# Fluctuating Paths of Least Time, Schrödinger Equation, Van der Waals torque
and Casimir Effect Mechanism

F. Ben Adda111Community College of Qatar, Department of math and Sciences,
Doha, Qatar

###### Abstract

The use of an infinity of fluctuating paths of least time that are compatible
with the quantum mechanics indeterminacy provides a new interpretation in
geometrical optic of the interference pattern of Young’s double slit
experiment, which suggests that the wave behavior of matter and radiation is
dictated by the space-time geodesics. Moreover, the association of a wave
function to each path of least time as a probability amplitude together with
an uncertainty for momentum and position allows to derive the Schrödinger’s
equation starting from the geodesic’s characteristics. A new insight is
obtained regarding the Van Der Waals torque as well as Casimir
attraction/repulsion mechanism.

PACS: 02.30.-f; 02.70.-c; 03.65.-w; 02.30.Jr; 12.20Ds; 42.25Hz.

###### Contents

  1. 1 Introduction
  2. 2 Graphical introduction of geodesics in an homogeneous, isotropic and expanding space-time
  3. 3 Infinity of fluctuating paths of least time and interference pattern in Young’s double-slit experiment
    1. 3.1 Observation and final state perturbation
  4. 4 Mathematical formulation of infinity of paths of least time
    1. 4.1 Infinity of paths of least time in 2D
    2. 4.2 Infinity of paths of least time in 3D
    3. 4.3 Properties
  5. 5 Action and wave function associated to the infinity of paths of least time
    1. 5.1 Action and wave function associated to each path of least time
    2. 5.2 Compatibility with the indeterminacy principle
    3. 5.3 Uncertainty relation, angles and action function
  6. 6 Derivation of Schrödinger Equation
    1. 6.1 Infinity of paths of least time in complex plan
    2. 6.2 Action function, wave function and uncertainty in the complex plan
    3. 6.3 Infinity of paths of least time and derivation of Schrödinger’s equation
  7. 7 Zero point energy, Van Der Waals torque and Casimir effect
    1. 7.1 Zero point energy and energy fluctuation
      1. 7.1.1 Variation of the energy magnitude of the physical system
      2. 7.1.2 Space minimal energy fluctuation
      3. 7.1.3 The zero point energy
    2. 7.2 Space minimal energy variation and torque effect interpretation
    3. 7.3 About the attractive repulsive Casimir force origin
    4. 7.4 Attractive or repulsive Casimir force causality interpretation
  8. 8 Conclusion

##  1 Introduction

The most challenging problem of modern physics is to find a coherent
geometrical representation of the space-time at all scales. The real geometry
and architecture of the space-time of our universe and the constraints that it
imposes to its content are still unknown (specially for the space-time at the
quantum scale ([1]) despite the fact that local descriptions of the space-time
geometry provide a suitable understanding of the behavior of several physical
systems on it to a certain limit. However, without knowing the real
constraints that the space-time geometry and characteristics impose to its
content, we may not be able to acquire a precise understanding of certain
physical systems behavior. At the quantum scale Heisenberg, Born and Jordan
have developed a matrix formalism of quantum mechanics in 1925
([18],[19],[20]) in which matrix was used as a physical property of elementary
particles. In 1926 Schrödinger introduced another approach of quantum
mechanics([21, 22, 23, 24]), where the physical properties of elementary
particles were described by a wave function solution of a differential
equation. The quantum formalism found its development on an infinitely
dimensional Hilbert space of functions where the quantum description of a
point particle was completely unusual for a trajectory basic understanding
since it prohibits the simultaneous determination of position and momentum
with accurate measurement, which suggests that the classical trajectory of a
particle has no physical meaning in the quantum mechanic’s interpretation.
Nevertheless, in 1948 Feynman introduced a different approach of quantum
mechanics using the path integral ([9]) as a sum over all possible
trajectories to calculate a total quantum amplitude compatible with the
Schrödinger’s equation. Feynman’s path integral conveys in its formalism a
connection between Lagrangian and quantum mechanics. Thus, despite the correct
multiple predictions of the Feynman’s path integral in physics, a lack of
mathematical rigor exists for the geometrical interpretation of the
integration over all possible paths.

In this case study we use the essence of the Feynman’s path integral in a
different way that involves the medium characteristics together with
Heisenberg uncertainty relation to provide a fully equivalent meaning
compatible with the Schrödinger’s equation in order to elaborate a new insight
for a better understanding of the Van Der Waals torque ([7],[10]) as well as
the Casimir effect ([5]) at the quantum scale. Many thoughts about the origin
and mechanism of the phenomenon of Casimir effect and Van Der Waals torque are
still under intense investigation experimentally and theoretically for their
utility in nano-mechanics, nano-technology ([6]). A controversy exists about
the origin of this phenomenon ([5],[11], [12], [13], [16]), and this case
study provides a new insight that might help to forge a clear understanding.

The plan of this paper is as follow: in section 2, a brief graphic
introduction is given about infinity of paths of least time. In section 3,
reproduction of interference pattern of Young’s double-slit experiment using
paths of least time and interpretation is given in geometrical optic. In
section 4, a mathematical formulation of paths of least time is presented in
2D, 3D, as well as properties. In section 5, action and wave function are
associated to each geodesic following the essence of Feynman’s path integral,
and formulation of an uncertainty relation. In section 6, the derivation of
Schrödinger Equation is obtained starting from characteristics of the infinity
of paths of least time in the complex plan. In section 7, approach to the zero
point energy, Van Der Waals torque and Casimir effect via Schrödinger Equation
are elaborated. As an outcome a new insight that concerns causality and
interpretation of Van Der Waals torque as well as mechanism of
attractive/repulsive Casimir force is figured out. In section 8, conclusion.

##  2 Graphical introduction of geodesics in an homogeneous, isotropic and
expanding space-time

Observation at large scale revealed confirmed characteristics of the space-
time: it is an homogeneous, isotropic, expanding and non homogeneous locally
space-time. A case study ([3]) based on a simulation of an homogeneous,
isotropic and expanding space time that expands via expansion of its basic
elements led to approach an interesting property of the space-time geodesics
at small and large scale; the geodesics (or paths of least time) in an
homogeneous, isotropic and expanding space-time that expands via expansion of
its basic elements cannot be straight line geodesics, rather it is an infinity
of fluctuating geodesics with non null curvature illustrated in (Fig.2). These
geodesics allow to approach the quantum scale.

Figure 1: Illustration of one geodesic in 2D for $N=6$ (alignment of 6
spheres) and arc radius $r=5\ mm$ between aligned antipodal points from A to
B.

Figure 2: Illustration of infinity of geodesics in 3D with identical length
for $N=6$ and arc radius $r=5\ mm$ between aligned antipodal points from A to
B.

Indeed, the infinity of fluctuating geodesics illustrated in (Fig.2) between
two distant locations have the same length, and are continuous,
differentiable, with a finite number of points of non differentiability
(antipodal points), where localization on it is impossible between any two
antipodal points due to the existence of an infinity of circle arcs with equal
length that prohibit localization, which makes these geodesics consistent with
the indeterminacy of quantum mechanics (Fig.4). Moreover, it is known that
within 1.1 mm there is an average of 2000 light wave lengths that correspond
to 4000 aligned spheres of radius $r_{0}=0.0001375mm$ using the geodesics
illustrated in (Fig.2). The aligned 4000 spheres of radius $r_{0}$ within 1.1
mm appear at the macroscopic observation as a straight segment (the
fluctuations are undiscernible), which fits the local description of a space-
time with null curvature (see Fig.4), meanwhile a microscopic view of them
reveals the existence of infinity of transverse fluctuating geodesics with
points of non differentiability. This characteristics might reveal new details
that concerns Young’s double-slit experiment.

Figure 3: Consistency with the indeterminacy of quantum mechanics. There
exists an infinity of geodesics with non null curvature between two distant
locations consistent with the indeterminacy of quantum mechanics

Figure 4: Alinement of spheres in $1.1mm$. A) 12 spheres of radius
$r_{A}\simeq 0,0458333mm$, B) 24 spheres of radius $r_{B}\simeq 0,0229166mm$,
C) 48 spheres of radius $r_{C}\simeq 0,0114583mm$, D) 4000 spheres of radius
$r_{D}\simeq 0,0001375mm$.

Indeed, It is known that the double-slit experiment is a corner stone in
quantum mechanics and plays a crucial role in our comprehension of the quantum
world mystery. It is also known that geometrical optic failed drastically to
explain it. Thus, the Young’s double-slit experiment ([14],[15]) conveys a
fundamental mystery in physics: how can light be emitted and absorbed as
corpuscular, and undergo interference pattern between source and detector
screen as a wave? The adoption of duality allows to circumvent this question
and provides flexible approach in our use of the corpuscular character or wave
nature of matter and radiation following the needs. Because of this mystery,
testing the infinity of fluctuating geodesics (Fig.2) with the interference
pattern seems to be the first step and might provide a new interpretation for
the phenomenon in geometrical optic. Meanwhile proving that these infinity of
geodesics convey a fully equivalent formalism compatible with the
Schrödinger’s equation will allow to approach energy fluctuation, zero point
energy, Van der Waals torque as well as the Casimir effect from a new
perspective.

##  3 Infinity of fluctuating paths of least time and interference pattern in
Young’s double-slit experiment

To simulate all possible paths of least time in phase that flare out from a
given slit $S_{1}$ and cover a chosen angle of $\theta=64^{\circ}$ beyond the
slit $S_{1}$ as illustrated in (Fig.6), we will use only one plan geodesic
from the infinity of geodesics illustrated in (Fig.2). The chosen angle can be
covered by 65 copies of the same plan geodesic represented in (Fig.2) for
$r_{0}=0.25\ mm$, N=102 (ie: on 102 aligned spheres) where the angle between
two successive plan geodesics is one degree using a rotation in the plan about
$S_{1}$ (interpretation of the diffraction using the fluctuating geodesics can
be found [2] for more details).

Figure 5: Geodesics diffraction: Illustration of 65 geodesics in phase, using
the geodesic given by (Fig.2) for $r_{0}=0.25\ mm$, N=102, that cover an angle
of $64^{\circ}$ from a narrow slit, to represent all possible geodesics that
diffract from the slit. The angle between two consecutive geodesic in phase is
$1^{\circ}$ .

Figure 6: The superposition of family of geodesics of radius $r_{0}=0.25mm$
from the $S_{1}$ with the family of geodesics with the same radius from the
slits $S_{2}$ generate fringes or bands in the geodesics intersection region.

The superimposition of two distant copies of the geodesic diffraction
illustrated in (Fig.6) simulates geodesics diffraction from two distant slits
where their superimposition reproduces fringes and bands pattern observed in
Young’s double-slit experiment not only on a given detector screen but also
within the whole geodesics intersection region between the slits and detector
screen (Fig.6). The interference patterns in the geodesics intersection region
appears because of the existence of sets of point intersections (arcs
intersection Fig.8). More precisely, if the local fluctuation centers
coincide, there are arcs intersection that form the interference (clear spots
in Fig.6). Otherwise, there is only single point intersection. The presence of
the physical system is doubled within the intersection circle arcs that form
the interference pattern. Thus, at the clear spots region the probability of
presence of the physical systems is doubled (in the 2D simulation) within
circle of arcs intersection, meanwhile the probability of presence of the
physical systems in the dark regions is doubled in single points of
intersection, which creates the appearance of fringes and interference pattern
within the intersection region as well as in the detector screen. Similarly in
3D, the clear spots mean that probability of presence of the physical systems
is multiple within spheres surface intersection when the spheres centers
coincide, meanwhile dark spots mean that the probability of presence of the
physical systems is multiple within arcs of spheres intersection. In both
cases the probability of presence of the physical systems in the intersection
region between slits as well as in the detector screen is more or less
important according to the existence of sets of geodesics intersection.
Moreover, the closer the slits are, the larger the arcs intersection is (see
Fig.8 and Fig.8) and the number of fringes increases. Their size decreases
when the distance between the slits $S_{1}$ and $S_{2}$ increases (see Fig.6
and Fig.10).

