On the Quantum Entanglement Reinterpretation Using the Time as Real Instantaneous Signal Field ()
1. Introduction
The direction of using the quantum entanglement in the measurement of time introduced by Don Page and Wooster who are argued that quantum entanglement can be used to measure time [5] , other theorist used quantum entanglement to explain the flow of time [6] . The current investigation of time using the quantum entanglement will take new direction in which we can represent the real time state of any physical system consisting of one or more matter particles at any space point P as single entangled state called hereinafter the real-time state with dimension equivalent to the number of constituent matter particles of the physical system and with components equivalent to time of each one of them at P, the author will investigate the translation of this real-time state as quantum entanglement phenomena in which the measurement of occupation of P by any one constituent matter particle of the physical system S immediately produce the equivalent measurement information of the part of lengths of all leaving epochs of P by the rest constituent matter particles of S that occupied and left P, this translation as we will see implied the existence of finite set of digital states which are: 1) representing a basis for Entanglement Translation of the real-time state at each spatial position, and 2) distributed into set of sequential digital levels in which the real-time state of the physical system is transits from one digital level to the next digital level equivalently to the orbital transition.
2. Basis Formulation
2.1. Why We Need for the Current Theory of Space and Time
In order to understand why we need to the current theory of time take for example the mechanism of forwarding the time in the analog clock, in which all three hands―second, minute and hour hand―are occupying and leaving their occupies space when the impulse system of the analog clock exerting a single impulse acting on all of them simultaneously, this single impulse is representing superposition of all electromagnetic wave that are reflecting by the surface’s points of underlying elementary constituent matter particle of the clock’s hands and its remainder parts at each exerting epoch of impulse system, however there is existing strong correlation between the number of exerted impulses, the number of occupation epochs of points in the space of motion of the clock’s hands by them and the measurable time by tracking the paths occupied by the analog clock’s hands, we can illustrating this correlation using the fact that the duration of each impulse is one second to write the time t at each point in the space of motion of the specified clock’s hand CH of this analog clock defined with respect to the join points of its hands by
during or after the nth occupation epoch of
by CH as the sequence function
![](//html.scirp.org/file/1-7503055x11.png)
seconds, such that
is initial leaving epoch of
started when the observer starting the tracking of the motion of CH and ending at starting of the first occupation epoch of
by CH,
for all
is representing the number of impulses exerted on CH by the analog clock’s impulse system during the kth occupation epoch of
by CH,
for all
is representing the summation of the number of impulses exerting after the time
on CH during the mth occupation epoch of
by it and the number of impulses exerted on it during the mth leaving epoch of
by it, and
is average of
and
fulfills
, thus
is representing the total lengths of all first
occupation epochs of
by CH after the time
,
is representing the total length of all first
leaving epochs of
by CH occurred between each two occupation epochs of
by it after the initial time
and
is the length of the leaving epoch of
by CH elapsed after the end of the nth occupation epoch of
by it, so if the analog clock is ideally perfect then
, 3600 or 216,000 in a case of
is representing the observable time measurable by tracking the motion second, minute or hour hand respectively, now if we have N identical perfect analog clocks such that for every
the point
is representing a point in the space of motion of the specified hand of the ith analog clock defined with respect to the join points of its hands,
is representing the number of occupation epochs of
by
―which is clock hand of the ith analog clock that we took under consideration,
for each
, is representing the number of impulses exerted on it by the impulse system of the ith analog clock during the kth occupation epoch of
by
,
for all
is representing the summation of the number of impulses exerting on
during the mth occupation epoch of
by it and the number of impulses exerting on it during the mth leaving epoch of
by it, and
is average of
,
and
fulfills
then we can define the real time state of this N analog clocks by
, such that
seconds where
is initial time elapsed during the epoch started when the observer starting the tracking of
and ending at starting of the first occupation epoch of
by it and
is the length of leaving epoch of
by
elapsed after the end of the
occupation epoch of
by it, now according to the classical mechanics the time is absolute at all occupation and leaving epochs of epoch of
by
and then if
for all
then all second, minute and hour hands of these identical clocks are synchronously occupy their corresponding points in their spaces of motion and synchronously leave them regardless of their spatial distribution or their surfaces orientation in space, and hence the real time state t according to classical mechanics should alwayslied at the equilibrium collinear set
, however in general relativity Einstein followed another direction and argued the existence of what is now called the gravitation time dilation [1] which implied that the gravitation field have different value of stress-energy-momentum tensor in different space points occupies by different hands of these analog clocks cause their running at different rates, so according to the general theory of relativity the real time state t may deviating from the equilibrium collinear set
as result of difference in gravitation field from one point of space to another point, thus we need mathematical formulation provide a measure to degree to which the real time state of any quantum system consisting of N matter Particles at any points inside their occupies paths from the equilibrium collinear set
and all another non-equilibrium collinear set in
, the author will approve that each equilibrium or non-equilibrium collinear set is representing vector subspace of
, and then if this vector subspaces endowed with usual dot product they will represent vector subspaces of n-dimensional Euclidian space which is representing an inner product Hilbert space.
