distances is improved, while
in the infrared (IR) regime (at large scales) they reproduce results of a
corresponding local theory Tomboulis (1997); Biswas  _et al._ (2012); Modesto
(2012); Biswas  _et al._ (2014); Shapiro (2015); Biswas  _et al._ (2010a,
2013). Main motivation for study such infinite derivative modifications of the
gravity equations is connected with attempts to solve long-standing problems
of cosmological and black hole singularities Biswas  _et al._ (2006); Koshelev
(2007); Biswas  _et al._ (2010b); Modesto  _et al._ (2011); Biswas  _et al._
(2012); Conroy  _et al._ (2015); Modesto and Rachwał (2017); Buoninfante  _et
al._ (2018); Koshelev  _et al._ (2018); Kilicarslan (2019a); Koshelev  _et
al._ (2019); Kilicarslan (2019b).

Complete equations of a modified gravity which include both nonlocality and
nonlinearity are quite complicated. Much easier for study is a linearized
version of the theory. However already study of linearized models allows one
to obtain several stimulating results: (i) The nonlocality removes
singularities of the field produced by a point-like sources Biswas  _et al._
(2012); de la Cruz-Dombriz  _et al._ (2019); Buoninfante and Mazumdar (2019);
Boos  _et al._ (2018a, 2020); (ii) There exists a mass gap for mini black hole
formation Frolov (2015); Frolov  _et al._ (2015); (iii) It allows one to
demonstrate a formation of the inner horizon for black hole creation in the
scattering of ultrarelativistic particles Frolov and Zelnikov (2016). A
standard technique for solving these problems is usage of nonlocal Green
functions which for the flat spacetime background can be found by means of the
Fourier transform.

Study of the nonlocal field in the presence of an external potential is a much
more complicated problem. There exist very special cases when it is possible
to find an explicit solution. For example, such a solution was obtained for
scattering of a nonlocal scalar field by a delta-like potential Boos  _et al._
(2018b); Frolov and Zelnikov (2018); Buoninfante  _et al._ (2019). However in
a general case in the presence of an arbitrary potential one needs to solve a
nonlocal linear equation with the space-dependent coefficients which is a very
non-trivial problem.

The purpose of this paper is to demonstrate that one can obtain asymptotic
solutions for this problem by using a standard WKB method. Its main idea is to
search for a solution $\Phi(x)$ of the field equation in the form
$\Phi(x)\approx u(x)\exp{[iS(x)/\hbar]}$. After substitution of this ansatz
into the field equation one collects terms of the expansion into the powers of
$\hbar$ which are of the same order of $\hbar$. In the leading order one gets
a first-order partial differential equation of the form $H(\nabla S,x)=0$
known as an eikonal equation. A sub-leading equation determines evolution of a
slowly changing field amplitude $u(x)$. The eikonal equation can be identified
with the Hamilton-Jacobi equation for the Hamiltonian obtained by the
substitution $\nabla S=p$ into it. Initial data for the Hamilton-Jacobi
equation specify a beam of trajectories in the phase space which forms a
Lagrangian submanifold (for details see e.g. a remarkable book Arnold  _et
al._ (2013)). Knowledge of this Lagrangian submanifold allows one not only to
construct the eikonal function $S(x)$ but also to find a solution of the
transport equation for the amplitude $u(x)$ by using the Liouville theorem.
This WKB method is widely used in the standard quantum mechanics and field
theory where the corresponding equations are second order partial differential
equations. However, it can be applied to a wider class of so called
quasilinear differential equations. A comprehensive presentation of these
results can be found in the book Maslov and Fedoriuk (1981). In these paper we
apply this method for study quasiclassical solutions of the linear nonlocal
scalar field equations in the presence of an external potential. A similar
approach for other higher and infinite order equations can be found in Lu  _et
al._ (2018).

This paper is organized as follows. In section II we describe a model of a
nonlocal scalar field with infinite number of derivatives which is analysed in
the paper. In section III we construct a WKB solution for such a field in the
presence of an external potential in any number of spacetime dimensions. In
section IV we consider a special case when the potential depends on only one
spatial Cartesian coordinate. Energy levels for the nonlocal scalar field
confined by the one-dimensional parabolic potential are calculated in section
V. Under-barrier propagation of the nonlocal field and barrier penetration
effect are discussed in section VI. Section VII contains discussion of the
obtained results. Additional technical details are collected in two
appendices.

##  II Nonlocal scalar field equation

We consider $N$-dimensional flat spacetime. Its metric in Cartesian
coordinates is

$ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu},\hskip
14.22636pt\mu,\nu=0,1,\ldots,N-1\,,$ (1)

where $\eta_{\mu\nu}=\mbox{diag}(-1,1,\ldots,1)$. Let us consider a scalar
massive field $\varphi$ obeying the Klein-Gordon equation

$(\hbar^{2}\Box\ -m^{2}-V)\varphi=0\,.$ (2)

Here $\Box=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$, $m$ is the mass of the
field and $V(x)$ is an external potential. In what follows we shall study
solutions of this equation and its nonlocal generalization in the WKB
approximation. For this reason we keep the Planck constant $\hbar$, while as
usual put $c=1$111In these units one has $[\hbar]=ML$, $[\Box]=L^{-2}$, and
$[m^{2}]=[V]=M^{2}$ .. The Klein-Gordon equation follows from the action

$W[\varphi]={1\over 2}\int d^{N}x\varphi(F(\Box)-V)\varphi\,,\
F(\Box)=\hbar^{2}\Box-m^{2}\,.$ (3)

We consider a generalization of the Klein-Gordon equation in which the
operator $F(\Box)$ is modified. Namely, we assume that it is a scalar operator
which may contain an arbitrary (finite or infinite) number of partial
derivatives $\partial/\partial x^{\mu}$. It is easy to check that the
covariance of the action requires that such an operator can be written as a
scalar function of the $\Box$-operator. In order to keep a proper
dimensionality of the action we introduce a parameter $\mu$ which has a
dimension of the mass and we write the operator $F(\Box)$ in the form

$F(\Box)=\mu^{2}\hat{F}(\hbar^{2}\Box/\mu^{2})\,.$ (4)

The corresponding generalized Klein-Gordon equation is

$[-\hat{F}(z)+\hat{V}]\varphi=0\,,$ (5)

where $z=\hbar^{2}\Box/\mu^{2}$ and $\hat{V}=V/\mu^{2}$.

We consider a class of theories for which the function $\hat{F}(z)$ has the
following properties:

  1. 1.

$\hat{F}(z)$ does not vanish anywhere on the complex plane of $z$ besides the
point $z=\hat{m}^{2}$;

  2. 2.

$\hat{F}|_{z=\hat{m}^{2}}=0$ and $d\hat{F}(z)/dz|_{z=\hat{m}^{2}}=1$;

  3. 3.

For a real value of its argument $z$ the function $\hat{F}(z)$ is real.

These conditions guarantee that the inverse of the operator $\hat{F}$ has a
single pole with its residue equal to 1. In other words, there is no new
unphysical (ghost) degrees of freedom in this theory. To satisfy the required
properties it is sufficient to choose $\hat{F}$ in the form

$\hat{F}(z)={\cal F}(\hat{z}),\hskip 14.22636pt{\cal
F}(\hat{z})=\hat{z}f(\hat{z}),\hskip 14.22636pt\hat{z}=z-\hat{m}^{2}\,,$ (6)

where $f(\hat{z})=\exp(g(\hat{z}))$ and $g(\hat{z})$ is an entire function of
the complex variable satisfying the condition $g(\hat{z}=0)=0$. We call
$f(\hat{z})$ a form factor.

In order to illustrate the results we shall use a special example of the form
factor. Namely we put $g(\hat{z})=\hat{z}^{2}$ then one has

${\cal F}(\hat{z})=X(\hat{z})\equiv\hat{z}\exp(\hat{z}^{2})\,.$ (7)

This function obeys all the conditions listed above. Besides this for real
$\hat{z}$ it is a monotonically increasing from $-\infty$ at $\hat{z}=-\infty$
to $+\infty$ at $\hat{z}=+\infty$ and is positive for $\hat{z}>0$. One also
has ${\cal F}(-\hat{z})=-{\cal F}(\hat{z})$. The function $X(x)$ is shown at
figure 1. Other properties of this function which are used later in the paper
can be found in appendix B.

Figure 1: Function $X(x)$.

One can expand the function ${\cal F}(\hat{z})$ into powers of $\hat{z}$. This
gives an infinite series of the powers of $\Box$ for the field operator
$\hat{F}(\hbar^{2}\Box/\mu^{2})$. This means that this operator is nonlocal.
Let us also mention that the mass parameter $\mu$ which enters this operator
determines a characteristic energy scale at which the effects of the
nonlocality become important. In the limit $\mu\to\infty$ the operator $F$
becomes $\hbar^{2}\Box-m^{2}$ and the theory is local.

