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1.2 Torque T

The torque

, (with

as force, as cross product, and as

position vector for the force) [1] is, generally spoken, a vector function of angle and

time . We have:

describing the magnitude and direction of the torque vector, describing an angle

vector in three-dimensional space

, and describing a point in time.

The torque can be applied in a jump manner to a particle or a body, see figure 2.

Hence, the torque function does not allow a conservation law.

Figure 2: Torque in Space and Time

The torque is hence defined by the units

.

2. Momentum p, Angular Momentum L, Overall Momentum, and Energy

E

2.1 Momentum p

We can integrate the force in time in order to obtain the momentum [1]:

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The momentum is a vector function which is dependent of space but not longer

of time , since the time dependence has disappeared by integrating the force in

the time interval

.

Due to the integration of a jump in the force function, the momentum function has no

jump any more, see figure 3. Hence, the momentum function allows a conservation

law.

Figure 3: Momentum in Space

One fundamental principle of physics is that the momentum is conserved in a

closed system. This can be derived from Newton's laws of motion [1].

Hence, in a closed system we have [1]:

describing the momentum before the interaction, and

after the interaction.

The momentum is hence defined by the units

.

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2.2 Angular Momentum L

When we turn from translational to rotational movements, 5 units have to be mapped

[1], see table 1.

Translational Unit

Rotational Unit

Table 1: Translational and Rotational Units ( defines the position vector of the force

;

define the starting and ending points, respectively, when integrating

the mass,

of the body; defines the distance of the body from the rotation axis.)

We can integrate the torque in time in order to obtain the angular momentum [1]:

The angular momentum is a vector function which is dependent of the angle but

not longer of time , since the time dependence has disappeared by integrating the

torque in the time interval

.

Due to the integration of a jump in the torque function, the angular momentum

function has no jump any more, see figure 4. Hence, the angular momentum function

allows a conservation law.

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Figure 4: Angular Momentum in the Angle

One fundamental principle of physics is that the angular momentum is conserved

in a closed system. This can be derived from Newton's laws of motion [1].

Hence, in a closed system we have [1]:

describing the angular momentum before the interaction, and

after the

interaction.

The angular momentum is hence defined by the units

.

2.3 Overall Momentum

The momentum describes the translational movements; the angular momentum

describes the rotational movements. Said two momenta superpose to the overall

momentum [1].

2.4 Energy E

We can integrate the force in space and the torque in the angle in order to obtain

the energy [1]:

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The energy is a scalar function which is dependent of time but not longer of

space and on the angle , since the space dependence has disappeared by

integrating the force in the space interval

, and since the dependence

on the angle has disappeared by integrating the torque in the angle interval

.

Due to the integration of a jump in the force and torque functions, the energy function

has no jump any more, see figure 5. Hence, the energy function allows a

conservation law.

One fundamental principle of physics is that the energy is conserved [1]. Hence,

we always have

describing the energy before the interaction, and

after the interaction.

The energy is defined by the units

.

Figure 5: Energy in Time

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3. Information I

We can integrate the force in space and time and the torque in the angle and in

time, or we can integrate the momentum in space and the angular momentum in the

angle , or we can integrate the energy in time to obtain the information :

The information is a scalar which is neither dependent on space nor on time, since

the space dependence has disappeared by integrating the force in the space

interval

and the torque in the angle interval

, and since

the time dependence has disappeared by integrating the force and the torque in

the time interval

.

Hence, information is described by a scalar (by a number) and not by a function,

like the force, momentum, angular momentum, or energy.

summarizes the momentum of the space interval

, the angular

momentum of the angle interval

and the energy of the time interval

in an index .

The information is defined by the units

.

**Corollary 1**: Hence, we have defined the information in a physical manner by using

a force and torque applied to a particle/body, and have connected this information to

bits used in information technology.

**Corollary 2**: Since information is not defined as a function of space and/or time but

by a number, we cannot formulate a law of conservation of information at this stage,

since we cannot speak of time or space before and after the interaction. Instead, I

summarizes the momentum, the angular momentum, and the energy of the

interaction process and represents the result as a number defining bits.

4. Planck units

In physics, we can use five universal constants (the Planck constants) in order to

define units of measurement [2].

The gravitational constant, ; the speed of light in vacuum, ; the Planck constant, ;

the Coulomb constant,

; the Boltzmann constant

. Since we are interested in

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the force, torque, momentum, angular momentum, energy, and information, but not

in electrical and thermodynamic processes, we focus on the first three Planck units.

