Currently at ]Max-Born-Institute, Max-Born-Str. 2A, 12489 Berlin, Germany

# Quantized relativistic time-of-arrival operators for spin-0 particles and
the quantum tunneling time problem

P.C.M. Flores<EMAIL_ADDRESS><EMAIL_ADDRESS>[  E.A. Galapon
Theoretical Physics Group, National Institute of Physics, University of the
Philippines Diliman, 1101 Quezon City, Philippines

###### Abstract

We provide a full account of our recent report [EPL, 141 (2023) 10001] which
constructed a quantized relativistic time-of-arrival operator for spin-0
particles using a modified Weyl-ordering rule to calculate the traversal time
across a square barrier. It was shown that the tunneling time of a
relativistic spin-0 particle is instantaneous under the condition that the
barrier height $V_{o}$ is less than the rest mass energy. This implies that
instantaneous tunneling is an inherent quantum effect in the context of
arrival times.

##  I Introduction

Tunneling is one of the most well-known quantum effects and has been a long
standing important subject of quantum mechanics. The simplest tunneling
phenomenon is demonstrated by a square potential barrier wherein the
Schrödinger equation predicts a non-zero probability that a particle initially
on the far left of the barrier is transmitted to the far right even if its
energy is less than the barrier height. However, tunneling becomes problematic
when one associates the time it takes a wavepacket to traverse the classically
forbidden region [1, 2] because it is compounded with the quantum time problem
(QTP), and superluminality. Standard quantum mechanics only treats time as a
parameter, as such, the quantum tunneling time problem may be ill-defined
because there is no canonical formalism in standard quantum mechanics to
answer questions regarding time durations [3, 4]. Moreover, a dynamical
treatment of time, e.g., a time operator, has been met with pessimism because
of Pauli’s no-go theorem [5] on the existence of a time operator. This has led
to several definitions of tunneling time using a parametric approach, e.g.
Wigner phase time [6], Büttiker-Landauer time [7], Larmor time [8, 9, 10],
Pollak-Miller time [11], dwell time [12], among many others [13, 14, 15, 16,
17, 18, 19, 20, 21, 22]. However, one of us has shown that Pauli’s no-go
theorem does not hold in the single Hilbert space formulation of quantum
mechanics [23] and constructed a corresponding barrier traversal time operator
to calculate the tunneling time [15]. By doing so, tunneling time was treated
as a dynamical observable which addresses any contentions on tunneling time
being an ill-defined problem.

There are still debates on the the validity of the various proposals and its
corresponding physical meaning when it predicts apparent superluminal
velocities [24]. Several experiments [25, 26, 27, 28, 29, 30, 31, 32, 33] to
measure the tunneling time have confirmed the superluminal behavior of a
tunneling particle but there is no consensus on whether the particle is
transmitted instantaneously or if it spends a finite time inside the barrier.
Moreover, the relation between these various proposed tunneling times is still
unclear but it has recently been argued that these tunneling times can be
classified into two distinct categories [34], i.e., arrival time and
interaction time. The former is concerned with the appearance of the tunneled
particle at the far side of the barrier while the latter determines the time
duration spent inside the barrier. Tunneling time as an “arrival time” is
demonstrated by attoclock experiments while “interaction time” by Larmor clock
experiments [34]. Some attoclock experiments have reported instantaneous [25,
26, 27, 28, 29, 30] tunneling while others reported finite tunneling times
[31, 32]. Moreover, a recent Larmor clock experiment has also reported finite
tunneling time [33]. Now, whether tunneling is instantaneous or not, the crux
of the problem is that both results imply that the particle exhibits
superluminal behavior below the barrier. This now raises the question on
whether the superluminality is a consequence of using non-relativistic quantum
mechanics, i.e., could there a fundamental difference if one uses a
relativistic theory?

There have been several studies to extend the analysis of tunneling times to
the relativistic case in order to adequately address the superluminal behavior
[35, 36, 37, 38]. It was shown by de Leo and Rotelli [35], then separately
again by de Leo [36] whom used the phase time via the Dirac equation in a step
potential to show that superluminal tunneling times is still present. Petrillo
and Janner [37] obtained similar results for a square barrier via the Dirac
equation. Krekora, Su, and Grobe [38] also used the Dirac equation for various
potential barriers of the form $V(x)=V_{o}e^{-(2x/w)^{n}}$ with an effective
width $w$, and defined an “instantaneous tunneling speed” to show superluminal
tunneling under the condition that the barrier height $V_{o}$ is less than
twice the rest mass energy. This apparent superluminal behavior despite the
relativistic treatment implies that the superluminal behavior is an inherent
quantum effect.

In this paper, we give a full account of our recent report [39] which proposed
a formalism on the construction of quantized relativistic TOA-operators for
spin-0 particles in the presence of an interaction potential. This was then
used to construct a corresponding barrier traversal time operator. By doing
so, the formalism can simultaneously addresses the compounding problems of
superluminality and the QTP in tunneling times. Now, it is well-known that
relativistic quantum mechanics is not a well-defined one-particle theory since
relativistic effects can lead to spontaneous pair-creation and annihilation
which might render the concept of TOA meaningless, i.e., we are not sure if
the particle that tunneled and arrived is the same particle we initially had.
To address this, we will impose the condition that the barrier height is less
than the rest mass energy.

The rest of the paper is structured as follows. In Sec. II we review the
construction of quantized non-relativistic TOA-operators in coordinate
representation using Weyl, Born-Jordan, and simple symmetric ordering [40]
which will then be modified to construct the corresponding relativistic
counterpart for spin-0 particles in Sec. III. The barrier traversal time
operator is then constructed in Sec. IV and will be shown to reduce to the
correct classical limit as $\hbar\rightarrow 0$ in Sec. V. Next, we establish
the expected barrier traversal time and show that tunneling is instantaneous
in Sec. VI, regardless of the ordering rule used. A single Gaussian wavepacket
is then used as an example in Sec. VII. Last, we conclude in Sec. VIII.

##  II Review of quantized non-relativistic TOA-operators

The rigorous mathematical framework of quantum mechanics was developed by von
Neumann using the Hilbert space $\mathcal{H}$ as its underlying linear
topological space wherein physical observables are generally identified with
maximally symmetric densely defined operators $\mathsf{\hat{A}}$ in
$\mathcal{H}$ while physical states are represented by the set of unit rays
$\ket{\psi}$ in $\mathcal{H}$. The eigenvalues of these operators then
represent the possible measurement outcomes of the corresponding observable
and its spectrum may be discrete, continuous, or a combination of both.
However, operators in quantum mechanics are generally unbounded with a
continuous spectrum corresponding to non-normalizable eigenfunctions, e.g. the
position and momentum operator whose eigenfunctions are the Dirac-delta
function $\delta(q-q_{o})$ and the plane wave
$\exp(ipq/\hbar)/\sqrt{2\pi\hbar}$, respectively.

In order to deal with these non-square integrable functions that are outside
the Hilbert space, one can use Dirac’s bra-ket notation which is made
mathematically rigorous by using the rigged Hilbert space (RHS) that utilizes
the theory of distributions [41, 42, 43, 44, 40]. In our case, we choose the
fundamental space of our RHS to be the space of infinitely continuously
differentiable complex valued functions with compact supports $\Phi$ such that
the RHS is $\Phi\subset L^{2}(\mathbb{R})\subset\Phi^{\times}$, where
$\Phi^{\times}$ is the space of all continuous linear functionals on $\Phi$.
The standard Hilbert space formulation of quantum mechanics is recovered by
taking the closures on $\Phi$ with respect to the metric of
$L^{2}(\mathbb{R})$.

In coordinate representation, a quantum observable $\mathsf{\hat{A}}$ is a
mapping from $\Phi$ to $\Phi^{\times}$, and is given by the formal integral
operator

$(\mathsf{\hat{A}}\varphi)(q)=\int_{-\infty}^{\infty}dq^{\prime}\matrixelement{q}{\mathsf{\hat{A}}}{q^{\prime}}\varphi(q^{\prime})$
(1)

where the kernel satisfies
$\matrixelement{q}{\mathsf{\hat{A}}}{q^{\prime}}=\matrixelement{q^{\prime}}{\mathsf{\hat{A}}}{q}^{*}$,
to ensure Hermiticity such that the eigenvalues of Eq. (1) are real-valued.
The integral Eq. (1) is interpreted in the distributional sense, i.e. it is a
functional on $\Phi$ wherein the kernel
$\matrixelement{q}{\mathsf{\hat{A}}}{q^{\prime}}$ is a distribution. As an
example, the position and momentum operators are now given as

$\displaystyle(\mathsf{\hat{q}}\varphi)(q)=$
$\displaystyle\int_{-\infty}^{\infty}dq^{\prime}\delta(q-q^{\prime})\varphi(q^{\prime})=q\varphi(q)$
(2) $\displaystyle(\mathsf{\hat{p}}\varphi)(q)=$
$\displaystyle\int_{-\infty}^{\infty}dq^{\prime}i\hbar\dfrac{d\delta(q-q^{\prime})}{dq^{\prime}}\varphi(q^{\prime})=-i\hbar\dfrac{d\varphi(q)}{dq}.$
(3)

There is still no consensus on how TOA-operators in the presence of an
interaction potetial are constructed [45, 46, 40]. One possible method is by
canonical quantization but it has been deemed not meaningful because the
classical TOA can be multiple and/or complex-valued. Moreover, canonical
quantization suffers from ordering ambiguities, obstruction to quantization
[47, 48], and circularity when imposing the correspondence principle [49, 50].
To overcome these problems, the method of “supraquantization” was proposed for
non-relativistic TOA-operators [50], i.e., constructing TOA-operators from
first principles of quantum mechanics. It turns out that for linear systems,
$V(q)=\alpha q^{2}+\beta q+\gamma$, the “supraquantized” TOA-operator is equal
to the canonically quantized TOA-operator using Weyl-ordering. Meanwhile, the
“supraquantized” TOA-operator for non-linear systems can be expressed as a
perturbation series wherein the leading term is the Weyl-ordered TOA-operator
[50, 40, 51]. This relation shows that canonical quantization is sufficient as
a leading order approximation of the canonical TOA-operator.

Now, the non-relativistic TOA-operators constructed by Galapon and Magadan
[40] quantized the corresponding classical non-relativistic TOA

$t_{x}(q,p)=-\text{sgn}(p)\sqrt{\dfrac{\mu_{o}}{2}}\int_{x}^{q}dq^{\prime}\left[\dfrac{p^{2}}{2\mu_{o}}+V(q)-V(q^{\prime})\right]^{-1/2}$
(4)

in coordinate representation. The function $\text{sgn}(p)$ is the sign of the
initial momentum $p$ which accounts for the particles moving from the left or
right. Meanwhile, $x$ is the arrival point and $\mu_{o}$ is the rest mass of
the particle. The expression Eq. (4) was obtained by treating energy as a
constant of motion and inverting the corresponding Hamilton equation of
motion. To quantize Eq. (4), it has been argued [40] that objections can be
addressed on physical grounds. First, the TOA of a quantum particle is always
real-valued because it can tunnel to the classically forbidden region. Second,
it is only meaningful to quantize the first TOA because the wavefunction will
collapse after a detector registers the TOA of the quantum particle. The
quantization of Eq. (4) was done by first expanding around the free TOA [40],
i.e.,

$t_{0}(q,p)=-\sum_{j=0}^{\infty}(-1)^{j}\mu_{o}^{j+1}\dfrac{(2j-1)!!}{j!p^{2j+1}}\int_{o}^{q}dq^{\prime}\left(V(q)-V(q^{\prime})\right)^{j}.$
(5)

It is then assumed that the potential is analytic at the origin wherein it
admits the expansion $V(q)=\sum_{n=0}\nu_{n}q^{n}$ such that

$\displaystyle\int_{o}^{q}dq^{\prime}\left(V(q)-V(q^{\prime})\right)^{j}=\sum_{n=1}^{\infty}a_{n}^{(j)}q^{n}.$
(6)

This then yields the local time of arrival (LTOA)

$t_{0}(q,p)=-\sum_{j=0}^{\infty}(-1)^{j}\mu_{o}^{j+1}\dfrac{(2j-1)!!}{j!}\sum_{n=1}^{\infty}a_{n}^{(j)}\dfrac{q^{n}}{p^{2j+1}},$
(7)

which is now amenable to quantization because it is single and real-valued in
its region of convergence in the phase space. Now, the LTOA converges
absolutely and uniformly only in some local neighborhood
$\omega=\omega_{q}\times\omega_{p}$ determined by
$\absolutevalue{V(q)-V(q^{\prime})}<p^{2}/2\mu_{o}$ for $p\neq 0$ and
continuous $V(q)$, and will diverge outside this region which signifies that
the particle has not classically arrived at $q=0$, i.e. the classically
forbidden region. However, the classical TOA Eq. (4) holds in the region
$\Omega=\Omega_{q}\times\Omega_{p}$ where $\omega\subset\Omega$. This means
that Eq. (4) is the analytic continuation of the LTOA in the region
$\Omega\backslash\omega$ [51].

