---
abstract: |
    The vacuum is proposed to be a dielectric medium constituted of neutral but polarizable vacuuons based on overall experimental observations in separate publications. In the present paper I formally develop the dielectric theory for this dielectric vacuum.\
    \
    \* This is an abbreviated but self-contained introduction to the dielectric theory I formulated as part of a unification scheme I developed since 2000, originally presented in full in book [*Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces*]{}, 1st & 2nd printins, Nova Science, NY, 2006.
author:
- |
    J. X. Zheng-Johansson\
    Institute of Fundamental Physics Research, 611 93 Nyköping, Sweden.\
    [ December, 2006]{}
title: ' Dielectric Theory of the Vacuum$^*$ '
---

[v]{}

ł

Introduction
============

The vacuum is induced to be a dielectric medium based on overall experimental observations in separate publications [@Unif1; @Unif2; @jxzj-vacstru]. In this paper I formally develop the dielectric theory for this dielectric medium, or vacuum dielectrics, with respect to an [**empty space**]{} (or free space)— [*the space left over after the removal of the dielectric vacuum*]{}. I will mainly address the questions that, under the influence of an external (vaculeon) charge placed in the vacuum, what is the electric polarization of the medium, the depolarization field, the polarization of each individual vacuuon, the local field acting on a vacuuon dipole, and the electric dipole moment of a vacuuon. These are the basis for expressing the interaction forces and potential energies between an external single vaculeon and its surrounding polarized vacuuons, and between the polarized vacuuons themselves presented in [@Unif1; @qts4sch], and furthermore for expressing the gravitational interaction force between masses[@Unif2; @qts4grv]. This paper is aimed to elucidate the formalism of the vacuum dielectrics mainly; the detail-filling equations for dipole and local fields for a three-dimensional vacuum, involved directly for obtaining vacuum interaction forces [@Unif1] are left out; these are given in [@Unif1].

A large part of the dielectric theory of the vacuum can be formulated in direct analogy to the well-established dielectric theory for ordinary dielectric materials (see e.g. [@Bottcher:1973], or see [@Unif1] for a brief review), given that vacuum is just one specific dielectric medium, and that we have practically based on Proposition 1 an explicit internal structure for the constituent vacuuon particles of the vacuum, from which the vacuum polarization and dielectric properties derive. Vacuum is actually a very well behaved dielectric by virtue of its isotropy and nonpolar nature, among others; although it has also particularities not present with the ordinary dielectric materials as I will discuss when in question.

Dipole Moment and Vacuum Polarization: Definitions {#Sec-App-dielec-muP}
==================================================

\[App-diele-VacPol\]

The electric [*polarization*]{} of an ordinary dielectric medium is defined as the [*dipole moment*]{} $\pdipb$ per unit volume, averaged over a macroscopic volume $\De V$: $\Pob= \frac{\pdipb}{ \De V}$, with respect to the vacuum. We similarly define the [**electric polarization of the dielectric vacuum**]{}, or the [**vacuum polarization**]{}, with respect to the empty space after the removal of the vacuum, over a macroscopic volume $\De V$ as: $$\begin{aligned}
\label{eq-app-dielce-x1}
\Polbvac= \frac{\pdipb_0}{ \De V},\end{aligned}$$ $\pdipb_0$ being the [*electric dipole moment*]{} of the vacuum in the specified volume $\De V$. By virtue of its structure proposed in [@Unif1] namely that, briefly, the vacuum consists of neutral but polarizable entities termed vacuuons, the vacuum presents no permanent dipoles. In other terms, the vacuuons have zero electric dipole moment in the absence of an applied field, and the vacuum is thus a [*nonpolar*]{} dielectric. In the presence of an applied electric field, the n- and p- vaculeon charges comprising a vacuuon will be displaced relative to each other, into dipole charges; accordingly the vacuum is produced with an electric polarization. The n- and p- vaculeons of a vacuuon are bound each other with a binding energy extraordinarily large, estimated [@Unif1] to be 0.5 GeV; therefore normally (except in a pair production) they can not be separated. The vacuum will therefore maintain [*electrically neutral*]{} even in the presence of applied fields. Hence the electric dipole moment in a macroscopic volume $\D V$ of the vacuum may be obtained by $$\begin{aligned}
\label{eq-app-mub}
\pdipb_0 = \sum_{s \ \in \ \De V} \Rb_s q_s, \end{aligned}$$ where $\Rb_s$ is the position vector of charge $q_s$. The subscript $0$ labels that a specified quantity, here $\pdipb_0$, pertains to the (dielectric) vacuum as opposed to an ordinary (dielectric) material medium. With (\[eq-app-mub\]), (\[eq-app-dielce-x1\]) thus writes $$\Polbvac = \frac{\pdipb_0}{\De V} =\frac{ \sum_{s \ {\rm in} \ \De V}     \Rb_s q_s    }{\De V }.  
                 \eqno({\rm \ref{eq-app-dielce-x1}a})$$

