

Ratio Mathematica Volume 37, 2019, pp. 5-23

Studies on the classical determinism
predicted by A. Einstein, B. Podolsky

and N. Rosen

Ruggero Maria Santilli∗

Abstract

In this paper, we continue the study initiated in preced-
ing works of the argument by A. Einstein, B. Podolsky and N.
Rosen according to which quantum mechanics could be “com-
pleted” into a broader theory recovering classical determin-
ism. By using the previously achieved isotopic lifting of ap-
plied mathematics into isomathematics and that of of quan-
tum mechanics into the isotopic branch of hadronic mechanics,
we show that extended particles appear to progressively ap-
proach classical determinism in the interior of hadrons, nuclei
and stars, and appear to recover classical determinism at the
limit conditions in the interior of gravitational collapse.
Keywords: EPR argument, isomathematics, isomechanics.
2010 AMS subject classifications: 05C15, 05C60. 1

∗Institute for Basic Research, Palm Harbor, FL 34683 U.S.A., research@i-b-r.org
1Received on September 21st, 2019. Accepted on December 20rd, 2019. Published on Decem-

ber 31st, 2019. ISSN: 1592-7415. eISSN: 2282-8214. doi: 10.23755/rm.v37i0.477 c©Ruggero
Maria Santilli

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Ruggero Maria Santilli

1. Introduction

1.1. The EPR argument

As it is well known, Albert Einstein did not consider quantum mechan-
ical uncertainties to be final, for which reason he made his famous quote
“God does not play dice with the universe.”

More particularly, Einstein accepted quantum mechanics for atomic
structures and other systems, but believed that quantum mechanics is an
“incomplete theory,” in the sense that it could be broadened into such a
form to recover classical determinism at least under limit conditions.

Einstein communicated his views to B. Podolsky and N. Rosen and
they jointly published in 1935 the historical paper [1] that became known
as the EPR argument.

Soon after the appearance of paper [1], N. N. Bohr published paper [2]
expressing a negative judgment on the possibility of “completing” quan-
tum mechanics along the lines of the EPR argument.

Bohr’s paper was followed by a variety of papers essentially support-
ing Bohr’s rejection of the EPR argument, among which we recall Bell’s
inequality [3] establishing that the SU(2) spin algebra does not admit limit
values with an identical classical counterpart.

We should also recall von Neumann theorem [4] achieving a rejection of
the EPR argument via the uniqueness of the eigenvalues of quantum me-
chanical Hermitean operators under unitary transforms.

The field became known as local realism and was centered on the rejec-
tion of the EPR argument via additional claims that hidden variables [5]
are not admitted by quantum axioms (see the review [6]).

1.2. The 1998 apparent proof of the EPR argument

In 1998, the author published paper [7] presenting an apparent proof of
the EPR argument based on the following main steps that we here outline
to render this paper minimally self-sufficient:

Step 1: The proof that Bell’s inequality, von Neumann’s theorem and
other similar objections against the EPR argument [6] are indeed correct,
but under the generally tacit assumptions of point-like particles moving
in vacuum under sole potential/Hamiltonian interactions (exterior dynam-
ical systems) when the systems are treated via quantum mechanics and
its underlying 20th century mathematics, including Lie’s theory and the

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Apparent proof of the EPR argument

Newton-Leibnitz differential calculus;

Step 2: The proof that the above treatments are not applicable for
extended, therefore deformable and hyperdense particles under condi-
tions of mutual penetration or entanglement occurring in the structure of
hadrons, nuclei, stars, and gravitational collapse such as for black holes,
with novel non-linear, non-local, and non-potential/non-Hamiltonian in-
teractions (interior dynamical systems);

Step 3: The treatment of interior systems via the axiom-preserving
lifting of 20th century applied mathematics known as isomathematics, whose
study was initiated by the author in the late 1970’s when he was at Har-
vard University under DOE support, Refs. [8] to [12] and then continued
by various mathematicians. Isomathematics is based on:

3-A) The axiom-preserving isotopy of the conventional associative prod-
uct between generic quantities a,b (numbers, functions, operators, etc.)
first introduced in Eq. (5), p. 71 of Ref. [11]

ab → a ? b = aT̂b, (1)

where T̂ is a positive-definite quantity called the isotopic element providing
a representation of the dimension, deformability and density of particles
and physical media in which they are immersed via realizations of the
type

T̂ = Diag.(
1

n21
,

1

n22
,

1

n23
,

1

n24
)e−Γ, (2)

where: n24 represents the density; n
2
k, k = 1, 2, 3 represents the deformable

share of particles; n2µ,µ = 1, 2, 3, 4, and Γ are solely restricted to be positive-
definite but otherwise admit a functional dependence on any needed local
variables, such as time t, coordinates r, momenta p, energy E, density d,
temperature τ, pressure π, wavefunctions ψ, their derivatives ∂ψ, etc.

nµ = nµ(t,r,p,E,d,τ,π,ψ,∂ψ,....) > 0, µ = 1, 2, 3, 4, (3)

Γ = Γ(t,r,p,E,d,τ,π,ψ,∂ψ,....) � 0. (4)

e− Γ(t,r,p,E,d,τ,π,ψ,∂ψ,....) � 1. (5)

3-B) The formulation of isoassociative algebras on an isofield F̂(n̂,?, Î)
first introduced in Ref. [13] (see also independent work [14]), with isounit

Î = 1T̂, (6)

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Ruggero Maria Santilli

and isoreal, isocomplex and isoquaternionic isonumbers n̂ = nÎ under isoprod-
uct (1), with ensuing isooperations such as the isosquare

n̂2̂ = n̂ ? n̂. (7)

Isofields also imply the lifting of functions into isofunctions [11] [20]

f̂(r̂) = [f(rÎ)]Î, (8)

among which we quote the isoexponentiation

êX = (eXT̂ )Î = Î (eT̂X), (9)

where X is a Hermitean operator.

3-C) The ensuing axiom-preserving lifting of Lie’s theory into a non-
linear, non-local and non-Hamiltonian form first introduced in Ref. [11]
(see also the recent paper [15] and independent work [16]), which theory is
today known as the Lie-Santilli isotheory, with isobrackets at the foundation
of Ref. [7]

[X,̂Y ] = X ? Y −Y ? X = XT̂Y −Y T̂X. (10)

3-D) The isotopic lifting of the Newton-Leibnitz differential calculus,
from its historical definition at isolated points, into a form defined on vol-
umes, first introduced in Ref. [17] (see Refs. [18] for vast independent
works) with isodifferential

d̂r̂ = T̂(r, ...)dr̂ =

= T̂(r, ...)d[rÎ(r, ...)] = dr + rT̂dÎ(r, ...),

(11)

and corresponding isoderivatives

∂̂f̂(r̂)

∂̂r̂
= Î

∂f̂(r̂)

∂r̂
. (12)

Step 4: The axiom-preserving lifting of quantum mechanics into the
isotopic branch of hadronic mechanics, or isomechanics for short, whose study
was initiated in Refs. [8] to [12] (see the 1995 monographs [19] [20] [21]
with 2008 upgrade [22] and independent studies [23][24]).

Isomechanics is formulated on a Hilbert-Myung-Santilli (HMS) isospace
[25] Ĥ over the isofield of isocomplex isonumbers Ĉ, and it is based on the

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Apparent proof of the EPR argument

iso-Heisenberg isoequations for the time evolution of a Hermitean operator
Q̂ in the infinitesimal form

î ? d̂Q̂
d̂t̂

= [Q̂,̂Ĥ] = Q̂ ? Ĥ − Ĥ ? Q̂ =

= Q̂T̂Ĥ − ĤT̂Q̂,
(13)

and the finite form

Q̂(t̂) = Û(t̂)† ? Q̂(0) ? Û(t̂) =

= êĤ?t̂?̂i ? Q̂(0) ? ê−î?t̂?Ĥ =

= eĤT̂tiQ(0)e−itT̂Ĥ,

(14)

with the following rules for the basic isounitary isotransforms

Û(t̂)† ? Û(t̂) = Û(t̂) ? Û(t̂)† = Î, (15)

where t̂ = tÎt is the isotime which is assumed hereon to coincide with con-
ventional time, Ît = 1. Dynamical equations (13) to (15) were first pre-
sented in Eq. (4.16.49), page 752 of Ref. [9] over conventional fields and
reformulated via the full use of isomathematics in Ref. [17]).