The monomials $q^{n}p^{-m}$ were then quantized by generalizing the Bender-
Dunne basis operators [52, 53],

$\mathsf{\hat{t}_{-m,n}}=\dfrac{\sum_{k=0}^{n}\beta_{k}^{(n)}\mathsf{\hat{q}}^{k}\mathsf{\hat{p}}^{-m}\mathsf{\hat{q}}^{n-k}}{\sum_{k=0}^{n}\beta_{k}^{(n)}},$
(8)

where, the coefficients satisfy the condition
$\beta_{k}^{(n)}=\beta_{n-k}^{(n)*}$ to ensure Hermiticity. Now, the most
well-studied [54, 55, 56, 57, 58, 59] ordering rules are Weyl, Born-Jordan,
and simple-symmetric with each having its own advantage . Specifically, Weyl
ordering preserves the covariant property of Hamiltonian dynamics with respect
to linear canonical transforms [57, 60] while Born-Jordan preserves the
equivalence of the the Schrödinger and Heisenberg formulation of quantum
mechanics [57, 61, 62]. On the other hand, simple-symmetric ordering just
provides the easiest possible ordering by using the “average rule” [54, 63].
These ordering rules are imposed on the basis operators
$\mathsf{\hat{t}_{-m,n}}$ by choosing the coefficients

$\displaystyle\beta_{k}^{(n)}=\begin{cases}\dfrac{n!}{k!(n-k)!}\quad&,\quad\text{Weyl}\\\
1\quad&,\quad\text{Born-Jordan}\\\
\delta_{k,0}+\delta_{k,n}\quad&,\quad\text{simple-symmetric}.\end{cases}$ (9)

It easily follows that in coordinate representation, the non-relativistic TOA-
operator admits the expansion

$(\mathsf{\hat{T}_{0}}\varphi)(q)=-\int_{-\infty}^{\infty}dq^{\prime}\sum_{j=0}^{\infty}(-1)^{j}\mu_{o}^{j+1}\dfrac{(2j-1)!!}{j!}\sum_{n=1}^{\infty}a_{n}^{(j)}\matrixelement{q}{\mathsf{\hat{t}_{-2j-1,n}}}{q^{\prime}}\varphi(q^{\prime}).$
(10)

wherein

$\displaystyle\matrixelement{q}{\mathsf{\hat{t}_{-m,n}}}{q^{\prime}}=\dfrac{i(-1)^{\frac{1}{2}(m-1)}}{2\hbar^{m}(m-1)!}P_{n}(q|q^{\prime})(q-q^{\prime})^{m-1}\text{sgn}(q-q^{\prime}),\quad
m=1,2,\dots$ (11) $\displaystyle
P_{n}(q|q^{\prime})=\begin{cases}\left(\dfrac{q+q^{\prime}}{2}\right)^{n}\quad&,\quad\text{Weyl}\\\
\\\ \dfrac{1}{n+1}\left(\dfrac{q^{n+1}-q^{\prime
n+1}}{q-q^{\prime}}\right)\quad&,\quad\text{Born-Jordan}\\\ \\\
\dfrac{q^{n}+q^{\prime n}}{2}\quad&,\quad\text{simple-symmetric}.\end{cases}$
(12)

The summation over $n$ in Eq. (10) is then evaluated using the following
identities

$\displaystyle\sum_{n=1}^{\infty}a_{n}^{(j)}P_{n}(q|q^{\prime})=\begin{cases}\int_{0}^{(q+q^{\prime})/2}ds\left[V\left(\dfrac{q+q^{\prime}}{2}\right)-V(s)\right]^{j}\quad&,\quad\text{Weyl}\\\
\\\
\int_{0}^{q}du\int_{0}^{s}(V(s)-V(u))^{j}-\int_{0}^{q^{\prime}}du\int_{0}^{s}(V(s)-V(u))^{j}\quad&,\quad\text{Born-
Jordan}\\\ \\\
\dfrac{1}{2}\int_{0}^{q}ds(V(q)-V(s))^{j}+\dfrac{1}{2}\int_{0}^{q^{\prime}}ds(V(q)-V(s))^{j}\quad&,\quad\text{simple-
symmetric},\end{cases}$ (13)

which follows from the assumed analyticity of the potential at the origin Eq.
(6). The resulting expression is further evaluated by taking the summation
over $j$.

Performing these operations yield the non-relativistic TOA-operators of the
form

$(\mathsf{\hat{T}_{0}}\varphi)(q)=\int_{-\infty}^{\infty}dq^{\prime}\dfrac{\mu_{o}}{i\hbar}T_{0}(q,q^{\prime})\text{sgn}(q-q^{\prime})\varphi(q^{\prime}),$
(14)

where $T(q,q^{\prime})$ is referred to as the time kernel factor (TKF) which
depends on the ordering rule used, i.e.,

$\displaystyle T_{0}^{(W)}(q,q^{\prime})=$
$\displaystyle\dfrac{1}{2}\int_{0}^{\frac{q+q^{\prime}}{2}}ds{{}_{0}}F_{1}\left[;1;\dfrac{\mu_{o}}{2\hbar^{2}}(q-q^{\prime})^{2}\left\\{V\left(\frac{q+q^{\prime}}{2}\right)-V(s)\right\\}\right]$
(15) $\displaystyle T_{0}^{(BJ)}(q,q^{\prime})=$
$\displaystyle\dfrac{1}{2(q-q^{\prime})}\int_{0}^{q}ds\int_{0}^{s}du{{}_{0}}F_{1}\left[;1;\dfrac{\mu_{o}}{2\hbar^{2}}(q-q^{\prime})^{2}\left\\{V(s)-V(u)\right\\}\right]$
$\displaystyle-\dfrac{1}{2(q-q^{\prime})}\int_{0}^{q^{\prime}}ds\int_{0}^{s}du{{}_{0}}F_{1}\left[;1;\dfrac{\mu_{o}}{2\hbar^{2}}(q-q^{\prime})^{2}\left\\{V(s)-V(u)\right\\}\right]$
(16) $\displaystyle T_{0}^{(SS)}(q,q^{\prime})=$
$\displaystyle\dfrac{1}{4}\int_{0}^{q}ds{{}_{0}}F_{1}\left[;1;\dfrac{\mu_{o}}{2\hbar^{2}}(q-q^{\prime})^{2}\left\\{V(q)-V(s)\right\\}\right]$
$\displaystyle+\dfrac{1}{4}\int_{0}^{q^{\prime}}ds{{}_{0}}F_{1}\left[;1;\dfrac{\mu_{o}}{2\hbar^{2}}(q-q^{\prime})^{2}\left\\{V(q^{\prime})-V(s)\right\\}\right]$
(17)

where ${{}_{0}}F_{1}(;a;z)$ is a specific hypergeometric function. The
superscripts “W”, “BJ”, and “SS” refer to the Weyl, Born-Jordan, and simple
symmetric ordering, respectively.

##  III Non-analytic quantization of the relativistic LTOA in coordinate
representation

Supposing that the relation between the non-relativistic “supraquantized” and
quantized TOA-operators also holds for the relativistic case, it should be
enough for now to consider the simplest approach by developing the method of
canonical quantization of relativistic TOA-operators, and leave the method of
“supraquantization” open for future studies. We follow the steps outlined in
Sec. II to construct the relativistic TOA-operator by quantizing the
corresponding “classical” relativistic time-of-arrival (CRTOA) obtained from
inverting the equation of motion from the Hamiltonian of special relativity
[64], i.e.,

$\displaystyle t_{x}(q,p)=$
$\displaystyle-\text{sgn}p\int_{x}^{q}\dfrac{dq^{\prime}}{c}\left(1-\dfrac{\mu_{o}^{2}c^{4}}{(H(q,p)-V(q^{\prime}))^{2}}\right)^{-1/2}$
(18)

wherein

$H(q,p)=\sqrt{p^{2}c^{2}+\mu_{o}^{2}c^{4}}+V(q)$ (19)

is the total energy of the positive energy solutions generated by the Klein-
Gordon equation. Similar to Eq. (4), the expression Eq. (18) was also obtained
by treating energy as a constant of motion and inverting the corresponding
Hamilton equation of motion. Without loss of generality, we assume the arrival
point to be the origin $x=0$ and impose that the potential is analytic at the
origin such that Eq. (18) has the expansion around the relativistic free TOA
given by

$\displaystyle t_{0}(q,p)=$
$\displaystyle-\mu_{o}\sum_{j=0}^{\infty}\sum_{k=0}^{j}\binom{-\frac{1}{2}}{j}\binom{j}{k}\dfrac{(2\mu_{o})^{j}}{(2\mu_{o}c^{2})^{j-k}}\sum_{n=1}^{\infty}a_{n}^{(2j-k)}\dfrac{\gamma_{p}^{k+1}}{p^{2j+1}}q^{n}$
(20)

where, $\gamma_{p}=\sqrt{1+p^{2}/\mu_{o}^{2}c^{2}}$. For consistency with Sec.
II, we shall also refer to Eq. (20) as the relativistic LTOA since it is also
single and real-valued within its region of convergence in the phase space.
That is, Eq. (20) will only converge absolutely and uniformly in some local
neighborhood $\omega=\omega_{q}\times\omega_{p}$ determined by
$\absolutevalue{V(q)-V(q^{\prime})}>\sqrt{p^{2}c^{2}+\mu_{o}^{2}c^{4}}-\mu_{o}c^{2}$
for $p\neq 0$ and continuous $V(q)$. Meanwhile Eq. (18) holds in the region
$\Omega=\Omega_{q}\times\Omega_{p}$ where $\omega\subset\Omega$ and is the
analytic continuation of the relativistic LTOA in the region
$\Omega\backslash\omega$.

Figure 1: Contours of integration for Eq. (25) for (a) $q-q^{\prime}>0$ and
(b) $q-q^{\prime}<0$.

The relativistic LTOA Eq. (20) is now amenable to quantization by promoting
the position and momentum $(q,p)$ into operators
$(\mathsf{\hat{q}},\mathsf{\hat{p}})$. There is still no consensus on the
existence of a position operator in relativistic quantum mechanics [65] but
the most suitable candidate is the Newton-Wigner position operator [66]. In
our case, we will only use the non-relativistic position operator
$\mathsf{\hat{q}}$ in quantizing Eq. (20) which is motivated by Razavi’s
relativistic free TOA operator [67, 68]

$\mathsf{\hat{T}_{Ra}}=-\dfrac{\mu_{o}}{2}\left(\mathsf{\hat{q}}\mathsf{\hat{p}^{-1}}\sqrt{1+\dfrac{\mathsf{\hat{p}^{2}}}{\mu_{o}^{2}c^{2}}}+\mathsf{\hat{p}^{-1}}\sqrt{1+\dfrac{\mathsf{\hat{p}^{2}}}{\mu_{o}^{2}c^{2}}}\mathsf{\hat{q}}\right).$
(21)

To quantize Eq. (20), we extend the Bender-Dunne basis operators [52, 53] to
separable classical function $f(q,p)=g(q)^{n}h(p)^{m}$, i.e.,

$f(q,p)\Rightarrow\mathsf{\hat{f}_{\hat{q},\hat{p}}}=\dfrac{\sum_{k=0}^{n}\alpha_{k}^{(n)}\mathsf{\hat{g}_{\hat{q}}}^{k}\mathsf{\hat{h}_{\hat{p}}}^{m}\mathsf{\hat{g}_{\hat{q}}}^{n-k}}{\sum_{k=0}^{n}\alpha_{k}^{(n)}}.$
(22)

where the coefficients $\alpha_{k}^{(n)}$ are given by Eq. (9). This now leads
to the quantization

$\displaystyle
Q\left[q^{n}p^{-2j-1}\gamma_{p}^{k+1}\right]=\begin{cases}\dfrac{1}{2^{n}}\sum_{r=0}^{n}\binom{n}{r}\mathsf{\hat{q}}^{r}\mathsf{\hat{p}}^{-2j-1}\mathsf{\gamma_{\hat{p}}^{k+1}}\mathsf{\hat{q}}^{n-r}\quad&,\quad\text{Weyl}\\\
\\\
\dfrac{1}{n+1}\sum_{r=0}^{n}\mathsf{\hat{q}}^{r}\mathsf{\hat{p}}^{-2j-1}\mathsf{\gamma_{\hat{p}}^{k+1}}\mathsf{\hat{q}}^{n-r}\quad&,\quad\text{Born-
Jordan}\\\ \\\
\dfrac{1}{2}\left(\mathsf{\hat{q}}^{n}\mathsf{\hat{p}}^{-2j-1}\mathsf{\gamma_{\hat{p}}^{k+1}}+\mathsf{\hat{p}}^{-2j-1}\mathsf{\gamma_{\hat{p}}^{k+1}}\mathsf{\hat{q}}^{n}\right)\quad&,\quad\text{simple-
symmetric}\end{cases}$ (23)

It follows from Eq. (23) that in coordinate representation, the quantized
relativistic TOA Eq. (20) now has the expansion

