# Semi-classical description of electrostatics and quantization of electric
charge

Kolahal Bhattacharya  St. Xavier’s College (Autonomous), Kolkata-700016, India
<EMAIL_ADDRESS>
###### Abstract

In this work, we present an explanation of the electric charge quantization
based on a semi-classical model of electrostatic fields. We claim that in
electrostatics, an electric charge must be equal to a rational multiple of the
elementary charge of an electron. However, the charge is quantized if the
system has certain boundary conditions that force the wavefunction
representing an electric field to vanish at specific surfaces. Next, we
develop the corresponding model for the electric displacement vector. It is
demonstrated that a number of classical results, e.g. bending of field lines
at the interface of two dielectric media, method of images, etc. are all
consistent with the predictions of this model. We also present the possible
form of Gauss’s law (or Poisson’s equation), to find the wavefunctions of the
field from a source charge distribution, in this model.

  *

Keywords: charge quantization, semi-classical methods, anti-Hermitian
operators, magnetic monopole.

##  1 Introduction

A prime example of the unresolved enigmas of theoretical physics is the
quantization of electric charge. Millikan’s famous oil drop experiment [1] in
1909 demonstrated that electric charge always appears as an integral multiple
of the elementary electric charge $e$. This result continues to hold even
today, with great experimental accuracy [2]. However, in spite of the
extraordinary success of Maxwell’s theory of electrodynamics and quantum
electrodynamics (QED), the theoretical justification of this empirical result
has remained unexplained until now. Moreover, the development of particle
physics in the last seventy years has demonstrated that quarks, the
constituent Fermions of Hadrons carry fractional charge $\pm e/3$, $\pm{2e}/3$
etc. There is no satisfactory explanation of these experimental results as
well. In this paper, we will explain the charge quantization in the non-
relativistic domain, as observed by Millikan.

In 1931, P. Dirac argued [3] that the charge quantization must happen if there
is a magnetic monopole. He showed that the unobservability of phase in the
quantum domain allows for singularities, manifested as sources of magnetic
fields. This leads to the condition that the product of electric and magnetic
charges should be an integral multiple of $\hbar c/2$, where $\hbar$
represents the reduced Planck’s constant and $c$ denotes the speed of light in
the vacuum. This marked the beginning of an organised search for a magnetic
monopole somewhere in the universe that continues even today because no
magnetic monopole has ever been located111Pierre de Maricourt of the
thirteenth century tried to separate the poles of a magnet by breaking magnets
into pieces [4]. So, this search is perhaps a millennium old..

There are other thought-provoking motivations to search for magnetic
monopoles. For example, the presence of a magnetic monopole will lead to a
symmetric extension of Maxwell’s laws for classical electrodynamics that are
invariant under duality transformation [5]. This symmetry will also suggest a
generalised Lorentz force:

$\vec{F}=q(\vec{\mathcal{E}}+\vec{v}\times\vec{\mathcal{B}})+g(\vec{\mathcal{B}}-\frac{\vec{v}}{c^{2}}\times\vec{\mathcal{E}})$
(1)

acting on a dyonic particle with electric charge $q$ and magnetic charge $g$
moving with a velocity $\vec{v}$ in the electric field $\vec{\mathcal{E}}$ and
magnetic field $\vec{\mathcal{B}}$. Now, if dyons exist, then space inversion
is no longer a valid symmetry in electrodynamics, as noted by Ramsey [6]. Now,
parity symmetry is broken maximally in weak interactions. This means that if
magnetic monopoles exist, due to the quantization of electric charges, this
will as well justify why parity is not a good symmetry of nature [7, 8, 9].
The magnetic monopole has also been proposed as a solution to the strong CP
problem [10]. So, the issue of the quantization of electric charge is deeply
connected to a large number of interesting questions in fundamental physics
and therefore, there have been consolidated experimental efforts to search for
magnetic monopoles. However, searches in the cosmic rays, bound matter, in
colliders via direct and indirect ways never ever found the existence of
monopoles [11]. The MoEDAL collaboration at CERN is an ongoing experiment and
they recently reported the result of their first run [12], where they ruled
out the existence of dyons carrying a magnetic charge up to five units of the
Dirac charge and an electric charge up to 200 times the electron’s charge for
dyons with a mass limits between 870 and 3120 GeV. Perhaps this hints that the
charge quantization problem can be approached in a different route which does
not require the existence of a magnetic monopole.

In this paper, we approach the problem using a quantum physical model of
electrostatics. In the recent past, there have been some works in
understanding the quantum or semi-classical nature of electrostatic fields
[13, 14]. Both authors appear to agree that electrostatic fields could be
described by non-travelling wavegroups, but otherwise, the frameworks are
different. The second one among these two demystified the nonlocality problem
of the Aharonov-Bohm effect which was a puzzle since 1959 [15]. After being
proved experimentally [16], this experiment was compared with the Michelson
and Morley experiment of recent times [17]. Apart from providing an
explanation of the nonlocality problem, the article [14] pointed out that
electric charge must be a rational multiple of the elementary electric charge
$e$. With this queue, the semi-classical model, introduced in [14] will be
used in our present quest in the hope that it can throw some light.

We first obtain the prediction of the model in the context of the charge
quantization problem. Then, we address another well-known piece of information
that the electrostatic field lines, in some cases, could exhibit features very
similar to the light rays in geometrical optics. For example, the bending of
electric field lines at the interface between two dielectric media is
analogous to the refraction of light. Similarly, in the examples of the method
of images (which is an elegant method [18, 19] to solve Laplace’s equation for
electrostatic potential under appropriate boundary conditions), the situations
are very much similar to the reflection of light by mirrors.

This model [14] asserts that the electrostatic field can exhibit semi-
classical nature when $e\Phi{t}\sim\hbar$ where $\Phi$ and $t$ represent
electrostatic potential and time - over which a charge is subjected to the
potential $\Phi$. If $e\Phi t\gg\hbar$, the classical nature of the fields
manifests. On the other hand, if $e\Phi{t}\sim\hbar$, then in a source-free
region, the wavefunctions of the electrostatic field satisfy a wave equation
that has the form of a homogeneous Helmholtz equation. An analogous situation
arises during the transition from ray optics to wave optics [20]. In the limit
where the wavelength of the light cannot be neglected, the wave nature of
light is manifested. Historically, Huygens contemplated plane and spherical
wavelets of light, envelopes of which proceed in the forward direction for the
propagation of light. In modern formalism, these waves are identified as the
solutions to the reduced wave equation [21]. Though this wave model of light
lacks the polarisation picture, it can be used to explain the reflection and
refraction of light. Therefore, the situations with optical analogies in
electrostatics may indicate an underlying semi-classical model for classical
electrostatics. Construction of such a mathematical model will be very
exciting and it may reveal some unknown features of electrostatic field
theory.

In [14], the wavefunctions of the electrostatic field have been introduced
merely as mathematical objects representing fields in the regions devoid of
source charges. In the current work, we extend this formalism in the presence
of source charges in section 2 to find the equations corresponding to the
wavefunctions that represent different components of the electrostatic field.
In this formalism, we shall observe the presence of an anti-Hermitian operator
that satisfies the spectral theorem and will investigate the classical limit
of the model. Next, in section 3, we present the proof of the quantization of
charge, as observed in experiments. To address the question of the similarity
between electrostatic field lines and light rays in geometrical optics, we
shall develop the semi-classical description for the electric displacement
vector $\vec{\mathcal{D}}$ in section 4. In the following section 5, we shall
describe how the semi-classical model of $\vec{\mathcal{E}}$ and
$\vec{\mathcal{D}}$ fields helps in understanding the refraction and
reflection of field lines across a boundary of two media of different
dielectric constants. In section 6, we compare the relation of this formalism
with the quantum limit of Gauss’s law (or Poisson’s equation), as presented in
[13]. Finally, we will conclude with a discussion of the implication of these
observations.

##  2 Wave equation in Electrostatics

One can conceive a variational principle for the electrostatic field (and
other curl-free vector fields) [22]:

$\delta\int_{P_{1}}^{P_{2}}{\mathcal{E}}\ ds=0.$ (2)

In Eq.(2), the integral is evaluated along a curve that is always superimposed
with the local direction of the field. We find that the field lines satisfy
the Euler-Lagrange equation:

$\nabla{\mathcal{E}}=\frac{d}{ds}\left(\mathcal{E}\frac{d{\bf r}}{ds}\right),$
(3)

exactly in a way similar to the light rays [23]. It has recently been shown
that the electrostatic field may exhibit a semi-classical behaviour if $e\Phi
t\sim\hbar$ [14]. Under source-free conditions, one can define the electric
field operator as a momentum conjugate to the position coordinates:

$\displaystyle\hat{\vec{p}}\psi_{E}=-i\bar{\gamma}\nabla\psi_{E}=(\hat{\bf
x}\hat{p}_{x}+\hat{\bf y}\hat{p}_{y}+\hat{\bf z}\hat{p}_{z})\psi_{E}=(\hat{\bf
x}\mathcal{E}_{x}+\hat{\bf y}\mathcal{E}_{y}+\hat{\bf
z}\mathcal{E}_{z})\psi_{E}=\vec{\mathcal{E}}\psi_{E},$ (4)

where ($\hat{p}_{x},\hat{p}_{y},\hat{p}_{z}$) denotes the momentum operators
in $(x,y,z)$ directions along which $\hat{\bf x},\hat{\bf y},\hat{\bf z}$ are
the unit vectors; $\bar{\gamma}\equiv\gamma/(2\pi)=\hbar/(e\cdot t)$ is a
scaling factor, which represents $1/(2\pi)$ times the minimum possible
electrostatic potential $\gamma$ in the problem. In a region devoid of source
charge density $\rho$ (where the conjugate momentum operators $\hat{p}_{x}$
etc. do not operate on the corresponding components of the field e.g.
$\mathcal{E}_{x}$ etc.), the non-travelling wavefunction $\psi_{E}$ satisfies:

$\bar{\gamma}^{2}\nabla^{2}\psi_{E}+\mathcal{E}^{2}\psi_{E}=0.$ (5)

We can readily verify that $\psi_{E}={\bf e}^{i{e\Phi t}/{\hbar}}$ is a
solution to this equation. The presence of the variable time factor ($t$) in
$\psi_{E}$ may appear contradictory in the context of standard electrostatics
problems, where one is interested in the time-averaged electric field or
potential at a test point, due to some given charge distribution. We will find
that the classical results follow from the boundary conditions on $\psi_{E}$
but they do not depend on $\psi_{E}$ itself. In certain special circumstances,
when we need to consider the electromagnetic communication between two bodies,
the time factor can be expressed as $L/c$ where $L$ is the distance between
the bodies, and $c$ is the speed of light in free space.

