NOTION OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE CHARACTERIZED AS A SOLITON HAVING A DIPOLE-LIKE FLUCTUATING CHARGE DISTRIBUTION

Abstract

It is shown how a smooth motion of the center of charge of an extended particle having a dipole-like fluctuating charge distribution can be described approximately in terms of the notions of real-valued 'proximity density' and complex-valued 'proximity amplitude'. This description is involved with an approximation procedure which is based on a particular time-slicing procedure applied to small time intervals, somewhat similar to what is done in Feynman's path integral approach to quantum mechanics.

Full text

CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
NOTION OF GAUSS PROXIMITY
APPLIED TO THE
INTRINSIC DYNAMICS OF A PARTICLE
CHARACTERIZED AS A
SOLITON HAVING A DIPOLE-LIKE FLUCTUATING CHARGE DISTRIBUTION
P R E P R I N T
Karl G. Kreuzer
kreuzer-dsr[at]magenta.de
2020-08-26 / revised 2023-08-26 / updated 2023-10-31
Abstract :
It is shown how a smooth motion of the center of charge of an extended particle with a dipole-like fluctuating charge distribution
can be described approximately in terms of the notions of real-valued ’proximity density’ and complex-valued ’proximity
amplitude’. This description involves an approximation procedure based on a special time-slicing procedure applied to small
time intervals, somewhat similar to what is done in Feynman’s path integral approach to quantum mechanics.
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
1. Particles Considered as Soliton Solutions of Fields
To specify the notion of ’position’ for a particle, we consider a particle from a field-theoretic viewpoint.
We assume that the traditional notions of ’time’ and ’space’ are appropriate, i.e. we describe ’time’ and ’space’ mathematically
as an ’event’-based 4-dimensional manifold, which is locally isomorphic to an affine linear space with an indefinite bilinear-form
embodying the ’causal structure’ ( Minkowski space ). Furthermore, we assume that physical objects and physical processes,
which are characterized by numerical attributes, are related to open compact non-denumerable subsets ( regions ) of ’events’
or to denumerable subsets ( sequences ) thereof, i.e. we assume that statements about ’measurable’ values of the numerical
attributes of a physical object can be made by means of certain functions of ’space-time coordinates’ of ’events’. The resulting
mathematical formalism is so-called classical field theory (CFT). CFT is associated with two characteristic formalisms, i.e. the
Lorentz transformation behavior machinery and the Lagrange-Noether conservation law machinery.1
To consider a free specific particle in the context of CFT means to assign one or several (number-valued) quantities ψn, called
field amplitudes, which are the values of functions fnof space-time coordinates xµand which are assumed to be non-vanishing
within a small tiny region of space, where the particle is assumed to exist for a time period [tI, tF] . These field amplitudes
embody the variables in mathematical expressions derived from a Lagrangian, which can be used to calculate the electric charge
and other supposed types of charges and all the characteristic properties of a particle can be calculated by means of integrals
and by means of a more or less sophisticated algebraic formalism. It is assumed that a Lagrangian can be chosen such that a
solution of the field equations yields the corresponding energy-momentum-density tensor and charge-current-density vector(s),
which describe the spatial structure of the particle ( as well as its kinematic and dynamic behavior ), vanish outside some
small tiny region of space. This means that we assume that the field equations corresponding to the chosen Lagrangian have
spatially localized solutions, i.e. soliton solutions. 2 3
1cf. A.O.Barut Electrodynamics and Classical Theory of Fields and Particles(1964) and B.Kosyakov Introduction to the Classical Theory of
Particles and Fields.
2In the case of the non-linear Klein-Gordon-Maxwell field a method of how to construct Lagrangians which have this desired property has been
proposed 1984 by the author.
3Soliton solutions are related to Schwartz functions and compressed Schwartz functions, i.e. Schwartz functions whose range are mapped onto to
a compact region of Rn.
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
By means of the energy-momentum-density tensor and charge-current-density vectors, the center of energy ~xCE(t) and the
centers of charges ~xCC,r(t), r = 1,...,K are specified for each instant of time.
The center of energy is basically given by the energy density, while the center of charge is basically given by the
square of charge density.4
Depending on the type of Lagrangian, for a soliton solution of the field equations these vectors are generally different from each
other.5It is assumed that a position vector ~xPOS(t) of the particle can be defined s a particular weighted sum of the
vectors ~xCE(t) and ~xCC,r (t), i.e.
~xPOS(t) := a ~xCE(t) + b1~xCC,1(t)+ b2~xCC,2(t)+ ... +bK~xCC,K (t), a, b1, b2,...,bKreal positive , a+b1+...+bK= 1 .(1)
It is assumed that this position vector is one of the ’measurable’ quantities of CFT.
In the case where ~xPOS (t) is considered with respect to an inertial frame of reference which yields a vanishing momentum
~
P(t) = ~
0, t ∈[tI, tF] and a vanishing center of energy ~xCE (t) = ~
0, t ∈[tI, tF] , ~xPOS (t) represents the motion of the weighted sum
of the centers of charge.
For simplicity, we will first assume that there is only one type of charge ( imagine electric charge ), is present in the Lagrangian.6
4The determination of the center of electric charge, as well as of any other kind of charge, is connected with certain difficulties, which arise from
the fact that the charge of a particle is a quantity which is characterized by an either positive or negative number of zero. Different definitions of
