On Discrete Cosmology, Gravitation and Mechanics

In this paper we will discretely reformulate the main fundamental magnitudes of mechanics and thermodynamics due to a new dynamic, discrete and irreversible nature for Time. The existence of a fundamental minimum time, implies that any physical system can only evolve discreetly according to this minimum time instead of a continuous evolution. Thus, the passage of Time must be considered a fundamental physical process and incorporated into Physics where the laws of Nature depend on a clear distinction between past, present and future. A time interval equals a loss of energy. The introduction of “dark matter”, “dark energy”, “ad hoc modifications of the laws of me-chanics” or “fundamental constants varying” will prove to be unnecessary in-asmuch as the view here to be developed will not require of a Universe provided with special properties. By considering that Universe can be expressed as the ensemble of N typical particles in motion of mass m, we will find possible solutions to some of the main problems of the current Physics, all from an existing deep connection between gravity, thermodynamics and quantum cosmology.


Introduction
One of the main problems in modern physics is the temporal irreversibility, that is, the difficulty of its fundamental theories in distinguishing between the past and the future. How is it possible that temporal irreversibility of macroscopic processes emerges from fundamental physical laws of reversible character?
As is well known, in the present physical laws of Newtonian Mechanics and character of Time is not distinguished. Quantum Mechanics also maintains this reversible character, although the role of the observer through the measurement process, makes it become irreversible through the "collapse" of the wave function. Thermodynamics through the second law is the one that most clearly indicates the need for a direction of Time: The loss of a part of unrecoverable energy in the form of heat, makes the physical processes irreversible and gives the Time a unique direction allowed for the future.
Also there exist in Nature the known examples of temporal irreversibility or so-called time's arrows 1 which clearly indicate a unique direction towards the future. In addition, recently in [1], the authors have revealed the observation of the non-Abelian Aharonov-Bohm effect that violates temporal symmetry and gives time a unidirectional character.
All of it rather suggests that passage of Time must be incorporated into Physics as a fundamental process by which Time is always advancing in a unique direction: in positive direction towards the future. We will raise this conjecture, the purport of which will hereafter be called the "Passage of Time Principle", to the status of a postulate. This postulate is sufficient to find possible solutions to some of the main problems of the current physics, because the difficulties being encountered today in the explanation of several cosmological and quantum phenomena, lie at the root of an insufficient consideration on concept of passage of Time and its implications in physical systems.

Kinematical Part
The following reflections are based on the principle previously mentioned and we define it as follows: Passage of Time Principle: The passage of Time is a fundamental physical process by which Time always is advancing in a unique direction: in positive direction towards the future.
This principle is valid for any reference system in Nature. Therefore we can conclude that:  Causality comes directly from the Passage of Time Principle.
 Any Time value determined with regard to any relative reference system is always positive and therefore no process associated with temporal reversibility can exist in the Universe: there are no causal loops or regions or phenomena where Time "evolves backward".
However, we know that all physical process is defined as the set of successive phases or states of a system, that is, the set of one or more state changes until a concrete state. We can never define a physical process at a given time because any physical process needs a time interval to be completed. The simpler the 1 The thermodynamic arrow of time that is provided by the second law of Thermodynamics, the cosmological arrow of time that points in the direction of the universe's expansion, the causal arrow of time: all cause precedes its effect, the quantum arrow of time due to wave function collapse, the psychological/perceptual arrow of time, etc. Journal of High Energy Physics, Gravitation and Cosmology process, the smaller the time interval. Thus, if a physical process consists of n successive subprocesses, every subprocess will need a minimum interval of time to be completed and therefore the total time interval of the physical process will be the sum of the minimum intervals of time of every subprocess.
According to this principle, the passage of Time is a fundamental physical process, therefore must exist a minimum interval of Time in Nature that will be the simplest physical process and which we will call "fundamental minimum time". Thus, the passage of Time will be done discreetly by successive advances of this fundamental minimum time and not continuously.
Let us denote this fundamental minimum time as τ . Therefore, any relative interval of time will consist in a sequence of this fundamental minimum time, that is, all time value t ∆ determined with regard to any reference system (proper time interval), can be expressed by the equation: With n + ∈  , that is: 1, 2, 3, n =  the number of intervals until n t .
By definition, this fundamental minimum time τ is constant for all relative reference system in Nature. According to Relativity, if for a given relative reference system K we have: t n τ ∆ = ⋅ ; then for a system K ′ that is in any state of motion with regard to K we will have: t n τ ′ ′ ∆ = ⋅ , that is, what varies with regard to every reference system is only the number of intervals. Faizal et al., in [2] described a discrete spectrum for time of several orders of magnitude greater than the Planck scale which are consistent with fundamental minimum time.