Figure 7: Zoom on arcs intersection that produce interference pattern in the
geodesics intersection region between slits screen and detector screen. Large
geodesics arcs intersection for reduced slits distance $d_{1}=d(S_{1},S_{2})$.

Figure 8: Zoom on arcs intersection that produce interference pattern in the
geodesics intersection region between slits screen and detector screen.
Reduced geodesics arcs intersection for larger slits distance
$d_{2}=d(S_{1},S_{2})$.

The superimposition of diffracted geodesics (Fig.6) from each slit allows to
explain in geometrical optic why and how certain regions can be privileged for
a physical system that follows these geodesics to build up fringe pattern, and
in particular it explains the observed fringe pattern of electron, atoms and
molecule when their motion follows paths of least time given by (Fig.2), which
suggests that the interference pattern is a pure manifestation of the space-
time geodesics rather than a wave nature. The wave behavior of matter and
radiation appears to be dictated by the space-time geodesics.

Figure 9: Superposition of geodesics of radius $r_{0}=0.25mm$ from the slit
$S_{1}$ with family of geodesics with radius $r_{0}=0.25mm$ from slit $S_{2}$
generate interference pattern in the geodesics intersection region. The number
of clear pattern increases when the distance between the slits increases,
$d(S_{1},S_{2})=2\ mm$.

Figure 10: The interference pattern disappears when we superimpose the family
of geodesics of radius $r_{0}=0.25mm$ from the slit $S_{1}$ with a family of
perturbed geodesics (paths with radius $r_{1}=0.625$ mm) from the slit
$S_{2}$.

###  3.1 Observation and final state perturbation

Perturbation of the space-time geodesics provides an explanation of how
observation affects the final state of the physical system. Indeed, the
interference pattern totally disappears when we superimpose a diffracted
family of geodesics with radius $r_{0}=0.25mm$ from the slit $S_{1}$ with a
diffracted family of perturbed geodesics with bigger radius $r_{1}=0.625$ mm
from the slit $S_{2}$. Their superimposition is illustrated in (Fig.10), and
it manifests absence of fringe in the intersection region. Perturbation in
(Fig.10) can be interpreted as follow: after interaction with light (since
experimental observations use light, and a photon or an electron cannot be
detected without interaction with photons), the physical system will follow
another path that is not a path with the lowest radius, it follows a geodesic
with bigger radius that represents another path for the physical system state
after interaction. The superimposition of two different families of geodesics
with different radius of fluctuation that flare out from two distant slits
will manifest absence of fringe in the intersection region as well as in the
detector screen. A more precise simulation of superimposition of geodesics
shows that total disorder occurs for a perturbation starting from
$r_{1}={3\over 2}r_{0}$ as indicated in table 1.

Table 1: Example of interference pattern from partial to total disorder Slit $S_{1}$ | Slit $S_{2}$ | $d(S_{1},S_{2})$ | Interference
---|---|---|---
$r_{0}=1mm$ | $r_{1}=1.1mm$ | 10mm | appearance of disorder
$r_{0}=1mm$ | $r_{1}=1.2mm$ | 10mm | important disorder
$r_{0}=1mm$ | $r_{1}=1.3mm$ | 10mm | more important disorder
$r_{0}=1mm$ | $r_{1}=1.5mm$ | 10mm | total disorder

##  4 Mathematical formulation of infinity of paths of least time

###  4.1 Infinity of paths of least time in 2D

The mathematical formulation of the geodesic illustrated in two dimensions
(Fig.2) and introduced in [3] as paths of least time in an homogenous,
isotropic and expanding space-time is given by the graph of the function
$\varphi(x)$ defined for all $x\in\hbox{\tenbb R}^{+}$ by:

$\varphi(x)=\sum_{n=0}^{N-1}g_{n}(x)$ (1)

where

$g_{n}(x)=\left\\{\begin{array}[]{ll}(-1)^{n}\
\sqrt{r^{2}-\Big{(}x-(2n+1)r\Big{)}^{2}}&\hbox{for}\quad x\in[2nr,2(n+1)r]\\\
0&\hbox{for}\quad x\not\in[2nr,2(n+1)r].\end{array}\right.$ (2)

For $n=0,\dots,N-1$, the graph of $g_{n}$ represents the geodesic between two
antipodal points on the circle of center $C_{n}=\Big{(}(2n+1)r,0\Big{)}$ and
radius $r$. For $n=0,\dots,N-1$ the function $g_{n}$ is continuous on the
closed interval $[2nr,2(n+1)r]$, differentiable on the open interval
$]2nr,2(n+1)r[$ and not differentiable at the points $x_{n}=2nr$ and
$x_{n+1}=2(n+1)r$.

###  4.2 Infinity of paths of least time in 3D

To provide a mathematical formulation of the infinity of paths of least time,
the parametric equations of (1) in 3D for $N$ aligned balls is given by:

$\left\\{\begin{array}[]{ll}x&=x\\\ y&=\varphi(x)\\\ z&=0\end{array}\right.$
(3)

and a rotation of the geodesic (3) about the $x$-axis with angle
$\theta\in[0,2\pi]$ is defined by the following parametric equations

$\left\\{\begin{array}[]{ll}x^{\prime}&=x\\\
y^{\prime}&=\varphi(x)\cos\theta\\\
z^{\prime}&=\varphi(x)\sin\theta\end{array}\right.$ (4)

that defines the parametric equations of any geodesics in three dimensions
illustrated in (Fig.2). Any section of the revolution surface (4) is a circle
of equation defined by $y^{\prime 2}+z^{\prime 2}=\varphi^{2}(x)$ that gives
for all $\theta\in[0,{2\pi}]$ the equation

$y^{\prime}=\pm\sqrt{\varphi^{2}(x)-z^{\prime 2}},$ (5)

and by substitution of $z^{\prime}$ in (5) we obtain

$y^{\prime}=\pm\sqrt{\varphi^{2}(x)-\varphi^{2}(x)\sin^{2}\theta}=\pm\varphi(x)\sqrt{1-\sin^{2}\theta}=\pm\varphi(x)\cos\theta$
(6)

that defines two opposite fluctuating curves for all $\theta\in[0,{2\pi}]$,
thus it is sufficient to use $y^{\prime}=\varphi(x)\cos\theta$ for all
$\theta\in[0,{2\pi}]$. The function (1) is a plan function, and it remains a
plan function for any rotation of angle $\theta\in[0,2\pi]$ with respect to
the x-axis. For the following, we denote for all $x\in\hbox{\tenbb R}^{+}$,
for all $\theta\in[0,{2\pi}]$

$X(x,\theta):=\varphi(x)\cos\theta=\sum_{n=0}^{N-1}g_{n}(x,\theta)$ (7)

where $\varphi(x)$ is given by (1), and for $n=0,\dots,N-1$

$g_{n}(x,\theta):=\left\\{\begin{array}[]{ll}(-1)^{n}\
\sqrt{r^{2}-\Big{(}x-(2n+1)r\Big{)}^{2}}\cos\theta&\hbox{for}\quad
x\in[2nr,2(n+1)r]\\\ 0&\hbox{for}\quad
x\not\in[2nr,2(n+1)r].\end{array}\right.$ (8)

###  4.3 Properties

To parameterize the function (7) with time it is sufficient to replace the
parameter $x$ in the function (7) by the function $x_{t}:=x(t)$ a continuous
and differentiable function on $\hbox{\tenbb R}^{+}$. The geodesics (7) are
then given for all $\theta\in[0,{2\pi}]$, for all $t\in\hbox{\tenbb R}^{+}$ by

$X(x_{t},\theta)=\sum_{n=0}^{N-1}g_{n}(x_{t},\theta)$ (9)

where

$g_{n}(x_{t},\theta)=\left\\{\begin{array}[]{ll}(-1)^{n}\
\sqrt{r^{2}-\Big{(}x_{t}-(2n+1)r\Big{)}^{2}}\cos\theta&\hbox{for}\quad
x_{t}\in[2nr,2(n+1)r]\\\ 0&\hbox{for}\quad
x_{t}\not\in[2nr,2(n+1)r]\end{array}\right.$ (10)

Consider a physical system of mass $m$ moving on the geodesics defined by (9),
assuming that for all $t\in[t_{a},t_{b}]\subset\hbox{\tenbb R}^{+}$,
$X(x(t_{a}),\theta)=a$ and $X(x(t_{b}),\theta)=b$ are two distant positions in
the space-time. The motion of the physical system is completely unknown
because of the existence of an infinity of paths of least time between the two
distant positions. Therefore for all $t\in[t_{a},t_{b}]$, there exists
$n\in\hbox{\tenbb N}$ such that for all $x_{t}\in]2nr,2(n+1)r[$, for all
$\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$ the
function $X(x(t),\theta)=g_{n}(x_{t},\theta)$ is differentiable. For
simplicity ,we denote $X_{x_{t}\theta}:=X(x(t),\theta)$ then for all
$t\in]t_{a},t_{b}[$ the instantaneous rate of change of (9) is given by

$V_{t}:={\partial X_{x_{t}\theta}\over\partial t}={\partial
g_{n}(x_{t},\theta)\over\partial
t}=(-1)^{n+1}{(x_{t}-(2n+1)r)\over\sqrt{r^{2}-\Big{(}x_{t}-(2n+1)r\Big{)}^{2}}}{\partial
x_{t}\over\partial t}\cos\theta,$ (11)

meanwhile the changes of $X_{x_{t}\theta}$ in $\theta$ gives

$V_{\theta}:={\partial X_{x_{t}\theta}\over\partial\theta}={\partial
g_{n}(x_{t},\theta)\over\partial\theta}=(-1)^{n+1}\sqrt{r^{2}-\Big{(}x_{t}-(2n+1)r\Big{)}^{2}}\sin\theta.$
(12)

Then for all $\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over
2}\\}$, and for $\beta:={\cos\theta\over\sin\theta}$, the equation (12)
becomes

$\beta{\partial X_{x_{t}\theta}\over\partial\theta}=-X_{x_{t}\theta},$ (13)

meanwhile the changes of $V_{\theta}$ in $\theta$ gives

${\partial
V_{t}\over\partial\theta}=(-1)^{n}{(x_{t}-(2n+1)r)\over\sqrt{r^{2}-\Big{(}x_{t}-(2n+1)r\Big{)}^{2}}}{\partial
x_{t}\over\partial t}\sin\theta,$ (14)

and then

$\beta{\partial V_{t}\over\partial\theta}=-V_{t}.$ (15)

##  5 Action and wave function associated to the infinity of paths of least
time

###  5.1 Action and wave function associated to each path of least time

The use of the infinity of paths of least time for the quantum physical system
motion between two different locations leads to associate to each path of
least time a probability amplitude (following the essence of Feynman’s path
integral) that corresponds to the probability of finding the physical system
in a certain location at a given angle
$\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$. For this
purpose we introduce the following: consider for all $t\in[t_{a},t_{b}]$, for
all $\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$ the
action function defined by

$\displaystyle S:\hbox{\tenbb R}\times[t_{a},t_{b}]$
$\displaystyle\rightarrow$ ℝ $\displaystyle\Big{(}X_{x_{t}\theta},t\Big{)}$
$\displaystyle\mapsto$ $\displaystyle S(X_{x_{t}\theta},t)$ (16)

related to the momentum, total energy and angle variation by

$\displaystyle{\partial{S}\over\partial{X_{x_{t}\theta}}}$ $\displaystyle=$
$\displaystyle m{V}_{t}={P}$ (17) $\displaystyle-{\partial{S}\over\partial t}$
$\displaystyle=$ $\displaystyle E$ (18)
$\displaystyle{\partial{S}\over\partial\theta}$ $\displaystyle=$
$\displaystyle{-\hbar\over 2\beta}.$ (19)

Consider for all $\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over
2}\\}$, for all $x_{t}\in[2nr,2(n+1)r]$ the wave function associated to each
path defined in (9) by