##  III WKB approximation

###  III.1 A solution of the nonlocal field equation in the WKB approximation

Let us write the nonlocal field equation (5) in the form222 In the nonlocal
theory it is convenient to use the parameter $\mu$ to define dimensionless
quantities. We specify such quantities by using a hat over them. Here and
later we use the following notations  $\hat{m}=m/\mu\,,\
\hat{\eta}=\eta/\mu\,,\ \hat{\eta}_{0}=\eta_{0}/\mu\,,\hat{V}=V/\mu\,,\
\hat{z}=\hat{\eta}_{0}^{2}-\hat{\eta}^{2}\,.$

$\hat{\cal O}\varphi=0,\hskip 5.69046pt\hat{\cal O}={1\over 2}(-\mu^{2}{\cal
F}(\hat{z})+\hat{V}),\hskip
5.69046pt\hat{z}=\hbar^{2}\Box/\mu^{2}-\hat{m}^{2}\,.$ (8)

We include a factor $1/2$ in the operator $\hat{\cal O}$ which evidently does
not change solutions but this simplifies the form of some relations. The
operator $\hat{\cal O}$ belongs to the class of equations (107) discussed in
the appendix. We are looking for a solution of the field equation (8) in the
WKB approximation and write it in the following form

$\varphi(x)=\exp\left({iS(x)\over\hbar}\right)\sum_{j=0}^{\infty}\left({\hbar\over
i}\right)^{j}u_{j}(x)\,.$ (9)

Here $S(x)$ is a fast changing phase, while $u_{j}(x)$ are slowly changing
functions. Let us emphasize that besides standard conditions of the validity
of the WKB approximation formulated in textbooks on quantum mechanics we also
assume that the parameter of nonlocality $\mu$ does not depend on $\hbar$. We
keep this parameter fixed in the limit $\hbar\to 0$. This allows us to discuss
effects of the nonlocality in the quasiclassical approximation.

In order to find the eikonal function $S(x)$ and the amplitudes $u_{j}(x)$ we
follow the steps described in appendix A. First of all we define a symbol
$H(x,p)$ of the operator $\hat{\cal O}$. Let $p_{\mu}$ be a covector and
denote $p^{2}=p_{\mu}p^{\mu}$. Then one should substitute $-\hbar^{2}\Box\to
p^{2}$ into (8). The corresponding symbol for the operator $\hat{\cal O}$ is

$H(x,p)={1\over 2}\left[-\mu^{2}{\cal F}(\hat{z})+V(x)\right],\hskip
5.69046pt\hat{z}=-{1\over\mu^{2}}(p^{2}+m^{2})\,.$ (10)

The eikonal equation (114) implies that the following constraint $H(x,p)=0$ is
valid.

The next step is to study a dynamical system in $2N$-dimensional phase space
satisfying the Hamilton’s equations

${dx^{\mu}\over d\tau}={\partial H\over\partial p_{\mu}},\hskip
14.22636pt{dp_{\mu}\over d\tau}=-{\partial H\over\partial x^{\mu}}\,.$ (11)

Since the Hamiltonian $H(x,p)$ is an integral of motion it is sufficient to
choose $H(x,p)=0$ at the initial ”time” $\tau=0$, then the constraint
$H(x,p)=0$ is valid for any later ”time”.

We define a $(N-1)$-dimensional surface $\Sigma$ by conditions

$x^{\mu}=f^{\mu}(y^{i}),\hskip 14.22636pti=1,\ldots,N-1\,,$ (12)

and choose an initial state for the field $\varphi$ in the form

$\left.\varphi(x)\right|_{\Sigma}=u_{0}(y)\exp\left[{i\over\hbar}S^{0}(y)\right]\,.$
(13)

After solving equations (119) we impose the initial conditions (120) for the
Hamiltonian equations (11). Solving this system we find

$x^{\mu}=x^{\mu}(\tau,y^{i}),\hskip 14.22636ptp_{\mu}=p_{\mu}(\tau,y^{i})\,.$
(14)

A sought solution $\varphi(x)$ of the nonlocal equation (8) in the leading
order of the WKB approximation is

$\varphi(x)=u_{0}(y)\sqrt{J(0,y)\over
J(\tau,y)}\exp\left[{i\over\hbar}\left(S^{0}(y)+\int_{0}^{\tau}p_{\mu}dx^{\mu}\right)\right]\,.$
(15)

This expression is obtained from (125) by using the relation

${\partial^{2}H(x,p)\over\partial x^{\mu}\partial p_{\mu}}=0$ (16)

valid for the Hamiltonian (10). The integral in the exponent in (15) is taken
over phase trajectories which belong to the Lagrangian submanifold determied
by given initial conditions. Here the function $J$ which enters into the
prefactor in (15) is defined by (124).

###  III.2 Remarks on the Hamilton’s equations

Let us discuss now the Hamilton’s equations in more detail. One has

${\partial H\over\partial p_{\mu}}={\cal F}^{\prime}p^{\mu}\,.$ (17)

Here a prime denotes a derivative of ${\cal F}(\hat{z})$ with respect to its
argument $\hat{z}$. Hence the Hamilton’s equations (11) take the form

$\displaystyle\dot{x}^{\mu}\equiv{dx^{\mu}\over d\tau}={\cal
F}^{\prime}p^{\mu}\,,$ (18) $\displaystyle\dot{p}_{\mu}\equiv{dp_{\mu}\over
d\tau}=-{1\over 2}V_{,\mu}\,.$ (19)

The first of these equations implies

$\dot{x}^{2}\equiv\eta_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=({\cal
F}^{\prime})^{2}p^{2}\,.$ (20)

Since $\hat{z}=-(p^{2}+m^{2})/\mu^{2}$ the right-hand side of this relation is
a function of $p^{2}$. We assume that one can solve equation (20) and find
$p^{2}$ as a function of $\dot{x}^{2}$. Then the Lagrangian of this dynamical
system is

$\displaystyle L(x,\dot{x})=\dot{x}^{\mu}p_{\mu}-H$ $\displaystyle={1\over
2}[\mu^{2}({\cal F}-2\hat{z}{\cal F}^{\prime})-2m^{2}{\cal
F}^{\prime}]-{1\over 2}V(x)\,.$ (21)

It is understood that in this expression for $L$ one should express $\hat{z}$
as a function of $\dot{x}^{2}$. Since $L$ does not contain explicitly the
parameter $\tau$ the corresponding ”energy” for this Lagrangian is conserved.
This ”energy” is nothing but the Hamiltonian in which $p^{2}$ is a function of
$\dot{x}^{2}$ defined by (20). The constraint equation in the dimensionless
form is (see footnote 2) )

$-{\cal F}(\hat{z})+\hat{V}(x)=0\,.$ (22)

If $\hat{V}(x)\geq 0$ then ${\cal F}(\hat{z})=\hat{z}f(\hat{z})$ should be
real and positive. The argument of this function
$\hat{z}=-(p^{2}+m^{2})/\mu^{2}$ is real. Since $f(0)=1$ and $f(\hat{z})$ does
not vanish this function is positive for real $\hat{z}$. This means that the
sign of ${\cal F}(\hat{z})$ coincides with the sign of $\hat{z}$. The equation
(22) implies that the classical motion is possible only in a domain where
$\hat{z}\geq 0$.

In the absence of the potential the constraint equation takes the form

$H_{0}(p)=-{1\over 2}\mu^{2}\hat{z}f(\hat{z})=0\,.$ (23)

Since $f(\hat{z})$ does not vanish this equation gives $p^{2}=-m^{2}$. The
condition $\mu^{2}{\cal F}^{\prime}(\hat{z}=0)=1$ implies that

$\left.{\partial H_{0}(p)\over\partial
p_{\mu}}\right|_{p^{2}+m^{2}=0}=p^{\mu}.$ (24)

This means that in the absence of the potential the Hamilton’s equations
(18)-(19) coincide with the equations for a free particle with the Hamiltonian
$(1/2)(p^{2}+m^{2})$. In other words, the effects of the nonlocality are
important only off-shell, that is in the presence on the external potential.

##  IV One-dimensional case

To illustrate an application of the general approach described in the previous
section we consider now a simple model. Namely, we assume that the external
potential $V(x)$ depends on only one spatial Cartesian coordinate. We denote
this coordinate by $q$ and use the following notations

$x^{\mu}=(t,q,x_{\perp}),\hskip 14.22636ptx_{\perp}=(x^{2},\ldots,x^{N})\,.$
(25)

We use index $a=2,\ldots,N$ to enumerate transverse coordinates $x_{\perp}$.
We write the eikonal function $S$ which enters the WKB solution (9) in the
form

$S=-\varepsilon t+(p_{\perp},x_{\perp})+{\cal S}(q),\hskip
5.69046pt(p_{\perp},x_{\perp})=\sum_{a=2}^{N}p_{a}x^{a}\,.$ (26)

We also assume that the functions $u_{j}$ depend only on the coordinate $q$.