The wave equation (5) is valid in the region of space devoid of source charge.
However, in the presence of source charge density $\rho\neq{0}$, the conjugate
momenta $\hat{p}_{x}$, etc. can operate on the components of the field e.g.
$\mathcal{E}_{x}$, etc. Not only that, the distribution of charges can be
different in different directions. For example, along the axis of the charged
plate capacitors, there is a non-zero divergence of the electric field at the
capacitor plates, due to the presence of source charge. However, in the
directions perpendicular to the axis, the field does not have divergence. For
such situations, it is more meaningful to contemplate different wavefunctions
of the electrostatic field along different directions:

$\displaystyle-i\bar{\gamma}\frac{\partial\psi_{E_{x}}}{\partial x}$
$\displaystyle=\mathcal{E}_{x}\psi_{E_{x}}$ (6a)
$\displaystyle-i\bar{\gamma}\frac{\partial\psi_{E_{y}}}{\partial y}$
$\displaystyle=\mathcal{E}_{y}\psi_{E_{y}}$ (6b)
$\displaystyle-i\bar{\gamma}\frac{\partial\psi_{E_{z}}}{\partial z}$
$\displaystyle=\mathcal{E}_{z}\psi_{E_{z}}$ (6c)

In a medium of permittivity $\epsilon_{0}$, wave equation for wavefunction
$\psi_{E_{z}}$ in $z$ direction take the form:

$\displaystyle-i\bar{\gamma}\frac{\partial^{2}\psi_{E_{z}}}{\partial z^{2}}$
$\displaystyle=\frac{\partial\mathcal{E}_{z}}{\partial
z}\psi_{E_{z}}+\mathcal{E}_{z}\frac{\partial\psi_{E_{z}}}{\partial z}$
$\displaystyle=\frac{\rho_{z}}{\epsilon_{0}}\psi_{E_{z}}+\mathcal{E}_{z}\frac{\mathcal{E}_{z}\psi_{E_{z}}}{-i\bar{\gamma}}$
$\displaystyle\implies$
$\displaystyle\bar{\gamma}^{2}\frac{\partial^{2}\psi_{E_{z}}}{\partial
z^{2}}+\mathcal{E}_{z}^{2}\psi_{E_{z}}=i\bar{\gamma}\frac{\rho_{z}}{\epsilon_{0}}\psi_{E_{z}}$
(7a) $\displaystyle\implies$
$\displaystyle\left[-\frac{\bar{\gamma}^{2}}{2\left(\frac{1}{\epsilon_{0}}\right)}\frac{\partial^{2}}{\partial
z^{2}}-\frac{\epsilon_{0}\mathcal{E}_{z}^{2}}{2}\right]\psi_{E_{z}}=i\left(-\frac{1}{2}\rho_{z}\bar{\gamma}\right)\psi_{E_{z}}$
(7b)

Here, we have denoted ${\partial{\mathcal{E}_{z}}}/{\partial z}=\rho_{z}$. For
$x$ and $y$ components, it is possible to have corresponding equations.
Eq.(7b) has the form of the time-independent Schr$\rm{\ddot{o}}$dinger’s
equation. The first and second terms on the left-hand side denote the kinetic
and potential energy density. We notice that the inverse of the permittivity
plays the role of mass of the $\psi_{E_{z}}$ field. The (imaginary) energy
density is given by the factor $-i\rho_{z}\bar{\gamma}/2$ on the right-hand
side of Eq.(7b). However, unlike Schr$\rm{\ddot{o}}$dinger’s equation, it
cannot be interpreted as an equation describing the evolution of a
wavefunction in an external potential barrier. Rather, this equation relates
the electric field component and the corresponding wavefunction, to the
related source charge distribution.

The presence of $i$ on the right-hand side of Eq.(7b) shows that the operator
on the left-hand side, acting on $\psi_{E_{z}}$ is an anti-Hermitian operator
(with imaginary eigenvalues), unlike the standard linear Hermitian operators
associated with physical observables. In fact, it is similar to the complex
scalar field operators in quantum field theory. From a mathematical point of
view, it is a normal operator. Such an operator $\hat{N}$ is defined on a
complex vector space $\mathcal{H}$, such that it commutes with its Hermitian
adjoint, i.e.: $\hat{N}\hat{N}^{\dagger}=\hat{N}^{\dagger}\hat{N}$. Both
Hermitian, as well as anti-Hermitian operators, are examples of normal
operators that assume the form of a diagonal matrix with respect to an
orthonormal basis, in accordance with the spectral theorem. However, concrete
examples of the latter are commonly not found. Naturally, the full potential
of these operators has not been fully explored in physics discourses. There
have been some pioneering works by Bender [24] on the possible use of non-
Hermitian operators in quantum field theory. The idea has also been supported
by R Penrose on p. 539 of ‘Road to Reality’ [25]. In this paper, we will find
tangible examples of these operators in the context of electrostatic field
theory.

It may be interesting to explore the physical meaning of the imaginary energy
density in Eq.(7b). Zhang [26] conjectured the electric charge as an imaginary
form of energy. Using this, he showed a pathway of unification of
gravitational and electrical forces classically. Similar or related ideas have
been expressed by other authors [27]. However, these ideas appear to be more
speculative. It seems that there may be some connection, but it will be
premature to say that one implies the other. In fact, it seems that the sense
in which they defined ‘imaginary energy’ of a charged particle is somewhat
different than the sense in which imaginary energy density appears in the
current framework. About the nature of the solutions, we comment that in the
absence of source charge, for which the right-hand-side of Eq.(7b) vanishes,
the equation is just a simple harmonic oscillator equation with a position-
dependent frequency, as pointed out in [14]. But in the presence of source
charge, the actual solution must be worked out using boundary conditions.

Before we go forward, it is worthwhile to investigate the combined three-
dimensional version of the problem. One way to accomplish that is by adding
Eq.(7b) with its $x$ and $y$ counterparts. However, this does not give new
mathematical insight into the system. On the contrary, if we choose to
represent the components of the electrostatic field ($\mathcal{E}_{z}$ etc.)
in terms of the partial derivatives of the logarithms of the corresponding
wavefunctions ($\psi_{E_{z}}$ etc.) on the basis of Eq.(6c) etc. then by
adding all the component equations, we can deduce an interesting result:

$\displaystyle-i\bar{\gamma}\left(\hat{\bf
x}\frac{\partial\ln\psi_{E_{x}}}{\partial x}+\hat{\bf
y}\frac{\partial\ln\psi_{E_{y}}}{\partial y}+\hat{\bf
z}\frac{\partial\ln\psi_{E_{z}}}{\partial z}\right)=(\hat{\bf
x}\mathcal{E}_{x}+\hat{\bf y}\mathcal{E}_{y}+\hat{\bf z}\mathcal{E}_{z})$
$\displaystyle\implies-i\bar{\gamma}\left(\hat{\bf x}\frac{\partial}{\partial
x}+\hat{\bf y}\frac{\partial}{\partial y}+\hat{\bf z}\frac{\partial}{\partial
z}\right)\cdot\left(\hat{\bf x}\frac{\partial\ln\psi_{E_{x}}}{\partial
x}+\hat{\bf y}\frac{\partial\ln\psi_{E_{y}}}{\partial y}+\hat{\bf
z}\frac{\partial\ln\psi_{E_{z}}}{\partial
z}\right)=\nabla\cdot\vec{\mathcal{E}}=\frac{\rho}{\epsilon_{0}}$
$\displaystyle\implies-i\bar{\gamma}\left(\frac{\partial^{2}\ln\psi_{E_{x}}}{\partial
x^{2}}+\frac{\partial^{2}\ln\psi_{E_{y}}}{\partial
y^{2}}+\frac{\partial^{2}\ln\psi_{E_{z}}}{\partial
z^{2}}\right)=\frac{\rho}{\epsilon_{0}}$
$\displaystyle\implies-i\bar{\gamma}\left(\hat{\bf
x}\frac{\partial^{2}}{\partial x^{2}}+\hat{\bf y}\frac{\partial^{2}}{\partial
y^{2}}+\hat{\bf z}\frac{\partial^{2}}{\partial z^{2}}\right)\cdot(\hat{\bf
x}\ln\psi_{E_{x}}+\hat{\bf y}\ln\psi_{E_{y}}+\hat{\bf
z}\ln\psi_{E_{z}})=\frac{\rho}{\epsilon_{0}}$ (8)

This equation demonstrates that in the vicinity of non-zero source charge, one
must talk about a vector of wavefunctions $(\hat{\bf
x}\ln\psi_{E_{x}}+\hat{\bf y}\ln\psi_{E_{y}}+\hat{\bf z}\ln\psi_{E_{z}})$ in
three-dimensional space, instead of $\psi_{E}$ which was used in the absence
of source charge. If there is a spherical symmetry, then Eq.(2) will assume a
simpler form. We are tempted to conjecture that this equation can be regarded
as the quantum mechanical version of Gauss’s law (or Poisson’s equation). It
must be noted that we had to introduce a three-dimensional vector of
wavefunctions and a new vector differential operator:

$\left(\hat{\bf x}\frac{\partial^{2}}{\partial x^{2}}+\hat{\bf
y}\frac{\partial^{2}}{\partial y^{2}}+\hat{\bf z}\frac{\partial^{2}}{\partial
z^{2}}\right)$ (9)

that operates on it. The current author is not aware of any examples where
such an operator has been applied.
Now, let us note that the complex conjugation of Eq.(7a) gives:

$\bar{\gamma}^{2}\frac{\partial^{2}\psi_{E_{z}}^{*}}{\partial
z^{2}}+\mathcal{E}_{z}^{2}\psi_{E_{z}}^{*}=-i\bar{\gamma}\frac{\rho_{z}}{\epsilon}\psi_{E_{z}}^{*}.$
(10)

Multiplying Eq.(7a) by $\psi_{E_{z}}^{*}$ and Eq.(10) by $\psi_{E_{z}}$, then
adding the resulting two equations, we get

$\displaystyle\bar{\gamma}^{2}\left(\psi_{E_{z}}^{*}\frac{\partial^{2}\psi_{E_{z}}}{\partial
z^{2}}+\psi_{E_{z}}\frac{\partial^{2}\psi_{E_{z}}^{*}}{\partial
z^{2}}\right)+2\mathcal{E}_{z}^{2}\psi_{E_{z}}^{*}\psi_{E_{z}}=0$
$\displaystyle\implies$
$\displaystyle\bar{\gamma}^{2}\left(\psi_{E_{z}}^{*}\frac{\partial^{2}\psi_{E_{z}}}{\partial
z^{2}}+\psi_{E_{z}}\frac{\partial^{2}\psi_{E_{z}}^{*}}{\partial
z^{2}}\right)+2\left(i\bar{\gamma}\frac{\partial\psi_{E_{z}}^{*}}{\partial
z}\right)\cdot\left(-i\bar{\gamma}\frac{\partial\psi_{E_{z}}}{\partial
z}\right)=0$ $\displaystyle\implies$
$\displaystyle\bar{\gamma}^{2}\left(\psi_{E_{z}}^{*}\frac{\partial^{2}\psi_{E_{z}}}{\partial
z^{2}}+\psi_{E_{z}}\frac{\partial^{2}\psi_{E_{z}}^{*}}{\partial
z^{2}}\right)+2\bar{\gamma}^{2}\frac{\partial\psi_{E_{z}}^{*}}{\partial
z}\frac{\partial\psi_{E_{z}}}{\partial z}=0$ $\displaystyle\implies$
$\displaystyle\frac{\partial^{2}}{\partial
z^{2}}\left|\psi_{E_{z}}\right|^{2}\equiv\frac{\partial^{2}}{\partial
z^{2}}\left(\psi_{E_{z}}^{*}\psi_{E_{z}}\right)=0,$ (11)

where, in the second equality, we used Eq.(6c) and its complex conjugate. One
can easily verify that in the absence of the source charge distribution
$\rho$, the corresponding equation becomes:

$\nabla^{2}|\psi_{E}|^{2}=0,$ (12)

which can perhaps be anticipated. As such, $\psi_{E}$ has a spherical symmetry
and we do not need to distinguish between the directions in the source-free
case. Invoking Born’s probability interpretation, we find that in the source-
free region, the probability density of finding $\psi_{E}$ at a point in space
is a Harmonic function (solution of Laplace’s equation), from Eq.(12). In the
presence of a source charge, we must be concerned with finding the probability
density of wavefunction of the electrostatic field in a given direction. That
still remains a Harmonic function, in that direction. Perhaps this suggests a
possible connection between the modulus square of the wavefunction of the
electric field and the electrostatic potential. If the boundary conditions on
the potential are the same as those on the wavefunction, (say, if both of them
are equal to zero on a boundary), then the uniqueness theorem states that the
modulus squared wavefunction is the same as the potential. In that case, one
can find the electrostatic potential, by solving Eq.(7b). We comment that if
only one component (say, $z$ component) of the curl of a vector field is zero
(or equivalently if the closed-loop line integral for that component of the
vector field is zero), then it is possible to find the semi-classical model
for that component.

We note that the Eq.(5) admits both positive and negative signed exponents as
basis wavefunctions, i.e. $\psi_{E}={\bf e}^{\pm i{q\Phi t}/{\hbar}}$. If
$q=-e$, the normalised solution of Eq.(5) will be $\Psi_{E}=\int
u(\vec{\mathcal{E}}){\bf e}^{\pm i{e\Phi t}/{\hbar}}d\vec{\mathcal{E}}$ where
$\Phi=-\int{\vec{\mathcal{E}}}\cdot d{\bf r}$. However, Eq.(7b) admits only
$\psi_{E_{z}}={\bf e}^{-i{q\int(\mathcal{E}_{z}dz)t}/{\hbar}}$ as the basis
wavefunctions. For $q=-e$, the corresponding normalizable solutions to Eq.(7b)
will be $\psi_{E_{z}}=\int u({\mathcal{E}}_{z}){\bf
e}^{i{e(\int{\mathcal{E}}_{z}dz)t}/{\hbar}}d{\mathcal{E}_{z}}$.

