In the following I shall perform a general rational quantization of some of the physical quantities and equations we meet in classical mechanics. This general quantization is a direct consequence of the quantization of space, time and mass. The existence of this basical quantization - and the cosmological consequences hereof - I have accounted for in my holistic quantum cosmology. The basis for the following - and my holistic quantum cosmology - is the existence of the following fundamental cosmological quantities:

1. The physically smallest cosmological quantum length r₀, also called elementary length
2. The physically smallest cosmological quantum time interval t₀, also called elementary time
   and
   
3. The total energy/matter mass of the Universe M₀

These cosmological quantities define an absolute and cosmological system of units, independent of specific local systems and the existence of living systems. (See chapter 5 in my treatise).
From the following it will be clear that:

1. All physical velocities can be given as a rational fraction of the velocity of light c₀.
2. All accelerations can be given as a rational fraction of a universal physically maximum acceleration.
3. The momentum of any system can be given as a rationel fraction of a universal maximum ’momentum’, or as a whole multiple of a physically smallest momentum. The physically smallest momentum in the Universe is equal to m_u ·  c₀, where m_u is the mass of the physically smallest matter/energy quantum in the actual Universe and c₀ the velocity of light. The physically smallest matter/energy quantum in the Universe I have given the name a uniton.
4. The content of energy of any system can be given as a rational fraction of the total energy of the Universe, or as a whole multiple of a physically smallest portion of energy. The smallest actual amount of energy in the Universe is identical to the kinetic energy of one uniton. I show, that the kinetic energy of one uniton is given by m_u· c₀².

As a consequence of the quantization of the physical quantities, all laws and formular connections in our description of the Universe will be quantized, viz. in the equations will appear characteristic rational quantum numbers. As a consequence hereof it will be possible to formulate the laws of nature as quantum number conditions.

The expression holistic refers to the fact, that in the physical expressions appear quantities as well from microcosmos, f.i. Planck's constant and the mass of an uniton, as also from macrocosmos, f.i. the total mass of the Universe.

As a consequence of the basical quantization of the fundamental physical quantities, all natural phenomena must - fundamentally seen - be described by a discrete mathematics and not, as it has been usual since Newton, by a continuous mathematics. As elementary length, and the corresponding elementary time, are extremely small quantities - when given by our practical measuring units - daily and macroscopic phenomena can still, adventageously and with great accuracy, be described by means of continuous mathematics. If you, on the contrary, wish a correct description and understanding of micro physical phenomena and the earliest states in the evolution of the Universe, then it is necessary to use a discontinuous mathematics. In an earlier chapter I have considered some consequences of what I call a physical quantum geometry. With the existence of a physically smallest ’distance standard’, we can not give a sophisticated description of nature by means of the continuous Euclidic geometry. In a physical quantum geometry, we must quantize the geometrical distance dimensions of  f.i. a ’circle’ so that the quotient between the 'quantum perimeter' and the 'quantum diameter' of the quantum circle defines a rational pi,   Likewise the pythagoreic relations will not be valid in a physical quantum geometry.

I shall show that the maximal work, which was carried out in the Universe, when it was 'born' within the first cosmic quantum time interval, is given by the total mass of the Universe multiplied by the square of the velocity of light. The magnitude of this maximal work is equal to the total energy, which broke out in the Universe, when it performed its first cosmic quantum jump over elementary length! Using my quantum cosmological equation, connecting the total mass of the Universe with the mass of a uniton, I can in a simple way derive the famous equation E = m · c₀² , where m is the mass of a system and c₀ is the velocity of light.

In the following I shall consider the quantum conditions of some special systems, a.o. the quantized motion of a particle, moving in a homogenous gravitational field. I shall show that the energy of a particle or a given system can either be given as a rational fraction of the total energy of the Universe, or as a whole multiple of the kinetic energy of a uniton.

Quantization of space, time and mass. Quantum kinematics and quantum dynamics

The physically smallest distance in the Universe - elementary length, r₀, and the physically smallest time interval - elementary time, t₀, are given by:

(1)

where h is Planck's constant, M₀ the total energy/matter mass of the Universe and c₀ the light constant.

Any finite physical distance is given by:

(2)

where is a natural number, the space quantum number.

Any finite physical time interval is given by:

(3)

where is a natural number, the time quantum number.

As a natural consequence of (2) and (3) also the physical quantities velocity and acceleration will be quantized.