![A neutral vacuuon is in the presence of an applied electric field $E_{q^*}$ polarized into a vacuuon dipole of dipole moment $\pdip_\v$, with a displacement $d$ between its two charges $+e$ and $-e$.[]{data-label="fig-dipole"}](ZhengJfigB1dipole_3.eps){width="50.00000%"}

For two opposite-signed charges ($q_p=e$, $q_n=-e$) on a polarized vacuuon $s$, mutually displaced by a distance vector $\db$ (Figure \[fig-dipole\]), equation (\[eq-app-mub\]) gives the (induced) [**electric dipole moment of a vacuuon**]{}: $$\displaylines{ 
\pdipb_{\v}  = \Rb_p (e) + \Rb_n (-e) 
\cr
\hfill = e(\Rb_p -\Rb_n ) = e \db. \hfill
\cr
\hfill 
({\rm \ref{eq-app-mub}}\arm)
}$$

![(a)-(b): An applied field $\Eb_{q^*}$ is produced by two parallel plates with free charges $+q $ and $-q$ on them, and with a dielectric vacuum medium in between. The vacuum is regarded explicitly as non-dielectric in (a), and as a dielectric in (b). (c) shows two charged plates as in (a)-(b), but with the vacuum in between removed, left with an empty space. In (d), without regard to the charged plates, the vacuum polarized as in (b) (or (a), or by the device in (c)), of a polarization $\Pob$ is represented as produced by a polarization charge $-q_p$ and $+q_p$ in the upper and lower surfaces. (e) Relationship between the applied-, macroscopic-, and polarization- field vectors $\Eb_{q^*}$, $\Eb_q$, and $\Eb_{p}$. In all graphs, $S$ indicates a chosen Gaussian surface enclosing charge $+q$ and similarly $-q_p$, and $\Areab$ the piece of area on $S$ with a normal parallel with $\Efb_q$. []{data-label="fig-dielec"}](ZhengJfigB2dielec_3.eps){width="70.00000%"}

Polarization Charge of a Dielectric Vacuum
==========================================

Consider here a region of the vacuum is subjecting to the influence of a uniform applied field, $\Efbo$ as measured in the vacuum. With no loss of generality let this $\Efbo$ be produced in between two parallel uniformly charged plates (Figure \[fig-dielec\]), of equal but opposite-signed free charges, $+q$ and $-q$. Figure \[fig-dielec\]a shows the traditional representation where one regards the vacuum as non-dielectric while, as one may recognize, a practically dielectric property is effectively reflected through a parameter, the permittivity of the vacuum.

Unless specified otherwise we below regard that [*the vacuum is a dielectric*]{} (Figure \[fig-dielec\]b). Suppose we have managed to remove the dielectric vacuum (i.e. all of the vacuuons composing it) in a region in question, say in a volume embracing the two parallel charged plates plus a certain vicinity region, and situate ourselves in the [*empty space*]{} left over as defined earlier. Then, a free charge in it will be $q^*$ if it was $q$ when measured in the dielectric vacuum; and we find by measurement the applied field produced by $q^*$ in this empty space is $\Efb_{q^*}$ (Figure \[fig-dielec\]c).