2.2. The Occupation Epoch Number of the Matter Particle
The occupation epoch number of the matter particle at each space point
donated by
is representing the number of occupation epochs of
by the matter particle with respect to some observer or measurement instrument observing the motion of matter particle during finite observation epoch.
2.3. The Time of the Matter Particle at Each Space Point
If
is the occupation epoch number of the matter particle P at the space point
then the time of the matter particle P at the space point
is defined as the total length of all the occupation and leaving epochs of
by the matter particle P and then is defined as following:
(1)
(2)
where:
is the length of the initial leaving epoch elapsed before the first occupation of
by the matter particle with respect to some observer or measurement instrument observing the motion of matter particle during finite observation epoch.
For each
:
is the length of the ith occupation epoch of
by the matter particle.
is the length of the ith leaving epoch of
by the matter particle elapsed aft the ith occupation epoch.
is the length of the nth occupation epoch of
by the matter particle.
is the length of the epoch elapsed after the end of the nth occupation epoch of
by the matter particle elapsed aft the nth occupation epoch.
is the average of the time periods:
![]()
Important note:
1) The temporal variable
is representing a signal indexed by the occupation epoch number.
2) The term
is representing a measure of the past temporal epoch before the nth occupation epoch of
by the matter particle, the term
is representing the a measure of the present nth occupation temporal epoch and the term
is representing a measure of the future temporal epoch after the nth occupation epoch of
by the matter particle.
2.4. Definition of the Infinitesimal Time as a Function of Infinitesimal Displacement of Space and Occupation Epoch Number
In order to write infinitesimal time as a function of infinitesimal displacement of space and occupation epoch number suppose we have matter particle with rest mass
move by speed v with respect to some local observer through some space point
at the nth occupation epoch of
by the matter particle, then according to the special relativity theory the momentum of the
particle is given by
, where c is the speed of light in vacuum, now
according of the wave-particle duality [6] if
is representing the wavelength of this matter particle at the nth occupation epoch of
by it then the momentum of this matter particle is also defined as following:
![]()
where h is the Planck’s constant.
→![]()
→![]()
→![]()
→
(3)
where
is representing infinitesimal displacement vector of matter
particle at space point
,
is representing infinitesimal displacement of time and
is the component of space metric tensor at the ith row and the jth column.
→![]()
→
(4)
Important notes:
From the Equation (3) the speed of matter particle is defined by:
(5)
Thus speed of matter particle should always bounded by the speed of light in vacuum c because:
(6)
For all matter particle possess non-zero mass
and occupies non-zero volume of space
. Thus we can conclude that the speed of light in vacuum is representing with respect to the current theory is unsurpassable limit for all matter particles possess non-zero mass and occupies non-zero volume of space.
2.5. The Real-Time State of Any Physical System Consisting of N Matter Particles at Specified Space Point
If we have a physical system consisting of N matter particles
then the real-time state of this physical system at each space point
is defined as following:
(7)
(8)
For all
:
where:
is the occupation epoch number of the matter particle
at
.
is the length of the initial leaving epoch elapsed before the first occupation of
by the matter particle
with respect to some observer or measurement instrument observing the motion of matter particle
during finite observation epoch.
is average time period of the first
time periods of
at
.
is the length of the
occupation time of
by the matter particle
.
is the time elapsed during the matter particle leaving the point
after the
occupation epoch.