$\displaystyle(\mathsf{\hat{T}_{c}}\varphi)(q)=-\mu_{o}\sum_{j=0}^{\infty}\sum_{k=0}^{j}$
$\displaystyle\binom{-\frac{1}{2}}{j}\binom{j}{k}\dfrac{(2\mu_{o})^{j}}{(2\mu_{o}c^{2})^{j-k}}$
$\displaystyle\times\sum_{n=1}^{\infty}a_{n}^{(2j-k)}\int_{-\infty}^{\infty}dq^{\prime}P_{n}^{(Q)}(q|q^{\prime})\matrixelement{q}{\mathsf{\hat{p}}^{-2j-1}\mathsf{\gamma_{\hat{p}}^{k+1}}}{q^{\prime}}\varphi(q^{\prime})$
(24)

where, $P_{n}^{(Q)}(q|q^{\prime})$ is given by Eq. (12) and the superscript
$(Q)$ refers the to quantization rule used. The momentum kernel
$\matrixelement{q}{\mathsf{\hat{p}}^{-2j-1}\mathsf{\gamma_{\hat{p}}^{k+1}}}{q^{\prime}}$
is evaluated by inserting the resolution of the identity
$\mathsf{1}=\int_{-\infty}^{\infty}dp\ket{p}\bra{p}$, and using the plane wave
expansion $\innerproduct{q}{p}=e^{iqp/\hbar}/\sqrt{2\pi\hbar}$, i.e.,

$\displaystyle\matrixelement{q}{\mathsf{\hat{p}}^{-2j-1}\mathsf{\gamma_{\hat{p}}^{k+1}}}{q^{\prime}}=$
$\displaystyle\int_{-\infty}^{\infty}\frac{dp}{2\pi\hbar}\exp[\frac{i}{\hbar}(q-q^{\prime})p]\frac{1}{p^{2j+1}}\left(\sqrt{1+\frac{p^{2}}{\mu_{o}^{2}c^{2}}}\right)^{k+1}.$
(25)

The integral on the right hand side of Eq. (25) diverges because of the pole
with order $2j+1$ at $p=0$. Moreover, it has branch points at $\pm i\mu_{o}c$
for even values of $k$. Now, this has already been evaluated [68] for the case
when $j=k=0$ and can be similarly evaluated as a distributional Fourier
transform using the contours shown in Fig. 1. The evaluation of Eq. (25) is
done by taking its complex extension and taking the average of the integrals
$\int_{\Gamma^{\pm}}dzf(z)z^{-2j-1}$, where the paths $\gamma^{+}$
($\gamma^{-}$) passes above (below) the pole at $z=0$. Performing this
integration assigns a value to Eq. (25) which coincides with the Hadamard
finite part [69], and is explicitly given as

$\displaystyle\matrixelement{q}{\mathsf{\hat{p}}^{-2j-1}\mathsf{\gamma_{\hat{p}}^{k+1}}}{q^{\prime}}=-\dfrac{1}{2i\hbar}(f_{j,k}(q,q^{\prime})+g_{j,k}(q,q^{\prime}))\text{sgn}(q-q^{\prime})$
(26)

where,

$\displaystyle f_{j,k}(q,q^{\prime})=$
$\displaystyle\dfrac{1}{(2j)!}\left(\dfrac{i}{\hbar}(q-q^{\prime})\right)^{2j}\int_{0}^{\infty}dye^{-y}\oint_{R}\dfrac{dz}{2\pi
i}\dfrac{1}{z}\sqrt{1+\dfrac{z^{2}}{\mu_{o}^{2}c^{2}}}^{k+1}\left(1-i\dfrac{\hbar}{q-q^{\prime}}\dfrac{y}{z}\right)^{2j}$
(27) $\displaystyle g_{j,k}(q,q^{\prime})=$
$\displaystyle\dfrac{(-1)^{j}i^{k}}{(\mu_{o}c)^{2j}}\left(\dfrac{1-(-1)^{k+1}}{2}\right)\dfrac{2}{\pi}\int_{1}^{\infty}dy\exp[-\dfrac{\mu_{o}c}{\hbar}\absolutevalue{q-q^{\prime}}y]\dfrac{\sqrt{y^{2}-1}^{k+1}}{y^{2j+1}}.$
(28)

The function $f_{j,k}(q,q^{\prime})$ is the contribution of the resiude $z=0$
and is rewritten in integral form using the residue theorem, wherein, the
contour $R$ is a circle in the complex plane with radius $r<\mu_{o}c$.
Meanwhile, $g_{j,k}(q,q^{\prime})$ is the contribution of the branch cut which
vanishes for odd values of $k$. Thus, the relativistic TOA-operator Eq. (24)
now has the expansion

$\displaystyle(\mathsf{\hat{T}_{c}}\varphi)(q)=\int_{-\infty}^{\infty}dq^{\prime}\dfrac{\mu_{o}}{i\hbar}T^{\\{Q\\}}(q,q^{\prime})\text{sgn}(q-q^{\prime})\varphi(q^{\prime})$
(29)

where, $T^{\\{Q\\}}(q,q^{\prime})$ is the relativistic TKF and has the
expansion

$\displaystyle T^{\\{Q\\}}(q,q^{\prime})=\dfrac{1}{2}\sum_{j=0}^{\infty}$
$\displaystyle\sum_{k=0}^{j}\binom{-\frac{1}{2}}{j}\binom{j}{k}\dfrac{(2\mu_{o})^{j}}{(2\mu_{o}c^{2})^{j-k}}(f_{j,k}(q,q^{\prime})+g_{j,k}(q,q^{\prime}))\sum_{n=1}^{\infty}a_{n}^{(2j-k)}P_{2j-k}^{\\{Q\\}}(q|q^{\prime}).$
(30)

An integral form factor for Eq. (30) is obtained by series rearrangement and
using the identities in Eq. (13).

### Modified Weyl-ordered TOA operator

Performing the summation yields

$\displaystyle
T^{\\{W\\}}(q,q^{\prime})=\dfrac{1}{2}\int_{0}^{\frac{q+q^{\prime}}{2}}ds\mathsf{W}_{s}(q,q^{\prime})$
(31)

where,

$\displaystyle\mathsf{W}_{s}(q,q^{\prime})=\mathsf{W}_{s}^{(1)}(q,q^{\prime})+\dfrac{2}{\pi}\int_{1}^{\infty}dz\exp[-\dfrac{\mu_{o}c}{\hbar}\absolutevalue{q-q^{\prime}}z]\dfrac{\sqrt{z^{2}-1}}{z}\mathsf{W}_{s,z}^{(2)}(q,q^{\prime})$
(32)

in which

$\displaystyle\mathsf{W}_{s}^{(1)}(q,q^{\prime})=\int_{0}^{\infty}dye^{-y}$
$\displaystyle\oint_{R}\dfrac{dz}{2\pi
i}\dfrac{1}{z}\sqrt{1+\dfrac{z^{2}}{\mu_{o}^{2}c^{2}}}$
$\displaystyle\times{{}_{0}}F_{1}\left[;1;\dfrac{\mu_{o}V_{s}^{(W)}(q,q^{\prime})}{2\hbar^{2}}\left(\left(q-q^{\prime}\right)-i\hbar\dfrac{y}{z}\right)^{2}\mathsf{P_{W}}(s,z,q,q^{\prime})\right]$
(33) $\displaystyle\mathsf{W}_{s,z}^{(2)}(q,q^{\prime})=$
$\displaystyle\dfrac{1}{2}\left[1-\dfrac{1}{z^{2}}\left(\dfrac{V_{s}^{(W)}(q,q^{\prime})}{\mu_{o}c^{2}}\right)^{2}+2i\dfrac{\sqrt{z^{2}-1}}{z^{2}}\left(\dfrac{V_{s}^{(W)}(q,q^{\prime})}{\mu_{o}c^{2}}\right)\right]^{-1/2}+f_{i\rightarrow-i}$
(34) $\displaystyle\mathsf{P_{W}}(s,z,q,q^{\prime})=$
$\displaystyle\left(\sqrt{1+\dfrac{z^{2}}{\mu_{o}^{2}c^{2}}}+\dfrac{V_{s}^{(W)}(q,q^{\prime})}{2\mu_{o}c^{2}}\right)$
(35) $V_{s}^{(W)}(q,q^{\prime})=V\left(\frac{q+q^{\prime}}{2}\right)-V(s).$
(36)

The factor ${{}_{0}}F_{1}(;a;z)$ in Eq. (33) is a specific hypergeometric
function, and the contour $R$ is a circle of radius $r<\mu_{o}c$ that encloses
the pole at $z=0$, while $f_{i\rightarrow-i}$ denotes changing $i$ to $-i$ of
the first term in Eq. (34). The TKF given by Eqs. (31)-(36) reduces to the
known kernel for Weyl-quantized non-relativistic TOA operator Eq. (15) in the
limit $c\rightarrow\infty$. See Appendix A for details.

### Modified Born-Jordan-ordered TOA-operator

Repeating the same steps yields

$\displaystyle T^{\\{BJ\\}}(q,q^{\prime})=$
$\displaystyle\dfrac{1}{2(q-q^{\prime})}\int_{0}^{q}ds\int_{0}^{s}du\mathsf{B}_{s,u}(q,q^{\prime})-\dfrac{1}{2(q-q^{\prime})}\int_{0}^{q^{\prime}}ds\int_{0}^{s}du\mathsf{B}_{s,u}(q,q^{\prime})$
(37)

where,

$\displaystyle\mathsf{B}_{s,u}(q,q^{\prime})=\mathsf{B}_{s,u}^{(1)}(q,q^{\prime})+\dfrac{2}{\pi}\int_{1}^{\infty}dz\exp[-\dfrac{\mu_{o}c}{\hbar}\absolutevalue{q-q^{\prime}}z]\dfrac{\sqrt{z^{2}-1}}{z}\mathsf{B}_{s,u,z}^{(2)}(q,q^{\prime})$
(38)

in which

$\displaystyle\mathsf{B}_{s,u}^{(1)}(q,q^{\prime})=\int_{0}^{\infty}dye^{-y}$
$\displaystyle\oint_{R}\dfrac{dz}{2\pi
i}\dfrac{1}{z}\sqrt{1+\dfrac{z^{2}}{\mu_{o}^{2}c^{2}}}$
$\displaystyle\times{{}_{0}}F_{1}\left[;1;\dfrac{\mu_{o}V_{s}^{(BJ)}(u)}{2\hbar^{2}}\left(\left(q-q^{\prime}\right)-i\hbar\dfrac{y}{z}\right)^{2}\mathsf{P_{BJ}}(s,u,z,q,q^{\prime})\right]$
(39) $\displaystyle\mathsf{B}_{s,u,z}^{(2)}(q,q^{\prime})=$
$\displaystyle\dfrac{1}{2}\left[1-\dfrac{1}{z^{2}}\left(\dfrac{V_{s}^{(BJ)}(u)}{\mu_{o}c^{2}}\right)^{2}+2i\dfrac{\sqrt{z^{2}-1}}{z^{2}}\left(\dfrac{V_{s}^{(BJ)}(u)}{\mu_{o}c^{2}}\right)\right]^{-1/2}+f_{i\rightarrow-i}$
(40) $\displaystyle\mathsf{P_{BJ}}(s,u,z,q,q^{\prime})=$
$\displaystyle\left(\sqrt{1+\dfrac{z^{2}}{\mu_{o}^{2}c^{2}}}+\dfrac{V^{(BJ)}(s,u)}{2\mu_{o}c^{2}}\right)$
(41) $V^{(BJ)}(s,u)=V(s)-V(u).$ (42)

The TKF given by Eqs. (37)-(42) also reduces to the known kernel for Born-
Jordan quantized non-relativistic TOA operator Eq. (16) in the limit
$c\rightarrow\infty$.