We now fill in between the charged parallel plates with a piece of dielectric vacuum medium. The field $\Efb_{q^*}$ produced by the free charges on the plates then polarizes the vacuuons constituting the vacuum in between into [**induced vacuuon dipoles**]{}, each of an electric dipole moment $\pdipb_{\v}$ aligned in the direction of $\Efb_{q^*}$. Through the volume between the two plates the displaced charges from each adjacent vacuuon dipoles are lined up head-to-tail in the direction perpendicular to the plates, and annul one another, giving a net zero charge in the interim of the chosen volume. Only on the surface $S$ of the enclosure, see Figure \[fig-dielec\]d, one finds net charges, referred to as [**polarization charge**]{} of the vacuum here, $\qpol$, on each face $\Area$ parallel with the plates; $\qpol$ has an opposite sign to the free charge $q$ on the plate just beside it.

Gauss’s Law in a Dielectric Vacuum {#App-dielec-GsDV}
==================================

The formulae of the ordinary form of Gauss’s law (for electricity here) for the vacuum and for an ordinary dielectric material of dielectric constant $\ke$ between the plates, respectively write, $$\displaylines{ \refstepcounter{equation} 
\label{App-dielec-6}
\hfill \ev_\osub \oint_{S} \Efbo \cdot d \Sb = q, \quad (\arm) \qquad
\qquad
 \ev_\osub \oint_{S} (\Efbo/\ke) \cdot d \Sb = q+q_{p'}, \quad (\brm) 
\hfill ({\rm \ref{App-dielec-6}})
}$$ with $\ev_\osub$ the permittivity of vacuum and $q_{p'}$ polarization charge. These are expressed by phenomenologically regarding the vacuum as “non-dielectric”. In other terms, in this form one does not regard the microscopic and dielectric property of the vacuum explicitly. (See [@Unif1] for a review of the derivation of these and related dielectric relations in terms of directly comparable denotations.)

In analogy to the ordinary form of Gauss’s law, we now establish the corresponding Gauss’s law applied to a general dielectric material with respect to the empty space. For illustration we choose the “dielectric material” between the two charged plates to be just the dielectric vacuum itself, as shown in Figure \[fig-dielec\]b. In a thought experiment it is easy to remove the vacuum outside the region enclosed between the plates, as in Figure \[fig-dielec\]d or b, and situate ourselves in the empty space and look at the interface between the vacuum and the empty space, the upper surface $\Area$ here; on $\Area$ we find the polarization charge to be $\qpol$, as specified earlier. And we will find by measurement the total charge on $\Area$ is $$\begin{aligned}
 \label{eq-App-dielec-qpq}
{q^*}' = q^*+q_p.\end{aligned}$$ We here do not presume any knowledge of the sign of $\qpol $, but rather leave it to be fixed through applying Gauss’s law, as by (\[App-dielec-72\]) below.

That is, as viewed in the empty space, $\Efb_q$ is effectively produced by ${q^*}'$ (the subscript $q$ indicates instead the charge as viewed in the vacuum), and $\Efb_{q^*}$ is produced by $q^* $ ($=q$ as we will show below).

Now with respect to the empty space, with the total charge as given by (\[eq-App-dielec-qpq\]), we apply Gauss’s law to the Gaussian surface $S$ enclosing the upper plate: $$\begin{aligned}
 \label{App-dielec-74}
\ev_\empt \oint \Efb_q \cdot d \Sb = q^* + \qpol. \end{aligned}$$ Where $\ev_\empt $ is the [**permittivity of the empty space**]{}; $\Efb_q$ is now the so-called macroscopic field produced by $q^*$ as measured in the dielectric vacuum. Note that since the empty space has no dielectric property, $\ev_\empt $ would merely represent a dimensional conversion factor. It is straightforward to prove [@Unif1] that (\[App-dielec-74\]) is equivalent to the ordinary form of Gauss’s law for vacuum, ([\[App-dielec-6\]a]{}).