Important notes:
(9)
(10)
(11)
(12)
where:
,
and
are called hereinafter past, present and future real- time state respectively.
2.6. The Entanglement Translation of Real-Time State of Any Physical System Consisting of N Matter Particles at Specified Space Point
If
is representing the real-time state
of some physical system consisting of N matter particles
at starting of the
occupation of
by the matter particle
for some
then the measurement process of the length of the
occupation epoch of
by the matter particle
that result the
is always transform the real-time state of physical system
according to the following Entangle-
ment Translation:
![]()
(13)
For all
.
Important note:
1) if the
is representing a set of N elementary matter particles then the occupation epoch is infinitesimal thus using the Equation (4) we can write the entanglement translation of
as following:
(14)
(15)
2)
is representing the normalized reciprocal function of
which is normalized by removing the infinity from the range of
reciprocal function
.
3)
or equi-
valently:
which
is always result binary digits indicate wither the matter particle
occupied
or not and then determine whether the υth component of the Real- time state
is non-zero covariant under Entanglement Translation or remain zero contravariant, thus the Entanglement Translation is:
I. Pure covariant transformation when
for all
.
II. Pure contravariant transformation when
for all
.
III. Mixed covariant and contravariant transformation when
and
for some
.
4) according to this transformation the measurement process of the occupation of
by the matter particle
that result
as observable quantity is the same to the measurement process of the part of leaving of
by the rest matter particles of the physical system that are occupied and left
, thus the time of each one of these matter particle at of
―which is representing the total length of all occupation and leaving epochs of
by the matter particle―should increase
immediately at the end of measurement epoch, however if some matter particle of the physical system does not occupy
from the starting of observation epoch until the starting
occupation of
by the matter particle
then this matter particle will never occupy
during the
occupation epoch of
by the matter particle
, and hence the time of this matter particle at
―which is representing of total length of all occupation and leaving epochs of
by it―will never change from zero during the measurement epoch.
5) For all
:
(16)
Which is representing tensor field that take the contravarinat vector
and covariant vector
![]()
and produce
components of the following matrix:
![]()
Such that
and
thus:
(17)
Such that:
is equivalent to the number of non-zero components of
and the Real-time state
.
Now:
![]()
→![]()
such that ![]()
→![]()
such that
is N × N identity matrix
→
(18)
Thus the Entanglement Translation:
is translational invariant with respect to the operator
.
2.7. The Real-Time Digital State of the N Matter Particles Physical System at Specified Space Point
If
is representing the real-time state
of some physical system consisting of N matter particles
at
then the Real-time digital state at
that is corresponding to
is defined as following:
(19)
Important note:
1) The value of
indicate wither the matter particle
occupied the point
or not with respect to observer or measurement instrument tracking its motion of
through
.
2) If
is representing the real-time
state of the physical system at starting of the
occupation of
by the matter particle
for some
and
is the result of the measurement process of the length of the
occupation epoch of
by the matter particle
then the Entanglement Translation of
is given as following:
![]()
Such that
(20)
Thus the Real-time digital state
is representing the base state of Entanglement Translation of the real-time state
.
However
which is
consisting of
N-tuples of binary digits thus the Real-time digital state
as well as Entanglement Translation
are quantifying the motion of the constituent matter particles of the physical system through
, this quantification allow these matter particles move exclusively at a finite sequential set of digital levels defined in the following section.