the center of charge can be given.
5This is one of the most striking features of CFT !
6A particle theory involving a Lagrangian with one electric charge-current density and a magnetic charge-current density, is one of the topics of
the author’s project Hypotron Theory.
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
2. Meeting Points of a Trajectory - Time-Slicing Procedure - Quasi-Stochastic Process
The following assumptions are made :
1. The energy-momentum-density tensor and charge-current-density vector are in such a way that the particle is character-
ized by a dipole moment which is fluctuating at a high rate, i.e. it is possible to identify a pair of poles with positions
~x+(t) and ~x−(t) which describe dipole which is permanently collapsing and re-extending at a ( more or less ) high rate
and with some ( more or less ) high degree of non-regularity of this process of collapsing and re-extending.
2. The location of the particle, i.e. its position vector ~xPOS(t), is permanently changing due to some self-interaction. The
motion, i.e. the way the position vector ~xPOS (t) varies with respect to time, is characterized by a continuous path ( and
corresponding trajectory ), which may be a regular path ( imagine the corresponding trajectory as an intersecting curve
with a ’simple nice shape’ ), or an irregular path ( imagine the corresponding trajectory as an intersecting curve with a
more or less ’chaotic curly clew-like shape’ ).
3. There are smallest intervals of time [t−τ, t] , and during such a time interval of time the corresponding motion is
characterized by a finite number of meeting points Pn(t, τ), n = 1,...,N(t, τ) , i.e. points in space where the trajectory
intersects itself or is close these points up to some tiny distance value. Each of the meeting points is characterized by the
relative meeting multiplicity wn(t, τ) , which is defined by
wn(t, τ) := Mn(t, τ ),the number how often the path meets the point Pn(t, τ )
M(t, τ),the total number of meetings , n = 1,...,N(t, τ),(2)
i.e. wn(t, τ) is the relative frequency of the point Pn(t, τ) being passed during the time interval [t−τ, t] . Obviously
N(t,τ)
X
n=1
wn(t, τ) = 1 .(3)
Note that the notation ’...’(t, τ) means the result of a mapping which maps from a set of sets of numbers, i.e. intervals [t−τ, t]
of R, to some set of values ’...’ !
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
The meetings during the time interval [t−τ, t] specify a fictitious quasi-stochastic process associated with the time interval
[t−τ, t] , in the following way :
Let tn,ν be those times when the position vector ~xPOS(t) of the particle coincides with position of the n-th meeting point , i.e.
~xPOS(tn,ν ) = ~
ξn(t, τ), n = 1,...,N(t, τ ), ν = 1,...,Mn(t, τ),(4)
and let
˜
tk, k = 1,...,
N(t,τ)
X
n=1
Mn(t, τ),(5)
be those times which are given by renumbering tn,ν by only one index only, and let
ˆ
tk, k = 1,...,
N(t,τ)
X
n=1
Mn(t, τ),(6)
be those times which are given by the permutation of ˜
tk, which ensures that
ˆ
tk<ˆ
tk+1 .(7)
The instants of time ˆ
t1,ˆ
t2,ˆ
t3,... specify the particular time slicing of the time interval [t−τ, t] .
If one connects the points specified by the vectors
~xPOS(ˆ
t1), ~xPOS(ˆ
t2), ~xPOS(ˆ
t3), . . . (8)
by straight line segments, one obtains a trajectory is obtained which has a zig-zag shape and which looks more or less irregular,
depending on the degree of irregularity of the particle’s motion. This is the typical feature of a Brownian motion, which can
be understood as a stochastic process. We call this particular motion a quasi-stochastic process, because by construction it
is the result from a smooth motion of the particle. 7
7The meeting points can alternatively be introduced by assuming that the position vector ~xPOS(t) of the particle often passes small spatial regions
Rnwhich are centered around ~
ξn.
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
3. Time-Averaged Position Vector - Proximity Density
It is important to note that (A) it is the hypothesis of a path being characterized by meeting points of the corresponding
trajectory, and (B) it is the non-zero duration τof the considered time interval [t−τ, t] , leading to the feature that the
location of the considered particle in time interval [t−τ, t] can be characterized in a statistical manner. Thus, the vague idea
of ( expressed in popular words ) ’probability of being somewhere’ can be given a precise meaning: It is the result of particular
mathematical approximation procedure which leads to an assignment of quasi-stochastic processes to the motion, the latter
being considered in small time intervals [t−τ, t] .
By means of the meeting points and the corresponding relative frequencies the notion of time averaged position of a particle
with regard to the time interval [t−τ, t] is defined as
~xTA(t, τ) :=
N(t,τ)
X
n=1
wn(t, τ)~
ξn(t, τ),(9)
where ~
ξn(t, τ) are the position vectors of the meeting points Pn(t, τ) .
Since τis considered to be a parameter with a fixed value and not as a variable, the dependence of ~xTA(t, τ ) on τcould be
omitted in the notation. This would make the time averaged position associated with a time interval [t−τ, t] formally look
like a position associated with an instant of time t, and omitting τin the notation would hide the statistical character inherent
in the definition of ~xTA(t, τ ) . This could lead to some confusion about the meaning of the notion of time-averaged position
vector, and therefore omitting of τin the notation should be avoided.
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
Due to the statistical character of the notion of time-averaged position vector, it is possible to introduce various statistical
quantities that are related to an average. In particular, the square variance of the time-averaged position vector can be
defined as
SQVAR(t, τ ) :=
N(t,τ)
X
