A quantum similarity discussion about Einstein–Podolsky–Rosen (EPR) paradox in Gaussian enfolded spaces

Abstract

The Einstein–Podolsky–Rosen discussion about the completeness of quantum mechanics is examined using Gaussian enfolded Euclidian spaces. The research is performed within a quantum similarity framework, forming Gaussian quantum polyhedra. The analysis is achieved through simplified descriptions of quantum systems using appropriate Minkowski normalized Gaussian density functions.

Full text

1 23
Journal of Mathematical Chemistry
ISSN 0259-9791
J Math Chem
DOI 10.1007/s10910-020-01158-7
A quantum similarity discussion about
Einstein–Podolsky–Rosen (EPR) paradox in
Gaussian enfolded spaces
Jing Chang & Ramon Carbó-Dorca

1 23
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Vol.:(0123456789)
Journal of Mathematical Chemistry
https://doi.org/10.1007/s10910-020-01158-7
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ORIGINAL PAPER
A quantum similarity discussion
aboutEinstein–Podolsky–Rosen (EPR) paradox inGaussian
enfolded spaces
JingChang1· RamonCarbó‑Dorca2
Received: 15 May 2020 / Accepted: 27 June 2020
© Springer Nature Switzerland AG 2020
Abstract
The Einstein–Podolsky–Rosen discussion about the completeness of quantum
mechanics is examined using Gaussian enfolded Euclidian spaces. The research is
performed within a quantum similarity framework, forming Gaussian quantum poly-
hedra. The analysis is achieved through simplified descriptions of quantum systems
using appropriate Minkowski normalized Gaussian density functions.
Keywords Einstein–Podolsky–Rosen (EPR) paradox· Gaussian enfolded spaces·
Quantum similarity· Gaussian functions· Quantum density functions· Quantum
polyhedra· Variance of a quantum polyhedron
1 Introduction
The Einstein, Podolsky, and Rosen (EPR) 1935 paper1 [1] on the completeness of
quantum mechanics originated myriads of discussions in the form of journal contribu-
tions, varied scientific essays, and books. Reviewing this extensive bibliography in a
very condensed way, the paper of Bell [2] could be quoted as a single summary of the
literature associated with the so-called EPR paradox in the mid of the 20th century.
Some modern 21st-century papers can also be mentioned [3–11], considering not only
the quality of these papers’ discussions on EPR and related experiments and problems
* Ramon Carbó-Dorca
ramoncarbodorca@gmail.com
Jing Chang
jchang5@ualberta.ca
1 Department ofChemical andMaterials Engineering, University ofAlberta, Edmonton,
ABT6G2V4, Canada
2 Institut de Química Computacional i Catàlisi, Universitat de Girona, 17071Girona, Catalonia,
Spain
1 Note a curious characteristic: the EPR publication possesses no references at all.
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but also the additional literature on the EPR problem, which can be obtained from the
included papers’ references.
Choosing to discuss the EPR paradox from the quantum similarity point of view
corresponds to a fourfold purpose. First, the interest of the EPR paradox has aroused
in one of the present authors [12, 13]. Second, the three research lines developed
by one of us within the framework of the enfolded Gaussian spaces [14–17]. Third,
the quantum similarity description of sets of quantum systems developed during the
last decades [20–40] in our laboratory. Finally, one can mention our interest in the
research about the time-like evolution of systems [41–44], as well as the point of
view of time description [45] for general purposes. The relevance of the present study
to the chemical and physical evolution of submicroscopic systems is thus obvious.
Accordingly, the present study will be organized as follows. First, Gaussian func-
tions as a quantum system straightforward descriptor set will be discussed.
Then, the Gaussian enfolded Euclidian spaces will be set up. Application of the
enfolded spaces to the description of a quantum system evolution will be given next.
The splitting of a quantum system in several subsystems will be presented in the
next section, providing the appearance of a Gaussian quantum polyhedron. Time-
like parameters and description of quantum polyhedra evolution will follow.
The definition of the similarity matrix of a Gaussian quantum polyhedron will
be presented, to illustrate and follow the evolution of the original quantum system,
once splits into a collection of subsystems.
The time-like evolution of the subsystem collection in the form of a Gaussian
quantum polyhedron will follow. Finally, the quantum polyhedron variance will be
discussed as an alternative way to study the EPR paradox.
2 Gaussian functions asdensity functions
A Gaussian function centered at a point
𝐑A
can be written through the following
expression:
where the exponential parameter
𝜃
is responsible for the Gaussian function form.
Smaller values of
𝜃
resultant function is Minkowski yield a flattened Gaussian struc-
ture, while larger values of the same parameter make the Gaussian dense at the func-
tion origin
𝐑A
. In such a way one can write the Gaussian function as a Dirac’s delta
function at the parameter infinite limit:
lim
𝜃→∞
exp