3) ![]()
where:
![]()
and:
![]()
Thus we can write the Entanglement Translation of
:
![]()
Such that
![]()
As following:
(21)
(22)
(23)
(24)
2.8. The Digital Levels of Real-Time Digital States of the Physical System Consisting of N Matter Particles
For any physical system S consisting of N matter particles
and for all
the digital level n of S is defined as the set of all possible real-time digital states of S at any space point
consisting of n components equivalent to one and rest components equivalent to zero, thus if
is representing the set of all subsets of
that are consisting of n ele-
ments, and for any set A and B
then the nth digital level of S
is defined as following:
(25)
which their element are defined as surjective function
such that for each
:
(26)
Important note:
1) If
is repre-
senting the real-time digital state of some physical system consisting of N matter particles
at
, then there exists:
![]()
and
fulfills:
![]()
2) If
is real-time digital state of some physical system consisting of the matter particles
at
then before the first occupation epochs of
by any matter particle belong to
―with respect to some observer or measurement instrument tracing the motion of the matter particles
through
― the real-time digital state of the of the physical system at
is equivalent to the equilibrium state
called herei-
nafter the falsehood digital state at
which is representing the unique element of the digital level 0 of the system, then this state either stay in the digital level 0 along the observation epoch of the physical system or change due the occupation of
by one matter particle belong to
to one state in the digital level 1 which is level consisting of all real-time digital states with one components equal one and rest components equal zero, then this state either stay in the digital level 1 or change due the occupation of
by new one matter particle belong to
to one state in digital level 2 which is level consisting of all digital states with two components equal one and rest components equal zero, and so on until the digital state of the physical system reach the stationary equilibrium digital
state
at the digital level N and then resisting at
this digital state for rest of observation epoch of the physical system. However we must keep in mind the impossibility of transition the Real-time digital state
to another Real-time digital state belong to the same or lower digital level or to another Real-time digital state belong to higher digital level with ith component equivalent to zero for all
fulfill
because each components of the digital state can only change from zero to one when some matter particle of the physical system start its first occupation epoch, Figure 1 representing an explanation of distribution of real-time digital states of any physical system consisting of 4 matter particles over their corresponding digital levels in addition to the possible transition of the Real-time digital states from different digital states to the Real-time digital states distributed in their near higher digital level.
3) If the constituent matter particles of the physical system
are distributed into set of finite orbits such as the distribution of electrons in atoms then for each one of these orbits the real-time digital state of the physical system at each space point belong to it will be the same when all matter particles at that orbit occupy all space points belong to it, however this symmetry of digital states at that orbit can break by the jumping of one matter particles to that orbit which can transit all points occupied by it at that orbit to the same digital state belong to next higher digital level, thus the distribution of the constituent
![]()
Figure 1. Illustration of digital levels and all possible transition between the real-time digital states belong to them for any physical system consisting of 4 matter particles.
matter particles into set of finite orbits is equivalent to the distribution of it into set of finite digital levels such that jumping of matter particle from initial orbit to the final orbit is equivalent to the transition of digital states at all points occupied by it at the final orbit to the same real-time digital state belong to next higher digital level.
2.9. The Real-Time Transition State
For any physical system S consisting of N matter particles
the real-time transition state of the physical system at
that is corresponding to its real-time digital state at
:
is defined as the superposition of all
real-time digital states at
that the physical system can transit to it at the of the next occupation epochs of
by one of its constituent matter particles which is defined as following:
(27)
Fulfills:
(28)
(29)
For all
fulfills
.
Such that:
(30)
(31)
fulfills
and
for all
fulfills
.
where:
is the tensor product (outer product) operation.
and
are normalized reciprocal
transpose of
and
respectively which are defined by taking the normalized reciprocal of the components of transpose of
and
respectively.
Important note:
1) When the physical system at the real-time digital state:
the next occupation epochs of
by one of its constituent matter particles P is either leaves the physical system at the real-time digital state
in a case that P occupied
at the previous occupation epoch or transit the real-time digital state to some state at the next higher digital level
such that
fulfills
, thus the real-time transition state is defined as superposition of its current real-time digital state and all real-time digital states at the next higher digital level that the physical system that can transit to them.
2) For all
the components of
and
at the ith row
and the jth column which are
and ![]()
respectively indicate wither both the matter particles
and
occupied
or not, thus
and
can play the role of observables operators that the Hermition matrices played in quantum mechanics because they are either leave the physical system at its current real-time digital state or transit it to one real-time digital state in next higher digital level when they act on the real-time transition state, however we use the normalized reciprocal as involution function instead of complex conjugate that used in quantum mechanics, so
and
are equal to their normalized reciprocal transposes in same way that the Hermition matrices equal to their complex conjugate transposes. In computation term
and
are representing irreversible gates that forward the time of each constituent particles of the physical system only at the direction of increasing the number of occupation and leaving epochs of space points by it.