Isomechanics is also based on the iso-Schrödinger isorepresentation char-
acterized by the fundamental representation of the isomomentum permitted
by the isodifferential isocalculus, Eq. (12),

p̂̂|ψ(t̂, r̂) >= −î ? ∂̂t̂,r̂|̂ψ(t̂, r̂) >=

= −iÎ∂r̂|̂ψ(t̂, r̂) >,
(16)

from which one can derive the iso-Schrödinger isoequation, [12] [17] [20]

î ? ∂̂t̂|ψ̂(t̂, r̂) >= Ĥ ? |ψ̂(t̂, r̂) >=

= Ĥ(r,p)T̂(t,r,p,E,d,τ,π,ψ,∂ψ,....)|ψ̂(t̂, r̂ >) =

= Ê ? |ψ̂(t̂, r̂) >= E|ψ̂(t̂, r̂) >

(17)

and the isocanonical isocommutation rules,

[r̂î,p̂j]|ψ̂ >= î ? δ̂i.j ? |ψ̂ >= iδij|ψ̂ > (18)

9


Ruggero Maria Santilli

[r̂î,r̂j]|ψ̂ >= [p̂î,p̂j]|ψ̂ >= 0. (19)

Note that the characterization of extended particles at mutual distances
smaller than their size requires the knowledge of two quantities, the conven-
tional Hamiltonian H for the representation of potential interactions, and
the isotopic element T̂ for the representation of dimension, shape, density
as well as of non-linear, non-local and non-potential interactions.

Step 5: The proof in Ref. [7] that the isotopic ŜU(2)-spin symmetry
for extended particles immersed within a dense hadronic medium admits
an explicit and concrete realization of hidden variables [5], e.g., of the type

T̂ = Diag.(λ, 1/λ), DetT̂ = 1. (20)

In particular, the isotopic ŜU(2)-spin isosymmetry admits limit condi-
tions with identical classical counterpart, Eq. (5.4) page 189 Ref. [7].

One aspect of isomathematics and isomechanics which is crucial for
this paper is that in all applications to date, the isotopic element T̂ has values
much smaller than 1, Eqs. (4) (5), as it has been the case for: the synthesis
of the neutron from the hydrogen in the core of stars; the representation of
nuclear magnetic moments and spin; new clean energies; and other appli-
cations [21].

It should be also noted that thanks to the new interactions represented
by T̂ , isomathematics and isomechanics have permitted the first known
identification of the attractive force between identical valence electron pairs in
molecular structures [26]. A significant confirmation of values |T̂| � 1 is
provided by the fact that exact representations of binding energies for the
hydrogen and water molecules have been achieved with isoseries based
on isoproduct (1) that are at least one thousand times faster than conventional
quantum chemical series [27] [28].

We should finally indicate that the numerical invariance of the isotopic
element T̂ and therefore, of the isounit Î = 1/T̂ , under isounitary time
evolutions (14) (15) was proved in Ref. [29]. Detailed reviews and up-
grades of isomathematics, isomechanics, and their applications to interior
problems which are specifically written for the EPR argument should soon
be available in Refs. [30] [31].

1.3. Aim of the paper

In this work, we shall attempt to complete the proof of the EPR argu-
ment of Ref. [7] by showing that extended particles in interior dynamical

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Apparent proof of the EPR argument

conditions appear to progressively recover classical determinism in inte-
rior dynamical conditions with the increase of the density and other char-
acteristics, as indicated at the end of Ref. [7].

It should be stressed that a technical understanding of this work re-
quires technical knowledge of hadronic mechanics, e.g., from Refs. [19]
[20] [21] or from the forthcoming reviews and upgrades [30] [31].