Before we conclude this section, let us check the classical limit of Eq.(7b).
In the investigation of the classical limit of quantum mechanics, a standard
approach is to write the wavefunction as $\psi=A{\bf e}^{iS/\hbar}$ where $A$
and $S$ are real quantities. If we substitute this into the
Schr$\rm{\ddot{o}}$dinger’s equation and take the limit, we find that $S$
(classical action) satisfies the Hamilton-Jacobi equation.

Now, in the context of Eq.(7b), we can proceed in a similar manner. Let us
write the wavefunction as $\psi_{E_{z}}=A{\bf
e}^{-i\frac{\Phi}{\bar{\gamma}}}$, noting that the electrostatic potential
plays a role similar to the classical action. Substituting this function into
Eq.(7b), we find:

$\displaystyle\bar{\gamma}^{2}\frac{\partial^{2}\psi_{E_{z}}}{\partial z^{2}}$
$\displaystyle=\bar{\gamma}^{2}\frac{d^{2}A}{dz^{2}}\left({\bf
e}^{-i\frac{\Phi}{\bar{\gamma}}}\right)-2i\bar{\gamma}\frac{dA}{dz}\frac{\partial\Phi}{\partial
z}\left({\bf
e}^{-i\frac{\Phi}{\bar{\gamma}}}\right)-i\bar{\gamma}\frac{\partial^{2}\Phi}{\partial
z^{2}}\left(A{\bf
e}^{-i\frac{\Phi}{\bar{\gamma}}}\right)-\mathcal{E}_{z}^{2}\left(A{\bf
e}^{-i\frac{\Phi}{\bar{\gamma}}}\right)$ (13a)
$\displaystyle=-\mathcal{E}_{z}^{2}\left(A{\bf
e}^{-i\frac{\Phi}{\bar{\gamma}}}\right)+i\bar{\gamma}\frac{\rho_{z}}{\epsilon_{0}}\left(A{\bf
e}^{-i\frac{\Phi}{\bar{\gamma}}}\right)$ (13b)

If we take the limit $\bar{\gamma}\rightarrow 0$ in Eq.(13a) and Eq.(13b),
then we have:

$\displaystyle\lim_{\bar{\gamma}\to
0}-i\bar{\gamma}\frac{\partial^{2}\Phi}{\partial z^{2}}\left(A{\bf
e}^{-i\frac{\Phi}{\bar{\gamma}}}\right)$ $\displaystyle=\lim_{\bar{\gamma}\to
0}i\bar{\gamma}\frac{\rho_{z}}{\epsilon_{0}}\left(A{\bf
e}^{-i\frac{\Phi}{\bar{\gamma}}}\right)$
$\displaystyle\implies\frac{\partial^{2}\Phi}{\partial z^{2}}$
$\displaystyle=-\frac{\rho_{z}}{\epsilon_{0}}$ (14)

This is Gauss’s law in one dimension. If the same is done in all directions,
we get the traditional differential form of Gauss’s law in three dimensions.

##  3 Proof of quantization of electric charge

In [14], it has been argued that the charge $q$ in an electrostatic system, in
general, should be a rational multiple of the elementary electric charge. This
can be proved in the following way. Consider a normalizable wavefunction
$\Psi_{E}$ in regions with non-zero field value. We demand that $\Psi_{E}$
must remain the same for a constant change in potential, just the way the
classical electric field remains unaffected by a constant change in
electrostatic potential. Referring to the form of $\Psi_{E}$, we notice that
the Fourier coefficient $u(\vec{\mathcal{E}})$ will remain invariant under the
transformation $\Phi\rightarrow\Phi+\Phi_{0}$ where $\Phi_{0}$ is a constant.
This leads to:

$\int u(-\nabla\Phi){\bf e}^{i\frac{q\Phi t}{\hbar}}d\vec{\mathcal{E}}=\int
u(-\nabla(\Phi+\Phi_{0})){\bf
e}^{i\frac{q(\Phi+\Phi_{0})t}{\hbar}}d\vec{\mathcal{E}}\\\ \implies{\bf
e}^{i\frac{q\Phi t}{\hbar}}={\bf e}^{i\frac{q(\Phi+\Phi_{0})t}{\hbar}}$ (15)

In Eq.(15), we can equate the integrands, because the integrals are equal for
arbitrary boundaries (any pair of upper and lower limits) and a constant
$\Phi_{0}$. Now, for an integer $n\in\mathbf{N}$, this leads to the following
condition:

$\displaystyle\frac{q\Phi t}{\hbar}$
$\displaystyle=2n\pi+\frac{q(\Phi+\Phi_{0})t}{\hbar}$ $\displaystyle\implies
q\Phi_{0}t$ $\displaystyle=-2n\pi\cdot\hbar=-nh$ $\displaystyle\implies q$
$\displaystyle=-\frac{nh}{\Phi_{0}t}=-\frac{n}{\Phi_{0}}(\gamma\cdot
e)=-\frac{n}{(\Phi_{0}/\gamma)}e=-\frac{n}{N}e,$ (16)

where in the last equation, we have used $\Phi_{0}=N\gamma$, where $N$ is an
integer. The potential $\Phi_{0}$ represents an area in the phase space
constituted by coordinates and conjugate momenta (electric field). This area
must be an integral multiple of the unit (minimum) potential $\gamma$. Since
$\Phi_{0}$ can be positive as well as negative, so $N$ can also be both
positive and negative. This result shows that the charge should be a rational
multiple of $e$, but does not explain why charges should be quantized. In the
following, we provide more direct proof of the said quantization. Let us
consider the problem in one dimension (say, in $z$ direction). If some charge
is distributed on a conductor, e.g. to a plate of a parallel plate capacitor
located at $z=a$ (whose another plate at $z=0$ is grounded), then the charge
density is given by $\rho=\sigma\delta_{D}(z-a)$, where $\delta_{D}$
represents Dirac delta function. Then, from Eq.(7b), we can say:

$\displaystyle-\frac{\hbar^{2}}{2\left(\frac{e^{2}t^{2}}{\epsilon_{0}}\right)}\frac{d^{2}\psi_{E_{z}}}{dz^{2}}-\frac{\epsilon_{0}}{2}\mathcal{E}_{z}^{2}\psi_{E_{z}}$
$\displaystyle=-i\bar{\gamma}\frac{\sigma}{2}\delta_{D}(z-a)\psi_{E_{z}}$ (17)

Let us check the boundary conditions of $\psi_{E_{z}}$ and $\mathcal{E}_{z}$.
Between the plates ($0<z<a$), $\mathcal{E}_{z}\neq 0$, and $\psi_{E_{z}}=\int
a(\mathcal{E}_{z}){\bf
e}^{i{e({\int\mathcal{E}}_{z}dz)t}/{\hbar}}d{\mathcal{E}_{z}}$. Exactly on the
surface of the conductor $z=a$, $\psi_{E_{z}}=0$, otherwise the RHS of Eq.(17)
will diverge. Assuming that the electric field between the plates is a
constant $\mathcal{E}_{0}$, we are led to the condition that
$\sin\left({e\mathcal{E}_{0}at}/\hbar\right)=0$ (or
$\cos\left({e\mathcal{E}_{0}at}/\hbar\right)=0$). If we adopt the $\sin$, we
get:

$\displaystyle\left(\frac{e\mathcal{E}_{0}at}{\hbar}\right)=n\pi$
$\displaystyle\implies\mathcal{E}_{0}=n\frac{\pi\hbar}{eat}=n\pi\frac{\bar{\gamma}}{a}$
(18)

Note that $\bar{\gamma}$ has the unit of electrostatic potential. Since the
electric field just outside the conductor is related to the charge density by
$\sigma=\epsilon_{0}\mathcal{E}_{0}$, it follows that the original charge
$Q(=\sigma A)$, given to the plate of area $A$ at $z=a$, must be quantized as
well. If we adopt $\cos$, even then the electric field - and the source charge
of that field - would be quantized.

There is no loss of generality in selecting a specific configuration of
conductors. We could as well choose a charged conductor of arbitrary shape.
Choose a point $P$ on the surface and call the outward normal direction the
$z$ direction. The value of the field just outside $P$ at a distance $\Delta$
($\rightarrow 0$) from the surface, would still be
$\mathcal{E}_{z}=\sigma/\epsilon_{0}$. The integral $\int\mathcal{E}_{z}dz$
will reduce to $\mathcal{E}_{z}\Delta$. This does not change any of the
arguments used in Eq.(3). We discuss the case of quantization of charge in
dielectric systems in the next section.

The eigenvalue at the right-hand side of Eq.(7b) i.e.
$-i{\rho_{z}\bar{\gamma}}/2$ must be quantized according to appropriate
boundary conditions, exactly in the same way the energy eigenvalues of a
particle in a potential well are quantized. This way, the boundary condition
requires the quantization of the charge in the source distribution. This
description, therefore, removes the need for magnetic monopoles due to the
quantization of electric charge.

The preceding discussion suggests that the quantization of electric charge has
two aspects. Fundamentally, it is not a quantized entity. In fact, it is a
rational multiple of the elementary electric charge $e$, as far as
electrostatics is concerned. However, it becomes quantized, when the boundary
condition requires it to be so (through Eq.(7b)).

##  4 Electric Displacement vector $\vec{\mathcal{D}}$

If we assume that the medium is a linear dielectric with the polarisation
vector $\vec{\mathcal{P}}=\epsilon_{0}\chi_{e}\vec{\mathcal{E}}$, where
$\chi_{e}$ represents the electric susceptibility, then the electric
displacement is given by
$\vec{\mathcal{D}}=\epsilon\vec{\mathcal{E}}=\epsilon_{0}(1+\chi_{e})\vec{\mathcal{E}}$.
In general, $\nabla\times\vec{\mathcal{D}}=\nabla\times\vec{\mathcal{P}}$.
Now, if in a given problem, $\nabla\times\vec{\mathcal{P}}={\bf 0}$ (or,
equivalently, $\oint\vec{\mathcal{P}}\cdot d\vec{l}=0$), the dielectric
potential $\Phi_{D}$ can be expressed in terms of free charge density
$\rho_{f}$ as:

$\Phi_{D}=\epsilon\Phi=\frac{1}{4\pi}\int\frac{\rho_{f}}{|{\bf r}-{\bf
r^{\prime}}|}d\tau^{\prime}\hskip 28.45274pt({\rm if}\hskip
7.11317pt\nabla\times\vec{\mathcal{P}}={\bf 0}).$

Based on [22], in this case also one can conceive $\delta\int\mathcal{D}ds=0$,
where this integral is evaluated along a curve, always superimposed with the
local direction of $\vec{\mathcal{D}}$. This can be used to obtain a semi-
classical model for the field $\vec{\mathcal{D}}$ in the limited cases where
$\nabla\times\vec{\mathcal{P}}={\bf 0}$. The eigenvalue equation for
$\vec{\mathcal{D}}$ field should be given by:

$-i\frac{\gamma_{D}}{2\pi}\nabla\psi_{D}=\vec{\mathcal{D}}\psi_{D},$ (19)

where ${\gamma_{D}}/(2\pi)(=\bar{\gamma}_{D}$, say) can be determined in the
following way: the change in the action of a charge $q$ introduced in a medium
of permittivity $\epsilon(\neq\epsilon_{0})$, if it is subjected to potential
$\Phi=\Phi_{D}/\epsilon$ is $\Delta S=-q\Phi t=-q(\Phi_{D}/\epsilon)t$. We
define $\bar{\gamma}_{D}$ as the minimum value of $\Phi_{D}$ corresponding to
the minimum action $\hbar$. If we take $q=-e$, then
$\bar{\gamma}_{D}={\epsilon\hbar}/({e\cdot t})=\epsilon\bar{\gamma}$.