3) we can calculate the coefficients
and
as following:
Since for all
and
:
![]()
the conditions:
![]()
![]()
Implied that:
![]()
![]()
And then:
![]()
![]()
So if
are representing the elements of
for some integer
then:
![]()
And:
![]()
→![]()
Now there is two possible cases of above linear system depend on the value of n defined as following:
A. if n = 0 then the linear system is reduced to:
![]()
And then:
![]()
B. If n > 0 then above linear system is defined in matrix form as following:
![]()
Thus using the Cramer’s rule [3] for solving the linear system we find:
![]()
And hence the real-time transition state is defined as following:
(32)
2.10. The Analog Occupies Path of the Physical System Consisting of N Matter Particles
The analog occupies path of N matter particles physical system is the path that consisting of all real-time states at all space point occupied or occupies at least by one of its constituent matter particles.
2.11. The Digital Occupies Path of the Physical System Consisting of N Matter Particles
The digital occupies path of N matter particles physical system is the path that consisting of all real-time digital states at all space point occupied or occupies at least by one of its constituent matter particles.
3. Mathematical Formulation: An Introduction to the Calculus of Fluctuation
3.1. The n-Dimensional Real Collinear Set
For each
the n-dimensional real collinear set of any two point
donated by
is defined as a set of all point in
lied at the line that contains p and q.
Example of real collinear set
The set of real numbers is a 1-dimesional collinear set of any two real number x, y. i.e.
,
.
3.2. The n-Dimensional Displacement Vector
For all
and
the n-dimensional displacement vector from p to q is defined as following:
(33)
Important note:
For all
the author will donate to the vector
by
such that
.
3.3. The Equilibrium Null Point and Vector
The equilibrium null point is the point is the point
and the
equilibrium null vector is vector
.
3.4. The Equilibrium Unity Point and Vector
The equilibrium unity point is the point
and the equili-
brium unity vector is the vector
.
3.5. Conditions of Positioning in n-Dimensional Collinear Set
For each
the n-dimensional real collinear set of any two point
such that
and
the conditions of positioning
in
are defined as following:
1) The tangent and cotangent of the angle between the line that connect x and q should be equal to the tangent and cotangent of the angle between the line that connect x and y, mathematically this condition is defined as following:
![]()
satisfy
and
.
2) The equivalent components of x and y should be equivalent to their corresponding components of q, mathematically this condition is defined as following: ![]()
fulfills
.
Important note:
From the first condition
satisfy
and ![]()
![]()
Also
![]()
From the second condition
satisfy
we find that
![]()
Also
![]()
→![]()
→
(34)
3.6. The n-Dimensional Collinear Vectors Set
If
is n-dimensional real collinear such that
then the collinear vectors set of
donated by
is the set of all displacement vectors in
that their heads are belonging to
and tails are equivalent to the equilibrium null point
, which is defined as following:
(35)
Important note:
For any
the exists displacement vector
with components equivalent to
thus for all
fulfill
:
such that
and
because:
and the components of
and
are equivalent to the components of
and
respectively.
![]()
3.7. Theorem (3.2)
For all
the n-dimensional displacement vector set
is representing vector subspace of
and
is repre-
senting subgroup of
.
Prove:
For all
, all
fulfills
, all
and all
such that:
![]()
![]()
and
:
1)
.
2)
(Closure under ad-
dition and scalar multiplication).
3)
(Associativity of addition).
4)
(Commutatively of
addition).
5)
(Identity element of addition).
6)
fulfills
(Inverse element of addition).
7)
(Compatibility of scalar multiplication with field
multiplication).
8)
(Distributivity of scalar multiplication with respect to vector addition).
9)
(Distributivity of scalar multiplication with respect to field addition).
10)
(Identity element of scalar multiplication).
3.8. The n-Dimensional Real State Space
For all
such that
the n-di- mensional real state space at x is the vector subspace of
defined by
and endowed with inner product
such that for all
and
:
![]()
Important note:
is representing vector subspace of n-dimensional Euclidean
space
thus is representing inner product Hilbert space.