−𝜃


𝐫−𝐑A


2

=𝛿

𝐫−𝐑A

.
The Minkowski norm of the Gaussian function (1) is easily calculated to be:
and
n
stands for the dimension of the space where the variable vector
𝐫
belongs.
(1)
g
A(𝐫

𝜃)=exp

−𝜃


𝐫−𝐑A


2
,

gA

=
∫D
gA(𝐫

𝜃)d𝐫=

𝜋
𝜃
n
2
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Therefore, using
𝜃=𝜋
in the Eq. (1), the resultant function is Minkowski
normalized:
g
A(𝐫

𝜋)=exp

−𝜋


𝐫−𝐑A


2

→

gA

=
1
.
Gaussian functions are positive definite functions, and the structure of the
Minkowski norms permits to describe exponent parameters, which can be associated
with several particles
Z
. Thus, one can write:

gA

=Z→

𝜋
𝜃
n
2
=Z→𝜃=
𝜋
Z
2
n
,
Therefore, one can consider a Minkowski normalized Gaussian function repre-
senting several particles
Z
and centered at the point
𝐑A
as:
Although the variable space might be generally described, the space holding
quantum systems with a finite number of particles can be taken as the usual three-
dimensional Euclidian one, and then the dimension parameter is chosen as:
n=3
.
Because of this above discussion, the Gaussian function of the Eq. (2) can be
considered as a simplified quantum density function of a system of
Z
particles, cen-
tered at the position
𝐑A
of a three-dimensional space. Such a Gaussian function used
as a simplified quantum density function has been recently used to describe the ele-
ments of the periodic table and discuss their relationships [46].
3 Gaussian enfolded Euclidean space
The Gaussian function in the Eq.(2) when written in a manner, such that no precise
specific position of the center is given, can be presented like:
and the location of the center might also be seen as a new variable vector
𝐑.
Then the Gaussian function (3) can be considered as a set of Gaussian functions
of the variable
𝐫
which are centered at any of the Euclidian space points
𝐑.
Looking at the previous definitions differently: the Gaussian functions of type (3)
enfold the Euclidian space in such a way that any point of it contains an associated
Gaussian function. Such a geometrical construct has been described, discussed, and
used in several papers [14–17]. This enfolding space structure can be traced in sev-
eral spaces, where each point is associated with a function [17].
If one considers the structure of the quantum vacuum, see for instance [18, 19], it
can be seen as the enfolding (3) with the parameter
Z=1
. Therefore, like:
which assigns a Gaussian function to every point in Euclidian space, as the Gaussian
(4), except for a convenient exponent, corresponds to the ground-state wave function
of the quantum harmonic oscillator.
(2)
g
A(𝐫

