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Articles and Publication  Physics Theoretical physics UNCERTAINTY RELATIONS AS A CONSEQUENCE OF THE LORENTZ TRANSFORMATIONS.
UNCERTAINTY
RELATIONS AS A CONSEQUENCE OF THE LORENTZ TRANSFORMATIONS.
© V. N.
Matveev and
O. V. Matvejev
Contact
to the author: matwad@mail.ru
A macroscopic object
consisting of a rod equipped with a pair of synchronized clocks is examined.
General physical relations are directly derived from Lorentz transformations for
the case of the rod's one-dimensional motion (along the X axis) – the
uncertainty relation of the object's x coordinate and the projection of its
impulse along the X axis, px, and the uncertainty relation of the
object's observation time, t, and its energy, E. The relations take the form:
ΔpxΔx ≥ H and ΔEΔt ≥ H. The H value
in the relation has action dimensions and is dependent upon the precision of the
rod's clocks and its mass.
It is shown that if the
macroscopic object in and of itself performs the function of an ideal physical
clock, the relations derived in the limiting case then take the form of ΔpxΔx
≥ h and ΔEΔt ≥ h, where h is the Planck constant.
_________________________________________________________________
Introduction
The uncertainty relation of a
microparticle coordinate and the projection of its impulse along the coordinate
axis, as well as the uncertainty relation of microparticle time and energy,
falls into the ranks of the most important quantum relations that express the
known uncertainty principle in a mathematical form. Quantum relations are not
observed in the macrocosm in the sense that they are vanishingly small with
respect to macrobodies and do not show themselves in practice. According to our
data, no general physical relations have been reported in extant literature that
extend to the macrocosm and that limit the simultaneous specification of the
impulse and the coordinate of physical bodies. In light of existing notions
concerning the properties of macroscopic bodies, the absence of such general
physical uncertainty relations seems natural, since uncertainty relations have
not been connected to measuring instrument errors, while it is possible to talk
about macrobody coordinate, impulse, and energy inaccuracies, as they seem just
as obvious today as measurement errors.
The objective of the work at hand
was to demonstrate the existence of general physical uncertainty relations that
extend to macrobodies. This objective was achieved over the course of solving a
problem that consisted of determining the velocity and the velocity-related
physical values of objects based on the degree of desynchronization of moving
clocks.
The relations ΔpxΔx
≥ H and ΔEΔt ≥ H were directly
derived from Lorentz transformations. The first relation connects the
uncertainty of the projection of the impulse, px, of the body
under consideration, Δpx, to the uncertainty of the x
coordinate, Δx, while the second relation connects the uncertainty
of the energy, E, of the object, ΔE, to the observation time,
t, uncertainty, Δt. The H value in the relations has
an action dimension and is dependent upon the precision of the rod's clocks and
its mass.
Taking into account the fact that
the recommendations formulated by metrologists for replacing the terms "error"
with the term "uncertainty" are not compulsory in nature, and bearing
in mind that these recommendations are still in the discussional stage [1-2], we
will use both terms in the work at hand.
We will take error to mean the
inaccuracy of measurement results that is due to purely metrological causes.
Increasing an instrument's measurement accuracy can reduce the error. The
absolute error of the distance between two points can serve as an example of an
error.
We will call the inaccuracy of
measurement results that cannot be eliminated by means of increasing measurement
instrument accuracy, and which may be due to terminological, conceptual, or
linguistic causes, the uncertainty of physical values. The uncertainty of the
distance between two spheres that are located close to one another can serve as
an example of this uncertainty. This distance remains uncertain with an accuracy
of up to the dimensions of the spheres, even in the presence of ideal
measurement accuracy, while the thing that remains unclear is what is meant by
the sought distance – the distance between the centers of mass of the spheres,
the distance between their geometric centers, the distance between the closest
points of the spheres, or something else. Construing uncertainty in this manner
is mentioned in extant literature, albeit in general terms. In this vein, for
example, taking into account the fact that it is only possible to specify the
location of a spatially extended body by determining the
position of a single solitary point that belongs to it with some degree of
uncertainty, the uncertainty of the position of a sphere determined by the
position of its center, which equals the radius of this sphere, is written about
in reference [3].