The gravitational constant, , is important when defining the force between two

particles/bodies [1]:

describing the mass of the first particle/body;

describing the mass of the

second particle/body; and describing the radius between the first and second

particle/body. The Force applied to matter is the key element in the introduction of

momentum, angular momentum, energy, and information. Hence, equation (1) shall

be defined the

**matter equation**.The speed of light in vacuum, , is important when defining the relationship between

matter (given by the mass ) and energy, [1]:

Hence, equation (2) shall be defined the

**energy equation**.The Planck constant, , is important when defining the relationship between energy,

, and frequency/time [1]:

defines the energy of the particle/wave and defines its frequency. We have

, with being the period of the wave. The Planck constant, , has the same

dimension as information:

. Indeed, the equations

known as Heisenberg's uncertainty principle [1] state that the information defined by

or

is at least . Hence, is the smallest possible information. Hence, we

shall call equation (3) as the

**information equation**.5. Petri nets

As already mentioned in corollary 2, summarizes the momentum, angular

momentum, and the energy of the interaction process and represents the result as a

number defining bits. This fact can be easily depicted by Petri nets. Petri nets are

defined in [3]. A possible Petri net is shown in figure 6.

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Figure 6: Petri Net

The circles

are called places and represent the state of a system. We can

identify the places with information as described above. Hence, the places

identify four information pieces

.

The rectangle is called transition and represents, together with the arrows, the

causal relationship between the Information pieces.

**Corollary 3**: Hence, Petri nets are causal nets representing the causal relationship

of information pieces.

**Corollary 4**: Information as defined above cannot be shown in space and/or time,

since said information is not a function of space and/or time. But said information can

be shown in a causal net.

6. Physical measurements

It is a well known principle of quantum theory that a measurement disturbs the

system which is measured [1]. Such an example is shown in figure 7.

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Figure 7: Physical Measurement

The triangle represents a photon used to perform a measurement, and the circle

represents an electron which is measured by the photon . The photon starts at

time , arrives at time at the electron, interacts with the electron from time to

time , is scattered from the electron at time , and arrives at the observer at time

. The same applies for the spatial coordinates , of course, and needs not to be

repeated.

The interaction between electron and photon is guided, of course, by the basic laws

of physics like the law of conservation of energy, and the law of conservation of

momentum and angular momentum (see the discussion above).

We have before the interaction, hence before the time :

describing the energy of the photon before the interaction, and

describing the

energy of the electron before the interaction.

We have after the interaction, hence after the time :

describing the energy of the photon after the interaction, and

describing the

energy of the electron after the interaction.

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According to the law of conservation of energy we have:

The energy

is transformed during the interaction (hence between and )

by an energy flow to the energy

; and the energy

is set up during the

interaction (hence between and ) by said energy flow from the energy

.

Hence, we can integrate equation

in time for the time period of interaction (from

to ), and obtain:

Equation

can be written as:

*Remark*: We could have derived the law of conservation of information also by

integrating the momentum in space and the angular momentum in the angle , and

by using the law of conservation of momentum and angular momentum. We could

also have derived the law of conservation of information by integrating the force in

space and time and the torque in the angle and in time, and by using the law of

conservation of momentum, angular momentum, and energy.

**Corollary 5**: The validity of the laws for conservation of energy, momentum, and

angular momentum leads to the law of conservation of information. No information

gets lost during a measurement.

**Corollary 6**: Whereas the laws of conservation of energy, momentum, and angular

momentum can be directly observed in the local reference frame of the interacting

particles, the law of conservation of information can only be observed during a

measurement by using the local reference frame of the particles and the local

reference frame of the observer.

**Corollary 7**: According to Albert Einsteins's special theory of relativity the reference

frames of particles and observer are connected by a Lorentz transformation [1].

Hence, space and time get transformed from one reference frame to another, and

the momentum, angular momentum, and energy also get transformed between both

reference frames [1]. Due to

and

the information gets transformed between

both reference frames connected by a Lorentz transformation.

**Corollary 8**: Information cannot be observed in a spacetime diagram like

momentum, angular momentum, and energy, since is not a function of spacetime.

But information can be observed in a causal net. Since the causal net consists of two

dimensions (places and transitions) [3], the observation of information adds two new

dimensions.

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7. Information and causality

As already discussed, information is naturally depicted by causal nets. A causal net

possesses two distinct elements: the places (represented by circles) and the

transitions (represented by rectangles). Figure 8 shows the two possibilities of an

elementary causal net [3].

Figure 8: The two Elementary Structures of a Causal Net

As shown in [3], every place has to be connected to a transition but not to

another place; and every transition has to be connected to a place but not to

another transition. Therefore, the two possibilities

or

define

the elementary net structures. It is proven in [3] that the places and the transitions

are dual entities, and that a causal net defines a continuum.

Carl Adam Petri shows in [3] that one can define a synchronic distance between

the places, and a translation distance between the transitions. The longer the

distances and are, the more places and transitions, respectively, are crossed.