### Modified simple-symmetric-ordered TOA-operator

Last, we have

$\displaystyle T^{\\{SS\\}}(q,q^{\prime})=$
$\displaystyle\dfrac{1}{4}\int_{0}^{q}ds\mathsf{S}(s,q)+\dfrac{1}{4}\int_{0}^{q^{\prime}}ds\mathsf{S}(s,q^{\prime})$
(43)

where

$\displaystyle\mathsf{S}(s,x)=\mathsf{S}^{(1)}(s,x)+\dfrac{2}{\pi}\int_{1}^{\infty}dz\exp[-\dfrac{\mu_{o}c}{\hbar}\absolutevalue{q-q^{\prime}}z]\dfrac{\sqrt{z^{2}-1}}{z}\mathsf{S}_{z}^{(2)}(s,x)$
(44)

in which

$\displaystyle\mathsf{S}^{(1)}(s,x)=\int_{0}^{\infty}dye^{-y}$
$\displaystyle\oint_{R}\dfrac{dz}{2\pi
i}\dfrac{1}{z}\sqrt{1+\dfrac{z^{2}}{\mu_{o}^{2}c^{2}}}$
$\displaystyle\times{{}_{0}}F_{1}\left[;1;\dfrac{\mu_{o}V^{(SS)}(s,x)}{2\hbar^{2}}\left(\left(q-q^{\prime}\right)-i\hbar\dfrac{y}{z}\right)^{2}\mathsf{P_{SS}}(s,z,x)\right]$
(45) $\displaystyle\mathsf{S}_{z}^{(2)}(s,x)=$
$\displaystyle\dfrac{1}{2}\left[1-\dfrac{1}{z^{2}}\left(\dfrac{V^{(SS)}(s,x)}{\mu_{o}c^{2}}\right)^{2}+2i\dfrac{\sqrt{z^{2}-1}}{z^{2}}\left(\dfrac{V^{(SS)}(s,x)}{\mu_{o}c^{2}}\right)\right]^{-1/2}+f_{i\rightarrow-i}$
(46) $\displaystyle\mathsf{P_{SS}}(s,z,x)=$
$\displaystyle\left(\sqrt{1+\dfrac{z^{2}}{\mu_{o}^{2}c^{2}}}+\dfrac{V^{(SS)}(s,x)}{2\mu_{o}c^{2}}\right)$
(47) $V^{(SS)}(s,x)=V(x)-V(s).$ (48)

The TKF given by Eqs. (43)-(48) also reduces to the known kernel for simple-
symmetric quantized non-relativistic TOA-operator Eq. (17) in the limit
$c\rightarrow\infty$.

In general, a closed form expression for the relativistic TKFs
$T^{\\{W\\}}(q,q^{\prime})$, $T^{\\{BJ\\}}(q,q^{\prime})$, and
$T^{\\{SS\\}}(q,q^{\prime})$ may be intractable because of how we assigned a
finite value to the divergent integral Eq. (26). It is possible that a
tractable form may be obtained using a different assignment to the divergent
integral. However, we justify the use of $T^{\\{W\\}}(q,q^{\prime})$,
$T^{\\{BJ\\}}(q,q^{\prime})$, and $T^{\\{SS\\}}(q,q^{\prime})$ because it
reduces to the non-relativistic time kernel[40].

##  IV Barrier traversal time operator

We use the measurement scheme shown in Fig. 2. Two detectors $D_{T}$ and
$D_{R}$ are placed at the arrival point $q=0$ and in the far left,
respectively. A square potential barrier of height $V(q)=V_{o}>0$ and length
$L=a-b$ is then placed between the detectors $a<q<b<0$. Next, a wavepacket
$\psi(q)$ initially centered at $q=q_{o}$ with momentum $p_{o}$ is placed
between $D_{R}$ and the barrier such that the tail of $\psi(q)$ does not
initially ‘leak’ into the barrier. The wavepacket is then launched at $t=0$
towards $D_{T}$ which records the arrival of the particle while the detector
$D_{R}$ does not record any data. This is done to avoid altering the
propagation of $\psi(q)$ and provide an indirect but accurately realistic way
of obtaining the TOA of the particle at the origin [15, 70, 71]. The same
measurement scheme is employed in the absence of the barrier.

The measurement is repeated several times for an ensemble of identically
prepared particles to obtain a TOA distribution at $D_{T}$. We assume that the
measured TOA distribution has an ideal distribution generated by the spectral
resolution of a corresponding TOA-operator $\mathsf{\hat{T}_{F}}$ and
$\mathsf{\hat{T}_{B}}$ in the absence and presence of the potential barrier,
respectively. In the succeeding expressions, the subscript $F$ ($B$) will
indicate the case when the barrier is absent (present). The traversal time
across the barrier is then deduced from the difference of the average value of
the measured TOA

$\Delta\bar{\tau}=\bar{\tau}_{F}-\bar{\tau}_{B}=\matrixelement{\psi}{\mathsf{\hat{T}_{F}}}{\psi}-\matrixelement{\psi}{\mathsf{\hat{T}_{B}}}{\psi}$
(49)

and is assumed to be the expectation value of the TOA-operator.

Figure 2: Measurement scheme for the traversal time of a particle in the (a)
absence of a barrier, and (b) presence of a barrier. The wavepacket
$\psi_{o}(q)$ is prepared between the detectors $D_{R}$ and $D_{T}$ such that
its tails does not extend to the barrier region.

In the absence of the barrier, the relativistic TKFs
$T^{\\{W\\}}_{F}(q,q^{\prime})$, $T^{\\{BJ\\}}_{F}(q,q^{\prime})$, and
$T^{\\{SS\\}}_{F}(q,q^{\prime})$ is obtained by substituting $V(q)=0$ into
Eqs. (31), (37) and (43), respectively. All ordering rules will yield the same
TKF

$\tilde{T}_{F}(\eta,\zeta)=\frac{\eta}{2}\mathsf{T}_{F}(\zeta),$ (50)

where,

$\mathsf{T}_{F}(\zeta)=1+\dfrac{2}{\pi}\int_{1}^{\infty}dz\dfrac{\sqrt{z^{2}-1}}{z}\exp(-\dfrac{\mu_{o}c}{\hbar}\absolutevalue{\zeta}z).$
(51)

The operator corresponding to the TKF $\tilde{T}_{F}(\eta,\zeta)$ coincides
with the Rigged Hilbert space extension of Razavi’s relativistic free TOA-
operator [68]

$\displaystyle(\mathsf{\hat{T}_{\text{Ra}}}\phi)(q)=\int_{-\infty}^{\infty}dq^{\prime}\matrixelement{q}{\mathsf{\hat{T}_{\text{Ra}}}}{q^{\prime}}\phi(q^{\prime})$
(52)

wherein, it was shown that the physical quantities associated with Eq. (52)
are consistent with special relativity. Now, the TKFs
$T^{\\{W\\}}(q,q^{\prime})$, $T^{\\{BJ\\}}(q,q^{\prime})$, and
$T^{\\{SS\\}}(q,q^{\prime})$ were derived under the assumption that the
interaction potential $V(q)$ is analytic. However, it can still be applied to
piecewise potentials such as the square barrier because the TKFs are in
integral form. We will justify this assumption later by establishing that in
the classical limit $\hbar\rightarrow 0$, the operator for the square
potential barrier corresponding to the TKFs reduce to its “classical”
relativistic TOA.

The TKF using the modified Wey ordering Eq. (31) may be obtained by mapping
the potential $V(q)$ from $(q,q^{\prime})$ coordinates into three non-
overlapping regions in the $(\eta,\zeta)$ coordinate wherein
$\eta=(q+q^{\prime})/2$ and $\zeta=q-q^{\prime}$. In this coordinate system,
the arrival point is now at $\eta=0$ and $V(\eta)=V_{o}$ for $a<\eta<b<0$ and
zero outside the interval $(a,b)$. For Region I, it is easy to see that
$V(\eta)=0$ for the entire integration region of Eq. (31). Meanwhile, for
Region II, we now have $V(\eta)=V_{o}$ and it is necessary to split the
integral Eq. (31) into two parts as $V(s)=0$ for $b<s<0$ while $V(s)=V_{o}$
for $\eta<s<b$. Last, for Region III we have $V(\eta)=0$ and split the
integral Eq. (31) into three parts as $V(s)=V_{o}$ for $a<s<b$ while $V(s)=0$
outside this interval. Performing these operations will yield

$\displaystyle\tilde{T}_{B}^{(I)}(\eta,\zeta)=$
$\displaystyle\dfrac{\eta}{2}\mathsf{T}_{F}(\zeta)$
$\displaystyle\tilde{T}_{B}^{(II)}(\eta,\zeta)=$
$\displaystyle\left(\dfrac{\eta+b}{2}\right)\mathsf{T}_{F}(\zeta)-\dfrac{b}{2}\mathsf{T}_{B}(V_{o},\zeta)$
(53) $\displaystyle\tilde{T}_{B}^{(III)}(\eta,\zeta)=$
$\displaystyle\left(\dfrac{\eta+L}{2}\right)\mathsf{T}_{F}(\zeta)-\dfrac{L}{2}\mathsf{T}_{B}(-V_{o},\zeta)$

in which $\mathsf{T}_{F}(\zeta)$ is given by Eq. (51) and

$\displaystyle\mathsf{T}_{B}(V_{o},\zeta)=$
$\displaystyle\mathsf{F}_{B}(V_{o},\zeta)+\dfrac{2}{\pi}\int_{1}^{\infty}dz\exp[-\dfrac{\mu_{o}c}{\hbar}\absolutevalue{\zeta}z]\dfrac{\sqrt{z^{2}-1}}{z}\mathsf{G}_{B}(V_{o},z).$
(54) $\displaystyle\mathsf{F}_{B}(V_{o},\zeta)=$
$\displaystyle\int_{0}^{\infty}dye^{-y}\oint_{R}\dfrac{dz}{2\pi
i}\dfrac{1}{z}\sqrt{1+\dfrac{z^{2}}{\mu_{o}^{2}c^{2}}}{{}_{0}}F_{1}\left[;1;\dfrac{\mu_{o}V_{o}}{2\hbar^{2}}\left(\zeta-i\hbar\dfrac{y}{z}\right)^{2}\left(\sqrt{1+\dfrac{z^{2}}{\mu_{o}^{2}c^{2}}}+\dfrac{V_{o}}{2\mu_{o}c^{2}}\right)\right]$
(55) $\displaystyle\mathsf{G}_{B}(V_{o},z)=$
$\displaystyle\dfrac{1}{2}\left\\{\left[1-\dfrac{1}{z^{2}}\left(\dfrac{V_{o}}{\mu_{o}c^{2}}\right)^{2}+2i\dfrac{\sqrt{z^{2}-1}}{z^{2}}\left(\dfrac{V_{o}}{\mu_{o}c^{2}}\right)\right]^{-1/2}+g_{i\rightarrow-i}\right\\}.$
(56)

We now work in the original $(q,q^{\prime})$-coordinate of our system to
evaluate the TKF $T^{\\{BJ\\}}(q,q^{\prime})$ given by Eqs. (37)-(41) and
later transform to the coordinates $(\eta,\zeta)$. For Region I, $V(q)=0$ for
the entire integration region of Eq. (37). Meanwhile, for Region II, it is
necessary to split the integral over $u$ of Eq. (37) into two parts as
$V(u)=0$ for $b<u<0$ while $V(u)=V_{o}$ for $s<u<b$ while $V(s)=V_{o}$ over
the whole region of $s$. We again repeat the same steps for Region III, and
split the integral over $u$ of Eq. (37) into three parts as $V(u)=V_{o}$ for
$a<u<b$ while $V(u)=0$ outside this interval. Then, $V(s)=V_{o}$ over the
whole region of $s$ in Eq. (37). Performing these operations and transforming
into the coordinates $(\eta,\zeta)$ will yield the same TKFs as Eq. (53).
Repeating the same procedure will yield the the same TKFs
$T^{\\{SS\\}}(q,q^{\prime})$. In the succeeding discussion, we shall only
refer to the modified Weyl-ordered operator since the same results will also
hold for the Born-Jordan and simple-symmetric case.

##  V Classical limit of the free and barrier TKFs

We now prove that the TKFs corresponding to the TOA-operator for the free and
barrier case are indeed the quantization of the CRTOA by taking their inverse
Weyl-Wigner transform

$\tilde{t}(q_{o},p_{o})=\dfrac{\mu_{o}}{i\hbar}\int_{-\infty}^{\infty}d\zeta
e^{-ip_{o}\zeta/\hbar}\tilde{T}(q_{o},\zeta)\text{sgn}(\zeta)$ (57)

where, $q_{o}$ and $p_{o}$ are the initial position and momentum,
respectively. For the free case, this is done by substituting Eq. (50) to Eq.
(57) which yields

$\displaystyle\tilde{t}_{F}=\dfrac{\mu_{o}}{i\hbar}\dfrac{q_{o}}{2}\int_{-\infty}^{\infty}d\zeta
e^{-ip_{o}\zeta/\hbar}\text{sgn}(\zeta)+\dfrac{\mu_{o}}{i\hbar}\dfrac{q_{o}}{2}\dfrac{2}{\pi}\int_{1}^{\infty}dz\dfrac{\sqrt{z^{2}-1}}{z}\int_{-\infty}^{\infty}d\zeta\exp[-\dfrac{\mu_{o}c}{\hbar}\absolutevalue{\zeta}z]e^{-ip_{o}\zeta/\hbar}\text{sgn}(\zeta).$
(58)

The first term of Eq. (58) is evaluated by taking the inverse of the
distributional Fourier transform [72]

$\int_{-\infty}^{\infty}dxx^{-m}e^{i\sigma
x}=i^{m}\dfrac{\pi}{(m-1)!}\sigma^{m-1}\text{sgn}\sigma.$ (59)

Meanwhile, the order of integration for the second term of Eq. (58) are
interchanged, and the inner integral is evaluated as a Laplace transform. The
resulting expression is further evaluated using the integral identity [68]

$\int_{1}^{\infty}dz\dfrac{\sqrt{z^{2}-1}}{z}\dfrac{a^{2}}{a^{2}+b^{2}z^{2}}=\dfrac{\pi}{2}\left(-1+\sqrt{1+\dfrac{a^{2}}{b^{2}}}\right).$
(60)

for all real $(a,b)$, which can also be obtained using the calculus of
residues. Thus, the classical limit of the free TOA-operator corresponding to
$\tilde{T}_{F}(\eta,\zeta)$ is

$\displaystyle\tilde{t}_{F}=$
$\displaystyle-\dfrac{\mu_{o}q_{o}}{p_{o}}\sqrt{1+\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}},$
(61)

which is the known free CRTOA obtained from directly integrating Eq.(18).