Z)=exp

−𝜋Z−
2
n


𝐫−𝐑A


2
.
(3)
g
(𝐫
|
𝐑
|
Z)=exp
(
−𝜋Z−2
n
|
𝐫−𝐑
|
2
)
(4)
g
(𝐫
|
𝐑)=exp
(
−𝜋
|
𝐫−𝐑
|2),
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If in the Eq.(3), the number of particles associated with the function tends to zero,
the exponent tends to infinity, and the function transforms into a Dirac’s delta function.
Thus, the Gaussian enfolded space corresponds to a Dirac function enfolded space; if
instead of a quantum vacuum, one prefers to describe the void space as a zero-particle
density function enfolding.
4 A quantum system description withinaGaussian enfolded space:
theinitial enfolded Euclidian space
Now suppose that the Euclidian space is Gaussian enfolded. In the absence of any sys-
tem, the vacuum could be constructed by Dirac’s functions assuming in this case that
it is void. Alternatively, one can admit an enfolding with Gaussian Minkowski normal-
ized functions, containing one virtual particle at each point. In both cases, the Dirac’s
delta function or the virtual particle Gaussian function transforms into a quantum den-
sity Gaussian function of the kind described in the Eq. (3), when a quantum system
appears at the space position
𝐑.
5 The appearance ofaquantum system of
Z
particles
According to the previous development, any quantum system with several
Z
particles
could be supposedly located at any point R in Euclidean space and might be approxi-
mated by a Gaussian function as depicted in the Eq.(3). Thus, at this point, the virtual
particle becomes a quantum system.
Moreover, due to space isotropy, it is indifferent to where the system is located.
Therefore, to start with, the origin of coordinates is chosen as the site where the quan-
tum system is located. That is:
𝐑=𝟎
and the Gaussian function acting as the quantum
system density function might be written:
6 Time asanatural sequence ofparameters
Now, to simply describe the time-like evolution of a system, time can be considered
as a natural sequence of time-like click values:
𝜏={0, 1, 2, …,t,…},
where at each
value, the whole enfolded space appears changed or unchanged. That is, at
𝜏=0
the
entire space is enfolded by one virtual particle Gaussian density functions, while at
𝜏=1
, for instance, a system defined by a density function like Eq.(5) does appear.
(5)
g
(𝐫
|
𝟎
|
Z)=exp
(
−𝜋Z−
2
n
|
𝐫
|
2
).
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7 Splitting oftheinitial quantum system intoaset ofseveral
subsystems
At a random time value:
𝜏=t
, one can suppose that the initial quantum system
ceases to be stationary and splits into a set of N subsystems, each one possessing
several particles arbitrarily different one from the other. That is, one deals a collec-
tion of subsystems with a specific number of particles:
{
Z
I|
I=1, N
}
and every one
of them is located into a set of points:
{
𝐑
I|
I=1, N
}
of the enfolded space.
Then, in the absence of particle creation or annihilation, the following relation
will hold:
and the set of subsystems will be described by a set of Gaussian density functions,
which can be written as:
At successive time-like clicks, the positions of the subsystems might change.
Thus, the positions of subsystems could be described as functions of the time-like
parameter, using the convention:
∀I=1, N∶𝐑I
⇒
𝐑I(𝜏)
.
The set of subsystem density functions
𝐏
can be easily recognized as a quantum
polyhedron,2 see references [47–56] to have more information on the definitions and
properties of this subject. Each subsystem Gaussian density function could be asso-
ciated with a vertex of the quantum polyhedron. The definition (6) can be particu-
larly named as a Gaussian quantum polyhedron.
8 Asynchronous evolution ofaquantum system toquantum
polyhedra
Evolution, as previously defined, from the original quantum system to a Gaussian
quantum polyhedron, has been performed as a synchronous phenomenon. A more
general situation might be associated with an asynchronous evolution encompass-
ing the synchronous one. In this case, the time-like natural parameter
𝜏
is trans-
formed into a time-like N-dimensional natural vector:

𝛕
K

=

𝜏
K;I
I=1, N

. In
such a way the successive polyhedra vertices are located at the enfolded space as:
∀
I=1, N∶𝐑
I
⇒𝐑
I(
𝜏
K;I)
, in this manner, each subsystem position depends on an
independent time-like parameter, a situation similar to several descriptions previ-
ously published in this journal [41–45]. The subindex K corresponds to an asynchro-
nous time-like vector click, composed of N sub-clicks.
Z
=
N
∑
I=1
Z
I
(6)
𝐏
=

g
I
𝐫


𝐑
I

Z
I
I=1, N
.
2 In the previous literature dealing with sets of density functions and in the present paper the term quan-
tum polyhedron is used instead of quantum polytope.
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To transform the time-like evolution of a quantum polyhedron from an asyn-
chronous process to a synchronous one, one just needs to take into account that
the time-like vector transforms into a time-like homothecy of the N-dimensional
unity vector
⟨𝟏�=(1, 1, 1 …,1
)
, that is:

𝛕
K

=𝜏
K
𝟏
.
Thus, one can see that, in any case, the evolution of the subsystem quantum
polyhedron could be described with a time-like vector. The structure of such a
vector will tell if the evolution process is synchronous or asynchronous.
9 Algebraic characterization oftheGaussian quantum polyhedron:
thesimilarity matrix
Transformation of the original point-like quantum system into a Gaussian quan-
tum polyhedron can be characterized, considering the nature of their vertices
at each time-like click. At every click, the quantum polyhedron vertices form a
set of functions which can supposedly interact via a Hermitian operator
𝛺
, say,
forming a symmetric
(N×N)
matrix
𝐙
so that one can write for every pair of
vertices:
Applying such a procedure to quantum polyhedra, in general, has been dis-
cussed and used to many quantum chemical problems, see [47–56] as examples.
It also corresponds to the basis of the so-called quantum similarity, for more
information see references [20–40].
The integral appearing in the Eq. (7) can be easily described through the
expression:
as it has been discussed from the quantum similarity initial times [20–25].
If the operator
𝛺
is positive definite, then theoretically, the matrix
𝐙
is also
positive definite. In case the operator coincides with a Dirac’s delta function,
that is:
𝛺(
𝐫
1
;𝐫
2)
=𝛿
(
𝐫
1
−𝐫
2)
, then the matrix
𝐙
corresponds to the overlap
between the pairs of Gaussian density functions associated with the vertices of
the quantum polyhedron.
In this circumstance, the interaction matrix can also be considered as the met-
ric matrix of the space subtended by the vertices of the quantum polyhedra, con-
structed in turn by the subsystem’s Gaussian density functions.
In quantum similarity lore, the matrix
𝐙
is named as the similarity matrix of
the quantum polyhedron.
Of course, the interaction or similarity matrix will depend on the time evolu-
tion of the quantum polyhedron. And this will be done via the position of the
polyhedron vertices within the Gaussian enfolded space.
(7)
∀
I,J=1, N∶z
IJ
=

g
I
𝛺g
J
→𝐙=

z
IJ 
I,J=1, N
,
∀
I,J=1, N∶

gI𝛺gJ

=
∬D
gI

𝐫1

𝛺

𝐫1;𝐫2

gJ

𝐫2

d𝐫1d𝐫
2
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10 The Einstein–Podolsky–Rosen paradox andthetime‑like
evolution ofaGaussian quantum polyhedron
The diagonal elements of the interaction matrix
𝐙
are stationary because they
only depend on the exponents, but not on the position in the enfolded space, that
is:
This point is interesting to note because even if the non-diagonal elements of
the interaction matrix become exactly null, then the interaction matrix could be
associated with the diagonal matrix defined by the elements of the Eq.(8).
However, the nature of the simplified description of the subsystem quantum
polyhedron in the form of Gaussian vertices allows saying that even at large, but
finite, distances the non-diagonal elements of the interaction matrix might be
small, but non-null.
One can measure the time-like evolution of the generated quantum polyhedron
via the expression:
where the trace of the interaction matrix
𝐙
corresponding to the first term in the
Eq.(9) is a stationary constant. As already commented, the second term will be non-
zero even at considerable but finite distances.
Therefore, within the time-like evolution of a quantum polyhedron, associated
with the subsystems that appeared from an initial quantum system, the interaction
among them will be non-zero throughout such time-like evolution.
It seems that from this point of view, using certainly a simplified model such
as the one proposed here, there is a steady interaction among all the parts that one
can suppose are split from a unique initial system.
No spooky action at a distance seems to be present. It merely appears the well-
defined sum of subsystem interactions.
11 Quantum polyhedron variance asameasure ofsubsystems’
interaction
The variance
m[2]
of a quantum polyhedron has been discussed in many instances.
See, for example, references [48, 49]. It could be considered as a measure of the
collective behavior of an indefinite number of subsystems. It must be kept in
mind the fact that being the variance positive definite by construction [56], its
value will be in any circumstance greater than zero in this case.
Initially, a function is needed to construct the polyhedron variance. It corre-
sponds to the so-called polyhedron centroid [55]:
(8)
∀
I=1, N∶z
II
=