Universe's Time and Fundamental Minimum Time
The age, cosmological Time or Time of the Universe t with regard to any relative reference system, is the elapsed time from the origin of the Universe 2 to any Time t. Therefore according Passage of Time Principle, Universe's Time t can be expressed with regard to any relative reference system, by the equation: With t n + ∈  , that is: 1, 2, 3, t n =  the number of intervals until t. In what follows, when we define the value of any physical quantity at Time t, we are referring to this Time of the Universe t.
We know Planck time is defined as: is the reduced Planck constant, G is the constant gravitation, c is the constant of the speed of light in the vacuum and P m is the so-called Planck Journal of High Energy Physics, Gravitation and Cosmology mass.
However, by definition, the fundamental minimum time τ is constant for all relative reference system in Nature, thus, since the Planck time P t is also always constant for all relative reference system because it is a function of the three fundamental constants, we can define τ as the product of a positive real constant k by Planck time, that is: Thus, we can now express Equation (2) Now, let us establish that with regard to any relative reference system, the total Mass of the Universe M at Universe's Time t can be expressed as the ensemble of N typical particles in motion of mass m. Thus: where N y m also are defined at Time t and ( N + ∈  ) is a positive integer with Therefore, since that any typical material particle of mass m in motion can be expressed at any Time t as a function of Planck mass P m as: We can replace this value in Equation (3) and we obtain for any reference system the equation: which is the Weinberg's relation [3] for any typical material particle of mass m in motion at Time t and where obviously: However, Equation (5) can be now expressed according Equation (4)  Gmt mc c where we have replaced the value Ω of Equation (5).
The number of intervals to determine the Time of the Universe t according to Equation (2), is the square root of the total number N of typical particles in motion of mass m in the Universe at Time t. 4) Universe Particles Number By Equation (4) and (6) we can express the total number N of typical particles in motion of mass m in the Universe at Time t.

Proper Time
According previous section, the Time of the Universe t can be expressed as: where N is the total number N of typical particles in motion of mass m in the Universe at Time t according Equation (8) and τ id the fundamental minimum time according Equation (7).
Let us now consider with regard to any relative reference system, an initial value of Universe Time i t and any later value t, with i t t > , where therefore an interval of time has elapsed i t t t ∆ = − . Thus, according to Equation (1) and Equation (9), all time interval t ∆ with regard to any relative reference system, that is its proper time interval, can be expressed as the interval of Universe's Time elapsed with regard to that reference system:

Proper Interval of Space. Fundamental Minimum Length
By Equation (7) for the fundamental minimum time τ , we verify that the product: Is a fundamental minimum length which we denote by l. Since τ is constant for any relative reference system and c is a fundamental constant, then l is equally constant for any relative reference system. Therefore, we can express any length i r r r ∆ = − (proper interval of space) with regard to any relative reference system, as: where r and i r are the particle's position at Time t and i t respectively and ( r n ∈  ), as number of steps to n r . Obviously this length will be positive + or negative − depending on the spatial direction with regard to the origin of the relative reference system that is defined.
Also according to Relativity, if for any relative reference system K we have a relative proper interval of space n r r n l = ⋅ , then for any system K ′ in any state of motion with regard to K we will have: n r r n l ′ ′ ′ = ⋅ .

Discrete Velocity of a Material Particle. Minimum Velocity
Let us now consider any relative reference system. We establish that with regard to this reference system, the velocity v of any material particle at Time t can be expressed of simple form according to Equation (12) and Equation (10) respectively, as: where obviously ( ) r n n v c ≤ ⇔ ≤ for Relativity. Therefore the equation of motion will be: where i r is the particle's initial position at Time i t . However, by Equation (57) we can express de speed of de light c at Time t as: where l is the fundamental minimum length according to Equation (11) and where we can establish min v as minimum velocity of any material particle at Time t. It's remarkable that min v is not constant as Time progresses. Therefore, any material particle only can have the discrete velocity v at Time t as: is a integer and will be positive + or negative − depending if velocity variation increases or decreases. Obviously if, at Time i t , for a relative reference system the material particle is at rest (

Fractional Acceleration of a Material Particle
Let us now consider any relative reference system. We establish that with regard to this reference system the acceleration a of any material particle at Time t can be expressed according to Equation (10) and Equation (17) as: where ( n + ∈  ) and ( n α ∈  ) is a integer and will be positive + or negative − depending on velocity variation if increases or decreases according to Equation (17).