$\displaystyle\Psi:\hbox{\tenbb R}\times[t_{a},t_{b}]$
$\displaystyle\rightarrow$ ℂ $\displaystyle(X_{x_{t}\theta},t)$
$\displaystyle\mapsto$
$\displaystyle\Psi(X_{x_{t}\theta},t)=e^{{i\over\hbar}{S}(X_{x_{t}\theta},t)},$
(20)

where $i$ is the imaginary unit, $\hbar={h\over 2\pi}$ and $h$ is the Plank’s
constant. The function $\Psi(X_{x_{t}\theta},t)$ represents the wave function
for a single particle where $X_{x_{t}\theta}$ is its position in space at the
time $t\in[t_{a},t_{b}]$ and the angle
$\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$ (this wave
function was used by Feynman in his postulate II of the path integral). Then
similarly to Feynman’s choice, the wave function (5.1) represents a
probability amplitude that corresponds to the probability of finding the
physical system in a certain location at the time $t$, and the angle
$\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$. The wave
function $\Psi(X_{x_{t}\theta},t)$ may take the approximate form
$\Psi(X_{x_{t}\theta},t)=Ae^{{i\over\hbar}{\mathcal{S}}(X_{x_{t}\theta},t)}$
where $A$ is the dimensional constant required for any physical
interpretation. In the following the amplitude $A$ of the used wave function
will be considered as equal to 1 for more simplicity and it can be
reintroduced if needed in any linear equation. The wave function (5.1) gives,
using equations (17) and (18), the correspondence principles of quantum
mechanics for momentum and total energy for all $x_{t}\in]2nr,2(n+1)r[$, for
all $\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$,

${P}=-i\hbar{\partial\Psi(X_{x_{t}\theta},t)\over\partial
X_{x_{t}\theta}}{1\over\Psi(X_{x_{t}\theta},t)},\qquad\hbox{and}\qquad
E=i\hbar{\partial\Psi(X_{x_{t}\theta},t)\over\partial
t}{1\over\Psi(X_{x_{t}\theta},t)}.$ (21)

###  5.2 Compatibility with the indeterminacy principle

The wave function (5.1) is interpreted as a probability amplitude that
represents the wave function for a single particle, in which $X_{x_{t}\theta}$
is its position in space at the time $t\in[t_{a},t_{b}]$, and angle
$\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$ and where
the square modulus of the wave function

$|\Psi(X_{x_{t}\theta},t)|^{2}=\Psi(X_{x_{t}\theta},t)\Psi^{*}(X_{x_{t}\theta},t)=\rho(X_{x_{t}\theta},t),$
(22)

for $\Psi^{*}$ the conjugate of $\Psi$, represents the probability density
that the physical system at the time $t\in[t_{a},t_{b}]$ and angle
$\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$ is located
at $X_{x_{t}\theta}$. If one measures the position of the physical system at
the time $t\in[t_{a},t_{b}]$ and the angle
$\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$, then the
location of the physical system is expressed by the probability distribution
as follow: the probability that the physical system’s position
$X_{x_{t}\theta}$ is between $X_{x_{t}\theta_{0}}=c$ and
$X_{x_{t}\theta_{1}}=d$, is given by the following integral of density

$P_{c\leq X_{x_{t}\theta}\leq
d}(t)=\int^{d}_{c}|\Psi(X_{x_{t}\theta},t)|^{2}dX_{x_{t}\theta}=\int^{\theta_{1}}_{\theta_{0}}|\Psi(X_{x_{t}\theta},t)|^{2}{\partial
X_{x_{t}\theta}\over\partial\theta}d\theta,$ (23)

where $t$ and $\theta$ are the time and angle at which the physical system is
measured. The normalization condition is then

$\int^{+\infty}_{-\infty}|\Psi(X_{x_{t}\theta},t)|^{2}dX_{x_{t}\theta}=\int^{2\pi}_{0}|\Psi(X_{x_{t}\theta},t)|^{2}{\partial
X_{x_{t}\theta}\over\partial\theta}d\theta=1.$ (24)

###  5.3 Uncertainty relation, angles and action function

The angle rate of change of the action function (5.1) for all
$\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$ given by
(19) conveys an uncertainty relation. Indeed, based on equation (19) we have
the following:

###### Lemma 1

Let $S$ be the action function defined by (5.1), and $\Psi$ the wave function
defined by (5.1). For all
$(t,\theta)\in[t_{a},t_{b}]\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$ we have

${\partial{S}\over\partial\theta}={-\hbar\over
2\beta}\quad\Longleftrightarrow\quad{P}\ X_{x_{t}\theta}={\hbar\over 2}$ (25)

where $X_{x_{t}\theta}$ is the position function, and ${P}=m{V}_{t}$ is the
momentum.

Proof. For all
$(t,\theta)\in[t_{a},t_{b}]\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$, $\exists n\in\hbox{\tenbb N}$ such that
$x_{t}\in]2nr,2(n+1)r[$, and $X_{x_{t}\theta}=g_{n}(x_{t},\theta)$. Thus,
using definition (5.1) of $\Psi$,
${S}(X_{x_{t}\theta},t)=-i\hbar\ln\Psi(X_{x_{t}\theta},t)$, and the chain rule
with (19) gives

${\partial S\over\partial\theta}={\partial S\over\partial
X_{x_{t}\theta}}{\partial X_{x_{t}\theta}\over\partial\theta}={-\hbar\over
2\beta},$ (26)

then by substitution of (17) and (12) in (26), we obtain
$mV_{t}V_{\theta}={-\hbar\over 2\beta}$, and the use of equation (13) gives
the result ${P}\ X_{x_{t}\theta}={\hbar\over 2}$.

##  6 Derivation of Schrödinger Equation

###  6.1 Infinity of paths of least time in complex plan

For the derivation of Schrödinger equation, the infinity of geodesics needs to
be defined in the complex set ℂ. Thus, consider for all $t\in[t_{a},t_{b}]$,
for all $\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$
the complex geodesics $u_{x_{t}\theta}:=kX_{x_{t}\theta}$ where
$k\in\hbox{\tenbb C}$ is a complex number that verifies $k^{2}=i$ and
$X_{x_{t}\theta}$ are the space-time geodesics (9) introduced above. Then
using (11) and (12) we have for all
$(t,\theta)\in[t_{a},t_{b}]\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$,

$\mathbb{V}_{t}:={\partial u_{x_{t}\theta}\over\partial
t}=kV_{t},\qquad\hbox{and}\qquad\mathbb{V}_{\theta}:={\partial
u_{x_{t}\theta}\over\partial\theta}=kV_{\theta},$ (27)

meanwhile using (13) for $\beta={\cos\theta\over\sin\theta}$,
$\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$, we have

$\beta{\partial u_{x_{t}\theta}\over\partial\theta}=-u_{x_{t}\theta},$ (28)

and the use of (15) gives

$\beta{\partial\mathbb{V}_{t}\over\partial\theta}=-\mathbb{V}_{t}.$ (29)

###  6.2 Action function, wave function and uncertainty in the complex plan

Consider for all
$(t,\theta)\in[t_{a},t_{b}]\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$ the action function defined by

$\displaystyle\mathbb{S}:\hbox{\tenbb C}\times[t_{a},t_{b}]$
$\displaystyle\rightarrow$ ℂ $\displaystyle(u_{x_{t}\theta},t)$
$\displaystyle\mapsto$
$\displaystyle\mathbb{S}(u_{x_{t}\theta},t)=S(kX_{x_{t}\theta},t),$ (30)

related to the momentum, total energy and angle variation by

$\displaystyle{\partial\mathbb{S}\over\partial u_{x_{t}\theta}}$
$\displaystyle=$ $\displaystyle m\ \mathbb{V}_{t}=\mathbb{P}$ (31)
$\displaystyle-{\partial\mathbb{S}\over\partial t}$ $\displaystyle=$
$\displaystyle\mathbb{E}$ (32)
$\displaystyle{\partial\mathbb{S}\over\partial\theta}$ $\displaystyle=$
$\displaystyle-{i\hbar\over 2\beta}.$ (33)

Consider for all
$(t,\theta)\in[t_{a},t_{b}]\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$, the wave function defined by

$\displaystyle\Psi:\hbox{\tenbb C}\times[t_{a},t_{b}]$
$\displaystyle\rightarrow$ ℂ $\displaystyle(u_{x_{t}\theta},t)$
$\displaystyle\mapsto$
$\displaystyle\Psi(u_{x_{t}\theta},t)=e^{{i\over\hbar}\mathbb{S}(u_{x_{t}\theta},t)}.$
(34)

Since $u_{x_{t}\theta}=kX_{x_{t}\theta}$, then the function
$\Psi(u_{x_{t}\theta},t)=\Psi(kX_{x_{t}\theta},t)$ represents the wave
function associated to the physical system of mass $m$ where $X_{x_{t}\theta}$
is its position at the time $t$ and the angle $\theta$. The squared module
$|{\Psi}(u_{x_{t}\theta},t)|^{2}$ represents the probability density for the
particle at time $t$ and the angle $\theta\in]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$ to be in the position $X_{x_{t}\theta}$. The
probability that the physical system’s position $X_{x_{t}\theta}$ is between
$X_{x_{t}\theta_{0}}=c$ and $X_{x_{t}\theta_{1}}=d$ is given by

$P_{c\leq X_{x_{t}\theta}\leq
d}(t)=\int^{d}_{c}|\Psi(u_{x_{t}\theta},t)|^{2}dX_{x_{t}\theta}=\int^{\theta_{1}}_{\theta_{0}}|\Psi(u_{x_{t}\theta},t)|^{2}{\partial
X_{x_{t}\theta}\over\partial\theta}d\theta,$ (35)

where $t$ and $\theta$ are the time and the angle at which the physical system
is measured. The normalization condition is given by

$\int^{+\infty}_{-\infty}|\Psi(u_{x_{t}\theta},t)|^{2}dX_{x_{t}\theta}=\int^{2\pi}_{0}|\Psi(u_{x_{t}\theta},t)|^{2}{\partial
X_{x_{t}\theta}\over\partial\theta}d\theta=1.$ (36)

The wave function (6.2) gives, using equation (31) and (32), the
correspondence principle of quantum mechanics for momentum and total energy

$\mathbb{P}=-i\hbar{\partial\Psi(u_{x_{t}\theta},t)\over\partial
u_{x_{t}\theta}}{1\over\Psi(u_{x_{t}\theta},t)},\qquad\hbox{and}\qquad\mathbb{E}=i\hbar{\partial\Psi(u_{x_{t}\theta},t)\over\partial
t}{1\over\Psi(u_{x_{t}\theta},t)}.$ (37)

The action functions (5.1) and (6.2) for all
$(t,\theta)\in[t_{a},t_{b}]\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$, verify the following equalities
${\partial\mathbb{S}\over\partial u_{x_{t}\theta}}=k\
{\partial{S}\over\partial X_{x_{t}\theta}}$, ${\partial\mathbb{S}\over\partial
t}=i\ {\partial{S}\over\partial t}$, and
${\partial\mathbb{S}\over\partial\theta}=i\ {\partial{S}\over\partial\theta}$.
Indeed from equation (31) and (27), we have ${\partial\mathbb{S}\over\partial
u_{x_{t}\theta}}=\mathbb{P}=m\mathbb{V}_{t}=mk{V}_{t}=k{\partial{S}\over\partial
X_{x_{t}\theta}}$. The chain rule and the use of
$u_{x_{t}\theta}=kX_{x_{t}\theta}$ give ${\partial\mathbb{S}\over\partial
t}={\partial\mathbb{S}\over\partial u_{x_{t}\theta}}{\partial
u_{x_{t}\theta}\over\partial t}=k{\partial{S}\over\partial
X_{x_{t}\theta}}k{\partial X_{x_{t}\theta}\over\partial
t}=i{\partial{S}\over\partial t}$. The equality
${\partial\mathbb{S}\over\partial\theta}=i\ {\partial{S}\over\partial\theta}$
is evident using equalities (33) and (19).