Let us assume first that the potential $V(q)$ vanishes. In order to solve the
corresponding field equation (8) one can put ${\cal S}(q)=\eta q$ where
$\eta=$const, $u_{0}$=const and $u_{j\geq 1}=0$. This is a plane wave solution
of the field equation (8). Denote $p_{\mu}=(-\varepsilon,\eta,p_{\perp})$,
where the components of the covector $p_{\perp}$ are $p_{a}$ with
$a=2,\ldots,N$. Hence $S=(p,x)=p_{\mu}x^{\mu}$. Other solutions of the field
equation can be obtained by superimposing the plane-wave solutions, which in
fact is just by using the Fourier representation. The constraint equation
$p^{2}+m^{2}=0$ gives

$-\varepsilon^{2}+p_{\perp}^{2}+m^{2}+\eta^{2}=0\,.$ (27)

In the presence of the potential we look for a solution of the nonlocal field
equation (8) in the form

$\varphi(x)=\exp\left[{i\over\hbar}\left(-\varepsilon
t+(p_{\perp},x_{\perp})\right)\right]\Phi(q)\,.$ (28)

Substituting this expression into (8) one obtains the following reduced
equation

$\displaystyle\hat{\cal O}_{1}\Phi=0\,,$ (29) $\displaystyle\hat{\cal
O}_{1}={1\over 2}[-\mu^{2}\hat{D}f(\hat{D})+V(q)]\,,$ (30)
$\displaystyle\hat{D}={1\over\mu^{2}}\left[\eta_{0}^{2}+\hbar^{2}{\partial^{2}\over\partial
q^{2}}\right],\hskip
5.69046pt\eta_{0}^{2}=\varepsilon^{2}-p_{\perp}^{2}-m^{2}\,.$ (31)

To obtain the symbol $H_{1}(q,\eta)$ of the operator $\hat{\cal O}_{1}$ we
substitute

${\hbar\over i}{\partial\over\partial q}\to\eta\,.$ (32)

The result is

$\displaystyle H_{1}(q,\eta)={1\over 2}\left[-\mu^{2}{\cal
F}(\hat{z})+V(q)\right]\,,$ (33) $\displaystyle{\cal
F}(\hat{z})=\hat{z}f(\hat{z}),\hskip
5.69046pt\hat{z}=\hat{\eta}_{0}^{2}-\hat{\eta}^{2}\,.$ (34)

For real $\eta$ the variable $\hat{z}$ is also real as well as the Hamiltonian
itself. We also require that for real $\hat{z}$ one has ${\cal
F}^{\prime}(\hat{z})>0$. For $f(\hat{z})=\exp(g(\hat{z}))$ this condition is
satisfied when

$1+\hat{z}{dg(\hat{z})\over d\hat{z}}>0.$ (35)

In particular, this is valid for the choice (7).

The Hamilton’s (18)-(19) in $(q,\eta)$-phase space reduce to

$\dot{q}={\partial H_{1}\over\partial\eta}=\mu^{2}{\cal F}^{\prime}\eta,\hskip
5.69046pt\dot{\eta}=-{\partial H_{1}\over\partial q}=-{dV\over dq}\,,$ (36)

while the constraint equation is $H_{1}(q,\eta)=0$. Let us denote by
$(q(t),\eta(t))$ a solution of these equations with the initial condition
$q(0)=q_{0}$ and $\eta(0)=\eta_{0}$. A solution of the field equation (29) in
the leading order of the WKB approximation is

$\displaystyle\Phi(q)$ $\displaystyle=$
$\displaystyle{1\over\sqrt{J}}\exp[{i\over\hbar}{\cal
S}(q)]\Phi_{0}(q_{0})\,,$ (37) $\displaystyle{\cal S}(q)$ $\displaystyle=$
$\displaystyle\int_{q_{0}}^{q}\eta
dq=\int_{0}^{\tau}\eta(\tau)\dot{q}(\tau)d\tau\,.$ (38)

In the expression for ${\cal S}(q)$ the integral is taken along the path
$q(\tau)$ connecting the initial point $q_{0}$ with a final point $q$. A
function $\Phi_{0}(q_{0})=\exp[{i\over\hbar}{\cal S}(q_{0})]u_{0}(q_{0})$ is
the initial value for $\Phi(q)$. The factor $J$ is

$J=\left|{dx\over d\tau}\right|=\mu{\cal
F}^{\prime}\sqrt{\hat{z}_{0}-\hat{z}},\hskip
14.22636pt\hat{z}_{0}=\hat{\eta}_{0}^{2}\,.$ (39)

The second equality in the expression for $J$ is obtained by using the first
Hamilton’s equation (36).

Since the Hamiltonian $H_{1}(q,\eta)$ is invariant under the reflection
$\eta\to-\eta$ one can write another WKB solution by a simple change of the
sign in the exponent of (37). Taking a superposition of these two WKB
solutions one can write $\Phi(q)$ in the following form

$\Phi(q)={C_{1}\over\sqrt{J}}\exp[{i\over\hbar}{\cal
S}(q)]+{C_{2}\over\sqrt{J}}\exp[-{i\over\hbar}{\cal S}(q)]\,.$ (40)

For $C_{2}=\bar{C}_{1}$ the WKB solution (40) is real.

##  V Bound motion: Energy levels

###  V.1 A general case

Let us assume that the potential $V(q)$ is nonnegative and at some point $a$
the following condition is valid

$\hat{\cal F}(\hat{\eta}_{0})=\hat{V}(a)\,.$ (41)

Then the constraint equation

$\hat{\cal F}(\hat{z})=\hat{V}(q)\,$ (42)

implies that the momentum $\eta$ vanishes at $a$. This means that $a$ is a
turning point. If $dV/dq(a)>0$ then the ”particle” comes to this point from
the left, that is from the domain where $q<a$ and $\eta>0$. After reaching
this turning point $\eta$ changes its sign and the ”particle” moves to the
left. If the potential $V(q)$ has another turning point $b$ where
$dV/dq(b)<0$, then the ”particle” comes to it from the right with $\eta<0$
changes its direction of motion at $b$ to the opposite and moves to the right
with $\eta>0$.

Let us assume that there exist two turning points $a<b$ such that the motion
is restricted by the interval $a\leq q\leq b$. Such a motion is called bound.
Let us consider a nonnegative potential $V(q)$ which has its minimum at
$q_{0}$ and $V(q_{0})=0$. At this point the absolute value of the momentum
$|\eta(q)|$ reaches its maximum which is equal to $\eta_{0}$. Thus for such a
bound motion the phase trajectory lies within a rectangle
$[a,b]\times[-\eta_{0},\eta_{0}]$. Let us emphasize that the Hamiltonian (33)
contains a free parameter $\eta_{0}$ which determines a size and a shape of
the phase trajectory. Since the parameter $\hat{z}$ depends on the square of
the momentum $\eta$, a phase trajectory on the plane $(q,\eta)$ is symmetric
with respect to a reflection $\eta\to-\eta$. For a given value $\eta_{0}>0$
constraint equation (42) defines a closed curve on $(q,\eta)$ phase plane (see
figure 2).

Figure 2: A phase trajectory

The energy levels of the states corresponding to the bound motion are
quantized. In the quasiclassical approximation these levels can be found by
using standard Bohr-Sommerfeld quantization condition Maslov and Fedoriuk
(1981)

$J=\oint\eta dq=\pi\hbar(2n+1)\,$ (43)

Here the integral is taken over a complete period. For a given $\eta_{0}>0$
this integral is equal to the surface area inside the phase-trajectory curve
corresponding to this parameter. Because of the reflection symmetry
$\eta\to-\eta$ this integral can be written as

$J=2\int_{a}^{b}\eta(q)dq\,.$ (44)

The integral is taken between the turning points $a$ and $b$.

Sometimes it is convenient to rewrite $J$ in another form. Using constraint
equation (42) one can express the coordinate $q$ as a function of $\eta$. This
function has two branches which we denote by $q_{\pm}(\eta)$ (see figure 2).
Then, integrating by parts (44) and taking into account that
$\eta(a)=\eta(b)=0$ one finds

$J=2\int_{0}^{\eta_{0}}(q_{+}(\eta)-q_{-}(\eta))d\eta\,.$ (45)

If the potential $V(x)$ is a symmetric function of $q$, $V(-q)=V(q)$, then one
has $q_{0}=0$ and $q_{-}(\eta)=-q_{+}(\eta)$. In this case the quantity $J$
can be written in the form

$J=4\int_{0}^{\eta_{0}}q_{+}(\eta)d\eta\,.$ (46)

###  V.2 Parabolic potential: Local theory

Let us consider a case when the potential $V(q)$ is parabolic

$V(q)=V_{1}^{2}q^{2}\,,$ (47)

where $V_{1}$ is a positive constant which has the dimension $[V_{1}]=M/L$.
For the local theory the form factor $f(\hat{z})=1$ and the constraint
equation takes the form

$\eta^{2}+V_{1}^{2}q^{2}=\eta_{0}^{2}\,.$ (48)

Solving (48) we find

$q=\pm V_{1}^{-1}\sqrt{\eta_{0}^{2}-\eta^{2}}\,.$ (49)

The integral (46) can be easily calculated. We denote the result by $J_{0}$.
Then one has

$J_{0}={\pi\ \eta_{0}^{2}\over V_{1}}\,.$ (50)

This gives the following expression for the energy levels

$\varepsilon_{n}^{2}=m^{2}+p_{\perp}^{2}+\hbar V_{1}(2n+1)\,.$ (51)

It is easy to see that the number of levels $\Delta n$ within the interval
$\Delta(\varepsilon^{2})$ is

$\Delta n={1\over 2\hbar V_{1}}\Delta(\varepsilon^{2})\,.$ (52)

The coefficient $1/(2\hbar V_{1})$ does not depend on $n$. In this sense, the
corresponding distribution of energy levels, $\Delta
n/\Delta(\varepsilon^{2})$, is equidistant.