The uncertainty of specifying the
moment in time of a short-term process that does not occur instantaneously, but
rather occupies a certain finite, let's say, very short time interval, can serve
as another example of an uncertainty of this type. This uncertainty can be
regarded as equal to half the duration of the process, if the moment in time of
its transit is called the moment into which the middle of the process falls.
These coordinate and time uncertainties, Δx and Δt, are
precisely the ones that play a part in the uncertainty relations we derived.
1. Single-time and single-coordinate data.
Determining the velocity of a rod based on the readings of the synchronized
clocks appurtenant to it
We will visualize a thin rod of
proper length, L, at two points, a and b, on which
synchronously running clocks, A and B, are installed at a distance
of d from one another.
Let's say that clocks A
and B, like the clocks appurtenant to the K0 reference
system, where the rod is at rest, show this system's time; i.e., the readings of
clocks A and B are always in agreement with the readings of the K0
system's clocks. The length, d, of section ab, which is located
between points a and b, may be equal to or less than the rod
length, L; i.e., the condition L ≥ d holds true in the
general case. If L > d, then the rod will look something like this:
----------A--------------------B-----------
Here, the A and B
characters conditionally designate clock A and
clock B, while the broken line shows the body of the rod.
If the distance, d, is
equal to the rod length, L – i.e., if L = d, clocks A and
B are then found at the ends of the rod.
We will call the arrangement that
consists of this rod and the two running clocks, A and B, situated
on it rod, R; i.e., clocks A and B will be
regarded as integral parts of rod R, and we will treat the readings of
clocks A and B as characteristic attributes of rod R.
Let's say that rod R,
which is positioned parallel to the Х0 axis of inertial
system K0, is at rest relative to this system and moves at a
constant velocity, V', along the Х' axis of another reference
system, K', remaining parallel to the Х' axis and its direction
of movement (the Х0 and Х' axes of the K0
and K' systems slide along one another over the course of their relative
motion). We will call this rod motion longitudinal motion (with respect to its
orientation in space), and it alone will be referred to in the future.
Pursuant to inverse Lorentz
transformations, at a moment in time of t' of the K' system, the readings,
τA,t' and τB,t', of clocks A
and B, which are in agreement with the readings of the K0
system's clocks, are determined by the relations
and  ,
where x'A,t'
and x'B, t' – the coordinates of clocks A
and B of rod R within system K' at a moment in tie of t',
while c – the speed of light in a vacuum.
As follows from the relations
presented above, the difference in the readings of clocks A and B
of rod R at a moment in time of t' for system K' equals
 .
(1)
Introducing the following
notation for the purpose of saving writing space
 ,
(2)
then from formula (1), with
allowance for notation (2), we obtain:
 .
(3)
Thus, having the data on the
coordinates and readings of clocks A and B at a moment in time of t',
formula (3) can be used to find the value of U', and this value can in
turn be used to find the velocity, V', of rod R in system K'.
We will call the data that characterize the object's spatially distributed
elements, but that relate to one and the same moment in time, t',
single-time data.
In addition to the possibility of
determining the rod's velocity using single-time data, the possibility also
exists of determining the velocity, V', of rod R in system K'
using single-coordinate data. We will call the data successively recorded at
different moments in time, but at one and the same point (with one and the same
coordinate), which characterize the elements of a spatially extended object at
the times that they are located at this (or near this) point, single-coordinate
data. Data of this type include, for example, the τA,x'
and τB,x' readings of clocks A and B of rod
R that are successively recorded at moments in time of t'A,x'
and t'B,x' at a point in system K' with a coordinate of
x', past which it is moving within this reference system.
According to inverse Lorentz
transformations, the single-coordinate τA,x' and τB,x'
readings of clocks A and B are connected to the moments in time, t'A,x'
and t'B,x', during which clocks A and B, at a
point with a coordinate of x', show the relations
 and  .
As follows from the relations
presented above, the difference in the readings of clocks A and B
of rod R at a point with a K' system coordinate of x'
equals
 . (4)
Introducing the notation  ,
then from formula (4), we obtain
 ,
(5)
from which the velocity, V',
of rod R can be learned as necessary.
The determination of velocity
using single-coordinate data is interesting in that it does not require distance
measurements, but rather is based on the measurement of the τB,x' –
τA,x' and t'B,x' – t'A,x'
time intervals.