###### Lemma 2

Let $\mathbb{S}$ be the action function defined by (6.2), and $\Psi$ the wave
function defined by (6.2), then for all
$(t,\theta)\in[t_{a},t_{b}]\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$, we have

${\partial\mathbb{S}\over\partial\theta}=-{i\hbar\over
2\beta}\quad\Longleftrightarrow\quad\mathbb{P}\ u_{x_{t}\theta}={i\hbar\over
2},$ (38)

where $u_{x_{t}\theta}$ is the virtual position function (the position
function multiplied by a complex number $k$ such that $k^{2}=i$), and where
$\mathbb{P}=m{\mathbb{V}}_{t}=k\ (mV_{t})$ the virtual momentum.

Proof: Using equality (33) together with
${\partial\mathbb{S}\over\partial\theta}=i\ {\partial{S}\over\partial\theta}$
for all $(t,\theta)\in[t_{a},t_{b}]\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$, gives
$i{\partial{S}\over\partial\theta}=-{i\hbar\over 2\beta}$, then from Lemma 1
we have ${P}\ X_{x_{t}\theta}={\hbar\over 2}$, and the multiplication by
$k^{2}$ allows to conclude.

###  6.3 Infinity of paths of least time and derivation of Schrödinger’s
equation

Using the wave function (6.2) we have the following:

###### Theorem 1

Let $\mathbb{S}$ be the action function defined by (6.2) that verifies
equations (31), (32), and (33), and let $\Psi$ be the wave function defined by
(6.2), then for all
$(t,\theta)\in[t_{a},t_{b}]\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$, the wave function $\Psi$ satisfies the time
evolution equation:

$i{\hbar}{\partial\Psi(u_{x_{t}\theta},t)\over\partial t}+{\hbar^{2}\over
2m}{\partial^{2}\Psi(u_{x_{t}\theta},t)\over\partial
u_{x_{t}\theta}^{2}}=\Phi(u_{x_{t}\theta},t)\Psi(u_{x_{t}\theta},t),$ (39)

where the potential function is given by
$\Phi(u_{x_{t}\theta},t)=-{\mathbb{P}^{2}\over 2m}$.

Proof. We denote $\Psi:=\Psi(u_{x_{t}\theta},t)$ for simplicity, for all
$t\in[t_{a},t_{b}]$ there exists $n\in\hbox{\tenbb N}$ such that
$x_{t}\in[2nr,2(n+1)r]$, then for all $x_{t}\in]2nr,2(n+1)r[$ and for all
$\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$ the
equation (37) for momentum gives

$\mathbb{V}_{t}=-{i\hbar\over m}\ {\partial\Psi\over\partial
u_{x_{t}\theta}}{1\over\Psi},$ (40)

and since $\mathbb{V}_{t}$ verify equation (29), then the substitution of
equations (40) in equation (29) gives

${i\hbar\over m}{\partial\Psi\over\partial
u_{x_{t}\theta}}{1\over\Psi}=-{i\hbar\over
m}\beta\Big{(}\frac{\partial}{\partial\theta}\Big{(}{\partial\Psi\over\partial
u_{x_{t}\theta}}{1\over\Psi}\Big{)}\Big{)}.$ (41)

The chain rule gives

${\partial\Psi\over\partial u_{x_{t}\theta}}={\partial\Psi\over\partial
t}{\partial t\over\partial u_{x_{t}\theta}}={\partial\Psi\over\partial
t}{1\over\mathbb{V}_{t}}$ (42)

and

$\frac{\partial}{\partial\theta}\Big{(}{\partial\Psi\over\partial
u_{x_{t}\theta}}{1\over\Psi}\Big{)}=\frac{\partial}{\partial
u_{x_{t}\theta}}\Big{(}{\partial\Psi\over\partial
u_{x_{t}\theta}}{1\over\Psi}\Big{)}\frac{\partial
u_{x_{t}\theta}}{\partial\theta}=\Big{(}{\partial^{2}\Psi\over\partial
u_{x_{t}\theta}^{2}}{1\over\Psi}-\Big{(}{\partial\Psi\over\partial
u_{x_{t}\theta}}\Big{)}^{2}{1\over\Psi^{2}}\Big{)}\frac{\partial
u_{x_{t}\theta}}{\partial\theta}.$ (43)

Using equations (42) and (43) in equation (41) gives

${i\hbar\over m}{1\over\Psi}{\partial\Psi\over\partial
t}{1\over\mathbb{V}_{t}}=-{i\hbar\over
m}\beta\Big{(}{\partial^{2}\Psi\over\partial
u_{x_{t}\theta}^{2}}{1\over\Psi}-\Big{(}{\partial\Psi\over\partial
u_{x_{t}\theta}}\Big{)}^{2}{1\over\Psi^{2}}\Big{)}\frac{\partial
u_{x_{t}\theta}}{\partial\theta}.$ (44)

Using equation (28) in equation (44), we obtain

${i\hbar\over m}{1\over\Psi}{\partial\Psi\over\partial
t}{1\over\mathbb{V}_{t}}=-{i\hbar\over m}\Big{(}{\partial^{2}\Psi\over\partial
u_{x_{t}\theta}^{2}}{1\over\Psi}-\Big{(}{\partial\Psi\over\partial
u_{x_{t}\theta}}\Big{)}^{2}{1\over\Psi^{2}}\Big{)}(-u_{x_{t}\theta}).$ (45)

The multiplication of both sides of the equation (45) by $m\Psi\mathbb{V}_{t}$
together with the consideration of equation (37) for momentum gives

$i\hbar{\partial\Psi\over\partial
t}=i\hbar\Big{(}{\partial^{2}\Psi\over\partial
u_{x_{t}\theta}^{2}}-{\mathbb{P}^{2}\over(i\hbar)^{2}}\Psi\Big{)}\mathbb{V}_{t}\
(u_{x_{t}\theta}).$ (46)

Lemma 2 gives

$i\hbar{\partial\Psi\over\partial
t}=i\hbar\Big{(}{\partial^{2}\Psi\over\partial
u_{x_{t}\theta}^{2}}-{\mathbb{P}^{2}\over(i\hbar)^{2}}\Psi\Big{)}({i\hbar\over
2m}),$ (47)

that is to say

$i{\hbar}{\partial\Psi\over\partial t}+{\hbar^{2}\over
2m}\Big{(}{\partial^{2}\Psi\over\partial
u_{\delta\varepsilon}^{2}}\Big{)}=-{\mathbb{P}^{2}\over 2m}\Psi,$ (48)

which concludes the proof.

##  7 Zero point energy, Van Der Waals torque and Casimir effect

To understand causality between energy variation and geometry variation of
paths of least time, consider a physical system that has a weak interaction
with matter, with non zero mass $m$, zero charge and that generates
interference pattern in the Youngs double-slit experiment following the
infinity of paths of least time (7) for all $\theta\in[0,{2\pi}]$ (as example
of physical system: neutrino). Let $\mathbb{S}$ be the action function defined
by (6.2) that verifies equations (31), (32), and (33), and let $\Psi$ be the
wave function defined by (6.2), that satisfies for all
$(t,\theta)\in[t_{a},t_{b}]\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$, the time evolution equation (39).

###  7.1 Zero point energy and energy fluctuation

The Schrödinger equation (39) provides interesting information about the zero
point energy. Indeed

$i{\hbar}{\partial\Psi(u_{x_{t}\theta},t)\over\partial t}+{\hbar^{2}\over
2m}{\partial^{2}\Psi(u_{x_{t}\theta},t)\over\partial
u_{x_{t}\theta}^{2}}=-{\mathbb{P}^{2}\over 2m}\Psi(u_{x_{t}\theta},t),$ (49)

using the correspondence principle of energy (37) it becomes

$E\Psi(u_{x_{t}\theta},t)=-{\hbar^{2}\over
2m}{\partial^{2}\Psi(u_{x_{t}\theta},t)\over\partial
u_{x_{t}\theta}^{2}}-{\mathbb{P}^{2}\over 2m}\Psi(u_{x_{t}\theta},t),$ (50)

where the partial derivative of the product gives

${\partial^{2}\Psi(u_{x_{t}\theta},t)\over\partial
u_{x_{t}\theta}^{2}}={i\over\hbar}{\partial\over\partial
u_{x_{t}\theta}}\Big{(}{\partial\mathbb{S}\over\partial u_{x_{t}\theta}}\
\Psi(u_{x_{t}\theta},t)\Big{)}=\Big{(}{i\over\hbar}{\partial^{2}\mathbb{S}\over\partial
u_{x_{t}\theta}^{2}}+({i\over\hbar})^{2}\Big{(}{\partial\mathbb{S}\over\partial
u_{x_{t}\theta}}\Big{)}^{2}\Big{)}\Psi(u_{x_{t}\theta},t).$ (51)

The substitution of (51) in (50) gives

$E\Psi(u_{x_{t}\theta},t)=\Big{(}{-i\hbar\over
2m}{\partial^{2}\mathbb{S}\over\partial u_{x_{t}\theta}^{2}}+{1\over
2m}\Big{(}{\partial\mathbb{S}\over\partial
u_{x_{t}\theta}}\Big{)}^{2}-{\mathbb{P}^{2}\over
2m}\Big{)}\Psi(u_{x_{t}\theta},t),$ (52)

and the substitution of equation (31) in (52) gives

$E\Psi(u_{x_{t}\theta},t)=\Big{(}{-i\hbar\over
2m}{\partial^{2}\mathbb{S}\over\partial
u_{x_{t}\theta}^{2}}+{\mathbb{P}^{2}\over 2m}-{\mathbb{P}^{2}\over
2m}\Big{)}\Psi(u_{x_{t}\theta},t)=\Big{(}{-i\hbar\over
2m}{\partial^{2}\mathbb{S}\over\partial
u_{x_{t}\theta}^{2}}\Big{)}\Psi(u_{x_{t}\theta},t),$ (53)

that is to say

$E={-i\hbar\over 2m}{\partial^{2}\mathbb{S}\over\partial
u_{x_{t}\theta}^{2}}.$ (54)

The chain rule as well as the use of equation (31), (28), (29), and (38) lead
to find

${\partial^{2}\mathbb{S}\over\partial
u_{x_{t}\theta}^{2}}=m{\partial\mathbb{V}_{t}\over\partial
u_{x_{t}\theta}}=m{\partial\mathbb{V}_{t}\over\partial\theta}{\partial\theta\over\partial
u_{x_{t}\theta}}={\mathbb{P}\over u_{x_{t}\theta}}={i\hbar\over
2u_{x_{t}\theta}^{2}}$ (55)

and the substitution of equation (55) in equation (54) gives

$E={\hbar^{2}\over 4mu_{x_{t}\theta}^{2}}={\hbar^{2}\over
4mk^{2}X_{x_{t}\theta}^{2}}=-i{\hbar^{2}\over 4mX_{x_{t}\theta}^{2}}=i|E|,$
(56)

where the energy magnitude of the physical system is given by

$|E|={\hbar^{2}\over 4mX_{x_{t}\theta}^{2}}.$ (57)

####  7.1.1 Variation of the energy magnitude of the physical system

The energy magnitude (57) is defined for all
$(x_{t},\theta)\in]2nr,(2n+2)r[\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$, varies as a convex function for
$x_{t}\in]2nr,(2n+2)r[$ (see Fig.12) and varies as a convex function for
$\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$ (see
Fig.12).

Figure 11: Variation of the function $|E|_{\theta}={k\over\cos^{2}\theta}$,
k=1, for all $\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over
2}\\}$. For a small constant $k={\hbar^{2}\over 4mr^{2}}$ the graph of
$|E|_{\theta}$ for all $\theta\in]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$ remain convex but the minimums will be more small.

Figure 12: Variation of the energy $|E|={k\over 2^{2}-(x_{t}-2)^{2}}$ with k=1
function of $x_{t}\in]0,4[$. The graph of $|E|$ defined in (57) for
$\theta=0$, $k={\hbar^{2}\over 4m}$ is convex function and its graph is
similar to this graph.