n=1
wn(t, τ)k~
ξn(t, τ)−~xTA(t, τ )k2.(10)
Obviously
SQVAR(t, τ )≥0.(11)
Hence, the square root of SQVAR(t, τ ) exists, and is used to define the variance of the time-averaged position :
VAR(t, τ ) := pSQVAR(t, τ).(12)
Square variance as well as variance represent a measure which can be understood as the degree of localization of the particle
with regard to the interval of time [t−τ, t] .
The time-averaging procedure above will lead to the notion of proximity density.
This notion is formed in a purely mathematical way as follows.
Using the Dirac delta function, the sum Pnwn(t, τ )~
ξn(t, τ) is replaced by a sum of integrals :
N(t,τ)
X
n=1
wn(t, τ)~
ξn(t, τ) =
N(t,τ)
X
n=1 ZR3
wn(t, τ)~x δ(~x −~
ξn(t, τ)) d3x . (13)
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
In the expression eq. (13) wn(t, τ )δ(~x −~
ξn(t, τ)) can be replaced by Gauss-like functions fn,ǫ,t,τ (~x), which represent approxi-
mately the Dirac delta function in the sense of distributions, i.e.
ZR3
wn(t, τ)~x δ(~x −~
ξn(t, τ)) d3x≈ZR3
fn,ǫ,t,τ (~x)~x d3x , (14)
where
ZR3
g(~x)fn,ǫ,t,τ (~x)d3x→ZR3
g(~x)δ(~x −~
ξn(t, τ)) d3xas ǫ→0,ZR3
fn,ǫ,t,τ (~x)d3x= 1 .(15)
Hence,
N(t,τ)
X
n=1
wn(t, τ)~
ξn(t, τ)≈ZR3
fǫ,t,τ (~x)~x d3x , (16)
where
fǫ,t,τ (~x) := 1
N(t, τ)
N(t,τ)
X
n=1
fn,ǫ,t,τ (~x),ZR3
fǫ,t,τ (~x)d3x= 1 .(17)
This means that ~xTA(t, τ ) can be expressed approximately as an integral, whose integrand contains a continuous function
fǫ,t,τ (~x) .
The use of Dirac delta function can be avoided by the concept of Gauss proximity. This numerical quantity will be the crucial
ingredient to express ~xTA(t, τ) as an integral.
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
Let
ft,τ (~x) :=
N(t,τ)
X
n=1
wn(t, τ)e−k~x−~
ξn(t,τ)k2/λ2, λ = some ultra-small length unit .(18)
The value of this function at some ~x is by definition the Gauss proximity8at ~x with respect to the points specified by ~
ξn(t, τ)
and their weights wn(t, τ ) .
Integrals of this quantity are
ZR3
e−k~x−~
ξn(t,τ)k2/λ2d3x=π3/2λ3,ZR3
e−k~x−~
ξn(t,τ)k2/λ2~x d3x=π3/2λ3~
ξn(t, τ).(19)
Hence, using a notation where ρTA(..., t, τ ) denotes the variable which adopts the values specified by the function ft,τ (...) , the
time-averaged position vector of the particle can be expressed as :
~xTA(t, τ ) = ZR3
ρTA(~x, t, τ)~x d3x , (20)
ρTA(~x, t, τ) := π−3/2λ−3
N(t,τ)
X
n=1
wn(t, τ)e−k~x−~
ξn(t,τ)k2/λ2.(21)
Formally the expression eq. (20) looks like the position vector specified by a field which describes a continuously distributed
bulk of matter characterized by a ’matter density’ ρTA(~x, t, τ) . However, ρTA(~x, t, τ) is not a ’matter density’, because it’s
definition is based on both a statistical concept and a metric concept.
8The Gauss proximity is numerical quantity which characterizes the nearness of point Pto an given set of points Pn. The Gauss proximity
serves as a weighting such that for given set of points Pnin the sum of the corresponding position vectors only those points contribute which are
close to a considered point P. Details will be presented in a separate paper.
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
The statistical meaning of ρTA(~x, t, τ ) emerges from the time-averaging procedure, i.e. the quantity ρTA(~x, t, τ ) has the dimension
of ’relative frequency’/’density’. Therefore one is attempted to call ρTA(~x, t, τ) ’probability density’. But that would be !!!
misleading because the entities to which ρTA(~x, t, τ ) refers are related by a well-defined dynamical process, and they cannot be
regarded as the ’events’ of ordinary probability theory, i.e. it makes no sense at all to try to apply ordinary probability theory
without further assumptions and statements about the interaction of particles during a ’measuring process’ ( cf. section 5 ). !!!
It is much more appropriate to call the quantity ρTA(~x, t, τ)statistical proximity density or just simply proximity density.
This name gives a clue to the way this quantity is constructed and avoids the misleading ideas involved in attempts to apply
ordinary probability theory. !!!
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
4. Time-Averaged Dipole-Moment-Weighted Position Vector - Proximity Amplitude
The time-averaged position vector of the particle ~xTA(t, τ ) is basically specified as the weighted sum of the positions of the
meeting points, and wn(t, τ ) plays the role of the weighting factors. This weighting procedure can be modified in order to take
into account the characteristics of a fluctuating dipole which is assumed to be inherent to the particle.
The basic idea here is that in the weighting procedure the contributions to the position ~xTA(t, τ ) will be decreased, if (1)
the dipole moments corresponding to the times tn,ν of the time slicing procedure are small, or if (2) the dipole moments
corresponding to tn,ν are partially compensated, so to speak, by dipole moments corresponding to some tn,µ pointing in almost
opposite directions. This idea leads to tuples of complex numbers associated with the degree of stretching of the dipole at
times tn,ν . From these complex numbers a complex-valued quantity called complex-valued normalized dipole amplitude will
emerge, which will serve as one of the basic ingredients in the definition of the proximity amplitude.
To start with the details, let
~
D(t) = ZR3
j0(t, ~x)~x d3x(22)
be the dipole moment of the particle, resulting from the charge density j0(t, ~x) at time t, and let
~
DREL(t) := 1
DMAX
~
D(t), DMAX := lowest upper limit of the amount of dipole moment of the fluctuating particle (23)
be the relative dipole moment. Obviously
k~
DREL(t)k ≤ 1.(24)
The relative dipole moment is a dimensionless quantity, which characterizes the degree of extension of the particle with respect
to it’s charge at time t.
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
Let ~eTAN(t) be the normalized tangent vector of the particle’s motion at time t, and let
~un,3(t, τ ) := 

