Extrait
2
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]:
3
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
.
4
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.
5
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]:
6
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
8
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.
9
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.
10
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.
11
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.
12
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
13
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:
14
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.
15
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
.
16
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
17
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.
Fin de l'extrait de 17 pages
- Citation du texte
- Alexander Mircescu (Auteur), 2016, On the Conservation of Momentum, Angular Momentum, Energy, and Information, Munich, GRIN Verlag, https://www.grin.com/document/343915
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