Indeed,

i) The variation of energy magnitude (57) for $x_{t}\in]2nr,(2n+2)r[$ is
related to the paths of least time fluctuation (it increases near the
antipodal points $(2nr,0,0)$ and $(2n+2)r,0,0)$ and is minimal at the center
of fluctuation $(2n+1)r,0,0)$ for all $n\in\hbox{\tenbb N}$). This energy
variation function of $x_{t}\in]2nr,(2n+2)r[$ characterizes the motion of the
physical system on the fluctuating paths of least time and since energy cannot
be created or destroyed, this variation is due to the continuous transfer of
part of energy of the physical system into changes of motion direction on each
path of least time in the path plan.

ii) The variation of energy magnitude (57) for
$\theta\in]2nr,(2n+2)r[\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$ is related to the change of paths of least time
function of $\theta$ (polarization). It is a transversal variation of the
energy magnitude (57) in the plan perpendicular to the paths of least time
principal axis.

Example of energy variation in 3D with respect to $(x_{t},\theta)$ is
illustrated in (Fig.14).

####  7.1.2 Space minimal energy fluctuation

The minimum of the energy magnitude (57) on $]2nr,(2n+2)r[$, for all
$\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$ can be
found. Indeed, since

Figure 13: 3D graph of $|E|={k\over(2^{2}-(x_{t}-2)^{2})\cos^{2}\theta}$ for
k=1, for all $x_{t}\in[0,4]$, and for all
$\theta\in[0,{2\pi}]\backslash\\{{\pi\over 2},{3\pi\over 2}\\}$. This is the
graph of (57) for $r=2$, $n=0$, $x_{t}\in[0,4]$ and ${\hbar^{2}\over
4mr^{2}}=k=1$. The graph of the energy (57) in 3D is similar to the graph for
a small radius $r$ and a small constant $k={\hbar^{2}\over 4mr^{2}}$.
Fluctuation of the energy magnitude $|E|$ in the $x_{t}$-axis and
$\theta$-axes.

Figure 14: Four imbedded families of fluctuating paths of least time between A
and B with radius $r_{1}<r_{2}<r_{3}<r_{4}$. The 4 families represent paths of
least time between A and B (with equal length 6$r_{1}\pi$). If $E_{0_{i}}$
i=1,..,4 is the zero point energy of the physical system following the
geodesics (9) of radius $r_{i}$ for i=1,..,4, then
$E_{0_{1}}>E_{0_{2}}>E_{0_{3}}>E_{0_{4}}$ for $r_{1}<r_{2}<r_{3}<r_{4}$.

$X_{x_{t}\theta}^{2}=\Big{(}r^{2}-(x_{t}-(2n+1)r)^{2}\Big{)}\cos^{2}\theta,$
(58)

then $0<X_{x_{t}\theta}^{2}\leq r^{2}\cos^{2}\theta$, which gives

$|E|_{\theta}:={\hbar^{2}\over
4mr^{2}\cos^{2}\theta}\leq|E|,\qquad\forall\theta\in]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\},$ (59)

the minimum $|E|_{\theta}$ is reached at $x_{t}=(2n+1)r$ (the local
fluctuation center of the paths of least time). The use of all paths of least
time by the physical system exhibits a periodic variation of the space minimal
energy $|E|_{\theta}$ of period $\pi$, see illustration of energy variation
(Fig.12). Indeed,

i) The space minimal energy $|E|_{\theta}$ is constant for each path of least
time,

ii) The space minimal energy $|E|_{\theta}$ is variable for all paths of least
time function of $\theta$.

iii) Since energy cannot be created or destroyed, the variation of space
minimum energy $|E|_{\theta}$ is consequence of a transfer of part of energy
to a change of direction in angle $\theta$ (change of path). A change of paths
of least time induces a change of the space minimum energy $|E|_{\theta}$.

####  7.1.3 The zero point energy

The magnitude $|E|_{\theta}$ define the space minimal energy for each path of
least time and by continuity of the limit of $|E|_{\theta}$ when $\theta$
tends to $0$ or $\pi$ or $2\pi$ with $\theta\neq{0,\pi,2\pi}$ gives

$\lim_{\theta\rightarrow{0,\pi,2\pi}}|E|_{\theta}={\hbar^{2}\over
4mr^{2}}=|E|_{0}.$ (60)

where $|E|_{0}$ is the ground state (the zero point energy) of the physical
system for all
$(t,\theta)\in[t_{a},t_{b}]\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$, and it is characterized based on this case study as
follow:

i) Since $|E|_{0}\neq 0$ for all
$(t,\theta)\in[t_{a},t_{b}]\times]0,{2\pi}[\backslash\\{{\pi\over
2},\pi,{3\pi\over 2}\\}$, then the ground state energy $|E|_{0}$ of the
physical system for all paths of least time can never be zero.

ii) The ground state energy $|E|_{0}$ for all paths of least time is always
constant, and it does not fluctuate. The ground state energy depends on the
fluctuation radius of paths of least time and the physical system mass.

iii) The greater the radius of paths of least time (9) is, the smaller the
zero point energy of the physical system is. That is to say, the smaller
$\lambda=4r$ (the periodic fluctuation length of each path of least time) the
bigger the zero point energy of the physical system is.

iv) The zero point energy can decrease if the physical system changes from an
infinity of fluctuating geodesics (9) with radius $r_{1}$ to another infinity
of fluctuating geodesics (9) with radius $r_{2}>r_{1}$ between two distant
locations under the action of an external form of energy (photon for example),
then the zero point energy of the physical system will decrease due to this
interaction by mean of part of the zero point energy transferred to increase
the fluctuation radius of paths of least time. However the lengths of the
paths of least time (9) between the two distant locations with radius $r_{1}$
or $r_{2}$ are equal with conservation of the least time (see Fig.14). The
variation of the zero point energy indicates a change of the physical system
fluctuation under the effect of external interaction (the photon). This is
what happens when observation is used in Young’s double-slit experiment.

###  7.2 Space minimal energy variation and torque effect interpretation

Investigation about the Van Der Waals torque between anisotropic parallel
plates can be elaborated using this formalism. Indeed a torque phenomenon is
normally generated in the presence of parallel plates with anisotropic optical
properties (such as birefringent crystals) causing a mechanical rotation
towards the alinement of the plates principal axes. Interpretation of the Van
Der Waals torque can be approached using the fluctuating paths of least time.

Figure 15: Positions where the Van der Waals torque effect can be measured for
anisotropic plates. If the plate is positioned on an antipodal point in
positions A, C, and E, then there is no torque effect. The Van Der Waals
torque can be measured at any position between plates A and C or between
plates C and E.

When anisotropic plate is placed (with oblique position of the optical axis)
in a fixed position $x_{t}\in]2nr,(2n+2)r[$ transversal to the paths of least
time principal axis direction of the physical system (the $x_{t}$-axes), the
energy (57) of the physical system in the transversal plan is a convex
function of $\theta$ (Fig.12) and constant for each
$\theta\in[0,{2\pi}]\backslash\\{{\pi\over 2},{3\pi\over 2}\\}$. The
phenomenon that generates the torque is a consequence of the variation of
energy on the plate plan for each $\theta\in[0,{2\pi}]\backslash\\{{\pi\over
2},{3\pi\over 2}\\}$ due to the birefringent property of the material plate
that creates a circular polarization (see circular polarization [4],[8],[17]).
Indeed, when the physical system enters the plate following a path of least
time for a given angle $\theta$ in a direction that is oblique with respect to
the optical axis of the plate, the birefringent material changes the path of
least time of the physical system by double refraction (due to the existence
of a variation in refractive index that is sensitive to direction in
anisotropic material) which changes the angle $\theta$ of the path of least
time in the plate plan from a fluctuating linear motion to helicoidal motion
(circular polarization) that generates variation of the energy magnitude (57)
of the physical system in the plate plan. This variation of energy for each
path is transferred into a mechanical rotation. Thus, if the motion of the
physical system is blocked, the plate will rotate back to the equilibrium
position that defines the stable initial energy at the position $x_{t}$ in the
plan plate.

The use of the infinity of paths of least time (illustrated in Fig.2) in the
interpretation of Van der Waals torque effect suggests that:

a) If the plate is positioned at an antipodal point ($x_{t}=2nr$, or
$x_{t}=(2n+2)r$) then there is no torque effect generated on the plate since
there is no possible change of paths of least time on those points (Fig.15).

b) The torque effect can be measured for all positions $x_{t}\in]2nr,(2n+2)r[$
for all $n\in\hbox{\tenbb N}$. Indeed, the presence of a metal plate optically
anisotropic in a transversal position to paths of least time principal axis
changes the energy magnitude (57) of the physical system function of $\theta$,
thus the rate of change of (57) with respect to $\theta$ using (58) gives

$F={\partial|E|\over\partial\theta}=\Big{(}{\hbar^{2}\over
4mX_{x_{t}\theta}^{2}\cos^{2}\theta}\Big{)}{\sin 2\theta}=2|E|\tan\theta$ (61)

which represents the amount of rotational energy gained by infinitesimal
rotation. The position where this amount of energy is extremum is determined
through the solution of equation ${\partial F\over\partial x_{t}}=0$ that is
to say

${\partial F\over\partial x_{t}}={\partial^{2}|E|\over\partial
x_{t}\partial\theta}=2{\partial|E|\over\partial
x_{t}}\tan\theta={\hbar^{2}\tan\theta\over
m\cos^{2}\theta}{(x_{t}-(2n+1)r)\over\Big{(}r^{2}-(x_{t}-(2n+1)r)^{2}\Big{)}^{2}}=0$
(62)

which gives that the amount of rotational energy gained by infinitesimal
rotation is minimal or maximal at the fluctuation center $x_{t}=(2n+1)r$
according to the sign of $\tan\theta$. Thus

i) For $\tan\theta>0$, the amount of rotational energy gained by infinitesimal
rotation is a minimum at the fluctuation center $x_{t}=(2n+1)r$ and equal to

$F=\Big{(}{\hbar^{2}\over 2mr^{2}\cos^{2}\theta}\Big{)}\tan\theta$ (63)

ii) For $\tan\theta<0$, the amount of rotational energy gained by
infinitesimal rotation is maximum at the fluctuation center $x_{t}=(2n+1)r$
and equal to (63)

###### Remark 1

a) For $\theta\in]0,{\pi\over 2}[\cup]\pi,{3\pi\over 2}[$ we have
$\tan\theta>0$, and for $\theta\in]0,{-\pi\over 2}[\cup]-\pi,{-3\pi\over 2}[$
we have $\tan\theta<0$, then the amount of rotational energy gained by
infinitesimal rotation at each geodesics local center of fluctuation
$x_{t}=(2n+1)r$ is minimal or maximal based on the rotation direction
clockwise or anticlockwise and is given by (63)

b) Within 1.1 mm there is an average of 2000 light wave lengths that
correspond to 4000 aligned spheres of radius r=0.0001375 mm using the
geodesics illustrated in (Fig.2) with 4001 antipodal points. Then based on
this case study within 1.1 mm there are 4001 positions where the Van der Waals
torque is null which explains the difficulty to measure Van der Waals torque
effect experimentally.