1
Mn(t, τ)
Mn(t,τ)
X
ν=1
~eTAN(tn,ν )
/k...k(25)
be the average of the tangent vectors ~eTAN(tn,ν ) corresponding to the times tn,1,...,tn,Mn(t,τ).
For each tfrom ~un,3(t, τ ) two vectors can be constructed, which are orthonormal w.r.t. ~un,3(t, τ) :9
~un,2(t, τ ) := 

Mn(t,τ)
X
ν=1 ~
D(tn,ν )−~
D(tn,ν )•~un,3(t, τ )~un,3(t, τ)
/k...k(26)
~un,1(t, τ ) := ~un,2(t, τ)×~un,3(t, τ ) (27)
Hence,
~un,i(t, τ )×~un,j(t, τ ) = ~un,k(t, τ ), i, j, k, = 1,2,3 cyclic order ,k~un,k(t, τ )k= 1 , k = 1,2,3.(28)
For each (t, τ) the vector ~un,3(t, τ ) specifies an oriented plane which is perpendicular to ~un,3(t, τ ) , and which contains the
meeting point Pn(t, τ ) .
The projection of the relative dipole moment vector onto this plane
~
A(tn,ν , t, τ ) := ~
DREL(tn,ν )−~
DREL(tn,ν )•~un,3(t, τ )~un,3(t, τ),k~
A(tn,ν, t, τ )k ≤ 1,(29)
is called the projected relative dipole moment. This is the quantity that plays the central role in the following considerations.
Similar to the relative dipole moment, the projected relative dipole moment is a dimensionless quantity, which characterizes
the degree of extension of the particle at times tn,ν with regard to its charge density.
9•denotes the scalar product of 3-vectors. ×denotes the cross product of 3-vectors.
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
Using the shorthand notations
~
An,ν := ~
An,ν (t, τ ) := ~
A(tn,ν , t, τ ),(30)
˜
φn,ν,µ := ˜
φn,ν,µ(t, τ ) := ˜
φ(tn,ν , tn,µ, t, τ ) := angle between ~
A(tn,ν, t, τ ) and ~
A(tn,µ, t, τ ),(31)
φn,ν := φn,ν (t, τ ) := φ(tn,ν, t, τ ) := angle from ~un,1(t, τ ) to ~
A(tn,ν , t, τ )dipole angles at meeting points Pn,(32)
the square of the norm of the vector PMn(t,τ)
ν=1 ~
An,ν , which is the double sum scalar products of the projected relative dipole
moment ~
A(tn,ν ) corresponding to tn,ν , can be expressed by means of complex numbers in the following way :
k
Mn(t,τ)
X
ν=1
~
An,ν k2=
Mn(t,τ)
X
ν,µ=1
~
An,ν•~
An,µ =
Mn(t,τ)
X
ν,µ=1
An,ν An,µ cos˜
φn,ν,µ(33)
=
Mn(t,τ)
X
ν,µ=1
An,ν An,µ cos(φn,ν −φn,µ) (34)
=
Mn(t,τ)
X
ν,µ=1
An,ν An,µ
1
2ei(φn,ν −φn,µ)+e−i(φn,ν −φn,µ )(35)
=
Mn(t,τ)
X
ν,µ=1
An,ν An,µ ei(φn,ν −φn,µ)(36)
=
Mn(t,τ)
X
ν,µ=1
An,ν An,µ ei φn,ν e−i φn,µ =

