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Quantum memory, gravity and the age of the universe
Wafik A. Wassef
Saskatchewan Institute of Applied Science and Technology
[email protected]
Abstract
The concept of quantum memory (Wassef, Can. J. Phys. 67, 493, 1989) asserts that the
state of a quantum system is not completely determined at a single point in space and at
the corresponding moment of time, but must include explicit information about its past
states during an interval of time called its memory span. This concept may be traced back
to the notion of duration introduced by Bergson. The overlap between past and present
states during a memory span is a consequence of the quantization of action. Therefore a
quantum system cannot be precisely represented by a single valued and continuous
function of space coordinate x(t) in terms of the continuous and independent time
variable t, although this may be considered a good approximation for macroscopic
systems (for convenience I will use one dimensional space x and two dimensional
spacetime). The two positions that represent a past and a present state of a particle coexist simultaneously during its memory span  and may be used to calculate its velocity
and acceleration. The particle occupies a position for an interval of time  then as it starts
fading away from this position it reappears at a neighboring point a distance x from its
previous location and so on. It is shown that these quantum increments in space and time
cannot be reduced to zero values. The quantization of spacetime is a necessary
consequence of the quantization of action defined by Planck’s constant (divided by 2) .
Action has dimensions of energy  time or force  distance  time. Consequently we
may add a third dimension perpendicular to the x-t plane that represents the magnitude of
a force F in order to create a space that has volume elements with dimensions of action.
For convenience I will call this space action space. This three dimensional action space is
divided into orthorhombic cells each of constant volume . The quantization of action
space implies that the volume of each cell in this space cannot be reduced to zero value.
Consequently each quantum system must have a non-zero memory span  and will move
with a non-zero quantum jump in space x. A necessary self-force acting on each
quantum system (between its past and present positions), even in the absence of external
forces, is required as well. Petkov investigated the self-force acting on an electron that
deviates from its geodesic path by disturbing the balance of the mutual repulsion of its
charge elements. Each volume element in action space is given by:
F.x.  
This equation may be considered a reinterpretation of Heisenberg’s uncertainty relations.
It indicates that a change in an external force F acting on a particle must result in changes
in both x and  in order to maintain the volume of each cell in action space constant and
equals to . Consequently the velocity and acceleration of this particle must change. Thus
Newton’s second law of motion in its discrete form is deduced. In the absence of external
forces an electron, for example, may be considered subject to two equal and opposite
forces: (i) a self-force due to the repulsive electrostatic force between its past and present
positions (ii) an inertial force equals in magnitude and opposite in direction to this selfforce. Thus the electron may be considered subject to a net of zero force. This satisfies
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both Newton’s first and third laws of motion. The introduction of this self - force reduced
Newton’s first law of motion into his second law of motion. The velocity of this electron
is shown to be:
e
v
2.mx
This electron, which has a classical radius r0, occupies a region of space during its
memory span . Consequently the minimum displacement of its center during this
interval of time must equal 2 r0. This minimum displacement corresponds to its maximum
velocity c. Thus the speed of light in vacuum is calculated and shown to be independent
of the frame of reference as the special theory of relativity predicts. We may as well
deduce from the same equation the relation: mc 2 = e 2 / (4  r0). It can be shown that in
a circular motion the centrifugal force may be attributed to the radial outward component
of the electrostatic repulsive self-force of each charged particle.
The memory span of the initial state of the universe, which is the longest memory
span to be found, must be related to the weakest force, namely, the gravitational force
between the two stable particles with the least masses, namely an electron and a proton,
when they are separated by the minimum distance between them. Therefore, we find:
2 2

GM 2 mc
Dirac investigated some numerical coincidences that link the age of the universe to some
of the atomic constants by relations similar to the last one. This formula gives the age of
the universe to be 13.825  10 9 years. This value is in a good agreement with the recent
measurements obtained by NASA’s Wilkinson Microwave Anisotropy Probe (WMAP)
which gave a value of 13.4 (  0.3)  10 9 years. In order to explain the increase of the
age of the universe in terms of the other constants, Dirac came to the conclusion that the
constant of gravity G must decrease with the increase of the age of the universe. This
conclusion is not supported by any observation. It is therefore more convenient to express
the same relation in terms of the spatial increments:
 x

GMm
Here in this equation we find an explicit expression for the expansion of all of space (in
terms of its increments) with the increase of the age of the universe. It implies as well that
the universe must be an open universe.
The persistence of the memory of the initial state of the universe introduces a
temporal interpretation of gravity. Gravity acts in such a way as to preserve the memory
of the initial state of the universe, when all matter was very close together in one place,
by diminishing the separation between all of its parts. As this memory started to fade
away, all parts of matter began to separate and started to drift apart. Thus space was
created and the universe started to expand. Gravity retained ties between all the fragments
of matter as a lingering memory of their common origin. The interplay between the
persistence of the memory of the initial state of the universe and its gradual fading away
produced Space and Time. Space is thus the separation between all parts of matter in the
universe due to the fading away of this memory, while Time is the memory span of the
partially persisting past states.