###  7.3 About the attractive repulsive Casimir force origin

Investigation about attractive-repulsive Casimir phenomena in vacuum between
two distant parallel uncharged plates can be elaborated using this formalism.
Indeed the placement of a first uncharged perfectly conducting plate 1 in a
position transversal to the paths of least time of the considered physical
system induces a change of its energy magnitude (57). The rate of change of
this energy (57) with respect to paths of least time $X_{x_{t}\theta}$ is
given by

${\partial|E|\over\partial X_{x_{t}\theta}}={-\hbar^{2}\over
2mX_{x_{t}\theta}^{3}}.$ (64)

Meanwhile, the placement of a second uncharged perfectly conducting plate 2,
closed and parallel to the plate 1, in a position transversal to the paths of
least time of the considered physical system induces a second changes of its
energy magnitude, and its rate of change with respect to paths of least time
$X_{x_{t}\theta}$ is given by

${\partial^{2}|E|\over\partial X_{x_{t}\theta}^{2}}={3\hbar^{2}\over
2mX_{x_{t}\theta}^{4}}.$ (65)

This last quantity indicates how fast the magnitude $|E|$ is increasing or
decreasing with respect to geodesics variation. Thus, equality (65) represents
an acceleration experienced by the physical system due to the change of energy
(consequences of the placement of the two closed plates). This local change of
energy (65) is transformed into a local force experienced by the physical
system given by

$F=m{\partial^{2}|E|\over\partial X_{x_{t}\theta}^{2}}={3\hbar^{2}\over
2X_{x_{t}\theta}^{4}},$ (66)

this force is also experienced by the plates and might be the origin of
Casimir effect. Indeed, the use in $X_{x_{t}\theta}$ of the parameter

$x_{t}=\pm\sqrt{r^{2}-{D^{2}\over\cos^{2}\theta}\sqrt{{360\hbar\over
c\pi^{2}}}}+(2n+1)r$ (67)

for all $\theta\in]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over 2}\\}$
gives

$X^{2}_{x_{t}\theta}=\Big{(}r^{2}-(x_{t}-(2n+1)r)^{2}\Big{)}\cos^{2}\theta=D^{2}\sqrt{{360\hbar\over
c\pi^{2}}}$ (68)

which gives in (66)

$F={3\hbar^{2}\over 2X_{x_{t}\theta}^{4}}={3\hbar^{2}c\pi^{2}\over
2{D^{4}360\hbar}}={\hbar c\pi^{2}\over 240{D^{4}}},$ (69)

which is the magnitude of Casimir force ([5]) felt by the plates by unity of
surface, where c is the speed of light , $\hbar$ is the reduced Planck
constant, and D is the distance that separates the plates.

###  7.4 Attractive or repulsive Casimir force causality interpretation

The magnitude energy $|E|$ defined in (57) of the physical system is a convex
function on the product
$]2nr,(2n+2)r[\times\Big{(}]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over
2}\\}\Big{)}$ for $n\in\hbox{\tenbb N}$ as illustrated in Fig.17.

Figure 16: Illustration of position and energy $|E|$ variation given by (57)
on $]2nr,(2n+2)r[\times(]0,{2\pi}[\backslash\\{{\pi\over 2},\pi,{3\pi\over
2}\\})$ for $n$ integer. The presence of plates A at $x_{t_{1}}$ and C at
$x_{t_{3}}$ induces energy variation, a transfer of part of energy into a
mechanical force (69) experienced by the plates A and C to move to a position
with lower energy (after transfer). The lower energy position is located in
the direction of each arrow. No attraction/repulsion at the positions
$x_{t_{2}}$ and $x_{t_{5}}$

Figure 17: Parallel plates: Based on illustration (Fig.17) the plate A cross
the $x_{t}$-axis at $x_{t_{1}}$, the plate B cross the $x_{t}$-axis at
$x_{t}=(2n+1)r$ , the plate C cross the $x_{t}$-axis at $x_{t_{3}}$ , the
plate D cross the $x_{t}$-axis at $x_{t_{4}}$, the plate E cross the
$x_{t}$-axis at $x_{t}=(2n+3)r$. All plates are parallel to the yz-plan. Only
two plates are considered for any interpretation of the attraction or
repulsion of the plates under Casimir force.

The magnitude energy (57) is determined at each $x_{t}\in]2nr,(2n+2)r[$, and
when a plate A is introduced in a position $x_{t_{1}}$ (Fig.17) parallel to
the $|E|z$-plan located at the left side of the fluctuation center
$x_{t}=(2n+1)r$ followed by the introduction of a second parallel plate C in
the the position $x_{t_{3}}$ at the right side of the fluctuation center
$x_{t}=(2n+1)r$ on paths of least time (Fig.17 and Fig.17), perturbation of
the energy state of the physical system is created and given by (65). Since
energy cannot be created or destroyed, rather it can only be converted into
another form of energy then the perturbation of the energy is manifested by a
transfer of part of the physical system energy into a mechanical force (69)
experienced by the plates to move from positions $x_{t_{1}}$ and $x_{t_{3}}$
to a new position with lower energy after transfer (see Fig.17). The lower
energy position of the plate A will be located to its right and a lower energy
position for the plate C will be located to its left based on the graph of the
energy magnitude illustrated in (Fig.17). The two plates will experience the
Casimir attractive force. However, if the considered plates are the plate C
and the plate D at the position $x_{t_{3}}$ and $x_{t_{4}}$ (Fig.17, and
Fig.17), the lower energy position for the plate C will be located to its left
and the lower energy position for the plate D will be located to its right
based on the graph of the energy magnitude. Therefore the two plates will
experience the Casimir repulsive force. The repulsive force or attractive
force are functions of the position of plates with respect to the fluctuation
centers of paths of least time $x_{t}=(2n+1)r$ in each local interval
$]2nr,(2n+2)r[$ for all $n\in\hbox{\tenbb N}$ and we have:

i) The repulsive or attractive Casimir force is due to the energy magnitude
variation of the physical system, consequence of the presence of closed
plates. This variation occurs by mean of a transfer of part of energy of (57)
at $x_{t}$ to repulsive or attractive motion. When a part of energy (57) at
$x_{t}$ is transferred, the plate will move to a position $x_{t^{\prime}}$
that fits the remaining energy (lower energy). The plates under Casimir force
always move to a new position with lower energy after transfer. Nevertheless,
the effect of Casimir force is local because the perturbation of the energy is
local.

ii) Attractive Casimir force: if the two positions are located at the left and
right side of the local fluctuation center $x_{t}=(2n+1)r$ then the two plates
will experience attraction. Attraction is also experienced for two plates when
one is positioned in the left side of a local fluctuation geodesic center, and
a second positioned in the right side of the next local fluctuation geodesic
center (Fig.17).

iii) Repulsive Casimir force: if one plate is located at the right side of the
center of the local interval $]2nr,(2n+2)r[$ and a second plate is located at
the left side of the center of the local interval $](2n+2)r,(2n+4)r[$ or
$](2n+4)r,(2n+6)r[$, then the two plates will experience a repulsive force
(Fig.17).

vi) if the two positions are located at two centers of fluctuation
$x_{t_{1}}=(2n+1)r$ and $x_{t_{2}}=(2n+3)r$ for all $n\in\hbox{\tenbb N}$,
then the plates won’t experience neither attraction nor repulsion. This case
can be justified. Indeed, we have ${\partial|E|\over\partial
X_{x_{t}\theta}}\Big{|}_{x_{t}=(2n+1)r}={\partial|E|\over\partial
x_{t}}{\partial x_{t}\over\partial X_{x_{t}\theta}}=0$ since
${\partial|E|\over\partial x_{t}}={\hbar^{2}(x_{t}-(2n+1)r)\over
2m\cos^{2}\theta(r^{2}-(x_{t}-(2n+1)r)^{2})^{2}}\Big{|}_{x_{t}=(2n+1)r}=0$.
Moreover at these positions the energy is minimal and it cannot be reduced to
a lower energy level, then there is no part of the energy that can be
transferred into another form (Fig.17).

##  8 Conclusion

The use of an infinity of fluctuating paths of least time that are compatible
with the quantum mechanics indeterminacy provides a new interpretation in
geometrical optic of the interference pattern of Young’s double-slit
experiment, not only in the detector screen but also in the whole geodesics
intersection region between the slits and detector screen, which suggests that
the wave aspect of matter and radiation (photons, electrons, neutrons, atoms
and molecules) is actually dictated by the space-time local geometry. Matter
wave nature is merely arising from interaction of the physical systems with
the space-time geodesics that induces a periodic fluctuating energy. Following
the essence of Feynman path integral, the association of a density of
probability of presence of the physical system to each used path of least time
(as well as a wave function) leads to derive the Schrödinger equation starting
from the geodesic’s characteristics together with an uncertainty relation
between position an momentum. The energy of the physical system extracted from
the Schrödinger equation has provided a valuable information about the origin
of the phenomenon of the Van Der Waals torque between anisotropic parallel
plates, as well as a clear mechanism of the attractive/repulsive Casimir
force. Energy variation appears to be a manifestation of the interaction of
the physical system with the space time geodesics, where the ground state
energy is a residual energy (the minimal energy limit) that cannot vanish. It
is the minimum energy conceived by the physical system that characterizes the
fluctuation radius of paths of least time in the medium. This is how the
space-time dictates the behavior of the physical system at the quantum scale
based on this case study.

## References

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# The repulsive Euler-Poisson equations with variable doping profile

Olga S. Rozanova<EMAIL_ADDRESS>Department of Mechanics and
Mathematrics, Moscow State University, Moscow 119991 Russia

###### Abstract

We prove that arbitrary small smooth perturbations of the zero equilibrium
state of the repulsive pressureless Euler-Poisson equations, which describe
the behavior of cold plasma, blow up for any non-constant doping profile
already in one-dimensional space. We also propose a numerical procedure that
allows one to find the blow-up time for any initial data and present examples
of such calculations for various doping profiles for standard initial data,
corresponding to the laser pulse.

###### keywords:

Euler-Poisson system , equations of cold plasma , singularity formation ,
variable doping profile

###### MSC:

35F55 , 35Q60 , 35B44 , 35B20 , 35L80

##  1 Introduction

The system of Euler-Poisson equations describing the behavior of cold plasma
in ${\mathbb{R}}^{d}$ in the repulsive case has the following form:

$\dfrac{\partial V}{\partial
t}+(V\cdot\nabla)V=-\nabla\Psi,\quad\dfrac{\partial n}{\partial t}+{\rm
div}\,(nV)=0,\quad\Delta\Psi=c(x)-n.$ (1)

The components of the solution $V$, $n>0$, $\Psi$ are the velocity, electron
density and electric field potential, respectively, they depend on the time
$t\geqslant 0$ and the point $x\in{\mathbb{R}}^{d}$. A fixed $C^{1}$ \- smooth
function $c(x)>0$ is the density background or the so-called doping profile.

An extensive literature is devoted to the Euler-Poisson equations. They
consider cases of repulsive and attractive forces, zero and non-zero
background density. The behavior of the solution in each of these cases is
significantly different. The case of repulsive force arises when modeling the
phenomena of plasma physics and semiconductors, that is, a medium consisting
of electrons. The attractive case refers to a medium in which the
gravitational force acts between particles, that is, to astrophysical models
(system (1) corresponds to this case after changing the sign in front of
$\Psi$ in the first equation). The Euler-Poisson system was considered both
with and without pressure. In all cases, there exist results about the local
existence of a solution to the Cauchy problem; it is known that there are
initial data at which the solution to the Cauchy problem loses its original
smoothness [16]; the possibility of constructing a global weak solution was
investigated (we refer to [3] for a recent review). An interesting topic is
the study of critical thresholds, that is, the most accurate separation of the
initial Cauchy data into two classes, one of which includes those that
correspond to globally smooth solutions, and the other to solutions that
develop a singularity over a finite time. This work began in [10], where
criteria were derived for basic one-dimensional and some more complex
situations involving the pressureless Euler-Poisson equations. The current
state of the problem can be found in [2].

In this paper we consider repulsive Euler-Poisson equations with a non-zero
constant density background describing cold plasma [8]. Recently, interest in
such models has increased significantly due to the possibility of creating
accelerators on the wake wave [11]. This is the most difficult case of the
Euler-Poisson equations, since the solutions are oscillating and even the zero
equilibrium is usually unstable due to the phenomenon of nonlinear resonance.

Since cold plasma is a very unstable medium, from a practical point of view,
questions about the possibility of choosing a regime that would guarantee a
smooth solution for as long as possible are relevant. However, in the space of
many spatial variables ${d}\geqslant 2$, ${d}\neq 4$, all physically
reasonable solutions lose smoothness in a finite time [14], [15]. However, for
$d=1$ and a constant background density $c(x)=C$, there is a sufficiently
large neighborhood of the zero equilibrium $V=V_{x}=\Phi_{x}=0$, $n=C$, such
that the solution with data from this neighborhood remains smooth for all
$t>0$ [10], [13] (in the case of pressure this also occurs [9]). It was
recently shown that for exceptional dimension $d=4$ there is also a
neighborhood of the zero equilibrium corresponding to a globally smooth
solution.