In the presence of the potential barrier, it easily follows from Eq. (61) that
the classical limit of the TKF for Region I $\tilde{T}_{B}^{(I)}(\eta,\zeta)$
is $\tilde{t}_{B}^{(I)}=\tilde{t}_{F}$. For Region II, the Weyl-Wigner
transform of the TKF $\tilde{T}_{B}^{(II)}(\eta,\zeta)$ is

$\displaystyle\tilde{t}_{B}^{(II)}=$
$\displaystyle-\dfrac{\mu_{o}(q_{o}+b)}{p}\sqrt{1+\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}}-\dfrac{b}{2}\dfrac{\mu_{o}}{i\hbar}\int_{-\infty}^{\infty}d\zeta
e^{-ip_{o}\zeta/\hbar}\mathsf{T}_{B}(V_{o},\zeta)\text{sgn}(\zeta),$ (62)

wherein

$\displaystyle\int_{-\infty}^{\infty}$ $\displaystyle d\zeta
e^{-ip_{o}\zeta/\hbar}\mathsf{T}_{B}(V_{o},\zeta)\text{sgn}(\zeta)$
$\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}d\zeta
e^{-ip_{o}\zeta/\hbar}\mathsf{F}_{B}(V_{o},\zeta)\text{sgn}(\zeta)+\left(\dfrac{2\hbar}{ip_{o}}\right)\dfrac{2}{\pi}\int_{1}^{\infty}dz\mathsf{G}_{B}(V_{o},z)\dfrac{\sqrt{z^{2}-1}}{z}\dfrac{p_{o}^{2}}{p_{o}^{2}+\mu_{o}^{2}c^{2}z^{2}}.$
(63)

The first term of Eq. (63) is evaluated by expanding the hypergeometric
function in $\mathsf{F}_{B}(V_{o},\zeta)$ using its power series
representation to perform a term-by-term integration. The resulting series
converges as long the initial energy of the particle is above the barrier
height, i.e.

$\displaystyle\int_{-\infty}^{\infty}$ $\displaystyle d\zeta
e^{-ip_{o}\zeta/\hbar}\mathsf{F}_{B}(V_{o},\zeta)\text{sgn}(\zeta)$
$\displaystyle=$
$\displaystyle\dfrac{2\hbar}{ip_{o}}\sum_{j=0}^{\infty}\dfrac{(2j)!}{j!j!}\left(\dfrac{-\mu_{o}V_{o}}{2\hbar^{2}}\right)^{j}\sum_{k=0}^{j}\binom{j}{k}\left(\dfrac{V_{o}}{2\mu_{o}c^{2}}\right)^{j-k}$
$\displaystyle\times\left\\{\sqrt{1+\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}}^{k+1}-\left(\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}\right)^{j+1}\binom{\frac{k+1}{2}}{j+1}{{}_{2}}F_{1}\left[1,\frac{1}{2}+j-\frac{k}{2};j+2;-\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}\right]\right\\}$
$\displaystyle=$
$\displaystyle\left(\dfrac{2\hbar}{ip_{o}}\right)\sqrt{1+\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}}\left[1+\dfrac{2\mu_{o}V_{o}}{p_{o}^{2}}\left(\sqrt{1+\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}}+\dfrac{V_{o}}{2\mu_{o}c^{2}}\right)\right]^{-1/2}$
$\displaystyle-\left(\dfrac{2\hbar}{ip_{o}}\right)\dfrac{2}{\pi}\int_{1}^{\infty}dz\mathsf{G}_{B}(V_{o},z)\dfrac{\sqrt{z^{2}-1}}{z}\dfrac{p_{o}^{2}}{p_{o}^{2}+\mu_{o}^{2}c^{2}z^{2}}$
(64)

The second line follows from using the integral representation of the Gauss
hypergeometric function

${{}_{2}}F_{1}(\alpha,\beta;\gamma;z)=\dfrac{\Gamma(\gamma)}{\Gamma(\beta)\Gamma(\gamma-\beta)}\int_{0}^{\infty}dt\dfrac{t^{c-b-1}(1+t)^{a-c}}{(t+1-z)^{a}}$
(65)

for $Re[\gamma]>\real[\beta]>0$ and $\absolutevalue{\text{Arg}(1-z)}<\pi$.
Combining Eqs. (62)-(64) thus yields

$\displaystyle\tilde{t}_{B}^{(II)}=$
$\displaystyle-\dfrac{\mu_{o}(q_{o}+b)}{p_{o}}\sqrt{1+\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}}+\dfrac{b}{c}\sqrt{\dfrac{1+\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}}{\left(\sqrt{1+\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}}+\dfrac{V_{o}}{\mu_{o}c^{2}}\right)^{2}-1}}.$
(66)

The first term of $\tilde{t}_{B}^{(II)}$ is the free CRTOA from the edge of
the barrier to the origin while the second term is the traversal time on top
of the barrier. Repeating the same steps, the Weyl-Wigner transform of
$\tilde{T}_{B}^{(III)}(\eta,\zeta)$ is

$\displaystyle\tilde{t}_{B}^{(III)}=$
$\displaystyle-\dfrac{\mu_{o}(q_{o}+L)}{p_{o}}\sqrt{1+\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}}+\dfrac{L}{c}\sqrt{\dfrac{1+\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}}{\left(\sqrt{1+\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}}-\dfrac{V_{o}}{\mu_{o}c^{2}}\right)^{2}-1}}.$
(67)

The first term of $\tilde{t}_{B}^{(III)}$ is the traversal time across the
interaction free region while the second term is the traversal time across the
barrier region. The Weyl-Wigner transforms $\tilde{t}_{B}^{(II)}$ and
$\tilde{t}_{B}^{(III)}$ also coincide with CRTOA obtained from directly
integrating Eq. (18).

In general, the classical limit of the TKF for a given quantization scheme is
obtained by

$\tilde{t}(q_{o},p_{o})=\lim_{\hbar\rightarrow
0}\dfrac{\mu_{o}}{i\hbar}\int_{-\infty}^{\infty}d\zeta
e^{-ip_{o}\zeta/\hbar}\tilde{T}^{\\{Q\\}}(q_{o},\zeta)\text{sgn}(\zeta),$ (68)

wherein the integral is understood in a distributional sense, provided that
the limit exists [40]. Notice that the Weyl-Wigner transform Eq. (57) does not
involve the vanishing of $\hbar$. Now, Eq. (68) implies that the classical
limit of the TKF for a given quantization scheme is, in general, dependent on
positive powers of $\hbar$. Such is the case for the Born-Jordan and simple-
symmetric ordering. Performing the limit $\hbar\rightarrow 0$ then reduces to
classical limits of the TKFs $T^{\\{BJ\\}}(q,q^{\prime})$ and
$T^{\\{SS\\}}(q,q^{\prime})$ into that equal to the Weyl-Wigner transform of
$T^{\\{W\\}}(q,q^{\prime})$.

##  VI Expected Barrier Traversal Time

We now assume that the average value of the measured TOA $\bar{\tau}$ at the
detector $D_{T}$ (see Fig. 2) is equal to the expectation value of the
operator $\mathsf{\hat{T}}$, i.e.

$\displaystyle\bar{\tau}=$
$\displaystyle\matrixelement{\psi}{\mathsf{\hat{T}}}{\psi}=\int_{-\infty}^{\infty}dq\psi^{*}(q)\int_{-\infty}^{\infty}dq^{\prime}\dfrac{\mu_{o}}{i\hbar}T(q,q^{\prime})\text{sgn}(q-q^{\prime})\psi(q^{\prime}).$
(69)

The incident wavefunction is assumed to be prepared in a pure state
$\psi(q)=\varphi(q)e^{ik_{o}q}$ with momentum expectation value $p_{o}=\hbar
k_{o}$, where $\matrixelement{\varphi}{\mathsf{\hat{p}}}{\varphi}=0$. We
further assume that $\varphi(q)$ is infinitely differentiable and impose the
condition that the support of $\varphi(q)$ is in Region III such that the tail
of $\varphi(q)$ does not ’leak’ into the barrier. To evaluate Eq. (69), it
will be convenient to perform a change of variables from $(q,q^{\prime})$ to
$(\eta,\zeta)$ such that $\bar{\tau}=\imaginary(\bar{\tau}^{*})$ wherein
$\bar{\tau}^{*}$ is the complex-valued TOA given by

$\displaystyle\bar{\tau}^{*}=-\dfrac{2\mu_{o}}{\hbar}\int_{-\infty}^{\infty}$
$\displaystyle d\eta\int_{0}^{\infty}d\zeta
e^{ik_{o}\zeta}\tilde{T}(\eta,\zeta)\varphi^{*}\left(\eta-\dfrac{\zeta}{2}\right)\varphi\left(\eta+\dfrac{\zeta}{2}\right).$
(70)

In the succeeding expressions, we indicate complex-valued quantities with an
asterisk ∗ wherein the imaginary component corresponds to the physical
quantity.

In the absence of the barrier, it easily follows from Eqs. (50) and (70) that
the complex-valued free TOA is

$\displaystyle\bar{\tau}_{F}^{*}=-\dfrac{\mu_{o}}{\hbar}$
$\displaystyle\int_{0}^{\infty}d\zeta
e^{ik_{o}\zeta}\mathcal{T}_{F}(\zeta)\int_{-\infty}^{\infty}d\eta\eta\varphi^{*}\left(\eta-\dfrac{\zeta}{2}\right)\varphi\left(\eta+\dfrac{\zeta}{2}\right).$
(71)

Meanwhile, in the presence of the barrier, we have

$\displaystyle\bar{\tau}_{B}^{*}=$
$\displaystyle-\dfrac{\mu_{o}}{\hbar}\int_{0}^{\infty}d\zeta
e^{ik_{o}\zeta}\int_{-\infty}^{\infty}d\eta\tilde{T}_{B}^{(III)}(\eta,\zeta)\varphi^{*}\left(\eta-\dfrac{\zeta}{2}\right)\varphi\left(\eta+\dfrac{\zeta}{2}\right)$
$\displaystyle=$
$\displaystyle\bar{\tau}_{F}^{*}-\dfrac{\mu_{o}L}{\hbar}\int_{0}^{\infty}d\zeta
e^{ik_{o}\zeta}(\mathsf{T}_{F}(\zeta)-\mathsf{T}_{B}(-V_{0},\zeta))\int_{-\infty}^{\infty}d\eta\varphi^{*}\left(\eta-\dfrac{\zeta}{2}\right)\varphi\left(\eta+\dfrac{\zeta}{2}\right).$
(72)

The measurable quantity for deducing the barrier traversal time is the TOA
difference between the free and barrier case
$\Delta\bar{\tau}=\imaginary(\Delta\bar{\tau}^{*})=\imaginary(\bar{\tau}_{F}^{*}-\bar{\tau}_{B}^{*})$,
which is explicitly given as

$\displaystyle\Delta\bar{\tau}^{*}=$
$\displaystyle\dfrac{\mu_{o}L}{p_{o}}\left(Q_{c}^{*}-R_{c}^{*}\right)$ (73)

wherein

$\displaystyle Q_{c}^{*}=$ $\displaystyle k_{o}\int_{0}^{\infty}d\zeta
e^{ik_{o}\zeta}\mathsf{T}_{F}(\zeta)\Phi(\zeta)$ (74) $\displaystyle
R_{c}^{*}=$ $\displaystyle k_{o}\int_{0}^{\infty}d\zeta
e^{ik_{o}\zeta}\mathsf{T}_{B}(-V_{0},\zeta)\Phi(\zeta)$ (75)
$\displaystyle\Phi(\zeta)=$
$\displaystyle\int_{-\infty}^{\infty}d\eta\varphi^{*}\left(\eta-\dfrac{\zeta}{2}\right)\varphi\left(\eta+\dfrac{\zeta}{2}\right).$
(76)

The complex-valued dimensionless quantities $Q_{c}^{*}$ and $R_{c}^{*}$
accounts for the contribution of the barrier and relativistic effects on the
non-relativistic free TOA $\mu_{o}L/p_{o}$. The physical content of the
quantities $Q_{c}$ and $R_{c}$ are investigated by taking the asymptotic
expansion in the high energy limit $k_{o}\rightarrow\infty$.