Since the vacuum is such a particular medium that it extends continuously throughout the space, in reality we can never actually find a real surface separating the vacuum and the empty space, and situate ourselves on the empty-space side of it. So, whenever making an actual measurement of the charge, we are only doing this in the interim of the vacuum continuum. In it at any point on any specified surface we make measurement, the polarization charges on its two opposite sides cancel. Consequently in realistic experiments we always measure the true value of the charge as in the empty space. That is $$\begin{aligned}
\label{eq-qq}
q^*\equiv q.\end{aligned}$$

Dielectric Constant. Susceptibility {#App-dielec-DCP}
===================================

Similarly as in the dielectric theory for ordinary materials, we introduce a [**dielectric constant of the (dielectric) vacuum**]{}, denoted by $\kvac$, with respect here to the empty space. $\kvac$ is defined such that the electric field measured in the dielectric vacuum $\Efb_q$, in relation to the field measured in the empty space $\Efb_{q^\emptr}$ is $$\displaylines{ \refstepcounter{equation} 
\label{App-dielec-70}
\hfill 
\Efb_q = \Efb_{q^\emptr} /\kvac.   \hfill ({\rm \ref{App-dielec-70}})
}$$ Accordingly the permittivity of the (dielectric) vacuum with respect to the empty space,[^1] in relation to that of the empty space is $$\displaylines{ \refstepcounter{equation} 
\label{App-dielec-71}
\hfill 
\evo= \kvac \ev_\empt.  
\hfill ({\rm \ref{App-dielec-71}})
}$$ A basic notion we maintain in the two alternative (the dielectric and the “non-dielectric”) representations of the vacuum is that, the $\Efb_q$ field as measured in a dielectric vacuum with respect to the empty space, is the same as one as actually measured in the real vacuum. —This must be so since, irrespective of which representation, we have one and the same vacuum and the force per charge (per unit volume) as represented by $\Efb_q$ in this vacuum must be of an unique value. The field produced at $R$ by a point charge $q$ located at $R=0$ is given by the Coulomb’s law $\Ef_q= \frac{q}{4\pi\evo R^2 }$; it thus follows that in the latter representation, $\evo$ is the same quantity as that actually measured for the vacuum through the Coulomb’s law.

Since the dielectric vacuum responds to $\Efb_{q^*}$ with a depolarization field (see below) which cancels a portion of $\Efb_{q^*}$, thus $\Efb_{q^*}>\Efb_q$. Combining this inequality with ([\[App-dielec-70\]]{}) and in turn the resultant with ([\[App-dielec-71\]]{}) we have $$\displaylines{
\hfill \kvac>1 \qquad\qquad         {\rm and} \quad \evo >\ev_\empt.  \hfill}$$

In further analogy to the dielectric theory for ordinary materials in a “non-dielectric vacuum”, we express the polarization charge of the dielectric vacuum in terms of the polarization of the vacuum via a corresponding Gauss’s law: $$\displaylines{ \refstepcounter{equation} 
\label{App-dielec-72}
\hfill 
\qpol = -\oint \Pob_0 \cdot d \Sb. 
\hfill ({\rm \ref{App-dielec-72}})
}$$ And we express $\Pob_0$ in relation to $\Efb_q$ through an [**electric susceptibility of the dielectric vacuum**]{}, $\Xvac$: $$\displaylines{ \refstepcounter{equation} 
\label{App-dielec-73}
\hfill 
 \Pob_0= \ev_\empt \Xvac \Efb_q. 
\hfill ({\rm \ref{App-dielec-73}})
}$$ Substituting with (\[App-dielec-73\]), (\[App-dielec-72\]) writes: $$\qpol = -\ev_\empt \Xvac  \oint \Efb_q   \cdot d \Sb. \eqno({\rm \ref{App-dielec-72}})'$$ Substituting (\[App-dielec-72\])$'$ into the right-hand side of (\[App-dielec-74\]), rearranging, we have: $$\refstepcounter{equation} \label{App-dielec-75}
     \ev_{\empt} \oint \Efb_q (1+ \Xvac)\cdot d \Sb = q^{\emptr}.  \eqno({\rm \ref{App-dielec-75}})$$