In this paper, we show that this phenomenon does not persist under any non-
constant background, including an arbitrarily small perturbation of a constant
one. In other words, if there is a point $x_{0}\in\mathbb{R}$ such that
$c^{\prime}(x_{0})\neq 0$, then any solution whose data are an arbitrarily
small perturbation of the equilibrium $V=0$, $\Psi=\rm const$ blows up in a
finite time.

The paper is organized as follows. In Sec.2 we rewrite (1) in terms of
velocity and electric field, which allows us to consider the dynamics of
solutions along characteristics and obtain a system of ODEs for this purpose.
We then linearize the above system of ODEs using Radon’s lemma and reduce the
blow-up problem to the problem of vanishing of a special scalar function,
which obeys a third order linear equation with time-dependent coefficients.
This problem can be studied numerically, and in Sec.4 we give examples of
computation of such kind for several doping profiles. In Sec.3 we consider a
perturbation of the zero equilibrium and choose the size of the perturbation
as a small parameter. Then we use the Floquet theory to prove that the
solution of the Cauchy problem with arbitrary data blow up in a finite time
due to nonlinear resonance. Sec.5 is devoted to discussion. In particular, we
outline a method for proving that, in the case of a damped Euler-Poisson
system, there exists a neighborhood of the zero equilibrium such that if the
initial data together with its derivatives are sufficiently small, then the
corresponding solution is globally smooth in time and tends to the zero
equilibrium at $t\to\infty$.

##  2 Analysis along characteristics

We introduce the electric field vector $E=\nabla\Psi$ and rewrite (1) in terms
of $V$ and $E$, see, e.g. [5] for details. In the case ${d}=1$ this results in

$\dfrac{\partial V}{\partial t}+V\dfrac{\partial V}{\partial
x}=-E,\quad\dfrac{\partial E}{\partial t}+V\dfrac{\partial E}{\partial
x}=c(x)V,\quad n=c(x)-\dfrac{\partial E}{\partial x}.$ (2)

We consider (2) with the Cauchy data

$(V,E)|_{t=0}=(V_{0}(x),E_{0}(x))\in{C^{2}}({\mathbb{R}}).$ (3)

The problem (2), (3) has a solution locally in time that is as smooth as the
initial data, and the formation of a singularity is associated with the blow-
up of the ether of the solution components themselves or their first
derivatives, e.g.[17], [1].

Thus, along characteristics $x=x(t)$, starting from a point
$x_{0}\in\mathbb{R}$ the solution $(V(x(t)),E(x(t)))$ obeys the system of ODEs

$\dot{x}=V,\quad\dot{V}=-E,\quad\dot{E}+V=c(x)V,$ (4)

with the initial data

$x(0)=x_{0},\quad V(0)=V_{0}(x_{0}),\quad E(0)=E_{0}(x_{0}).$ (5)

Next, let us denote $V_{x}=v$, $E_{x}=e$. Then, differentiating (2) with
respect to $x$, we obtain the following system along the characteristic
$x=x(t)$:

$\dot{v}=-v^{2}-e,\quad\dot{e}=-sv+c(x)v+c^{\prime}(x)V,$ (6)

with the initial data

$v(0)=V^{\prime}_{0}(x_{0}),\quad s(0)=E^{\prime}_{0}(x_{0}).$ (7)

We see that if $c(x)=C=\rm const$, then system (6) splits from (4) and can be
easily integrated. In this way, it is possible to obtain an analytical
criterion for the formation of a singularity in terms of the initial data
[13], [10]. It implies that if the initial data (3) are such that for any
$x_{0}\in\mathbb{R}$ the inequality

$\displaystyle\left(V^{\prime}_{0}(x_{0})\right)^{2}+2\,E^{\prime}_{0}(x_{0})-C<0$
(8)

holds, then the solution is periodic in $t$ and retains initial smoothness for
all $t>0$. We see that for $C>0$ there is a neighborhood of equilibrium
$V=E=0$ in the $C^{1}$ \- norm such that a solution starting from this
neighborhood preserves smoothness.

Obviously, if $x_{0}\in\mathbb{R}$ is such that $c^{\prime}(x_{0})=0$, then
from the previous consideration it follows that the solution does not blow up
along the specific characteristic $x(t)=V$, $x(0)=x_{0}$.

In the case $c^{\prime}(x_{0})\neq 0$ we can consider (6) as a system for $v$
and $s$ with known from (4) time-dependent coefficients. Using the Radon
theorem [12], [7], we can linearize this system. For convenience, we present
the statement of this theorem here.

######  Theorem 2.1.

[The Radon lemma] A matrix Riccati equation

$\dot{W}=M_{21}(t)+M_{22}(t)W-WM_{11}(t)-WM_{12}(t)W,$ (9)

($W=W(t)$ is a matrix $(n\times m)$, $M_{21}$ is a matrix $(n\times m)$,
$M_{22}$ is a matrix $(m\times m)$, $M_{11}$ is a matrix $(n\times n)$,
$M_{12}$ is a matrix $(m\times n)$) is equivalent to the homogeneous linear
matrix equation

$\dot{Y}=M(t)Y,\quad M=\left(\begin{array}[]{cc}M_{11}&M_{12}\\\
M_{21}&M_{22}\end{array}\right),$ (10)

($Y=Y(t)$ is a matrix $(n\times(n+m))$, $M$ is a matrix $((n+m)\times(n+m))$ )
in the following sense.

Let on some interval ${\mathcal{J}}\in\mathbb{R}$ the matrix-function
$\,Y(t)=\left(\begin{array}[]{c}{Q}(t)\\\ {P}(t)\end{array}\right)$ (${Q}$ is
a matrix $(n\times n)$, ${P}$ is a matrix $(n\times m)$)  be a solution of
(10) with the initial data

$Y(0)=\left(\begin{array}[]{c}{\mathbb{I}}\\\ W_{0}\end{array}\right)$

($\mathbb{I}$ is the identity matrix $(n\times n)$, $W_{0}$ is a constant
matrix $(n\times m)$)  and $\det{Q}\neq 0$ on ${\mathcal{J}}$. Then
$W(t)={P}(t){Q}^{-1}(t)$ is the solution of (9) with $W(0)=W_{0}$ on
${\mathcal{J}}$.

We see that system (6) can be written as (9) with

$\displaystyle W=\begin{pmatrix}v\\\ e\end{pmatrix},\quad
M_{11}=\begin{pmatrix}0\\\ \end{pmatrix},\quad M_{12}=\begin{pmatrix}1&0\\\
\end{pmatrix},$ $\displaystyle M_{21}=\begin{pmatrix}0\\\
c^{\prime}(x(t))V(x(t))\end{pmatrix},\quad M_{22}=\begin{pmatrix}0&-1\\\
c(x(t))&0\\\ \end{pmatrix}.$

Then, according Theorem 2.1 the solition of (6) is

$\displaystyle\ \quad W(t)=\frac{P(t)}{Q(t)},$

where $P(t)=(p_{1}(t),p_{2}(t))^{T}$ and $Q(t)$ solves the linear system

$\displaystyle\left(\begin{array}[]{c}\dot{Q}\\\ \dot{p}_{1}\\\
\dot{p}_{2}\end{array}\right)=M\left(\begin{array}[]{c}Q\\\ p_{1}\\\
p_{2}\end{array}\right),\quad M=\begin{pmatrix}0&1&0\\\ 0&0&-1\\\
c^{\prime}(x)V&c(x)&0\\\ \end{pmatrix}.$

subject to the initial data

$\displaystyle\left(\begin{array}[]{c}Q(0)\\\
P(0)\end{array}\right)=\left(\begin{array}[]{c}1\\\
W_{0}\end{array}\right),\quad W_{0}=\left(\begin{array}[]{c}v(x_{0})\\\
e(x_{0})\end{array}\right).$

Thus, if in a point $t_{*}>0$ we have $Q(t_{*})=0$, then the solution of (6)
blows in the point $t_{*}$.

One can check that the system for $(Q,p_{1},p_{2})$ can be reduced to one
linear equation for $Q(t)$:

$Q^{\prime\prime\prime}+c(x(t))Q^{\prime}+c^{\prime}(x(t))V(x(t))Q=0,$ (13)

with the initial conditions

$Q(0)=1,\quad Q^{\prime}(0)=V^{\prime}_{0}(x_{0}),\quad
Q^{\prime\prime}(0)=-E^{\prime}_{0}(x_{0}),$ (14)

found from (7).

For arbitrary initial data and an arbitrary doping profile $c(x)$ the system
(4), (13) can be solved numerically with the initial data (5), (14) for
$x_{0}\in\mathbb{R}$. This procedure can be performed in reasonable increments
with respect to $x$ to cover any desired segment $[x_{-},x_{+}]\in\mathbb{R}$.
Note that it is much easier to determine numerically whether some component of
the solution vanishes than to investigate where this component of the solution
goes to infinity or not. We provide examples of such calculations in Sec.4.

##  3 Small perturbations of the zero equilibrium

We are going to prove the following theorem.

######  Theorem 3.1.

An arbitrary small smooth perturbation of the zero solution $V=E=0$ of system
(2) blows up in a finite time for any non-constant doping profile $c(x)>0$.

In other words, if there is a point $x_{0}\in\mathbb{R}$ such that
$c^{\prime}(x_{0})\neq 0$, then a solution with data
$(V_{0}(x_{0}),E_{0}(x_{0}))$ such that $\|(V_{0}(x_{0}),E_{0}(x_{0}))\|\ll
1$, blows up in a finite time along the characteristic, starting from $x_{0}$.

Let us choose $V_{0}(x_{0})=\varepsilon$ as a small parameter. System (4)
implies

$\ddot{V}+c(x(t))V=0,\quad\dot{x}(t)=V,$ (15)

therefore if we put $V(0)=\epsilon$, $\dot{V}(0)=-E(0)=0$, then the solution
can be sought in the form of a series in $\epsilon$. From (15) we get

$\displaystyle V(t)=\varepsilon\cos\omega t+o(\varepsilon),\quad
x(t)=x_{0}+\varepsilon\frac{\sin\omega t}{\omega}+o(\epsilon),$ (16)
$\displaystyle c(x(t))=c(x_{0})+\varepsilon c^{\prime}(x_{0})\frac{\sin\omega
t}{\omega}+o(\varepsilon),\quad\omega=\sqrt{c(x_{0})}.$ (17)

Then we can substitute (16), (17) to (13) as coefficients and apply Floquet’s
theory (e.g. [4], section 2.4), which has been successfully used to study
other problems, associated with the repulsive Euler-Poisson system. (e.g. [6],
[15]). The Floquet theory deals with linear systems and equations with
periodic coefficients. According to this theory, for the fundamental matrix
$\Psi(t)$ ($\Psi(0)={\mathbb{I}}$, where ${\mathbb{I}}$ is the identity
matrix) there is a constant matrix $M$, possibly with complex coefficients
such that $\Psi(T)=e^{TM}$, where $T$ is the period of the coefficients. The
eigenvalues of the matrix of monodromy $e^{TM}$ are called the characteristic
multipliers of the system. If among the characteristic multipliers there are
those whose modulus is greater than one, then the zero solution of the linear
system under study is Lyapunov unstable ([4], Theorem 2.53).

The fundamental matrix can be found in the form of the series
$\Psi(t)=\Psi_{0}(t)+\Psi_{1}(t)\varepsilon+\Psi_{2}(t)\varepsilon^{2}+o(\varepsilon^{2}),$
$\Psi_{0}(0)=\mathbb{I}$, $\Psi_{i}(0)=0$, $i\in\mathbb{N}$. In our case

$\displaystyle\Psi(t)=\begin{pmatrix}Q_{1}(t)&Q_{2}(t)&Q_{3}(t)\\\
Q^{\prime}_{1}(t)&Q^{\prime}_{2}(t)&Q^{\prime}_{3}(t)\\\
Q^{\prime\prime}_{1}(t)&Q^{\prime\prime}_{2}(t)&Q^{\prime\prime}_{3}(t),\\\
\end{pmatrix}$

where $Q_{1}(t)$, $Q_{2}(t)$, $Q_{3}(t)$ is the fundamental system of
solutions, such that

$\displaystyle Q_{1}(0)=1,\,Q_{1}^{\prime}(0)=0,\,Q_{1}^{\prime\prime}(0)=0,$
$\displaystyle Q_{2}(0)=0,\,Q_{2}^{\prime}(0)=1,\,Q_{2}^{\prime\prime}(0)=0,$
$\displaystyle Q_{3}(0)=0,\,Q_{3}^{\prime}(0)=0,\,Q_{3}^{\prime\prime}(0)=1.$

We are looking for $Q_{i}$, $i=1,2,3,$ as a series in $\varepsilon$. At each
step, we have to solve a linear inhomogeneous system with constant
coefficients; the period $T=\frac{2\pi}{\sqrt{c(x_{0})}}$ depends on the
starting point of the characteristic.