g
I
𝛺g
I
=f

Z
I.
(9)
J
=
N
∑
I=1
zII +1
2
N
∑
I=1
N
∑
J=1
𝛿(I
≠
J)z
IJ
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then, once described the Gaussian quantum polyhedron, as explained in the present
work and defined in the Eq.(6), leading to a well-defined interaction or similarity
matrix
𝐙
, this also permits to compute a scalar acting as the variance of the polyhe-
dron. The variance can also be easily computed as:
and the trace of the similarity matrix corresponds to the first term sum appearing in
the Eq.(9). In contrast, the complete sum of the elements of a matrix noted as:
⟨𝐙⟩
,
corresponds to the second part of the variance expression.
In fact, the Eq.(11) is somehow related to a mean value of the interaction of
the quantum polyhedron, as expressed in the Eq.(9). To see such a relation, one
might split the interaction matrix into the sum of two matrices, the diagonal
𝐙D
and the off-diagonal
𝐙O
parts, according to:
Then, using the fact that the complete sum of any matrix is a linear operator,
one can write:
and considering that:
Tr
[𝐙]=
⟨
𝐙
D⟩
, the quantum polyhedron variance can also be
written as:
An expression, which can be associated with an average over the elements of
the interaction or similarity matrix. As in Eq.(9), the first term of the variance is
a sum involving the diagonal elements, invariant upon the polyhedron evolution.
While the second term, associated with the sum of off-diagonal interaction matrix
elements, corresponds to a term diminishing as the subsystems become more dis-
tant in the enfolded Euclidian space.
At the light of its definition and considering its connection with the interaction
matrix, the quantum polyhedron variance can also be taken as a measure of the
(10)
g
C=N−1
N
∑
I=1
gI
,
(11)
m
[2]=N−1
N

I=1∫D
gI

2d𝐫−∫D
gC

2d𝐫
=N−1
N

I=1
zII −N−2
N

I=1
N

J=1
zIJ =N−1Tr[𝐙]−N−2

𝐙

𝐙=𝐙
D
+𝐙
O
⇒
𝐙
D={zII |I=1, N}∧
𝐙
O=𝐙−𝐙D=
{
𝛿(I≠J)z
IJ |
I,J=1, N
}

𝐙
O
=

𝐙−𝐙
D
=

𝐙

−

𝐙
D
⇒

𝐙

=

𝐙
D
+

𝐙
O
m[2]
=N
−1
Tr [𝐙]−N
−2
𝐙=

N
−1
−N
−2
𝐙
D
−N
−2
𝐙
O
=N−2

(N−1)