###  V.3 Parabolic potential: Nonlocal theory

####  V.3.1 Phase trajectories

In order to discuss the effect of the nonlocality on the distribution of the
energy levels for the trapped nonlocal field we consider the same parabolic
potential (47) as in the previous subsection, but modify the kinetic part of
the effective Hamiltonian. We use dimensionless units defined in the footnote
2 and denote

$\hat{q}=V_{1}q/\mu,\hskip
14.22636pt\hat{z}_{0}=\hat{\eta}_{0}^{2}={1\over\mu^{2}}(\varepsilon^{2}-m^{2}-p_{\perp}^{2})\,.$
(53)

Then the constraint (42) gives

$\hat{q}=\pm\hat{\cal F}(\hat{z})^{1/2}\,.$ (54)

Let us denote

$\hat{\eta}=\hat{\eta}_{0}\sin\phi,\hskip
14.22636pt\hat{z}=\hat{z}_{0}\cos^{2}\phi\,.$ (55)

Then relations (54) and (55) allow one to write the equation for phase
trajectories in the parametric form $(q(\phi),\eta(\phi))$. The phase
trajectories for the special choice of the form factor (7) are shown at figure
3.

Figure 3:  Phase trajectories for the form factor $f=\exp(\hat{z}^{2})$.
Different curves correspond to the different values of $\hat{z}_{0}$
parameter: $0.14$ for line 1; $0.7$ for line 2; $1.0$ for line 3 and $1.1$ for
line 4.

####  V.3.2 Energy levels

Using dimensionless variables the action integral (46), which enters the Bohr-
Sommerfeld relation (43) for the energy levels, can be written as follows

$J={2\mu^{2}\over
V_{1}}\int_{0}^{\hat{z}_{0}}\,{d\hat{z}\sqrt{z}\over\sqrt{\hat{z}_{0}-\hat{z}}}f^{1/2}(\hat{z})\,.$
(56)

We use here the expression for $\hat{\cal F}$ in terms of the form factor $f$,
$\hat{\cal F}=\hat{z}f(\hat{z})$. After change of variables,
$\hat{z}=\hat{z}_{0}\xi$, relation (56) takes the form

$J=\sigma(\hat{z}_{0})J_{0}\,.$ (57)

Here $J_{0}$ is the value of the action integral for the local theory (50) and
the factor $\sigma(\hat{z}_{0})$ is

$\sigma(\hat{z}_{0})={2\over\pi}\int_{0}^{1}{d\xi\sqrt{\xi}\over\sqrt{1-\xi}}f^{1/2}(\hat{z}_{0}\xi)\,.$
(58)

For a fixed value of $\eta_{0}$ and $\mu\to\infty$ the parameter
$\hat{z}_{0}\to 0$. Since $f(0)=1$ the integral (58) simplifies. It can be
calculated analytically and the result is $\sigma(0)=1$. Hence in this limit
$J$ coincides with $J_{0}$. This is not surprising since in the limit
$\mu\to\infty$ the theory becomes local. For a numerical calculation of
$\sigma(\hat{z}_{0})$ it is convenient to put $\xi=\cos^{2}\phi$ and rewrite
(58) in the form

$\sigma(\hat{z}_{0})={4\over\pi}\int_{0}^{\pi/2}d\phi\,\cos^{2}\phi
f^{1/2}(\hat{z}_{0}\cos^{2}\phi)\,.$ (59)

Figure 4: Function $\sigma(\hat{z}_{0})$ for the form factor
$f(\hat{z})=\exp(\hat{z}^{2})$.

The function $\sigma(\hat{z}_{0})$ for the choice of the form factor
$f(\hat{z})=\exp(\hat{z}^{2})$ is shown at figure 4. For this form factor it
is possible to show that for large $\hat{z}_{0}$ the function
$\sigma(\hat{z}_{0})$ has the follows asymptotic form

$\sigma(\hat{z}_{0})\sim{1.13\over\hat{z}_{0}}\exp(\hat{z}_{0}^{2}/2)\,.$ (60)

Figure 5 which plots the ratio $\sigma(\hat{z}_{0})/\sigma_{0}(\hat{z}_{0})$
demonstrates this property.

Figure 5: Ratio $\sigma/\sigma_{0}$ as a function of $\hat{z}_{0}$.

The parameter $\hat{z}_{0}$ which enters $J$ depends on the energy
$\varepsilon$. The condition of quantization

$J=\pi\hbar(2n+1)$ (61)

allows one to find the corresponding quasiclassical energy levels
$\varepsilon_{n}$. Since for small $\hat{z}_{0}$ the coefficient
$\sigma(\hat{z}_{0})$ is close to 1, the number of the energy levels $n$ in
the interval $\Delta\varepsilon^{2}$ has the form (52). However for large $n$
this distribution of the number of levels is quite different. For the form
factor $f(\hat{z})=\exp(\hat{z}^{2})$ one can find it by using the asymptotic
form of the coefficient $\sigma(\hat{z}_{0})$

$J=1.13{\pi\mu^{2}\over V_{1}}\exp(\hat{z}_{0}^{2}/2)\,.$ (62)

The condition of quantization (61) gives

$\exp(\hat{z}_{0}^{2}/2)=0.88{\hbar V_{1}\over\mu^{2}}(2n+1)\,.$ (63)

For $n\gg 1$ one gets

$\hat{z}_{0}^{2}\approx 2\ln n\,.$ (64)

Since $\hat{z}_{0}=\eta_{0}^{2}/\mu^{2}$, one finds

$\varepsilon^{2}\approx m^{2}+p_{\perp}^{2}+\sqrt{2}\mu^{2}\sqrt{\ln n}\,.$
(65)

This relation implies

$\Delta n\approx{\sqrt{2}\over\mu^{2}}n\sqrt{\ln
n}\,\Delta(\varepsilon^{2})\,.$ (66)

The obtained result means that the equidistance of the energy level
distribution (52) of the local theory is broken by nonlocality. In the limit
of large $n$ and given interval $\Delta(\varepsilon^{2})$ the number of the
corresponding levels $\Delta n$ grows with $n$ as $n\sqrt{\ln n}$.

##  VI Barrier penetration

###  VI.1 WKB approximation for under-barrier ”motion”

Let us consider a potential which has the form shown at figure 6 and assume
that the effective energy of the ”particle” is such that its trajectory has a
turning point at $q=a$. At this point one has

${\cal F}(\hat{z}_{0})=\hat{V}(a),\hskip
14.22636pt\hat{z}_{0}=\hat{\eta}_{0}^{2}\,.$ (67)

Let us write the constraint equation (42) in the form

${\cal F}(\hat{z}_{0})-{\cal F}(\hat{z})=\hat{V}(a)-\hat{V}(q)\,.$ (68)

In the domain to the left from the turning point $a$ one has
$\hat{V}(q)<\hat{V}(a)$. We assume that for real $\hat{z}$ the function ${\cal
F}(\hat{z})$ is monotonically growing. Then one has $\hat{z}<\hat{z}_{0}$. In
the region $q>a$ where $\hat{V}(q)>\hat{V}(a)$ one should have
$\hat{z}>\hat{z}_{0}$. This condition implies that $\eta^{2}<0$ which is
impossible for a real value of $\eta$.

Figure 6: Potential $V(q)$.

It is well known that in the local theory a quasiclassical solution for a wave
function in the classically forbidden domain can be obtained by considering
complex trajectories. Namely, one should change $\eta\to i\eta$. The same
trick does work for the non-local theories considered in this paper. After the
change $\hat{z}\to\hat{\eta}_{0}^{2}+\hat{\eta}^{2}$ the value of $\hat{z}$
can be larger than $\hat{z}_{0}$ for real $\hat{\eta}$ and the equation (68)
has a solution in the region where $q>a$, that is in the classically forbidden
domain.

A quasiclassical solution in the underbarrier domain can be obtained by the
change $\eta\to i\eta$ in (40) and it is of the form

$\displaystyle\Phi(q)={C_{-}\over\sqrt{J}}\exp[-{1\over\hbar}{\cal
S}(q)]+{C_{+}\over\sqrt{J}}\exp[{1\over\hbar}{\cal S}(q)]\,,$ (69)
$\displaystyle{\cal S}(q)=\int_{q_{0}}^{q}\eta
dq=\int_{0}^{\tau}\eta(\tau)\dot{q}(\tau)d\tau\,.$ (70)

Here

$J=\mu{\cal F}^{\prime}\sqrt{\hat{z}-\hat{z}_{0}},\hskip
14.22636pt\hat{z}_{0}=\hat{\eta}_{0}^{2}\,.$ (71)

For real coefficients $C_{\pm}$ the WKB solution (69) is real.