2. Relation of rod velocity and
coordinate uncertainties calculated using single-time data
In talking about the
determination of the velocity of rod R using the difference in the
readings of clocks A and B, we tacitly proceeded on the basis of
the fact that clocks A and B of rod R run ideally; i.e.,
the readings of clocks A and B are in absolutely precise agreement
with the ideally accurate readings of the K0 system clocks. We
speculate that a certain maximum absolute error exists in the time readings of
each of clocks A and B, which are appurtenant to rod R. We
will designate the absolute error of these clocks as Δτ. Here,
adhering to the generally accepted assumption, we will presume that all the
clocks appurtenant to any inertial reference system, including the K0
system, run with ideal accuracy.
The existence of an absolute
error, Δτ, in the readings of each of clocks A and B
means that, at a given moment in time, t0, within the K0
system, the readings of each of these clocks of rod R can differ from the
readings of the K0 system clocks located close to them and
running with ideal accuracy by a value that does not exceed Δτ. In
this regard, the remarks made above concerning the agreement of the reading of
the K0 system clocks with those of clocks A and B
must be interpreted with allowance for the finite accuracy of the latter.
If the rate of the K'
system clocks, the measured x'B, t' – x'A,t'
value, and the speed of light c value are as accurate as desired, the
error in the U' value calculated using formula (3) will then be solely
due to the existence of an absolute error, Δ(τB,t' – τA,t'),
in the difference of the τB,t' – τA,t' readings
of clocks A and B.
In this instance, the ΔU' error, with
allowance for formula (3), is expressed by the equality
(6)
In instances when the maximum
absolute error of the difference in the readings of clocks A and B
consists of the Δτ errors of each of these clocks – i.e., when
Δ(τB,t' – τA,t') = 2Δτ,
it follows from equality (6) that:
 .
(7)
We note that the Δτ
error is not dependent upon reference system selection, since it consists of the
maximum possible difference between the readings of each of clocks A and B
and the readings of the К0 system clocks located close to
them, where rod R is at rest. It is clear that this difference is not
dependent upon the reference system within which it is picked up.
We will now imagine that, in
addition to the condition of the single-time nature of the readings of clocks A
and B within system K', the requirement of the single-coordinate
nature of the specification of the location where rod R is situated at a
moment in time of t' is satisfied. Let's say the essence of this
requirement consists of using a single solitary x'R coordinate
to specify the location of the projection of rod R on the X' axis.
This requirement can be satisfied if the rod length is ignored and it is
regarded as a point. But if, due to the necessity of taking the property of the
rod's spatial extent into account, it is impossible to ignore its length, the
requirement of the single-coordinate nature of the specification of the position
of rod R can only be satisfied in part. For example, the coordinate of
any point appurtenant to rod R can be specified as its x'R
coordinate and a reference to its uncertainty can accompany the specification of
this coordinate. In particular, the coordinate of the geometric center of rod R,
or the coordinate of its center of mass, can serve as the coordinate of this
point. In such cases, the distance from the point with a coordinate of x'R
to the point of the rod's projection on the X' axis farthest away from it
can be regarded as the uncertainty, Δx'R, of the x'R
coordinate.
When the position of rod R
is specified in this manner, the x'R coordinate indicates the
location of one of a set of points of the projection of rod R that lies
on the X' axis. If we give this point preference for one reason or
another, the Δx'R uncertainty will then determine a range
of point coordinates that, in the presence of other considerations, could also
be regarded at the point coordinates of rod R.
For example, if the coordinate, x'ab,
of the geometric center (the midpoint) of the ab section of rod R
parallel to the X' axis is selected as this section's coordinate, the
uncertainty, Δx'ab, of the x'ab
coordinate of the rod's ab section can by definition be regarded as a
value equal to half the length of the section of the moving (within the K'
system) rod.
Since the length,  ,
of the rod's moving ab section within the K' system equals x'B,t'
– x'A,t' , the uncertainty, Δx'ab, of
the x'ab coordinate of section ab of rod R then equals
½(x'B,t' – x'A,t'). Thus, formula (7) can
be presented in the form
 .
(8)
Within the framework we have
provided when introducing the determination of uncertainty, the Δx'ab
value is absolutely an uncertainty and not an error, since it is dependent upon
the length of the rod's ab section and cannot be reduced by means of
increasing measurement accuracy.