Hence, and define two possible axes, one for the places and one for the

transitions, respectively. In [3] and are dimensionless, since they are applied to

the abstract structure of causal nets.

We have already shown in corollary 3 that we can identify the places with information

pieces . In this case, the synchronic distance defines an information axis, ,

defining distances among the information pieces , bearing in mind that several

information pieces can occupy the same location on the information axis defined by

. In the case of information, is not longer dimensionless (like for abstract causal

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nets), but possesses the dimension of information, namely

.

The transitions define the interaction of the information pieces according to

equation

above. Hence, we have information pieces

before the

interaction, such that said information pieces are connected by the transition to

information pieces

after the interaction, and such that

due to

the law of conservation of information.

Otherwise stated, we have a vector of information pieces before the interaction:

, and we have a vector of information pieces after the interaction:

. The transition acts as a matrix projecting to . The matrix has hence

the structure

.

Hence, we have the equation

showing how the information before the interaction is projected on the information

after the interaction. Hence, the matrix showing the interaction described by a

transition is dimensionless. We identify the matrix with causality, and conclude that

causality is dimensionless, contrary to information.

We can use the translation distance to define a causality axis, , defining distances

among the matrices

, bearing in mind that several matrices can occupy the same

location on the causality axis defined by . In the case of causality, remains

dimensionless like for abstract causal nets.

The dimensions for information and for causality are the two additional

dimensions mentioned in corollary 8.

8. Joint consideration of momentum, angular momentum, energy, and

information in space, time, and causality

In physics, the conservation of momentum, angular momentum and energy are

fundamental principles as discussed above. Furthermore, we have derived a

conservation of information in this paper. Hence, we have four conservation laws:

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**Corollary 9**: Since momentum and angular momentum are quantities describing the

features of particles, momentum and angular momentum can be viewed as features

of matter. Therefore, we have conservation laws describing how matter, energy, and

information are conserved.

In quantum theory, Werner Heisenberg has shown that the uncertainty principle is

valid when measuring the momentum range

and spatial range

of a particle, or

the energy range

and the temporal location range

of a particle. The same

applies, of course, for the measurement of the angular momentum range

of a

particle and the angle location range

of said particle.

Carl Adam Petri shows in [3] that the causal nets also possess an uncertainty

relation. We therefore conclude that the measurement of the information range

and the measurement of the causality range

are also uncertain. Therefore, we

have:

The uncertainty of information in causality can be formally derived as follows. is the

smallest possible information (see the last paragraph of point 4 above). Hence,

cannot be smaller than . Hence,

has to be the identity element in this case.

The first uncertainty relation is explained by Werner Heisenberg in the following

manner. The more accurate the measurement of a spatial position of a particle, the

smaller the wavelength of the measuring wave must be. But the smaller the

wavelength of the measuring wave is, the bigger the momentum of the measuring

wave is, such that the impact on the momentum of the measured particle is big,

leading to a big uncertainty of said momentum. The more accurate the measuring of

a momentum of a particle, the bigger the wavelength of the measuring wave must

be, in order not to influence the particle momentum. But the bigger the wavelength of

the measuring wave is, the less precise the measurement of the spatial position of

the particle is, leading to a big uncertainty is said position. Hence, momentum and

space cannot be determined with big accuracy at the same time.

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The second uncertainty relation can be explained in a corresponding manner. The

more accurate the measurement of a rotation angle of a particle, the smaller the

wavelength parallel to the rotational movement of the measuring wave must be. But

the smaller the wavelength parallel to the rotational movement of the measuring

wave is, the bigger the angular momentum of the measuring wave is, such that the

impact on the angular momentum of the measured particle is big, leading to a big

uncertainty of said angular momentum. The more accurate the measuring of an

angular momentum of a particle, the bigger the wavelength parallel to the rotational

movement of the measuring wave must be, in order not to influence the particle

angular momentum. But the bigger the wavelength parallel to the rotational

movement of the measuring wave is, the less precise the measurement of the

rotation angle of the particle is, leading to a big uncertainty is said rotation angle.

Hence, angular momentum and rotation angle cannot be determined with big

accuracy at the same time.

The third uncertainty relation is explained by Werner Heisenberg in a similar manner.

The more accurate the measurement of a temporal position of a particle, the bigger

the frequency of the measuring wave must be. But the bigger the frequency of the

measuring wave is, the bigger the energy of the measuring wave is, such that the

impact on the energy of the measured particle is big, leading to a big uncertainty of

said energy. The more accurate the measuring of an energy of a particle, the smaller

the frequency of the measuring wave must be, in order not to influence the particle

energy. But the smaller the frequency of the measuring wave is, the less precise the

measurement of the temporal position of the particle is, leading to a big uncertainty is

said position. Hence, energy and time cannot be determined with big accuracy at the

same time.