If only one (say, $z$) component of the electric displacement vector is
irrotational i.e. $\left({\partial\mathcal{D}_{y}}/{\partial
x}-{\partial\mathcal{D}_{x}}/{\partial y}\right)=0$ [or, equivalently
$\oint\mathcal{D}_{z}dl=0$ along a chosen closed contour], then we can get a
semi-classical model with wavefunction $\psi_{D_{z}}$ that represents only the
$z$ component of $\vec{\mathcal{D}}$:

$\displaystyle-i\frac{\gamma_{D}}{2\pi}\frac{\partial\psi_{D_{z}}}{\partial
z}$ $\displaystyle=\mathcal{D}_{z}\psi_{D_{z}}$ (20a)
$\displaystyle\implies\bar{\gamma}_{D}^{2}\frac{\partial^{2}}{\partial
z^{2}}\psi_{D_{z}}+{\mathcal{D}_{z}}^{2}$
$\displaystyle\psi_{D_{z}}=i\bar{\gamma}_{D}\rho_{f}\psi_{D_{z}}.$ (20b)

Using the value of $\bar{\gamma}_{D}$, Eq.(20b) (for linear dielectrics)
reduces to:

$\displaystyle\bar{\gamma}^{2}\frac{\partial^{2}\psi_{D_{z}}}{\partial
z^{2}}+{\mathcal{E}_{z}}^{2}\psi_{D_{z}}=i\frac{\rho_{f}}{\epsilon}\bar{\gamma}\psi_{D_{z}}$
$\displaystyle\implies\left[-\frac{\bar{\gamma}^{2}}{2\left(\frac{1}{\epsilon}\right)}\frac{\partial^{2}}{\partial
z^{2}}-\frac{{\mathcal{D}_{z}}{\mathcal{E}_{z}}}{2}\right]\psi_{D_{z}}$
$\displaystyle=i\left(-\frac{1}{2}\rho_{f}\bar{\gamma}\right)\psi_{D_{z}}.$
(21)

The normalised solution of Eq.(4) can be constructed as $\Psi_{D_{z}}=\int
v({\mathcal{D}_{z}}){\bf
e}^{-i{\Phi_{D_{z}}}/{\bar{\gamma}_{D}}}d{\mathcal{D}_{z}}=\int
v({\mathcal{D}_{z}}){\bf
e}^{-i{\Phi_{z}}/{\bar{\gamma}}}d{{\mathcal{D}_{z}}}$. Comparing with the form
of the solution to Eq.(7b) (discussed just before section 3), we find that the
plane wave basis of the semi-classical wavefunction of the fields
$\vec{\mathcal{E}}$ and $\vec{\mathcal{D}}$ are identical. But the
coefficients are different, as expected.

In principle, one can always do the same exercise for the vector field
$\vec{\mathcal{D}}-\vec{\mathcal{P}}$ (=$\epsilon_{o}\vec{\mathcal{E}}$),
whose curl is zero. Not surprisingly, one finds the total charge density as
the sum of free charge density $\rho_{f}$ and bound charge density $\rho_{b}$
(from $-\nabla\cdot\vec{\mathcal{P}}$ term).

We make the following observation about the quantization of electric charge in
the case of dielectrics. Unlike the conductors, here we may have free and
bound charges and they might reside within the body as well as on the
interface. In addition, the electrostatic field and the displacement vectors
do not remain perpendicular to the interface. But the second point does not
pose a serious problem, because typically we would still deal with the same
basis wavefunctions in a given direction, as found just before (we shall see
an example in the context of dielectric half-plane image problem). So, for
surface charge distribution, which can be represented by a delta function, one
can predict the existence of quantized charges. That conclusion will not hold
for the smooth continuous volume charge distributions. Most likely, such
configurations will not have quantized charges. However, if the volume charge
is made up of many individual point charges embedded in the medium, then the
source term is composed of a summation over delta functions located at those
points. These point charges must then be quantized.

##  5 Electrostatic refraction

Let us consider two halves of the full space filled with linear dielectric
materials with dielectric constants $\epsilon_{1}$ and $\epsilon_{2}$. We
consider the oblique incidence of the electric field line at the boundary (see
the following Figure 1). We would like to approach this problem from the semi-
classical description of the fields. To accomplish that, we make several
observations:

(a)

(b)

Figure 1: (a) Direction change of electric field lines at the interface
between two dielectric media, (b) closed loop line integral of
$\vec{\mathcal{P}}$ vector on both sides of the interface.

(a) On the interface, the free charge density $\rho_{f}=0$, but the total
charge density $\rho$ which is defined as the sum of free charge density and
the bound charge density, is not zero. In this context, boundary conditions
are written as:

$\displaystyle\bar{\gamma}^{2}\frac{d^{2}}{dx^{2}}\psi_{E_{x}}+\mathcal{E}_{x}^{2}\psi_{E_{x}}$
$\displaystyle=0$ (22a)
$\displaystyle\bar{\gamma}^{2}\frac{d^{2}}{dy^{2}}\psi_{E_{y}}+\mathcal{E}_{y}^{2}\psi_{E_{y}}$
$\displaystyle=0$ (22b)
$\displaystyle\bar{\gamma}^{2}\frac{d^{2}}{dz^{2}}\psi_{E_{z}}+\mathcal{E}_{z}^{2}\psi_{E_{z}}=i\bar{\gamma}\frac{\sigma_{b}}{\epsilon_{0}}$
$\displaystyle\delta_{D}(z)\psi_{E_{z}},$ (22c)

where we denoted the wavefunctions corresponding to individual components of
the electric field with separate subscripts. Referring to Eq.(22c), we note
that the existence of the delta function at $z=0$ implies discontinuity in the
first $z$ derivative of $\psi_{E_{z}}$, which (from Eq.(6c)) also implies
discontinuity in the $z$ component of the electric field. However, along the
tangential direction, the field is continuous, since there is no infinite jump
in these directions. So, one has $\mathcal{E}_{1x,y}=\mathcal{E}_{2x,y}$ at
the interface.

(b) At the interface, we can show that $\oint\mathcal{P}_{z}dl=0$ (which also
implies $\oint\mathcal{D}_{z}dl=0$) along a closed rectangular contour, going
into and turning back from both sides of the interface, as can be seen with
reference to Fig. 1(b):

$\displaystyle\oint\mathcal{P}_{z}dl$
$\displaystyle=\int_{A}^{M}P_{z}dl+\int_{M}^{B}P_{z}dl+\cancelto{0}{\int_{B}^{C}P_{z}dl}+\int_{C}^{N}P_{z}dl+\int_{N}^{D}P_{z}dl+\cancelto{0}{\int_{D}^{A}P_{z}dl}$
$\displaystyle=\epsilon_{0}\left(\chi_{e}^{(1)}\mathcal{E}_{z}^{1}\cdot
AM+\chi_{e}^{(2)}\mathcal{E}_{z}^{2}\cdot
MB-\chi_{e}^{(2)}\mathcal{E}_{z}^{2}\cdot
CN-\chi_{e}^{(1)}\mathcal{E}_{z}^{1}\cdot ND\right)=0.$

This allows us to develop the semi-classical description for $\mathcal{P}_{z}$
and hence for $\mathcal{D}_{z}$. Since $\rho_{f}=0$ on the interface, in this
case, Eq.(20b) has a simple form:

$\bar{\gamma}_{D}^{2}\frac{d^{2}}{dz^{2}}\psi_{D_{z}}+\mathcal{D}_{z}^{2}\psi_{D_{z}}=0.$
(23)

As before, we can argue in favour of continuity of $\mathcal{D}_{z}$ across
the interface due to the absence of infinite jump in the first derivative of
$\psi_{D}$. So, we have $D_{1z}=D_{2z}$. Using
$\vec{\mathcal{D}}=\epsilon\vec{\mathcal{E}}$, and the continuity of the
tangential component of the electric field, we deduce the relation:

$\frac{\epsilon_{1}}{\epsilon_{2}}=\frac{\tan\theta_{1}}{\tan\theta_{2}}.$
(24)

Finally, we evaluate $\oint\mathcal{P}_{x}dl=0$ along the same contour for
completeness:

$\displaystyle\oint\mathcal{P}_{x}dl$
$\displaystyle=\cancelto{0}{\int_{A}^{M}P_{x}dl}+\cancelto{0}{\int_{M}^{B}P_{x}dl}+\int_{B}^{C}P_{x}dl+\cancelto{0}{\int_{C}^{N}P_{x}dl}+\cancelto{0}{\int_{N}^{D}P_{x}dl}+\int_{D}^{A}P_{x}dl$
$\displaystyle=\epsilon_{0}\left(-\chi_{e}^{(2)}\mathcal{E}_{z}^{2}\cdot
BC+\chi_{e}^{(1)}\mathcal{E}_{z}^{1}\cdot DA\right)\neq 0.$

Thus, the corresponding semi-classical method is not possible for the
tangential components $\vec{\mathcal{P}}_{x,y}$ (and for
$\vec{\mathcal{D}}_{x,y}$).

####  5.0.1 Dielectric half-plane image problem

Let us now consider the Dielectric half-plane image problem which is covered
in standard texts [19, 18]. The electric field $\vec{\mathcal{E}}_{1}$
originates from a charge $q$ located at a distance $-z_{0}$ in the medium with
dielectric constant $\epsilon_{1}$. The problem seeks to find the potential
function that will satisfy the boundary conditions. The standard approach to
the problem is to assume that there are two image charges. One of them
($q^{\prime}$) is located within the medium with dielectric constant
$\epsilon_{2}$ at a distance $z_{0}$ from the boundary and the other
($q^{\prime\prime}$) is placed at the same location as the original charge
$q$. Solving the boundary conditions, the values of the image charges are:

$q^{\prime}=\frac{\epsilon_{1}-\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}q\hskip
71.13188ptq^{\prime\prime}=\frac{2\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}q.$
(25)

We intend to approach the problem from the semi-classical model of the
electric fields. The wavefunction of the electric displacement vector
$\Psi_{D}^{q}$, due to the real charge, will $reflect$ from the interface and
there will be also some $transmission$. Hence, the net wavefunction at $z<0$
is the sum of the wavefunctions due to the real charge and the wavefunction
that is $reflected$:

$\Psi_{D}^{1}(z)=\int v(\mathcal{D}_{1z}){\bf
e}^{i\frac{\int\mathcal{E}_{1z}dz}{\bar{\gamma}}}d{\mathcal{D}_{1z}}+\mathcalligra{r}\,\int
v(\mathcal{D}_{1z}){\bf
e}^{-i\frac{\int\mathcal{E}_{1z}dz}{\bar{\gamma}}}d{\mathcal{D}_{1z}},$ (26)

where $\mathcalligra{r}\,$ denotes the amplitude of reflection back into the
material with dielectric constant $\epsilon_{1}$. On the other hand, the
wavefunction at $z>0$ can be written as:

$\Psi_{D}^{2}(z)=\mathcalligra{t}\,\int v(\mathcal{D}_{2z}){\bf
e}^{i\frac{\int\mathcal{E}_{2z}dz}{\bar{\gamma}}}d{\mathcal{D}_{2z}},$ (27)

where $\mathcalligra{t}\,$ denotes the transmission amplitude. To investigate
the boundary conditions, we must notice the values of $\Psi_{D}^{1,2}$ at a
distance $|\Delta|\rightarrow 0$ on either side of $z=0$. We have seen that
$D_{z}$ is continuous across the boundary, i.e. $D_{1z}=D_{2z}$. So, we can
drop the coefficients while comparing the wavefunctions at both sides close to
the boundary. We can also assume that $\mathcal{E}_{1z}$ and
$\mathcal{E}_{2z}$ vary sufficiently slowly with respect to $z$ near the
boundary. This implies that
$\int\mathcal{E}_{1z,2z}dz\approx\mathcal{E}_{1z,2z}z$. Therefore, from
Eq.(26), at $z=-\Delta(<0)$ the functional dependence of $\Psi_{D}^{1}$ is
given as:

$\displaystyle\psi_{D}^{1}(z):={\bf
e}^{i\frac{\mathcal{E}_{1z}z}{\bar{\gamma}}}+\mathcalligra{r}\,{\bf
e}^{-i\frac{\mathcal{E}_{1z}z}{\bar{\gamma}}}.$ (28)

And from Eq.(27), we have at $z=\Delta(>0)$:

$\psi_{D}^{2}(z):=\mathcalligra{t}\,{\bf
e}^{i\frac{\mathcal{E}_{2z}z}{\bar{\gamma}}}.$ (29)

The boundary conditions at $z=0$ are:

$\displaystyle\left(\psi_{D}^{1}\right)_{z=0^{-}}$
$\displaystyle=\left(\psi_{D}^{2}\right)_{z=0^{+}}$ (30a)
$\displaystyle\left(\frac{\partial\psi_{D}^{1}}{\partial z}\right)_{z=0^{-}}$
$\displaystyle=\left(\frac{\partial\psi_{D}^{2}}{\partial z}\right)_{z=0^{+}}$
(30b)

The first condition yields:

$\displaystyle 1+\mathcalligra{r}\,$ $\displaystyle=\mathcalligra{t}\,.$ (31)

The second condition implies:

$\displaystyle\left(\frac{i}{\bar{\gamma}}\mathcal{E}_{1z}{\bf
e}^{\frac{i}{\bar{\gamma}}\mathcal{E}_{1z}z}-\mathcalligra{r}\,\cdot\frac{i}{\bar{\gamma}}\mathcal{E}_{1z}{\bf
e}^{-\frac{i}{\bar{\gamma}}\mathcal{E}_{1z}z}\right)_{z=0^{-}}$
$\displaystyle=\mathcalligra{t}\,\left(\frac{i}{\bar{\gamma}}\mathcal{E}_{2z}{\bf
e}^{\frac{i}{\bar{\gamma}}\mathcal{E}_{2z}z}\right)_{z=0^{+}}$
$\displaystyle\implies\mathcal{E}_{1z}-\mathcalligra{r}\,\cdot\mathcal{E}_{1z}$
$\displaystyle=\mathcalligra{t}\,\mathcal{E}_{2z}.$ (32)

In Eq.(5.0.1), $-\mathcalligra{r}\,\cdot\mathcal{E}_{1z}$ and
$\mathcalligra{t}\,\mathcal{E}_{2z}$ are reflected and transmitted components
of the electric field, respectively. From Eq.(31) and Eq.(5.0.1), the
reflection amplitude $\mathcalligra{r}\,$ at the interface can be calculated
as:

$\displaystyle\frac{\mathcal{D}_{1z}}{\epsilon_{1}}-\mathcalligra{r}\,\cdot\frac{\mathcal{D}_{1z}}{\epsilon_{1}}$
$\displaystyle=(1+\mathcalligra{r}\,)\frac{\mathcal{D}_{2z}}{\epsilon_{2}}$
$\displaystyle\implies\frac{1}{\epsilon_{1}}-\frac{\mathcalligra{r}\,}{\epsilon_{1}}$
$\displaystyle=\frac{1}{\epsilon_{2}}+\frac{\mathcalligra{r}\,}{\epsilon_{2}}$
$\displaystyle\implies\left(\frac{1}{\epsilon_{1}}-\frac{1}{\epsilon_{2}}\right)$
$\displaystyle=\mathcalligra{r}\,\left(\frac{1}{\epsilon_{1}}+\frac{1}{\epsilon_{2}}\right)$
$\displaystyle\implies\mathcalligra{r}\,$
$\displaystyle=-\frac{\epsilon_{1}-\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}.$
(33)

The corresponding transmission amplitude $\mathcalligra{t}\,$ can be
calculated as:

$\mathcalligra{t}\,=1+\mathcalligra{r}\,=\frac{2\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}.$
(34)

It is worth pointing out that $\mathcalligra{r}\,$ and $\mathcalligra{t}\,$
are the amplitudes for the wavefunctions of the fields at the interface and
are not the coefficients of the image charges required to solve the problem.
They scale $\psi_{D}^{1,2}$ by a constant. In the Hilbert space, the resulting
states do not represent any new state. Using the reflected and transmitted
wavefunctions in Eq.(5) and Eq.(7b) does not reproduce the values of the image
charges, since the eigenvalue equation does not allow that. These scaled
wavefunctions are the results of the boundary conditions and the bound charge
present at the interface.

However, non-vanishing values of these amplitudes suggest that the reflected
classical electric field at $z<0$ could be assumed to be due to an $image$
charge $q^{\prime}$ located at the mirror image position $+z_{0}$, and the
transmitted classical electric field at $z>0$ could be assumed to be due to an
$image$ charge $q^{\prime\prime}$ that is located exactly at the position
coincident with the original charge. Referring to Eq.(5.0.1), we could
identify that the reflected electric field $\mathcal{E}_{1z}^{q^{\prime}}$ due
to $q^{\prime}$ is:

$\mathcal{E}_{1z}^{q^{\prime}}=-r\mathcal{E}_{1z}^{q}\implies
q^{\prime}=\frac{\epsilon_{1}-\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}q.$ (35)

From the same equation, the transmitted electric field
$\mathcal{E}_{1z}^{q^{\prime\prime}}$ due to $q^{\prime\prime}$ is:

$\mathcal{E}_{1z}^{q^{\prime\prime}}=t\mathcal{E}_{2z}^{q}\implies
q^{\prime\prime}=\frac{2\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}q.$ (36)

###  5.1 Infinite grounded conducting plane

Let us discuss the relevance of this concept in the context of the infinite
grounded conducting plane image problem which can be thought of as the
limiting case of the previous problem, in which
$\epsilon_{2}\rightarrow\infty$. In this case, Eq.(35) shows that one needs to
assume an image charge $-q$ inside the conductor. The charge
$q^{\prime\prime}$ is non-zero, however, its contribution to electric
potential is vanishing as explained in [19].

We observe an important connection between $\psi_{E_{z}}$ and potential $\Phi$
in this problem. Since the induced charge density on the surface can be
expressed by a delta function, therefore, Eq.(7b) dictates that
$\psi_{E_{z}}=0$ on $z=0$. Classically, in this problem, we have
$\nabla^{2}\Phi=0$ (Laplace’s equation), along with the boundary condition
$\Phi=0$ on the conductor. This is exactly parallel to
${\partial^{2}}|\psi_{Ez}|^{2}/{\partial z^{2}}=0$ (Eq.(2)), with the boundary
condition $\psi_{Ez}=0$ (implying $|\psi_{Ez}|^{2}=0$) at $z=0$. Therefore, we
conclude that in this problem, $\Phi\propto|\psi_{Ez}|^{2}$. This argument is
also applicable for the grounded conducting sphere image problem [28] where
one is asked to calculate the image charge and location when a real charge is
placed in front of a grounded conducting sphere.

It may be noted that the superposition of $\psi_{Ez}$ must be done in the
quantum mechanical sense taking into account the phase, as demonstrated in
[14]. However, in the square of the modulus of $\psi_{Ez}$, relevant in
several electrostatics problems, the sensitivity to the phase is washed out.
No wonder that electrostatic potential obeys the classical superposition
principle, where one adds up the potentials algebraically, without any
reference to phase.

##  6 Semi-classical limit of Gauss’s law or Poisson’s equation

At this point, it is perhaps good to note the difference between the two
frameworks that discuss the quantum theory of electrostatics and the possible
expression of Gauss’s law in this limit. First of all, we are not interested
in the derivation of the classical version of Gauss’s law or its higher-order
corrections in the non-relativistic limit of QED which have been addressed
[29]. We are specifically interested in the form of Gauss’s law (or wave
equation) in which the non-travelling wavefunctions (or non-travelling
wavefunctions (or ‘electrostatic coherent state—a notion which involves (non-
dynamical) longitudinal photons’) of the electric field can be calculated from
a given source charge distribution. The framework presented in [13] assumes
that Gauss’s law should hold in quantum theory as
$(\nabla\cdot\vec{\mathcal{E}})\Psi=\rho\Psi$, or as its expectation value
thereof (see the discussion at pp.3-4). In this equation, the electric field
is an operator. On the other hand, the framework on electrostatic field
theory, presented in [14] and extended in the present work, the electrostatic
field is taken as a conservative vector field which can have a semi-classical
nature in a limit. We called this framework semi-classical, because the main
features of the quantum wavefunction are illuminated by classical physics.
This framework seems to be consistent with classical physics. This framework
provides another form of Gauss’s law, given in Eq.(2). Most likely, the
difference arises due to the fact that the latter is a semi-classical model,
as opposed to the former which derives from quantum field theory. However, we
comment that to check consistency with the classical results, the semi-
classical model should be a better starting point. In this case, we must deal
with a vector of wavefunctions representing the three components of the
electrostatic field. Eq.(2), when solved, would give the wavefunctions
representing three different components of the electrostatic field due to a
source charge distribution $\rho$, just like solving the Gauss’s law (or
Poisson’s equation) in electrostatics can be used to evaluate the electric
field in a problem.

In principle, there can be a spherical symmetry in the source charge
distribution, for which Eq.(6a), Eq.(6b), and Eq.(6c) will fuse into the basic
Eq.(4). In that case, the semi-classical version of Gauss’s law (or Poisson’s
equation) takes a simpler form:

$\displaystyle-i\bar{\gamma}\frac{1}{\psi_{E}}\nabla\psi_{E}$
$\displaystyle=-i\bar{\gamma}\nabla{(\ln\psi_{E})}=\vec{\mathcal{E}}$
$\displaystyle\implies-i\bar{\gamma}\nabla^{2}(\ln\psi_{E})$
$\displaystyle=\nabla\cdot\vec{\mathcal{E}}=\frac{\rho}{\epsilon_{0}}$
$\displaystyle\implies\nabla^{2}(\ln\psi_{E})$
$\displaystyle=i\frac{\rho}{\bar{\gamma}\epsilon_{0}}.$ (37)

This equation should be interpreted as a method to find $\psi_{E}$, the
wavefunction of an electrostatic field due to a spherically symmetric $\rho$.
As expected, symmetry leads to considerable simplification of the problem. We
comment that the eigenvalue equation Eq.(4) can be thought of as the analogue
of the gradient equation: $\vec{\mathcal{E}}=-\nabla\Phi$, because when the
divergence operator is applied to it, we get the Gauss’s law (or the Poisson’s
equation).

##  7 Summary and Discussions

In this paper, we gave an explanation of the quantization of electric charge
without requiring the existence of magnetic monopole, based on a semi-
classical model of curl-free vector fields [14] developed in the context of
resolving the nonlocality problem of the Aharonov-Bohm effect [15]. Through
this exercise, we resolved an open problem in physics that remained a mystery
for about the last hundred years. Apart from charge quantization, the semi-
classical model of static conservative fields has been found to resolve the
nonlocality problem of the Aharonov-Bohm effect (which was also an open
problem since 1959), Aharonov-Casher effect and He-McKellar-Wilkens effect;
for deriving the magnetic flux quantum, etc. All these effects, except for the
electrostatic Aharonov-Bohm effect have been experimentally validated. The
reason why the electrostatic Aharonov-Bohm effect could not be observed is
usually attributed to the difficulty in reducing the electric field to zero in
the experimental setup, in contradiction with the preamble of the original
thought experiment. On the other hand, the consistency of this model with
known results from classical physics further emphasises its validity. The
plane waves of the electrostatic fields are found to be similar to the plane
waves used in geometrical optics. The author was thrilled to find that the
electrostatic potential and the modulus square of the wavefunctions
representing an electrostatic field could be related if they are subjected to
the same boundary condition. Its implication for the difference between the
quantum and classical superposition principles (discussed at the end of
subsection 5.1) must be appreciated. It should be possible to deduce the
equation corresponding to Eq.(24) in the context of the passage of a magnetic
field line between two media with different magnetic permeabilities, based on
a semi-classical model on magnetostatics.

In the literature, we found another paper on the quantum physical model of
electrostatics proposed by Kay [13], based on his previous paper [30]. These
works are based on quantum electrodynamics and describe the wavefunctions of
electric fields as coherent states. The author talks about two frameworks in
which Gauss’s law in electrostatics holds as an operator equation
($(\nabla\cdot\vec{\mathcal{E}})\Psi=\rho\Psi$) or its expectation. However,
it is not clear if this model can be used to achieve the tasks we performed.

At this point, it may be interesting to note the relationship between the
semi-classical model with QED, in which electrostatic force arises due to the
interaction between bodies mediated by the virtual photons that exist for a
very short time scale determined by the uncertainty principle [31]. Though QED
is undoubtedly the most successful theory, it is perhaps safe to say that the
time-invariant picture of classical electrostatics is not intuitive from the
dynamical description of the fields in quantum electrodynamics [see the
discussion after Eq.(1) in the introduction of [14]]. In addition,
quantization of electromagnetic Hamiltonian by treating it effectively as a
harmonic oscillator may be difficult in a frame where there is only an
electrostatic field, but no magnetic field. This is perhaps the key difference
between QED and the current formulation. To reproduce the present model from
QED, one needs to come out of the traditional photon picture, by transforming
to a reference frame where only an electrostatic field is present. It may not
be possible to directly take a limit and reproduce the model. This is because
the variational principle $\delta\int_{P_{1}}^{P_{2}}\mathcal{E}ds=0$ [22]
which is the basis of the semi-classical model, is an independent component
that was not known explicitly to be a part of the standard electrostatics
framework. This principle looks like Fermat’s principle, but cannot be derived
from it. The standard theory of QED does not explicitly have this piece of
information. However, this relationship is definitely worth exploring. It is
possible that combining this model with QED will lead to a more complete
theory.