###  VI.2 Nonlocal field in a linear potential

For a linear potential

$V(q)=U_{0}+U_{1}(q-a)\,,$ (72)

one-dimensional nonlocal field equation (29) can be solved exactly. For this
purpose we denote $q=a+x$ and write $\Phi(x)$ in the form

$\Phi(x)=\int_{-\infty}^{\infty}d\eta\,e^{i\eta
x/\hbar}\,\tilde{\Phi}(\eta)\,.$ (73)

Using the inverse Fourier transform one gets

$\tilde{\Phi}(\eta)={1\over 2\pi\hbar}\int_{-\infty}^{\infty}dx\,e^{-i\eta
x/\hbar}\,\Phi(x)\,.$ (74)

The field equation (29) implies the following differential equation for
$\tilde{\Phi}(\eta)$

$\left[U_{1}{\hbar\over i}{\partial\over\partial\eta}+U_{0}+\mu^{2}{\cal
F}(\hat{z})\right]\tilde{\Phi}(\eta)=0\,.$ (75)

A solution of this equation is

$\displaystyle\tilde{\Phi}(\eta)=C\exp\left[-{i\over\hbar U_{1}}{\cal
S}(\eta)\right]\,,$ (76) $\displaystyle{\cal
S}(\eta)=U_{0}\eta+\mu^{2}\int_{0}^{\eta}d\eta{\cal F}(\hat{z})\,.$ (77)

Substitution of this expression into (73) gives

$\Phi(x)=C\int_{-\infty}^{\infty}d\eta\ \exp\left[{i\over\hbar}\left(\eta
x-{1\over U_{1}}{\cal S}(\eta)\right)\right]\,.$ (78)

This is a desired solution of the nonlocal field equation (29)

###  VI.3 Connection of WKB solutions at a turning point

Consider a potential of the form shown at figure 6 and denote by $a$ a
coordinate of a left turning point. One has $V_{,q}(a)>0$. To the left of $a$
and at some distance from it one can use WKB solution (40). For the real field
$\Phi(q)$ we denote

$C_{1}={1\over 2}C\exp(-i\psi),\hskip 14.22636ptC_{2}={1\over
2}C\exp(i\psi)\,,$ (79)

where $C$ and $\psi$ is real and write the WKB solution (40) in the form

$\displaystyle\Phi(q)={C\over\sqrt{J}}\cos\left({\cal S}/\hbar-\psi\right)\,,$
(80) $\displaystyle{\cal S}=\int_{a}^{q}\eta(q)dq,\hskip 14.22636ptJ=\mu{\cal
F}^{\prime}\sqrt{\hat{z}_{0}-\hat{z}}\,.$ (81)

To the right of the turning point $a$ and at some distance from it we use a
decreasing in the under-barrier domain WKB solution

$\displaystyle\Phi(q)={C_{-}\over\sqrt{J}}\exp(-{\cal S}/\hbar)\,,$ (82)
$\displaystyle{\cal S}=\int_{a}^{q}\eta(q)dq,\hskip 14.22636ptJ=\mu{\cal
F}^{\prime}\sqrt{\hat{z}-\hat{z}_{0}}\,.$ (83)

At the turning point the WKB approximation is not valid. In order to establish
a relation between (80) and (82) solutions one can use the following trick
Landau and Lifshitz (1965) which can be easily adapted to our problem. For
this purpose we consider formally $\Phi(q)$ as a function of a complex
variable $q$ and find it along a path on the complex plane of $q$ connecting
positive and negative values of $q-a$. We choose this path to be a half of the
circle of constant radius $\rho$ with the center at $q=a$ and take $\rho$ so
that the conditions of the validity of the WKB approximation are satisfied
along the path. Let us denote

$q-a=\rho e^{\pm\phi}\,.$ (84)

We choose a sign $+$ for a semi-circle in the upper half of the complex
$q$-plane, and $-$ for a semi-circle in the lower half of the complex
$q$-plane. The angle $\phi$ changes from $0$ to $\pi$. At the turning point
$\hat{z}=\hat{z}_{0}=\hat{\eta}_{0}^{2}$ and $q=a$. We assume that $\rho$ is
small so that $|\hat{z}-\hat{z}_{0}|$ and $|q-a|$ are also small and use the
following linearized form of the constraint equation

${\cal F}^{\prime}_{0}(\hat{z}-\hat{z}_{0})=\hat{V}_{,q}(a)(q-a)\,,\ \
\left.{\cal F}^{\prime}_{0}=(d{\cal F}/d\hat{z})\right|_{\hat{z}_{0}}\,.$ (85)

Let us denote $B=\sqrt{\hat{V}_{,q}(a)/{\cal F}^{\prime}_{0}}$. One also has
$\hat{z}-\hat{z}_{0}=\hat{\eta}^{2}$. Then (84) and (85) imply

$\eta=\mu B\rho^{1/2}e^{\pm i\phi/2}\,.$ (86)

For the analytically continued WKB solution (82) we get

$\displaystyle{\cal S}^{\pm}={2\over 3}\mu B\rho^{3/2}\exp(\pm 3i\phi/2)\,,$
(87) $\displaystyle J^{\pm}=\mu B{\cal F}^{\prime}_{0}\rho^{1/2}\exp(\pm
i\phi/2)\,.$ (88)

Here $\pm$ stands for the quantities calculated along a semi-circle in the
upper ($+$) and lower ($-$) half of the $q$-plane. Using these relations at
$\phi=\pi$ where $a-q=\rho$ one obtains

$C={1\over 2}C_{-},\hskip 14.22636pt\psi=-\pi/4\,.$ (89)

This establishes a relation between the WKB solution (82) in the domain to the
right of the turning point and the WKB solution (80) to the left from it. This
gives a required connection formula. It should be emphasized that for the
operator $F(\Box)$ with the properties specified in section II this connection
relation is the same as for a standard Schrödinger equation.

We assume now that the potential has two turning points $a<b$ and the domain
between these points is forbidden for the classical motion. We suppose that in
the region $q<a$ the field $\Phi(q)$ has the form (80) and its amplitude is
$C$. Then in the region $q>b$ it has the same form with a different amplitude
$\tilde{C}$. Similarly to the local case the ratio $|\tilde{C}/C|^{2}$ can be
found by using the connection formula (89) twice Landau and Lifshitz (1965)

$\displaystyle R=\left({\tilde{C}\over C}\right)^{2}=\exp(-{2\over\hbar}P)\,,$
(90) $\displaystyle P=\mu\int_{a}^{b}\sqrt{\hat{z}-\hat{z}_{0}}\ dq\,.$ (91)

Here $\hat{z}_{0}=\hat{\eta}_{0}^{2}$ and
$\hat{z}=\hat{z}_{0}^{2}+\hat{\eta}^{2}$. In the expression for $P$ the
quantities $\hat{z}$ and $q$ satisfy the constraint (42). The quantity $R$
gives a probability of the barrier penetration. It should be emphasized that
these relations are obtained in the WKB approximation which is valid when
$R\ll 1$.

###  VI.4 Barrier penetration

####  VI.4.1 Inverse parabolic potential: Local theory

For the local theory the effective Hamiltonian is

$H(q,\eta)={1\over 2}[-(\eta_{0}^{2}-\eta^{2})+V(q)]\,,$ (92)

where $\eta_{0}^{2}=\varepsilon^{2}-p_{\perp}^{2}-m^{2}$. Let the potential
$V(q)$ be of the form

$V(q)=\begin{cases}V_{0}-V_{1}^{2}q^{2}\,,&\text{if\ \ \ }|q|\leq q_{*}\,,\\\
0\,,&\text{if\ \ \ }|q|>q_{*}\,.\end{cases}$ (93)

where $V_{0}>0$ and $q_{*}={\sqrt{V_{0}}/V_{1}}$. This potential is shown at
figure 7. It is called an inverse parabolic potential.

We assume that $\eta_{0}^{2}<V_{0}$ so that there exit two turning points at
$q=\pm a$, where

$a={\sqrt{V_{0}-\eta_{0}^{2}}\over V_{1}}\,.$ (94)

For the motion in the under-barrier domain we put $\eta=i\tilde{\eta}$ where

$\tilde{\eta}=\pm\sqrt{V-\eta_{0}^{2}}=\pm V_{1}\sqrt{a^{2}-q^{2}}\,.$ (95)

Figure 7: The inverse parabolic potential

We denote by $P_{0}$ the following integral

$\displaystyle P_{0}$ $\displaystyle=$
$\displaystyle\int_{-a}^{a}|\tilde{\eta}|dq$ (96) $\displaystyle=$
$\displaystyle V_{1}\int_{-a}^{a}dq\sqrt{a^{2}-q^{2}}={1\over 2}\pi
a^{2}V_{1}\,.$

Using relation (94) we can also write $P_{0}$ in the form

$P_{0}={\pi(V_{0}-\eta_{0}^{2})\over 2V_{1}}\,.$ (97)

Let us emphasize that $P_{0}$ depends only on the difference
$V_{0}-\eta_{0}^{2}$ but it does not depend on the value of $\eta_{0}$ itself.
In other words, the penetration probability is sensitive only to the fact how
close the effective energy $\eta_{0}^{2}$ is to the top of the potential
$V_{0}$. Let us also remind that the adopted WKB approximation is valid only
when the dimensionless quantity $P_{0}$ is large, that is when the probability
of the penetration $R_{0}=\exp(-2P_{0}/\hbar)$ is small (see e,g, Landau and
Lifshitz (1965)).