3.9. Equilibrium and Non-Equilibrium Classification of n-Dimensional Real State Space
For all
we can classify the n-dimensional real state
space
as:
1) Equilibrium n-dimensional real state space in a case of
for all
.
2) Non-Equilibrium n-dimensional real state space in a case of
for some
.
Important notes:
If
is equilibrium n-dimensional real state space then:
![]()
3.10. The Normalized Reciprocal of Real Scalar
For any
the normalized reciprocal of x is defined as following:
![]()
Important note:
1) is called normalized reciprocal because the usual reciprocal of x which is
equal
is contain infinity in his range when
, so it is normalized by
removing this infinity from its range:
2)
is representing binary digit.
3) The Dirac’s delta function is related to normalized reciprocal as following:
(36)
Which is equal zero at all
and one at
.
3.11. The Normalized Reciprocal Transpose of the Matrix
For any matrix
the normalized reciprocal of
is defined as following:
(37)
3.12. Signal Tensor Field
At any point
the signal tensor field is second order ten-
sor that take
from
and its corresponding dual vector
from
such that
and produce
components
defined as following:
(38)
Or in tensor notation:
(39)
For all
.
Important notes:
1) For all
the μth diagonal component of
which is equal
is corresponding to the μth components of the binary digital state binary state
so the diagonal components of
are representing the digital components of it, and all the rest components of
for all
are representing the analog normalized ratio between the components of
at different indices.
2) For all
if
then
for all
so in the signal tensor field the present and absent of the digital signal is restricted by the present and absent of its corresponding analog signals and vice versa.
3)
at all point
because for all
fulfill
and ![]()
![]()
thus:
(40)
4) ![]()
→
( 41)
or in tensor notation:
(42)
For this reason
will play in the digital matter particle Physics the similar role that Herniation matrix or in general adjoin operator plays in quantum physics.
5) For all
the vector
:
![]()
→
(43)
Such that
is equivalent to the digital level of the digital state
.
In tensor notation this equation is given as following:
(44)
3.13. The Fluctuation Tensor Field
The fluctuation tensor field is representing bilinear map
―such that
is the space of all
square matrix―defined for all
and
as following:
(45)
Such that
is the outer product (tensor product) operation [2] defined as following:
(46)
and
(47)
Thus If
and
then the fluctuation tensor field of x and y donated by
is the set of
ordered real numbers
called its components indexed by
such that
and defined as following:
(48)
The Properties of commutator tensor field:
a Anti-symmetric because:
![]()
b Alternating because:
![]()
c Non-degenerate because:
For every
there exists
and
fulfills
,
and
implied:
![]()
Because ![]()
d Bilinear because for all
,
,
,
and all
:
.
Thus
.
Also
.
Thus
.
3.14. Theorem 3.3
If
,
then the fluctuation tensor
is representing a measure of degree to which the point x and y deviate from belonging to the collinear set
and
respectively because:
1)
for all
vanished when there is no deviation.
2)
for all
and
i.e. the fluctuation tensor field
is a invariant under parallel translation of
with respect to the line that all elements of
lie or parallel translation of
with respect to the line that all elements of
lie. Thus all element lie at single line parallel to the line that all elements of some collinear set S lie have a same measure of deviation from belonging to S defined as fluctuation tensor field.
Prove:
1) for all
and all
fulfill
:
![]()
and hence:
![]()
→![]()
2) For all
,
and all
:
![]()
![]()
Thus if
and
then there exist four points
,
,
,
fulfill
,
and for all
fulfill
and
and all
fulfill
and
:
![]()
![]()
![]()
![]()
→![]()
→
For all
:
![]()
![]()
→![]()
3.15. The Spin’s Fluctuation Tensor Field at Each Matter Particle’s Surface Point
The spin fluctuation tensor field is defined at each surface point of any matter
particle resisting with respect to an arbitrary observer at the position ![]()
as following:
(49)
where:
is linear and momentum vector of the matter particle.