Because L ≥ d
in the general case, the Δx' uncertainty of the x' coordinate
of rod R within the arbitrary positioning of clocks A and B
thereon (Δx' = ½L') can then generally both
equal and exceed the Δx'ab value. And since the relation
Δx' ≥ Δx'ab holds true, then in
the general case of the arbitrary positioning of the clocks on the rod, formula
(8) takes the form:
 .
(9)
Relation (9) pertains to the
general case of the arbitrary positioning of the clocks on the rod and makes the
transition to the equality ΔU'Δx' = c2Δτ
in the special and most favorable case (as far as the determination of the U'
value using single-time data) of clock positioning on the ends of the rod (Δx'=
Δx'ab).
The product of the error, ΔU',
of the U' value and the uncertainty, Δx', of the x'
coordinate of rod R during an instantaneous observation will only be
dependent upon the error in the readings of clocks A and B of
rod R. Therefore, relation (9) remains unchanged in any inertial
reference system and can be written for an arbitrary system in the form:
(10)
If the mass of rod R with
clocks A and B is known without a doubt and equals MR
(here and further on, the concepts of Lorentz-invariant mass [4] will be used),
relation (10) can be transformed into the relation MRΔUxΔx
≥ MRc2Δτ by multiplying its
left-hand and right-hand members times MR, whence, taking into
account the fact that MRUx = px,
we obtain:
 .
(11)
Introducing the notation
 ,
(12)
then from formula (11), we obtain
 .
(13)
The Δτ error of clocks A
and B of rod R is one of the internal parameters of rod R
and is not a part of the parameters of the measuring devices that are external
relative to rod R. Therefore, the ΔUx and
Δpx errors in relations (10) and (13) can tentatively be
referred to as external (relative to rod R) uncertainties. The internal
properties of rod R itself (a salient feature of clocks A and B
appurtenant to it) determine the external uncertainties ΔUx
and Δpx, and in the presence of a specific
Δx uncertainty, they cannot be eliminated by means of increasing the
accuracy of the measuring instruments located beyond rod R.
3. Relation of rod energy and
time uncertainties calculated using single-coordinate data
The error, ΔΓ',
of the Γ' value can be derived from formula (5). Only the τB,x'
– τA,x' value error due to the Δτ
error in the readings of clocks A and B determines the ΔΓ'
error of the Γ' value, since the t'B,x' – t'A,x'
time interval at point x', according to our initial assumption, can
be measured by the K' system clocks with an ideal accuracy. Therefore,
from formula (5), we obtain:
 ,
or with allowance for the fact that Δ(τB,x'
– τA,x') = 2Δτ,
 ,
whence it follows that:
 .
(14)
We will now assume that, in
addition to the condition of the single-coordinate nature of the observation
clock readings, the requirement of the single-time nature of the specification
of the observation time of the ab section of rod R, moving past point x',
is imposed. Let's say the requirement consists of using a single solitary moment
in time, t'ab,x', to specify the section ab observation
time at point x'.
Since the observation at a point
with a coordinate of x' is carried out over a time interval of t'B,x'
– t'A,x', it is then only possible to
approximately specify the observation time by indicating, for example, the
moment in time, t'ab,x', of the middle of the time interval, t'B,x'
– t'A,x', that goes toward observation. Here,
the uncertainty, Δt'ab,x', of the moment in the
observation time that equals half the t'B,x' – t'A,x'
observation time can be specified; i.e., it can be assumed that Δt'ab,x'
= ½(t'B,x' – t'A,x').
Then formula (14) can be presented in the form:
 .
(15)
For the arbitrary positioning of
the clocks on the rod, the Δt'x' observation time for the
entire rod, R, with a length of L proves to be greater than the
observation time, Δt'ab,x', for the ab rod section;
i.e., the condition Δt'x' ≥ Δt'ab,x'
is satisfied, as a result of which it follows from formula (15) that
 .
(16)
During an accurate observation,
the product of the uncertainty, Δt' x', of a moment in
time, t' x', of the observation of rod R and the ΔΓ'
error of the Γ' value will only be dependent upon the Δτ
error of the readings of clocks A and B of rod R. Therefore,
relation (16) remains unchanged within any inertial reference system and can be
written for an arbitrary system in the form:
 .