We should indicate that the words “completion of quantum mechan-
ics” is used in Einstein’s sense for the intent of honoring his memory. For
instance, the conventional associative product ab of Eq. (1), which is at the
foundation of quantum mechanics, admits a “completion” into the equally
associative, yet more general isoproduct aT̂b. Under no conditions Ein-
stein’s word “completed theory” should be confused with a ’final theory,’
that is a theory admitting no additional Einstein’s “completions.” In fact,
the time-reversal invariant, Lie-isotopic isomathematics and isomechanics
studied in this work admit the “completion” into the covering, irreversible
Lie-admissible genomathematics and genomechanics (in which T̂ is no longer
Hermitean) which, in turn, admit a covering via the most general math-
ematics and mechanics conceived by the human mind, the multi-valued
hypermathematics and hypermechanics [32] [33], with additional “coverings”
remaining possible in due time [19] [20] [21] .

The reader should be finally aware that the isotopic element T̂ and
isounit Î = 1/T̂ are inverted in some of the early quoted literature not deal-
ing with determinism without affecting their consistency. An important
aim of this paper has been that of achieving the final selection of isotopic
element and isounit which is compatible with studies on determinism.

2. Recovering of determinism in interior conditions?

2.1. Heisenberg uncertainty principle

Consider an electron in empty space represented with the 3-dimensional
Euclidean space E(r,δ,I), where r represents coordinates, δ = Diag.(1, 1, 1)
represents the Euclidean metric and I = Dian(1, 1, 1, ) is the space unit.

Let the operator representation of said electron be done in a Hilbert
space H over the field of complex numbers C with states Ψ(r) and familiar
normalization

< Ψ(r)| |Ψ(r) >=
∫ +∞
−∞

Ψ(r)†Ψ(r)dr = 1. (21)

As it is well known, the primary objections against the EPR argument

11


Ruggero Maria Santilli

[2] [3] [4] were based on the uncertainty principle formulated by Werner
Heisenberg in 1927, according to which the position r and the momentum p of
said electron cannot both be measured exactly at the same time.

By introducing the standard deviations ∆r and ∆p, the uncertainty prin-
ciple is generally written in the form (see, e.g., [5])

∆r∆p ≥
1

2
~, (22)

easily derivable via the vacuum expectation value of the canonical com-
mutation rule

∆r∆p ≥ |
1

2i
< Ψ| [r,p] |Ψ > | =

1

2
~. (23)

The standard deviations have the known form [34] (with ~ = 1)

∆r =
√
< Ψ(r)|[ r − (< Ψ(r)| r |Ψ(r) >)]2|Ψ(r) >,

∆p =
√
< Ψ(p)| [p− (< Ψ(p)| p |Ψ(p) >)]2|Ψ(p) >,

(24)

where Ψ(r) and Ψ(p) are the wavefunctions in coordinate and momentum
spaces, respectively.

2.2. Particle in interior conditions

We consider now the electron, this time, in the core of a star classically
represented with the iso-Euclidean isospace Ê(r̂, δ̂, Î) [17] with basic isounit
Î = 1/T̂ > 0, isocoordinates r̂ = rÎ, isometric

δ̂ = T̂δ, (25)

and isotopic element of type (2) under conditions (3) to (5).
Besides being immersed in the core of a star, the electron has no Hamil-

tonian interactions. Consequently, we can represent the electron in the
HMS isospace Ĥ [25] over the isofield of isocomplex isonumbers Ĉ [13],
and introduce the time independent isoplanewave [20]

Ψ̂(r̂) = ψ̂(r̂)Î =

= N̂ ? (êî?k̂?r̂)Î = N(eikT̂r̂)Î,

(26)

where N̂ = NÎ is an isonormalization isoscalar, k̂ = kÎ is the isowavenumber,
and the isoexponentiation is given by Eq. (9).

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Apparent proof of the EPR argument

The corresponding representation in isomomentum isospace is given
by

Ψ̂(p̂) = M̂ ? êî?n̂?p̂, (27)

where M̂ = MÎ is an isonormalization isoscalar and n̂ = nÎ is the isowavenum-
ber in isomomentum isospace.