Mn(t,τ)
X
ν=1
An,ν ei φn,ν 


Mn(t,τ)
X
µ=1
An,µ ei φn,µ 
(37)
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
Eq. (34) results from the fact that the projected relative dipole moments corresponding to tn,ν are coplanar vectors, and from
the symmetry properties of the cosine function. Eq. (35) is a consequence an elementary property of the complex-valued
exponential function. Eq. (36) results from a splitting of the double sum and exchanging the indices νand µ. Eq. (37) results
from properties of complex numbers.
Because of eq. (37), it is convenient to introduce abbreviations and names for the following quantities :
An,ν (t, τ ) := k~
An,ν(t, τ )kso-called dipole amplitude at time tn,ν ,(38)
ˆ
An,ν (t, τ ) := An,ν(t, τ )/v
u
u
t
N
X
n=1
Mn(t,τ)
X
ν,µ=1
~
An,ν(t, τ )•~
An,µ(t, τ ) so-called normalized dipole amplitude at time tn,ν ,(39)
ψn,ν (t, τ ) := ˆ
An,ν (t, τ )ei φn,ν (t,τ )so-called complex-valued normalized dipole amplitude at time tn,ν ,(40)
ψn(t, τ) :=
Mn(t,τ)
X
ν=1
ψn,ν (t, τ ) (41)
so-called summed-up complex-valued normalized dipole amplitude at meeting point Pn(t, τ ) .
Obviously,
0≤An,ν (t, τ )≤1,
N(t,τ)
X
n=1
ψn(t, τ)ψn(t, τ ) = 1 ,0≤ψn(t, τ)ψn(t, τ )≤1,0≤ˆ
An,ν (t, τ )≤1.(42)
The definition in eq. (39) requires that the denominator √... does not vanish. This represents a condition which must be
fulfilled by the vectors ~
A(tn,ν, t, τ ) .
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
Finally, the time-averaged dipole-moment-weighted position vector is defined as
~xTADA(t, τ) := 