(17)
Multiplying the left-hand and
right-hand members of relation (17) times MRc2,
bearing in mind that ΔΓMRc2 = ΔE,
and using notation (12), we obtain
 .
(18)
If the Δτ error is
regarded as a rod parameter that is not a part of the measuring equipment
parameters, the ΔΓ and ΔE values can then be
referred to as uncertainties.
4. A physical clock. Relation
of the impulse and coordinate uncertainties of a spatially extended body
We will assume that each of
clocks A and B discretely changes its reading with frequency,
ν. The maximum absolute Δτ error in the readings of the moments
in time of each of the clocks will equal 1/ν. We will also assume that this
change in the clock readings occurs synchronously; i.e., not only are the clock
readings at a given moment in time synchronized, but also the times of the
change in readings.
In this instance, the maximum
error of the difference in the readings of clocks A and B will
equal not the sum of the two Δτ errors, but rather a single
Δτ error. Then relations (13) and (18), deduced from the condition of
the equality of the clock A and B difference error to the two
Δτ errors, take the form:
(19)
and
 .
(20)
Let's imagine that each of clocks
A and B consists of two components, one of which, being artificial
("manmade"), performs the function of a display and provides discrete
time readings, changing them in a keeping with external signals, while the other
one – we will call it a physical clock – generates these signals in a
natural, easy manner and controls the change in the display's readings.
For the sake
of clarity, we will visualize the second part of the physical clock as a piece
of radioactive material with a long half-life (the material's radiant power can
be regarded as constant for a sufficiently long time frame).
If the display reacts to a
specific portion, ε, of the physical clock material's absorbed
gamma-radiation energy by changing its readings, then in the presence of a
material radiant power than equals P, the frequency, ν, of the
change in the display readings will equal P/ε.
We will call the ε portion
of the absorbed energy that leads to a change in the display's reading the
energy of the perceived (by the display) physical clock signals.
We will assume that, regardless
of the quantity of the physical clock material, the display absorbs all of the
energy radiated, and that each ε portion of the energy absorbed performs
the function of a reading change signal that is perceived by the display. The
frequency, ν, of the physical clock materials signals perceived by
the display, and accordingly the frequency of the change in the physical clock's
readings, will then be proportional to the quantity of the physical clock's
material. This means that if the frequency of the perceived signals equals ν0
in the presence of a physical clock material unit mass of m0,
it will then equal M0ν0/m0 in the
presence of a mass of M0; i.e.,
 ,
(21)
Since the maximum absolute
Δτ error of the readings of each of the clocks equals 1/ν, the
following expression can then be derived from equality (21):
 .
(22)
Taking the mass of the rod
without the clocks to be negligible as compared to the mass of the physical
clock material concentrated in clocks A and B, or assuming that
the rod R consists entirely of the physical clock's material, the 2M0
value (since the mass of the material in both clocks equals 2M0)
can be set to equal the mass, MR, of rod R with clocks A
and B. Taking this into account, it follows from formulas (12) and (22)
that
(23)
The H value is dependent
upon physical clock type and display sensitivity, and is not a physical constant.
The H becomes different when the display's sensitivity changes, or when
the physical clock's radioactive material is replaced with a radioactive
material that has a different radiant power.
In the context of the physical
clock, relations (19) and (20) pertain to the case of the arbitrary distribution
of the physical clock's material along the rod and make the transition to
equalities in the special case when the physical clock's material is
concentrated at the ends of rod R.
At first glance, the relations
derived only hold true for the techniques of instantaneous and point
observations of an object, and the uncertainties going into these relations
consist exclusively of the uncertainties inherent in these techniques. In
actuality, however, it is impossible to measure even the constant velocity of
this rod R, equipped with clocks A and B, with absolute accuracy
using conventional methods (based on the path traversed and the time), if the
word combination "rod R with clocks A and B" is
taken to mean a specific object, the complement of characteristic traits of
which includes the difference in the readings of clocks A and B (or
the events that characterize this object). In such cases, the velocity and the
values derived from it (the impulse and energy) prove to be tied not to an
extraneous reference system, but rather to this object's characteristics traits,
for example, to time or event marks [5].