####  VI.4.2 Inverse parabolic potential: Nonlocal theory

Let us assume again that the potential $V(q)$ has the form (93). We denote by
$\hat{z}_{0}$ and $\hat{z}_{1}$ solutions of the following equations

${\cal F}(\hat{z}_{0})=\hat{V}_{0}-\hat{V}_{1}^{2}a^{2},\hskip 14.22636pt{\cal
F}(\hat{z}_{1})=\hat{V}_{0}\,.$ (98)

Then the WKB probability of the barrier penetration is

$\displaystyle R=\exp\left(-{2\over\hbar}P\right)\,,$ (99) $\displaystyle
P={\mu\over\hat{V}_{1}}\int_{\hat{z}_{0}}^{\hat{z}_{1}}{d\hat{z}\over\sqrt{\hat{z}-\hat{z}_{0}}}\sqrt{{\cal
F}(\hat{z}_{1})-{\cal F}(\hat{z})}\,.$ (100)

Let us calculate the value of $P$ for the case when ${\cal F}$ is of the form
(7). When both parameters $V_{0}$ and $\eta_{0}^{2}$ are much smaller than
$\mu^{2}$, the function $X(\hat{z})\approx\hat{z}$ and the integral in (100)
can be easily taken with the following result

$P={\pi\mu(\hat{z}_{1}-\hat{z}_{0})\over
2\hat{V}_{1}}={\pi(V_{0}-\eta_{0}^{2})\over 2V_{1}}\,.$ (101)

In this regime the effects of the nonlocality are not important and we
reproduce the expression for $P_{0}$ obtained in the local case (see equation
(97)).

The effects of the nonlocality play an important role in the different regime,
when $\hat{V}_{0}\gg 1$ and $\hat{\eta}_{0}^{2}\gg 1$. To demonstrate this we
consider again the theory with the form factor $f(\hat{z})=\exp(\hat{z}^{2})$.
In order to compare the expression for $P$ with $P_{0}$ we fix as earlier the
value of the difference $V_{0}-\eta_{0}^{2}$, but now we assume that
$\hat{V}_{0}\gg 1$. Then one has $\hat{z}_{1}-\hat{z}_{0}\ll 1$. In the
appendix B it is shown that one can write the following approximate expression
for $P$ in this case

$P=\kappa(\hat{V}_{0})P_{0},\hskip
14.22636pt\kappa(\hat{V}_{0})={1\over\sqrt{X^{\prime}_{1}}}.$ (102)

Here $\left.X^{\prime}_{1}=dX/dx\right|_{x=\hat{z}_{1}}$ and
$X(\hat{z}_{1})=\hat{V}_{0}$. Using (B) one can write

$\displaystyle
X^{\prime}_{1}(\hat{z}_{1})=(1+2\hat{z}_{1}^{2})\exp(\hat{z}_{1}^{2})\,,$
(103) $\displaystyle\hat{V}_{0}=\hat{z}_{1}^{2}\exp(\hat{z}_{1}^{2})\,.$ (104)

These relations give a parametric representation of
$X^{\prime}_{1}(\hat{z}_{1})$ as a function of $\hat{V}_{0}$. The plot of the
function $\kappa(\hat{V}_{0})$ is shown at figure 8. It shows that this
function is equal to 1 at $\hat{V}_{0}=0$ and it is less than 1 and
monotonically decreases for positive $\hat{V}_{0}$.

Figure 8: Parameter $\kappa$ as a function of $\hat{V}_{0}$.

Since $R=\exp(-2P/\hbar)$ the relation (102) can be rewritten as follows

$R=(R_{0})^{\kappa(\hat{V}_{0})}\,.$ (105)

Since $R_{0}<1$ the value of $R$ is always greater that $R_{0}$. This means
that in the nonlocal theory the probability of the barrier penetration is
larger that in the local case. In other words nonlocality makes the barrier
more transparent in the high-energy regime.

##  VII Discission

Let us discuss the obtained results. In this paper we study quasiclassical
solutions of the nonlocal massive scalar field equations. The kinetic part of
the action for such a theory contains a function of the $\Box$-operator, which
is chosen so that the theory does not contain new unphysical degrees of
freedom. In section II we formulated conditions imposed on the form factor of
such a theory. We also described a special model which is used later in the
paper for the illustration. In section III we described an ansatz used in the
WKB approximation for a solution of the nonlocal field equation. In the
leading order of the $1/\hbar$-expansion one obtains the equation for the
eikonal function $S(x)$ which is of the form $H(x,\nabla S)=0$. This is a
first order partial differential equation which has the form of the Hamilton-
Jacobi equation for the Hamiltonian $H(x,p)$, obtained by substitution $\nabla
S=p$. To calculate the eikonal function $S(x)$ it is sufficixent to construct
the Lagrangian submanifold in the phase space $(x,p)$ by using the initial
conditions for $S$. Such a Lagrangian submanifold is formed by phase
trajectories which are solutions of the Hamiltonian equations of motion. Since
the system is conservative this submanifold belongs to the subspace
$H(x,p)=0$. Let us emphasize that this procedure is very similar to the
standard one adopted for the quantum mechanics and local field theory. When
the field operator is of the second order the corresponding effective
Hamiltonian $H(x,p)$ is a quadratic polynomial in momentum $p$. For the
nonlocal theory $H(x,p)$ is a non-polynomial function of $p$ and this is a
main difference with the local case. As a result the construction of the WKB
solutions becomes technically more complicated.

WKB solutions presented in section III are valid for a wide class of nonlocal
scalar field theories with an arbitrary external potential $V(x)$ in a flat
spacetime with any number of spatial dimensions. These relations are greatly
simplified for the case when the potential $V$ depends only on one spatial
Cartesian coordinate. For this case the phase space is two-dimensional and the
Hamilton’s equations are completely integrable. The corresponding WKB
solutions are described in section IV. In section V we used the WKB
approximation for the calculation of the energy levels of the nonlocal field
trapped by a one-dimensional potential. To illustrate general formulas we
considered the case when the form factor of the nonlocality is of the special
form (7) and the potential is parabolic. The energy levels $\varepsilon_{n}$
of the field depend on a discrete integer parameter $n$. For the local case
the number $\Delta n$ of the levels in the interval $\Delta(\varepsilon^{2})$
is $\Delta n=\Delta(\varepsilon^{2})/(2\hbar V_{1})$ and it does not depend on
the value of $n$. In this sense the levels are equidistant. In the nonlocal
case for large $n$ this dependence is different

$\Delta n\approx{\sqrt{2}\over\mu^{2}}n\sqrt{\ln
n}\,\Delta(\varepsilon^{2})\,.$ (106)

This means that for a fixed value of $\hbar V_{1}/\mu^{2}$ and very high
energy there are much more energy levels in a given interval of
$\Delta(\varepsilon^{2})$ than in the local case.

Finally, we discussed the under-barrier motion and obtained an expression for
the probability of the barrier penetration. This probability has the form
$\sim\exp(-2P/\hbar)$. We calculated the factor $P$ for the the special form
of the nonlocality (7) in the presence of the inverse parabolic potential. For
the local theory this factor is $P_{0}=\pi(V_{0}-\eta_{0}^{2})/(2V_{1})$. In
other words, it depends on how close is the energy of the state $\eta_{0}^{2}$
to the top of the potential $V_{0}$, but it does not depend on the height
$V_{0}$ of the potential itself. The situation in the nonlocal case is quite
different. Namely, one has $P=\kappa P_{0}$, where the factor $\kappa$ depends
on the height $V_{0}$ of the potential. This factor is always less than 1 and
it decreases when $V_{0}$ grows. This means that the nonlocality ”helps” the
field to penetrate the potential barrier.

The examples considered in this paper are rather simple. It would be
interesting to apply the developed in the paper method for study the effects
of the nonlocality in the early cosmology and in the black holes interior.

##  Appendix A WKB approximation

We collected here useful information concerning construction of solutions of
the nonlocal field equations in the WKB approximation. Further details and
required mathematical foundations can be found in the book Maslov and Fedoriuk
(1981).

Let $(x^{1},\ldots,x^{N})$ be coordinates of a point $x$ in $R^{N}$. We denote
by $\alpha$ a set of $N$ non-negative integer numbers
$\alpha=(\alpha_{1},\ldots,\alpha_{N})$. We consider the following operator in
this space

$\hat{\cal O}=\sum_{\alpha}a_{\alpha}\left({\hbar\over
i}{\partial\over\partial x^{1}}\right)^{\alpha_{1}}\ldots\left({\hbar\over
i}{\partial\over\partial x^{N}}\right)^{\alpha_{N}}\,.$ (107)

The sum is taken over all different combinations of $\alpha$. We do not assume
that the number of terms in the sum is finite, so that this expression can be
used as a formal series in derivatives representation of nonlocal operators.

Let us make a substitution

${\hbar\over i}{\partial\over\partial x^{\mu}}\to p_{\mu}$ (108)

in the expression (107). Then we obtain a function of $2N$ variables which we
denote by $H=H(x,p)$

$H=H(x,p)=\sum_{\alpha}a_{\alpha}p_{1}^{\alpha_{1}}\ldots
p_{N}^{\alpha_{N}}\,.$ (109)

Here $p=(p_{1},\ldots,p_{N})$. This object is called a symbol of the operator
$\hat{\cal O}$.