2.3. Isodeterministic isoprinciple

The isopropability isofunction is given by [20]

P̂ = <̂|? |>̂ =< Ψ̂(r̂)| T |Ψ̂(r̂) > I =

= [
∫ +∞
−∞ Ψ̂(r̂)

† ? Ψ̂(r̂) ? d̂r̂]Î =

= [
∫ +∞
−∞ ψ̂(r̂)

†ψ̂(r̂)d̂r̂]Î,

(28)

where one should keep in mind that the isodifferential d̂r̂ is now given by
Eqs. (11).

The isoexpectation isovalues of a Hermitean operator Q̂ are then given
by [20]

<̂|? Q̂ ? |>̂ =< Ψ̂(r̂)|? Q̂ ? |Ψ̂(r̂) > Î =

= [
∫ +∞
−∞ Ψ̂(r̂)

† ? Q̂ ? Ψ̂(r̂)d̂r̂]Î =

= [
∫ +∞
−∞ ψ̂(r̂)

†Q̂ψ̂(r̂)d̂r̂]Î,

(29)

with corresponding expressions for the isoexpectation isovalues in isomo-
mentum isospace.

We now introduce, apparently for the first time in this paper, the iso-
topic operator

T̂ = T̂ Î = I, (30)
that, despite its seemingly irrelevant value, is indeed the correct operator
formulation of the isotopic element for the transition of the isoproduct
from its scalar form (1) into the isoscalar form

n̂2̂ = n̂ ? n̂ = n̂ ? T̂ ? n̂ = n2Î. (31)

Since the identity I can be inserted anywhere in the expectation values
of quantum mechanics without altering the results, realization (33) illus-
trates the central feature of the isotopies, namely, the property that the ab-
stract axioms of quantum mechanics admit a “hidden” realization broader

13


Ruggero Maria Santilli

than that of the Copenhagen School whose degrees of freedom have been
used in Ref.[7] for the proof of the EPR argument [1].

We now introduce the isoexpectation isovalue of the isotopic operator

<̂|? T̂ ? |>̂ =< Ψ̂(r̂)|? T̂ ? |Ψ̂(r̂) > Î =

= [
∫ +∞
−∞ ψ̂(r̂)

†T̂ψ̂(r̂)d̂r̂]Î,

(32)

and assume the isonormalization

<̂|? T̂ ? |>̂ =

=
∫ +∞
−∞ ψ̂(r̂)

†T̂ψ̂(r̂)d̂r̂ = T̂.

(33)

We then introduce, in this paper apparently for the first time, the iso-
standard isodeviation for isocoordinates ∆r̂ = ∆rÎ and isomomenta ∆p̂ =
∆pÎ, where ∆r and ∆p are the standard deviations in our space.

By using isocanonical isocommutation rules (18), we obtain the expres-
sion

∆r̂ ? ∆p̂ = ∆r∆pÎ ≈ 1
2
| < Ψ̂(r̂)|? [r̂̂,p̂] ? Ψ̂(r̂) > | =

= 1
2
| < Ψ̂(r̂)|T̂ [r̂̂,p̂]T̂|Ψ̂(r̂) > .

(34)

By eliminating the common isounit Î, we then have the desired isode-
terministic isoprinciple here proposed apparently for the first time

∆r∆p ≈ 1
2
| < Ψ̂(r̂)|? [r̂̂,p̂] ? |Ψ̂(r̂) >=

= 1
2
| < Ψ̂(r̂)|T̂[r̂̂,p̂]T̂|Ψ̂(r̂) >=

∫ +∞
−∞ ψ̂(r̂)

†T̂ψ̂(r̂)d̂r̂ = T � 1

(35)

where the property ∆r∆p � 1 follows from the fact that the isotopic ele-
ment T̂ has always a value smaller than 1 (Section 1.2).

It is now necessary to verify isoprinciple (35) by proving that the iso-
standard isodeviations tend to null values when T̂ → 0.