N(t,τ)
X
n=1
Mn(t,τ)
X
ν,µ=1
~
An,ν (t, τ )•~
An,µ(t, τ )

−1N(t,τ)
X
n=1
Mn(t,τ)
X
ν,µ=1
~
An,ν(t, τ )•~
An,µ(t, τ )~xPOS(tn,ν ),(43)
i.e. it is basically an average over position vectors where sums of scalar products of projected dipole moments serve as
weighting factors.
By means of the normalized dipole amplitude and the complex-valued normalized dipole amplitude, and because of
~xPOS(tn,ν ) = ~
ξn(t, τ) , the time-averaged dipole-moment-weighted position vector can be expressed as
~xTADA(t, τ) =
N(t,τ)
X
n=1 

Mn(t,τ)
X
ν=1
ˆ
An,ν(t, τ )ei φn,ν(t,τ )


Mn(t,τ)
X
ν=1
ˆ
An,ν (t, τ )e−i φn,ν (t,τ )
~
ξn(t, τ),(44)
and
~xTADA(t, τ) =
N(t,τ)
X
n=1 

Mn(t,τ)
X
ν=1
ψn,ν (t, τ )


Mn(t,τ)
X
ν=1
ψn,ν(t, τ )
~
ξn(t, τ),(45)
and
~xTADA(t, τ) =
N(t,τ)
X
n=1
ψn(t, τ)ψn(t, τ )~
ξn(t, τ).(46)
Note that eq. (46) is formally similar to eq. (9) , the expression for the time-averaged position vector ~xTA(t, τ ) .
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
In the case of the central rest frame of the particle, ~xTADA (t, τ ) and ~xTA(t, τ) are expressions for the time-averaged motion of
the center of charge. The difference between the time-averaged dipole-moment-weighted position vector ~xTADA(t, τ ) and the
time-averaged position vector ~xTA(t, τ) is :
~xTADA(t, τ)−~xTA(t, τ ) =
N(t,τ)
X
n=1 ψn(t, τ)ψn(t, τ )−wn(t, τ)~
ξn(t, τ) (47)
=
N(t,τ)
X
n=1 


Mn(t,τ)
X
ν=1
ˆ
An,ν (t, τ )ei φn,ν (t,τ )


Mn(t,τ)
X
µ=1
ˆ
An,µ(t, τ )e−i φn,µ(t,τ )
−Mn(t, τ)
M(t, τ)
~
ξn(t, τ).
(48)
To pass from a tuple of complex numbers ψn(t, τ ) to a proximity density ρTADA(~x, t, τ) , the same procedure is performed as in
the case of the time-averaged position vector. This leads to
~xTADA(t, τ) = ZR3
ρTADA(~x, t, τ)~x d3x(49)
where in this case
ρTADA(~x, t, τ) := π−3/2λ−3
N(t,τ)
X
n=1
ψn(t, τ)ψn(t, τ )e−k~x−~
ξn(t,τ)k2/λ2.(50)
This normalized quantity is called proximity density or statistical proximity density, which is a normalized quantity, i.e.
ZR3
ρTADA(~x, t, τ)d3x= 1 .(51)
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
To express the vector ~xTADA (t, τ ) as an integral
ZR3
Ψ(~x, t, τ) Ψ(~x, t, τ )~x d3x , (52)
the complex-valued quantity Ψ(~x, t, τ) is defined as follows :
Ψ(~x, t, τ ) :=
N(t,τ)
X
n=1
ψn(t, τ) ˆχn(~x, t, τ ) =
N(t,τ)
X
n=1
χn(~x, t, τ),(53)
where
χn(~x, t, τ ) := ψn(t, τ ) ˆχn(~x, t, τ),ˆχn(~x, t, τ) := √π−3/2λ−3e−1
2k~x−~
ξn(t,τ)k2/λ2(54)
is the complex-valued proximity at some point with respect to the meeting point Pn, and | k...k | denotes the norm of the
function PN(t,τ)
n=1 χn, i.e.
| k
N(t,τ)
X
n=1
χn(~x, t, τ)k | =
ZR3
N(t,τ)
X
n=1
χn(~x, t, τ )
N(t,τ)
X
m=1
χm(~x, t, τ)d3x