We are looking for a solution $\varphi$ of the equation

$\hat{\cal O}\varphi=0$ (110)

in the form of the following formal series

$\varphi(x)=\exp\left({iS(x)\over\hbar}\right)\sum_{j=0}^{\infty}\left({\hbar\over
i}\right)^{j}u_{j}(x)\,.$ (111)

Usually it is sufficient to consider only first few terms in the series. This
gives what is known as the WKB approximation for the solution $\varphi$. To
obtain the functions $S(x)$ and $u_{j}(x)$ that enter (111) the following
procedure is used (see e.g. Arnold  _et al._ (2013); Maslov and Fedoriuk
(1981) and references therein).

If one substitute expression (111) into the equation (110) then one finds that
in the leading order $j=0$ the following condition is to be satisfied

$H\left(x,{\partial S\over\partial x}\right)u_{0}=0\,,$ (112)

where

${\partial S\over\partial x}=\left({\partial S\over\partial
x^{1}}\ldots,{\partial S\over\partial x^{N}}\right)\,.$ (113)

For a nonvanishing function $u_{0}(x)$ the following Hamilton-Jacobi equation
is valid

$H\left(x,{\partial S\over\partial x}\right)=0\,.$ (114)

Let us consider a $(2N)$-dimension space $\Gamma^{2N}$ with coordinates
$(x,p)=(x^{1},\ldots,x^{N},p_{1},\ldots,P_{N})$ and denote by $\Omega$ a
symplectic two-form in it

$\Omega=\sum_{\mu=1}^{N}dp_{\mu}\wedge dx^{\mu}\,.$ (115)

An $N$-dimensional surface $\Lambda^{N}$ in $\Gamma^{2N}$ is called a
Lagrangian submanifold if the $\Omega$ vanishes on $\Lambda^{N}$. If a
Lagrangian submanifold can be uniquely projected onto the $x$-space, then
there exists such a function $S(x)$ that $p_{\mu}=\partial S/\partial_{\mu}$
on $\Lambda^{N}$.

The Hamiltonian $H(x,p)$ given by the relation (109) implies the following
Hamilton’s equations

${dx^{\mu}\over d\tau}={\partial H(x,p)\over\partial p_{\mu}},\hskip
14.22636pt{dp_{\mu}\over d\tau}=-{\partial H(x,p)\over\partial x^{\mu}}\,.$
(116)

These equations determine the Hamiltonian flow in the phase space
$\Gamma^{2N}$ preserving the symplectic form $\Omega$ and the Hamiltonian
itself is an integral of motion.

Let $\Sigma^{N-1}$ be a $(N-1)$-dimensional surface in the coordinate space.
Its embedding is defined by the equations

$x^{\mu}=f^{\mu}(y^{i}),\hskip 14.22636pti=1,\ldots,N-1\,.$ (117)

One can consider $y^{i}$ as coordinates on $\Sigma^{N-1}$. We associate with
$\Sigma^{N-1}$ a $(2N-2)$-dimensional phase space $\Gamma^{2N-2}$ and denote
its canonical coordinates by
$(y,v)=(y^{1},\ldots,y^{N-1},v_{1},\ldots,v_{N-1})$. The symplectic 2-form
$\omega$ in $\Gamma^{2N-2}$ is

$\omega=\sum_{i=1}^{N-1}dv_{i}\wedge dy^{i}\,.$ (118)

Denote by $S^{0}=S^{0}(y)$ a function on $\Sigma^{N-1}$ and $\nabla
S^{0}=(S^{0}_{,y^{1}},\ldots,S^{0}_{,y^{N-1}})$, then a $(N-1)$-dimensional
surface $(y,\nabla S)$ of $\Gamma^{2N-2}$ is a Lagrangian submanifold
$\Lambda^{N-1}$, so that $\omega|_{\Lambda^{N-1}}=0$.

Let us define a covector $p(y)=p_{\mu}(y)$ on $\Sigma^{N-1}$ by two conditions

$p_{\mu}(y)dx^{\mu}=p_{\mu}(y){\partial f^{\mu}\over\partial y^{i}}=0\,,\
H(x(y),p(y))=0\,.$ (119)

We assume that this system of $N$ equations has a solution which determines
$p_{\mu}(y)|_{\Sigma^{N-1}}$.

Let us consider the following initial conditions for the Hamilton’s equations
(116)

$x^{\mu}|_{\tau=0}=f^{\mu}(y^{i}),\hskip
14.22636ptp_{\mu}|_{\tau=0}=p_{\mu}(y)\,.$ (120)

A family of solutions of the Hamilton’s equations (116) with these initial
conditions, $(x(\tau,y),p(\tau,y))$, forms a $N$-dimensional surface in
$\Gamma^{2N}$ which we denote by ${\cal B}$. It is possible to show that
${\cal B}$ is a Lagrangian submanifold in $\Gamma^{2N}$ and that $H(x,p)=0$ on
it. This implies that there exists such a function $S(x)$ that

$p_{\mu}|_{{\cal B}}={\partial S\over\partial x^{\mu}}\,.$ (121)

This function is a solution of the Hamilton-Jacobi equation (114) satisfying
the initial condition $S(x)|_{\Sigma^{N-1}}=S^{0}(y)$.

This function $S(x)$ can be found as follows. Denote a 1-form
$\theta=\sum_{\mu}p_{\mu}dx^{\mu}$, then $\Omega=d\theta$. Since the
symplectic form $\Omega$ vanishes on ${\cal B}$, the integral

$\int p_{\mu}dx^{\mu}$ (122)

calculated along a path on ${\cal B}$ between its two points does not depend
on the choice of the path. The required solution $S(x)$ of the Hamilton-Jacobi
equation satisfying the initial condition $S|_{\Sigma^{N-1}}=S(y)$ can be
written in the form

$S(x)=S^{0}(y)+\int_{0}^{\tau}p_{\mu}dx^{\mu}\,.$ (123)

Here $x^{\mu}=x^{\mu}(\tau,y)$ and $p_{\mu}=p_{\mu}(\tau,y)$.

Let us note that the map $x^{\mu}=x^{\mu}(\tau,y)$ may be considered as a
transformation between the coordinates $(\tau,y$ and $x^{\mu}$. Let us denote
by $J$ a Jacobian of this transformation

$J(\tau,y)=\left|\det{\partial x(\tau,y)\over\partial(\tau,y)}\right|\,.$
(124)

Then the leading term of the asymptotic solution (111) is Maslov and Fedoriuk
(1981)

$\displaystyle\varphi(x)$ $\displaystyle=$ $\displaystyle
u_{0}(y)\sqrt{J(0,y)\over
J(\tau,y)}\exp\left[{i\over\hbar}\left(S^{0}(y)+\int_{0}^{\tau}p_{\mu}dx^{\mu}\right)\right.$
(125) $\displaystyle-$ $\displaystyle\left.{1\over
2}\int_{0}^{\tau}\sum_{\mu=1}^{N}{\partial^{2}H(x,p)\over\partial
x^{\mu}\partial p_{\mu}}d\tau\right]\,.$

For $\tau=0$ the right-hand side of this expression reduces to
$u_{0}(y)\exp\left[{i\over\hbar}S^{0}(y)\right]$ and it should be found from
the initial condition for the function $\varphi(x)$ at $\Sigma$.

##  Appendix B Function $X(x)=x\exp(x^{2})$ and its properties

In this appendix we discuss some properties of the function

$X(x)=x\exp(x^{2})\,.$ (126)

The derivatives of this function are

$\displaystyle X^{\prime}=(1+2x^{2})\exp(x^{2})\,,$ $\displaystyle
X^{\prime\prime}=6x(1+{2\over 3}x^{2})\exp(x^{2})\,,$ (127)
$\displaystyle\alpha={1\over 2}{X^{\prime\prime}\over
X^{\prime}}=x\left(1+{2\over 2x^{2}+1}\right)\,.$

We denote by $Y$ an inverse function of $X$ such that the following relations
are valid

$y=X(x),\hskip 14.22636ptx=Y(y)\,.$ (128)

The inverse function can be written as follows

$Y(y)=y\exp\left(-{1\over 2}W(2y^{2})\,.\right)$ (129)

Here $W$ is Lambert $W$ function defined by the relation

$W(z)\exp(W(z))=z\,.$ (130)

For a real positive argument $W$ is a real single-valued function.