For this purpose, we introduce the following simple isotopy of Eqs.
(24) (where we ignore the common multiplication by the isounit)

∆r =

√
< Ψ̂(r̂)|[ r̂− < Ψ̂(r̂)|? r̂ ? |Ψ̂(r̂) >]2̂̂|Ψ(r̂) >,

∆p =

√
< Ψ̂(p̂)| [p̂− < Ψ̂(p̂)|? p̂ ? |Ψ̂(p̂) >]2̂|Ψ̂(p̂) >,

(36)

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Apparent proof of the EPR argument

where the differentiation between the isotopic elements for isocoordinates
and isomomenta is ignored for simplicity.

It is then easy to see that the isosquare (7) implies the covering forms
of the isostandard isodeviations

∆r =

√
T̂ < Ψ̂(r̂)|[ r̂− < Ψ̂(r̂)|? r̂ ? |Ψ̂(r̂) >]2|Ψ̂(r̂) >,

∆p =

√
T̂ < Ψ̂(p̂)| [p̂− < Ψ̂(p̂)|? p̂ ? |Ψ̂(p̂) >]2|Ψ̂(p̂) >,

(37)

that indeed approach null value under the limit conditions

LimT̂=0∆r = 0,

LimT̂=0∆p = 0,
(38)

thus confirming isodeterministic isoprinciple (35).

2.4. Particles under pressure

To illustrate the above expressions, we consider an electron in the cen-
ter of a star, thus being under extreme pressures π from the surrounding
hadronic medium in all radial directions, while ignoring particle reactions
in first approximation or under a sufficiently short period of time.

These conditions are here rudimentarily represented by assuming that
the Γ > 0 function of the the isotopic element (2) is a constant linearly de-
pendent on the pressure π, resulting in a realization of the isotopic element
of the type

T̂ = e−wπ � 1, Î = e+wπ � 1, (39)
where w is a positive constant.

The isodeterministic isoprinciple for the considered particle is then given
by

∆r∆p ≈
1

2
e−wπ � 1, (40)

and tends to null values for diverging pressures.
The above example illustrates the consistency of isorenormalization

(33) because, a constant isotopic element implies the consistent expression

<̂ψ̂(r̂)|T̂|ψ̂(r̂) > Î =

T < ψ̂(r̂)| |ψ̂(r̂) > Î =

< ψ̂(r̂)| |ψ̂(r̂) >,

(41)

15


Ruggero Maria Santilli

while, by contrast, the following alternative isonormalization

<̂ψ̂(r̂)|T̂|ψ̂(r̂) > Î = Î, (42)

would imply the expression

< ψ̂(r̂)||ψ̂(r̂) > Î = Î, (43)

which is manifestly inconsistent since < ψ̂(r̂)||ψ̂(r̂) > is an ordinary num-
ber while Î is a matrix with integro-differential elements.

Note that we have considered a free particle immersed in a hadronic
medium, rather than a bound state of extended particles in condition of
mutual penetration. Consequently, in our view, isotopic element (2) repre-
sents a subsidiary constraint caused by the pressure of the hadronic medium
encompassing the particle considered, by therefore restricting the values
of the isostandard isodeviations for isocoordinates and isomomenta.

Illustrations of the isodeterministic isoprinciple in specific structure
models of hadrons and related aspects have been studied in Ref. [21]
and their interpretation in terms of the isodeterministic isoprinciple will
be studied in future works.

2.5. Gravitational example

To provide a gravitational illustration, recall that isotopic element (2)
contains as particular cases all possible symmetric metrics in (3+1)-dimensions,
thus including the Riemannian metric [20].

We then consider the 3-dimensional sub-case of isotopic element (2)
and factorize the space component of the Schwartzchild metric gs(r) ac-
cording to isotopic rule introduced in Refs. [35] [36]

gs(r) = T̂(r)δ, (44)

where δ is the Euclidean metric.
We reach in this way the following realization of the isotopic element

T̂ =
1

1 − 2M
r

=
r

r − 2M
, (45)

where M is the gravitational mass of the body considered, with ensuing
isodeterministic isoprinciple

∆r̂∆p̂ ≈ T̂ =
r

r − 2M
⇒r→0= 0, (46)

16


Apparent proof of the EPR argument

which confirms the statement in page 190 of Ref. [7], on the possible recov-
ering of full classical determinism in the interior of gravitational collapse
(see Ref. [37], Chapter 6 in particular, for a penetrating critical analysis of
black holes).