It is easy to see that if we substitute Eq. (51) to Eq. (74), then it follows
that the quantity $(\mu_{o}L/p_{o})Q_{c}$ is just the expectation value of the
free relativistic TOA-operator calculated by Flores and Galapon [73]. Thus,

$Q_{c}\sim\sqrt{1+\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}}.$ (77)

which is the relativistic correction to the non-relativistic free TOA
$\mu_{o}L/p_{o}$. Now, the quantity $R_{c}^{*}$ is a Fourier integral with
respect to the asymptotic parameter $k_{o}$. We use the same steps outlined in
Sec. IV for the calculation of the Weyl-Wigner transform of the TKF
$\tilde{T}_{B}^{(III)}(\eta,\zeta)$, and perform repeated integration-by-parts
to collect powers of $\hbar$. Taking the imaginary part of $R_{c}^{*}$ thus
yields

$\displaystyle\imaginary[R_{c}^{*}]\sim$
$\displaystyle\sum_{m=0}^{\infty}\Phi^{(2m)}(0)\dfrac{(-1)^{m}\hbar^{2m}}{p_{o}^{2m}}\sum_{j=0}^{\infty}\dfrac{(2j)!}{(1)_{j}j!}\left(\dfrac{\mu_{o}V_{o}}{2p_{o}^{2}}\right)^{j}\sum_{k=0}^{j}\binom{j}{k}\left\\{\left(-\dfrac{V_{o}}{2\mu_{o}c^{2}}\right)^{j-k}\right.$
$\displaystyle\times\left.\sum_{l=0}^{j}\binom{\frac{k+1}{2}}{l}\binom{2m+2j-2l}{2j-2l}\left(\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}\right)^{l}\right\\}+\dfrac{2}{\pi}\int_{1}^{\infty}dz\dfrac{\left(\frac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}\right)}{z^{2}+{\left(\frac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}\right)}}\dfrac{\sqrt{z^{2}-1}}{z}\mathsf{G}(V_{o},z)$
(78) $\displaystyle\sim$
$\displaystyle\sum_{j=0}^{\infty}\dfrac{(2j)!}{(1)_{j}j!}\left(\dfrac{\mu_{o}V_{o}}{2p_{o}^{2}}\right)^{j}\sum_{k=0}^{j}\binom{j}{k}\left(-\dfrac{V_{o}}{2\mu_{o}c^{2}}\right)^{j-k}\left\\{\sqrt{1+\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}}^{k+1}\right.$
$\displaystyle-\left.\left(\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}\right)^{j+1}\binom{\frac{k+1}{2}}{j+1}{{}_{2}}F_{1}\left[1,\frac{1}{2}+j-\frac{k}{2};j+2;-\dfrac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}\right]\right\\}$
$\displaystyle+\dfrac{2}{\pi}\int_{1}^{\infty}dz\dfrac{\left(\frac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}\right)}{z^{2}+{\left(\frac{p_{o}^{2}}{\mu_{o}^{2}c^{2}}\right)}}\dfrac{\sqrt{z^{2}-1}}{z}\mathsf{G}(V_{o},z)$
(79)

The second line Eq. (79) follows from the classical limit $\hbar\rightarrow 0$
in which only the terms with $m=0$ will not vanish, wherein we used the
normalization conditions $\Phi(0)=1$. The integral representation of the Gauss
hypergeometric function, Eq. (65), is again used to perform the summation
which yields

$R_{c}\sim\dfrac{p_{o}}{\mu_{o}c}\sqrt{\dfrac{E_{p}^{2}}{(E_{p}-V_{o})^{2}-\mu_{o}^{2}c^{4}}}$
(80)

where $E_{p}=\sqrt{p^{2}c^{2}+\mu_{o}^{2}c^{4}}$. Thus, $R_{c}$ is just the
ratio of the energy of the incident particle and its energy above the barrier.
This leads us to the interpretation that $R_{c}$ is the effective index of
refraction (IOR) of the barrier with respect to the wavepacket. The same
interpretation was made in the non-relativistic case for the square potential
barrier and well [15, 74]. This implies that the traversal time across the
barrier is given by $\bar{\tau}_{\text{trav}}=(\mu_{o}L/p_{o})R_{c}$.

Figure 3: Contour of integration for Eq. (86) leading to the interchange of
the order of integration in Eq. (87) Figure 4: Contours of integration of Eq.
(90) for the (a) the first integral when $V_{o}/\hbar c<\mu_{o}c/\hbar$, and
(b) the second integral

We now establish the expected traversal time across the potential barrier and
use the same notations as that of Galapon [15] for consistency. To evaluate
the complex-valued IOR Eq. (75), we introduce the inverse Fourier transform of
the wavepacket
$\varphi(q)=(2\pi)^{-1}\int_{-\infty}^{\infty}d\tilde{k}e^{i\tilde{k}q}\phi(\tilde{k})$
such that

$\Phi(\zeta)=\int_{-\infty}^{\infty}d\tilde{k}|\phi(\tilde{k})|^{2}e^{i\tilde{k}\zeta}.$
(81)

Substituting Eq. (81) to Eq. (75), and performing a change of variable
$\tilde{k}=k-k_{o}$ yields

$\dfrac{R_{c}^{*}}{k_{o}}=\int_{0}^{\infty}d\zeta\mathsf{T}_{B}(-V_{0},\zeta)\int_{-\infty}^{\infty}dk|\phi(k-k_{o})|^{2}e^{ik\zeta}$
(82)

Notice that $\phi(k-k_{o})$ is the Fourier transform of the full incident
wavefunction $\psi(q)=e^{ik_{o}q}\varphi(q)$, i.e.

$\phi(k-k_{o})=\tilde{\psi}(k)=\dfrac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dqe^{-ikq}\psi(q)$
(83)

Thus, we have

$\displaystyle\dfrac{R_{c}^{*}}{k_{o}}=\int_{0}^{\infty}$ $\displaystyle
d\zeta\mathsf{T}_{B}(-V_{0},\zeta)\int_{-\infty}^{\infty}dke^{ik\zeta}\absolutevalue{\tilde{\psi}(k)}^{2}$
(84) $\displaystyle=\int_{0}^{\infty}$ $\displaystyle
d\zeta\mathsf{F}_{B}(-V_{0},\zeta)\int_{-\infty}^{\infty}dke^{ik\zeta}\absolutevalue{\tilde{\psi}(k)}^{2}+\dfrac{2}{\pi}\int_{1}^{\infty}dy\dfrac{\sqrt{y^{2}-1}}{y}\mathsf{G}_{B}(V_{o},y)\int_{-\infty}^{\infty}dk\dfrac{\frac{\mu_{o}c}{\hbar}y}{k^{2}+\frac{\mu_{o}^{2}c^{2}}{\hbar^{2}}y^{2}}\absolutevalue{\tilde{\psi}(k)}^{2}.$
(85)

The last line follows from interchanging the order of integration in the
second term of Eq. (85) but the same cannot be done on the first term.
Specifically, if we use the same steps outlined in Sec. IV to perform a term-
by-term integration on the first term of Eq. (85), then this will lead to an
infinite sum of divergent integrals whose values may be assigned using
analytic continuation, regularization, and many others. However, it was
recently shown by one of us that this naive interchange in the ordering of
integrals leading to divergent integrals sometimes miss significant terms [75,
76]. This was shown to have physical significance in the traversal time of a
non-relativistic particle across a potential well [74].

Table 1: Numerical verification of $\tilde{R}_{c}$ for spatially narrow Gaussian wavepackets $\sigma=0.5$ when there are above and below barrier components | Integral: Eq. (75) | Summation: Eq. (100) | Evaluated: Eq. (94)
---|---|---|---
$k_{o}=2.00;V_{o}=0.2$ | 1.32442 | 1.32442 | 1.32442
$k_{o}=2.00;V_{o}=0.3$ | 1.38141 | 1.38141 | 1.38141
$k_{o}=2.00;V_{o}=0.5$ | — | 1.48255 | 1.48255
$k_{o}=2.00;V_{o}=0.6$ | — | 1.52350 | 1.52350
$k_{o}=0.90;V_{o}=0.3$ | 0.99882 | 0.99888 | 0.99888
$k_{o}=3.00;V_{o}=0.3$ | 1.24812 | 1.24812 | 1.24811
$k_{o}=5.00;V_{o}=0.3$ | 1.09394 | 1.09394 | 1.09393
$k_{o}=0.15;V_{o}=0.3$ | 0.18996 | 0.18996 | 0.18996
$k_{o}=0.20;V_{o}=0.3$ | 0.25253 | 0.25253 | 0.25253
$k_{o}=0.25;V_{o}=0.3$ | 0.31446 | 0.31446 | 0.31446
Table 2: Numerical verification of $\tilde{R}_{c}$ for spatially wide Gaussian wavepackets $\sigma=9.0$ and $V_{o}=0.3$ when there are only below barrier components  | Integral: Eq. (75) | Evaluated: Eq. (94)
---|---|---
$k_{o}=0.19$ | $2.23294\times 10^{-16}$ | $1.34410\times 10^{-29}$
$k_{o}=0.25$ | $1.77061\times 10^{-14}$ | $2.01917\times 10^{-24}$
$k_{o}=0.28$ | $1.84479\times 10^{-16}$ | $5.06286\times 10^{-22}$

To make the the interchange in the orders of integration on the the first term
of Eq. (85) valid, we use the methods of Pablico and Galapon [74] and use the
contour shown in Fig. 3. We let $p(z)=|\tilde{\psi}(z)|^{2}$ and assume that
$\tilde{\psi}(z)$ does not have any poles in the complex plane, i.e.

$\int_{-\infty}^{\infty}e^{ix\zeta}p(x)=\int_{-\infty}^{\infty}dxe^{-(\epsilon-
ix)\zeta}p(x+i\epsilon).$ (86)

This now makes

$\displaystyle\int_{0}^{\infty}$ $\displaystyle
d\zeta\mathsf{F}_{B}(-V_{0},\zeta)\int_{-\infty}^{\infty}dke^{ik\zeta}\absolutevalue{\tilde{\psi}(k)}^{2}=\int_{-\infty}^{\infty}dkp(k+i\epsilon)\int_{0}^{\infty}d\zeta\mathsf{F}_{B}(-V_{o},\zeta)e^{-(\epsilon-
ik)\zeta}.$ (87)

The interchange is valid provided that $\epsilon>k$. We can now use the series
representation of the hypergeometric function in
$\mathsf{F}_{B}(-V_{o},\zeta)$ and use the same methods outlined in Sec. IV.
This turns the first term of Eq. (85) into

$\displaystyle\int_{0}^{\infty}d\zeta$
$\displaystyle\mathsf{F}_{B}(-V_{0},\zeta)\int_{-\infty}^{\infty}dke^{ik\zeta}\absolutevalue{\tilde{\psi}(k)}^{2}$
$\displaystyle=i\sum_{n=0}^{\infty}$
$\displaystyle\dfrac{(2n)!}{(1)_{n}n!}\left(\dfrac{\mu_{o}V_{o}}{2\hbar^{2}}\right)^{n}\int_{-\infty}^{\infty}dkp(k+i\epsilon)\text{csgn}(k+i\epsilon)$
$\displaystyle\times\left(\sqrt{1+\dfrac{\hbar^{2}(k+i\epsilon)^{2}}{\mu_{o}^{2}c^{2}}}\right)^{n+1}\left((k+i\epsilon)^{2}+\dfrac{V_{o}^{2}}{\hbar^{2}c^{2}}\right)^{-n-\frac{1}{2}}$
$\displaystyle-\dfrac{2i}{\pi}$
$\displaystyle\int_{1}^{\infty}dy\dfrac{\sqrt{y^{2}-1}}{y}\mathsf{G}_{B}(V_{o},y)\int_{-\infty}^{\infty}dkp(k+i\epsilon)\dfrac{\frac{\hbar^{2}}{\mu^{2}c^{2}}(k+i\epsilon)}{y^{2}+\frac{\hbar^{2}}{\mu^{2}c^{2}}(k+i\epsilon)^{2}}$
(88)

where $\text{csgn}(z)$ is the complex signum function

$\displaystyle\text{csgn}(z)=\begin{cases}1\quad&,\text{Re}(z)>0\\\
-1\quad&,\text{Re}(z)<0\\\
\text{sgn}(\text{Im}(z))\quad&,\text{Re}(z)=0.\end{cases}$ (89)

To understand the physical content of Eq. (88), we consider the following
integral in the complex plane,

$\displaystyle\oint
dzp(z)\left(\sqrt{1+\dfrac{\hbar^{2}z^{2}}{\mu_{o}^{2}c^{2}}}\right)^{n+1}\left(z^{2}+\dfrac{V_{o}^{2}}{\hbar^{2}c^{2}}\right)^{-n-\frac{1}{2}}\quad\text{and}\quad\oint
dzp(z)\dfrac{\frac{\hbar^{2}}{\mu^{2}c^{2}}z}{y^{2}+\frac{\hbar^{2}}{\mu^{2}c^{2}}z^{2}},$
(90)

wherein the first integral has four branch points at $z=\\{\pm
i\frac{\mu_{o}c}{\hbar},\pm i\frac{V_{o}}{\hbar c}\\}$ while the second
integral has poles at $z=\pm i\frac{\mu c}{\hbar}$. We assume that the branch
points satisfy $V_{o}/\hbar c<\mu_{o}c/\hbar$ which is equivalent to the
condition $V_{o}<\mu_{o}c^{2}$. The integrals Eq. (90) are then evaluated
using the contours in Fig. 4 (see Appendix B for details) and the resulting
expressions are substituted to Eq. (85) which yields