Let us denote the eigenvalues of the matrix $\Psi(T)$ as $\lambda_{i},$
$i=1,2,3,$, and the eigenvalues of the matrix
$\Psi_{k}(T)=\Psi_{0}(T)+\Psi_{1}(T)\varepsilon+\cdots+\Psi_{k}(T)\varepsilon^{k}$,
$k\in\mathbb{N}$ as $\bar{\lambda}_{ik}$. Calculating the eigenvalues at each
step is quite cumbersome, but it can be done analytically using a computer
algebra package (for example, MAPLE). To prove instability, we need to find an
expansion up to order $k$ in $\varepsilon$ such that among
$\bar{\lambda}_{ki}$ there is a value greater than one in absolute value.

It can be readily shown that $\bar{\lambda}_{i0}=\bar{\lambda}_{i1}=1$.
However, for $k=2$ we get

$\displaystyle\bar{\lambda}_{12}=1-\frac{12^{1/3}\pi}{6}\left(\frac{(c^{\prime}(x_{0}))^{1/3}}{(c(x_{0}))^{1/2}}\right)^{5}\,\varepsilon^{5/3},$
$\displaystyle\bar{\lambda}_{22}=1+\frac{6^{1/3}\pi}{2^{5}3^{3}}\left(\frac{(c^{\prime}(x_{0}))^{1/3}}{(c(x_{0}))^{1/2}}\right)^{5}\,(1+\sqrt{3}\,\mathbbm{i})\varepsilon^{5/3},$
$\displaystyle\bar{\lambda}_{32}=1+\frac{6^{1/3}\pi}{2^{5}3^{3}}\left(\frac{(c^{\prime}(x_{0}))^{1/3}}{(c(x_{0}))^{1/2}}\right)^{5}\,(1-\sqrt{3}\,\mathbbm{i})\,\varepsilon^{5/3}.$

Since $\lambda_{i}=\bar{\lambda}_{i2}+o(\varepsilon^{2})$, we see that for any
combination of signs of $\varepsilon$ and $c^{\prime}(x_{0})$ there always
there exists an eigenvalue greater than one in absolute value. The theorem is
proved. $\Box$

######  Remark 1.

We can see that the value

$m(x)=\Big{|}{\frac{(c^{\prime}(x))^{1/3}}{(c(x))^{1/2}}}\Big{|}$ (18)

can be considered as a measure of instability for the specific doping profile.
Thus, to find the most ”dangerous” point, we have to find the point $x=m_{*}$,
the maximum of $m(x)$.

##  4 Numerical study

For the numerical examples we choose the initial data in the form of a
standard laser pulse [11], [5]:

$E_{0}=a\,x\,e^{-\frac{x^{2}}{2}},\qquad V_{0}=0.$ (19)

1\. The first test is the case of the constant background profile $c(x)=1$.
From (13), (14) we find

$Q(x)=1-2E^{\prime}_{0}(x_{0})\sin^{2}\frac{t}{2},$

and for $E^{\prime}_{0}(x_{0})<\frac{1}{2}$ the function $Q(t)$ does not
vanish at all, and therefore the solution does not blow up along the
characteristic starting from $x_{0}$, what corresponds to criterion (8). For
the initial data (19) this condition holds for all $x_{0}\in\mathbb{R}$ for
$a<\frac{1}{2}$. Otherwise, if $E^{\prime}_{0}(x_{0})\geqslant\frac{1}{2}$ for
a point $x_{0}$, then the solution blows up within a time $t_{*}<2\pi$.

2\. The second example is

$c(x)=1+\frac{0.3}{1+x^{2}}.$ (20)

In the data (19) we choose $a=0.3$, so for any constant $c(x)=C\in[1,1.3]$ the
solution of the Cauchy problem (2), (19) does not blow up.

We present the dependence of the blow-up time on $x_{0}$ for
$x_{0}\in[0.3,1.3]$ computed from (4), (13), (5), (14) by the Fehlberg fourth-
fifth order Runge-Kutta method with the space step $h=0.01$. The function
$Q(t)$ oscillates with an increasing amplitude, the blow-up time is about
$75$, it attains on the characteristic starting from the points
$x_{0}\approx\pm 0.6$. See Fig.1, left. For the starting points
$x_{0}\in(0,0.3)$ the blow-up time is greater than $130$. The picture is
symmetrical about the origin. For this profile, the maximum $m_{*}$ found from
(18) is approximately $0.636$, which is very close to the minimum point of
blow-up. The initial profile (19) cannot be considered as a small perturbation
of the zero equilibrium, therefore we can conclude that for this case the
shape of the initial data does not have a significant effect on the position
of the initial point $x_{0}$ of the characteristic along which the derivatives
go to infinity first, which determines the blow-up time of the entire solution
of the Cauchy problem (i.e. the minimum of the blow-up time over all
$x_{0}\in\mathbb{R}$).

The stepwise nature of the dependence of the blow-up time on the initial point
for a specific characteristic is explained by the oscillating nature of
$Q(t)$. Namely, a jump in the graph occurs when $Q(t)$ crosses the $Q=0$ axis
during the next oscillation.

2\. The third example is for the periodic symmetric profile

$c(x)=1+0.3\cos 2x.$ (21)

In the data (19) we choose again $a=0.3$ and present the computations on the
period $T=\pi$. The derivatives remain bounded for the characteristics
starting from the points, where $c^{\prime}(x)=0$ (i.e.
$x=0,\,\frac{\pi}{2},\,\pi$). Near these points the derivatives go to infinity
within a time greater than $250$. The blow-up time for the solution to the
Cauchy problem is less than $25$. See Fig.1, right. For this profile the
maximal points $m_{*}$ on the period, found from (18) are approximately $1$
and $2$.

Figure 1: The blow-up time along characteristics starting from points of the
axis $x$ for the initial data (19), $a=0.3$. Left: profile (20), right:
profile (21).

2\. The fourth example is

$c(x)=\frac{1}{(K|x|+1)^{2}},\quad K>0,$ (22)

which is interesting since $m(x)=2K=\rm const$. After examples 2 and 3, one
may get the impression that when determining the time of formation of a
singularity, it is not so much the form of the initial data that is important
as the doping profile. With this example we show that this is, generally
speaking, not the case. Fig.2, made for $K=1$ for the interval
$x_{0}\in(0.2,2.85)$, illustrates that the blow-up time for the solution in
this case is minimal for the points closer to the maximum of
$s(x)=E_{0}^{\prime}(x)$. In the data (19) we choose again $a=0.3$.

Figure 2: Left: The blow-up time along characteristics starting from points of
the axis $x$ for the initial data (19), $a=0.3$, for the profile (22), $K=1$.
Right: the behavior of $s(x)=E_{0}^{\prime}(x)$ in the same interval.

##  5 Discussion

1.A natural question is whether there is a globally smooth solution in time
for a non-constant doping profile far from the zero equilibrium, and a first
candidate is a stationary solution $(V(x),E(x))$ other than the zero
equilibrium. However, this solution cannot be smooth for all $\mathbb{R}$.
Indeed, from the second equation of (2) it follows that if $V(x)\neq 0$, then
$E_{x}=c(x)>0$, $\Psi_{xx}=c(x)>0$ and $\Psi$ is strictly concave. Therefore,
the identity $\frac{1}{2}V^{2}=-\Psi+\rm const$, following from the first
equation of (2), cannot be satisfied for all $x\in\mathbb{R}$. Moreover, this
solution has no physical meaning, since for it $n=0$.

Note that the traveling wave solution, depending on the self-similar variable
$x-Wt$, $W=\rm const$, does not exist for a variable doping profile, in
contrast to the case $c(x)=C=\rm const$. However, even for a constant density
background, a smooth traveling wave exists only for sufficiently large $W$
(see [13]).

Note also that for the attractive Euler-Poisson equations there exists
globally smooth solutions for a variable background density $c(x)$, see [1],
Theorem 2.2, where a periodic $c(x)$ was considered.

2\. Similar to the model (1), we can consider the Euler-Poisson equations with
damping

$\dfrac{\partial V}{\partial t}+(V\cdot\nabla)V=-\nabla\Psi-
qV,\quad\dfrac{\partial n}{\partial t}+{\rm
div}\,(nV)=0,\quad\Delta\Psi=c(x)-n.$ (23)

In this case, the technique presented in this article allows us to reduce the
question on the blow up of a solution of (23) in a finite time to the analysis
of the possibility of vanishing of $Q(t)$ in a point $t_{*}>0$. The function
$Q$ solves the equation

$Q^{\prime\prime\prime}+qQ^{\prime\prime}+c(x(t))Q^{\prime}+c^{\prime}(x(t))V(x(t))Q=0,$

with the initial data (14), $V(t)$ and $x(t)$ are the solutions of

$\ddot{V}+q\dot{V}+c(x(t))V=0,\quad\dot{x}(t)=V,$

with the initial data

$x(0)=x_{0},\quad V(0)=V_{0}(x_{0}),\quad
V^{\prime}(0)=-E_{0}(x_{0})-qV_{0}(x_{0}).$

Standard calculations show that if the initial data $V_{0}(x),E_{0}(x)$ are
such that
$V_{0}(x_{0}),E_{0}(x_{0}),V^{\prime}_{0}(x_{0}),E^{\prime}_{0}(x_{0})$ are
sufficiently small in absolute value for each $x_{0}\in\mathbb{R}$, then
$Q(t)>0$ along each characteristic, and therefore the respective solution of
(23), with the initial data

$(V,\Psi^{\prime})|_{t=0}=(V_{0}(x),\Psi^{\prime}_{0}(x))\in{C^{2}}({\mathbb{R}}),$

keeps the initial smoothness. In other words, in the case $q>0$ there is a
small neighborhood of the zero equilibrium in the $C^{1}$ \- norm such that
the solution with initial data from this neighborhood remains smooth.
Moreover, it can be shown that this solution asymptotically tends to zero
equilibrium as $t\to\infty$.

3\. For the repulsive Euler-Poisson equations in many spatial dimensions we
can consider solutions with radial symmetry and a variable radially symmetric
density background. This problem was considered for the constant density
background in [14], where it was found that for $d\neq 1$ and $4$ any non-
trivial solution other than a simple wave (i.e. $E=E(V)$) blows up. Since
simple waves do not exist for a variable doping profile, it is natural to
expect that any non-trivial solution blows up.

Note that a thorough analysis of the blow-up conditions was carried out for
the radially symmetric case for the attractive case and the case of zero
background, as well as for the repulsive case with a constant non-zero
background at $d=4$ was made in [2]. In fact, for the repulsive case the
existence of a global smooth solution requires the period of oscillations
along every characteristic to be identical. This means that the oscillations
are isochronous (i.e. the period does not depend on the amplitude). As follows
from [14], for the case of a constant background, the oscillations are
determined by the function $F(t)$, which is a solution of a nonlinear Liénard
type equation

$F^{\prime\prime}+(2+d)\,F\,F^{\prime}+F+d\,F^{3}=0,$

and the Sabatini criterion [18] implies that the oscillations are isochronous
if and only if $d=1$ and $d=4$.

In [3], a global weak solution was constructed for the radially symmetric case
of the repulsive Euler-Poisson equations with pressure with a variable doping
profile.

## Acknowledgements

Supported by RSF grant 23-11-00056 through RUDN University.

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