For a particle, moving in a given reference system, we define the quantized velocity v - in a given direction - as the travelled distance divided by the elapsed time interval. Using the basically quantized quantities in (2) and (3) we get for v:

(4)

where is a rational velocity quantum number, which can take rational values between 0 and 1.
For a particle, with a velocity changing in time, the velocity quantum number is a function of the time quantum number.
For a particle, changing velocity, we define the magnitude of the quantized acceleration a - in a given direction - by the following:

(5)

where is a rational acceleration quantum number, which can take rational values between 0 and 1.   a_max is equal to the quotient between the velocity of light and elementary time and is thus a maximal greatest physical acceleration in the Universe. The maximum acceleration occurred when the Universe, with its matter and motions, was 'born' within the first cosmic quantum jump over a distance equal to elementary length!

It must be assumed that the mass m of any system is a rational fraction of the total mass of the Universe M₀, viz.:

(6)

where can be called the cosmic mass quantum number.

In a given reference system we shall define the quantized momentum p of a particle - in a given direction - by the following:

(7)

where is the rational momentum quantum number, which can take rational value between 0 and 1.
The quantity M₀· c₀ gives a maximum ’momentum of the Universe’.

Quantum force and quantum work

If a particle in a given reference system accelerates, viz. changes velocity, we say that the particle is influenced by one or more forces. The magnitude F_res of the total influence of forces on the particle, also called the resulting force, we define as the change in the momentum of the particle divided by the corresponding time interval, viz.:

(8)

In equation (8) is a rational force quantum number, which can take rational values between 0 og 1.
The quantity M₀· a_max is the maximum physical force, which may work in the Universe, and this maximum force was applied when the Universe was 'born' and performed its first cosmical quantum jump. See later.

The work, A_F, done by a constant force over a distance can - if the directions of force and motion are parallel- be defined as:

(9)

defines a rational work quantum number, or energy quantum number, which can take rational values between 0 og 1.

The quantity M₀· c₀² is - as I shall show in the following - the total energy in the Universe! The content of energy in any subsystem is thus equal to a rational fraction of the total energy of the Universe!

A work done on a system means that there has been an exchange of energy between the systems exchanging forces. The magnitude of the work, A_res, done by the resulting force on a particle, is equal to the increase of the kinetic energy of the particle,    From (9) we get for the work done by the resulting force:

(10)

The embryonic work of the Universe and its total content of energy

When the Universe was 'born' and performed its first cosmical quantum jump over a distance equal to elementary length, the total energy, existing in the Universe, came into being. This total energy has since then been distributed as kinetic energy of all the smallest matter/energy quanta - the unitons - of which everything in the Universe consist. We can calculate this total cosmic energy, E_kin,cos, by calculating the maximum physical work A_max, which was done over elementary length r₀:

(11)

We see that the total energy of the Universe is equal to the product of its total mass and the velocity of light squared, thus an expression corresponding to what Einstein found in his special theory of relativity from 1905. Here, however, I arrived to the expression by a quantum physical method, and thereby in a fundamentally different way!

Derivation of the formula E = m·c₀² and a uniton-mechanical explanation

In my holistic quantum cosmology (see this) I show the existence of the following connection:

(12)

where N³ is the cosmic evolution quantum number, which controls the evolution of the Universe and is a discrete parameter, ’ticking’ up through the natural numbers. N is equal to the actual quotient between the electrostatic and the gravitostatic forces between two electrons.   m_u is the mass of a uniton.   From (12) we see that in any state of the Universe N³ unitons exist. In our epoch N³ = 7,2 · 10¹²⁷, meaning an extremely high number of unitons. Any limited amount of matter consists of a definite number of unitons. Inserting the expression from (12) in equation (11) we get:

(13)

From equation (13) we see that the kinetic energy of one uniton E_kin,u is determined by the product of the mass of a uniton and the velocity of light squared! This uniton-kinetic energy is the physically smallest amount of energy - elementary energy - in the actual Universe.
As the mass m of a limited amount of matter is determined by the sum of the masses of the unitons of which the amount of matter consists, the following is valid:

(14)

The total content of energy E_mat of a definite amount of matter is thus given by the product of its mass and the velocity of light squared!
The above considerations give us the following new recognition and explanation of the famous equation E = m· c₀² :
The total energy in a given amount of matter is a measure of the total kinetic energy of the unitons, of which the matter consists!! By this recognition of our physical world it is not difficult to understand that all matter, regardless of its chemical appearance contains a total energy as given by equation (14)!