Equation of Motion
By Equation (18) any variation of velocity i v v v ∆ = − during Time elapsed i t t t ∆ = − with regard to any relative reference system can be expressed as: According to Equation (14) and where , i i r v are the particle's initial position and velocity at Time i t respectively.

Velocity-Distance Law. Fractional H(t) Parameter
Let us now consider any relative reference system. According Equation (12) for any length r ∆ (proper interval of space), we can now express Equation (16) as: And therefore: where ( ) Time and as a consequence of discrete velocities of the material particles. It is remarkable that this equation is not only applicable to great distances but has a general character at any distance.

Fractional Acceleration as a Function of H(t)
Let us now consider any relative reference system. We establish that at Time i t , the particle's initial velocity is: 0 Therefore, according to Equation (19) for the velocity: i v v a t = + ∆ and Equation (20), the velocity and acceleration of material particle at Time t can be expressed as: where obviously . Therefore, by Equation (13), we can also express the fractional acceleration of any material particle at any Time t as: where obviously ( n + ∈  ) and ( r n ∈  ).

On passage of Time as Irreversible Physical Process
Let us consider any relative reference system. According Equation (7) for the fundamental minimum time we obtain: We also establish that this reference system is isolated, that is, on the material particle does not act any force and that the material particle is at rest. According Passage of Time Principle and as m t cte ⋅ ≡ by the constancy of the fundamental minimum time τ for any reference system, the mass m of material particle must decrease as Time progresses. Thus is, if we consider any interval of proper time i t t t ∆ = − according Equation (10), the variation of Equation (23) with regard to this proper time interval t ∆ will be: And therefore finally we obtain: where obviously 2 E mc = is the energy of the particle at Universe's Time t by Equation (26).
Therefore, according to Passage of Time Principle we can conclude that any material particle of the Universe must lose an unrecoverable quantity of mass/energy as Time progresses according Equation (24) converting the passage Journal of High Energy Physics, Gravitation and Cosmology of Time in a irreversible physical process, the irreversibility in a fundamental property in Nature and establishing a clear distinction between past, present and future. Thus, all the physical magnitudes of any system in Nature must be specified with regard to Universe's Time t. One way to verify the Passage of Time is to determine this lost mass/energy.

A Time Interval Equals a Loss of Energy
It's remarkable that from Equation (24) we can obtain the following relationships: where η + ∈  is a constant. Therefore any time interval equals a loss of energy:

Energy and Frequency of a Material Particle
Consider any relative reference system. By selecting the first equation in Equation (7) for the fundamental minimum time, Equation (5) for Ω , 2 h π =  and Equation (9) for the Time of the Universe, we can then express the energy 2 E mc = of any material particle at Time t as: with ν as a fundamental oscillation frequency associated with every particle m at Time t. Thus, we will have for any material particle of the universe in any state of motion, the relation: However, we know by wave mechanics the relationship between the frequency of a wave ν , its wavelength λ and the propagation velocity or phase velocity p v , that is: Therefore, according to Equation (26) we can write this previous equation as: According Equation (8), we obtain for the wavelength associated to the particle at Time t: That is the famous De Broglie wavelength and where is the velocity of the material particle m at Universe Time t.

Minimum Energy and Mass
According to Equation (26) we can also express the energy E of any material Journal of High Energy Physics, Gravitation and Cosmology particle at Time t as: where min E is the minimum energy at Time t. Therefore, now we can express the minimum mass in nature min m or "bit" (the minimum information) at Time t as: where also we have considered Equation (6) for the Universe's Mass at Time t.
According to both equations, obviously we can express minimum energy as:

Discrete Energy Variation
According to Equation (29), any mass/energy variation of material particle o system (set of them), that is: where ( q n ∈  ), since q n can be positive or negative if particle or system (set of them) increases or loses energy. For example: By Equation (24) any material particle of the Universe must lose an unrecoverable quantity of mass/energy as Time progresses, which we can express it according to Equation (9), Equation (10) and Equation (29) as:

Minimum Frequency
By Equation (26) we can express now the minimum frequency at Time t as:

Discrete Linear Momentum. Minimum Linear Momentum
Let us now consider any relative reference system. The linear momentum p for any material particle at Time t is defined as: m v are the mass and velocity of material particle respectively at Time t. However, according to Equation (16) for discrete velocity, we can express the discrete linear momentum p on any material particle m at Time t as: where ( v n + ∈ Z ) is a positive integer. Therefore, the discrete minimum linear momentum min p of any material particle m at Time t will can be expressed as: With ( p n ∈  ) as a integer which will be positive + or negative − depending if linear momentum variation increases or decreases and where

Fractional Orbital Angular Momentum
In a simple way, quantum mechanics establishes that the angular orbital momentum L of a material particle of mass m is an integer multiple of Planck constant, since the circumference's length must be multiple of the wavelength h mv λ = associated with particle Equation (27), that is: where r is the distance of the particle from the axis and p mv = the linear momentum at Time t according to Equation (32). Let us now consider any relative reference system. According to Equation (11) for the fundamental minimum length l and Equation (12) for any proper interval of space r ∆ , we must express the wavelength and the length of the circumference as multiples of the fundamental minimum length, that is: are positive integers. Therefore, if we take the variable l from one any equation and replacing it in another, we obtain: where we verify that orbital angular momentum of any material particle at Time t, including photons of light, is not an integer but a fraction of reduced Planck's constant, as has already been demonstrated in [4] by Kyle E. Ballantine et al., that describe that in reduced dimensions, photons can have a half-integer total angular momentum.

Fractional Force F Acting on a Material Particle
Let us now consider any relative reference system. The force F acting on a Journal of High Energy Physics, Gravitation and Cosmology material particle at Time t can be expressed as: where m = p v is the linear momentum at Time t according to Equation (32).
By Equation (24) any material particle of the Universe must lose an unrecoverable part of mass/energy as Time progresses, thus we obtain: is the fractional acceleration at Time t of the material particle according to Equation (18). However, we can consider 0 by the relation between acceleration and velocity at Time t is: according to Equation (21), we finally obtain the Force F acting on a material particle at Time t as: Thus, fractional force F can be generally expressed according to Equation (18) for the fractional acceleration a at Time t of the material particle, as: where ( n + ∈  ) and ( n α ∈  ).

Universe Energy and Mass
Let us now consider any relative reference system. According to Equation (4) and Equation (29) And therefore the Universe's Energy at Time t as: where N N ℵ = is the total minimum masses number or can be also understood as the total information bits number. Also, according to Equation (26) and Equation (31), we can now the Universe's Energy at Time t as: That is, Universe's Energy as an ensemble of N N ℵ = quantum oscillators. Journal of High Energy Physics, Gravitation and Cosmology

Universe's Area and Holographic Principle
According to Equation (8) and Equation (29) we can express at Time t the number ℵ of information bits or minimum masses as: where the quantity ( min t m  ) has dimensions of Area (m 2 ) according to holographic principle proposed by Professor Gerard 't Hooft [5]. Therefore, by Equation (8) and Equation (5) we can express the Universe's Area A as a function of N N information bits or minimum masses as: Obviously, we can also express N N ℵ = at Time t as a set of n subsystems, as:

Expanding Universe as Consequence of a Particle Creation Process as an Alternative to Dark Energy
Consider with regard to a relative reference system any interval of proper time i t t t ∆ = − according to Equation (10). By Equation (6) where we can establish that However, we know that t n τ ∆ = ⋅ by Equation (10), then by Equation (7) Therefore we obtain by equalizing: The temperature or thermal radiation T associated to the increase of Energy

Accelerated Universe
According to Mechanics we know that the work done by any force F on a material body produces a variation of its energy, where the work can be expressed as the product of force for the displacement done, that is: However, according to Equation (12)  Therefore we can obtain the force F as:

Universe's Radius
Let us now consider any relative reference system.
where we have applied the principle of simplicity. Thus, if we replace the value for the fundamental minimum length ( l cτ = ) in Equation (9) we obtain the equation: which is the well-known Weyl-Eddington relationship.

Universe's Entropy and Time's Arrow
We can also express the increase of Universe Energy U E T S ∆ = ∆ during elapsed interval of proper time t ∆ by Equation (52) and Equation (53), as: which is in perfect agreement with the universal bound on the entropy.