Finally, we comment that the semi-classical model lacks the important aspect
of spin. This aspect arises in the QED based-model [13] naturally. We did not
need it for the problems discussed in this paper. A more complete model,
however, can be formulated if spin can be taken into account. It may be
possible to accomplish this using the square-root operator, as shown in the
context of light rays in optics [32].

##  8 Acknowledgement

The author acknowledges the anonymous reviewers for providing useful
suggestions; Prof. Anwesh Mazumdar and Prof. Sudipto Roy for their support and
encouragement.

##  9 Data availability statement

No new data were created or analysed in this study.

##  10 References

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# Sharpness of Bernoulli percolation via couplings

Hugo Vanneuville CNRS and Institut Fourier (Université Grenoble Alpes)

###### Abstract

In this paper, we consider Bernoulli percolation on a locally finite,
transitive and infinite graph (e.g. the hypercubic lattice $\mathbb{Z}^{d}$).
We prove the following estimate, where $\theta_{n}(p)$ is the probability that
there is a path of $p$-open edges from $0$ to the sphere of radius $n$:

$\forall p\in[0,1],\forall m,n\geq 1,\quad\theta_{2n}(p-2\theta_{m}(p))\leq
C\frac{\theta_{n}(p)}{2^{n/m}}.$

This result implies that $\theta_{n}(p)$ decays exponentially fast in the
subcritical phase. It also implies the mean-field lower bound in the
supercritical phase. We thus provide a new proof of the sharpness of the phase
transition for Bernoulli percolation.

Contrary to the previous proofs of sharpness, we do not rely on any
differential formula. The main novelty is a stochastic domination result which
is inspired by [Russo, 1982].

We also discuss a consequence of our result for percolation in high
dimensions, where it can be seen as a near-critical sharpness estimate.

##  1 Introduction

###  1.1 Sharpness of the phase transition

In this paper, we consider Bernoulli bond percolation on a locally finite,
vertex-transitive and countably infinite graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ (e.g. the hypercubic lattice
$\mathbb{Z}^{d}$). This model is defined as follows: each edge
$e\in\mathcal{E}$ is open with some probability $p\in[0,1]$ and closed
otherwise, independently of the other edges. We let $\mathbb{P}_{p}$ denote
the corresponding product probability measure on $\\{0,1\\}^{\mathcal{E}}$,
where $1$ means open and $0$ means closed.

We fix a vertex $0\in\mathcal{V}$ once and for all and we let
$\textup{dist}:\mathcal{V}^{2}\rightarrow\mathbb{N}$ denote the graph distance
on $\mathcal{G}$. Also, for every $n\geq 1$, we let $S_{n}\subset
B_{n}\subset\mathcal{V}$ denote the sphere and ball of radius $n$ around $0$,
i.e.111Here and in all the paper, $n$ is an integer. $m,i,j,k$ will also
always be integers.

$S_{n}=\\{v\in\mathcal{V}:\textup{dist}(0,v)=n\\}\quad\text{and}\quad
B_{n}=\\{v\in\mathcal{V}:\textup{dist}(0,v)\leq n\\}.$

We let $\mathcal{E}_{n}$ be the set of edges in $B_{n}$, i.e.

$\mathcal{E}_{n}=\\{\\{v,w\\}\in\mathcal{E}:v,w\in B_{n}\\}.$

If $W\subset\mathcal{V}$ and $v\in\mathcal{V}$, we let $\\{v\leftrightarrow
W\\}$ denote the event that there is a path of open edges from $v$ to some
site of $W$ and we let $\\{v\nleftrightarrow W\\}=\\{v\leftrightarrow
W\\}^{c}$. We also let

$\theta_{n}(p)=\mathbb{P}_{p}[0\leftrightarrow
S_{n}]\quad\text{and}\quad\theta(p)=\lim_{n\rightarrow+\infty}\theta_{n}(p).$

(And we use the convention that $\theta_{n}(p)=0$ for every $n\geq 1$ if
$p<0$.) The critical point is

$p_{c}=\inf\\{p\in[0,1]:\theta(p)>0\\}.$

We refer to [Gri99, BR06] for more about percolation. Our main result is the
following estimate.

######  Theorem 1.1.

Let $C_{0}=4\log 2$. Then, for every $p\in[0,1]$ and $m,n\geq 1$,

$\theta_{2n}(p-2\theta_{m}(p))\leq C_{0}\frac{\theta_{n}(p)}{2^{n/m}}.$

######  Remark 1.2.

One can actually replace $p-2\theta_{m}(p)$ by $p(1-2\theta_{m}(p))$ (cf. the
proof at the beginning of Section 2). This can be useful when $p_{c}$ is
small. (But we have chosen to minimize the number of parentheses in the
statement.)

######  Remark 1.3.

If $\theta(p_{c})=0$ then Theorem 1.1 applied to $p=p_{c}$ give some “slightly
subcritical” information. In the case of percolation in high dimensions,
Theorem 1.1 applied to $p=p_{c}$ is sharp in some sense. See Section 4, where
we discuss this further and give a new proof of a near-critical sharpness
bound from the recent paper [HMS21].

Theorem 1.1 implies the following theorem that is often referred to as
sharpness of the phase transition (or more precisely subcritical sharpness and
the mean-field lower bound). The first item – i.e. subcritical sharpness – has
originally been proven independently in [Men86] and [AB87] and the second item
– i.e. the mean-field lower bound – has originally been proven in [CC87]. We
refer to [DT16, DRT19] for more recent proofs.

######  Theorem 1.4.

Let $p\in[0,1]$.

  * •

If $p<p_{c}$, then there exist $c,C>0$ such that, for every $n\geq 1$,
$\theta_{n}(p)\leq Ce^{-cn}$;

  * •

If $p>p_{c}$, then $\theta(p)\geq(p-p_{c})/2$.

###### Proof.

Let us first prove exponential decay in the subcritical phase: Let $p<p_{c}$.
We observe that, for every $p^{\prime}\in(p,p_{c})$, there exists $m\geq 1$
such that $p\leq p^{\prime}-2\theta_{m}(p^{\prime})$. We fix such
$(p^{\prime},m)$. Theorem 1.1 applied to $p^{\prime}$ (with the crude bound
$\theta_{n}(p^{\prime})\leq 1$) implies that

$\forall n\geq 1,\quad\theta_{2n}(p)\leq C_{0}\exp\Big{(}-\frac{\log
2}{m}n\Big{)}.$

This implies the result for $n$ even, so also for $n$ odd since
$\theta_{n}(p)$ is decreasing in $n$.

Let us now prove the mean field lower bound: Let $p\in[0,1]$ and $m\geq 1$.
Theorem 1.1 implies that $\theta(p-2\theta_{m}(p))=0$. As a result,
$p-2\theta_{m}(p)\leq p_{c}$. If we let $m$ go to $+\infty$, we obtain that
$\theta(p)\geq(p-p_{c})/2$. ∎

######  Remark 1.5.

The following result was proven in [CC87]: $\theta(p)\geq(p-p_{c})/p$. We will
also obtain this result, see the discussion below Corollary 1.6. However, we
will not obtain the bound $\theta(p)\geq\frac{p-p_{c}}{p(1-p_{c})}$ proven in
[DT17].

###  1.2 A stochastic domination result

The main novelty of this paper is a stochastic domination result. The quantity
that will appear in this result is in some sense more $1-\theta_{n}$ than
$\theta_{n}$ itself, so we let

$\pi_{n}(p)=\mathbb{P}_{p}[0\nleftrightarrow
S_{n}]=1-\theta_{n}(p)\quad\text{and}\quad\pi(p)=\lim_{n\rightarrow+\infty}\pi_{n}(p)=1-\theta(p).$

Before stating our result, let us state a corollary. We let $\preceq$ denote
stochastic domination between probability measures (i.e. the existence of an
increasing coupling). We recall that
$\mathcal{E}_{n}=\\{\\{v,w\\}\in\mathcal{E}:v,w\in B_{n}\\}$.

######  Corollary 1.6.

For any $m,n\geq 1$ and $p\in(0,1)$,

$\mathbb{P}_{p\pi_{m}(p)}\big{[}\omega_{|\mathcal{E}_{n}}\in\cdot\big{]}\preceq\mathbb{P}_{p}\big{[}\omega_{|\mathcal{E}_{n}}\in\cdot\;\big{|}\;0\nleftrightarrow
S_{n+m}\big{]}.$

By applying this to the decreasing event $\\{0\nleftrightarrow S_{n}\\}$, we
obtain that

$\pi_{n}(p\pi_{m}(p))\geq\frac{\pi_{n}(p)}{\pi_{n+m}(p)}.$ (1)

One can interpret Corollary 1.6 as follows in the case $n\gg m\gg 1$: up to a
small error (that corresponds to what happens in the thin annulus
$B_{n+m}\setminus B_{n}$), multiplying $p$ by $\pi_{m}(p)$ (which is very
close to $1$ if $p<p_{c}$) has more effect than conditioning on
$\\{0\nleftrightarrow S_{n+m}\\}$.

Inequality (1) implies the inequality $\theta(p)\geq(p-p_{c})/p$ from Remark
1.5:

###### Proof.

Let $p\in(p_{c},1)$ and $m\geq 1$. Inequality (1) implies that
$\pi(p\pi_{m}(p))=1$. Therefore, $p\pi_{m}(p)\leq p_{c}$. If we let $m$ go to
$+\infty$, we obtain that $\pi(p)\leq p_{c}/p$, or equivalently that
$\theta(p)\geq(p-p_{c})/p$. ∎

When the reader looked at Inequality (1), they may have wanted to make a
telescopic product. So let us do this right now.

######  Lemma 1.7.

For any $m\geq 1$ and $p\in(0,1)$,

$\prod_{i\geq 1}\pi_{im}(p\pi_{m}(p))\geq\pi_{m}(p).$

###### Proof.

By (1),

$\prod_{i\geq 1}\pi_{im}(p\pi_{m}(p))\geq\prod_{i\geq
1}\frac{\pi_{im}(p)}{\pi_{(i+1)m}(p)}=\frac{\pi_{m}(p)}{\pi(p)}\geq\pi_{m}(p).\qed$

######  Remark 1.8.

Even if we will use Lemma 1.7 rather than the present remark, let us note that
a similar telescopic product already implies an interesting sharpness-type
result. More precisely, let us prove in this remark that

$\forall p<p_{c},\quad\sum_{n\geq 1}\theta_{n}(p)<+\infty.$ (2)

###### Proof of (2).

Let $p\in(0,p_{c})$. We first note that (2) is equivalent to $\prod_{n\geq
1}\pi_{n}(p)>0$, so let us prove this. Let $p^{\prime}\in(p,p_{c})$ and $m\geq
1$ such that $p\leq p^{\prime}\pi_{m}(p^{\prime})$. We have

$\prod_{n\geq 1}\pi_{n}(p)\geq\prod_{n\geq
1}\pi_{n}(p^{\prime}\pi_{m}(p^{\prime}))\overset{\eqref{eq:cor}}{\geq}\prod_{n\geq
1}\frac{\pi_{n}(p^{\prime})}{\pi_{n+m}(p^{\prime})}=\prod_{n=1}^{m}\pi_{n}(p^{\prime})>0.\qed$

In order to prove Theorem 1.1, we need a result that is analogous to Corollary
1.6 but that involves inhomogeneous percolation. Let
$\textbf{{p}}\in[0,1]^{\mathcal{E}}$. In inhomogeneous percolation of
parameter p, each edge $e$ is open with probability $\textbf{{p}}_{e}$ and
closed otherwise, independently of the other edges. We let
$\mathbb{P}_{\textbf{{p}}}$ denote the corresponding product probability
measure on $\\{0,1\\}^{\mathcal{E}}$ and we let

$\pi_{n}(\textbf{{p}})=\mathbb{P}_{\textbf{{p}}}[0\nleftrightarrow S_{n}].$

Moreover, for every $e=\\{v,w\\}\in\mathcal{E}$ and $n\geq 1$, we choose to
measure the probability that $e$ is not connected to $S_{n}$ by using the
following quantity:

$\pi_{n}(e,\textbf{{p}})=\min\\{\mathbb{P}_{\textbf{{p}}}[v\nleftrightarrow
S_{n}],\mathbb{P}_{\textbf{{p}}}[w\nleftrightarrow S_{n}]\\}.$

For every $m,n\geq 1$, we define
$\textbf{{q}}_{n}^{m}:(0,1)^{\mathcal{E}}\rightarrow(0,1)^{\mathcal{E}}$ by

$\textbf{{q}}_{n}^{m}(\textbf{{p}})_{e}=\begin{cases}\textbf{{p}}_{e}\pi_{n+m}(e,\textbf{{p}})\text{
if $e\in\mathcal{E}_{n}$,}\\\ \text{$\textbf{{p}}_{e}$ otherwise}.\end{cases}$

(The values of $\textbf{{q}}_{n}^{m}$ outside of $\mathcal{E}_{n}$ have no
importance in the present section, but this choice will be useful later.)