𝐙D

−

𝐙O

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action at a distance between the collection of subsystems, which has evolved from
an original quantum system.
Such a relationship will never fade away, whenever one does accept the possi-
bility of very large, but finite, distances between the subsystems.
12 Description oftheinitial quantum system alongwithits time‑like
evolution
As described in the previous sections of this study, the time-like evolution of
subsystems seems to do not consider the original quantum system. Practically,
the evolution of the subsystem set has been studied, which has been supposedly
formed by an arbitrary number of elements, to discuss the problem in the middle
of a very general situation.
However, in the geometrical structure of the Gaussian quantum polyhedron
subsystem set does appear, along with the time-like evolution, the essence of the
original quantum system. Indeed, one can again think about the density functions
of a set of subsystems, as the quantum polyhedron described in the Eq.(6) is used
here. A superposition similar to the Eq.(10) but scaled by the number of quantum
subsystems might be constructed:
such that the Minkowski norm of the function
gO
can be easily calcu-
lated:
⟨
gO
⟩
=
N
∑
I=1⟨
gI
⟩
=
N
∑
I=1
ZI=
Z
,
Therefore, the superposition (12) has the same Minkowski norm as the original
quantum system function (3). The meaning of this can be associated to the fact
that an initial quantum system splitting into
N
quantum subsystems, whatever his
time-like evolution might be, remains entangled.
12.1 Convex evolution ofthequantum subsystem set
Alternatively, starting from the entanglement shown in the analysis of Eq.(12),
one can take the vertices of the quantum polyhedron
𝐏
as a basis set to construct a
quantum system Gaussian density function. An original quantum system density
might be reconstructed at every time-like click, as a convex linear combination:
which must be Minkowski renormalized, because:
(12)
g
O=NgC=
N
∑
I=1
gI
,
(13)
g
c(𝐫

𝜃)=
N

I=1
cIgI𝐫


𝐑I


ZI←
N

I=1
cI=1∧cI⊂[0, 1]
,
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The renormalization described in the Eq. (14) could be avoided in case that,
instead of using density functions, one uses shape functions, see, for example, Ref.
[57]. However, for the present study, it is enough to show the possibility of describ-
ing the original system in the manner of Eqs.(13) and (14), along with the time-like
evolution of the vertices of the subsystem quantum polyhedron.
Such a possibility is, of course, connected with the fact that all the successive
Gaussian quantum polyhedra vertices remain interacting in their future time-like
evolution.
As a final detail, one can ask about the nature of the convex coefficients used in
the Eq.(13). A possible simple solution that might connect all the particle numbers
involved in the problem, is to define a convex set of coefficients like:
Then the time-like description of the evolution of the original system position in
the enfolded Euclidian space can be written as:
The convex coefficient set, defined in this case as weights involving the number
of particles, also demands a renormalization of the original system density function.
13 Conclusions
Perhaps, if the insight of Born [58] and later the intuition of Kohn [59–61], about
the role of density functions in atomic and molecular quantum mechanics, could
have surfaced in the paper of Ref. [1] instead of 10years later, then a considerable
deal of paperwork could be spared.
Maybe, some of the ideas about quantum mechanics and Hilbert spaces expressed
by Anastopoulos [62] at the turn of the century, will agree with some points of view
of the present paper. Moreover, the recent presentation of Chen [63], promoting the
density function as the origin of information in the quantum universe, provides good
support to the present discussion. Even more recently, a paper by Perelman [64]
describes how mass can be associated to the quantum mechanical density function
and the consequences of this connection into the space–time structure and gravity.
In this paper has been shown, indeed in a very naïve manner, that instead of wave
functions, quantum mechanical density functions might be employed to describe and
(14)
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∀
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Z
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∑
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N
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I=1
ZI=Z−1
Z=1
𝐑
c=
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I=1
cI𝐑I=Z−1
N
∑
I=1
ZI𝐑
I
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follow the time-like evolution of a quantum system, after splitting into a collection
of subsystems. This will result in a possible non-null interaction between the subsys-
tems along with their time-like evolution. Therefore, any se of quantum subsystems
issued from a previous system might be considered entangled.
Thus, the interminable debate from 1935 up to present times about spooky inter-
action at a distance between subsystems arising from a unique quantum system,
could have been dramatically shortened.
Compliance with ethical standards
Conict of interest The authors state that there is no conflict of interest related to this work.
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