Entropy-Action Equivalence Law
Let us now consider any relative reference system. According to Equation (48) any particle (or system) of total mass m at Time t increase its entropy during the proper time interval elapsed i t t t ∆ = − as: which we can establish as law of the entropy increase for any material particle during proper time interval elapsed t ∆ or also entropy-action equivalence law, since as we verify the increase of entropy S ∆ is proportional to the action E t ⋅ ∆ .

Energy vs. Time Interval Relationship
An important result is the relationship between the energy 2 E mc = of a material particle at Universe's Time t and the time interval t ∆ in which can be measured. By Equation (7) for fundamental minimum time and Equation (54) we have: Now, multiplying by n + ∈  for any proper time interval according to Equation (10) we obtain: which explains the impossibility of measuring the energy of a material particle at any given time interval t τ ∆ < .

Discrete Entropy and Time's Arrow
According to Equation (61) and Equation (62) we can also obtain discrete increase of entropy S ∆ as: Thus, discrete entropy increase is a multiple of 2 B k . Obviously 2 B k represents the minimum production of entropy. Equation (63) also can be expressed as: ⇒ ∆ > Journal of High Energy Physics, Gravitation and Cosmology which represents the arrow of time for any material particle or set of them (system).

Special Relativistic Entropy-Action Equivalence Law
According to Equation (4), the total Mass of the Universe M at Universe's Time t can be expressed as the ensemble of N typical particles in motion of mass m. If the elapsed proper time interval t ∆ is small enough, we can consider the velocity v of material particle in motion as constant. Thus, according to special relativity we know that: as lineal momentum at Time t. Therefore, by Equation (61) we can now express the special relativistic entropy-action equivalence law as: Therefore, if we denote as the kinetic entropy due to motion as entropy at rest during elapsed proper time interval t ∆ , we can express also the special relativistic entropy-action equivalence law as:

Fractional Temperature
In classical Thermodynamics, for a system (particle or set of them) of mass m, where entropy S is a function of its internal energy E, the temperature T at Time t is given by: However, according to Equation (5) which represents the fractional temperature T at Time t where ( q n ∈  ) and Journal of High Energy Physics, Gravitation and Cosmology

Temperature as a Function of Acceleration
We can also express Equation (65) as: However, according to Equation (18) for fractional acceleration of any particle or system, we now obtain: where obviously the term q n c n t is equal to an acceleration according to Equation (18). Therefore, any material particle with acceleration a have associated a temperature o thermal radiation T according to Equation (66).

Universe Gravitational Potential
By Equation (6)  Thus, we can relate the origin of inertia of any material particle to its interactions with the whole universe, according to the non-local potential of the whole universe, Φ , acting on any material particle of the world ensemble as: is the total energy of the particle at Universe Time t according to Equation (26). Therefore also we can now express the origin of ν as a fundamental frequency of oscillation associated with every particle m at Time t due its interactions with the whole universe, according to the non-local potential of the whole universe, Φ , acting on any material particle as:

On the Origin of Gravity
Let us now consider any relative reference system. By Equation (52) , we obtain: ( ) which is according to Equation (66) for temperature as a function of acceleration.
Then, by substitution of m T in Equation (67), by Equation (4) and simplifying, we obtain: However, by Equation (9), fundamental minimum length l cτ = , Equation (12) for any length and Equation (29) for minimum mass, this previous equation can also be expressed as: where obviously ( n + ∈ Z ). Therefore, a consequence of that all N typical particles in the universe are "gravitationally entangled" is that the work done by the gravitational force ( , that is, the needed energy to displace a material particle of the world ensemble to the distance r ∆ . Or also we can express that every unrecoverable lost minimum mass min m is due the work done by ( ) the Universe gravitational force acting on any material particle at every τ , that is, as the energy needed to displace a particle of the world ensemble to the fundamental minimum length l.

Negative Acceleration
By Equation (64) we can now express the quantity of energy lost at Time t during elapsed proper time interval t ∆ according to Equation (68) as: Obviously since we have considered the Universe as an entropic system Equa-Journal of High Energy Physics, Gravitation and Cosmology tion (45), this quantity of energy lost for every material particle at Time t can be expressed as: E T S ∆ = ∆ . Therefore according to Equation (66) for temperature as a function of acceleration and Equation (63) for discrete entropy, we obtain: Therefore, we obtain that the work done by the Universe gravitational force causes a negative acceleration P c a t = − on every material particle due to the loss of every minimum mass min m as Time progresses. According to Equation (20) we can express as: , v r n n are any positive integers. This negative acceleration should be perceived in those objects or particles not subjected to any other force.