Our more precise stochastic domination result is the following proposition.

######  Proposition 1.9.

For any $m,n\geq 1$ and $\textbf{{p}}\in(0,1)^{\mathcal{E}}$,

$\mathbb{P}_{\textbf{{q}}_{n}^{m}(\textbf{{p}})}\big{[}\omega_{|\mathcal{E}_{n}}\in\cdot\big{]}\preceq\mathbb{P}_{\textbf{{p}}}\big{[}\omega_{|\mathcal{E}_{n}}\in\cdot\;\big{|}\;0\nleftrightarrow
S_{n+m}\big{]}.$

By applying this to the decreasing event $\\{0\nleftrightarrow S_{n}\\}$, we
obtain that

$\pi_{n}(\textbf{{q}}_{n}^{m}(\textbf{{p}}))\geq\frac{\pi_{n}(\textbf{{p}})}{\pi_{n+m}(\textbf{{p}})}.$
(3)

Corollary 1.6 is a consequence of this proposition:

###### Proof of Corollary 1.6.

Let $p\in(0,1)$ and define $\textbf{{p}}\in(0,1)^{\mathcal{E}}$ by $\forall
e\in\mathcal{E},\textbf{{p}}_{e}=p$. Let $e\in\mathcal{E}_{n}$. Then, each
endpoint of $e$ is at distance at least $m$ from $S_{n+m}$, so
$\pi_{n+m}(e,\textbf{{p}})\geq\pi_{m}(p)$. As a result,
$\textbf{{q}}_{n}^{m}(\textbf{{p}})_{e}\geq p\pi_{m}(p)$. ∎

###  1.3 Further remarks

  * •

Contrary to the other proofs of sharpness (see [Men86, AB87, DT16, DRT19]), we
do not use any differential formula (such as Russo’s formula).222In this
spirit, it seems worth mentioning here the recent paper [DMT21] that proposes
a new proof – without differential formulas – of some scaling relations in the
plane. One could argue that this should make the proof easier to extend to
some models with dependence. Indeed, the link between the derivative of the
probabilities and the pivotal set can be very complicated in dependent models.
Unfortunately (and contrary to [DRT19]), we have not succeeded in extending
the present proof to other models (such as FK percolation or some continuous
percolation model). Moreover, our proof of sharpness is short but longer than
the one of [DT16] (see also [DT17]). However, we believe that our approach
(and more generally the fact that sharpness can be proven via couplings),
intermediate results such as Proposition 1.9, and our main result Theorem 1.1,
may be useful. (See Section 4 for a consequence of Theorem 1.1.)

  * •

The proof of Proposition 1.9 is inspired by [Rus82, Section 3]. The main
difference is that (even if we will not say this explicitly in order to
simplify the exposition) we have used a decision tree. Decision trees are
central objects in computer science that have been shown to be very useful to
prove sharpness results in [DRT19]. See [GS14] for more about the application
of decision trees to percolation.

#### Organization of the paper.

Theorem 1.1 is proven in Section 2 by using Lemma 1.7 and (3). Proposition 1.9
is proven in Section 3. In Section 4, we discuss a consequence of Theorem 1.1
for percolation in high dimensions.

#### Acknowledgments.

I would like to thank Barbara Dembin and Tom Hutchcroft for inspiring
discussions. Moreover, I would like to thank Vincent Beffara and Stephen
Muirhead for a simplification of the proof of some lemmas. I am also grateful
to Stephen Muirhead for his comments on a previous version of the paper.

##  2 Proof of Theorem 1.1

Theorem 1.1 essentially follows by applying (3) inductively and by using Lemma
1.7 to say that, in this procedure, $p$ has not decreased too much. We first
note that it is sufficient to prove the following:

For every $p\in(0,1)$ and $i,m,n\geq 1$,

$\pi_{n+im}(p\pi_{m}(p)^{2})\geq\pi_{n}(p)^{1/2^{i}}.$ (4)

###### Proof of Theorem 1.1 by using (4).

Let $p\in(0,1)$ and $m,n\geq 1$. We observe that Theorem 1.1 is obvious if
$\theta_{m}(p)\geq 1/2$ or $m\geq n$, so we assume that $\theta_{m}(p)<1/2$
and $n>m$. Next, we note that

$p\pi_{m}(p)^{2}\geq p-2\theta_{m}(p).$

As a result, (4) applied to $i=i_{0}:=\lfloor n/m\rfloor$ implies that

$\theta_{2n}(p-2\theta_{m}(p))\leq\theta_{n+i_{0}m}(p-2\theta_{m}(p))\leq
1-(1-\theta_{n}(p))^{1/2^{i_{0}}}.$

We conclude by using that $1-(1-x)^{a}\leq(2\log 2)ax$ for every $x\in[0,1/2]$
and $a\geq 0$ and that $i_{0}\geq n/m-1$. ∎

So let us prove (4). Let $\textbf{{p}}\in(0,1)^{\mathcal{E}}$ and $m,n\geq 1$.
We first compare $\pi_{n+m}(\textbf{{q}}_{n}^{m}(\textbf{{p}}))$ with
$\pi_{n}(\textbf{{p}})$. To this purpose, we observe that
$\pi_{n+m}(\textbf{{q}}_{n}^{m}(\textbf{{p}}))$ is larger than or equal to
both $\pi_{n+m}(\textbf{{p}})$ and
$\pi_{n}(\textbf{{q}}_{n}^{m}(\textbf{{p}}))$. As a result, (3) implies that

$\pi_{n+m}(\textbf{{q}}_{n}^{m}(\textbf{{p}}))\geq\sqrt{\pi_{n}(\textbf{{p}})}.$
(5)

For every $i,m,n\geq 1$, we let

$\textbf{{q}}_{n}^{m,(i)}=\textbf{{q}}_{n+(i-1)m}^{m}\circ\dots\circ\textbf{{q}}_{n+m}^{m}\circ\textbf{{q}}_{n}^{m}.$

By applying (5) inductively, we obtain that, for every
$\textbf{{p}}\in(0,1)^{\mathcal{E}}$ and $i,m,n\geq 1$,

$\pi_{n+im}(\textbf{{q}}_{n}^{m,(i)}(\textbf{{p}}))\geq\pi_{n}(\textbf{{p}})^{1/2^{i}}.$
(6)

In order to apply (6), we need a lower bound on
$\textbf{{q}}_{n}^{m,(i)}(\textbf{{p}})$.

######  Lemma 2.1.

Let $p\in(0,1)$, define $\textbf{{p}}\in(0,1)^{\mathcal{E}}$ by
$\textbf{{p}}_{e}=p$ for every $e\in\mathcal{E}$ and let $i,m,n\geq 1$. Then,

$\forall e\in\mathcal{E},\quad\textbf{{q}}_{n}^{m,(i)}(\textbf{{p}})_{e}\geq
p\prod_{j=1}^{i}\pi_{jm}(p).$

###### Proof.

First step: $e\in\mathcal{E}_{n}$. Let $p,i,m,n$ as in the statement. We
observe that

$\textbf{{q}}_{n}^{m,(i)}=\textbf{{q}}_{n+(i-1)m}^{m}\circ\textbf{{q}}_{n}^{m,(i-1)}$

(where $\textbf{{q}}_{n}^{m,(0)}:=id$). In this step, we consider some
$e\in\mathcal{E}_{n}$. Since $e\in\mathcal{E}_{n+(i-1)m}$, we have

$\textbf{{q}}_{n}^{m,(i)}(\textbf{{p}})_{e}=\textbf{{q}}_{n}^{m,(i-1)}(\textbf{{p}})_{e}\times\pi_{n+im}(e,\textbf{{q}}_{n}^{m,(i-1)}(\textbf{{p}}))\\\
\geq\textbf{{q}}_{n}^{m,(i-1)}(\textbf{{p}})_{e}\times\pi_{im}(p),$

where in the inequality we have used that
$\textbf{{q}}_{n}^{m,(i-1)}(\textbf{{p}})\leq p$ and that the endpoints of $e$
are at distance at least $im$ from $S_{n+im}$. So the result in the case
$e\in\mathcal{E}_{n}$ follows by induction on $i$.

Second step: $e\notin\mathcal{E}_{n}$. Let $p,i,m,n$ as in the statement. In
this step, we consider some $e\notin\mathcal{E}_{n}$. If
$e\notin\mathcal{E}_{n+(i-1)m}$, then
$\textbf{{q}}_{n}^{m,(i)}(\textbf{{p}})_{e}=\textbf{{p}}_{e}=p$. If
$e\in\mathcal{E}_{n+(i-1)m}\setminus\mathcal{E}_{n}$, then we let
$j\in\\{1,\dots,i-1\\}$ be the number such that
$e\in\mathcal{E}_{n+jm}\setminus\mathcal{E}_{n+(j-1)m}$ and we observe that

$\textbf{{q}}_{n}^{m,(i)}(\textbf{{p}})_{e}=\textbf{{q}}_{n+(i-1)m}^{m}\circ\dots\circ\textbf{{q}}_{n+jm}^{m}(\textbf{{p}})_{e}=\textbf{{q}}_{n+jm}^{m,(i-j)}(\textbf{{p}})_{e}.$

We conclude by applying the first step to $i-j$ and $n+jm$ instead of $i$ and
$n$. ∎

We are now in shape to prove (4).

###### Proof of (4).

Let $i,m,n\geq 1$ and $p\in(0,1)$, and define
$\textbf{{p}}_{m}\in(0,1)^{\mathcal{E}}$ by
$(\textbf{{p}}_{m})_{e}=p\pi_{m}(p)$ for every $e\in\mathcal{E}$. By Lemma 2.1
applied to $p=p\pi_{m}(p)$ and then by Lemma 1.7,

$\forall
e\in\mathcal{E},\quad\textbf{{q}}_{n}^{m,(i)}(\textbf{{p}}_{m})_{e}\geq
p\pi_{m}(p)\prod_{j\geq 1}\pi_{jm}(p\pi_{m}(p))\geq p\pi_{m}(p)^{2}.$

This together with (6) applied to $\textbf{{p}}=\textbf{{p}}_{m}$ implies that

$\pi_{n+im}(p\pi_{m}(p)^{2})\geq\pi_{n}(p\pi_{m}(p))^{1/2^{i}}\geq\pi_{n}(p)^{1/2^{i}}.\qed$

##  3 Proof of Proposition 1.9

Let $m,n\geq 1$ and $\textbf{{p}}\in(0,1)^{\mathcal{E}}$. In order to simplify
the notations, we write

$A=\\{0\nleftrightarrow S_{n+m}\\},\quad E=\mathcal{E}_{n+m},\quad
N=\text{Card}(E)\quad\text{and}\quad F=\mathcal{E}_{n}.$

In this section, we prove Proposition 1.9. To this purpose, we construct a
coupling $(X,Y)=(X_{e},Y_{e})_{e\in E}$ of

$\mu_{\textbf{{p}}}:=\mathbb{P}_{\textbf{{p}}}\big{[}\omega_{|E}\in\cdot\;\big{|}\;A\big{]}\quad\text{and}\quad\mathbb{P}_{\textbf{{q}}_{n}^{m}(\textbf{{p}})}\big{[}\omega_{|E}\in\cdot\big{]}$

such that $X_{|F}\geq Y_{|F}$. (It would have been sufficient to construct the
coupling in $F$ but it will be easier to construct it in $E$.)

Let $(U_{e})_{e\in E}$ be i.i.d. uniform random variables in $[0,1]$. First,
we define $Y$ by $Y_{e}=1$ if
$U_{e}\leq\textbf{{q}}_{n}^{m}(\textbf{{p}})_{e}$ and $Y_{e}=0$ otherwise, for
every $e\in E$.

We use notations similar to those in [DRT19]: we let $\vec{E}$ denote the set
of all $N$-uples $e=(e_{1},\dots,e_{N})\in E^{N}$ with $e_{j}\neq e_{k}$ for
all $j\neq k$ (i.e. this is the set of all possible orderings of $E$). We fix
some $e^{0}\in\vec{E}$ that we will just use to make some arbitrary decisions:
when we refer to the “first edge”, we always refer to the ordering given by
$e^{0}$.