The quantity on the right-hand side of (\[App-dielec-75\]), $q^{\emptr}$, is the free charge measured in the empty space. According to the principle of Gauss’s law applied here to the charge in empty space, the quantity on the left-hand side must amount to the total electric flux produced by the charge $q^{\emptr}$. This corresponds to a so-called electric [**displacement vector**]{} of here the dielectric vacuum: $$\displaylines{  \refstepcounter{equation} 
\label{App-dielec-77}
\hfill \Dbvac = \evo  \Efb_q, \hfill    ({\rm \ref{App-dielec-77}})
}$$ so defined such that $$\displaylines{
\hfill \oint \Dbvac \cdot d \Sb = q^*. \hfill ({\rm \ref{App-dielec-74}})'
}$$ ([\[App-dielec-74\]]{})$'$ and ([\[App-dielec-75\]]{}) give $$\refstepcounter{equation} \label{App-dielec-77b}
\Dbvac=     \ev_{\empt} \Efb_q (1+ \Xvac).  
\eqno ({\rm \ref{App-dielec-77b}a})$$ Canceling $\Dbvac$ between ([\[App-dielec-77b\]]{}a) and ([\[App-dielec-77\]]{}) further gives $$\displaylines{
\hfill \qquad \evo  \Efb_q =    \ev_{\empt} \Efb_q (1+ \Xvac). 
           \hfill  ({\rm \ref{App-dielec-77b}b})
}$$ Substituting into ([\[App-dielec-77b\]]{}b) with (\[App-dielec-71\]) we have $\kvac  \ev_\empt  \Efb_q =\ev_\empt  \Efb_q (1+ \Xvac) $. Or, $$\refstepcounter{equation} 
\label{App-dielec-78}
\kvac  =1+ \Xvac. \eqno({\rm \ref{App-dielec-78}})$$ This gives the dielectric constant of vacuum in relation to susceptibility. Substituting with ([\[App-dielec-73\]]{}), ([\[App-dielec-77b\]]{}b) alternatively writes: $$\evo  \Efb_q= \ev_{\empt} \Efb_q + \Pob_0.    \eqno({\rm \ref{App-dielec-77b}}\crm)$$

Depolarization Field
====================

Applying Gauss’s law for the Gaussian surface about the upper charged plate and enclosing $\qpol (<0)$ only (Figure \[fig-dielec\]d), we get the [**electric polarization field of the vacuum**]{}, produced by the charge displacements of the constituent vacuuons under the influence of the applied field $\Efb_{q^*}$. That is $$\begin{aligned}
\label{App-dielec-17b}
\Efbpol = - \frac{|\qpol|}{\permitempty} = -\frac{\Polbvac}{\permitempty}.\end{aligned}$$ Canceling $\Polbvac$ between (\[App-dielec-73\]) and (\[App-dielec-17b\]) gives: $$\begin{aligned}
 \label{eq-App-Efbpol2}
 \Efbpol 
= - \Xvac  \Efbo. \end{aligned}$$ $\Efbpol $ is explicitly expressed by (\[eq-App-Efbpol2\]) to be antiparallel to $\Efb_{q^*}$ and similarly to $\Efb_q$, and therefore is termed the [**depolarization field**]{} of the vacuum. Substituting (\[App-dielec-17b\]) for $\Pob_0$ into (\[App-dielec-77b\]c) we have $$\displaylines{
\evo \Efb_q = \ev_\empt \Efb_q -  \ev_\empt  \Efb_\pol.
\cr
{\rm Sorting \ gives}\hfill 
\cr\refstepcounter{equation}\label{App-dielec-18}
\hfill
\qquad  \Efb_\pol = -\Efb_q(\frac{\evo }{\ev_\empt }-1) = -\Efb_q(\kvac-1), \hfill ({\rm \ref{App-dielec-18}})
}$$ where the last expression used ([\[App-dielec-71\]]{}). From (\[App-dielec-18\]), we obtain the so-called [**macroscopic field**]{} in the dielectric vacuum: $$\begin{aligned}
\Efb_{q^\emptr} + \Efb_\pol=\Efb_q. \end{aligned}$$ This is just the “applied” field $\Efb_q$ (with respect to a “non-dielectric” vacuum) actually as measured in the vacuum (cf. the remark on the “basic notion” after Eq. [\[App-dielec-71\]]{}).