Quantum Gravity
According to §5.1 we found that exists a non-local collective gravitational interaction of all particles within the Universe's Horizon, as a consequence of which all N typical particles of mass m in motion in the Universe are "gravitationally entangled" and form a unified statistical ensemble.
Thus, according to Equation (37) and Equation (41) we will consider any local subsystem within the Universe of i ℵ minimum masses or i N material particles of total mass i M , that is: , that form a closed surface of area i A at Time t. For simplicity, we will consider that the total mass i M is concentrated in the center of the surface. We will also consider that a material particle or body of mass m is located on the surface at a distance r from the center of the surface at Time t and that there is no external force acting on the subsystem or on the material particle, thus the total local system is isolated.

Equivalence's Principle as a Consequence of Discrete Space
Now, according to Equation (12) for the discrete space, we can now express at Time t the distance r from the center of surface which is located the material particle or body of mass m as: Obviously, this length r also can be expressed by Equation (57) Thus, we prove that a gravitational acceleration is equal to an any inertial acceleration, that is, the complete physical equivalence of a gravitational acceleration and a corresponding acceleration of the reference system.

Emergent Gravity
The local system is isolated, therefore there is no external force acting on the surface or on the material particle and therefore system entropy S is a function of its internal energy E and the work T S ∆ done by the internal force during any elapsed proper time interval t ∆ can be expressed according to Mechanics as: F r T S ∆ = ∆ where T is the temperature o thermal radiation at Time t associated to the material particle and F is now a entropic force.
Thus we can express, according to Equation (4)  ) acting on the material particle m during elapsed proper time interval t ∆ , which is located at a distance r from the center of surface, causes an unrecoverable loss of energy is, the needed energy to displace a material particle to the distance r ∆ .
However, this quantity of lost energy by every material particle at Time t can be expressed as: E T S ∆ = ∆ . Therefore according to Equation (66) [10] where this range represents the smallest non-zero mass for any particle quanta in the entropic gravity framework.

Gravitational Entropy
According to Equation (59), Equation (60), Equation (41) and by Equation (42), the total entropy of any system within the Universe with total mass i M can be expressed as a function of its total minimum masses number i ℵ or its Area i A at Time t as:

On the Observed Flattening of Rotation Curves in Galaxies
The concept of "Dark Matter" was proposed by Prof. F. Zwicky to explain the anomalous rotation curves of the galaxies. The problem was that, according to Newtonian dynamics, the velocities of any body of mass m at a distance r from the center of the galaxy, must be expressed by assuming a circular orbit, as: which can be considered as the radius upper limit of a gravitational system at Time t, as for example a rotating Galaxy, that is: where s r α = is the deflection angle at Time t, that is, the identical result predicted by General Relativity in [15].

Curvature of a Ray of Light by Sun's Gravity
If we take the radius and solar mass for example: ( ) We have obtained the value according to astronomical observations (Eddington 1920), that is, a light ray grazing the surface of the Sun is deflected by 1.75 arc seconds.

Statistical Interpretation of Entropy-Action Equivalence Law
We consider that with regard to any relative reference system, a particle of mass m experiences during any interval of proper time i t t t ∆ = − con ( i t t > ) by Equation (10), an increase of entropy E m c = are the entropy, mass and energy of the particle at Time i t , because as we proved, any material particle must be lost, with regard to any relative reference system, at every interval of proper time t ∆ an irrecoverable part of mass/energy as Time progresses.
Obviously the entropy ( )

Probability
We consider now the total volume of possible states for any material particle m at Universe Time t, that is:

( )
, W E t . Let us ask now: What is the probability of finding to the material particle at Time t in a unique state within the total volume of possible configurations where every state has the same probability? In this case, the situation is similar to having at Time t a closed box full of "n" balls every one of them labeled as "1" to "n" and find the probability of extracting at Time t any number, for example number "14". Obviously probability is: 1 p n = .
Therefore, according Equation (87) where obviously the interval of proper time is:

Discrete Wave Function
According to previous section, we consider a material particle m with regard to any relative reference system. The following reflections will be made according to the Wave Mechanics. By Equation (26) we can write: where w is the angular velocity at Time t. However, by Equation (27) we can also express the energy of the particle at Time t as: We now consider m = = p v k  by Equation (27)