1/2
.(55)
Obviously the norms of ˆχnand χnare
| kˆχnk |2= 1 ,| kχnk |2=ZR3
χn(~x, t, τ )χn(~x, t, τ )d3x=ψn(t, τ)ψn(t, τ),(56)
and because of the definiton of ψn
ZR3
N(t,τ)
X
n=1
χn(~x, t, τ )χn(~x, t, τ )d3x=
N(t,τ)
X
n=1
ψn(t, τ)ψn(t, τ ) = 1 .(57)
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
Furthermore :
ZR3
χn(~x, t, τ)χm(~x, t, τ )d3x≈0 provided that k~
ξn−~
ξmkis sufficiently large compared to λ , (58)
ZR3
χn(~x, t, τ)χm(~x, t, τ )~x d3x≈0 provided that k~
ξn−~
ξmkis ...see above... ,(59)
| k
N(t,τ)
X
n=1
χn(., t, τ)k | = 1 +
N(t,τ)
X
n,m=1,n6=mZR3
χn(~x, t, τ)χm(~x, t, τ )d3x≈1 provided that k~
ξn−~
ξmkis ...see above... .(60)
Hence, the following approximations
ZR3X
n
χn(~x, t, τ )χn(~x, t, τ )~x d3x , (61)
≈ZR3 X
n
χn(~x, t, τ )! X
m
χm(~x, t, τ)!~x d3x , (62)
=ZR3
Ψ(~x, t, τ) Ψ(~x, t, τ )~x d3x(63)
can be done, which finally results in
~xTADA(t, τ)≈ZR3
Ψ(~x, t, τ ) Ψ(~x, t, τ )~x d3x , ρTADA(~x, t, τ )≈Ψ(~x, t, τ ) Ψ(~x, t, τ).(64)
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
Thus we have expressed ~xTADA(t, τ ) approximately as an integral with an integrand which contains complex-valued quantity,
which is related to the proximity density ρTADA(~x, t, τ ) . Therefore it makes sense to call Ψ(~x, t, τ )proximity amplitude or more
precisely statistical proximity amplitude.
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
5. Relationship between Proximity Density and Proximity Amplitude and the Statistics of Position Measurements
What remains to be done is to establish a link between the quasi-stochastic process and the statistics in experiments which
are thought to represent a ’measurement’ of the position of a particle. For this purpose we assume that it is possible to refer
to an intertial frame of reference, which yields a vanishing momentum of the particle ~
P(t) = ~
0, t ∈[tI, tF] and a vanishing
center of energy of the particle ~xCE (t) = ~
0, t ∈[tI, tF] , which by definition is the central rest system of the particle. In this case
~xPOS(t) represents the motion of the center of charge, and ~xTA(t, τ ) and ~xTADA(t, τ) represent approximate descriptions thereof.
If it were possible to perform a measurement of the position vector ~xPOS(t) of the particle in the period of time
[t−τ, t] , then it is quite natural to expect that one of those vectors is obtained as a possible result where the particle is most
likely present in the period of time [t−τ, t] , which is one of the vectors ~
ξn(t, τ) corresponding to the meeting points Pn(t, τ ) .
Hence, the assumption to be made then is that -broadly speaking- the statistics resulting from the relative frequency of the
meeting points agrees with the statistics of the results obtained in the measurement of position. Thereby, expressed in popular
terms, the ’probability of the particle of being somewhere’ is transformed into a ’probability of finding the particle somewhere’.
Defining
~xBORN(t, τ ) := ZR3
Ψ(~x, t, τ) Ψ(~x, t, τ )~x d3x(65)
and assuming that
~xPOS(t)≈~xTADA(t, τ )≈~xBORN(t, τ ) (66)
are ’good’ approximations and assuming that it is possible to measure ~xTADA(t, τ ) in the period of time [t−τ, t] , then it makes
sense to state that the particle’s position vector ~xPOS (t) at the instant of time tis determined in an approximate manner by
the complex-valued proximity amplitude Ψ(~x, t, τ) associated with the time period [t−τ , t] .
Note that ~xBORN (t, τ ) can be expressed explicitly in terms of the vectors of the meeting points and the corresponding normalized
dipole amplitudes and dipole angles.
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
Conversely it can be argued that some given complex-valued function Φ(~x, t, τ) can be considered as a possible candidate for
representing a definite proximity amplitude, and can be used to calculate approximately the vector ~xTADA(t, τ) according to
eq. (78), and assuming that a best approximation can be achieved if Φ(~x, t, τ ) is normalized to unity.
This finally leads to the idea to consider
ZR3
Ψ(~x, t, τ) Ψ(~x, t, τ )~x d3x(67)
as the value of the position of the center of charge and to assume that this is a quantity which is measurable in appropriate
experiments.
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
6. Summary and Proposal for Future Investigations
The following tables summarize the quantities which are involved with the definition of the proximity amplitude eq. (53). A
notation is used in which the dependence of these quantities on (t, τ) is suppressed.
Pn, n = 1,...N meeting points
wnrelative meeting multiplicity of meeting point Pnsee eq. (2)
~
ξnposition vector meeting point Pnsee eq. (4)
~un,3average tangent vector at meeting point Pnsee eq. (25)
~