$\displaystyle R_{c}^{*}=$ $\displaystyle i\dfrac{\hbar k_{o}}{\mu
c}\int_{0}^{\infty}dk\left(\absolutevalue{\tilde{\psi}(k)}^{2}-\absolutevalue{\tilde{\psi}(-k)}^{2}\right)\sqrt{\dfrac{\tilde{E}_{k}^{2}}{(\tilde{E}_{k}-V_{o})^{2}-\mu_{o}^{2}c^{4}}}$
$\displaystyle+k_{o}\dfrac{2}{\pi}\int_{1}^{\infty}dy\dfrac{\sqrt{y^{2}-1}}{y}\mathsf{G_{B}}(V_{o},y)\int_{-\infty}^{\infty}dk\absolutevalue{\tilde{\psi}(k)}^{2}\dfrac{\frac{\mu
c}{\hbar}y}{k^{2}+\frac{\mu^{2}c^{2}}{\hbar^{2}}y^{2}}$ (91)

in which, $\tilde{E}_{k}=\sqrt{\hbar^{2}k^{2}c^{2}+\mu_{o}^{2}c^{4}}$. It is
easy to see that the first term of Eq. (91) is generally complex-valued while
the second term is always real-valued. Thus, taking the imaginary component of
the IOR yields

$\imaginary[R_{c}^{*}]=\dfrac{\hbar k_{o}}{\mu_{o}c}\tilde{R}_{c}=\dfrac{\hbar
k_{o}}{\mu_{o}c}\text{Re}\left\\{\int_{0}^{\infty}dk\left(\absolutevalue{\tilde{\psi}(k)}^{2}-\absolutevalue{\tilde{\psi}(-k)}^{2}\right)\sqrt{\dfrac{\tilde{E}_{k}^{2}}{(\tilde{E}_{k}-V_{o})^{2}-\mu_{o}^{2}c^{4}}}\right\\}$
(92)

The right-hand side of Eq. (92) is only real-valued when
$\absolutevalue{k}>\kappa_{c}$, where

$\kappa_{c}=\sqrt{\dfrac{2\mu_{o}V_{o}}{\hbar^{2}}\left(1+\frac{V_{o}}{2\mu_{o}c^{2}}\right)}$
(93)

provided that $V_{o}<\mu_{o}c^{2}$. Thus, Eq. (92) becomes

$\tilde{R}_{c}=\tilde{R}_{c}^{(+)}-\tilde{R}_{c}^{(-)}=\int_{\kappa_{c}}^{\infty}dk\absolutevalue{\tilde{\psi}(+k)}^{2}\sqrt{\dfrac{\tilde{E}_{k}^{2}}{(\tilde{E}_{k}-V_{o})^{2}-\mu_{o}^{2}c^{4}}}-\int_{\kappa_{c}}^{\infty}dk\absolutevalue{\tilde{\psi}(-k)}^{2}\sqrt{\dfrac{\tilde{E}_{k}^{2}}{(\tilde{E}_{k}-V_{o})^{2}-\mu_{o}^{2}c^{4}}}$
(94)

It easily follows that the barrier traversal time now has the form

$\bar{\tau}_{\text{trav}}=\dfrac{\mu_{o}L}{p_{o}}\imaginary[R_{c}^{*}]=t_{c}\tilde{R}_{c},$
(95)

where, $t_{c}=L/c$ is the time it takes a photon to traverse the barrier
length. The term $\tilde{R}_{c}^{(+)}$ ($\tilde{R}_{c}^{(-)}$) characterizes
the contribution of the positive (negative) components of the energy
distribution of $\tilde{\psi}(k)$ with $\absolutevalue{k}>\kappa_{c}$ to the
effective IOR $\tilde{R}_{c}$. Clearly, the quantity

$\bar{\tau}_{\text{trav}}^{(\pm)}=t_{c}\tilde{R}_{c}^{(\pm)}=\int_{\kappa_{c}}^{\infty}dk\bar{\tau}_{\text{top}}(k)|\tilde{\psi}(\pm
k)|^{2}$ (96)

is the weighted average of the classical above barrier traversal time

$\bar{\tau}_{\text{top}}(k)=t_{c}\sqrt{\dfrac{\tilde{E}_{k}^{2}}{(\tilde{E}_{k}-V_{o})^{2}-\mu_{o}^{2}c^{4}}}$
(97)

with weights $|\tilde{\psi}(\pm k)|^{2}$. The effective IOR Eq. (94) shows
that the contribution of the below barrier energy components of
$\tilde{\psi}(k)$ with $\absolutevalue{k}<\kappa_{c}$ vanishes, which leads us
to the same conclusion as that of Galapon [15]. That is, the below barrier
energy components of $\tilde{\psi}(k)$ are transmitted instantaneously which
implies that tunneling, whenever it occurs, is instantaneous.

Thus, the instantaneous tunneling time predicted in Ref. [15] is not a mere
consequence of using a non-relativistic theory but is an inherent quantum
effect in the context of “arrival times” as it still manifests even with a
relativistic treatment. However, there is a specific configuration in a
tunneling experiment such that this instantaneous tunneling time can be
observed. Specifically, it is implied from Eq. (94) that the initial incident
wavepacket $\psi(q)$ must be sufficiently spatially wide so that the spread in
momentum is narrow. This will ensure that $\tilde{\psi}(k)$ only has below
barrier components. Additionally, Eq. (94) rests on the assumption that
$\psi(q)$ does not initially ‘leak’ inside the barrier region, as such, the
initial incident wavepacket must be placed very far from the barrier.

Figure 5: Momentum density distribution $|\tilde{\psi}(k)|^{2}$ of spatially
wide Gaussian wavepackets for the parameters $\mu_{o}=c=\hbar=1$ with
$k_{o}=1.3$. The red line represents $\kappa_{c}=1.7025$ with $V_{o}=0.99$.

##  VII Barrier traversal time of Gaussian wavepackets

We consider an incident Gaussian wavepacket , i.e.

$\varphi(q)=\dfrac{1}{\sqrt{\sigma\sqrt{2\pi}}}\exp[-\dfrac{(q-q_{o})^{2}}{4\sigma^{2}}].$
(98)

that is initially centered at $q=q_{o}$ with a position variance $\sigma^{2}$.
In momentum representation, this leads to

$\displaystyle\absolutevalue{\tilde{\psi}(\pm k)}^{2}=$
$\displaystyle\sqrt{\dfrac{2\sigma^{2}}{\pi}}\exp[-2\sigma^{2}(k\mp
k_{o})^{2}].$ (99)

For completeness, we first numerically verify the equivalence of Eqs. (94) and
Eq. (75). However, Eq. (75) is numerically taxing and unstable as the
potential $V_{o}$ increases, such that $\mathsf{T}_{B}(-V_{o},\zeta)$ must be
represented in an equivalent expression. This is done by using the power
series representation of the hypergeometric function in Eq. (55) to perform a
term-by-term integration which yields

$\displaystyle\mathsf{T}_{B}(-V_{o},\zeta)=$
$\displaystyle\sum_{l=0}^{\infty}\dfrac{(2l)!}{l!l!}\left(\dfrac{-\mu_{o}V_{o}}{2\hbar^{2}}\right)^{l}\sum_{m=0}^{l}\binom{l}{m}\left(\dfrac{-V_{o}}{2\mu_{o}c^{2}}\right)^{l-m}\sum_{n=0}^{l}\binom{\frac{m+1}{2}}{n}\left(\dfrac{-\hbar^{2}}{\mu_{o}^{2}c^{2}}\right)^{n}\dfrac{\zeta^{2l-2n}}{(2l-2n)!}$
$\displaystyle+\dfrac{2}{\pi}\int_{1}^{\infty}dz\exp[-\dfrac{\mu_{o}c}{\hbar}\absolutevalue{\zeta}z]\dfrac{\sqrt{z^{2}-1}}{z}\mathsf{G}_{B}(V_{o},z)$
(100)

Eq. (100) is then substituted to Eq. (75). This series will converge as long
as the initial energy of the particle is above the barrier height for
$V_{o}<\mu_{o}c^{2}$. text

The equivalent expressions for the effective IOR given by Eqs. (75), (94), and
(100) were numerically evaluated using Wolfram Mathematica 12 - Student
edition. The computer used has the following specifications: an Intel Core
i5-9300H CPU @ 2.40 GHz, 8.0 GB Ram, and a 64-bit operating system $\times$
64-based processor. Table 1 compares the values of $\tilde{R}_{c}$ for
spatially narrow Gaussian wavepackets, i.e. the wavepackets have a wide spread
in momentum such that it can have both below and above barrier components. The
evaluation of Eq. (75) is numerically taxing for the computer as the potential
increases but the equivalent expression Eq. (100) converges to the same value
as that of Eq. (94). Moreover, it can be seen that for the parameters wherein
Eq. (75) is evaluated, the equivalent expressions Eq. (100) and (94) all
converge to the same value. Table 2 compares the values of $\tilde{R}_{c}$ for
spatially wide Gaussian wavepackets, i.e. the wavepackets have a narrow spread
in momentum such that it only has below barrier components. Eq. (100) will not
converge so we only compare Eqs. (75) and (94). It can be seen that the the
values become numerically zero, which supports our earlier conclusion. This
gives us confidence in the final expression of the barrier traversal time Eq.
(95).

Figure 6: The effective IOR $\tilde{R}_{c}$ of spatially wide Gaussian
wavepackets for the parameters $\mu_{o}=c=\hbar=1$ with $\sigma=6$. The red
line represents $\kappa_{c}=1.7025$ with $V_{o}=0.99$. The area below the blue
line represents the superluminal region for the traversal time when
$\tilde{R}_{c}<1$.  Figure 7: Momentum density distribution
$|\tilde{\psi}(k)|^{2}$ for the parameters $\mu_{o}=c=\hbar=1$ with
$\sigma=6$. The red line represents $\kappa_{c}=1.7025$ with $V_{o}=0.99$.

To further appreciate the importance of distinguishing the below and above
barrier components, consider Fig. 5. The components on the right (left) of the
red line $\kappa_{c}$ are the above (below) barrier components. It can easily
be seen from Fig. 5 that all the components of $|\tilde{\psi}(k)|^{2}$ for the
cases $\sigma=4.0$ and $\sigma=6.0$ are below $\kappa_{c}$ which will tunnel
instantaneously through the barrier $V_{o}=0.99$. This is easily verified by
evaluating Eq. (94) for these parameters, which will yield $\tilde{R}_{c}\sim
0$. Fig. 6 shows the effective IOR $\tilde{R}_{c}$ for spatially wide Gaussian
wavepackets as the initial momentum $k_{o}$ increases. It can be seen that
there is a region where the traversal time $\bar{\tau}_{\text{trav}}$ becomes
superluminal as $k_{o}$ increases such that the spread of
$|\tilde{\psi}(k)|^{2}$ starts to go beyond $\kappa_{c}$. This is shown in
Fig. 7. We can thus estimate that if the initial momentum
$k_{o}<\kappa_{c}-\sigma_{k}$, where $\sigma_{k}$ is the momentum variance,
then the traversal time becomes superluminal because
$\int_{\kappa_{c}}^{\infty}dk|\tilde{\psi}(k)|^{2}$ is small which effectively
leads to $\tilde{R}_{c}<1$ or equivalently $\bar{\tau}_{\text{trav}}<t_{c}$.
Moreover, the traversal time becomes subliminal when the initial momentum
$k_{o}>\kappa_{c}-\sigma_{k}$, wherein the peak of $\tilde{R}_{c}$ is roughly
at $k_{o}=\kappa_{c}+\sigma_{k}$. The effective IOR $\tilde{R}_{c}$ then
eventually plateaus to some value as all the components of
$|\tilde{\psi}(k)|^{2}$ are above $\kappa_{c}$.

##  VIII Conclusion

In this paper, we have given a full account of [EPL, 141 (2023) 10001]. The
general form of the quantized relativistic TOA-operators in the presence of an
interaction potential were also obtained using a modified Weyl, Born-Jordan,
and simple symmetric ordering rule. These were then used to investigate the
traversal time of a relativistic quantum particle across a square barrier. We
have shown that tunneling is still instantaneous for the three ordering rules
despite a relativistic treatment of time as a dynamical observable, provided
that the barrier height is less than the rest mass energy. This result is
similar to the earlier work of Galapon [15] for a non-relativistic particle.
That is, tunneling is instantaneous and that only the above barrier energy
components of the initial wavepacket’s momentum distribution contribute to the
barrier traversal time.