Consider a following integral

$I=\int_{x_{0}}^{x_{1}}{dx\over\sqrt{x-x_{0}}}\sqrt{X(x_{1})-X(x)}\,.$ (131)

Let us assume that $x_{1}-x_{0}$ is small. To obtain an approximate value of
the integral $I$ in this case we use the Taylor expansion

$X(x)=X_{1}-X^{\prime}_{1}(x_{1}-x)+{1\over
2}X^{\prime\prime}_{1}(x_{1}-x)^{2}+\ldots\,.$ (132)

Here $X_{1}=X(x_{1})$, $X^{\prime}_{1}=X^{\prime}(x_{1})$,
$X^{\prime\prime}_{1}=X^{\prime\prime}(x_{1})$. Skipping the higher order
terms denoted in (132) by dots one has

$I=\sqrt{X^{\prime}_{1}}\int_{x_{0}}^{x_{1}}{dx\over\sqrt{x-x_{0}}}\sqrt{(x_{1}-x)-\alpha(x_{1}-x)^{2}}\,,$
(133)

where $\alpha={1\over 2}(X^{\prime\prime}_{1}/X^{\prime}_{1})$. This integral
can be calculated exactly. Let us denote $\beta=x_{1}-x_{0}$, then for
$\alpha\beta<1$ the answer is

$I={2\sqrt{X^{\prime}_{1}}\over
3\alpha}\left[(2\alpha\beta-1)E(\sqrt{\alpha\beta})-(\alpha\beta-1)K(\sqrt{\alpha\beta})\right]\,.$

Here $E$ and $K$ are complete elliptic integrals. Forthe small value of
$\alpha\beta$ one has

$I={\pi\beta\over 2}\sqrt{X^{\prime}_{1}}\left(1-{3\over
8}\alpha\beta\right)\,.$ (134)

For $\alpha\beta\ll 1$ one can neglect the second term. Keeping the leading
term one gets

$I={\pi\beta\over 2}\sqrt{X^{\prime}_{1}}={\pi\over
2}{X_{1}-X_{0}\over\sqrt{X^{\prime}_{1}}}\,.$ (135)

For small $x_{1}$ one has $X^{\prime}_{1}\approx 1$ and

$I\approx{\pi\over 2}(X_{1}-X_{0})\,.$ (136)

## Acknowledgments

The author thanks the Natural Sciences and Engineering Research Council of
Canada and the Killam Trust for their financial support.

## References

  * Wataghin (1934) G. Wataghin, Z. Phys. 88, 92 (1934).
  * Yukawa (1950a) H. Yukawa, Phys. Rev. 77, 219 (1950a).
  * Yukawa (1950b) H. Yukawa, Phys. Rev. 80, 1047 (1950b).
  * Efimov (1968) G. V. Efimov, Comm. Math. Phys. 7, 138 (1968).
  * Efimov (1972) G. V. Efimov, Annals Phys. (NY) 71, 466 (1972).
  * Efimov (1974) G. V. Efimov, Int. J. Theor. Phys. 10, 19 (1974).
  * Alebastrov and Efimov (1974) V. A. Alebastrov and G. V. Efimov, Commun. Math. Phys. 38, 11 (1974).
  * Witten (1986) E. Witten, Nucl. Phys. B268, 253 (1986).
  * Frampton and Okada (1988) P. H. Frampton and Y. Okada, Phys. Rev. D37, 3077 (1988).
  * Eliezer and Woodard (1989) D. A. Eliezer and R. P. Woodard, Nucl. Phys. B325, 389 (1989).
  * Kostelecky and Samuel (1990) V. A. Kostelecky and S. Samuel, Phys. Rev. D42, 1289 (1990).
  * Harms and Leblanc (1995) B. Harms and Y. Leblanc, Annals Phys. 242, 265 (1995), arXiv:hep-th/9307042 [hep-th] .
  * Tseytlin (1995) A. A. Tseytlin, Phys. Lett. B363, 223 (1995), arXiv:hep-th/9509050 [hep-th] .
  * Calcagni and Modesto (2014) G. Calcagni and L. Modesto, Journal of Physics A: Mathematical and Theoretical 47, 355 (2014).
  * Calcagni and Modesto (2015) G. Calcagni and L. Modesto, Physical Review D 91, 124059 (2015).
  * Hossenfelder (2013) S. Hossenfelder, Living Rev. Rel. 16, 2 (2013), arXiv:1203.6191 [gr-qc] .
  * Tomboulis (1997) E. Tomboulis, (1997), arXiv:hep-th/9702146 [hep-th] .
  * Biswas  _et al._ (2012) T. Biswas, E. Gerwick, T. Koivisto, and A. Mazumdar, Phys. Rev. Lett. 108, 031101 (2012), arXiv:1110.5249 [gr-qc] .
  * Modesto (2012) L. Modesto, Phys. Rev. D86, 044005 (2012), arXiv:1107.2403 [hep-th] .
  * Biswas  _et al._ (2014) T. Biswas, A. Conroy, A. S. Koshelev, and A. Mazumdar, Class.Quant.Grav. 31, 015022 (2014), arXiv:1308.2319 [hep-th] .
  * Shapiro (2015) I. L. Shapiro, Phys. Lett. B744, 67 (2015), arXiv:1502.00106 [hep-th] .
  * Biswas  _et al._ (2010a) T. Biswas, T. Koivisto, and A. Mazumdar, JCAP 1011, 008 (2010a), arXiv:1005.0590 [hep-th] .
  * Biswas  _et al._ (2013) T. Biswas, T. Koivisto, and A. Mazumdar, in  _Proceedings, Barcelona Postgrad Encounters on Fundamental Physics_ (2013) pp. 13–24, arXiv:1302.0532 [gr-qc] .
  * Biswas  _et al._ (2006) T. Biswas, A. Mazumdar, and W. Siegel, Journal of Cosmology and Astroparticle Physics 0603, 009 (2006).
  * Koshelev (2007) A. S. Koshelev, Journal of High Energy Physics 0710, 029 (2007).
  * Biswas  _et al._ (2010b) T. Biswas, T. Koivisto, and A. Mazumdar, Journal of Cosmology and Astroparticle Physics 1011, 008 (2010b).
  * Modesto  _et al._ (2011) L. Modesto, J. W. Moffat, and P. Nicolini, Physics Letters B 695, 397 (2011).
  * Conroy  _et al._ (2015) A. Conroy, A. Mazumdar, S. Talaganis, and A. Teimouri, Phys. Rev. D92, 124051 (2015), arXiv:1509.01247 [hep-th] .
  * Modesto and Rachwał (2017) L. Modesto and L. Rachwał, Int. J. Mod. Phys. D26, 1730020 (2017).
  * Buoninfante  _et al._ (2018) L. Buoninfante, A. S. Koshelev, G. Lambiase, and A. Mazumdar, JCAP 1809, 034 (2018), arXiv:1802.00399 [gr-qc] .
  * Koshelev  _et al._ (2018) A. S. Koshelev, J. Marto, and A. Mazumdar, Phys. Rev. D98, 064023 (2018), arXiv:1803.00309 [gr-qc] .
  * Kilicarslan (2019a) E. Kilicarslan, Turk. J. Phys. 43, 126 (2019a), arXiv:1811.00843 [gr-qc] .
  * Koshelev  _et al._ (2019) A. S. Koshelev, J. Marto, and A. Mazumdar, Journal of Cosmology and Astroparticle Physics 2019, 020 (2019).
  * Kilicarslan (2019b) E. Kilicarslan, Phys. Rev. D99, 124048 (2019b), arXiv:1903.04283 [gr-qc] .
  * de la Cruz-Dombriz  _et al._ (2019) l. de la Cruz-Dombriz, F. J. Maldonado Torralba, and A. Mazumdar, Phys. Rev. D99, 104021 (2019), arXiv:1812.04037 [gr-qc] .
  * Buoninfante and Mazumdar (2019) L. Buoninfante and A. Mazumdar, Phys. Rev. D100, 024031 (2019), arXiv:1903.01542 [gr-qc] .
  * Boos  _et al._ (2018a) J. Boos, V. P. Frolov, and A. Zelnikov, Phys. Rev. D97, 084021 (2018a), arXiv:1802.09573 [gr-qc] .
  * Boos  _et al._ (2020) J. Boos, J. Pinedo Soto, and V. P. Frolov, Phys. Rev. D101, 124065 (2020), arXiv:2004.07420 [gr-qc] .
  * Frolov (2015) V. P. Frolov, Phys. Rev. Lett. 115, 051102 (2015), arXiv:1505.00492 [hep-th] .
  * Frolov  _et al._ (2015) V. P. Frolov, A. Zelnikov, and T. de Paula Netto, JHEP 06, 107 (2015), arXiv:1504.00412 [hep-th] .
  * Frolov and Zelnikov (2016) V. P. Frolov and A. Zelnikov, Phys. Rev. D93, 064048 (2016), arXiv:1509.03336 [hep-th] .
  * Boos  _et al._ (2018b) J. Boos, V. P. Frolov, and A. Zelnikov, Phys. Lett. B782, 688 (2018b), arXiv:1805.01875 [hep-th] .
  * Frolov and Zelnikov (2018) V. P. Frolov and A. Zelnikov, Phys. Rev. D98, 084035 (2018), arXiv:1809.00417 [hep-th] .
  * Buoninfante  _et al._ (2019) L. Buoninfante, A. Mazumdar, and J. Peng, Phys. Rev. D 100, 104059 (2019), arXiv:1906.03624 [gr-qc] .
  * Arnold  _et al._ (2013) V. Arnold, K. Vogtmann, and A. Weinstein,  _Mathematical Methods of Classical Mechanics_ (Springer New York, 2013).
  * Maslov and Fedoriuk (1981) V. Maslov and M. Fedoriuk,  _Semi-Classical Approximation in Quantum Mechanics_ , Mathematical Physics and Applied Mathematics (Springer Netherlands, 1981).
  * Lu  _et al._ (2018) F. Lu, B. Lv, P. Wang, and H. Yang, Nucl. Phys. B937, 502 (2018), arXiv:1701.00395 [gr-qc] .
  * Landau and Lifshitz (1965) L. D. Landau and E. M. Lifshitz,  _Quantum Mechanics_ , 2nd ed. (Pergamon Press, Bristol, UK, 1965).