The smallest values of momentum, angular momentum and energy of our Universe

Using the connection given by equation (12) we can express the content of energy of different systems as a whole multiplum of the kinetic energy of one uniton. As a particle we shall understand a limited uniton- dynamical system, consisting of a definite number of unitons. The uniton defined mass m of the particle is given by:

(15)

where n_m is a natural number, the mass quantum number.
If the mass of a system is given in cosmological units (see chapt.5), then this is equal to the value of the mass quantum number!
The quantity in (7) can be written as:

(16)

where n_p is a natural number, the natural momentum quantum number.
The quantity p_u, determined by:

(17)

gives the smallest physical momentum - elementary momentum - in the Universe. If this elementary momentum is multiplied by the extension R of the Universe, we get the physical quantity called angular momentum. The product of p_u and R gives Planck's constant h, viz. the following is valid:

(18)

It will be seen that Planck's constant is not as fundamental as presumed in established physics. Elementary momentum and the extension of the Universe must be considered more fundamental! The expression in equation (18) is a limit case of Heisenberg's uncertainty relations regarding momentum and position, as it gives the context between minimum uncertainty for momentum and maximum uncertainty for position. Multiplying the elementary momentum by elementary length, r₀, we can find the value of the physically smallest angular momentum in the Universe, elementary angular momentum l_u:

(19)

We see that the value of elementary angular momentum is extremely small given by SI-units, as defined by men. If we, on the contrary, give elementary angular momentum in cosmological units, then the value is equal to 1 (masson · spaton²)/tempon.
Elementary angular momentum must be interpreted as being equal to the angular momentum of one uniton, viz. it can also be designated as the spin of the uniton.
The numerically total angular momentum of the Universe can be calculated by multiplying elementary momentum, first by the number N³ of unitons in the Universe, and thereupon again by N³, as the last mentioned multiplication takes into account the extension of the Universe. Taking into account the direction of all angular momenta, the total angular momentum of the Universe will be 0.
The value of elementary momentum p_u is also equal to 1, when given in cosmological units!
The magnitude of a resulting force, F_res, can be written as:

(20)

where and give the corresponding increases in momentum quantum number and time quantum number, respectively.

Equations (10) and (11) can be transformed to:

(21)

(22)

where n_A and n_A,res can be considered as natural work quantum numbers.   E_kin,u is the kinetic energy of one uniton, and this is the physically smallest amount of energy in the Universe - elementary energy. In cosmological units the value of elementary energy is equal to 1.
Analog to the expression in (18) we can also express Planck's constant h by the age of the Universe, T, as follows:

(23)

viz. as a product of elementary energy and the age of the Universe. Equation (23) is, like equation (18), a limit case of Heisenberg's uncertainty relations for time and energy.

Let us analyse the state of energy of a particle, moving in a homogenous gravitational field, using the quantized expressions of energy.

The kinetic energy E_kin of a particle can be written:

(24)

where m is the mass of the particle, equal to a natural number multiplied by the mass of a uniton.   v is the rationally quantized velocity, and n_Ek is a natural number - the kinetic energy quantum number.   From (24) we see that the kinetic energy of the particle is given by a whole multiple of the elementary energy.
For a particle moving in a homogenous gravitational field the gravitational potential energy, E_pot, can be calculated as:

(25)

where g is the acceleration of gravity at the position, equal to a rational acceleration quantum number multiplied by the universal maximum acceleration - the quotient between the velocity of light and elementary time.   H is the vertical height measured from a chosen origo and determined by a space quantum number multiplied by the elementary length.   n_Ep is the energy quantum number, corresponding to the potential energy - the potential energy quantum number. We see from (25), that also the gravitational potential energy can be calculated as a whole multiple of the elementary energy.

A particle, moving in the homogenous gravitational field, has within each quantum time interval a total socalled mechanical energy, E_mek, given by:

(26)

where n_mek is the mechanical energy quantum number.
We can now express the conservation of the mechanical energy of the particle as an energy quantum number condition. During the motion of the particle it must be valid that the increase of the mechanical energy of the particle must be zero, viz.:

(27)

During the motion of the particle we thus have:

(28)

Expressed in words equation (28) means: The kinetic energy quantum number will increase just as must as the potential energy quantum number decreases! If for instance the potential energy quantum number decreases by 10, it means that the kinetic energy quantum number will increase by 10. In practical life the change of these numbers will be extremely great. But fundamentally seen, it must be assumed that nature functions as I have described!

© Louis Nielsen, July 1998
E-mail: louis44nielsen@gmail.com

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