Important note:
Each components of the spin’s fluctuation tensor field
is corresponding to either positive or negative components of angular momentum vector:
and vice versa, so the
components of the spin’s fluctuation tensor field
are vanishes― becomes zero―when all component of
are vanishes and vice versa, also the components of the spin’s fluctuation tensor field
are conserved when the components of
are conserved and vice versa, those two strong correlated feature between the components of the spin’s fluctuation tensor field
and the components of
allow the author to introduce the following theorem.
3.16. Theorem 3.5: Spin’s Fluctuation Theorem
Any matter particle P possesses a non-zero mass m can possess:
1) A non-zero spin angular momentum at arbitrary time interval
if and only if all surface points of P are not moving in the same direction of its position vector during this time interval.
2) A conserved spin angular momentum at any two arbitrary time intervals
and
if and only if any surface points of P donated by s located with respect to an arbitrary observer or measurement instruement at
the position
at the time t fulfills:
(50)
where:
i)
is infinitesimal displacement of s at the time interval
.
ii)
is infinitesimal displacement of s at the time interval
.
Prove:
1) Suppose we have some arbitrary surface point of P resisting with respect to
an arbitrary observer at the position
at the time t.
Now the spin angular momentum of P at
is defined as a cross product of
and the linear momentum vector
as following:
![]()
In other hand the spin’s fluctuation tensor field
is defined as following:
![]()
Thus every component of
is equivalent to either positive or negative value of one component of
, so all components of
vanishes―becomes zero―when all components of
vanishes.
Now the infinitesimal displacement vector
, thus the fluctuation
tensor field of
and
is defined using bilinearity property of fluctuation tensor field as following:
![]()
→![]()
Because
is representing infinitesimal non-zero change of time, the spin’s fluctuation tensor field
vanishes―and then the spin angular momentum vector
―when all components of the fluctuation tensor field
vanishes so according to the theorem 3.2
vanishes when
or equivalently when
. Thus the matter
particle P possesses a non-zero spin angular momentum if and only if all surface points of P are not moving in the same direction of their position vectors.
2) Lets represents a surface point of P resisting with respect to an arbitrary ob-
server at the position
at the time t, let
is
representing the infinitesimal displacement of the matter particle’s surface
points during the time interval
,
is the position
of the matter particle’s surface point s at the time
, and let
is representing the infinitesimal displacement of
the matter particle’s surface points during the time interval
, now the spin angular momentum vectors of the matter particle at s during the time interval
and
and donated by
and
respectively are given as following:
![]()
![]()
Also the spin’s fluctuations tensor fields of the matter particle at s during the time interval
and
and donated by
and
respectively are given as following:
![]()
![]()
Thus every component of
is equivalent to either positive or negative value of one component of
and every component of
is equivalent to either positive or negative value of one component of
, so
if and only if
and then if and only if:
![]()
or equavently if and only if:
![]()
or equivently if and only if:
Because
![]()
and
,
![]()
if and only if:
![]()
Important Notes:
1) The first part of the above theorem implied that if the matter particle possesses non-zero mass and non-zero spin angular momentum then it’s all surface points should move in different direction of all position vectors defined with respect to all observers observing them, thus the first part of the above theorem approve that the non-zero spin angular momentum of any matter particle possesses non-zero mass is intrinsic property independent from the observer’s localization with respect to its surface point.
2) The second part of the above theorem illustrate the relation between spatial and temporal coordinates of the surface points of any matter particle possesses non-zero mass and conserved spin angular momentum.
3.17. Theorem 3.6
If
,
, ![]()
and
such that
,
,
is not parallel to
and
then
and
are intersected at the point
fulfills:
1)
(51)
For all
and all
fulfill
.
2) if
and
for some
fulfill
and
, and some
and
fulfill
,
,
for all
―i.e.
is not parallel to
―and either
or
, then:
(52)
For all
and all
fulfill
.
Prove:
Suppose we have
,
,
and
such that
,
,
and
is not parallel to
, suppose that
is intersection point of
and
: Now for all
fulfills
,
and either
or
―i.e.
:
(a)
and
(b)
→![]()
![]()
→![]()
![]()
→![]()
![]()
→
(c)
(d)
Now by substituting
from Equation (c) into Equation (d) we find:
![]()
→![]()
→![]()
→![]()
(e)
Now from Equation (a):
![]()
For all
.