The results in this section are the direct dielectric-theoretical basis for the depolarization radiation force considered as a cause of gravity[@qts4grv].

Vacuuon Polarizability. Local Field. Dipole Field {#App-dielec-Polty}
=================================================

We now represent the dielectric response of the vacuum, under the influence of the applied uniform field $\Efb_{q^*}$, in terms of the individual constituent vacuuons. $\Efb_{q^*}$ polarizes the initially neutral vacuuons into [**vacuuon dipoles**]{} aligned in the direction of $\Efb_{q^*}$, each having a dipole charge $|e|$, charge displacement $d$, and an electric dipole moment $\pdipb_\v$ expressed earlier in (\[eq-app-mub\]) in terms of $d$. Assuming $\Efb_{q^*}$ is small so that the resulting charge displacement is linear, accordingly $\pdipb_{\v} $ will be in linear proportion to a so-called [**local field**]{}, $\Efblocalempty$, produced by the applied field at the vacuuon. That is $$\begin{aligned}
\label{eq-app-dielec-20}\label{eq-App-dielec-50}
\pdipb_{\v}  (\Rb_s)= \adip_\v  \Efblocalempty (\Rb_s),\end{aligned}$$ where $\adip_\v$ is the [**polarizability of vacuuon**]{}.

The local field at the site $\Rb_s$ of vacuuon $s$, is given by the sum of the applied field $\Efb_{q^*}$ and the total dipole field $\Efbdipoleempty $ produced at it by all other vacuuon dipoles from the surrounding. That is $$\begin{aligned}
\label{eq-app-dielec-21}
\Efblocalempty (\Rb_s)=\Efb_{q^*} (\Rb_s) + \Efbdipoleempty (\Rb_{s}).\end{aligned}$$

The electric field at $\Rb_s$ as produced by a dipole located at $\Rb_{s'}$, having an electric moment $\pdipb_{\v j}(\Rb_{s'} )$, is given by the standard equation of electrostatics: $$\displaylines{ \refstepcounter{equation} 
\label{eq-app-dielec-22}
\hfill \Efbdipoleempty{}_{,s'} (\Rb_{s}) 
= \frac{3 ({\pdipb_{\v}}_{j} \cdot \lb_j ) \lb_j - \l_j^2 {\pdipb_{\v}}_{j} }{ \l_j^5 },       \hfill ({\rm \ref{eq-app-dielec-22}})
}$$ where $\lb_j=\Rb_s -\Rb_{s'}   $ is the distance vector from vacuuon $s'$ to $s$, $j=s-s'= 1,2,\ldots$ for $\Rb_{s'}>0$; or $\lb_j=\Rb_s +\Rb_{s'}   $ and $j=s+s'= 1,2,\ldots$ for $\Rb_{s'}<0$. The total electric field at the $s$th dipole’s location $\Rb_s$ as produced by all the other dipoles, or the [**total dipole field**]{}, is the vector sum $$\refstepcounter{equation}  \label{eq-dielecX1}\label{eq-app-dielec-23}
\Efbdipoleempty (\Rb_s)= \sum_{j=s-s' (\ne 0)} \frac{3 ({\pdipb_{\v}}_j \cdot \lb_j ) \lb_j - \l_j^2 {\pdipb_{\v}}_j }{ \l_j^5 }.   \eqno({\rm \ref{eq-app-dielec-23}})$$ $\Efbdipoleempty$, and accordingly $\Efb_{\local}$ and $\pdipb_{\v} $ are generally dependent on the geometry of the dielectric specimen and the orientation of $\Efb_{q^\emptr}$ relative to the specimen.