We define $X$ together with a random variable e with values in $\vec{E}$
inductively as follows, where we use the notations
$e_{[k]}=(e_{1},\dots,e_{k})$ and $x_{e_{[k]}}=(x_{e_{1}},\dots,x_{e_{k}})$:

  * •

We let $\textbf{{e}}_{1}$ be first edge adjacent to $0$;

  * •

We let $X_{\textbf{{e}}_{1}}=1$ if
$U_{\textbf{{e}}_{1}}\leq\mu_{\textbf{{p}}}[\omega_{\textbf{{e}}_{1}}=1]$ and
$X_{\textbf{{e}}_{1}}=0$ otherwise;

  * •

Let us assume that $\textbf{{e}}_{[k-1]}$ and $X_{\textbf{{e}}_{[k-1]}}$ have
been defined. If there exists an edge $\\{v,w\\}\in
E\setminus\\{\textbf{{e}}_{1},\dots,\textbf{{e}}_{k-1}\\}$ such that $v$ or
$w$ is connected to $0$ by edges in
$\\{\textbf{{e}}_{1},\dots,\textbf{{e}}_{k-1}\\}$ that are open in $X$, then
we let $\textbf{{e}}_{k}$ be the first such edge. Otherwise, we just let
$\textbf{{e}}_{k}$ be the first edge in
$E\setminus\\{\textbf{{e}}_{1},\dots,\textbf{{e}}_{k-1}\\}$;

  * •

We let $X_{\textbf{{e}}_{k}}=1$ if
$U_{\textbf{{e}}_{k}}\leq\mu_{\textbf{{p}}}[\omega_{\textbf{{e}}_{k}}=1\mid\omega_{\textbf{{e}}_{[k-1]}}=X_{\textbf{{e}}_{[k-1]}}]$
and $X_{\textbf{{e}}_{k}}=0$ otherwise.

Let us show that $X$ has the desired law (see [DRT19, Lemma 2.1] for a more
general result – we include a proof here for completeness).

######  Lemma 3.1.

In the construction above, we have $X\sim\mu_{\textbf{{p}}}$.

###### Proof.

Let $x$ in the support of $X$ (that is included in that of
$\mu_{\textbf{{p}}}$ by construction) and let $e$ be the unique element of
$\vec{E}$ such that $\textbf{{e}}=e$ on $\\{X=x\\}$. We have

$\mathbb{P}[X=x]=\prod_{k=1}^{N}\mathbb{P}\big{[}X_{e_{k}}=x_{e_{k}}\;\big{|}\;X_{e_{[k-1]}}=x_{e_{[k-1]}}\big{]}.$

By construction of $(X,\textbf{{e}})$,
$\\{X_{e_{[k-1]}}=x_{e_{[k-1]}}\\}=\\{X_{e_{[k-1]}}=x_{e_{[k-1]}}\\}\cap\\{\textbf{{e}}_{[k]}=e_{[k]}\\}$.
As a result, the above equals

$\displaystyle\prod_{k=1}^{N}\begin{cases}\mathbb{P}\big{[}U_{e_{k}}\leq\mu_{\textbf{{p}}}\big{[}\omega_{e_{k}}=1\;\big{|}\;\omega_{e_{[k-1]}}=X_{e_{[k-1]}}\big{]}\;\big{|}\;X_{e_{[k-1]}}=x_{e_{[k-1]}}\big{]}\text{
if $x_{e_{k}}=1$}\\\
\mathbb{P}\big{[}U_{e_{k}}>\mu_{\textbf{{p}}}\big{[}\omega_{e_{k}}=1\;\big{|}\;\omega_{e_{[k-1]}}=X_{e_{[k-1]}}\big{]}\;\big{|}\;X_{e_{[k-1]}}=x_{e_{[k-1]}}\big{]}\text{
if $x_{e_{k}}=0$}\end{cases}$
$\displaystyle=\prod_{k=1}^{N}\mu_{\textbf{{p}}}\big{[}\omega_{e_{k}}=x_{e_{k}}\;\big{|}\;\omega_{e_{[k-1]}}=x_{e_{[k-1]}}\big{]}=\mu_{\textbf{{p}}}[x],$

where we have used that $U_{e_{k}}$ is independent of $X_{e_{[k-1]}}$ (because
the latter is measurable with respect to $U_{e_{[k-1]}}$). ∎

We are now in shape to prove Proposition 1.9.

###### Proof of Proposition 1.9.

It is sufficient to show that $X_{|F}\geq Y_{|F}$. As a result, it is
sufficient to show that, if $(x,e)\in\\{0,1\\}^{E}\times\vec{E}$ is in the
support of $(X,\textbf{{e}})$, then for every $k$ such that $e_{k}\in
F=\mathcal{E}_{n}$, we have

$\mu_{\textbf{{p}}}\big{[}\omega_{e_{k}}=1\;\big{|}\;\omega_{e_{[k-1]}}=x_{e_{[k-1]}}\big{]}\geq\textbf{{q}}_{n}^{m}(\textbf{{p}})_{e_{k}}=\textbf{{p}}_{e_{k}}\pi_{n+m}(e_{k},\textbf{{p}}).$

Let us fix such $(x,e)$ and $k$. We recall that we say that an edge is pivotal
for $A$ if changing the state of this edge changes $1_{A}$. We first observe
that, under $\mathbb{P}_{\textbf{{p}}}$, $\omega_{e_{k}}$ is independent of
$A\cap\\{\omega_{e_{[k-1]}}=x_{e_{[k-1]}}\\}\cap\\{\text{$e_{k}$ is not piv.\
for $A$}\\}$ (because the latter is measurable with respect to
$\omega_{|E\setminus\\{e_{k}\\}})$. By using this in the second equality
below, we obtain that

$\mu_{\textbf{{p}}}\big{[}\omega_{e_{k}}=1\;\big{|}\;\omega_{e_{[k-1]}}=x_{e_{[k-1]}}\big{]}\\\
=\mathbb{P}_{\textbf{{p}}}\big{[}\omega_{e_{k}}=1,\text{$e_{k}$ is not piv.\
for $A$}\;\big{|}\;A,\omega_{e_{[k-1]}}=x_{e_{[k-1]}}\big{]}\\\
=\textbf{{p}}_{e_{k}}\mathbb{P}_{\textbf{{p}}}\big{[}\text{$e_{k}$ is not
piv.\ for $A$}\;\big{|}\;A,\omega_{e_{[k-1]}}=x_{e_{[k-1]}}\big{]}.$

We deduce that it is sufficient to prove that

$\mathbb{P}_{\textbf{{p}}}\big{[}\text{$e_{k}$ is not piv.\ for
$A$}\;\big{|}\;A,\omega_{e_{[k-1]}}=x_{e_{[k-1]}}\big{]}\geq\pi_{n+m}(e_{k},\textbf{{p}}).$
(7)

In order to prove this, we distinguish between two cases, where $\tau$ is the
last $j\in\\{1,\dots,N\\}$ at which at least one endpoint of $e_{j}$ is
connected to $0$ by edges in $\\{e_{1},\dots,e_{j-1}\\}$ that are open in $x$.

  * •

If $k>\tau$, then knowing that $\omega_{e_{[k-1]}}=x_{e_{[k-1]}}$ is
sufficient to know whether $A$ holds or not, so the left-hand-side of (7)
equals $1$;

  * •

If $k\leq\tau$, then we can write $e_{k}=\\{v_{k},w_{k}\\}$ where $w_{k}$ is
connected to $0$ by edges in $\\{e_{1},\dots,e_{k-1}\\}$ that are open in $x$.
We observe that, if $e_{k}$ is pivotal for $A$ and if
$\omega_{e_{[k-1]}}=x_{e_{[k-1]}}$, then there is an open path from $v_{k}$ to
$S_{n+m}$ that does not use any edge from $\\{e_{1},\dots,e_{k}\\}$. We let
$B$ denote this event. By this observation and then the Harris–FKG inequality
(see [Gri99, Theorem 2.4] or [BR06, Lemma 3]) applied to
$\mathbb{P}_{\textbf{{p}}}[\,\cdot\mid\omega_{e_{[k-1]}}=x_{e_{[k-1]}}]$, we
have

$\displaystyle\mathbb{P}_{\textbf{{p}}}\big{[}\text{$e_{k}$ is piv.\ for
$A$}\;\big{|}\;A,\omega_{e_{[k-1]}}=x_{e_{[k-1]}}\big{]}$
$\displaystyle\leq\mathbb{P}_{\textbf{{p}}}\big{[}B\;\big{|}\;A,\omega_{e_{[k-1]}}=x_{e_{[k-1]}}\big{]}$
$\displaystyle\leq\mathbb{P}_{\textbf{{p}}}\big{[}B\;\big{|}\;\omega_{e_{[k-1]}}=x_{e_{[k-1]}}\big{]}\text{
(this is where we have used Harris--FKG)}$
$\displaystyle=\mathbb{P}_{\textbf{{p}}}[B]\leq\mathbb{P}_{\textbf{{p}}}[v_{k}\leftrightarrow
S_{n+m}]\leq 1-\pi_{n+m}(e_{k},\textbf{{p}}).$

This shows (7) and ends the proof. ∎

##  4 A consequence for percolation in high dimensions

Consider bond percolation on the hypercubic lattice $\mathbb{Z}^{d}$ for some
$d\geq 11$. The following near-critical sharpness estimate has recently been
proven in [HMS21].

######  Theorem 4.1 ([HMS21, Theorem 1.2]).

There exist $c,C,\varepsilon_{0}>0$ such that, for every
$\varepsilon\in(0,\varepsilon_{0})$ and $n\geq 1$,

$\frac{c}{n^{2}}\exp(-C\varepsilon^{1/2}n)\leq\theta_{n}(p_{c}-\varepsilon)\leq\frac{C}{n^{2}}\exp(-c\varepsilon^{1/2}n).$

In this subsection, we give a new (shorter) proof of the upper bound from
Theorem 4.1. (The techniques of the present paper seem quite useless in order
to prove the lower bound.) As in [HMS21], we need the following result from
[KN11] (that relies on [HS94] for $d\geq 19$ and on [FvdH17] for the extension
to $d\geq 11$):

There exist $c,C>0$ such that, for every $n\geq 1$,

$\frac{c}{n^{2}}\leq\theta_{n}(p_{c})\leq\frac{C}{n^{2}}.$ (8)

(Actually, we just need the upper bound.)

###### Proof of the upper bound from Theorem 4.1.

We fix a constant $C$ such that (8) holds. Also, we fix some
$\varepsilon_{0}>0$ such that

$(2C/\varepsilon_{0})^{1/2}\geq 1.$

Let $\varepsilon\in(0,\varepsilon_{0})$ and write
$m=\lceil(2C/\varepsilon)^{1/2}\rceil$. We have

$\forall n\geq
1,\quad\theta_{n}(p_{c}-\varepsilon)\overset{\eqref{eq:KN11}}{\leq}\theta_{n}(p_{c}-2\theta_{m}(p_{c}))\overset{\text{\eqref{eq:KN11}
and Thm.\
\ref{thm:quant}}}{\leq}\frac{C_{0}C}{n^{2}}\times\frac{1}{2^{n/m}}\\\
\leq\frac{C_{0}C}{n^{2}}\exp\Big{(}-\frac{\log
2}{2\times(2C)^{1/2}}\varepsilon^{1/2}n\Big{)},$

where in the last inequality we have used that $m\leq
2\times(2C/\varepsilon)^{1/2}$ (because $\varepsilon\leq\varepsilon_{0}$). ∎

######  Remark 4.2.

The reason why Theorem 1.1 is sharp (in some sense) for percolation in high
dimensions is that the mean-field equality $\eta_{1}\nu=1$ holds for these
graphs. Here, $\eta_{1}$ is the one-arm exponent and $\nu$ is the correlation
length exponents – see e.g. the recent work [DM21] for the definition of
$\eta_{1}$ and $\nu$, some inequalities involving these exponents, and further
references. For percolation on $\mathbb{Z}^{d}$ with $d\geq 11$, it has been
proven that these exponents exist and satisfy $\eta_{1}=1/\nu=2$ [HS94, KN11,
FvdH17]. Let us note that, as one can see for instance by using Theorem 1.1,
if $\eta_{1}$ and $\nu$ exist, then they satisfy $\eta_{1}\nu\leq 1$.

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