→![]()
→
(f)
Now by substituting
from Equation (e) into Equation (f) we find:
![]()
→![]()
→![]()
→![]()
For all
and all
fulfill
,
and either
or
.
This proves the first part of the theorem, now to prove the second part we can use the first proved part as following:
![]()
→![]()
→![]()
→![]()
Thus if
and
for some
fulfill
and
, and some
and
fulfill
,
,
for all
―i.e.
is not parallel to
―and either
or
, then:
![]()
→![]()
→![]()
For all
and all
fulfill
,
and either
or
.
This equation is representing the fundamental fluctuation tensor field equations which are invariant under any arbitrary change of
,
and
by any
,
and
respectively
such that
for all
, thus:
![]()
For all
and all
fulfill
,
and either
or
.
3.18. Orthogonal n-Dimensional Real Collinear Sets
For each
the two collinear sets
and
are called orthogonal if and only if:
1)
.
2)
, for all
and
.
Such that
donate to the dot product of
and
.
3.19. N-Dimensional Real Collinear Sets Space
The n-dimensional real collinear sets local space at
donated by
is the space of all n-dimensional real collinear sets intersected at the point t, which is given as following:
![]()
Important note
Any
is representing the origin of
fulfills for any
two collinear sets
and
belong to
, such that
,
, fulfill
for all
―i.e.
is not parallel to
―and
fulfill
and
the equations:
![]()
For all
and all
fulfills
,
and either
or
, these equations are invariant under any arbitrary change of
and
.
3.20. The n-Dimensional Real Collinear Sets Bundle
The n-dimensional real collinear sets bundle is the union of all n-dimensional real collinear sets spaces at all points in
which is given as:
(53)
3.21. The n-Dimensional Real Coplanar Set
For all
the n-dimensional coplanar set of p, q and r donated by
is a set of all point in
lied at the n-dimensional plane contains p, q and r, which is defined as following:
(54)
3.22. The n-Dimensional Real Coplanar Space
The n-dimensional real coplanar space at each point
donated by
is defined as following:
(55)
Important notes:
is defined as the space of all coplanars
that are containing some point
represents intersection point of collinear sets then using the theorem (3.6) we find:
(56)
For all
and all
fulfills
,
and
.
3.23. The n-Dimensional Real Coplanar Bundle
The n-dimensional real coplanar bundle is union of all n-dimensional real coplanar spaces at all space point in
which is defined as following:
(57)
3.24. The N-Dimensional Real Space-Time
For any physical system consisting of N matter particles, the N-dimensional real space-time is the section of N-dimensional real coplanar bundle that consisting of all real-time states of the physical systems at each all space point occupied by one or more constituent matter particles of the physical system during finite epoch with respect to an arbitrary observer. Thus if
then at each space point
occupied by one or more constituent matter particles of the physical system the real-time state of the physical system is representing an element of the section of
that defined as following:
(58)
Important notes:
If the real-time state of the physical system at the space point
occupied by one or more constituent matter particles of it is defined according to the equation 9 as following:
,
then there exist at least in principle
fulfill
,
,
and
(59)
4. Conclusion
As the spatial coordinates x, y and z which are representing the lengths between origin (
) and (
), (
) and (
) respectively, the time coordinate is representing the total length of all occupation and leaving epochs of space point including the length of the initial leaving epoch elapsed before the first occupation of space point by the matter particle during finite observation epoch, this implied that the direction of time of any matter particle at each space point P occupied by it is the direction of increasing the number of occupation and leaving epochs of P by it, and the measurement of occupation of space point by one constituent matter particles of the physical system produce the same measurement of the time of the rest matter particles of the physical system that occupied and left the space point during finite observation epoch regardless of their distribution in space, this give simple reinterpretation of quantum entanglement. The motion of the constituent matter particles on separated orbits is equivalent to their motion in separated digital levels, and their transitions from one orbit to another one is equivalent to their transition from one digital level to another.
Acknowledgements
Thanks for my father who supported all my education levels and for my wife Ayaat Ahmed Osman for here incorporeal support for me in scientific papers publications.