Explicit expressions of ([\[eq-app-dielec-23\]]{}) for a linear chain of vacuuon dipoles and for a three dimensional vacuum polarized by a point charge are given in [@Unif1] and not elaborated here.

Acknowledgements {#acknowledgements .unnumbered}
================

The author thanks scientist P-I. Johansson of Uppsala Univ. for his continued support of the research. Prof. R. Lundin has given appreciative review of the unification scheme conceived and developed by the author, and has reflected this in the Forward to the two books [@Unif1; @Unif2] by the author and collaborator P-I. Johansson.

[150]{}

J. X. Zheng-Johansson and P-I. Johansson, with Foreword by Prof. R. Lundin, [*Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces. In: Horizons in World Physics*]{}, 2nd printing, (Nova Science Publishers, New York, 2006) ISBN 1-59454-260-0; arxiv:physics/0412168 (an early brief outline containing terms out of date).

(updated title), (Nova Science Publishers, Inc., New York, 2006) ISBN 1-59454-261-9.

J. X. Zheng-Johansson, “Vacuum Structure”, one chapter in [@Unif1]a; separate paper submitted for publication (2006).

J. X. Zheng-Johansson and P-I. Johansson, “Inference of Schrödinger Equation from Classical-Mechanics Solution,” [*Quantum Theory and Symmetries IV*]{}, [**2**]{}, A Supplement to the Bulgarian Journal of Physics [**33**]{}, ed. V.K. Dobrev, (Heron Press, Bulgaria, 2006) 763-770; preprint: arxiv:physics/0411134.

J. X. Zheng-Johansson, P-I. Johansson, and R. Lundin, “Depolarization Radiation force in a Dielectric Medium. Its Analogy with Gravity,” [*Quantum Theory and Symmetries IV*]{}, [**2**]{}, A Supplement to the Bulgarian Journal of Physics [**33**]{}, ed. V.K. Dobrev, (Heron Press, Bulgaria, 2006) 771-780; “Prediction of Universal Gravity between Charges in a Dielectric Medium,” arxiv:physics/0411245.

C. J. F. Böttcher, [*Theory of electric Polarization*]{} [**I**]{}, (Elsevier Scientific Publishing Company, Amsterdam, 1973); N.W. Ashcroft and D. Mermin, [*Solid State Physics*]{}, (W.B. Saunders College, 1976) Chapter 27. C. Kittel, [*Introduction to Solid State Physics*]{}, 6ed, (John Wiley & Sons, Inc N.Y., 1986) Chapter 13.

[^1]: For the variables $\Xvac$, $\kvac$, and $q^*$ of the dielectric vacuum as measured with respect to the empty space, and also the applied field $E_{q^*}$ in the empty space, we have used a subscript $*$ to distinguish from their counterparts $\Xcal_0$ ($\equiv 0$), $\ke_0$ ($\equiv 1$) and $q$ ($=q^*$ numerically but having a distinct physics), for a “non-dielectric vacuum” with respect to the vacuum itself, and also $E_q$. The latter typically take trivial values only but are ordinarily expressed as variables. By this denotation method all of the variables including ${{{\bf P}\hspace{-0.25cm}-}}_{0} $, $\pdipb_0$, $\pdipb_\v$, $q_p$, ${\ev_\empt}$ $\Efbpol$, $\Efb_{\dipole}$, $\Efb_{\local}$, $\a_\v$ for the dielectric vacuum with respect to the empty space should also be suffixed by $*$, as ${{{\bf P}\hspace{-0.22cm}-}}_{0*}$, $\pdipb_{0^*}$, $\pdipb_{\v^*}$, $q_p^*$, ${\ev_{\empt^*}}$, $\Efb_{p^*}$, $\a_{\v^*}$ in cases where explicitness is needed. We have in this text dropped the $*$ for the latter variables given that no confusion is being caused.