It should perhaps be indicated that Refs. [35] [36] introduced the fac-
torization of a full Riemannian metric g(x), x = (r,t) in (3 + 1)-dimensions

g(x) = T̂gr(x)η, (47)

where T̂gr is the gravitational isotopic element, and η is the Minkowski metric
η = Diag.(1, 1, 1,−1).

Refs. [35] [36] then reformulated the Riemannian geometry via the
transition from a formulation over the field of real numbers R to that over
the isofield of isoreal isonumbers R̂ where the gravitational isounit is evi-
dently given by

Îgr(x) = 1/T̂gr(x). (48)

The above reformulation turns the Riemannian geometry into a new
geometry called iso-Minkowskian isogeometry, which is locally isomorphic to
the Minkowskian geometry, while maintaining the mathematical machinery
of the Riemannian geometry (covariant derivative, connection, geodesics,
etc.) us fully maintained, although reformulated in terms of the isodiffer-
ential isocalculus [38].

The apparent advantages of the identical iso-Minkowskian reformula-
tion of Riemannian metrics and Einstein’s field equations (see, e.g., Eqs.
(2.9), page 390 of Ref. [38]) are:

1) The achievement of a consistent operator form gravity in terms of
relativistic hadronic mechanics [39] whose axioms are those of quantum me-
chanics, only subjected to a broader realization;

2) The achievement of a universal symmetry of all non-singular Rie-
mannian metrics, which symmetry is locally isomorphic to the Lorentz-
Poincaré symmetry, today known as the Lorentz-Poincaré-Santilli (LPS) isosym-
metry [40], and it is notoriously impossible on a conventional Riemannian
space over the reals;

3) The achievement of clear compatibility of Einstein’s field equation
with 20th century sciences, such as a clear compatibility of general relativ-
ity with special relativity via the simple limit Îgr = I implying the tran-
sition from the universal LPS isosymmetry to the Poincaré symmetry of
special relativity with ensuing recovering of conservation and other spe-
cial relativity laws [41] [42]; the achievement of axiomatic compatibility
of gravitation with electroweak interactions thanks to the replacement of
curvature into the new notion of isoflatness with the ensuing, currently

17


Ruggero Maria Santilli

impossible, foundations for a grand unification [43]; and other intriguing
advances.

3. Concluding remarks

t In this paper, we have continued the study of the EPR argument [1]
conducted in Ref. [7] and preceding works, with particular reference to
the study of the uncertainties for extended particles immersed within hy-
perdense medias with ensuing linear and non-linear, local and non-local
and Hamiltonian as well as non-Hamiltonian interactions.

This study has been conducted via the use of isomathematics and isome-
chanics characterized by the isotopic element T̂ of Eq. (1) which represents
the non-linear, non-local and non-Hamiltonian interactions of the particles
with the medium [19] [20] [21].

The main result of this paper is that the standard deviations of coor-
dinates and momenta for particles within hyperdense media are charac-
terized by the isotopic element that, being always very small, T̂ � 1, re-
duces the uncertainties in a way inversely proportional to a non-linear
increase of the density, pressure, temperature, and other characteristics of
the medium, while admitting the value T̂ = 0 under extreme/limit con-
ditions with ensuing recovering of full determinism as predicted by A.
Einstein, B. Podolsky and N. Rosen [1].

We can, therefore, tentatively summarize the content of this paper with
the following:

ISODETERMINISTIC ISOPRINCIPLE: The product of isostandard isodeviations
for isocoordinates ∆r̂ and isomomenta ∆p̂, as well as the individual isodevia-
tions, progressively approach classical determinism for extended particles in the
interior of hadrons, nuclei, and stars, and achieve classical determinism at the
extreme densities in the interior of gravitational collapse.

Acknowledgments

Sincere thanks are due to Thomas Vougiouklis, Svetlin Georgiev and
Jeremy Dunning Davies for penetrating critical comments. Additional
thanks are also due to Mrs. Sherri Stone for an accurate proofreading of
the manuscript.

18


Apparent proof of the EPR argument

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