In
order to grasp the essence of the latter comment, we will address the concepts
of relativity and uncertainty.
It was shown in reference [5]
that many of the questions arising in the quagmire of physical relativism are
erased if attention is directed to the presence of uncertainty in physical
relativity. Rather than talking about the relativity of the velocity of a body
and about the fact that the velocity of a body without specifying a reference
system makes no physical sense, it is proposed in reference [5] that the term
"uncertain velocity" be used with respect to the irrelative velocity
of a body. In this case, the velocity of a body that is not tied to a specific
reference system should be referred to as uncertain velocity. The uncertainty of
the irrelative velocity of a body equals the c constant. It can be said
of the irrelative velocity of this body, for example, that it equals zero, while
its uncertainty equals c. This specification of velocity does not differ
qualitatively in any way from the conventional specification of a velocity value
supplemented by the specification of its error.
We draw attention to two methods
that make it possible to avoid the uncertainty of the velocity of a body and to
give it certainty.
The first generally accepted
method consists of specifying the reference system within which the velocity of
a body is examined. After the reference system is specified, the velocity of
this body becomes certain.
The second method, despite its
obviousness, is scarcely mentioned at all in physics. This method consists of
individualizing an object, which in turn consists of describing it in greater
detail, and under certain conditions, it is capable of replacing the selection
of a reference system.
In the example we examined, the
object with respect to which relations (19) and (20) hold true is not a rod, R,
equipped with clocks A and B, but rather a rod, R, equipped
with clocks A and B, and possessing a predetermined difference,
τB – τA, in the readings, τA
and τB, of clocks A and B.
If one and the same rod, R,
equipped with clocks A and B, is by definition regarded as a set
of more concrete subobjects, each of which has an inherent difference, τB
– τA, in the readings of clocks A and B,
then these different objects will possess different velocities. The possibility
of thus dividing an object into more concrete subobjects eludes relativists,
since it puts an end to the objective nature of physical relativism. Relativists
are incapable of understanding something that was clear to Heracles, who saw the
difference between one and the same river and particular concrete "subobjects"
of this river that differed from one another. At the same time, relativists
themselves note that one and the same object differs in different reference
systems due to the relativity of its single-time nature; i.e., it breaks down
into subobjects of sorts. It is strange that they don't comprehend the fact that,
having described a subobject, its velocity can often be determined based on
"external appearance" without specifying a reference system.
For example, in the case we
examined, a subobject such as a rod, R, that has a specific difference,
τB,t' – τA,t', in the readings,
τA,t' and τB,t', of clocks A
and B, moves at a specific longitudinal velocity, Vx.
When the clocks run with ideal accuracy, the Vx
velocity value is functionally dependent upon the τB,t' – τA,t'
difference value. A subobject such as a rod, R, with a difference, τB,t'
– τA,t', in the readings, τA,t'
and τB,t', of clocks A and B that equals
zero is at rest. Its longitudinal velocity, Vx, equals
zero and cannot differ from zero in any reference system, since subobjects with
a difference, τB,t' – τA,t', in
the readings of clocks A and B that equals zero do not exist in
any reference system where this rod moves longitudinally. This rod, R,
with clocks A and B, has different longitudinal velocities in
different reference systems, but the subject rod, R, with a difference,
τB,t' – τA,t', in the
readings of clocks A and B that equals zero, being a concrete
subobject, only has a longitudinal velocity that equals zero. This fact is
entirely independent of the manner in which the velocity is measured – based
on the clock readings or using the path and distance traversed.
In addition to this, if clocks A
and B provide discrete readings, the τB,t' – τA,t'
difference will also have a certain discreteness. Here, the longitudinal
velocity value, Vx, will have an uncertainty of ΔVx.
In this case, a rod, R, with a given difference in the readings of clocks
A and B, being a subobject, may be found in a certain set of
reference systems in a state of motion at various longitudinal velocities that
differ from one another by a magnitude that does not exceed a certain value,
which is the ΔVx uncertainty. This uncertainty in
accuracy equals the uncertainty that occurs during the single-time determination
of the Vx velocity of a given subobject.