The fourth uncertainty relation can be explained as follows. The more accurate the

measurement of the information of a particle, the more isolated the particle must be,

in order to be able to measure its energy, momentum, and angular momentum

states. But the more isolated said particle is, the more interactions with other

particles are destroyed, leading to a big uncertainty of causality. The more accurate

the measurement of causality shall be, the more particle interactions shall be

observed. But the more particle interactions shall be observed, the less isolated

single particles shall be, leading to a big uncertainty of the information of a single

particle. Hence, information and causality cannot be determined with big accuracy at

the same time.

**Corollary 10**: The substances matter (as defined by its momentum and angular

momentum), energy, and information lead to an uncertainty relation in their existence

forms, space, time, and causality.

**Corollary 11**: In the physical view of a single local reference frame, there are 8

dimensions: 3 dimensions of space,

, 3 complementary dimensions of space

given by the overall momenta (

) [1], 1 dimension

of time

, and 1 complementary dimension of time given by the energy

[1]. In

the physical view of several local reference frames, there are 10 dimensions: the 8

dimensions of the single local reference frame, 1 dimension of causality

, and 1

complementary dimension of causality given by the information

.

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9. Final remarks on the nature of information

The derivation of information achieved in this paper, and especially the derivation of

the law of conservation of information, appear astonishing viewed from the point of

view of information technology. Indeed, in information technology we have

Shannon's definition of information by using probabilities. Furthermore, information

can be always destroyed and created, such that a law of conservation of information

makes no sense in information technology.

This apparent discrepancy can be best resolved by analysing the nature of energy in

physics and engineering. The law of conservation of energy is a very old principle in

physics, and is the basis for many physical derivations. Nevertheless, when

performing engineering, like constructing an electric power station, one arrives at the

conclusion that not all energy forms are suitable for the set task. Indeed, when

converting the mechanical energy of the turbine into electrical energy by the

generator, one obtains the equation:

This equation depicts the thermodynamic principle that the mechanical energy of the

turbine cannot be entirely transformed to electrical energy; instead heat is produced

during the energy transformation process [1].

Physically spoken, no energy is lost, since the energy before the interaction (the

mechanical energy) is exactly the same as the energy after the interaction (the

electrical and the heat energy). But from the point of view of an engineer, there is a

loss, since the heat energy cannot be used and therefore defines a technological

loss. Hence, in engineering, the heat energy is minimized as much as possible. For

this reason, an energy efficiency factor

is defined in engineering:

This factor is between 0 and 1, and shall be as much as possible near 1, but can

never achieve 1 due to thermodynamic principles [1]. This shows that the

fundamental physical principle of the conservation of energy is not always of interest

in engineering; instead physically equivalent energy types are rated in engineering,

defining some energy types as desired and others as not desired, such that energy

losses might appear in engineering.

The same reasoning as for energy applies also for information. Indeed, it has been

shown in this paper that information is conserved in physics. Hence, from a physical

point of view, information is never lost. But from the point of view of information

technology, some information pieces

might be desired, since they represent

information which can be used for a given purpose, whereas other information

pieces

might be undesired, since they cannot be used for said given

purpose.

Hence, information pieces are rated in information technology with respect to their

usability. Therefore, in information technology an information loss is possible, due to

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the fact of the presence of desired and undesired information. One could define an

information efficiency factor for information technology:

This factor is between 0 and 1, and shall be as much as possible near 1, but can

never achieve 1 due to uncertainty principles between information and causality.

This shows that the fundamental physical principle of the conservation of information

is not always of interest in information technology; instead physically equivalent

information pieces are rated in information technology, defining some information

pieces as desired and others as not desired, such that information losses might

appear in information technology.

Shannon's definition of information by using probabilities is then another form of

defining desired and not desired information by weighting some information pieces

more than others. This is in line with the point of view of engineering which rates

energy, information, and momenta (although momenta have not been addressed in

this discussion of point 9). But the ratings performed in engineering do not contradict

the physical conservation laws for momentum, angular momentum, energy, and

information.

Bibliography

[1]: Carlo Maria Becchi; Massimo D'Elia: Introduction to the Basic Concepts of

Modern Physics; Special Relativity, Quantum and Statistical Physics; Third Edition;

Springer; 2016.

[2]:

https://en.wikipedia.org/wiki/Planck_units#Cosmology

[3]: C. A. Petri: Nets, time and space; Theoretical Computer Science 153; 1996.

17 of 17 pages

- Quote paper
- Alexander Mircescu (Author), 2016, On the Conservation of Momentum, Angular Momentum, Energy, and Information, Munich, GRIN Verlag, https://www.grin.com/document/343915

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