An,ν , ν = 1,...,Mnprojected relative dipole moment at meeting point Pnat time tn,ν see eq. (29)
φn,ν , ν = 1,...,Mndipole angles at meeting point Pnat times tn,ν see eq. (32)
ˆ
An,ν , ν = 1,...,Mnnormalized dipole amplitudes at meeting point Pnat times tn,ν see eqs. (38,39)
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
ψn,ν , ν = 1,...,Mncomplex-valued normalized dipole amplitudes at meeting point Pnat times tn,ν see eq. (40)
ψnsummed-up complex-valued normalized dipole amplitude at meeting point Pnsee eq. (41)
χn(~x) complex-valued weighted proximity at point specified by ~x w.r.t. the meeting point Pnsee eq. (54)
Ψ(~x, t, τ) complex-valued proximity amplitude at point specified by ~x w.r.t. the set of all meeting points see eq. (53)
The proximity density and proximity amplitude constructed above are characteristics of the set of the meeting points, which
can be considered as an ’ensemble’ of points as well as a stochastic process. It is also a characteristic property of the motion of
the particle during some small time interval. This means that, with regard to questions of interpretation of traditional ( and
questionable ) axiomatic ( and dogmatic ) quantum mechanics, the proximity density and proximity amplitude constructed
above support, so to speak, both, i.e. parts of Bohr’s vague -but not convincing- Copenhagen interpretation as well as parts
of the strict -but also not convincing- statistical interpretation introduced by Weyl and von-Neumann. In retrospect, Born’s
intuitive ’probability rule’ of traditional axiomatic quantum mechanics seems to be justified provided that the physical meaning
of the ’wave function’ of quantum mechanics is understood in terms of proximity amplitude.
The present considerations also complement Feynman’s approach to traditional quantum mechanics, which rests on the idea
of a stochastic zig-zag motion and the corresponding mathematical formalism of path integrals for ’probability amplitudes’.
However, Feynman’s approach to traditional quantum mechanics does not answer the question of the concrete physical origin
and background of ’probability amplitudes’. Feynman’s time slicing procedure does not reflect any possible physical processes
between the individual time stamps. If one understands ’probability amplitude’ in the sense of proximity amplitude, then it
becomes clear what is assumed to be going on between the individual time stamps, and why, from a physical and conceptual
point of view, Feynman’s approach to traditional quantum mechanics can make any sense at all.
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CONCEPT OF GAUSS PROXIMITY APPLIED TO THE INTRINSIC DYNAMICS OF A PARTICLE
The basic problem for future research concerning the present topic is to find out whether or not it is possible to establish a
foundation of some modified quantum mechanics in a non-dogmatic and non-mystical manner in terms of classical field theory
and in terms of some new statistical concepts, which reflect the assumption that a particle has an inherent ( more or less regular
or irregular or chaotic or quasi-stochastic ) fluctuation of distributions of electric charge and of perhaps alleged magnetic charge
and of perhaps alleged other charge-like quantities.
Because all the quantities considered in the present work emerge from the mathematical constructs of classical field theory, the
way in which the proximity density and the proximity amplitude are constructed supports the idea that classical field theory
is the ultimate bedrock of theoretical particle physics. Classical field theory, when applied to objects on the most lowest scale
sketched above, can also be understood as mathematical machinery which, from the very beginning of this physical theory, is
able to avoid the introduction of ’point-like’ structures. Instead ’distributions’ of charge and energy play the dominant role.
Soliton solutions of the field equations are the means that most vividly represent the notion of a particle. However, even
classical field theory has a conceptual deficiency and requires a particular generalization, which encompasses the transition
from ’sharp’ ( rational or real ) numbers to some kind of ’fuzzy’ numbers, thus reflecting aspects of some ’uncertainty’ in a
natural manner. Preliminary investigations have shown that there is a close relationship between a particular type of fuzzy
numbers and of the mathematical theory of Hilbert spaces ( cf. the author’s project B-numbers ). It is expected that this
transition will eliminate mathematical inconsistencies of classical field theory, and will also lead to new ontological aspects of
this subject.
∗ ∗ ∗
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