The results of this paper implies that instantaneous tunneling time, or
generally superluminal tunneling times, across a square barrier is not a
consequence of using a non-relativistic theory but is an inherent quantum
effect in the context of arrival times. However, this instantaneous tunneling
can only be observed if the following conditions are satisfied: (i) the
initial incident wavepacket $\psi(q)$ must be spatially wide to ensure that
all the momentum components are below the barrier; and (ii) the initial
incident wavepacket must be placed very far from the barrier to prevent any
‘leaking’ into the barrier.

It remains to be explored the case when $V_{o}>\mu_{o}c^{2}$, which can be
done by modifying the contour in Fig. 4. By doing so, it is expected that one
should be able to extract a non-zero value for the below-barrier contributions
to the effective IOR of the barrier. The caveat is that the effects of
spontaneous pair creation and annihilation may be significant in this regime
such that the concept of TOA loses its meaning. That is, the particle that
arrived may not be the same initial particle that tunneled through the barrier
such that the concept of TOA-becomes ill-defined.

It should then be enough to use a non-relativistic theory and investigate the
effects of the shape of the barrier to the measured tunneling times. It is
well-known that non-linear systems such as the square barrier suffers from
obstructions to quantization [50]. In the non-relativistic case, the
correction terms to the TKF for non-linear systems, such as the square
barrier, has been recently obtained [51]. Applying these correction terms to
the non-relativistic case may lead to non-zero tunneling times.

We leave the problem for spin-$1/2$ particles open for future studies. Earlier
studies done by Bunao and Galapon [77, 78], where the TOA-operators were
obtained by solving the time-energy canonical commutation relation, have shown
that $\mathsf{\hat{T}_{S-1/2}}=\mathsf{\hat{T}_{S-0}}+\mathsf{\hat{T}_{E}}$ in
which, $\mathsf{\hat{T}_{S-1/2}}$ and $\mathsf{\hat{T}_{S-0}}$ are the free-
particle TOA-operator for spin-$1/2$ and spin-$0$ particles, respectively.
Meanwhile, $\mathsf{\hat{T}_{E}}$ is an extra term which is not invariant
under parity transforms, and commutes with the Dirac-Hamiltonian. The term
$\mathsf{\hat{T}_{E}}$ was then thrown away because it does not contribute
anything to the conjugacy relation implying
$\mathsf{\hat{T}_{S-1/2}}=\mathsf{\hat{T}_{S-0}}$, but $\mathsf{\hat{T}_{E}}$
may provide for other characteristics, roles, and physical interpretations of
a time observable [79]. We expect the same behavior when extending the
formalism to spin-$1/2$ particles, i.e., there will be an extra term to the
barrier traversal time operator for spin-$1/2$ particles. However, the
quantization prescription does not impose conjugacy between the Hamiltonian
and TOA-operators, as such, it is possible that this extra term cannot be
simply thrown away and may lead to non-zero tunneling times.

###### Acknowledgements.

P.C.M. Flores would like to thank D.A.L. Pablico and C.D. Tica for fruitful
discussions regarding the evaluation of the divergent integrals in term-by-
term integration. P.C.M. Flores acknowledges the support of the Department of
Science and Technology – Science Education Institute through the ASTHRDP-NSC
graduate scholarship program.

## Data Availability Statement

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

##  Appendix A Non-relativistic limit of the time kernel factors

For completeness, we show how the relativistic TKFs in Sec. III reduces to the
known TKFs of the non-relativistic TOA operator constructed by Galapon and
Magadan [40]. We first evaluate the modified Weyl-ordered relativistic TKF
operator as follows

$\displaystyle\lim_{c\rightarrow\infty}$ $\displaystyle
T^{\\{W\\}}(q,q^{\prime})$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\int_{0}^{\frac{q+q^{\prime}}{2}}ds\lim_{c\rightarrow\infty}\mathsf{W}_{s}(q,q^{\prime})$
$\displaystyle=$
$\displaystyle\dfrac{1}{2}\int_{0}^{\frac{q+q^{\prime}}{2}}ds\lim_{c\rightarrow\infty}\left\\{\mathsf{W}_{s}^{(1)}(q,q^{\prime})+\dfrac{2}{\pi}\int_{1}^{\infty}dz\exp[-\dfrac{\mu_{o}c}{\hbar}\absolutevalue{q-q^{\prime}}z]\dfrac{\sqrt{z^{2}-1}}{z}\mathsf{W}_{s,z}^{(2)}(q,q^{\prime})\right\\}$
(101)

It can easily be seen that the second term of Eq. (101) vanishes
exponentially. Meanwhile, the first term of Eq. (101) reduces into

$\displaystyle\lim_{c\rightarrow\infty}$
$\displaystyle\mathsf{W}_{s}^{(1)}(q,q^{\prime})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dye^{-y}\oint_{R}\dfrac{dz}{2\pi
i}\dfrac{1}{z}\lim_{c\rightarrow\infty}\sqrt{1+\dfrac{z^{2}}{\mu_{o}^{2}c^{2}}}{{}_{0}}F_{1}\left[;1;\dfrac{\mu_{o}V_{s}^{(W)}(q,q^{\prime})}{2\hbar^{2}}\left(\left(q-q^{\prime}\right)-i\hbar\dfrac{y}{z}\right)^{2}\mathsf{P_{W}}(s,z,q,q^{\prime})\right]$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}dye^{-y}\oint_{R}\dfrac{dz}{2\pi
i}\dfrac{1}{z}{{}_{0}}F_{1}\left[;1;\dfrac{\mu_{o}V_{s}^{(W)}(q,q^{\prime})}{2\hbar^{2}}\left(\left(q-q^{\prime}\right)-i\hbar\dfrac{y}{z}\right)^{2}\right].$
(102)

The right-hand side of Eq. (102) is further evaluated by taking the series
representation of the hypergeometric function to perform a term-by-term
integration, i.e.,

$\displaystyle\lim_{c\rightarrow\infty}$
$\displaystyle\mathsf{W}_{s}^{(1)}(q,q^{\prime})$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}\dfrac{1}{(1)_{m}m!}\left(\dfrac{\mu
V_{s}^{(W)}(q,q^{\prime})}{2\hbar^{2}}\right)^{m}\sum_{n=0}^{2m}\binom{2m}{n}(q-q^{\prime})^{2m-n}\left(-i\hbar\right)^{n}\int_{0}^{\infty}dye^{-y}y^{n}\oint_{R}\dfrac{dz}{2\pi
i}\dfrac{1}{z^{n+1}}$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}\dfrac{1}{(1)_{m}m!}\left(\dfrac{\mu
V_{s}^{(W)}(q,q^{\prime})}{2\hbar^{2}}(q-q^{\prime})^{2}\right)^{m}$
$\displaystyle=$ $\displaystyle{{}_{0}}F_{1}\left[;1;\dfrac{\mu
V_{s}^{(W)}(q,q^{\prime})}{2\hbar^{2}}(q-q^{\prime})^{2}\right]$ (103)

Thus, we now have the non-relativistic limit of the Weyl-ordered TKF given by

$\lim_{c\rightarrow\infty}T^{\\{W\\}}(q,q^{\prime})=\dfrac{1}{2}\int_{0}^{\frac{q+q^{\prime}}{2}}ds{{}_{0}}F_{1}\left[;1;\dfrac{\mu
V_{s}^{(W)}(q,q^{\prime})}{2\hbar^{2}}(q-q^{\prime})^{2}\right]$ (104)

The same process is applied to obtain the non-relativistic limit of the Born-
Jordan and simple-symmetric ordered TKFs.

##  Appendix B Further details on the evaluation of the complex-valued IOR
$R_{c}^{*}$

Here, we provide the details on the evaluation of the contour integrals Eq.
(90). Let us first consider the following integral

$\oint
dzp(z)\left(\sqrt{1+\dfrac{\hbar^{2}z^{2}}{\mu_{o}^{2}c^{2}}}\right)^{n+1}\left(z^{2}+\dfrac{V_{o}^{2}}{\hbar^{2}c^{2}}\right)^{-n-\frac{1}{2}},$
(105)

which is separately evaluated using the left and right box contours in Fig.
4(a). It is straightforward to show that the integral Eq. (105) will vanish
along the paths $z=\pm r+iy$ since $p(z)=\absolutevalue{\tilde{\psi}(z)}^{2}$
vanishes as $r\rightarrow\infty$. Moreover, Eq. (105) also vanish along the
semicircular paths around the branch points $z=\delta
e^{i\theta}+i(\mu_{o}c/\hbar)$ and $z=\delta e^{i\theta}+i(V_{o}/\hbar c)$ as
$\delta\rightarrow 0$. Taking the difference of the non-vanishing terms of the
right and left box contours will then yield

$\displaystyle\int_{-\infty}^{\infty}dx$ $\displaystyle
p(x+i\epsilon)\text{csgn}(x+i\epsilon)\left(\sqrt{1+\dfrac{\hbar^{2}(x+i\epsilon)^{2}}{\mu_{o}^{2}c^{2}}}\right)^{n+1}\left((x+i\epsilon)^{2}+\dfrac{V_{o}^{2}}{\hbar^{2}c^{2}}\right)^{-n-\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx\left(p(x)-p(-x)\right)\left(\sqrt{1+\dfrac{\hbar^{2}x^{2}}{\mu_{o}^{2}c^{2}}}\right)^{n+1}\left(x^{2}+\dfrac{V_{o}^{2}}{\hbar^{2}c^{2}}\right)^{-n-\frac{1}{2}}$
$\displaystyle-i\left(1-(-1)^{n+1}\right)\left(-i\dfrac{\hbar^{2}}{\mu^{2}c^{2}}\right)^{n}\int_{1}^{\infty}dyp\left(i\dfrac{\mu
c}{\hbar}y\right)\sqrt{\dfrac{y^{2}-1}{y^{2}-\frac{V_{o}^{2}}{\mu^{2}c^{4}}}}\left(\dfrac{\sqrt{y^{2}-1}}{y^{2}-\frac{V_{o}^{2}}{\mu^{2}c^{4}}}\right)^{n}$
(106)

We can similarly evaluate the integral

$\displaystyle\oint
dzp(z)\dfrac{\frac{\hbar^{2}}{\mu^{2}c^{2}}z}{y^{2}+\frac{\hbar^{2}}{\mu^{2}c^{2}}z^{2}}$
(107)

using the contour in Fig. 4(b). It is also straightforward to show that the
integral Eq. (107) will vanish along the paths $z=\pm r+iy$ since
$p(z)=\absolutevalue{\tilde{\psi}(z)}^{2}$ vanishes as $r\rightarrow\infty$.
Using the residue theorem, it is easy to show that

$\displaystyle\int_{-\infty}^{\infty}dx$ $\displaystyle
p(x+i\epsilon)\dfrac{\frac{\hbar^{2}}{\mu^{2}c^{2}}(x+i\epsilon)}{y^{2}+\frac{\hbar^{2}}{\mu^{2}c^{2}}(x+i\epsilon)^{2}}=\int_{-\infty}^{\infty}dxp(x)\dfrac{\frac{\hbar^{2}}{\mu^{2}c^{2}}x}{y^{2}+\frac{\hbar^{2}}{\mu^{2}c^{2}}x^{2}}-\pi
ip\left(i\dfrac{\mu c}{\hbar}y\right)$ (108)

We then substitute both Eqs. (106) and (108) into Eq. (88) which yields

$\displaystyle\int_{0}^{\infty}d\zeta$
$\displaystyle\mathsf{F}_{B}(-V_{0},\zeta)\int_{-\infty}^{\infty}dke^{ik\zeta}\absolutevalue{\tilde{\psi}(k)}^{2}$
$\displaystyle=$ $\displaystyle i\dfrac{\hbar}{\mu
c}\int_{0}^{\infty}dk\left(\absolutevalue{\tilde{\psi}(k)}^{2}-\absolutevalue{\tilde{\psi}(-k)}^{2}\right)\sqrt{\dfrac{\tilde{E}_{k}^{2}}{(\tilde{E}_{k}-V_{o})^{2}-\mu_{o}^{2}c^{4}}}$
(109)

Last, we combine Eqs. (109) and (85) to obtain

$\displaystyle R_{c}^{*}=$ $\displaystyle i\dfrac{\hbar k_{o}}{\mu
c}\int_{0}^{\infty}dk\left(\absolutevalue{\tilde{\psi}(k)}^{2}-\absolutevalue{\tilde{\psi}(-k)}^{2}\right)\sqrt{\dfrac{\tilde{E}_{k}^{2}}{(\tilde{E}_{k}-V_{o})^{2}-\mu_{o}^{2}c^{4}}}$
$\displaystyle+k_{o}\dfrac{2}{\pi}\int_{1}^{\infty}dy\dfrac{\sqrt{y^{2}-1}}{y}\mathsf{G_{B}}(V_{o},y)\int_{-\infty}^{\infty}dk\absolutevalue{\tilde{\psi}(k)}^{2}\dfrac{\frac{\mu
c}{\hbar}y}{k^{2}+\frac{\mu^{2}c^{2}}{\hbar^{2}}y^{2}}.$ (110)

Notice that the first term of (110) is generally complex-valued while the
second term is always real-valued.

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