If clocks A and B
do not run, but rather stand still, continuously indicating an identical time,
then the ΔVx uncertainty of the longitudinal
velocity of rod, R, with a difference of τB,t'
– τA,t' in the readings of clocks A and B
that equals zero, will equal the speed of light, c. Such a rod, R,
with a difference of τB,t' – τA,t'
in the readings of clocks A and B that equals zero, can be found
in all reference systems and possesses any velocity over a range of zero to the
speed of light, c.
Conclusion
The purpose of the work at hand
consisted not of finding areas of common interest between Lorentzian physics and
quantum mechanics, but rather of demonstrating the existence of general physical
uncertainty relations in the physics of the macrocosm. The general physical
relations derived are externally reminiscent of the known uncertainty relations
of quantum mechanics; however, the physical essence of the values that go into
the relations and that contain the relations themselves are different than those
in quantum mechanics. In relation (13), the Δx uncertainty is taken
to mean the uncertainty in specifying the position of a projection with a length
of Δx of a spatially extended object using a single solitary
coordinate, x, while in the Heisenberg relation, Δx is a
probabilistic characteristic of the position of a microparticle described by the
root-mean-square deviation from the mean value.
The Δpx
value in relation (13) is the px impulse error, which in the
presence of a predetermined Δx uncertainty, generally speaking, can
be reduced by using another type of physical clock, while Δpx
in the Heisenberg relation is an uncertainty that it is fundamentally impossible
to reduce in the presence of a predetermined Δx value.
Different
meanings are placed on the H and h
values.
The H value in relations
(13), (18), (19), and (20) is dependent upon display sensitivity and physical
clock type. If the sensitivity of the display is changed or the physical clock's
radioactive material is replaced with a radioactive material that has a
different mass radiation frequency, the H value will be different.
However, the fundamental Planck constant, h, is unrelated to the physical
properties of any specific material. The fact that these relations prove to be
connected to the Heisenberg relation, not only externally, but also internally,
seems especially strange.
First, there is the question of
the minimum value, Hmin, of an H parameter with an
action dimension. Can one assume that this value may be as small as desired in
the macrocosm and may correspond , for example, to the condition Hmin
<< h?
And second, relations (19) and
(20) are formally transformed into the relations
and 
via the simple substitution of
the energy, hν0, of a photon with a frequency of ν0
in place of the unit energy, m0c2, of the physical
clock in formula (23); i.e., by taking the unit mass of a photon, the ν0
frequency of which numerically equals the ν0 frequency of the
signals of a hypothetical change in the clock readings as the physical clock. It
would be more correct to proceed on the basis of the concept of
Lorentz-invariant mass and the equality of photon mass to zero, then a pair of
photons coming from opposite directions with equal energies of E = hν
and with a resultant impulse, P, that equals zero, should be regarded as
a physical clock of a unit mass of m0. Since the unit mass, m0,
of this clock equals  =2E/c2,
while the ν0 summation frequency of the electromagnetic
oscillations equals 2ν, then taking the ν0
frequency of the unit mass physical clock to equal the 2ν summation
frequency of the photon pair, it follows from formula (23) that H = Hmin
= h.
Another approach is also possible.
In considering formula (23), if
it is assumed that a certain Hmin value exists that is common
to all types of physical clocks, then 2m0c2/ν0
= Hmin. The latter equality only occurs when m0c2
= ½ Hminν0. If the lowest energy
of a hypothetic signal of a change in clock readings is taken as m0c2,
then Hmin must be equal to the Planck constant h, since
the ½hν0 value is the minimum possible energy of
the zero-point oscillations of an oscillator with a frequency of ν0.
The relations expressed by
formula (23), if we move to them from relations (19) and (20), do not reflect
the statistical nature of the generation of the clock reading change signals;
thus, when using this approach, reference can only be made to the order of the
parameter present in the right-hand member of the relations and not its precise
value.
REFERENCES
- Тарбеев Ю.В., Слаев В.А.,
Чуновкина А.Г. Проблемы применения в
России международного Руководства по
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техника, 1997, №1. С
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69 - 72. {Yu. V. Tarbeyev, V. A. Slayev, and A. G. Chunovkina. Problems With
Using the International Guide to the Expression of Uncertainty in Measurement
in Russia. Izmeritelnaya Tekhnika [Measurement Techniques], No. 1,
1997, pp. 69-72}.
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"Количественное описание
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Publishing date: July 7, 2010
Source: SciTecLibrary.ru
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